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THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY MOMENTUM-SPACE DYNAMICS OF RUNAWAY ELECTRONS IN PLASMAS Adam Stahl Department of Physics Chalmers University of Technology Gothenburg, Sweden, 2017 MOMENTUM-SPACE DYNAMICS OF RUNAWAY ELECTRONS IN PLASMAS Adam Stahl © Adam Stahl, 2017 ISBN 978-91-7597-532-0 Doktorsavhandlingar vid Chalmers tekniska högskola Ny serie nr 4213 ISSN 0346-718X Subatomic and Plasma Physics Department of Physics Chalmers University of Technology SE-412 96 Gothenburg Sweden Telephone +46-(0)31-772 10 00 Printed by Reproservice Chalmers tekniska högskola Gothenburg, Sweden, 2017 MOMENTUM-SPACE DYNAMICS OF RUNAWAY ELECTRONS IN PLASMAS Adam Stahl Department of Physics Chalmers University of Technology Abstract Fast electrons in a plasma experience a friction force that decreases with increasing particle speed, and may therefore be continuously accelerated by sufficiently strong electric fields. These so-called runaway electrons may quickly reach relativistic speeds. This is problematic in tokamaks – devices aimed at producing sustainable energy through the use of thermonuclear fusion reactions – where runaway-electron beams carrying strong currents may form. If the runaway electrons deposit their kinetic energy in the plasma-facing components, these may be seriously damaged, leading to long and costly device shutdowns. Crucial to the runaway phenomenon is the behavior of the runaway electrons in two-dimensional momentum space. The interplay between electric-field acceleration, collisional momentum-space transport, and radiation reaction determines the dynamics and the growth or decay of the runaway-electron population. In this thesis, several aspects of this interplay are investigated, including avalanche multiplication rates, synchrotron radiation reaction, modifications to the critical electric field for runaway generation, rapidly changing plasma parameters, and electron slide-away. Two numerical tools for studying electron momentum-space dynamics, based on an efficient solution of the kinetic equation, are presented and used throughout the thesis. The spectrum of the synchrotron radiation emitted by the runaway electrons – a useful diagnostic for their properties – is also studied. It is found that taking the electron distribution into account properly is crucial for the interpretation of synchrotron spectra; that a commonly used numerical avalanche operator may either overestimate or underestimate the runawayelectron growth rate, depending on the scenario; that radiation reaction modifies the critical electric field, but that this modification often is small compared to other effects; that electron slide-away can occur at significantly weaker electric fields than expected; and that collisional nonlinearities may be significant for the evolution of runaway-electron populations in disruption scenarios. Keywords: fusion-plasma physics, tokamak, runaway electrons, synchrotron radiation, critical electric field, slide-away, non-linear collision operator i Publications This thesis is based on the work contained in the following papers: A A. Stahl, M. Landreman, G. Papp, E. Hollmann, and T. Fülöp, Synchrotron radiation from a runaway electron distribution in tokamaks, Physics of Plasmas 20, 093302 (2013). http://dx.doi.org/10.1063/1.4821823 http://arxiv.org/abs/1308.2099 B M. Landreman, A. Stahl, and T. Fülöp, Numerical calculation of the runaway electron distribution function and associated synchrotron emission, Computer Physics Communications 185, 847-855 (2014). http://dx.doi.org/10.1016/j.cpc.2013.12.004 http://arxiv.org/abs/1305.3518 C A. Stahl, O. Embréus, G. Papp, M. Landreman, and T. Fülöp, Kinetic modelling of runaway electrons in dynamic scenarios, Nuclear Fusion 56, 112009 (2016). http://dx.doi.org/10.1088/0029-5515/56/11/112009 http://arxiv.org/abs/1601.00898 D A. Stahl, E. Hirvijoki, J. Decker, O. Embréus, and T. Fülöp, Effective critical electric field for runaway electron generation, Physical Review Letters 114, 115002 (2015). http://dx.doi.org/10.1103/PhysRevLett.114.115002 http://arxiv.org/abs/1412.4608 E A. Stahl, M. Landreman, O. Embréus, and T. Fülöp, NORSE: A solver for the relativistic non-linear Fokker-Planck equation for electrons in a homogeneous plasma, Computer Physics Communications 212, 269-279 (2017). http://dx.doi.org/10.1016/j.cpc.2016.10.024 http://arxiv.org/abs/1608.02742 F A. Stahl, O. Embréus, M. Landreman, G. Papp, and T. Fülöp, Runaway-electron formation and electron slide-away in an ITER post-disruption scenario. Journal of Physics: Conference Series 775, 012013 (2016). http://dx.doi.org/10.1088/1742-6596/775/1/012013 http://arxiv.org/abs/1610.03249 iii Statement of contribution In Papers A, C, E and F, I performed all simulations and associated analysis, and produced all the figures. I wrote all the text in Papers C, E and F, and a majority of that in Paper A. In addition I did all of the programming associated with the code SYRUP used in Paper A and the tool NORSE described in Paper E, as well as most of the programming related to the capabilities of CODE described in Paper C. I performed the majority of the analytical calculations required for Papers A, C and E. In Paper B, I was mainly responsible for the work presented in Section 6, but in addition contributed to the remainder of the text. To a lesser extent, I was also involved in the development of the tool CODE described in the paper. In Paper D, I performed the numerical simulations, together with the associated implementation and analysis, produced the majority of the figures and text, and did some of the analytical calculations. iv Additional publications, not included in the thesis G O. Embréus, A. Stahl, and T. Fülöp, Effect of bremsstrahlung radiation emission on fast electrons in plasmas, New Journal of Physics 18, 093023 (2016). http://dx.doi.org/10.1088/1367-2630/18/9/093023 http://arxiv.org/abs/1604.03331 H J. Decker, E. Hirvijoki, O. Embréus, Y. Peysson, A. Stahl, I. Pusztai, and T. Fülöp, Numerical characterization of bump formation in the runaway electron tail, Plasma Physics and Controlled Fusion 58, 025016 (2016). http://dx.doi.org/10.1088/0741-3335/58/2/025016 http://arxiv.org/abs/1503.03881 I E. Hirvijoki, I. Pusztai, J. Decker, O. Embréus, A. Stahl, and T. Fülöp, Radiation reaction induced non-monotonic features in runaway electron distributions, Journal of Plasma Physics 81, 475810502 (2015). http://dx.doi.org/10.1017/S0022377815000513 http://arxiv.org/abs/1502.03333 J O. Embréus, S. Newton, A. Stahl, E. Hirvijoki, and T. Fülöp, Numerical calculation of ion runaway distributions, Physics of Plasmas 22, 052122 (2015). http://scitation.aip.org/content/aip/journal/pop/22/5/10.1063/1.4921661 http://arxiv.org/abs/1502.06739 K G. I. Pokol, A. Kómár, A. Budai, A. Stahl, and T. Fülöp, Quasi-linear analysis of the extraordinary electron wave destabilized by runaway electrons, Physics of Plasmas 21, 102503 (2014). http://dx.doi.org/10.1063/1.4895513 http://arxiv.org/abs/1407.5788 v Conference contributions L A. Tinguely, R. Granetz, and A. Stahl, Analysis of Runaway Electron Synchrotron Emission in Alcator C-Mod, Proceedings of the 58th Annual Meeting of the APS Division of Plasma Physics 61, 18, TO4.00007 (2016). http://meetings.aps.org/Meeting/DPP16/Session/TO4.7 M C. Paz-Soldan, N. Eidietis, D. Pace, C. Cooper, D. Shiraki, N. Commaux, E. Hollmann, R. Moyer, R. Granetz, O. Embréus, T. Fülöp, A. Stahl, G. Wilkie, P. Aleynikov, D. P. Brennan, and C. Liu, Synchrotron and collisional damping effects on runaway electron distributions, Proceedings of the 58th Annual Meeting of the APS Division of Plasma Physics 61, 18, CO4.00010 (2016). http://meetings.aps.org/Meeting/DPP16/Session/CO4.10 N O. Ficker, J. Mlynar, M. Vlainic, V. Weinzettl, J. Urban, J. Cavalier, J. Havlicek, R. Panek, M. Hron, J. Cerovsky, P. Vondracek, R. Paprok, J. Decker, Y. Peysson, O. Bogar, A. Stahl, and the COMPASS Team, Long slide-away discharges in the COMPASS tokamak, Proceedings of the 58th Annual Meeting of the APS Division of Plasma Physics 61, 18, GP10.00101 (2016). http://meetings.aps.org/Meeting/DPP16/Session/GP10.101 O Y. Peysson, G. Anastassiou, J.-F. Artaud, A. Budai, J. Decker, O. Embréus, O. Ficker, T. Fülöp, K. Hizanidis, Y. Kominis, T. Kurki-Suonio, P. Lauber, R. Lohner, J. Mlynar, E. Nardon, S. Newton, E. Nilsson, G. Papp, R. Paprok, G. Pokol, F. Saint-Laurent, C. Reux, K. Sarkimaki, C. Sommariva, A. Stahl, M. Vlainic, and P. Zestanakis, A European Effort for Kinetic Modelling of Runaway Electron Dynamics, Theory and Simulation of Disruptions Workshop (2016). http://tsdw.pppl.gov/Talks/2016/Peysson.pdf P T. Fülöp, O. Embréus, A. Stahl, S. Newton, I. Pusztai, and G. Wilkie, Kinetic modelling of runaways in fusion plasmas, Proceedings of the 26th IAEA Fusion Energy Conference, Kyoto, Japan, TH/P4–1 (2016). Q O. Embréus, A. Stahl, and T. Fülöp, Effect of bremsstrahlung radiation emission on fast electrons in plasmas, Europhysics Conference Abstracts 40A, O2.402 (2016). http://ocs.ciemat.es/EPS2016PAP/pdf/O2.402.pdf vi R A. Stahl, O. Embréus, E. Hirvijoki, I. Pusztai, J. Decker, S. Newton, and T. Fülöp, Reaction of runaway electron distributions to radiative processes, Proceedings of the 57th Annual Meeting of the APS Division of Plasma Physics 60, 19, PP12.00103 (2015). http://meetings.aps.org/link/BAPS.2015.DPP.PP12.103 S O. Embréus, A. Stahl, and T. Fülöp, Conservative large-angle collision operator for runaway avalanches, Proceedings of the 57th Annual Meeting of the APS Division of Plasma Physics 60, 19, PP12.00107 (2015). http://meetings.aps.org/link/BAPS.2015.DPP.PP12.107 T S. Newton, O. Embréus, A. Stahl, E. Hirvijoki, and T. Fülöp, Numerical calculation of ion runaway distributions, Proceedings of the 57th Annual Meeting of the APS Division of Plasma Physics 60, 19, CP12.00118 (2015). http://meetings.aps.org/link/BAPS.2015.DPP.CP12.118 U I. Pusztai, E. Hirvijoki, J. Decker, O. Embréus, A. Stahl, and T. Fülöp, Non-monotonic features in the runaway electron tail, Europhysics Conference Abstracts 39E, O3.J105 (2015). http://ocs.ciemat.es/EPS2015PAP/pdf/O3.J105.pdf V G. Papp, A. Stahl, M. Drevlak, T. Fülöp, P. Lauber, and G. Pokol, Towards self-consistent runaway electron modelling, Europhysics Conference Abstracts 39E, P1.173 (2015). http://ocs.ciemat.es/EPS2015PAP/pdf/P1.173.pdf W O. Embréus, S. Newton, A. Stahl, E. Hirvijoki, and T. Fülöp, Numerical calculation of ion runaway distributions, Europhysics Conference Abstracts 39E, P1.401 (2015). http://ocs.ciemat.es/EPS2015PAP/pdf/P1.401.pdf X A. Stahl, E. Hirvijoki, M. Landreman, J. Decker, G. Papp, and T. Fülöp, Effective critical electric field for runaway electron generation, Europhysics Conference Abstracts 38F, P2.049 (2014). http://ocs.ciemat.es/EPS2014PAP/pdf/P2.049.pdf Y G. Pokol, A. Budai, J. Decker, Y. Peysson, E. Nilsson, A. Stahl, A. Kómár, and T. Fülöp, Interaction of oblique propagation extraordinary electron waves and runaway electrons in tokamaks, Europhysics Conference Abstracts 38F, P2.042 (2014). http://ocs.ciemat.es/EPS2014PAP/pdf/P2.042.pdf vii Z viii A. Stahl, M. Landreman, T. Fülöp, G. Papp, and E. Hollmann, Synchrotron radiation from runaway electron distributions in tokamaks, Europhysics Conference Abstracts 37D, P5.117 (2013). http://ocs.ciemat.es/EPS2013PAP/pdf/P5.117.pdf Contents Abstract i Publications iii 1 Introduction 1 2 Runaway-electron generation and loss 2.1 The runaway region of momentum space . . . . . . . . . . . . . 2.2 Runaway-generation mechanisms . . . . . . . . . . . . . . . . . 2.3 Damping and loss mechanisms for runaways . . . . . . . . . . 7 7 14 17 3 Simulation of runaway-electron momentum-space dynamics 3.1 The kinetic equation . . . . . . . . . . . . . . . . . . . . 3.2 Collision operator . . . . . . . . . . . . . . . . . . . . . 3.3 Avalanche source term . . . . . . . . . . . . . . . . . . 3.4 CODE . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 NORSE . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 21 24 26 29 30 . . . . . . . . . . . . . . . . . . . . 4 Synchrotron radiation 33 4.1 Emission and power spectra . . . . . . . . . . . . . . . . . . . . 34 4.2 Radiation-reaction force . . . . . . . . . . . . . . . . . . . . . . 40 5 Nonlinear effects and slide-away 45 5.1 Ohmic heating . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Electron slide-away . . . . . . . . . . . . . . . . . . . . . . . . 46 6 Concluding remarks 49 6.1 Summary of the included papers . . . . . . . . . . . . . . . . . 49 6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Bibliography 55 Included papers (A-F) 69 ix Acknowledgments The life of a doctoral student has sometimes been described as a stumbling journey down a seemingly endless dark tunnel. Not so in the Plasma Theory Group at Chalmers, where the guidance of Professor Tünde Fülöp provides a map, as well as the light by which to read it. I am fortunate and thankful to have had her support throughout my doctoral studies. In fact, I wish every doctoral student the privilege of having Tünde as their supervisor (although I recognize the logistical nightmare this would entail)! I also owe a lot to my co-supervisors Dr. Matt Landreman and Dr. István Pusztai. Matt in particular has been involved on some level in most projects I have undertaken during these years and have identified (and often solved) many a problem I didn’t even know I had. István has provided the stability (and ability) closer to home, always contributing knowledgeable input on all things plasma physics. The plasma theory group would be nothing without its current and former members, of which there are many. Thanks for the marvelous physics and non-physics discussions, salty lunches, pizza parties and ski trips. I will miss you all. In particular I would like to thank my long-time roommate and close collaborator Ola Embréus for his remarkable insight and investigative journalism, and my short-time roommate and long-time friend Dr. Gergely Papp for his eagle eye and grounded feet. Finally; the reason I leave the office in the evening with a smile on my face. Thank you Hedvig for providing distractions and focus, cookies and salads, insight and bogus, odd meters and ballads. Without these, no map in the world would see me through to the end of the tunnel. Thank you all xi 1 Introduction In plasma physics, many interesting phenomena occur that are outside of our everyday experience. One of these is the generation of so-called runaway electrons (or simply runaways) – electrons that under certain conditions are continuously accelerated by electric fields [1, 2]. The dynamics of the process is such that the runaways quickly reach relativistic energies; they move with speeds very close to that of light. The study of runaway electrons therefore combines two fascinating areas of physics: Einstein’s special relativity [3], and plasma physics; giving rise to interesting dynamics (as well as complicated mathematics). As we shall see, runaways appear in a variety of atmospheric, astrophysical and laboratory contexts. Apart from their intrinsic interest, these highly energetic particles are also a cause for concern in the context of fusion-energy experiments [4]. Generating electric power using controlled thermonuclear fusion reactions is a promising concept for a future sustainable energy source [5–7], but stable and controllable operating conditions are required for a successful fusion power plant. The presence of runaway electrons in the plasmas of fusion reactors under certain circumstances is one of the main remaining hurdles on the road to realization of fusion power production [8], as the runaways have the potential to severely damage the machine when they eventually leave the plasma and strike the wall [9]. In order to accurately assess the frequency of such events, as well as the resulting damage in a given situation, there is a great need to improve the understanding of the mechanisms that generate and suppress runaway electrons, and to better describe their dynamics [10]. In order for runaways to be generated, a comparatively long-lived electric field is needed. Due to the natural tendency of the plasma particles to rearrange in order to screen out such fields, they are not normally present in unmagnetized plasmas. However in certain situations, for instance if a current running through the plasma changes quickly or if the magnetic field lines in a magnetized plasma reconnect, an electric field is induced which may be sufficient to lead to runaway formation. Runaway electrons do form in atmospheric plasmas – they have been linked to for instance lightning discharges 1 Chapter 1. Introduction Figure 1.1: A tokamak plasma (pink), together with various common terms and concepts. [11], impulsive radio emissions [12], and terrestrial gamma-ray flashes [13] – and in the mesosphere [14]. In astrophysical plasmas, they are expected to form in for instance solar flares [15] and large-scale filamentary structures in the galactic center [16]. Under certain circumstances, other plasma species may also run away. Both ion and positron runaway have been investigated in recent work (see Refs. [17–19], as well as Paper J, not included in the thesis). Our main interest in this thesis is however electron runaway in the context of magnetic-confinement thermonuclear fusion. The most common type of fusion device is called a tokamak (see for instance [7, 20]). It uses strong magnetic fields to confine a plasma in which the fusion reactions between hydrogen-isotope ions take place. The charged particles in the plasma follow helical orbits (spirals) around the magnetic field lines due to the Lorentz force [21, 22], and are thus (to a first approximation) prevented from reaching the walls of the device. In a tokamak, the “magnetic cage” (and thus the plasma) has the form of a torus (a doughnut), as shown in Fig. 1.1. The torus shape can be thought of as being formed from a cylinder, bent around so that its two ends connect. The direction along the axis of this cylinder is referred to as toroidal (and the axis itself the magnetic axis), while 2 Introduction the direction around the circumference of the cylinder is called poloidal and its radius is the minor radius. The radius of the circle defined by the magnetic axis is called the major radius, and the relation between the major and minor radii is referred to as the aspect ratio. The plasma temperature and density are maximized close to the magnetic axis, and this is also where the runaways predominantly form. Due to the loop-like toroidal geometry, the magnetic field acts as an “infinite racetrack” for the runaways, which make millions of toroidal revolutions of the tokamak each second. In order to achieve satisfactory plasma confinement, it is necessary to drive a strong current in the plasma. The poloidal magnetic field induced by the current introduces a helical twist to the magnetic field lines, and it can be shown that each field line covers a toroidal surface of constant pressure. The tokamak plasma can therefore be viewed as being made up of a series of such nested flux surfaces. The plasma current is generated using transformer action: a changing current is driven through a conducting loop interlocked with the tokamak vessel, and the change in this current induces a voltage (the so-called loop voltage) which drives a current in the plasma. Since a hot plasma is a very good conductor, the loop-voltage does not need to be particularly strong during normal operation (it is usually of order 1 V), and tokamak plasmas (discharges or “shots”) are routinely maintained for several seconds, and sometimes for several minutes or more. Inductive current drive does not enable continuous (steady-state) operation, however, and the tokamak is fundamentally a pulsed device (in the absence of auxiliary current-drive systems). During the start-up of a tokamak discharge, a plasma is formed by the ionization of a gas. For this process, a strong electric field is usually needed. Runaways may form in this situation [23–25], however their formation can usually be avoided by maintaining a high enough gas/plasma density. The case of a changing current is more problematic. Abrupt changes in plasma current occur during so-called disruptions, in which the plasma becomes unstable, rapidly cools down due to a loss of confinement, and eventually terminates [8, 26, 27]. As the plasma cools, the resistivity increases drastically (since it is proportional to T −3/2 ), and a large electric field is induced which tries to maintain the current (in accordance with Lenz’s law [28]). Near the magnetic axis, this field is often strong enough to lead to runaway generation, and runaway beams in the center of the plasma have been observed during disruptions in many tokamaks (for instance JET [29–31], DIII-D [32, 33], Alcator C-Mod [34], Tore Supra [35], KSTAR [36], COMPASS [37], ASDEX Upgrade [38] and TCV [39]). Runaways can also be generated in so-called sawtooth crashes [40], and even during normal stable operation if the density 3 Chapter 1. Introduction is low enough [39, 41, 42] (the accelerating field in this case is the normal loop voltage). Some auxiliary plasma-heating schemes produce an elevated tail in the electron velocity distribution which can lead to increased runaway production, should a disruption occur [39, 43]. The runaways predominantly form close to the magnetic axis of the tokamak, where the flux-surface radius is small compared to the major radius. Many aspects of the fundamental runaway dynamics can therefore be studied in the large–aspect-ratio limit where transverse spatial effects can be neglected. Since the plasma is essentially homogeneous in the direction along the magnetic field, the spatial dependence can be neglected entirely, and for an understanding of the basic mechanisms it is sufficient to treat the runaway process purely in momentum-space. As discussed in Sec. 3.1, one of the momentumspace coordinates (the angle describing the gyration around the field lines), can be averaged over, reducing the problem to two momentum-space dimensions. These simplifications are done throughout this thesis, except for parts of Paper A (where a radial dependence is included). In practice, the situation is more complicated, however, and spatial effects are often important, as will be discussed in Sec. 2.3. Several numerical tools that take magnetic-trapping and radial diffusive-transport effects into account (such as LUKE [44–46] and CQL3D [47, 48]) also exist. The main reason for the interest in runaway research is that the runaways pose a serious threat to tokamaks. During disruptions, a large fraction of the initial plasma current (which is often several megaamperes) can be converted into runaway current. The runaways are normally well-confined in the tokamak, but a variety of mechanisms (such as instabilities or a collective displacement of the entire runaway beam) can transport them out radially. Unless their generation is successfully mitigated, or runaway-beam stability can be externally enforced, the runaways eventually escape the plasma and strike the wall where they can destroy sensitive components or degrade the wall material [49]. In present-day tokamaks, runaways are a nuisance, but usually not a serious threat (although there are exceptions, see for instance Ref. [31]). However, the avalanche multiplication (see Sec. 2.2.3) of a primary runaway seed is predicted to scale exponentially with plasma current [50], and it is believed that in future devices which will have a larger current (such as the International Thermonuclear Experimental Reactor ITER [51, 52] and eventually commercial fusion reactors), the problem will be much more severe. In these devices, disruptions can essentially not be tolerated at all and much effort is devoted to research on runaway and disruption mitigation techniques [10, 33, 49, 53]. 4 Introduction This thesis focuses on the dynamics of the runaway electrons in momentum space, their generation and loss, and the forces that affect them. Most of the results presented herein are not specific to fusion plasmas or a tokamak magnetic geometry, however we will make use of fusion-relevant plasma parameters throughout. The majority of the work described in this thesis was conducted using two numerical tools developed as part of the thesis work: CODE, described in Papers B and C; and NORSE, described in Paper E. With these tools, all three major runaway-generation mechanisms (Dreicer generation, hot-tail generation and avalanche generation through knock-on collisions – to be discussed in Sec. 2.2), as well as two important energy-loss mechanisms (synchrotron and bremsstrahlung radiation emission) can be studied in detail. Synchrotron radiation is particularly important, and will be discussed in Chapter 4. It is also the main subject of Papers A and D – as a diagnostic for the runaway distribution and as a damping mechanism for runaway growth, respectively. Let us now turn to discussing the basic mechanisms responsible for the runaway phenomenon, and the quantities characterizing the runaway dynamics. 5 2 Runaway-electron generation and loss In short, a runaway electron is an electron in a plasma which experiences a net accelerating force during a substantial time – enough to give it a momentum significantly larger than that of the electrons in the thermal population. The accelerating force on the electron is supplied by an electric field E, so that FE = −eE, where e is the elementary charge. Meanwhile, Coulomb interaction with the other particles in the plasma (commonly referred to as collisions) introduces a friction force FC (v). The origin of the runaway phenomenon is that FC,k , the component of the friction force parallel to the electric (or magnetic, if the plasma is magnetized) field, is a nonmonotonic function of the particle velocity, with a maximum around the electron thermal speed (vth ) [54], as illustrated in Fig. 2.1. Therefore ∂FC,k /∂v < 0 for particles that are faster than vth – the friction force on these particles decreases with increasing particle velocity. The physical origin of this effect is that the faster particles spend less time in the vicinity of other particles in the plasma; as the particle speed increases, the impulse delivered to the fast particle in each encounter decreases more rapidly than the number of encounters increases, leading to a reduction in the friction [54]. This implies that if the accelerating force is sufficient to overcome the friction at the current velocity v0 of the particle, |FE | > FC,k (v0 ), it will be able to accelerate the particle for all v > v0 , i.e. the particle will be continuously accelerated to relativistic energies – it will run away – as long as the electric field persists. 2.1 The runaway region of momentum space For an electric field of intermediate strength (Ec < E < Esa , with the critical field Ec and slide-away field Esa to be defined in this section), there exist some velocities for which the acceleration by the electric field overcomes the 7 Chapter 2. Runaway-electron generation and loss F FC,|| vth c v Figure 2.1: Friction force on an electron due to collisions, as a function of its velocity (schematic). collisional friction. These velocities constitute the runaway region Sr in velocity space, indicated in Fig. 2.2. If only collisional friction is considered, this region is semi-infinite in momentum space, extending from some lowest critical momentum pc to arbitrarily high momenta, due to the dependence of the friction force on velocity, as discussed above. The particles that are not in the runaway region are only to a lesser extent affected by the electric field, since the collisions dominate their dynamics. Due to the directivity of the electric field, the force balance is not homogeneous in velocity. Several other forces also affect the dynamics, in particular radiation-reaction forces associated with synchrotron and bremsstrahlung emission. In these cases, the force balance is altered, ultimately preventing the electrons from reaching arbitrarily high momenta (this is the motivation for the use of the slightly vague definition of runaway at the beginning of this chapter). How to determine the runaway region in these more general situations is discussed in Sec. 2.1.3. Furthermore, the picture is complicated by the fact that not only friction due to Coulomb collisions (collisional slowing down) contributes to the dynamics. The collisions also lead to diffusion of the electrons in velocity space: both parallel to the particle velocity and perpendicular to it (referred to as pitch-angle scattering). The diffusion is caused by velocity-space gradients in the distribution of particle velocities (see Sec. 3.1), parallel and perpendicular to v for the two effects, respectively. Pitch-angle scattering does not directly affect the energy of the electron, but is important for the behavior in two-dimensional velocity space (see for instance [54] for a comprehensive introduction to collisional phenomena). In general, the full evolution of the runaway population can only be obtained using numerical simulations. 8 2.1 The runaway region of momentum space |F| eEsa Fee,|| eE eE>Fee,|| eEc vth vc Sr c v Figure 2.2: Forces corresponding to collisional friction against electrons (Fee,k ), the critical field (Ec ) and the slide-away field (Esa ). The runaway region in velocity space (Sr ) associated with the electric field E is shown as the gray part of the velocity axis. 2.1.1 Critical electric field and critical momentum The critical electric field for runaway electron generation, Ec , is the weakest field at which runaway is possible, see Fig. 2.2. The accelerating force due to Ec is simply equal (and opposite) to the sum of all the forces acting to slow the down, P particle P at the speed vmin where they are minimized: eEc = min F (v) = i f,i i Ff,i (vmin ). In the simplest case, the only retarding force is due to collisions with electrons (due to the mass difference, the energy lost by the electrons in collisions with ions is neglected, as are all other forces): eEc = Fee,k (v = vmin ). In this case, it is easy to obtain an expression for Ec . The friction force on a highly energetic electron is given by [54]: Fee,k (v) = 1 me c3 νrel , v2 (2.1) where v is the speed of the particle, c is the speed of light, me is the electron rest mass and νrel = ne e4 ln Λ 4πε20 m2e c3 (2.2) is the collision frequency for a highly relativistic electron. Here ne is the number density of electrons, ln Λ is the Coulomb logarithm (see for instance Refs. 9 Chapter 2. Runaway-electron generation and loss [54, 55]) and ε0 is the vacuum permittivity. The collision frequency is defined such that 1/ν is (approximately) the average time for a particle to experience a 90◦ deflection due to an accumulation of small-angle Coulomb interactions (which are much more frequent than large-angle collisions in fusion plasmas). The friction force in Eq. (2.1) is minimized as v → c (we have already mentioned that Fee,k is monotonically decreasing for large velocities)1 . We thus have that the critical field is Ec = Fee,k (v → c)/e, or Ec = ne e3 ln Λ me c νrel = , e 4πε20 me c2 (2.3) which was obtained by Connor and Hastie in 1975 [56]. As discussed in Paper D and Sec. 4.2.2, synchrotron radiation reaction leads to an increase in the critical field as the minimum of the friction force is effectively raised. Since the synchrotron radiation-reaction force vanishes along the parallel axis, this is however an effect of dynamics in 2D momentum-space and cannot be easily accounted for in the simple model considered here. For any E > Ec , there exists some speed vc above which the electric field overcomes the friction force. Particles with a velocity greater than this critical speed will run away, and vc thus marks the lower boundary of the runaway region (in the parallel direction), as illustrated in Fig. 2.2. It is customary to study runaways in terms of momentum rather than velocity. The critical momentum is a simple function of the electric field strength p if expressed in terms of the normalized momentum p = γv/c, where γ = 1/ 1 − v 2 /c2 is the relativistic mass factor: 1 pc = p , E/Ec − 1 (2.4) if the electron is assumed to move parallel to the electric field [56]. Similarly, p the corresponding critical γ is γc = (E/Ec )/(E/Ec − 1). 2.1.2 Dreicer and slide-away fields The critical field Ec corresponds to the field balancing the minimum of the collisional friction force. The Dreicer field, ED [1, 2], on the other hand, 1 Like much of plasma physics, this result assumes that the Coulomb logarithm ln Λ is a constant. In reality, it is energy dependent and increases logarithmically with p for large p. The minimum of the friction force is correspondingly found at v somewhat below c, however the value of Ec is only moderately affected. 10 2.1 The runaway region of momentum space approximately balances the p maximum of the friction force, which is located around v = vth , with vth = 2Te /me the electron thermal speed and Te the electron temperature2 . The critical and Dreicer fields are related by the ratio of the thermal energy to the electron rest energy: ED = ne e3 ln Λ me c2 Ec = . Te 4πε20 Te (2.5) For fields stronger than the slide-away field Esa ' 0.214ED [1], the accelerating force overcomes the friction at all particle velocities, and the whole electron population thus runs away. This phenomenon is called slide-away [57]. In practice, the electric field in a fusion plasma is almost always much smaller than the Dreicer and slide-away fields. Therefore runaway dynamics can usually be studied in the regime where Ec < E Esa is fulfilled, in which case the runaways can be treated as a small perturbation to a velocity distribution that is close to local thermal equilibrium (i.e. a Maxwellian). If the electric field is comparable to Esa , however; the distribution will deviate strongly from a Maxwellian shape. This will in turn lead to a reduction in the friction in the bulk and a corresponding decrease in the slide-away field. Thus – as discussed in Sec. 5.2 – even though E < Esa initially, a transition to slide-away can quickly occur due to the distortion of the distribution caused by the strong electric field. 2.1.3 General calculation of the runaway region Let us now discuss how to define the runaway region in the full two-dimensional momentum space. The two momentum-space dimensions are conveniently parametrized by the coordinates (p, ξ), where p = γv/c is the normalized momentum and ξ = p|| /p is the cosine of the particle pitch angle (which characterizes the pitch of the helix that describes the particle orbit around a magnetic field line). These coordinates are suitable for analytical as well as numerical calculations, and will therefore be used throughout much of the remainder of this thesis. In the definition of ξ, pk is the component of the momentum parallel to the magnetic field and similarly p⊥ is the perpendicular component. The coordinates (pk , p⊥ ) are convenient for visualizing the distribution and will therefore be used in several figures throughout the thesis. Note also that the relativistic mass factor is related to p through γ 2 = p2 + 1. 2 It is customary in plasma physics to let Te ≡ kB Te , so that the “temperature” actually is the thermal energy, and to express it in eV. 11 Chapter 2. Runaway-electron generation and loss In general, the object of interest when studying runaway-electron dynamics is the distribution function f of electron momenta, which will be thoroughly introduced in Sec. 3.1. For the following discussion, we just note that the runaways normally form a narrow tail in the distribution function, centered around the parallel axis (i.e. at ξ ≈ 1), whereas the contours of the equilibrium (Maxwellian) distribution form concentric half circles in the (pk , p⊥ )-plane, as illustrated in Fig. 2.3. The lower boundary of the runaway region in 2D momentum-space is often called the separatrix, as it separate two regions with distinct dynamical properties. It is well-defined in the parallel direction where it takes the value p = pc [56], however there are several ways to express its dependence on the pitch angle. In the following discussion, an unambiguous separatrix in momentum space is obtained by neglecting the effect of collisional diffusion. A common definition of the separatrix divides momentum space into two regions based on whether the accelerating electric field or the friction from Coulomb collisions dominate. The boundary between these regions is described by p = ps , with −1 p2s (ξ) = (ξE/Ec − 1) (2.6) (see for instance Paper H). At the parallel axis (ξ = 1), we find ps (1) = pc , as expected. Equation (2.6) does not take into account the fact that in 2D, the electric field can accelerate a particle into the runaway region, even though it experiences a net slowing-down force (i.e. is not in the runaway region initially). This is possible because of the anisotropy of ps , and the use of Eq. (2.6) leads to an underestimation of the fraction of particles that will run away. A more pertinent definition can be obtained by looking at particle trajectories in phase space [58]. The trajectory which terminates at ξ = 1 and p = pc is the separatrix, since particles on it neither end up in the bulk population nor reach arbitrarily high energies. This trajectory is given by [59] 2 ps,traj (ξ) = −1 ξ+1 E −1 . 2 Ec (2.7) The two separatrices in a typical scenario are shown in Fig. 2.3. In many cases, the runaways form a narrow beam close to the parallel axis, in which case ps and ps,traj give similar results. In fact, in these cases an isotropic runaway region (ps,iso (ξ) = pc ) also is a good approximation (as discussed in Paper C), 12 2.1 The runaway region of momentum space Figure 2.3: Runaway-region separatrices ps , ps,traj and ps,iso at an electric field E/Ec = 800, overlayed on contours of a Maxwellian distribution with Te = 5.1 eV and ne = 5 · 1019 m−3 . F is the distribution f normalized to its maximum value (F = f / max[f ]). and such a separatrix has been included in the figure as well. In certain cases, such as when hot-tail generation dominates (see Sec. 2.2.2), the details of the separatrix are however of importance for the size of the runaway population. The separatrices discussed so far are valid in the limit where the bulk of the distribution is well described by a nonrelativistic Maxwellian, and when including only friction due to Coulomb collisions. As is suggested by the results in Paper D, however; synchrotron radiation reaction may have a significant impact on the separatrix (see also Refs. [60–62]). In general, the separatrix for an arbitrary electron distribution can be obtained by considering the forces that affect a test particle: dp p = FEp − FCp − FSyn , dt dξ ξ = FEξ − FCξ − FSyn , dt (2.8) (2.9) where the expressions for the force associated with the electric field, FEi (with i i ∈ {p, ξ}), and the synchrotron radiation-reaction force FSyn are discussed in 13 Chapter 2. Runaway-electron generation and loss Sec. 3.1 and Sec. 4.2, respectively. The expressions for the collisional electronelectron friction FCi are given in Paper E (see also Sec. 5.1 of that paper). Although other reaction forces (such as bremsstrahlung) may also contribute to the force balance, here we include only the synchrotron radiation-reaction force as it is the dominant contribution in most plasmas of interest. The critical momentum in the parallel direction can be determined from dp/dt = 0 at ξ = 1, since the separatrix becomes purely perpendicular to the parallel axis as ξ → 1. The separatrix can then be traced out by numerically integrating the above equations from ξ = 1 to ξ = −1. In the appropriate limit, the result agrees with ps,traj (ξ). In general, the separatrix depends on the distribution through the terms FCi and should be updated as the distribution changes. It should be noted that in certain situations, additional regions in momentum space emerge which cannot be characterized as either bulk or runaway regions. The example of bump-on-tail formation induced by synchrotron or bremsstrahlung emission, where relativistic electrons accumulate around a certain multi-MeV energy, is discussed in Papers G, H and I (not included in the thesis). The momentum space dynamics leading to the formation of the bump is complicated and involves the interaction between acceleration, pitch-angle scattering and subsequent synchrotron or bremsstrahlung emission back-reaction, forming structures akin to convection cells in the high-energy electron distribution [Paper H]. Such phenomena are not described by the above definition for the separatrix (although generalized models exist that do take them into account, see for instance [60, 63]). 2.2 Runaway-generation mechanisms There are two main mechanisms for generating runaways, referred to as Dreicer [1, 2] and avalanche [50, 64–66] generation. In the former, initially thermal electrons become runaways by a gradual diffusion through momentum space until they reach a velocity where they run away. Dreicer generation is an example of a primary mechanism, as it generates runaways without the need for a pre-existing fast population. Once some runaways exist, one of them may impart a large fraction of its momentum to a thermal electron in a single event, known as a knock-on collision. This generates a second runaway if both electrons are in the runaway region after the collision. This process is also called secondary runaway generation, as it requires the presence of a seed, and leads to an exponential growth of the runaway population (hence the name avalanche). In this section, we will look more closely at these two 14 2.2 Runaway-generation mechanisms mechanisms. We will also discuss another primary runaway mechanism: hottail generation, which can dominate if the plasma temperature decreases on a short timescale. 2.2.1 Dreicer generation Due to momentum-space transport processes, new particles steadily diffuse into the runaway region, increasing the runaway density. This is known as Dreicer generation, and is caused by the gradient ∂f /∂p that develops across the separatrix as particles in the runaway region are accelerated to higher energies. An approximate expression [56, 67–69] for the steady-state growth rate of the runaway population due to this effect is " # r dnr 1 + Zeff 1 −3(1+Zeff )/16 = Cne νth exp − − , (2.10) dt 4 where = E/ED , nr is the runaway number density, νth = ne e4 ln Λ 3 4πε20 m2e vth (2.11) is the electron-electron collision frequency of thermal particles, C is a constant of order unity [4, 56] (not determined by the analytical model), and Zeff is the effective ion charge, which is a measure of the plasma composition (Zeff = 1 is a plasma consisting of pure hydrogen, or otherwise singly charged ions). Equation 2.10 is valid when the distribution is close to a Maxwellian, i.e. when is small; and for E Ec . Closer to the critical field, correction factors are introduced in the exponents so that the growth rate vanishes for E ≤ Ec [56], but these are neglected here for simplicity. Note that the growth rate depends on (and is exponentially small in) E/ED , not E/Ec . This means that even if the field is significantly larger than Ec , the runaway production rate may be very small if E ED . This effect, which is in essence a temperature dependence, can partly explain recent observations indicating that E/Ec & 10 is required for runaway acceleration [41, 42]. Its importance is discussed and quantified in Paper D. 2.2.2 Hot-tail generation Primary runaways can also be produced by processes other than momentumspace diffusion, for instance by highly energetic γ-rays through pair produc- 15 Chapter 2. Runaway-electron generation and loss tion, or in tritium decay (in fusion plasmas). In a typical fusion plasma, these two processes are usually insignificant, however if the conditions are right they may provide a sufficient seed for avalanche multiplication [70]. There is however an additional primary-runaway mechanism – hot-tail generation [47, 71] – which relies on a rapid cooling of the plasma. If the plasma-cooling timescale is significantly shorter than the collision time at which particles equilibrate, electrons that initially constituted the high-energy part of the bulk distribution can remain as a drawn-out tail at the new lower temperature. This is because they take a longer time to equilibrate, as their collision time is significantly longer than that of the slow particles. If an electric field is also present, some of these tail electrons may belong to the runaway region of momentum space and will therefore be accelerated. Under certain circumstances, hot-tail generation can be the dominating runaway-generation mechanism – albeit for a short time – and may provide a strong seed for multiplication by the avalanche mechanism. This is particularly the case in disruptions in tokamaks, where the plasma quickly loses essentially all stored thermal energy due to a sudden degradation in confinement. Approximately, hot-tail generation dominates over Dreicer generation in a disruption if √ 3/2 1 3 π µ0 ene vth,f qa R (2.12) νth,0 tT < 3 4 Ba is fulfilled [59], where νth,0 is the initial thermal collision frequency, tT is the temperature-decay time, µ0 is the vacuum permeability, vth,f is the thermal speed at the final temperature, R is the major radius of the tokamak, and Ba and qa are the magnetic field and so-called safety factor [20] on its magnetic axis. This estimate was obtained by assuming the temperature drop to be described by Te (t) = Te,0 (1 − t/tT )2/3 , with Te,0 the initial temperature. The hot-tail mechanism is discussed in Paper C. See also Refs. [59, 72–74] for a more in-depth discussion of hot-tail growth rates in various cooling scenarios. 2.2.3 Secondary generation Secondary runaways are formed when existing runaways collide with thermal electrons, if the collision imparts enough momentum to the thermal electron to kick it into the runaway region while the incoming (primary) electron remains a runaway itself [50, 64–66]. Such events are referred to as close, large-angle or 16 2.3 Damping and loss mechanisms for runaways knock-on collisions, and are normally rare in a fusion plasma (their contribution to the collisional dynamics is a factor ln Λ smaller than that of small-angle collisions). In the context of runaway generation they become important due to the special characteristics of the runaway region, since once a particle is a runaway it can quickly gain enough energy to cause knock-on collisions of its own. For it to be able to contribute to the avalanche process, the kinetic energy of the incoming runaway must be at least twice as large as the critical energy: γ − 1 > 2(γc − 1). The avalanche growth rate was calculated by Rosenbluth & Putvinski [50], who also derived an approximate operator for avalanche generation (see Paper C and Sec. 3.3 for a detailed discussion of avalanche operators and their influence on runaway dynamics). In a cylindrical plasma, the growth rate takes the form −1/2 4(Z + 1)2 (E − 1) dnr ' nr νrel 1 − E −1 + 2 eff2 , (2.13) dt cz ln Λ cz (E + 3) p where E = E/Ec and cz = 3(Zeff + 5)/π. In the limit where E Ec and Zeff = 1, this simplifies to r (E − 1) dnr π ' nr νrel . (2.14) dt 2 3 ln Λ The growth rate is proportional to the runaway density nr , meaning that the growth is exponential (hence the name avalanche). We also note that the dependence on E is linear in Eq. (2.14), and nearly so in the more general expression (2.13), whereas it is exponential in Eq. (2.10) for the Dreicer growth rate. Therefore, avalanche generation tends to dominate for weak fields (as long as there is some runaway population to start with), but for strong fields primary generation becomes more important. 2.3 Damping and loss mechanisms for runaways The discussion so far has focused on the interplay between the electric field and elastic Coulomb collisions in a quiescent, homogeneous, fully ionized plasma. In practice, runaway electrons do not reach arbitrarily high energies or persist indefinitely. Many processes contribute to the damping of their growth, slowing them down, or transporting them out of the plasma. Of particular importance when it comes to limiting the energy achieved by the runaways are radiative processes: synchrotron-radiation emission due to 17 Chapter 2. Runaway-electron generation and loss (predominantly) the gyro motion, and fast-electron bremsstrahlung due to inelastic scattering off the much heavier ions. The emitted radiation takes away momentum, and the electron must therefore lose a corresponding amount. This radiation reaction effectively introduces an additional force which can counteract the accelerating electric field. Synchrotron radiation is discussed in more detail in Chapter 4 and the effect of bremsstrahlung emission was studied in Paper G. Another factor that can increase the effective friction compared to the classical estimate is partially ionized atoms. The highly energetic runaway electrons may penetrate (parts of) the electron cloud surrounding the nucleus and thus effectively scatter off a charge larger than the net charge of the ion. For heavy ions such as argon or tungsten, which are often present during or after disruptions, this can have a significant effect on the runaway slowing-down [75–79]. The runaways may also lose energy in ionizing collisions. The above mechanisms are pure momentum-space effects. The runaway beam will however eventually occupy a sizable fraction of the tokamak cross-section, and magnetic trapping effects may become important. They typically lead to a reduction in both the Dreicer and avalanche growth rates, which can be as large as 50% already at r/R = 0.1, with r and R the minor and major radii [46]. Additionally, stochastic field-line regions (caused by for instance overlapping magnetic-island structures) can lead to radial transport of the runaways towards the edge of the plasma, where they are eventually lost to the wall [49]. This can be both beneficial (if it occurs early in the acceleration process, before the runaways have reached high energies) and detrimental (if a fully formed, substantial runaway beam is transported into the wall). By applying external magnetic fields with a well-defined periodicity, so-called Resonant Magnetic Perturbations (RMPs), the stochasticity of the edge region of the plasma can be purposefully increased. This can lead to a more rapid radial transport of the runaways, resulting in a reduction in their kinetic energy upon impact with the wall, however since the runaways predominantly form in the center of the plasma, efficient mitigation can be hard to achieve [80–82]. The runaways are also subject to outward radial transport because of another, more fundamental effect: the acceleration of a runaway particle implies a change in its angular momentum with respect to the symmetry axis of the torus. This causes a shift of the runaway orbit away from the flux surface, as the canonical angular momentum of the particle should be conserved [83, 84]. 18 2.3 Damping and loss mechanisms for runaways At high enough particle energies, the runaways will simply drift out of the plasma and into the wall. Another effect not captured by a pure momentum-space treatment is the interaction of the runaways with various waves in the plasma. There is evidence that existing waves, such as toroidal Alfvén eigenmodes, can disperse the runaway beam [85–88]. Due to their highly anisotropic momentum distribution, the runaways may also destabilize and act as a drive for plasma waves, such as the whistler [89–91] and EXtraordinary ELectron (EXEL) waves [92, Paper K], which in turn can affect the runaway distribution and reduce the runaway growth. The picture is thus complicated in practice, however even the basic dynamics of the runaway process are not always well understood. Significant experimental and theoretical effort is spent on improving that understanding and it is the aim of this thesis to contribute to this endeavor. 19 3 Simulation of runaway-electron momentum-space dynamics Although the single-particle estimates considered in Chapter 2 can be useful in describing some of the phenomena associated with runaways, a complete and thorough understanding of their dynamics can only be gained through a treatment of the full kinetic problem. In some idealized situations the equations can be solved analytically, however in general the interplay between the various processes involved in the momentum-space transport of electrons must be studied using numerical tools. The runaways often comprise a small fraction of the total number of electrons, and features in the distribution of electrons many orders of magnitude smaller than the bulk population must be accurately resolved. Continuum discretization methods (i.e. finite difference, element, and volume methods) are well adapted for problems of this type, whereas Monte Carlo methods become inefficient and have problems with numerical noise. In this thesis, two finite-difference tools for studying runaway-electron dynamics are described: CODE (Papers B and C, Sec. 3.4) and NORSE (Paper E, Sec. 3.5). Microscopic Coulomb interaction between particles (collisions) are very important for the runaway dynamics and we will discuss the treatment of both small (Sec. 3.2) and large-angle (Sec. 3.3) collisions. Another important effect is synchrotron radiation reaction, however we postpone the description of the corresponding operator to Sec. 4.2.3. We begin by discussing the equations governing the evolution of the electron population. 3.1 The kinetic equation When it comes to describing plasma phenomena, several theoretical frameworks of varying degrees of complexity (and explanatory power), have been 21 Chapter 3. Simulation of runaway-electron momentum-space dynamics developed. Fluid theories, although tractable, numerically efficient, and useful in other contexts, are based on the assumption that the plasma particles are everywhere in local thermal equilibrium and can be described by nearMaxwellian distributions. In order to treat the runaway-electron phenomenon, such a model is inadequate1 , as the runaways by definition constitute a highenergy (non-thermal) tail of the particle distribution. It is therefore necessary to use kinetic theory, where the distribution of particle positions and velocities is the prime object of study. The so-called kinetic equation describes the evolution of a distribution of plasma particles of species a, fa (x, p, t), according to X ∂ ∂ ∂fa + (ẋfa ) + (ṗfa ) = Cab {fa , fb } + S, ∂t ∂x ∂p (3.1) b where x and p denote the position and momentum, respectively, and ṗ describes the macroscopic equations of motion (given for instance by the Lorentz force due to the presence of macroscopic electric and magnetic fields). The collision operator Ca describes microscopic interactions between the plasma particles (collisions), which are normally treated separately from the macroscopic equations of motion. In general, the collision operator depends on the distributions of all the particle species b in the plasma and includes contributions from both elastic and inelastic Coulomb collisions. In the latter (which are often neglected), photons are emitted and carry away some of the energy and momentum – this radiation is referred to as bremsstrahlung (see for instance Paper G). S represents any sources or sinks of particles or heat, such as ionization and recombination of neutral atoms, fueling in laboratory plasmas or heat lost from the plasma, due to radiative processes. Under certain conditions, the collisions can be neglected, in which case Eq. (3.1) (with S = 0) is known as the Vlasov equation. With a two-particle collision operator valid for arbitrary momentum transfer (or equivalently collision distance), it is called the Boltzmann equation, although in practice several simplifications must be made to be able to treat the collisions. Under the assumption that the momentum transfer in each collision is small, the Boltzmann collision operator simplifies to the Fokker-Planck collision operator, and Eq. (3.1) is correspondingly called the Fokker-Planck equation [93, 94]. This operator is sufficient to treat primary runaway generation, but is not able to describe the 1 Interestingly, in his seminal papers on electron runaway, Dreicer derived the basic runaway dynamics using a two-fluid treatment [1]. He did however recognize the limitations of this description and the follow-up paper uses a kinetic approach [2]. 22 3.1 The kinetic equation avalanche process in which the momentum transfer to the secondary particle in a knock-on collision is significant. Avalanche generation is instead treated by including a special source term Sava , discussed in Sec. 3.3. In general, the distribution fa is defined on a six-dimensional phase-space, and is very demanding to treat in its entirety. Various approximations are routinely employed to reduce the kinetic equation to a manageable number of dimensions (see for instance Ref. [54]). In many situations, the fundamentals of the runaway problem can be studied in a homogeneous plasma, so that the spatial dependence can be ignored. In addition, one of the momentumspace dimensions (describing the rapid gyro motion around the magnetic field lines) can be averaged over if a sufficiently strong magnetic field is present (so that the gyro radius is ignorable in comparison to the typical length scale of the gradients in the plasma and the gyration time is short compared to the timescales of other processes). The tools developed here therefore solve the kinetic equation in two momentum-space dimensions only, allowing for fast calculation while most of the relevant physics is retained. The kinetic equation implemented in CODE and NORSE can be expressed as ∂fe eE ∂fe ∂ − · + · (Fsyn fe ) = Cee {fe } +Cei {fe } +Sava +Sp +Sh , (3.2) ∂t me c ∂p ∂p where the second term describes the acceleration due to the electric field, the third term describes the effects of synchrotron radiation reaction (see Sec. 4.2), Cee and Cei describe collisions with electrons and ions, respectively, and Sp and Sh are sources of particles and heat. The two momentum-space dimensions are conveniently described by the coordinates (p, ξ) introduced in Sec. 2.1.3. In these coordinates, the electric-field term becomes eEk eE ∂fe ∂fe 1 − ξ 2 ∂fe · = + ξ . (3.3) me c ∂p me c ∂p p ∂ξ Equation (3.2) is then solved for the electron distribution. Both CODE and NORSE can calculate the time evolution of fe , starting from some initial (often Maxwellian) distribution, and CODE is also able to determine the (quasi) steady-state distribution directly (in the absence of an avalanche source). In general, parameters such as the electric field, effective charge, temperature, and density may vary in time, and both tools have the ability to model this. Such capability is necessary in order to describe hot-tail runaway generation and other dynamic scenarios. 23 Chapter 3. Simulation of runaway-electron momentum-space dynamics Throughout the rest of this thesis, we will omit the subscript e, and let f denote the distribution function of electrons. We will also assume an implicit minus sign in the electric field, so that the runaway electrons are accelerated in the positive-p direction. The evolution of the distribution function in a typical runaway case is shown in Fig. 3.1. Starting from a Maxwellian distribution, the electric field pulls out a high-energy tail centered around p⊥ = 0 (but with significant spread in p⊥ due to pitch-angle scattering). In 500 thermal collision times, the tail of the distribution reaches pk ≈ 8, which corresponds to a kinetic energy of 3.6 MeV. 3.2 Collision operator As discussed in the previous section, the two tools CODE and NORSE have many of the same capabilities. The main difference between them lies in the treatment of the electron-electron Coulomb collisions. CODE uses a collision operator linearized around a Maxwellian, taking advantage of the fact that the runaways in many cases constitute a small part of the electron distribution, so that the collisions between runaways may be neglected. This approach allows for very efficient numerical evaluation of the problem as long as the plasma parameters remain constant. NORSE, on the other hand, uses a fully nonlinear relativistic collision operator [95–97], which makes it possible to treat distributions of arbitrary shape. Thus, NORSE can be used in situations where the runaways make up a sizable fraction of the distribution, or where the electric field is strong enough that the electron population is in the slideaway regime (E > Esa ). For a thorough discussion of collision operators in general, see Ref. [54]. The generally valid collision operator in NORSE accurately treats the elastic electron-electron collisions in the Fokker-Planck limit. However, in the linearization procedure used to derive the operator in CODE, some properties of the full operator are compromised. In particular, the linearized operator is often written as a sum of so-called test-particle and field-particle terms: l tp fp Cee {f } ' Cee {f } = Cee + Cee , (3.4) tp where the test-particle term Cee = Cee {f1 , fM } describes collisions of the fp perturbation f1 with the bulk plasma (fM ), and the field-particle term Cee = Cee {fM , f1 } describes the reaction of the bulk to the perturbation. Here, f = fM + f1 with f1 fM , and thus collisions between particles represented 24 3.2 Collision operator Figure 3.1: Evolution of the electron distribution under a constant electric field of E = 0.4 V/m (corresponding to E/Ec = 9.6 and E/ED = 0.056). Contours of the distribution (F = f / max[f ]) in 2D momentum space are shown at a) the initial time, b) τth = 167, c) τth = 333, and d) τth = 500 thermal collision times. The parameters were: T = 3 keV, n = 5 · 1019 m3 , Zeff = 2 and B = 0, and the results were obtained using CODE with avalanche generation disabled. 25 Chapter 3. Simulation of runaway-electron momentum-space dynamics by the perturbation (Cee {f1 , f1 }) have been neglected as they are second order in the small quantity f1 .2 It is common in runaway studies to neglect the fieldparticle term, as it only affects the bulk plasma and complicates the problem. If this is done, however, the conservation properties of the linearized operator l are compromised as the test-particle term only conserves particles, not Cee momentum or energy. This does not significantly affect the runaway dynamics, but is important for the accurate determination of properties of the bulk (such as the conductivity). CODE includes both test-particle and field-particle terms, as discussed in Paper C, however unlike in NORSE, the latter term is nonrelativistic (i.e. the bulk temperature is assumed to be small compared to the electron rest energy). The test-particle term in CODE, which is valid for arbitrary energies, was derived in Ref. [80]. Electron-ion collisions can be described by a much simpler operator, due to the mass difference between the species involved in the collision (and assuming the ions to be immobile on the timescales of interest). Both CODE and NORSE use an electron-ion collision operator which describes pitch-angle scattering, but neglects the energy transfer to the much heavier ion [54]. As part of the work on Paper G (not included in this thesis), an operator for inelastic (bremsstrahlung) Coulomb collisions was developed, and is available in CODE [98]. 3.3 Avalanche source term The avalanche process due to large-angle Coulomb collisions between existing runaways and thermal electrons cannot be captured using the Fokker-Planck formalism, and a special source term – derived from the Boltzmann collision operator – must be included to treat this process. In a linearized formulation, several avalanche sources describing the creation of the secondary runaway particles can be formulated (using various assumptions, as will be discussed shortly), however in the case of a strongly non-Maxwellian distribution function f , no such easily tractable operator is available. For this reason, the avalanche operators described below are included in CODE but not in NORSE. 2 Note that for the runaway problem, it is not required that f1 (p, ξ) fM (p, ξ) for all p and ξ – only that the perturbation is small in a global sense, so that collisions between runaway particles can be ignored. In the tail, the perturbation is usually many orders of magnitude larger than the Maxwellian at the corresponding momentum. 26 3.3 Avalanche source term The generally-valid Boltzmann collision operator is notoriously difficult to handle numerically, and an efficient solution of the runaway problem requires the use of reduced models. Work on a simplified conservative treatment of the avalanche process is ongoing [Conf. Contrib. S], but no practical such operator is yet available. From the kinematics of a single large-angle collision, a source term for the generated secondary particles can however be derived, taking the energy distribution of the incoming electrons into account by utilizing the full Møller scattering cross-section [99]. This was demonstrated by Chiu et al. in 1998 [47]. This operator obeys the kinematics of the problem (in the sense that the momentum of the generated particle is restricted by that of the incoming particle), however since no sink of particles is included, it violates the conservation properties of the full Boltzmann operator. No modification to the momentum of the incoming particle is made and no particle is removed from the thermal population. Nevertheless, the operator in Ref. [47] is able to accurately capture the exponential growth of the avalanche. The source term at a point (p,ξ) takes the form SCh (p, ξ) = 1 νrel p4in f˜(pin )Σ(γ, γin ) , 2 ln Λ γpξ (3.5) where pin and γin are the normalized momentum and relativistic mass factor of the incoming primary runaway, γ is the relativistic mass factor for the generated secondary runaway, f˜ is the angle-averaged electron distribution (i.e. all incoming particles are assumed to have vanishing pitch-angle), and Σ is the Møller cross section. Note that due to the kinematics of the problem, the coordinates are related in such a way that only primary particles with a single pin can contribute to the source at a point (p, ξ). By taking the high-energy limit of the Møller cross section, i.e. assuming that the incoming primaries are all highly relativistic, the source term can be simplified further. The resulting operator, first derived by Rosenbluth and Putvinski the year before [50], is widely used and given by nr νrel 1 ∂ 1 SRP (p, ξ) = δ(ξ − ξs ) 2 . (3.6) 4π ln Λ p ∂p 1 − γ Due to the assumptions used, the kinematics are restricted further, and all secondary runaways are generated on a parabola given by ξs = p/(1 + γ). However, since every runaway is assumed to have very large momentum, secondary particles can be generated with momenta larger than that of any of the particles in the actual distribution. This is illustrated in Fig. 3.2, where 27 Chapter 3. Simulation of runaway-electron momentum-space dynamics Figure 3.2: Electron distribution (F = f / max[f ]) after 10 thermal collision times with a) the source in Eq. (3.5) and b) the source in Eq. (3.6). This early in the evolution, the result with only primary generation included is indistinguishable from that in a). The parameters were: T = 300 eV, n = 5 · 1019 m3 , E = 0.5 V/m (corresponding to E/Ec = 12 and E/ED = 0.07), Zeff = 1 and B = 0, and the results were obtained using CODE. an unphysical horn-like structure is created, extending to large momenta. In some situations, the Rosenbluth-Putvinski model therefore tends to overestimate the avalanche growth rate, compared to the source term of Chiu et al. However, as is shown in Paper C; in certain parameter regimes, the opposite tendency is seen. This is due to the non-trivial dependence of the Møller cross section on the momenta of the colliding particles. If secondary generation dominates, the quasi-steady-state runaway distribution function can be calculated analytically (assuming a growth rate consistent with the Rosenbluth–Putvinski source), and is given by fava (pk , p⊥ ) = pk nr Ê p2 exp − − Ê ⊥ , πcz pk ln Λ cz ln Λ pk (3.7) p where Ê = (E/Ec − 1)/2(1 + Zeff ) and cz = 3(Zeff + 5)/π [89]. Equation (3.7), which is valid when γ 1 and E/Ec 1, was used extensively in the calculation of synchrotron spectra in Paper A, and also as a benchmark in Paper B. An example distribution is plotted in Fig. 3.3. 28 3.4 CODE Figure 3.3: Contour plot of the tail of the analytical avalanche distribution in Eq. (3.7) for Te = 1 keV, ne = 1 · 1020 m−3 , Zeff = 1.5 and E/Ec = 15. The distribution is not valid for the bulk plasma, and is therefore cut off at low momentum (in this case at pk = 5). 3.4 CODE CODE (COllisional Distribution of Electrons) was developed to be a lightweight tool dedicated to the study of the properties of runaway electrons. It solves the kinetic equation (3.2) using a finite-difference discretization of p together with a Legendre-mode decomposition of ξ, and an implicit time-advancement scheme. The discretization is advantageous as the Legendre polynomials are eigenfunctions of the collision operator, allowing for a straight-forward implementation and an efficient numerical treatment. In particular, time advancement can be performed at low computational cost, as it is sufficient to build and invert the matrix representing the system only once, provided that the plasma parameters are independent of time. The system can then be advanced in time using just a few matrix operations in each time step. For small to moderately-sized problems, such as the scenario considered in Fig. 3.1, CODE runs in a couple of seconds on a standard desktop computer. More involved set-ups involving time-dependent plasma parameters or low temperatures in combination with large runaway energies execute in minutes or sometimes hours. Memory requirements range from a few hundred MB (or less) to tens of GB, depending on resolution. CODE, which is written in Matlab, has contributed to a number of studies and is used at several fusion sites around the world. The original version, described in Paper B, included the relativistic testparticle collision operator [80] and the Rosenbluth-Putvinski avalanche operator in Eq. (3.6), as well as the ability to find both time-dependent and steady-state solutions for f . Subsequent extensions include: time-dependent 29 Chapter 3. Simulation of runaway-electron momentum-space dynamics plasma parameters, the field-particle term of the collision operator and an associated heat sink, the Chiu et al. avalanche operator (all described in Paper C), and operators for synchrotron (Paper D, Sec. 4.2.3) and bremsstrahlung (Paper G) radiation reaction. Various other technical features, such as efficient non-uniform automatically extending finite-difference grids, have also been added [Conf. Contrib. V]. 3.5 NORSE Building on the experience gained from the work on CODE, NORSE (NOnlinear Relativistic Solver for Electrons) was developed to extend the range of applicability of runaway-modeling tools to the regime where the runaway population becomes substantial; i.e. the regime of greatest concern in practice. The fully nonlinear treatment also makes it possible to consistently study phenomena such as heating by the electric field, which is discussed in Paper F and Sec. 5.1. The numerical implementation of NORSE – described in detail in Paper E – differs substantially from that in CODE. In particular, a linearly implicit timeadvancement scheme is used to treat the nonlinear problem. The electronelectron collision operator implemented in NORSE is formulated in terms of five potentials (analogous to the two so-called Rosenbluth potentials [94] in the nonrelativistic case), which are functions of the distribution [96, 97]. By calculating the potentials explicitly in each time step from the known distribution, the remainder of the kinetic equation can be formulated as a matrix equation, which can be solved implicitly using standard techniques. Unlike in CODE, this scheme requires the formation and inversion of a matrix in each timestep, and is therefore in general more computationally demanding. If time-dependent parameters are used, however; NORSE is marginally faster than CODE due to an improved numerical implementation. NORSE, also written in Matlab, uses a discretization scheme different to that in CODE, as required by the nonlinear problem. Momentum space is discretized on a two-dimensional non-uniform finite-difference grid, which allows for improved numerical efficiency as the grid points can be chosen in a way suited to the problem at hand (i.e. smaller grid spacing close to ξ = 1 where the runaway tail forms, but larger grid spacing around ξ = 0 where the distribution lacks fine-scale features). For the explicit calculation of the five potentials, however; a mixed finite-difference–Legendre-mode representation is employed (similar to CODE), as in this basis the potentials become one-dimensional in- 30 3.5 NORSE tegrals over p. This representation is only used to calculate the potentials – which are integral moments of f – and a small number of Legendre modes are often sufficient for an accurate treatment. The mapping between the two representations can be performed to machine precision at little computational cost. NORSE includes time-dependent plasma parameters, an operator for synchrotron radiation reaction, a runaway region determined from the distribution in accordance with Eqs. (2.8) and (2.9), and elaborate heat and particle sinks. 31 4 Synchrotron radiation Charged particles in accelerated motion emit radiation [100]. In the presence of a magnetic field, the particles in a plasma follow helical orbits as a consequence of the Lorentz force [21, 22]; in other words, they are continuously accelerated “inwards” – perpendicular to their velocities. The radiation emitted by electrons due to this motion is known as cyclotron radiation if the particles are nonrelativistic (or mildly so) and synchrotron or betatron radiation if they are highly relativistic (the names come from the types of devices where the radiation was first observed [101]). Synchrotron radiation has many applications in the study of samples in condensed matter physics, materials science, biology and medicine, where it is used in for instance scattering and diffraction studies, and for spectroscopy and tomography [102]. The radiation is usually produced using dedicated facilities (synchrotrons), but is also emitted in some natural processes, in particular in astrophysical contexts (where it can be used as a source of information about the processes in question). From the distinction between cyclotron and synchrotron radiation, it is evident that in a nonrelativistic plasma (with a temperature significantly below 511 keV), only the far tail of the electron distribution may emit synchrotron radiation. The only plasma particles that reach highly relativistic energies are the runaway electrons; the study of the synchrotron emission from a plasma is thus a very important source of information about the runaways, and their dynamics. Synchrotron radiation plays an important role in several of the papers included in this thesis. In Papers A and B, it is used as a source of information about the runaway population – as a “passive” diagnostic not affecting the electron distribution. This is discussed in Sec. 4.1. In Paper D (and also Papers H and I, not included in this thesis), the impact of the emission on the distribution is analyzed – i.e. the synchrotron radiation plays an “active” role through the radiation reaction associated with its emission, as discussed in Sec. 4.2. 33 Chapter 4. Synchrotron radiation 4.1 Emission and power spectra The theory of synchrotron radiation was first derived by Schott in 1912 [103], but was rediscovered, and to a large extent reworked, by Schwinger in the 40’s [104]. More recent, detailed discussions of the properties of synchrotron radiation can be found in Refs. [100, 102, 105, 106]. In the rest frame of the particle, the synchrotron radiation is emitted almost isotropically, however the transformation to the lab frame introduces a strong forward-beaming effect. Since the motion of the particle is predominantly parallel to the magnetic field lines (the runaways are accelerated by Ek ), the synchrotron radiation will be emitted in this direction as well, even though it is the perpendicular motion of the particle that is the cause of the emission. Since the synchrotron radiation is directed, its observation requires detectors in the right location and with the right field of view. In many tokamak experiments this necessitates the use of dedicated cameras for the study of synchrotron emission from runaways, and the number of such set-ups around the world is limited but has seen an increase in recent years. Synchrotron emission from runaways has now been observed in a number of tokamaks, including TEXTOR [107, 108], DIII-D [41, 109], ASDEX Upgrade [39], Alcator C-Mod [Conf. Contrib. L], FTU [110], EAST [111], KSTAR [25], and COMPASS [37]. Another important question is the emission spectrum, since it determines the detector type to use. As part of the work on Paper A, the numerical tool SYRUP (SYnchrotron emission from RUnaway Particles) was developed to calculate the synchrotron spectra of both single electrons and runaway distributions. As we shall see, the emission has a distinct peak; for runaways in tokamaks it is often located in the near infrared, at wavelengths of a few µm. Electrons with energies above about 20–25 MeV do however also emit a substantial amount of radiation in the visible range. The majority of the observations above were done using fast visual cameras, due to the availability and good performance of the technology compared to infrared detectors (although several IR systems are also in use). Visual cameras also provide a more sensitive diagnostic of the highest-energy part of the runaway distribution, which is often considered to be of most interest. In this section, we will first examine the synchrotron spectra emitted by single electrons, followed by a generalization to a distribution of runaway electrons. This topic is also discussed in more detail in Paper A. 34 4.1 Emission and power spectra 4.1.1 Single-particle spectrum in a straight magnetic field The frequency of the cyclotron or synchrotron radiation emitted by a particle is a multiple of the frequency with which it orbits the magnetic field line (the gyro or cyclotron frequency ωc = eB/γme ). In the case of cyclotron emission, the fundamental and the first few harmonics dominate completely, whereas for the high-energy synchrotron emission, the high harmonics (up to some cut off) are dominant. Since these are spaced very close together (compared to the scale of the fundamental frequency), synchrotron radiation essentially has a continuous frequency spectrum [105]. The emission can span a large part of the electromagnetic spectrum, from microwaves to hard x rays, depending on the frequency of the gyro motion and the electron energy. In terms of quantities convenient for plasma physics [105], the synchrotron power spectrum emitted by a fast electron can be expressed as Z ∞ 1 ce2 P(λ) = √ K5/3 (l) dl , (4.1) 3 ε0 λ3 γ 2 λc /λ where Kν (x) is the modified Bessel function of the second kind (of order ν), and λc is a critical wavelength given by λc = 4π cme γk 4π c = , 3 ωc γ 3 3 eBγ 2 (4.2) with γk = (1 − vk2 /c2 )−1/2 the relativistic γ factor due to the motion parallel to the magnetic field, and B the magnetic field strength. The spectrum, which peaks at λ ' 0.42λc , is plotted in Fig. 4.1 for a few different parameter sets. There is a sharp cutoff at short wavelengths, but a much slower decay towards longer wavelengths. Note that both the peak wavelength and the emitted power are sensitive to both the particle energy and pitch. In particular, it is possible to produce similar synchrotron spectra using significantly different parameter sets, which makes it difficult to treat the inverse problem of finding the parameters of a particle (or distribution) that produced a certain spectrum. 4.1.2 Single-particle spectrum in a toroidal magnetic field Equation (4.1) considers the radiation emitted due to pure gyro motion around a straight field line. In a tokamak, particle orbits are more complicated since 35 Chapter 4. Synchrotron radiation Figure 4.1: Synchrotron power spectrum for a single electron with kinetic energy Ek and pitch described by tan θ = v⊥ /vk (with θ the pitch angle). both the motion around the torus, that due to the helicity of the field lines, and various drifts contribute. The synchrotron power spectrum for a particle trajectory including the gyro motion, the motion along a toroidal magnetic field, and vertical centrifugal drift was derived by I. M. Pankratov in 1999 [112], and is Z ∞ ce2 P(λ) = g(y) J0 aζy 3 sin (h(y)) dy 3 2 ε0 λ γ 0 Z ∞ π , (4.3) −4a y J00 aζy 3 cos (h(y)) dy − 2 0 where a = η/(1 + η 2 ), g(y) = y −1 + 2y, h(y) = 3ζ y + y 3 /3 /2, ζ= η= 4π R p , 3 3 λγ 1 + η2 eBR v⊥ ωc R ' tan θ, 2 γme vk c (4.4) (4.5) R is the tokamak major radius, Jν (x) is the Bessel function, Jν0 (x) its derivative with respect to the argument, and θ is the pitch angle. The parameter η 36 4.1 Emission and power spectra is the ratio between the perpendicular and drift velocities of the particle and determines how much the radius of curvature (and thus the synchrotron emission) varies along the particle orbit [112, 113]. The parameter ζ is proportional to the ratio λmax /λ, where λmax is the wavelength where the spectrum peaks (the exact expression for which depends on the parameter regime [113]). The integrands in Eq. (4.3) are products of Bessel functions and trigonometric functions and are highly oscillatory with respect to the variable of integration (y). Because of this, numerical integration – although possible – is not straight-forward. For wavelengths shorter than the maximum, the oscillations become particularly rapid, as ζ (which appears in the arguments of both the Bessel and trigonometric functions) becomes large. In Ref. [112], two asymptotic forms of Eq. (4.3) are also derived. These use approximations for the integrals, meaning that they are more suited for numerical implementation. In Paper A, the three formulas of Ref. [112], together with Eq. (4.1), are studied and compared for a variety of tokamak parameters and it is concluded that the cylindrical limit (Eq. 4.1) is a good approximation to Eq. (4.3) in large devices, whereas in devices with small major radius, one of the asymptotic expressions is more suitable in terms of approximating Eq. (4.3). In general, however, the power spectra from the various expressions are similar. 4.1.3 Spectrum from a runaway distribution The total synchrotron power emitted by an electron in circular motion is [104] Ptot = e2 ω⊥ 3 4 β γ , 6πε0 ρ ⊥ (4.6) where ω⊥ is the angular velocity, ρ is the radius of curvature and β⊥ = v⊥ /c. In a homogeneous magnetized plasma, the angular velocity and curvature radius are the Larmor frequency and radius, respectively: ωc and rL = v⊥ /ωc . This gives Ptot = e4 e2 ωc2 3 4 e4 2 2 2 β⊥ γ = B β γ = B 2 p2⊥ . ⊥ 6πε0 v⊥ 6πε0 m2e c 6πε0 m2e c (4.7) The total emitted power thus scales as p2⊥ = γ 2 (v⊥ /c)2 = γ 2 sin2 θ (v/c)2 ≈ γ 2 θ2 , with θ the particle pitch angle, meaning that the most energetic particles with the largest pitch angles emit most strongly. It has therefore been 37 Chapter 4. Synchrotron radiation assumed that the emission from these particles completely dominates the spectrum, and when interpreting synchrotron spectra and emission patterns, the simplification of considering a mono-energetic beam of electrons with a single pitch – referred to here as the “single-particle approximation” – has frequently been employed (see for instance Refs. [109, 114–116]). Less approximate synchrotron spectra can be calculated by using the average emission from the entire runaway distribution, according to Z Z 2π f (p|| , p⊥ ) P p⊥ dp⊥ dp|| = P̄ (p|| , p⊥, λ) dp⊥ dp|| , (4.8) P (λ) = nr Sr Sr where P = P(p|| , p⊥, λ) is one of the single particle emission formulas (i.e. Eqs. 4.1 and 4.3), Sr is the runaway region in momentum space, and P̄ is the integrand – the contribution to the emitted synchrotron power from a given region of momentum space. Spectra calculated from runaway distributions using the above equation are studied in detail in Papers A and B. The validity of the single-particle approximation can be assessed by examining the contribution to the total synchrotron emission at a given wavelength from different parts of momentum space. This is done in Fig. 4.2. The figure shows a runaway distribution and the corresponding contribution to the emission (the quantity P̄ ) for two different wavelengths1 . In this case, the spectrum (calculated using Eq. 4.1) peaks at around 15 µm. At short wavelengths (Fig. 4.2b), the emission is localized to the particles of largest energy and pitch angle, as expected from the simple argument above. At somewhat longer wavelengths (Fig. 4.2c), however; particles in a larger part of momentum space contribute (and the emission is much stronger in general). In addition, for this wavelength, the main contribution is not from the particles with the largest pitch-angle, but P̄ instead peaks closer to the parallel axis. Both these effects lead to the single-particle approximation becoming a poor estimate in general (at the very least, different particle energies and pitches would have to be used to approximate the distribution at different wavelengths). This fact is also evident from the comparison of single-particle and runawaydistribution synchrotron spectra, where the spectra generally differ both qualitatively and quantitatively, as shown in Papers A and B. Including the full runaway distribution in the calculation is thus absolutely necessary to obtain 1 The parameters used were T = 300 eV, n = 5 · 1019 m−3 , E = 1 V/m, Zeff = 1 and B = 5.3 T, and the distribution was obtained by running CODE for 17000 collision times, including synchrotron-radiation back-reaction, but neglecting avalanche runaway generation. 38 4.1 Emission and power spectra -10 9 -15 6 3 -20 0 -25 9 2 6 1 3 0 0 1 9 6 0.5 3 0 0 0 5 10 15 20 25 Figure 4.2: a) Electron distribution in 2D momentum space and b)–c) contribution to the corresponding synchrotron emission at wavelengths b) λ = 700 nm and c) λ = 7 µm. The plotted quantity in b) and c) is P̄norm = P̄ / max[P̄7µm ], i.e. P̄ normalized to its maximum value for λ = 7 µm. Note the difference in emission amplitude for the two wavelengths. 39 Chapter 4. Synchrotron radiation accurate results, and in this sense SYRUP constitutes a significant improvement over previous methods. The synchrotron spectrum is very sensitive to changes in the plasma parameters and electric-field strength. This sensitivity is due to the dependence on the exact shape of the runaway distribution, something which cannot be captured by the mono-energetic approximation. It is clear from the discussion above that particles of many different energies and pitches contribute to the observed synchrotron spectrum. This is also true for the spatial shape of the detected radiation spot, where particles at different spatial locations and with different momenta contribute to overlapping regions on the detector plane. This added complication makes the task of extracting information about the runaway distribution from the synchrotron image highly non-trivial. In the analysis in Paper A, analytical avalanche distributions were used. The analytical formula (Eq. 3.7) represents a steady-state limit, however, and is not able to capture dynamical effects or describe the synchrotron emission in the early stages of the runaway-population evolution. In addition, it does not include the effect of radiation-reaction losses, which can have a significant effect when E/Ec is small. In Paper B, numerical distributions from CODE were used to study both dynamic phenomena and distributions where the Dreicer mechanism was dominant. Excellent agreement was also found between the numerical distribution and Eq. (3.7) at sufficiently late times. 4.2 Radiation-reaction force As an electron emits radiation, it receives an impulse in the opposite direction due to the conservation of momentum. There is therefore a radiation-reaction force Fsyn associated with the emission of synchrotron radiation, which acts to slow the particle down (and reduce its pitch angle). Synchrotron emission only becomes important at relativistic energies, however; contrary to collisional friction, the radiation reaction force increases with particle speed (in accordance with the estimate in the previous section). This completely changes the force balance for runaways at high momentum and non-vanishing pitch angles2 . In the following discussion it is convenient to consider only the onedimensional single-particle force balance to qualitatively illustrate the impact of the synchrotron radiation reaction on the dynamics. In practice, however; 2 The force balance exactly on the parallel axis remains unchanged, as non-vanishing p⊥ is required for the emission of synchrotron radiation. 40 4.2 Radiation-reaction force |F| Ff eE Fsyn eE>Ff FC eE*c eEc vth v1 vmin Sr v2 c v Figure 4.3: Schematic representation of the forces associated with total friction (Ff ), collisional friction (FC ), synchrotron radiation reaction (Fsyn ), the (classical) critical field (Ec ) and the critical field including synchrotron radiation reaction (Ec∗ ). The two critical velocities v1 and v2 corresponding to the electric field E are also shown, together with the runaway region (Sr ), and the speed at which the total friction force is minimized (vmin ). See Fig. 2.2 for comparison. Note that the velocity scale is chosen for clarity – in practice both vmin and v2 lie close to c. the problem involves transport processes in two-dimensional momentum space and must in general be treated using numerical tools, see Chapter 3. The force balance in the presence of synchrotron radiation reaction is depicted in Fig. 4.3. Two effects are of particular interest in the figure: firstly, the radiation-reaction force effectively prevents runaways from reaching arbitrary energies, and secondly, it raises the critical field for runaway generation. We will discuss these two effects in turn in the following sections. 4.2.1 A limit on the achievable runaway energy As can be seen in Fig. 4.3; for a given electric field E, there are two speeds (v1 and v2 ) larger than the thermal speed for which the total friction force equals the accelerating force: Ff (v1,2 ) = |eE|. Runaway is only possible for v1 < v < v2 , meaning that particles are unlikely to reach energies significantly higher than that corresponding to v2 . This has been suggested as a possi- 41 Chapter 4. Synchrotron radiation ble mechanism limiting the achievable runaway energy, both in the case of synchrotron and bremsstrahlung radiation reaction [60, 66, 117, 118]. An additional consequence of the existence of an upper bound to the runaway region is that particles tend to accumulate in the vicinity of the speed v2 in velocity space. Provided certain conditions are satisfied, momentumspace transport mechanisms associated with this process can even lead to the formation of a non-monotonic feature – a “bump” – on the runaway tail, as discussed in Papers G, H and I (not included in the thesis). 4.2.2 Effect on the critical field As indicated by Fig. 4.3, the minimum of the friction force for any nonvanishing ξ is no longer found at v = c (see Sec. 2.1.1), but at some intermediate speed vmin satisfying v1 < vmin < v2 . (Note that, since the synchrotron emission is a relativistic effect, vmin will nevertheless be close to c.) Therefore, the minimum field necessary for runaway generation to occur is raised accordingly: |eEc∗ (ξ)| = Ff (vmin , ξ), (4.9) where Ec∗ (ξ) is the critical field at a given ξ in the presence of radiation reaction forces and Ec∗ > Ec for all non-vanishing ξ, as depicted in Fig. 4.3. Note that the force balance on the parallel axis (ξ = 1) remains unaffected by the radiation reaction, and thus the critical field Ec is unchanged if collisional diffusion is neglected. In practice, however; diffusive and dynamic processes play an important role in determining the effective critical field. The full problem is studied in Paper D using simulations in CODE, and it is found that the synchrotron radiation reaction can reduce the Dreicer growth rate by orders of magnitude for E/Ec close to unity, corresponding to an increase in the effective critical field. 4.2.3 Operator for synchrotron radiation reaction To understand the full role of radiation-reaction effects on plasma dynamics, the single-particle treatment considered above is insufficient. Fully kinetic simulations are necessary to capture the interplay between the various processes affecting the momentum-space transport. Such calculations can be 42 4.2 Radiation-reaction force performed using CODE or NORSE, and for this purpose an operator describing the radiation reaction is needed. The radiation-reaction force can be calculated from the Abraham-LorentzDirac force affecting an electron [119], e2 γ 2 3γ 2 γ2 3γ 2 2 Frad = v̈ + 2 (v · v̇) v̇ + 2 v · v̈ + 2 (v · v̇) v , (4.10) 6πε0 c3 c c c where v is the velocity of the particle. Assuming that the magnetic force dominates, so that the particle is predominantly accelerated perpendicular to its velocity (v · v̇ = 0), the expression can be simplified to γp 1 − ξ 2 p Fsyn = − (4.11) τ p r pξ 1 − ξ 2 ξ Fsyn =− , (4.12) γτr where τr = 6πε0 (me c)3 /e4 B 2 is the radiation-damping timescale. The radiationreaction force enters the kinetic equation (3.2) as an operator of the form (∂/∂p) · (Fsyn f ), and using Eqs. (4.11) and (4.12), the explicit form is ∂ 1 − ξ2 ∂f 2 ∂f · (Fsyn f ) = − −ξ + 4p2 + f . (4.13) γ2p ∂p γτr ∂p ∂ξ 1−ξ 2 The force acts to limit both the particle energy and pitch, which is to be expected as the emitted synchrotron power is proportional to Ptot ∼ γ 2 θ2 (see Sec. 4.1.3). Fig. 4.4 shows the effect on the distribution: its width in p⊥ and extension in pk is reduced in the presence of a magnetic field. Equations (4.11) and (4.12) were derived in Ref. [117], the first paper to properly consider the role of radiation reaction in runaway dynamics, but in that paper a high-energy limit was used which lead to an incomplete expression in place of Eq. (4.13), as pointed out in [120]. 43 Chapter 4. Synchrotron radiation Figure 4.4: Electron distribution functions in CODE after 700 thermal collision times, a) without and b) with synchrotron radiation reaction included. The parameters were T = 3 keV, n = 1 · 1019 m−3 , E = 0.05 V/m, Zeff = 1 and B = 6 T. 44 5 Nonlinear effects and slide-away When studying the behavior of runaway electrons, a very common approximation is to treat the runaways as a perturbation to an electron population in local thermal equilibrium. This linearization of the distribution function f around a Maxwellian significantly simplifies the collision operator Cee in Eq. (3.2), as well as the numerical method needed to solve the kinetic equation. However, this approach is not valid when the runaway population becomes substantial, or the linearization becomes invalid for other reasons (such as when a strong electric field E & Esa shifts the bulk of the electron population). There is also a risk that the linearized treatment fails to accurately describe subtle phenomena such as feedback loops (which may for instance cause rapid depletion of the thermal population). The tool NORSE was developed to be able to treat such situations, as discussed in Chapter 3. Here we will introduce two particular nonlinear effects considered in Papers E and F: Ohmic heating and slide-away. 5.1 Ohmic heating The accelerating force of the parallel electric field affects all electrons, not only runaways. However, for most particles, the energy gained from the electric field is quickly redistributed into random motion through collisions with other plasma particles. As a consequence, the electric field acts as a source of heat, primarily affecting the thermal bulk of the electron population – this is commonly referred to as Ohmic or Joule heating. The associated change in the energy W of the electron population is given by the energy moment of the electric-field term in the kinetic equation: Z dW E ∂f 3 2 = d p (γ − 1)me c − · , (5.1) dτ Ec ∂p with τ = νrel t the time in units of the collision time for relativistic electrons. If the electric field is sufficiently strong, the heating of the bulk can be sub- 45 Chapter 5. Nonlinear effects and slide-away stantial. This can have consequences for electron slide-away, as discussed in the next section. The Ohmic heating is automatically accounted for in a nonlinear treatment of the kinetic equation, making NORSE able to consistently model this process. This is less straight-forward to do in a linearized tool such as CODE, since the heating effectively involves a change to the Maxwellian around which the distribution is linearized; either the heat must be removed, or the properties of the Maxwellian modified. The heat supplied by the electric field does not always stay in the plasma, however. Spatial temperature gradients lead to heat transport (a fundamental problem in fusion reactors), although this process may or may not be relevant on the runaway acceleration timescale. More important in a cold post-disruption fusion plasma may be the energy radiated away via atomic transitions in partially ionized impurity ions (the so-called line radiation), or spatial transport of particles and energy due to unstable modes (sometimes driven by the runaways themselves – see for instance [90] and Paper K). As a consequence, the plasma temperature may stay constant, or even decrease, despite the existence of a strong electric field. 5.2 Electron slide-away As discussed in Sec. 2.1.2: if the electric field is stronger than the collisional friction in the entire momentum space, all electrons are accelerated, which is known as the slide-away regime [1, 57]. Recalling that the slide-away field Esa ∼ n/T (see Eq. 2.5), where these quantities are those of the bulk (since the maximum of the friction force is located at around v = vth ), a transition to the slide-away regime can be induced through either of the following four mechanisms: I: E > Esa – For electric fields stronger than the classical slide-away field Esa , slide-away is immediate II: E . Esa – Although the field is slightly weaker than Esa , it distorts the distribution, which lowers the friction. This process is a positive feedback loop, since the reduced friction makes acceleration easier, leading to further distortion of the distribution, reduced friction, etc. Eventually, the friction is sufficiently reduced that the electric-field acceleration dominates everywhere, leading to slide-away. This process happens on 46 5.2 Electron slide-away timescales comparable to the thermal collision time, and is thus very quick. III: E < Esa , no or inadequate heat sink – The field is significantly weaker than Esa but supplies heat to the distribution. This heat is not efficiently removed, which leads to Ohmic heating of the bulk electrons. The increase in temperature reduces the slide-away field, eventually leading to slide-away as the electric field becomes comparable to Esa (t). The process can be slow or quick, depending on the initial value of E/Esa and the efficacy of the heat sink. IV: E < Esa , adequate heat sink – The heat supplied by the electric field is removed and the temperature is kept constant. The electric field causes prolonged substantial runaway generation, which eventually leads to depletion of the bulk population of electrons. This reduces the slideaway field, and eventually slide-away is reached. This process is generally the slowest. Mechanisms I–III are discussed in Paper E. In Paper F, the focus is the influence on the transition to slide-away of the properties of the heat sink. Both mechanisms III and IV are observed in a tokamak-relevant scenario, depending on the details of the heat sink. The feedback loop related to mechanism II is also discussed, as mechanisms III and IV exhibit the behavior of mechanism II just before slide-away is reached. In an idealized situation, slide-away should eventually occur in all systems with a persistent electric field. In practice, however; this is not observed. If E Esa , the timescale of the slide-away transition is slow compared to most processes in the plasma. In this case, cold electrons are supplied to the thermal population, compensating for the particles running away and thus preventing a transition in accordance with mechanism IV. Also in the absence of efficient cooling (mechanism III), slide-away may be prevented by the feedback between the current and the electric field. The electric field in a tokamak disruption is generated as a consequence of a reduction in the plasma current (due to a quick cooling of the plasma), however if runaway generation becomes strong enough to significantly affect the total current, the electric field will be reduced. This may interrupt a transition through mechanisms III or IV before slide-away is reached. Fields sufficiently strong to cause slide-away through mechanisms I–II are uncommon in fusion experiments. Nevertheless, as demonstrated in Paper F, non-linear simulations can be vital for the understanding of some realistic tokamak scenarios where a linearized approach quickly becomes invalid. 47 6 Concluding remarks In order to reduce the threat posed by runaway electrons to future fusionenergy devices, progress on several fronts are necessary. The understanding of runaway generation and loss mechanisms needs to be improved – both theoretically and experimentally – so that reliable operation scenarios and efficient mitigation schemes can be developed. At the core of the runaway phenomenon lies dynamics in momentum space, which is the topic of this thesis. The most important mechanisms affecting these dynamics are summarized schematically in Fig. 6.1. In this thesis, these mechanisms (apart from bremsstrahlung radiation reaction) have been investigated using purpose-built numerical tools, leading to several new insights concerning runaway dynamics. The present Chapter provides a summary of the papers that constitute this work, as well as a short outlook. 6.1 Summary of the included papers An important component in increasing the understanding of runaway dynamics is to improve the capability to analyze experimentally observed runaway beams and to extract the information available in the few diagnostic measurements that are sensitive to the runaway parameters. To this end, Paper A is focused on calculating the synchrotron spectrum emitted by a distribution of runaway electrons: a significant improvement over previous methods which typically interpret the observed spectra using a single-particle approximation for the runaway population (as discussed in connection with Fig. 4.2). As shown in the paper, this approximation fails to capture both qualitative and quantitative features of the synchrotron spectrum, and the use of distributionintegrated spectra is essential to accurately infer the runaway parameters. The paper analyses the spectra obtained using analytical avalanche distributions (Eq. 3.7). Together with the possibility to easily obtain runaway-electron distribution functions numerically using CODE or NORSE, the work in this thesis makes a significant contribution to the interpretation of synchrotron spectra, and thereby to the experimental analysis of runaway-electron parameters. 49 Chapter 6. Concluding remarks Synchrotron radiation reaction Large-angle collisions Pitch-angle scattering E-field Collisional friction Parallel collisional diffusion Bulk heating and cooling Bremsstrahlung radiation reaction Figure 6.1: Schematic presentation of the various effects of importance for runaway-electron momentum-space dynamics and their qualitative effect on the distribution function. Bremsstrahlung radiation reaction is included for completeness, even though it has not been discussed in detail in this thesis. Paper B describes and validates the development of CODE, a tool for calculating the time-evolution of the electron distribution function (or its steady-state shape) in the presence of electric-field acceleration and Coulomb collisions. In the paper, the obtained distributions are used to calculate synchrotron spectra (in accordance with Paper A), giving access to parameter regimes and evolution times not properly modeled by the analytical avalanche distribution. Paper C expands the capability of CODE by introducing time-dependent plasma parameters (enabling the modeling of dynamic scenarios such as disruptions); a conservative collision operator essential for calculating e.g. the plasma conductivity; and the avalanche source term in Eq. (3.5). A scenario dominated by hot-tail runaway generation is investigated, and the effect on the avalanche 50 6.1 Summary of the included papers growth rate of the choice of avalanche operator is quantified. It is found that the commonly used Rosenbluth-Putvinski avalanche operator can both overestimate and underestimate the avalanche growth rate significantly, depending on the parameter range. Apart from acting as an observable diagnostic for the runaway distribution, the emission of synchrotron radiation also affects the distribution itself, as discussed in Sec. 4.2. To model this behavior, Paper D introduces an operator describing synchrotron radiation reaction into CODE and uses it to investigate the effect of the radiation reaction on the critical field for runaway generation. The paper also explains that a large part of the observed modification to the critical field (initially attributed to so-called “anomalous losses”, which include synchrotron radiation reaction) is in fact likely to be just a manifestation of the temperature dependence of the Dreicer runaway-generation rate (Eq. 2.10), so that the parameter determining the runaway growth is E/ED rather than E/Ec . The paper also shows that redistribution of particles in velocity space can give the impression (because of reduced synchrotron emission) that the runaway population is decaying, when in fact both the number and total kinetic energy of the runaways keep increasing. Again, these insights impact the interpretation of experimental observations and thus contribute to the understanding of runaway parameters in practice. Paper E describes the development of NORSE. The motivation behind this new Fokker-Planck tool was to investigate the impact of collisional nonlinearities on the runaway dynamics. Specifically, NORSE makes it possible to study situations where the runaways constitute a substantial part of the electron distribution, or the electric field is significant compared to ED . This had not previously been done in a framework allowing for relativistic particle energies. The paper highlights the fact that a transition to the slide-away regime can be initiated for electric fields well below the traditional slide-away field Esa . This line of investigation is then continued in Paper F, which uses an ITERdisruption scenario to explore the importance of Ohmic heating of the bulk electron population. In addition, the paper studies the impact on the slideaway dynamics of the efficiency of the available heat-loss mechanisms. It is found that in the absence of spatial-transport and current–electric-field feedback effects, the electron population in an ITER disruption should eventually transition to a slide-away regime, however the time scale of this transition depends strongly on the heat-loss rate. This could potentially have large consequences for the understanding of runaway dynamics in ITER, although further investigations are needed. 51 Chapter 6. Concluding remarks 6.2 Outlook In this thesis, the focus has been on modeling of the dynamics of runaway electrons in momentum space. These dynamics have sometimes been misunderstood or misinterpreted, as in the case of observed modifications to the critical electric field, which motivated the work on Paper D. The tools developed in this thesis can help avoid future such misunderstandings – as they facilitate the investigation of runaway dynamics – and make more accurate interpretation of experimental data possible. All results presented in this thesis were obtained using predefined electricfield evolutions, i.e. the applied electric field was not affected by the electron distribution in any way and was therefore not calculated self-consistently. As long as the runaway population and current are small, this approximation is adequate, however in cases where the runaways contribute a substantial part of the plasma current (or otherwise significantly affect the plasma evolution), a self-consistent treatment is essential. This point is of particular concern in the scenario considered in Paper F. To build on the work presented here, a logical step forward is to include a self-consistent electric field, taking the evolution of the electron distribution into account. Another area for further research is to extend the numerical treatment to include one spatial (radial) dimension. This would make it possible to capture magnetic trapping effects, as well as collisional radial transport of the runaways. Naturally, such a development is not without complications, both analytically and numerically, and the increased dimensionality puts much higher demands on the computational resources. Although Fokker-Planck tools (such as LUKE [44, 45]) that include a radial coordinate do exist, they are not primarily focused on runaway research. Neither do they necessarily include all the relevant effects (for instance LUKE includes trapping effects, but not consistent radial transport). Some progress towards including the two effects mentioned above have been made, and is described in Conf. Contrib. V. The adopted approach is to couple a Fokker-Planck solver (in this case CODE) to the 1D fluid code GO [72, 121–123], which evolves the plasma parameters and current, and handles radial electric-field diffusion. This approach could serve as a first step towards a self-consistent model, however further work is needed since CODE does not include trapping effects. 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Fülöp, Synchrotron radiation from a runaway electron distribution in tokamaks, Physics of Plasmas 20, 093302 (2013). http://dx.doi.org/10.1063/1.4821823 http://arxiv.org/abs/1308.2099 PHYSICS OF PLASMAS 20, 093302 (2013) Synchrotron radiation from a runaway electron distribution in tokamaks € p1 € lo A. Stahl,1 M. Landreman,2 G. Papp,1,3 E. Hollmann,4 and T. Fu 1 Department of Applied Physics, Nuclear Engineering, Chalmers University of Technology and Euratom-VR Association, SE-412 96 G€ oteborg, Sweden 2 Plasma Science and Fusion Center, MIT, Cambridge, Massachusetts 02139, USA 3 Department of Nuclear Techniques, Budapest University of Technology and Economics, Association EURATOM, H-1111 Budapest, Hungary 4 Center for Energy Research, University of California, San Diego, La Jolla, California 92093-0417, USA (Received 9 August 2013; accepted 16 August 2013; published online 19 September 2013) The synchrotron radiation emitted by runaway electrons in a fusion plasma provides information regarding the particle momenta and pitch-angles of the runaway electron population through the strong dependence of the synchrotron spectrum on these parameters. Information about the runaway density and its spatial distribution, as well as the time evolution of the above quantities, can also be deduced. In this paper, we present the synchrotron radiation spectra for typical avalanching runaway electron distributions. Spectra obtained for a distribution of electrons are compared with the emission of mono-energetic electrons with a prescribed pitch-angle. We also examine the effects of magnetic field curvature and analyse the sensitivity of the resulting spectrum to perturbations to the runaway distribution. The implications for the deduced runaway electron parameters are discussed. We compare our calculations to experimental data from DIII-D and estimate the maximum observed runaway energy. [http://dx.doi.org/10.1063/1.4821823] I. INTRODUCTION Understanding the process of runaway beam formation and loss in tokamaks is of great importance, due to the potentially severe damage these electrons may cause in disruptions. In present tokamaks, runaway electrons have energies between a few hundred keV and tens of MeV, and in a next-step device like ITER, they are projected to reach a maximum energy of up to 100 MeV.1 Runaway electrons emit synchrotron radiation,2–5 the spectrum of which depends on the velocity-space distribution of the radiating particles. Therefore, the spectrum can be used to obtain information about the departure of the velocity distribution from isotropy and about the energy of the particles. The emitted radiation can also be an energy loss mechanism,6 although in tokamaks this loss is not appreciable unless the electrons have very large energies, above 70 MeV.3 Many theoretical studies of the synchrotron radiation of the energetic population have been done before, either using approximate electron distribution functions or assuming straight magnetic field lines.7–9 In several studies, the synchrotron emission from a single particle is used as an approximation for the entire runaway distribution,4,5 using a specific momentum and pitch-angle for the electrons, often identified as the maximum momentum and pitch-angle of the electrons in the runaway beam. In the present work, we use an electron distribution function typical of avalanching runaway electron populations in tokamak disruptions. As we will show, taking into account the whole distribution is important, since synchrotron radiation diagnostics based on single particle emission can give misleading results. Furthermore, we will illustrate that synchrotron radiation can be used to detect signs of modification of the electron distribution, which can occur due to for instance wave-particle interaction. 1070-664X/2013/20(9)/093302/9/$30.00 The structure of the paper is as follows. In Sec. II, we give several expressions for the radiated synchrotron power, including the effect of field-curvature. We also discuss the applicability of these expressions in different contexts. Section III is devoted to the analysis of the synchrotron radiation spectrum from an avalanching runaway electron distribution. We will describe the parametric dependences on magnetic field, density, temperature, effective charge, and electric field. In Sec. IV, we discuss the potential use of synchrotron radiation as a diagnostic. We also present a comparison between the synchrotron spectrum calculated for the avalanching runaway electron distribution and an experimentally measured synchrotron spectrum from DIII-D. Our conclusions will be summarized in Sec. V. II. SYNCHROTRON EMISSION FORMULAS The power radiated by an electron with Lorentz factor c 1 at wavelength k in the case of straight magnetic field lines is10 ð 1 ce2 1 P cyl ðkÞ ¼ pffiffiffi 3 K5=3 ðlÞdl ; (1) 3 0 k c2 kc =k where e is the electron charge, c is the speed of light, 0 is the vacuum permittivity, kc ¼ ð4pcme ck Þ=ð3eBc2 Þ; cjj ¼ 1= qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v2jj =c2 ; me is the electron rest mass, B is the magnetic field, k denotes the component along the magnetic field, and K ðxÞ is the modified Bessel function of the second kind. The radiation is emitted in a narrow beam in the parallel direction due to relativistic effects.10 In a tokamak, the effects of magnetic field line curvature and curvature drift have to be taken into account. This has been done in Ref. 11, where the following expression was obtained: 20, 093302-1 093302-2 P full ðkÞ ¼ Stahl et al. (ð Phys. Plasmas 20, 093302 (2013) dy 3 1 ð1 þ 2y2 ÞJ0 ðay3 Þsin n y þ y3 2 3 0 y ð 4g 1 3 1 p dyyJ 0 0 ðay3 Þcos n y þ y3 ; 2 1þg 0 2 3 2 (2) ce2 e0 k3 c2 1 where a ¼ ng=ð1 þ g2 Þ, 4p R pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3 kc3 1 þ g2 (3) eBR v? xc R v? ’ ; cme v2k cc vk (4) n¼ g¼ R is the tokamak major radius, J ðxÞ is the Bessel function, and J0 ðxÞ its derivative. The integrands in Eq. (2) are highly oscillatory and the calculation of synchrotron spectra can become computationally heavy. This motivates examining more approximate formulas which are less complex, especially when considering possible diagnostic applications. In Eqs. (21) and (26) of Ref. 11, two limits of Eq. (2) are given. These two limits are obtained by first expanding in n 1, which can be translated pffiffiffiffiffiffiffiffiffiffiffiffiffito a condition for the wavelength k ð4p=3ÞR=ðc3 1 þ g2 Þ. Then, to obtain the first of the two expressions, Eq. (2) is also expanded in the smallness of the argument of the Bessel functions, leading to the condition ng ⱗ 1 þ g2 . The resulting approximative formula is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ce2 2 1 þ g2 n 4g I ðaÞ þ I ðaÞ ; (5) P as1 ðkÞ e 0 1 40 1 þ g2 k5 Rc where I ðxÞ is the modified Bessel function. P as1 was the expression used to calculate the synchrotron radiation of an avalanching population of positrons in Ref. 12 and in fitting of the synchrotron spectrum in the optical range in DIII-D in Ref. 5. The two conditions required for validity of Eq. (5) can be summarized as g=ð1 þ g2 Þⱗ1=n 1, which leads to a rather narrow validity range for P as1 . Figures 1(a) and 1(b) show the range of wavelengths for which P as1 is valid (kl ⱗ k ku ) for different runaway momenta in DIII-D-size and ITER-size tokamaks, respectively. Note that the wavelength should be much smaller than the solid line(s) in the figure for P as1 to be valid. It is clear that for wavelengths in the 0.1–1 lm range (as in the measurements described in Ref. 5), the approximative formula P as1 is only valid for particles with large normalized momenta p ¼ cv=c, and not necessarily for all values of v? =vk . To obtain the second limit of Eq. (2) (Eq. (26) in Ref. 11), k ð4p=3ÞRg=½c3 ð1 þ gÞ3 has to be fulfilled. Equation (2) then simplifies to pffiffiffi 2 4p R 1 3 ce c ð1 þ gÞ2 : exp P as2 ðkÞ ¼ p ffiffi ffi 3 kc3 1 þ g g 8p 0 k2 R (6) (7) The condition in Eq. (6) is more strict than the one stemming from n 1; it is only necessary to fulfill Eq. (6) for Eq. (7) to be valid. Figures 2(a) and 2(b) show the upper bound for the wavelength given by Eq. (6). We conclude that for the visible part of the spectrum, P as2 could be a suitable approximative formula for runaway electron beams with p < 50 and v? =vjj < 0:1. In the opposite case, when p and v? =vjj are large then either the full expression P full , or in some cases P as1 , can be used. In general, the difference between the emitted power given by P cyl (valid in the cylindrical limit) and P full (including field line curvature) is not very large if we consider only emission by a single particle. Single particle synchrotron spectra calculated by P cyl and P full , as well as the approximate formulas P as1 and P as2 are shown in Figs. 3(a) and 3(b) for particles with normalized momentum p ¼ 50 (corresponding to a particle energy of roughly 25 MeV) and v? =vk ¼ 0:1 in two different tokamaks. For such particles, the peak emission is for wavelengths of a few lm (the near infrared part of the spectrum). Figure 3(a) shows that for medium-sized tokamaks (such as DIII-D), P full is closely approximated by P as2 . This is not surprising, as P as2 is valid in most of the wavelength range considered (especially for shorter wavelengths), whereas P as1 is only valid for longer wavelengths for these parameters. For large tokamaks (such as ITER), P full is best approximated by P cyl , as the effects of field curvature become small for such large major radii. Figure 3(b) shows that P as2 is not a good approximation in this case, which is expected, since P as2 is not valid in this region. Figures 3(c) and 3(d) investigate the energy dependence of the above conclusions. The quantity plotted is log10 ðP i ðkÞÞ. Figure 3(c) confirms that P as2 is a good approximation to P full in DIII-D for a wide range of runaway energies. For the highest energies, agreement is still very good FIG. 1. Upper and lower bounds on the wavelength k for which P as1 is valid. Note the logarithmic scale on the vertical axis. The parameters used are (a) B ¼ 2:1 T and R ¼ 1:67 m and (b) B ¼ 5:3 T and R ¼ 6 m. 093302-3 Stahl et al. Phys. Plasmas 20, 093302 (2013) FIG. 2. Upper bounds on the wavelength k for which P as2 is valid. Note the logarithmic scale on the vertical axis. The parameters used are (a) B ¼ 2:1 T and R ¼ 1:67 m and (b) B ¼ 5:3 T and R ¼ 6 m. for short wavelengths, but less so for longer wavelengths. This agrees with Fig. 2(a), which indicates that P as2 is no longer valid for high energies and long wavelengths. Figure 3(c) also shows that for a tokamak this size, the difference between P cyl and P full increases with p, and using P cyl is not recommended if quantitative agreement is sought. In an ITER-like device, however, Fig. 3(d) indicates that P cyl approximates P full very well over the whole energy range considered. Formally, P full reduces to P cyl when R ! 1 and ck ’ c/v? (where this latter relation is equivalent to cv? / c 1). III. SPECTRUM FROM RUNAWAY ELECTRON DISTRIBUTIONS In Refs. 4 and 5, the synchrotron spectrum is calculated by multiplying the single particle spectrum by the number of runaways with a specific pitch-angle and momentum. In this section, we investigate how the synchrotron spectrum changes if we take into account the whole runaway electron distribution instead of the single particle approximation considered above. We calculate the synchrotron emission integrated over a runaway electron distribution using PðkÞ ¼ 2p nr ð fRE ðp; vÞ P i ðp; v; kÞ p2 dp dv ; (8) Rr where fRE is the runaway distribution function, P i is one of the single particle emission formulas discussed in Sec. II, v ¼ pk =p is the cosine of the pitch-angle, and nr is the runaway electron density. The runaway region of momentum space Rr is defined by a separatrix ps ¼ ðE 1Þ1=2 such that all particles with p > ps are considered runaways.13 Here, E ¼ Ejj =Ec is the parallel electric field Ejj normalized to the critical field Ec ¼ me c=ðesÞ, with s ¼ ð4pre2 ne c ln KÞ1 the collision time for relativistic electrons, re the classical electron radius, ne the electron density, and lnK the Coulomb logarithm. As we normalize to nr ; PðkÞ is the average emission per runaway. The alternative choice of normalizing by the runaway current Ir was also considered, and it was found that all results presented below are essentially unchanged aside from an overall scale factor, since the speed of all runaways is nearly c. In large tokamak disruptions, secondary runaway generation is expected to dominate over primary generation, in which case the runaway distribution will grow approximately FIG. 3. Single particle synchrotron emission from different emission formulas. (a) and (b) show emitted spectra for particles with v? =vk ¼ 0:1 and p ¼ 50 and tokamak parameters corresponding to (a) DIII-D and (b) ITER. The solid (blue) line corresponds to the expression including the field-line curvature, P full . The dotted (black) line is the cylindrical limit, P cyl . The dashed-dotted (red) and dashed (green) lines correspond to the approximative expressions P as1 and P as2 , respectively. (c) and (d) show contours of log10 ðP i ðkÞÞ (with P i in units of W=lm) for various particle momenta and compares (c) P cyl ; P full , and P as2 and (b) P cyl and P full . 093302-4 Stahl et al. exponentially in time: @fRE =@t / fRE . In this case of exponential growth, the electron distribution can be approximated by14 ! ^ 2 pk nr E^ Ep ? exp fRE ðpk ; p? Þ ¼ ; (9) cz lnK 2pk 2pcz pk lnK where E^ ¼ ðE 1Þ=ð1 þ Zeff Þ; Zeff is the effective ion pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi charge and cz ¼ 3ðZeff þ 5Þ=p, and the momentum space coordinates are related to p and v through pjj ¼ pv and pffiffiffiffiffiffiffiffiffiffiffiffiffi p? ¼ p 1 v2 . Derivation of Eq. (9) assumes strong anisotropy (p? pjj ) and high electric field (E 1). In addition to the lower boundary p ¼ ps of the runaway region, an upper cut-off p ¼ pmax of the distribution will be introduced. This cut-off is physically motivated by the finite life-time of the accelerating electric field and the presence of loss mechanisms, such as radiation and radial transport. As it was shown in Sec. II, the inclusion of field curvature effects via the use of P full rather than P cyl had little effect on the synchrotron emission of a single particle in an ITERsized device. The effect is larger in smaller devices. When the complete runaway distribution is taken into account, these conclusions still hold. Figure 4 shows synchrotron spectra calculated using Eq. (8) together with the distribution in Eq. (9) and the emission formulas P cyl ; P full ; P as1 , and P as2 . The calculation was performed for both a DIII-D-size and an ITER-size device, as the field curvature is what separates the different formulas. The parameters used in the calculation in Fig. 4 are maximum normalized momentum pmax ¼ 100 (corresponding to a maximum runaway energy of roughly 50 MeV), parallel electric field Ek ¼ 2 V=m, effective charge Zeff ¼ 1, background electron density ne ¼ 3 1020 m3 , and background plasma temperature T ¼ 10 eV. The relatively low temperature is what is expected after a thermal quench in a disruption. In DIII-D, the post thermal-quench temperature is estimated to be as low as T ¼ 2 eV.15 Figure 4(a) shows that in DIII-D, P full is well approximated by P as2 , especially in the short wavelength slope region of the spectrum. In ITER, P cyl is a good approximation, as shown in Fig. 4(b). This is expected since the field curvature is much smaller here. These results are consistent with the conclusion regarding single particles in Fig. 3. For simplicity, throughout the remainder of this paper, we will use P cyl when calculating synchrotron spectra (except for the comparison with DIII-D data in Sec. IV B). Synchrotron spectra calculated by P cyl and P full are qualitatively similar Phys. Plasmas 20, 093302 (2013) for both small and large machines and are also often quantitatively similar for large machines. The single particle synchrotron emission formulas are independent of the plasma temperature, effective charge, density, and the strength of the electric field. These quantities do however affect the shape of the runaway distribution, which in turn affects the synchrotron emission. Figure 5 shows scans in these parameters, the magnetic field, and maximum momentum pmax of the distribution. The baseline scenario corresponds to the parameters used in Fig. 4 together with B ¼ 3 T. Since P cyl is used, there is no dependence on R. Figure 5 shows that the average synchrotron emission increases with B; T; Zeff ; ne , and pmax , but decreases with increasing electric field strength. The dependence on ne and E is particularly strong, and we note that the average emission can vary over several orders of magnitude. This variation is completely missing from the single particle approximation used in Sec. II. If, as a disruption mitigation technique, a large amount of material is injected into the plasma (for instance in the form of a massive gas injection), the increase in density would lead to increased synchrotron emission from the runaways (if the mitigation is unsuccessful). This could give the impression of an increase in the number of runaways even though this is not necessarily the case. The figure also shows that the wavelength of peak emission shifts appreciably with varying parameter values. Generally, an increased average emission is accompanied by a shift of the peak emission towards shorter wavelengths. The total synchrotron emission of a single particle scales roughly as ðcv? =vk Þ2 .4 Thus, the most strongly emitting particles are highly energetic with large pitch-angle. These particles emit at shorter wavelengths, so the shift of the wavelength of peak emission with increased total emission is expected. In light of the particle energy dependence of the emitted synchrotron power, the decrease in emission with increasing electric field strength may seem a little surprising, as a stronger accelerating field leads to more highly energetic particles. The explanation can be found in the shape of the runaway beam. Figure 6 shows the runaway distribution, Eq. (9), in (pk ; p? )-space for three of the parameter sets in the electric field scan in Fig. 5(b). The figure shows that the distribution, in addition to being extended in pk , becomes more narrow in p? as the electric field strength increases. This leads to lower average-per-particle emission by virtue of the pitch-angle dependence of P cyl , despite the presence of a greater number of highly energetic particles. FIG. 4. Comparison of the synchrotron spectrum from a runaway distribution (Eq. (9)), as calculated using P cyl ; P full ; P as1 , or P as2 . Normalizing the emitted power by the runaway current Ir instead of by nr gives negligible difference in these figures or any figures below (the curves are not even distinguishable), since most runaways move at speed c. 093302-5 Stahl et al. Phys. Plasmas 20, 093302 (2013) FIG. 5. Synchrotron spectra calculated using Eq. (8) together with P cyl and Eq. (9). Note that the spectra are normalized to the runaway density. Unless otherwise noted, the parameters are pmax ¼ 100; Ek ¼ 2 V=m; Zeff ¼ 1; ne ¼ 3 1020 m3 ; T ¼ 10 eV, and B ¼ 3 T. For this scenario, Ec ¼ 0:15 V=m. FIG. 6. Shape of the analytical avalanche distribution (Eq. (9)) for three of the parameter sets in Fig. 5(b). The plot shows contours of the quantity log10 jfRE =nr j. 093302-6 Stahl et al. Phys. Plasmas 20, 093302 (2013) FIG. 7. Synchrotron spectra (average emission per particle) calculated using the runaway distribution in Eq. (9) and P cyl for DIII-D-like and ITER-like cases. The synchrotron spectrum from a single particle with p ¼ 100 and v? =vjj ¼ 0:15 is also shown. Note that the single particle spectra have been multiplied by a small factor to fit on the same scale. The parameters used for the distributions are pmax ¼ 100 and A: Ek ¼ 2 V=m; Zeff ¼ 1; ne ¼ 5 1019 m3 ; T ¼ 2 eV; B : Ek ¼ 10 V=m; Zeff ¼ 1:5; ne ¼ 1 1020 m3 ; T ¼ 2 eV; C : Ek ¼ 2 V=m; Zeff ¼ 1; ne ¼ 5 1020 m3 ; T ¼ 10 eV; D : Ek ¼ 10 V=m; Zeff ¼ 2; ne ¼ 1 1021 m3 ; T ¼ 10 eV. Figure 7 shows a comparison of the average synchrotron spectrum calculated for the runaway distribution Eq. (9) and for a single particle. The figure clearly shows that using the single-particle emission overestimates the synchrotron emission per particle by several orders of magnitude. (Note that the values for the emitted power per particle were divided by a large number to fit in the same scale.) The overestimation is caused by the fact that the single-particle approximation assumes that all particles emit as much synchrotron radiation as the most strongly emitting particle in the actual distribution, as discussed in Sec. I. Furthermore, the wavelength of peak emission is shifted towards shorter wavelengths when using this approximation. Using the single-particle approximation can thus give misleading results regarding both the spectrum shape and the total emission strength. IV. SYNCHROTRON RADIATION AS A RUNAWAY ELECTRON DIAGNOSTIC The interest in the synchrotron emission of runaways is primarily motivated by its potential as a runaway diagnostic. In principle, the distribution can be determined by acquiring an experimental synchrotron spectrum and comparing it to calculations using Eq. (8) for a range of pmax , provided all other relevant parameters are known. There are however several problems with this approach. First, the complete synchrotron spectrum is not known. Detectors are only sensitive in a limited wavelength range, which is likely to also contain contaminating radiation from other sources in the plasma. Second, the relevant plasma parameters are not always well known, especially during disruptions. This can lead to significant uncertainty in the computed synchrotron spectrum, as the parameter scans in Fig. 5 indicated. Using a single particle approximation for the runaway distribution seemingly avoids the second issue, but as we have seen, it also ignores factors that can influence the emission by orders of magnitude. A. Spectrum slope and maximum runaway energy Simple measurements of the synchrotron power for different wavelengths on the steep slope of the spectrum have been used to estimate the runaway energy,4 using the single particle emission formulas and assuming mono-energetic runaways with well-defined pitch-angle. In this case, there is a monotonic relationship between the slope and the particle energy (as the wavelength of peak emission decreases monotonically with increasing p). The slope can be obtained through a relative measurement of the synchrotron power at two wavelengths, S ¼ Pðk1 Þ=Pðk2 Þ. However, as the runaway distribution is sensitive to the plasma parameters, when taking it into account there is in general no such simple relationship between the slope of the spectrum and the maximum runaway energy in the distribution. If all other parameters are fixed the relation still holds, as is shown in Fig. 8(a). This follows naturally from the relation for single particles, as when pmax is increased, more particles that emit at short wavelengths are included, and the average emission correspondingly shifts towards shorter wavelengths, affecting the slope. But if the plasma parameters are uncertain, the slope FIG. 8. Spectra calculated using the analytical avalanche distribution Eq. (9) and P cyl . In (a), the parameters used are the same as the baseline scenario in Fig. 5, but with different maximum particle momenta. All the curves in panel (b) have the same slope S, as calculated with k1 ¼ 1:5 lm, k2 ¼ 2:8 lm. The plasma parameters that differ between the curves are indicated in the figure. The remaining parameter values are ne ¼ 3 1020 m3 and B ¼ 3 T. 093302-7 Stahl et al. Phys. Plasmas 20, 093302 (2013) can be misleading. Figure 8(b) shows multiple spectra with the same slope S for k1 ¼ 1:5 lm and k2 ¼ 2:8 lm. Using only a measurement of S in the above range, they cannot be distinguished, despite the appreciable difference in average emission. This type of two-point slope measurement can be performed using physical wavelength filters placed in front of the detector,4 in which case measurements are constrained to specific k1 and k2 that cannot be easily changed. The pmax of the different spectra in Fig. 8(b) range from 50 to 90, with only modest variation of the plasma parameters E; Zeff , and T (all of which are hard to estimate during disruptions). Thus, if the plasma properties are uncertain, there is no clear correlation between S and pmax of the distribution. Another weakness of using the slope is the difficulty in asserting that both measurement points are actually located on the approximately linear part of the spectrum. As the plasma parameters change, the peak of the spectrum may shift (as discussed in connection with Fig. 5). Choosing k1 and k2 that are suitable for a wide range of different conditions (as when using physical filters) is not easy. Instead of using the slope directly, one should calculate the emission for an assumed beam-like distribution function (e.g., similar to Eq. (9)), and iteratively find the pmax, which fits the synchrotron spectrum best. B. Synchrotron emission in DIII-D It is interesting to investigate how a synchrotron spectrum calculated for an avalanching distribution compares with an experimentally measured synchrotron spectrum from DIII-D. In the specific experimental scenario we consider (shot number 146 704 and time t ¼ 2290 ms16), the loop voltage is 7 V, the density 3:9 1019 m3 , and the plasma current Ip ¼ 0:15 MA, measured near the end of a runaway plateau phase. The runaway density can be estimated from the current using nr ¼ Ip =ðecAre Þ, where Are is the area of the runaway beam. The runaway beam radius in this case was around 20 cm. The temperature is assumed to be 1:5 eV and Zeff ¼ 1. For synchrotron emission by mono-energetic runaway electrons, the conversion to the measured brightness can be done using Eq. (2) in Ref. 5 Bðk; h; cÞ ¼ Pðk; h; cÞ 2R nr ; ph (10) where R is the major radius (of the runaway beam) and h ¼ v? =vk is the tangent of the particle pitch-angle. Taking into account the runaway distribution, we calculate the brightness as ð vmax ð pmax 1 BðkÞ ¼ 4R Pðk;hðvÞ;cðpÞÞf ðp;vÞp2 dpdv; (11) vmin pmin hðvÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi where hðvÞ ¼ tanðarccosðvÞÞ ¼ 1 v2 =v and cðpÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p2 þ 1; pmin ¼ ðE 1Þ1=2 and the integration limits for the pitch-angle are vmin ¼ 0, vmax ¼ 1. Since we consider the visible part of the spectrum, all pmin below p ¼ 50 produce identical results, as only the highest energy particles emit in this range. Equation (10) is strictly valid for 1=c h.5 As we are interested in the complete distribution with both small c and small h, we use instead the effective qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi viewing aperture heff h2 þ c2 þ ðrlens =r0 Þ2 . Here, rlens ¼ 2 cm is the lens aperture of the detector and r0 ’ 2 m is the distance between the detector and the runaway beam. Introducing heff into Eq. (11), we find ð 1 Pðk; v; pÞ f ðp; vÞ p2 dpdv : BðkÞ ¼ 4R (12) Rr heff ðp; vÞ Figure 9 shows a comparison of spectra calculated using Eq. (12) together with P as2 and Eq. (9), and the experimentally measured spectra for different runaway beam radii (the beam is assumed to have circular cross-section), pmax, loop voltages, and densities. The good agreement for rre ¼ 20 cm and pmax ¼ 130 leads us to estimate the maximum runaway electron energy to be around 65 MeV. This is much larger than the mean energy of several MeV estimated from other diagnostics.16 For comparison, we also fit the experimental data with synchrotron spectra from a mono-energetic runaway population (using Eq. (10)), for different particle energies and pitch-angles. As in Ref. 5, we assume that 1% of the runaway population (calculated with rre ¼ 20 cm) has the specific energy considered. The results are shown in Fig. 10. This fitting procedure gives a lower estimate for the maximum runaway energy, at about 40–50 MeV, depending on pitch-angle. C. Effect of wave-particle interaction Another instance where the synchrotron spectrum from a complete runaway distribution is useful is in investigations of mechanisms that affect the shape of the distribution itself. One such mechanism is resonant wave-particle interactions, and here we consider their effect on the synchrotron spectrum through a modification of part of the distribution given in Eq. (9). A runaway distribution is normally strongly peaked around the parallel direction (v ¼ 1), i.e., it has a high degree of anisotropy in momentum space (see for instance Fig. 6). Wave-particle interaction tends to drive the distribution towards isotropy through pitch-angle scattering of electrons with resonant momenta.17 A simple way to simulate the decrease in anisotropy is to introduce a flat profile in part of momentum space, as indicated in Fig. 11. The usual integral for the total emitted power, Eq. (8), is split up into three regions in momentum-space. The first and third parts remain unmodified, with the usual distribution function fRE. In the second (middle) part, the distribution function is assumed to be flat. We denote the lower and upper boundaries of this region pL and pU, respectively. The momentum space volume of the shaded block in the figure should be the same as that of the part of the distribution it replaces, which gives us a condition from which to calculate the appropriate height of the block. The integration of the normal distribution is taken over the entire v-range (v 2 ½0; 1). As the distribution decreases exponentially with decreasing v, the contribution from particles with low v is very small. When the modifications are introduced, however, the contribution could be substantial, and we need to restrict the extent of the block for the modified part of the 093302-8 Stahl et al. Phys. Plasmas 20, 093302 (2013) FIG. 9. Measured visible spectrum in DIII-D during the runaway plateau at t ¼ 2290 ms in shot 146 704. The data are a superposition of synchrotron radiation from runaways and line radiation from the background plasma. Theoretical synchrotron spectra are also shown for various (a) runaway beam radii, (b) maximum normalized momenta pmax , (c) loop-voltages Vloop , and (d) densities n. Unless otherwise noted the parameters are pmax ¼ 130; rre ¼ 0:2 m, n ¼ 3:9 1019 m3 , and Vloop ¼ 7 V, which are indicated by the red (dashed-dotted) lines. distribution in v (v 2 ½vmin ; 1). The introduction of vmin can be seen as a compensation for the fact that in reality the pitch-angle scattered particles are not evenly distributed in v. Letting fc ðp; vÞ ¼ h be a constant distribution where h represents the height of the block, and equating the momentum space volume of the block with that of the part of the distribution it replaces, we have ð 1 ð pU ð 1 ð pU f ðp;vÞp2 dpdv ¼ 2p fc ðp;vÞp2 dpdv V ¼ 2p 0 pL vmin pL 2p ¼ h ð1 vmin Þðp3U p3L Þ: 3 (13) We may solve this for h, and obtain FIG. 10. Measured visible spectrum in DIII-D during the runaway plateau at t ¼ 2290 ms in shot 146 704. Spectra from several mono-energetic populations calculated using P as2 are also shown. The number of runaways used to obtain the spectra was 1% of nr calculated from the runaway current (assuming rre ¼ 20 cm). ð 1 ð pU 3 f ðp; vÞp2 dpdv h¼ 0 pL ð1 vmin Þðp3U p3L Þ (14) as the block height that conserves the total number of particles. We emphasize that the above modification represents a “worst case scenario” in terms of the effect on the spectrum. In a more realistic case, the modifications would be less severe. The analytical avalanche distribution (Eq. (9)) was modified according to the above, with pL ¼ 25 and pU ¼ 35 since this is a typical range where wave-particle interactions manifest.17 The maximum pitch-angle in the modified region was set to p? =pk ¼ 0:2 (vmin ¼ 0:98), which is qualitatively consistent with experimental estimates of the maximum runaway pitch-angle.4,5 In Fig. 12, modified distributionintegrated synchrotron spectra are shown and compared with those of unmodified distributions. From the figure, it is clear that there is an appreciable increase in the average emission of the runaways as a result of the modifications to the distribution. Again, this increase is related to the pitch-angle dependence of P cyl . The isotropization broadens the distribution in pitch-angle which leads to a higher average emission. Due to the difference in the synchrotron spectrum, the onset of a particle-wave resonance should be detectable. However, as we have seen before, there are also other changes in plasma parameters that could have a similar effect on the synchrotron emission. Our goal in this exercise is not to explore the parameter space of artificially modified distributions—the modifications introduced above are too crude to lead to quantitative conclusions—but rather to illustrate the sensitivity of the synchrotron spectrum to the details of the runaway distribution. The analysis here shows that the spectrum from a 093302-9 Stahl et al. Phys. Plasmas 20, 093302 (2013) FIG. 11. Schematic runaway distribution with modifications emulating the effects of wave-particle interaction. also illustrated that using the slope of the spectrum for estimating the runaway energy can be misleading, and in general one should calculate the emission from an assumed approximative distribution and iteratively find the maximum runaway energy to fit the synchrotron spectrum. Finally, through a comparison with an experimental synchrotron spectrum from DIII-D, we have estimated the maximum runaway electron energy in that particular experimental scenario to be around 65 MeV. ACKNOWLEDGMENTS FIG. 12. Synchrotron spectra from unmodified and modified runaway distributions for different electric field strengths. The parameters used are pmax ¼ 50; pL ¼ 25; pU ¼ 35; Zeff ¼ 1:6; ne ¼ 3 1020 m3 ; T ¼ 10 eV, and B ¼ 3 T. For these parameters, the critical field is Ec ¼ 0:15 V=m. The maximum pitch-angle for the particles in the modified region was set to p? =pk ¼ 0:2. distribution modified by particle-wave interaction can imply runaway parameters distinctly different from those that are actually present, especially if only a limited part of the spectrum is considered. Failure to include such effects can thus lead to incorrect conclusions regarding the runaway beam properties. V. CONCLUSIONS The synchrotron emission spectrum can be an important diagnostic of the runaway electron population. In some previous work, synchrotron spectra have been interpreted under the assumption that all runaways have the same energy and pitch-angle. In practice, however, runaway electrons have a wide distribution of energies and pitch-angles. When taking into account the full distribution, the most suitable approximative emission formula may not be the one that has been used in previous work (P as1 ). Instead, depending on the major radius of the device and the actual runaway electron distribution, either P cyl (for large devices) or P as2 (for medium-sized devices) are more suitable. Although the single particle synchrotron emission formulas do not depend on the plasma temperature, effective charge, density or electric field strength, the total synchrotron emission is sensitive to these parameters, as they determine the shape of the runaway distribution. We have shown that the single-particle emission overestimates the synchrotron emission per particle by orders of magnitude, and the wavelength of the peak emission is shifted to shorter wavelengths compared with the spectrum from an avalanching runaway electron distribution. We have The authors are grateful to Y. Kazakov for fruitful discussions. This work was funded by the European Communities under Association Contract between EURATOM, HAS, and Vetenskapsrådet. The views and opinions expressed herein do not necessarily reflect those of the European Commission. M.L. was supported by the United States Department of Energy’s Fusion Energy Postdoctoral Research Program administered by the Oak Ridge Institute for Science and Education. E.H. was supported in part by the United States Department of Energy under DE-FG02-07ER54917. 1 G. Papp, M. Drevlak, T. F€ul€op, and G. I. Pokol, Plasma Phys. Controlled Fusion 53, 095004 (2011). D. Winske, Th. Peter, and D. A. Boyd, Phys. 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Fülöp, Numerical calculation of the runaway electron distribution function and associated synchrotron emission, Computer Physics Communications 185, 847-855 (2014). http://dx.doi.org/10.1016/j.cpc.2013.12.004 http://arxiv.org/abs/1305.3518 Computer Physics Communications 185 (2014) 847–855 Contents lists available at ScienceDirect Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc Numerical calculation of the runaway electron distribution function and associated synchrotron emission Matt Landreman a,b,∗ , Adam Stahl c , Tünde Fülöp c a University of Maryland, College Park, MD, 20742, USA b Plasma Science and Fusion Center, MIT, Cambridge, MA, 02139, USA c Department of Applied Physics, Nuclear Engineering, Chalmers University of Technology and Euratom-VR Association, Göteborg, Sweden article info Article history: Received 15 May 2013 Received in revised form 11 September 2013 Accepted 3 December 2013 Available online 9 December 2013 Keywords: Fokker–Planck Runaway electrons Relativistic Plasma Kinetic Synchrotron emission abstract Synchrotron emission from runaway electrons may be used to diagnose plasma conditions during a tokamak disruption, but solving this inverse problem requires rapid simulation of the electron distribution function and associated synchrotron emission as a function of plasma parameters. Here we detail a framework for this forward calculation, beginning with an efficient numerical method for solving the Fokker–Planck equation in the presence of an electric field of arbitrary strength. The approach is continuum (Eulerian), and we employ a relativistic collision operator, valid for arbitrary energies. Both primary and secondary runaway electron generation are included. For cases in which primary generation dominates, a time-independent formulation of the problem is described, requiring only the solution of a single sparse linear system. In the limit of dominant secondary generation, we present the first numerical verification of an analytic model for the distribution function. The numerical electron distribution function in the presence of both primary and secondary generation is then used for calculating the synchrotron emission spectrum of the runaways. It is found that the average synchrotron spectra emitted from realistic distribution functions are not well approximated by the emission of a single electron at the maximum energy. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Due to the decrease in the Coulomb collision cross section with velocity, charged particles in an electric field can ‘‘run away’’ to high energies. In tokamaks, the resulting energetic particles can damage plasma-facing components and are expected to be a significant danger in the upcoming ITER experiment. Electrons are typically the species for which runaway is most significant [1,2], but runaway ions [3] and positrons [4,5] can also be produced. Relatively large electric fields are required for runaway production, and in tokamaks these can arise during disruptions or in sawtooth events. Understanding of runaway electrons and their generation and mitigation is essential to planning future large experiments such as ITER. Runaway electrons emit measurable synchrotron radiation, which can potentially be used to diagnose the distribution function, thereby constraining the physical parameters in the plasma. The runaway distribution function and associated synchrotron ∗ Correspondence to: Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, MD 20742, USA. Tel.: +1 651 366 9306. E-mail addresses: [email protected], [email protected] (M. Landreman). 0010-4655/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cpc.2013.12.004 emission depend on the time histories of the local electric field E, temperature T , average ion charge Z , and density n. To infer these quantities (and the uncertainty in these quantities) inside a disrupting plasma using the synchrotron emission, it is necessary to run many simulations of the runaway process, scanning the various physical parameters. To make such a scan practical, computational efficiency is important. To this end, in this work we demonstrate a framework for rapid computation of the runaway distribution function and associated synchrotron emission for given plasma parameters. The distribution function is computed using a new numerical tool named CODE (COllisional Distribution of Electrons). Physically, the distribution function is determined by a balance between acceleration in the electric field and collisions with both electrons and ions. The calculation in CODE is fully relativistic, using a collision operator valid for both low and high velocities [6] and it includes both primary and secondary runaway electron generation. If primary runaway electron generation dominates, CODE can be used in both time-dependent and time-independent modes. The latter mode of operation, in which a long-time quasi-equilibrium distribution function is calculated, is extremely fast in that it is necessary only to solve a single sparse linear system. Due to its speed and simplicity, CODE is highly suitable for coupling within larger more 848 M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 expensive calculations. Besides the inverse problem of determining plasma parameters from synchrotron emission, other such applications could include the study of instabilities driven by the anisotropy of the electron distribution function, and comprehensive modeling of disruptions. Other numerical methods for computing the distribution function of runaways have been demonstrated previously, using a range of algorithms. Particle methods follow the trajectories of individual marker electrons. Deterministic particle calculations [7] can give insight into the system behavior but cannot calculate the distribution function, since diffusion is absent. Collisional diffusion may be included by making random adjustments to particles’ velocities, an approach which has been used in codes such as ASCOT [8] and ARENA [9]. For a given level of numerical uncertainty (noise or discretization error), we will demonstrate that CODE is more than 6 orders of magnitude faster than a particle code on the same computer. Other continuum codes developed to model energetic electrons include BANDIT [10], CQL3D[11,12] and LUKE [13,14]. These sophisticated codes were originally developed to model RF heating and current drive, and contain many features not required for the calculations we consider. For example, CQL3D contains ∼90,000 lines of code and LUKE contains ∼118,000 lines, whereas CODE contains <1200 lines (including comments). While future more elaborate modeling may require the additional features of a code like CQL3D or LUKE, for the applications we consider, we find it useful to have the nimble and dedicated tool CODE. For calculations of non-Maxwellian distribution functions in the context of RF heating, an adjoint method [15] can be a useful technique for efficient solution of linear inhomogeneous kinetic equations. However, the kinetic equation we will consider is nonlinear (if avalanching is included) and homogeneous, so the adjoint method is not applicable. In several previous studies, a single particle with a representative momentum and pitch-angle is used as an approximation for the entire runaway distribution [16,17] when computing the synchrotron emission. In this paper, we present a computation of the synchrotron radiation spectrum of a runaway distribution in various cases. By showing the difference between these spectra and those based on single particle emission we demonstrate the importance of taking into account the entire distribution. The remainder of the paper is organized as follows. In Section 2 we present the kinetic equation and the collision operator used. Section 3 details the discretization scheme and calculation of the primary runaway production rate, with typical results shown in Section 4. The avalanche source term and its implementation are described in Section 5. In this section we also demonstrate agreement with an analytic model for the distribution function [18]. Computation of the synchrotron emission spectrum from the distribution function is detailed in Section 6, and comparisons to single-particle emission are given. We conclude in Section 7. 2. Kinetic equation and normalizations We begin with the kinetic equation ∂f − eEb · ∇p f = C {f } + S . (1) ∂t Here, −e is the electron charge, E is the component of the electric field along the magnetic field, b = B/B is a unit vector along the magnetic field, ∇p is the gradient in the space of relativistic  momentum p = γ mv, γ = 1/ 1 − v 2 /c 2 , v = |v | is the speed, m is the electron rest mass, c is the speed of light, C is the electron collision operator, and S represents any sources. All quantities refer to electrons unless noted otherwise. Eq. (1) is the largeaspect-ratio limit of the bounce- and gyro-averaged Fokker–Planck equation (Eq. (2) in [19]). Particle trapping effects are neglected, which is reasonable since runaway beams are typically localized close to the magnetic axis. We may write b · ∇p f in (1) in terms of scalar variables using 1 − ξ 2 ∂f ∂f + (2) ∂p p ∂ξ where p = |p|, and ξ = p · b/p is the cosine of the pitch angle relative to the magnetic field. The distribution is  function defined such that the density n is given by n = d3 p f , so f has dimensions of (length × momentum)−3 , and we assume the distribution function for small momentum to be approximately √ the Maxwellian fM = nπ −3/2 (mve )−3 exp(−y2 ) where ve = 2T /m is the thermal speed, and y = p/(mve ) = γ v/ve is the normalized b · ∇p f = ξ momentum. We use the collision operator from Appendix B of Ref. [6]. This operator is constructed to match the usual nonrelativistic testparticle operator in the limit of v ≪ c, and in the relativistic limit it reduces to the operator from Appendix A of Ref. [20]. The collision operator is C {f } = 1 ∂ p2 ∂ p  p2 CA  ∂f CB ∂ ∂f + CF f + 2 (1 − ξ 2 ) ∂p p ∂ξ ∂ξ (3) where Γ Ψ (x), v   Γ δ 4 x2 CB = Z + φ(x) − Ψ (x) + , 2v 2 CA = CF = Γ T Ψ (x), (4) (5) (6)  δ = ve /c, x = v/ve = y/ 1 + δ 2 y2 , Z is the effective ion charge, √ Γ = 4π ne4 ln Λ = (3 π /4)νee ve3 m2 (7) √ 4 is identical √ to the Γ defined in Refs. [6,20,21], νee = 4 2π e n ln Λ/(3 mT 3/2 ) is the usual Braginskii electron collision frex quency, φ(x) = 2π −1/2 0 exp(−s2 ) ds is the error function, and   1 dφ Ψ (x) = 2 φ(x) − x (8) 2x dx is the Chandrasekhar function. In the nonrelativistic limit δ → 0, then y → x, and (3) reduces to the usual Fokker–Planck testparticle electron collision operator. The collision operator (3) is approximate in several ways. First, it originates from the Fokker–Planck approximation in which small-angle collisions dominate, which is related to an expansion in ln Λ ≫ 1. Consequently, the infrequent collisions with large momentum exchange are ignored, so the secondary avalanche process is not included at this stage, but will be addressed later in Section 5. Also, the modifications to the Rosenbluth potentials associated with the high-energy electrons are neglected, i.e. collisions with high-energy field particles are ignored. The kinetic equation is normalized by multiplying through with m3 ve3 π 3/2 /(νee n), and defining the normalized distribution function F = (π 3/2 m3 ve3 /n)f (9) so that F → 1 at p → 0. We also introduce a normalized electric field Ê = −eE /(mve νee ) (10) which, up to a factor of order unity, is E normalized by the Dreicer field. The normalized time is t̂ = νee t and the normalized source is Ŝ = Sm3 ve3 π 3/2 /(νee n). We thereby obtain the dimensionless M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 equation 849 Then the operation ∂F ∂F + Ê ξ ∂y ∂ t̂ √ 3 π − 4 √ 3 π − 4 + Ê 1 − ξ 2 ∂F 2L + 1 ∂y   1 ∂ 2 Ψ (x) ∂ F y + 2 Ψ ( x ) F y2 ∂ y x ∂y   1 δ 4 x2 ∂ ∂F Z + φ(x) − Ψ (x) + (1 − ξ 2 ) 2 2xy 2 ∂ξ ∂ξ y = Ŝ . 2 Notice that this equation has the form of a linear inhomogeneous 3D partial differential equation: (12) for a linear time-independent differential operator M. If a timeindependent equilibrium solution exists, it will be given by F = M −1 Ŝ. Since both the electric field acceleration term and the collision operator in the kinetic equation (1) have the form of a divergence of a flux in velocity space, the total number  of particles  is constant in time in the absence of a source: (d/dt ) d3 p f = d3 p S. However, runaway electrons are constantly gaining energy, so without a source at small p and a sink at large p, no time-independent distribution function will exist. From another perspective, a nonzero source is necessary to find a nonzero equilibrium solution of (11), because when Ŝ = 0, (11) with ∂/∂ t̂ = 0 is a homogeneous equation with homogeneous boundary conditions. (The boundary conditions are that F be regular at y = 0, ξ = −1, and ξ = 1, and that f → 0 as y → ∞.) Thus, the solution of the time-independent problem F = M −1 Ŝ for Ŝ = 0 would be F = 0. To find a solution, we must either consider a time-dependent problem or include a nonzero S. In reality, spatial transport can give rise to both sources and sinks, and a sink exists at high energy due to radiation. When included, secondary runaway generation (considered in Section 5) also introduces a source. To avoid the added complexity of these sinks and sources and simultaneously avoid the intricacies of time dependence, when restricting ourselves to primary generation we may formulate a time-independent prob−y2 lem as follows. We take Ŝ = α e for some constant α , representing a thermal source of particles. Eq. (11) for ∂/∂ t̂ = 0 may be divided through by α and solved for the unknown F /α . Then α may be determined by the requirement F (p = 0) = 1, and F is then obtained by multiplying the solution F /α by this α . The constant α represents the rate at which particles must be replenished at low energy to balance their flux in velocity space to high energy. Therefore, α is the rate of runaway production. As we do not introduce a sink at high energies, F will have a divergent integral over velocity space. CODE can also be run in time-dependent mode. Once the velocity space coordinates and the operator M are discretized, any implicit or explicit scheme for advancing a system of ordinary differential equations (forward or backward Euler, Runge–Kutta, trapezoid rule, etc.) may be applied to the time coordinate. (Results shown in this paper are computed using the trapezoid rule.) Due to the diffusive nature of M, numerical stability favors implicit timeadvance schemes. 3. Discretization We first expand F in Legendre polynomials PL (ξ ): F (y, ξ ) = ∞  L=0 FL (y)PL (ξ ). 1 PL (ξ )(·)dξ (14) −1 is applied to the kinetic equation. Using the identities in the Appendix, we obtain (11) ∂F + MF = Ŝ ∂ t̂  (13)  ∞   L+1 ∂ FL  L ∂ + Ê δL+1,ℓ + δL−1,ℓ 2L + 3 2L − 1 ∂y ∂ t̂ ℓ=0   (L − 1)L Ê (L + 1)(L + 2) δL+1,ℓ − δL−1,ℓ + y 2L + 3 2L − 1 √ √   3 π Ψ (x) ∂2 3 π 2Ψ (x) dx dΨ − δL,ℓ 2 − + δL,ℓ 4 x ∂y 2 y dy dx √   3 π 1 dx dΨ 2Ψ (x) Ψ (x) dx ∂ − + − 2 + 2Ψ (x) δL,ℓ 4 x dy dx xy x dy ∂y √    3 π δ 4 x2 + Z + φ(x) − Ψ (x) + L(L + 1)δL,ℓ Fℓ = ŜL (15) 2 8xy 2 2 2 −3/2 where dx/dy = (1 + δ y ) and ŜL = (2L + 1) 2 −1 1 −1 , dΨ /dx = 2π −1/2 e−x − (2/x)Ψ (x), 2 Ŝ dξ is the appropriate Legendre mode of Ŝ (y, ξ ) = L=0 ŜL (y)PL (ξ ). Note that the collision operator is diagonal in the L index, and the electric field acceleration term is tridiagonal in L. It is useful to examine the L = 0 case of (15), which corresponds to (half) the integral of the kinetic equation over ξ : ∞  ∂ F0 Ê 1 ∂ − 2 −y2 F1 y ∂y 3 ∂ t̂ √   3 π 2 Ψ (x) ∂ F0 + y + 2Ψ (x)F0 = Ŝ0 . (16) 4 x ∂y ∞ Applying 4π −1/2 y dy y2 (·) for some boundary value yb , and b assuming the source is negligible in this region, we obtain 1 dnr 4  Ê −y2 F1 νee n dt̂ 3 π √   3 π 2 Ψ (x) ∂ F0 y + 2Ψ (x)F0 + 4 x ∂y = −√ (17) y=yb where nr is the number of runaways,  meaning the number  ∞ of electrons with y > yb , so that nr = y>y d3 p f = 2π mv y dp p2 1 −1 b e b dξ f . The runaway rate calculated from (17) should be independent of yb in steady state (as  ylong as yb is in a region of Ŝ0 = 0), which can be seen by applying y b2 dy y2 (·) to (16). We find in pracb1 tice it is far better to compute the runaway production rate using (17) than from the source magnitude α , since the latter is more sensitive to the various numerical resolution parameters. To discretize the equation in y, we can apply fourth-order finite differences on a uniform grid. Alternatively, for greater numerical efficiency, a coordinate transformation can be applied so grid points are spaced further apart at high energies. The y coordinate is cut off at some finite maximum value ymax . The appropriate boundary conditions at y = 0 are dF0 /dy = 0 and FL = 0 for L > 0. For the boundary at large y, we impose FL = 0 for all L. This boundary condition creates some unphysical grid-scale oscillation at large y, which may be eliminated by adding an artificial diffusion c1 y−2 (∂/∂ y)y2 exp(−[y − ymax ]/c2 )∂/∂ y localized near ymax to the linear operator. Suitable values for the constants are c1 = 0.01 and c2 = 0.1. This term effectively represents a sink for particles, 850 M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 a b log10 Fig. 1. (Color online) Typical results of CODE, obtained for δ = 0.1, Ê = 0.1, and Z = 1. (a) Normalized distribution function F for p⊥ = 0. Results are plotted for two different sets of numerical parameters ({Ny = 300, ymax = 20, Nξ = 20} and {Ny = 1200, ymax = 40, Nξ = 40}). The results overlap completely, demonstrating excellent convergence. A Maxwellian is also plotted for comparison. (b) Contours of F at values 10z for integer z. Bold contours indicate F = 10−5 and 10−10 . which must be included in the time-independent approach due to the particle source at thermal energies. Since this diffusion term is exponentially small away from ymax , the distribution function is very insensitive to the details of the ad-hoc term except very near ymax . All results shown hereafter are very well converged with respect to doubling the domain size ymax , indicating the results are insensitive to the details of the diffusion term. 4. Results for primary runaway electron generation Fig. 1 shows typical results from a time-independent CODE computation. To verify convergence, we may double Nξ (the number of Legendre modes), double Ny (the number of grid points in y), and double the maximum y (ymax ) at fixed y grid resolution (which requires doubling Ny again). As shown by the overlap of the solid red and dashed blue curves in Fig. 1a, excellent convergence is achieved for the parameters used here. Increasing the ad-hoc diffusion magnitude c1 by a factor of 10 for the parameters of the red curve causes a relative change in the runaway rate (computed using (17) for yb = 10) of |dnr /dt̂ (c1 = 0.1) − dnr /dt̂ (c1 = 0.01)|/[dnr /dt̂ (c1 = 0.01)] < 10−9 , demonstrating the results are highly insensitive to this diffusion term. As expected, the distribution function is increased in the direction opposite to the electric field (p∥ > 0). While the distribution function is reduced in the direction parallel to the electric field (p∥ < 0) for y < 5, F is actually slightly increased for y > 5 due to pitch-angle scattering of the high-energy tail electrons, an effect also seen in Fokker–Planck simulations of RF current drive [22]. The pitchangle scattering term can be artificially suppressed in CODE, in which case F is reduced in the direction parallel to the electric field for all y. Fig. 2 compares the distribution functions obtained from the time-independent and time-dependent approaches. At sufficiently long times, the time-dependent version produces results that are indistinguishable from the time-independent version. For comparison with previously published results, we show in Fig. 3 results by Kulsrud et al. [23], who considered only the nonrelativistic case δ → 0. The agreement with CODE is exceptional. The runaway production rate in CODE is computed using (17) for yb = 10. (Any value of yb > 5 gives indistinguishable results.) Ref. [23] uses a different normalized electric field EK which is re√ lated to Ê by EK = 2(3 π )−1 Ê, and in Ref. [23] the runaway √ rate is also normalized by a different collision frequency νK = 3 π/2 νee . It should also be noted that the Kulsrud computations are timedependent, with a simulation run until the flux in velocity space reaches an approximate steady state. Each CODE point shown in Fig. 3 took approximately 0.08 s on a single Dell Precision laptop with Intel Core i7-2860 2.50 GHz CPU and 16 GB memory, running in MATLAB. Faster results could surely be obtained using a lowerlevel language. Fig. 2. (Color online) The distribution function from time-dependent CODE at various times. At t = 1000/νee , the distribution function is indistinguishable from the solution obtained using the time-independent scheme (t = ∞) over the momentum range shown. Fig. 3. (Color online) Benchmark of CODE in the nonrelativistic limit δ → 0 against data in Table 1 of Ref. [23]. To emphasize the speed of CODE, we have directly compared it to the ARENA code [9] for computing the runaway rate using the parameters considered in [23]. ARENA is a Monte Carlo code written in Fortran 90 and designed specifically to compute the runaway distribution function and runaway rate. Detailed description of the current version of ARENA is given in Refs. [24,25]. Both codes were run on a single thread on the same computer with an Intel Xeon 2.0 GHz processor. ARENA required 49,550 s to reproduce the left square point in Fig. 3, and 5942 s to reproduce the top-right square point. 50,000 particles were required for M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 reasonable convergence. For comparison, at a similar level of convergence, time-independent CODE required 0.00106 s and 0.000696 s for the two respective points, and time-dependent CODE required 0.0307 s and 0.00082 s respectively. Thus, for these parameters, both time-independent and time-dependent CODE require less than 1.5 × 10−7 as many cpu-hours as ARENA for the same hardware. 5. Secondary runaway electron generation In the previous sections we used the Fokker–Planck collision operator, which includes ‘‘distant’’ (large impact parameter) collisions but not ‘‘close’’ (small impact parameter) collisions in which a large fraction of energy and momentum are transferred between the colliding particles. Close collisions are infrequent compared to distant collisions, and are therefore neglected in the Fokker–Planck operator. However, close collisions may still have a significant effect on runaway generation, since the density of runaways is typically much smaller than the density of thermal electrons which may be accelerated in a close collision. The production of runaways through close collisions is known as secondary production, or as avalanche production since it may occur with exponential growth. To simulate secondary generation of energetic electrons, we use a source term derived in [19], starting from the Møller scattering cross-section in the w ≫ 1 limit, with w = p/(mc ) = δ y a normalized momentum. In this limit, the trajectories of the primary electrons are not much deflected by the collisions. The source then takes the form ∂ δ(ξ − ξ2 ) 2 S= 4π τ ln Λ w ∂w nr 1   1 √ 1− 1 + w2 , (18) where 1/τ = 4π ne4 ln Λ/(m2 c 3 ) is the collision frequency for relativistic electrons, nr is the density of the fast electrons and √ ξ2 = w/(1 + 1 + w 2 ) is the cosine of the pitch angle at which the runaway is born. (Our Eq. (18) differs by a factor m3 c 3 compared to the source our distribution function  in Ref. [19] since we normalize  as n = d3 p f instead of n = d3 w f . There is also a factor of 2π difference due to the different normalization of the distribution function.) Due to the approximations used to derive S, care must be taken in several regards. First, to define nr in (18), it is not clear where to draw the dividing line in velocity space between runaways and non-runaways. One possible strategy for defining nr is to compute the separatrix in velocity space between trajectories that will have bounded and unbounded energy in the absence of diffusion, and to define the runaway density as the integral of f over the latter region [26]. This approach may somewhat overestimate the true avalanche rate, since it neglects the fact that some time must elapse between an electron entering the runaway region and the electron gaining sufficient energy to cause secondary generation. As most runaways have ξ ≈ 1, we may approximate the separatrix by setting dw/dt = 0 where dw/dt = eE /(mc ) − (1 + 1/w 2 )/τ defines the trajectory of a particle with ξ = 1, neglecting diffusion in momentum and pitch angle. The runaway region is therefore w > wc where wc = [(E /Ec ) − 1]−1/2 and Ec = mc /(eτ ) is the 1 ∞ critical field, and so we take nr = 2π m c −1 dξ w dw w f . (We c cannot define nr by the time integral of (17), since (17) is no longer valid when S is nonzero away from p ≈ 0.) A second deficiency of (18) is that S is singular at w → 0, so the source must be cut off below some threshold momentum. Following Ref. [12], we choose the cutoff to be wc . Neither of the cutoffs discussed here would be necessary if a less approximate source term than (18) were used, but derivation of such an operator is beyond the scope of this paper. Normalizing and applying (14) as we did previously for the other terms in the kinetic equation, the source included in CODE 3 3 2 851 becomes ŜL = nr 3πδ 5 2L + 1 n 16 ln Λ 2 PL (ξ2 ) (1 − √ 1 1 + w 2 )2 √ 1 + w2 y . (19) When secondary generation is included, CODE must be run in timedependent mode. To benchmark the numerical solution of the kinetic equation including the above source term by CODE, we use the approximate analytical expression for the avalanche distribution function derived in Section II of Ref. [18]:  k faa (w∥ , w⊥ ) = w∥ exp γ̃ t −  − E /Ec − 1  Z +1 γ̃ τ E /Ec − 1 2 w⊥ 2w∥ w∥  (20) where k is a constant. The quantity γ̃ is the growth rate γ̃ = (1/f )∂ f /∂ t, which must be independent of both time and velocity for (20) to be valid. Eq. (20) is also valid only where p∥ ≫ p⊥ and in regions of momentum space where S is negligible. (This restriction is not a major one since S = 0 everywhere except on the ξ = ξ2 curve.) If most of the runaway distribution function is  accurately described by (20), then we may approximate nr ≈ d3 p faa =   ∞ ∞ 2π m3 c 3 −∞ dw∥ 0 dw⊥ w⊥ faa , giving γ̃ = (1/nr )dnr /dt and k = nr e−γ̃ t τ (21) 2 π m3 c 3 ( 1 + Z ) where nr e−γ̃ t is constant. (Eq. (21) may be inaccurate in some situations even if (20) is accurate in part of velocity-space, because (21) requires (20) to apply in all of velocity-space.) Figs. 4 and 5 show comparisons between distributions from CODE and (20)–(21) for two different sets of parameters. More precisely, the quantity plotted in Fig. 4–5 is log10 (m3 c 3 f /nr ). To generate the figures, CODE is run for a sufficiently long time that (1/f )∂ f /∂ t becomes approximately constant. The resulting numerical value of (1/nr )dnr /dt is then used as γ̃ when evaluating (20)–(21). For a cleaner comparison between CODE and analytic theory in these figures, we minimize primary generation in CODE in these runs by initializing f to 0 instead of to a Maxwellian. For both sets of physical parameters, the agreement between CODE and (20) is excellent in the region where agreement is expected: where p∥ ≫ p⊥ and away from the curve ξ = ξ2 . 6. Synchrotron emission Using the distribution functions calculated with CODE, we now proceed to compute the spectrum of emitted synchrotron radiation. Due to the energy dependence of the emitted synchrotron power, the emission from runaways completely dominates that of the thermal particles. The emission also depends strongly on the pitch-angle of the particle. In a cylindrical plasma geometry, the emitted synchrotron power per wavelength at wavelength λ from a single highly energetic particle is given by [27] 4π P (γ , γ∥ , λ) = √ ce2 3 λ γ 3  2 ∞ K5/3 (l) dl, (22) λc /λ where the two-dimensional  momentum of the particle is determined by γ and γ∥ = 1/ 1 − v∥2 /c 2 , Kν (x) is a modified Bessel function of the second kind, and λc = 4π mc 2 γ∥ 3 eBγ 2 , (23) where B is the magnetic field strength. Using CODE we will demonstrate that the synchrotron radiation spectrum from the entire runaway distribution is substantially 852 M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 (1 − w 2 ξ 2 /(1 + w2 ))−1 , and integrating (22) over the runaway region R in momentum space, we obtain the total synchrotron emission from the runaway distribution. Normalizing to nr , we find that the average emitted power per runaway particle at a wavelength λ is given by P (λ) = Fig. 4. (Color online) Contour plots of the long-time distribution function from CODE (shown in two different coordinate systems), obtained for E /Ec = 40 (Ê = 0.532), Z = 3, δ = 0.1 and t = 5000/νee . Results are plotted for the numerical parameters Ny = 1500, ymax = 1500 and Nξ = 100, with time step dt = 10/νee . The analytical distribution in (20)–(21) for the same physical parameters is also plotted for comparison, together with part of the curve where avalanche runaways are created (ξ = ξ2 ). Fig. 5. (Color online) Contour plots of the long-time distribution function from CODE (shown in two different coordinate systems), obtained for E /Ec = 100 (Ê = 0.332), Z = 1, δ = 0.05 and t = 6000/νee . Results are plotted for the numerical parameters Ny = 1500, ymax = 3000 and Nξ = 180, with time step dt = 25/νee . The analytical distribution in (20)–(21) is also plotted for comparison, together with part of the curve where avalanche runaways are created (ξ = ξ2 ). different from the spectrum obtained from a single particle approximation. By transforming to the more suitable coordinates w and ξ , related to γ and γ∥ through γ 2 = 1 + w 2 and γ∥2 = 2π nr  f (w, ξ ) P (w, ξ , λ) w 2 dw dξ . (24) R Up to a factor ecA, where A is the area of the runaway beam, normalization by nr is equivalent to normalization by the runaway current, since the emitting particles all move with velocity ≈ c. The per-particle synchrotron spectra generated by the CODE distributions in Figs. 4 and 5 were calculated using this formula, and are shown in Fig. 6, together with the spectra radiated by electron distributions for other electric field strengths. For the physical parameters used, we note that the peak emission occurs between 7 and 25 µm. The synchrotron spectra show a decrease in per-particle emission with increasing electric field strength. Even though a stronger electric field leads to more particles with high energy (and thus high average emission), it also leads to a more narrow distribution in pitch-angle. This reduction in the number of particles with large pitch-angle leads to a decrease in average emission. Both figures confirm that the average emission is reduced for higher electric fields, implying that the latter mechanism has the largest impact on the spectrum. In calculating the spectra, the runaway region of momentum space, R, was defined such that the maximum particle momentum was wmax = 50 (which translates to ymax = 500 and ymax = 1000 respectively for the cases shown in Figs. 4 and 5), corresponding to a maximum particle energy of ≃25 MeV. Physically the cutoff at large energy can be motivated by the finite life-time of the accelerating electric field and the influence of loss mechanisms such as radiation. Since the radiated synchrotron power increases with both particle energy and pitch, this truncation of the distribution is necessary to avoid infinite emission, although the precise value for the cutoff depends on the tokamak and on discharge-specific limitations to the maximum runaway energy. For the low-energy boundary of R, wmin = wc = [(E /Ec ) − 1]−1/2 was used, and all particles with ξ ∈ [0, 1] were included. Although no explicit cutoff was imposed in ξ , the distribution decreases rapidly as this parameter decreases from 1 (as can be seen in Figs. 4 and 5) and there are essentially no particles below some effective cutoff value. Fig. 6 also shows the synchrotron spectrum from single particles with momentum corresponding to the maximum momentum of the distributions (w = 50), and several values of pitch-angle y⊥ /y∥ . This single-particle ‘‘approximation’’ is equivalent to using a 2D δ -function model of the distribution, as was done in Refs. [28,16] (and with some modification in [17]). The figure shows that this approximation significantly overestimates the synchrotron emission per particle. Note that in the figure, the values for the emitted power per particle were divided by a large number to fit in the same scale. The overestimation is not surprising, since the δ function approximation effectively assumes that all particles emit as much synchrotron radiation as one of the most strongly emitting particles in the actual distribution. The figure also shows that the δ -function approximation leads to a different spectrum shape, with the wavelength of peak emission usually shifted towards shorter wavelengths. In order to obtain an accurate runaway synchrotron spectrum, it is thus crucial to use the full runaway distribution in the calculation. In the cases shown in Fig. 6, the runaway electron distribution is dominated by secondary generation. For comparison, in Fig. 7–8 we show a case where the distribution is dominated by primary generation. Fig. 7a shows contours of a distribution from primaries only, together with a distribution obtained with the avalanche M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 a 853 Emitted Power [10–15 W/µm] Emitted Power [10–15 W/µm] b Wavelength [µm] Wavelength [µm] Fig. 6. (Color online) Synchrotron spectra (average emission per particle) for the runaway distributions in (a) Fig. 4 and (b) Fig. 5. Emission spectra from the CODE distributions in Figs. 4 and 5 are shown in solid black, together with spectra from distributions with varying electric field strength but otherwise identical physical parameters. A magnetic field of B = 3 T was used. The synchrotron spectra from single particles with w = 50 and various pitch angles are also shown. (These single-particle spectra are the same in figures a and b, as the particle parameters are independent of simulation settings.) Emitted Power [10–16 W/µm] a b Wavelength [µm] Fig. 8. (Color online) Synchrotron spectra (average emission per particle) for the runaway distributions shown in Fig. 7 for magnetic field B = 3 T. Fig. 7. (Color online) (a) Contour plots of the distribution function from primaries only (black solid line), together with a distribution obtained with the avalanche source enabled (dashed red line), for E /Ec = 10 (Ê = 0.523), Z = 1 and δ = 0.2 at t = 45/νee . The quantity plotted is log10 (F ). Results are plotted for the numerical parameters Ny = 20, ymax = 100 and Nξ = 130, with time step dt = 0.02/νee . (b) Contour plots of the distribution function at different times with the avalanche source enabled, using the above parameters. source enabled, and confirms that the distribution is dominated by primaries, except for a small number of secondary runaways generated along the curve ξ = ξ2 . Fig. 7(b) shows contour plots with the avalanche source enabled, for three different times. The physical parameters used in Fig. 7 are temperature T = 10 keV, density n = 5 × 1019 m−3 , effective charge Z = 1, and electric field E = 0.45 V/m. The collision time in this case is 0.39 ms, so the times shown in the figure correspond to 5.9, 11.8 and 17.6 ms, which correlates well with the time-scale of the electric field spike for a typical disruption in DIII-D (see e.g. Fig. 2 in [29]). Fig. 8 compares the synchrotron spectra from the distributions shown in Fig. 7. The main difference compared to the case dominated by secondary generation (Fig. 6) is the generally longer wavelengths in the spectrum. The reason is the low runaway electron energy (w . 10) in the runaway electron distribution in this case. The small peak at short wavelengths in the spectra including the avalanche source stems from the secondary runaways generated at ξ = ξ2 (visible in Fig. 7(a)). In principle, we may also use the synchrotron spectra from distributions calculated through CODE to estimate the maximum energy of the runaways in existing tokamaks. However, due to the region of sensitivity of the available detectors, there is only a limited wavelength range in which calculated spectra can be fitted to experimental data in order to determine the maximum runaway energy. The available range often corresponds to the short wavelength slope of the spectrum, where the emitted power shows an approximately linear dependence on wavelength. Indeed, the short-wavelength spectrum slope has been used to estimate the maximum runaway energy in experiments [16]. If the runaway distribution function is approximated by a δ -function at the maximum available energy and pitch angle, there is a monotonic relationship between the short-wavelength spectrum slope and the maximum particle energy (at fixed pitch angle). Such a relationship holds because increasing the particle energy leads to more emission at shorter wavelengths, resulting in a shift of the wavelength of peak emission towards shorter wavelengths, and a corresponding change in the spectrum slope. 854 M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 Using an integrated synchrotron spectrum from a CODE distribution is much more accurate than the single particle approximation, but it also introduces additional parameters (Ê, δ , Z ). If the physical parameters are well known, a unique relation still holds between the spectrum slope and the maximum particle energy. During disruptions however, many parameters (like the temperature and the effective charge) are hard to measure with accuracy. As the shape of the underlying distribution depends on the values of the parameters, the synchrotron spectrum will do so as well. This complexity is apparent in Fig. 6, where the single particle approximation produces identical results in the two cases, whereas the spectra from the complete distributions are widely different. The dependence on distribution shape makes it possible in principle for two sets of parameters to produce the same spectrum slope for different maximum energies. Given this insight, using the complete runaway distribution when modeling experimentally obtained spectra is necessary for an accurate analysis and reliable fit of the maximum particle energy. In this context, CODE is a very useful tool with the possibility to contribute to the understanding of runaways and their properties. 7. Conclusions In this work, we have computed the synchrotron emission spectra from distribution functions of runaway electrons. The distribution functions are computed efficiently using the CODE code. Both primary (Dreicer) and secondary (avalanche) generation are included. A Legendre spectral discretization is applied to the pitch-angle coordinate, with high-order finite differences applied to the speed coordinate. A nonuniform speed grid allows high resolution of thermal particles at the same time as a high maximum energy without a prohibitively large number of grid points. If secondary generation is unimportant, the long-time distribution function may be calculated by solving a single sparse linear system. The speed of the code makes it feasible to couple to other codes for integrated modeling of complex processes such as tokamak disruptions. CODE has been benchmarked against previous analytic and numerical results in appropriate limits, showing excellent agreement. In the limit of strong avalanching, CODE demonstrates agreement with the analytic distribution function (20) from Ref. [18]. The synchrotron radiation spectra are computed by convolving the distribution function with the single-particle emission. We find that the radiation spectrum from a single electron at the maximum energy can differ substantially from the overall spectrum generated by a distribution of electrons. Therefore, experimental estimates of maximum runaway energy based on the single-particle synchrotron spectrum are likely to be inaccurate. A detailed study of the distribution-integrated synchrotron spectrum and its dependences on physical parameters can be found in [30]. In providing the electron distribution functions (and thus knowledge of a variety of quantities through its moments), the applicability of CODE is wide, and the potential in coupling CODE to other software, e.g. for modeling of runaway dynamics in disruptions, is promising. For a proper description of the runaways generated in disruptions it is important to take into account the evolution of the radial profiles of the electric field and fast electron current self-consistently. This can be done by codes such as GO, initially described in Ref. [31] and developed further in Refs. [32,33]. GO solves the equation describing the resistive diffusion of the electric field in a cylindrical approximation coupled to the runaway generation rates. In the present version of GO, the runaway rate is computed by approximate analytical formulas for the primary and secondary generation. Using CODE, the analytical formulas can be replaced by a numerical solution for the runaway rate which would have several advantages. One advantage would be that Dreicer, hot-tail and secondary runaways could all be calculated with the same tool, avoiding the possibilities for doublecounting and difficulties with interpretations of the results. Also, in the present version of GO, it is assumed that all the runaway electrons travel at the speed of light, an approximation that can be easily relaxed using CODE, which calculates the electron distribution in both energy and pitch-angle. Most importantly, the validity region of the results would be expanded, as the analytical formulas are derived using various assumptions which are often violated in realistic situations. The output would be a selfconsistent time and space evolution of electric field and runaway current, together with the electron distribution function. This information can then be used for calculating quantities that depend on the distribution function, such as the synchrotron emission or the kinetic instabilities driven by the velocity anisotropy of the runaways. Acknowledgments This work was supported by US Department of Energy grants DE-FG02-91ER-54109 and DE-FG02-93ER-54197, by the Fusion Energy Postdoctoral Research Program administered by the Oak Ridge Institute for Science and Education, and by the European Communities under Association Contract between EURATOM and Vetenskapsrådet. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The authors are grateful to J. Rydén, G. Csépány, G. Papp, P. Helander, E. Nilsson, J. Decker, and Y. Peysson for fruitful discussions. Appendix. Integrals of Legendre polynomials Here we list several identities for Legendre polynomials which are required for the spectral pitch-angle discretization. To evaluate the ξ integral of the ξ ∂ F /∂ y term in (11), we use the recursion relation ξ PL (ξ ) = L+1 2L + 1 PL+1 (ξ ) + L 2L + 1 PL−1 (ξ ) (A.1) where PL−1 is replaced by 0 when L = 0. Applied to the relevant integral in (11), and noting the orthogonality relation (2L + 1 1)2−1 −1 PL (ξ )Pℓ (ξ )dξ = δL,ℓ , we find 2L + 1  2 1 dξ ξ PL (ξ )Pℓ (ξ ) = −1 L+1 2L + 3 δℓ,L+1 + L 2L − 1 δℓ,L−1 . (A.2) Similarly, to evaluate the ξ integral of the ∂ F /∂ξ term in (11), we use the recursion relation (1 − ξ 2 )(dPL /dξ ) = LPL−1 (ξ ) − Lξ PL (ξ ) (A.3) to obtain 2L + 1  2 1 dξ PL (ξ )(1 − ξ 2 ) −1 dPℓ dξ (L + 1)(L + 2) (L − 1)L = δℓ,L+1 − δℓ,L−1 . 2L + 3 2L − 1 (A.4) Finally, the pitch-angle scattering collision term gives the integral 2L + 1 2  1 dξ PL (ξ ) −1 ∂ ∂ (1 − ξ 2 ) Pℓ (ξ ) = −(L + 1)LδL,ℓ . ∂ξ ∂ξ References [1] H. Dreicer, Phys. Rev. 117 (1960) 329. [2] J.W. Connor, R.J. Hastie, Nucl. Fusion 15 (1975) 415. [3] H.P. Furth, P.H. Rutherford, Phys. Rev. Lett. 28 (1972) 545. (A.5) M. Landreman et al. / Computer Physics Communications 185 (2014) 847–855 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] P. Helander, D. Ward, Phys. Rev. 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Bekefi, Radiation Processes in Plasmas, Wiley, 1966. [28] K.H. Finken, J.G. Watkins, D. Rusbuldt, W.J. Corbett, K.H. Dippel, D.M. Goebel, R.A. Moyer, Nucl. Fusion 30 (1990) 859. [29] E.M. Hollmann, M.E. Austin, J.A. Boedo, N.H. Brooks, N. Commaux, N.W. Eidietis, D.A. Humphreys, V.A. Izzo, A.N. James, T.C. Jernigan, A. Loarte, J. Martin-Solis, R.A. Moyer, J.M. Munoz-Burgos, D.L. Rudakov, E.J. Strait, C. Tsui, M.A.V. Zeeland, J.C. Wesley, J.H. Yu, Nucl. Fusion 53 (2013) 083004. [30] Stahl A., Landreman M., G. Papp, E. Hollmann, T. Fülüp, Phys. Plasmas 20 (2013) 093302. [31] H. Smith, P. Helander, L.-G. Eriksson, D. Anderson, M. Lisak, F. Andersson, Phys. Plasmas 13 (2006) 102502. [32] K. Gál, T. Fehér, H. Smith, T. Fülöp, P. Helander, Plasma Phys. Controll. Fusion 50 (2008) 055006. [33] T. Fehér, H.M. Smith, T. Fülöp, K. Gál, Plasma Phys. Controll. Fusion 53 (2011) 035014. Errata, Paper B In Eq. (11), the partial derivative with respect to y in the third term should be replaced by a partial derivative with respect to ξ, so that the first row reads ∂F ∂F 1 − ξ ∂F + Ê ξ + Ê ∂y y ∂ξ ∂ t̂ On the left-hand side of Eq. (17), the collision frequency should be removed, so that the expression starts with 1 dnr = ... n dt̂ Paper C A. Stahl, O. Embréus, G. Papp, M. Landreman and T. Fülöp, Kinetic modelling of runaway electrons in dynamic scenarios, Nuclear Fusion 56, 112009 (2016). http://dx.doi.org/10.1088/0029-5515/56/11/112009 http://arxiv.org/abs/1601.00898 Nuclear Fusion International Atomic Energy Agency Nucl. Fusion 56 (2016) 112009 (10pp) doi:10.1088/0029-5515/56/11/112009 Kinetic modelling of runaway electrons in dynamic scenarios A. Stahl1, O. Embréus1, G. Papp2, M. Landreman3 and T. Fülöp1 1 Department of Physics, Chalmers University of Technology, Göteborg, Sweden Max Planck Institute for Plasma Physics, Garching, Germany 3 University of Maryland, College Park, MD, USA 2 E-mail: [email protected] Received 5 January 2016, revised 4 March 2016 Accepted for publication 29 March 2016 Published 22 July 2016 Abstract Improved understanding of runaway-electron formation and decay processes are of prime interest for the safe operation of large tokamaks, and the dynamics of the runaway electrons during dynamical scenarios such as disruptions are of particular concern. In this paper, we present kinetic modelling of scenarios with time-dependent plasma parameters; in particular, we investigate hot-tail runaway generation during a rapid drop in plasma temperature. With the goal of studying runaway-electron generation with a self-consistent electric-field evolution, we also discuss the implementation of a collision operator that conserves momentum and energy and demonstrate its properties. An operator for avalanche runaway-electron generation, which takes the energy dependence of the scattering cross section and the runaway distribution into account, is investigated. We show that the simplified avalanche model of Rosenbluth and Putvinskii (1997 Nucl. Fusion 37 1355) can give inaccurate results for the avalanche growth rate (either lower or higher) for many parameters, especially when the average runaway energy is modest, such as during the initial phase of the avalanche multiplication. The developments presented pave the way for improved modelling of runaway-electron dynamics during disruptions or other dynamic events. Keywords: runaway electrons, Fokker–Planck equation, avalanche generation, hot-tail generation, linearized collision operator (Some figures may appear in colour only in the online journal) 1. Introduction Kinetic simulation is the most accurate and useful method for investigating runaway-electron dynamics, and we recently developed a new tool called code (collisional distribution of electrons [5]) for fast and detailed study of these processes. code solves the spatially homogeneous kinetic equation for electrons in 2D momentum space, including electric-field acceleration, collisions, avalanche runaway generation and synchrotron-radiation-reaction losses [5–7]. In code, momentum space is discretized using finite differences in momentum and a Legendre-mode decomposition in pitchangle cosine. Often, the time evolution of the distribution is the desired output, but a (quasi-)steady-state solution can also be efficiently obtained through the inversion of a single sparse system (in the absence of an avalanche source). code has been used to study the spectrum of the synchrotron radiation emitted by runaways [5], the corresponding influence of Runaway electrons, a phenomenon made possible by the decrease of the collisional friction with particle energy [1, 2], are common in plasmas in the presence of strong external electric fields or changing currents. The tightly focused beam of highly relativistic particles can be a serious threat to the first wall of a fusion reactor, due to the possibility of localized melting or halo-current generation [3]. In the quest for avoidance or mitigation of the harmful effects of runawayelectron losses, a greater understanding of the runawayelectron phenomenon is required [4]. Improved knowledge of runaway-electron formation mechanisms, dynamics and characteristics will benefit the fusion community and contribute to a stable and reliable operation of reactor-scale tokamaks. 0029-5515/16/112009+10$33.00 1 © 2016 EURATOM Printed in the UK A. Stahl et al Nucl. Fusion 56 (2016) 112009 κ = m3v˜e 3π 3 / 2 /n˜, we have also defined the distribution function F = F ( y, ξ ) = κ f (normalized so that F( y = 0) = 1 for a Maxwellian with T = T˜ and n = n˜), time tˆ = ν˜ee t, and electric field Eˆ = −eE /mv˜eν̃ee, as well as the normalized operators ˜ /3m2v˜e 3 Cˆ = C κ /ν˜ee and Sˆ = Sκ /ν˜ee, with ν˜ee = 16 π e4n˜ ln Λ the reference electron thermal collision frequency,−e, m and v the charge, rest mass and speed of the electron, and γ the relativistic mass factor. Note that |Eˆ | = (3 π /2)E /ED, with ED the Dreicer field [1]. C is the Fokker–Planck collision operator and S an operator describing close (large-angle) Coulomb col lisions. These operators will be discussed more thoroughly in sections 3 and 4, respectively; for now we just state the formulation of the collision operator employed in [5] using the normalizations above: the emission on the distribution function [6–8], and the factors influencing the critical electric field for runaway-electron generation [6, 9]. In this paper we describe improvements to code which enable us to investigate the effect of hot-tail runaway generation on the distribution (section 2). This process can be the dominant mechanism in rapidly cooling plasmas. We also discuss the implementation of a full linearized collision operator, and demonstrate its conservation properties (section 3). The use of this operator is necessary in cases where the correct plasma conductivity is required, and our implementation indeed reproduces the Spitzer conductivity [10] for weak electric fields. In addition, an improved model for the largeangle (knock-on) Coulomb collisions leading to avalanche multiplication of the runaway population [11], is described in section 4. This model takes the energy dependence of the runaway distribution into account, and uses the complete energydependent Møller scattering cross section [12]. We find that its use can in some cases lead to significant modifications to the avalanche growth rate, compared to the more simplified model of Rosenbluth and Putvinskii [13]. The improvements described in this work enable the detailed study of runaway processes in dynamic situations such as disruptions, and the conservative collision operator makes self-consistent calculations of the runaway population and current evolution in such scenarios feasible [14]. ⎛∂ ⎡ ⎛1 ∂ 2 ⎞ ⎤ tp ⎢ y2Ψ ⎜ + 2 ⎟F ⎥ Cˆ = cC v¯e 3y−2 ⎜⎜ ∂ ∂ y x y v ⎝ ¯ ⎣ e ⎠ ⎦ ⎝ + cξ ∂ ∂F ⎞ ⎟. (1 − ξ 2 ) ∂ξ ⎟⎠ 2x ∂ξ (2) The superscript tp denotes that this is the test-particle part of the linearized collision operator Cl discussed in section 3. Here (and throughout the rest of this paper), a bar denotes a quantity normalized to its reference value (i.e. v¯e = ve /v˜e), x = y /γ = v /v˜e is the normalized speed, cC = 3 π ν¯ee /4, cξ = Z eff + Φ − Ψ + v¯e 2δ 4x 2 /2, Z eff is the effective ion charge, Φ = Φ(x /v¯e ) and Ψ = Ψ(x /v¯e ) = v¯e 2[Φ − v¯e−1x dΦ/d(x /v¯e )]/2x 2 are the error and Chandrasekhar functions, respectively, and δ = v˜e /c (with c the speed of light) is assumed to be a small parameter (i.e. the thermal population is assumed to be non-relativistic). Changes to the plasma temperature manifest as shifts in the relative magnitude of the various terms in equation (2) (through δ and the quantities with a bar), as well as a change in the overall magnitude of the operator, whereas changes in density only have the latter effect. In both cases, the distribution is effectively colliding with (and relaxing towards) a Maxwellian different from the one native to the reference momentum grid. Heat or particles are introduced to (or removed from) the bulk of the distribution when using this scheme, as all changes to plasma parameters are described by changes to the Maxwellian. This provides a powerful way of simulating rapid cooling, for instance associated with a tokamak disruption. 2. Time-dependent plasma parameters To be able to investigate the behavior of the electron population in dynamic scenarios such as disruptions or sawtooth crashes, it is necessary to follow the distribution function as the plasma parameters change. To this end, code has been modified to handle time-dependent background-plasma parameters. Since the kinetic equation is treated in linearized form, the actual temperature and density of the distribution are determined by the background Maxwellian used in the formulation of the collision operator. This allows for a scheme where the kinetic equation is normalized to a reference temperature T̃ and number density ñ, so that the discretized equation can be expressed on a fixed reference grid in momentum space (throughout this paper, we will use a tilde to denote a reference quantity). By changing the properties of the Maxwellian equilibrium around which the collision operator is linearized, the evolution of the plasma parameters can be modelled on the reference grid without the need for repeated interpolation of the distribution function to new grids. Analogously to [5], the kinetic equation in 2D momentum space for the electron distribution function f experiencing an electric field E (parallel to the magnetic field) and collisions, can be expressed as 2.1. Hot-tail runaway-electron generation If the time scale of the initial thermal quench in a disruption event is short enough—comparable to the collision time— the tail of the initial Maxwellian electron distribution will not have time to equilibrate as the plasma cools. The particles in this supra-thermal tail may constitute a powerful source of runaway electrons, should a sufficiently strong electric field develop before they have time to reconnect with the bulk electrons. This process is known as hot-tail generation, and can be the dominant source of runaways ⎛ ∂F 1 − ξ 2 ∂F ⎞ ∂F + + Eˆ ⎜ξ ⎟ = Cˆ {F} + Sˆ{F}. (1) ˆ y ∂ξ ⎠ ∂t ⎝ ∂y Here we have introduced a convenient normalized momentum y = γv /v˜e, where v˜e = 2T˜ /m is the reference electron thermal speed, and the cosine of the pitch angle ξ = y∥ / y. Using 2 A. Stahl et al Nucl. Fusion 56 (2016) 112009 Figure 1. (a) Temperature and electric-field evolution in equations (3) and (4). (b) Parallel (ξ = 1) electron distributions (solid) and corresponding Maxwellians (dashed) at several times during the temperature drop in (a). A momentum grid with a fixed reference temperature T˜ = 100 eV was used and the distributions are normalized to F(y = 0) in the final time step to facilitate a comparison. For the temperature evolution in equation (3), analytical results for the hot-tail runaway generation were obtained in [18]. Assuming the background density to be constant, the runaway fraction at time t can be written as under certain conditions [15, 16]. It has previously been investigated analytically or using Monte-Carlo simulations [16, 17] or purpose-built finite-difference tools [17, 18]. Using code to model a temperature drop enables the efficient study of a wider range of scenarios, and allows full use of other capabilities of code, such as avalanche generation or synchrotron radiation reaction. Here, we restrict ourselves to a proof-of-principle demonstration, and leave a more extensive investigation to future work. To facilitate a comparison to the theoretical work by Smith and Verwichte [18], we will model a rapid exponential temper ature drop, described by ∞⎡ n r,dir (u3 − 3τ )2 / 3 ⎤ −u2 2 4 ⎢1 − 3c ⎥ e u du, = (5) n (u − 3τ )2 / 3 ⎦ π uc ⎣ ∫ where τ (t ) = (3 π /4)νee(t − t ) = (3 π /4)(tˆ − tˆ ) is a nor malized time, u(t ) = x[0] + 3τ (t ), x[0] is the speed nor malized to the initial thermal speed, and uc is related to the critical speed for runaway generation: uc(t ) = x[c0] + 3τ (t ). Equation (5), which corresponds to equation (18) in [18], is only valid when a significant temperature drop has already taken place (as manifested by the appearance of the cooling time scale t as a ‘delay’ in the expression for τ, see [18]). Equation (5) is derived in the absence of an electric field; only an exponential drop in the bulk temperature is assumed. The electric field shown in figure 1(a) is only used to define a runaway region, so that the runaway fraction can be calculated. In other words, it is assumed that the electric field does not have time to influence the distribution significantly during the temperature drop. The runaway fraction calculated using equation (5) includes only the electrons in the actual runaway region, i.e. particles whose trajectories (neglecting collisional momentum-space diffusion) are not confined to a region close to the origin. In this case, the lower boundary of the runaway region is given in terms of the limiting (non-relativistic) momentum y for a given ξ: ycξ = (δ 2[(ξ + 1)E /2Ec − 1])−1/2 [17], where Ec = 4πe3n ln Λ/mc 2 is the critical electric field for runaway generation [19]. The temperature drop does however lead to an isotropic high-energy tail (in the absence of an electric field). By defining the runaway region as y > yc = (δ 2[E /Ec − 1])−1 / 2, thereby including all particles with v > vc, equation (5) can be simplified to T (t ) = Tf + (T0 − Tf )e−t / t, (3) with T0 = 3.1 keV the initial temperature, Tf = 31 eV the final temperature, and t = 0.3 ms the cooling time scale. We also include a time-dependent electric field described by E (t ) ⎛ E ⎞ T0 =⎜ ⎟ , (4) ⎝ ED ⎠0 T (t ) ED with (E /ED )0 = 1/530 the initial normalized electric field. The temperature and electric-field evolutions are shown in figure 1(a) and are the same as those used in figure 5 of [18], as are all other parameters in this section. Figure 1(b), in which the additional parameters n = 2.8 ⋅ 1019 m−3, and Z eff = 1 were used, illustrates the distribution-function evolution during the temperature drop. The figure shows that as the temperature decreases, most of the electrons quickly adapt. At any given time t, the bulk of the distribution remains close to a Maxwellian corresponding to the current temperature T (t). The initially slightly more energetic electrons, although part of the original bulk population, thermalize less efficiently. On the short cooling time-scale, they remain as a distinct tail, and as the thermal speed decreases they become progressively less collisional. This process is evident in the first three time steps shown (t = 0.025–0.83 ms). In the final time step, the electric field has become strong enough to start to affect the distribution, and a substantial part of the high-energy tail is now in the runaway region. This can be seen from the qualitative change in the tail of the distribution, which now shows a positive slope associated with a strong flow of particles to higher momenta. 2 2 nr = uc e−uc + erfc(uc ), (6) n π where erfc(x) is the complementary error function. By default, code uses such an isotropic runaway region, which is a good 3 A. Stahl et al Nucl. Fusion 56 (2016) 112009 Figure 2. Hot-tail runaway density obtained using code—with (blue, dashed) and without (yellow, dash–dotted; red, dotted) an electric field included during the temperature drop—and the analytical estimates equations (5) and (6) (black, solid), for the temperature and E-field evolution in figure 1(a). An (a) ξ-dependent and (b) isotropic lower boundary of the runaway region was used. The collision operator in equation (2) was used for the blue and yellow lines, whereas its non-relativistic limit was used for the red and black lines. approximation in the case of only Dreicer and avalanche generation (especially once the runaway tail has become substantial); however, in the early stages of hot-tail-dominated scenarios, the isotropic runaway region significantly overestimates the actual runaway fraction, and the lower boundary ycξ must be used. Figure 2 compares the runaway density evolution computed with code, using both ξ-dependent and isotropic runaway regions, to equations (5) and (6), respectively. The parameters of the hot-tail scenario shown in figure 1 were used, and no avalanche source was included in the calcul ation. The collision operator used in [18] is the non-relativistic limit of equation (2), with cξ = 0 (since the distribution is isotropic in the absence of an electric field). code results using both this operator (red, dotted) and the full equation (2) (yellow, dash–dotted) are plotted in figure 2, with the latter producing ∼ 50% more runaways in total. This difference can likely be explained by the relatively high initial temperature (3 keV) in the scenario considered, in which case the non-relativistic operator is not strictly valid for the highest-energy particles. Good agreement between code results and equations (5) and (6) (black, solid) is seen for the saturated values in the figure. A code calculation where the electric-field evolution is properly included in the kinetic equation (corresponding to the distribution evolution in figure 1(b)) is also included (blue, dashed), showing increased runaway production. With the isotropic runaway region (figure 2(b)), the increase is smaller than a factor of 2, and neglecting the influence of the electric field can thus be considered reasonable for the parameters used, at least for the purpose of gaining qualitative understanding. With the ξ-dependent runaway region (figure 2(a)), the change in runaway generation is more pronounced, and the inclusion of the electric field leads to an increase by almost an order of magnitude. Note that the final runaway density with the electric field included is very similar in figures 2(a) and (b), indicating that the details of the lower boundary of the runaway region become unimportant once the tail is sufficiently large. Throughout the remainder of this paper we will make use of the isotropic runaway region. We conclude that, in order to obtain quantitatively accurate results, the electric field should be properly included, and a relativistic collision operator should be used. This is especially true when modelling ITER scenarios, where the initial temperature can be significantly higher than the 3 keV used here. 3. Conservative linearized Fokker–Planck collision operator Treating the runaway electrons as a small perturbation to a Maxwellian distribution function, the Fokker–Planck operator for electron–electron collisions [20, 21] can be linearized and written as C{ f } C l{ f } = C tp + C fp. The so-called test-particle term, C tp = C nl{ f1 , fM}, describes the perturbation colliding with the bulk of the plasma, whereas the field-particle term, C fp = C nl{ fM , f1}, describes the reaction of the bulk to the perturbation. Here C nl is the non-linear Fokker–Planck–Landau operator, fM denotes a Maxwellian, and f1 = f − fM the perturbation to it ( f1 fM). Collisions described by C{ f1 , f1 } are neglected since they are second order in f1. The full linearized operator Cl conserves particles, momentum and energy. Since it is proportional to a factor exp(−y 2 ), the field-particle term mainly affects the bulk of the plasma, and is therefore commonly neglected when studying runaway-electron kinetics. The test-particle term in equation (2) only ensures the conservation of particles, however, not momentum or energy. Under certain circumstances, it is necessary to use a fully conservative treatment also for the runaway problem, in particular when considering processes where the conductivity of the plasma is important. In the study of runaway dynamics during a tokamak disruption using a self-consistent treatment of the electrical field, accurate plasma-current evolution is essential, and the full linearized collision operator must be used. A non-linear collision operator valid for arbitrary particle (and bulk) energy has been formulated [22, 23]. The col lision operator originally implemented in code is the result of an asymptotic matching between the highly relativistic limit of the test-particle term of the linearized version of that operator, with the usual non-relativistic test-particle operator [24], and is given in equation (2). The relativistic field-particle term is significantly more complicated, however, and its use would be computationally more expensive. Here we instead 4 A. Stahl et al Nucl. Fusion 56 (2016) 112009 Figure 3. (a) Parallel momentum and (b) energy moments of the distribution function in code, using different collision operators. Initially, E = 50 V m−1 and Z eff = 1 were used, but for t > t0, the electric field was turned off and the ion charge set to Z eff = 0. Using two Legendre modes for the field-particle term was sufficient to achieve good conservation of energy and parallel momentum. implemented the non-relativistic field-particle term, as formulated in [25, 26]. As will be shown, this operator (together with the non-relativistic limit of equation (2)) accurately reproduces the Spitzer conductivity for sufficiently weak electric fields and temperatures where the bulk is non-relativistic. Using the normalization in section 2, the field-particle term is is modest (the operator ∇2v is proportional to l2, with l the Legendre mode index, and G and H therefore decay rapidly with increasing l). The conservation properties of the full non-relativistic col lision operator (equations (2) and (7)), as well as the relativistic test-particle operator in equation (2), are shown in figure 3. As an electric field is applied to supply some momentum and energy to the distribution, the parallel momentum (figure 3(a)) quickly reaches a steady-state value corresponding to the plasma conductivity, which differs by about a factor of two for the two operators (see below). The electric field is turned off at t = t0 = 100 collision times (and Z eff = 0 is imposed to isolate the behavior of the electron–electron col lision operator), at which point the parallel momentum for the operator in equation (2) (blue, dashed) is lost on a short time scale as the distribution relaxes back towards a Maxwellian. In contrast, the full linearized operator (black, solid) conserves parallel momentum in a pure electron plasma, as expected. The electric field continuously does work on the distribution, a large part of which heats the bulk electron population, but the linearization of the collision operator breaks down if the distribution deviates too far from the equilibrium solution. As long as a non-vanishing electric field is used together with an energy conserving collision operator, an adaptive sink term removing excess heat from the bulk of the distribution must be included in equation (1) to guarantee a stable solution. Physically this accounts for loss processes that are not included in the model, such as line radiation, bremsstrahlung and radial heat transport. The magnitude of the black line in figure 3(b) therefore reflects the energy content of the runaway population—not the total energy supplied by the electric field—since a constant bulk energy is enforced. The energy sink is not included for t > t0 (since E = 0), however, and the energy conservation observed is due to the properties of the collision operator itself. Again, the use of the collision operator in equation (2) is associated with a quick loss of kinetic energy as soon as the electric field is removed. The electrical conductivity of a fully ionized plasma subject to an electric field well below the Dreicer value—the Spitzer conductivity—can be expressed as ⎡ 2x 2 ∂ 2G ⎤ 2 c −2 2 fp − 2 H + 4πF ⎥ , Cˆ = 3C/ 2 e−v¯e x ⎢ 4 (7) ⎣ v¯e ∂x 2 ⎦ v¯e π where G and H are the Rosenbluth potentials, obtained from the distribution using v˜e 2∇2v H = −4πF , v˜e 2∇2v G = 2H . (8) The system of equations composed of equations (7) and (8), together with the non-relativistic limits of equations (1) and (2) (y → x and δ → 0), is discretized (see [5]) and solved using an efficient method described in [27]. The equations are combined into one linear system of the form ⎛ M11 M12 M13 ⎞⎛ F ⎞ ⎛ Si ⎞ ⎜M M 0 ⎟⎟⎜⎜ G ⎟⎟ = ⎜⎜ 0 ⎟⎟, (9) 22 ⎜ 21 ⎝ 0 M31 M33 ⎠⎝ H ⎠ ⎝ 0 ⎠ where the first row describes the kinetic equation (1) (with Si representing any sinks or sources), and the second and third rows correspond to equation (8). This approach makes it possible to consistently solve for both the Rosenbluth potentials and the distribution with a single matrix operation. Since there is no explicit need for the Rosenbluth potentials, however, G and H can be eliminated by solving the block system analytically: 1 −1 ((10) M11 − [M12 − M13 M− 33 M32] M 22 M21)F ≡ MF = Si. If only the test-particle operator (equation (2)) is used, M reduces to M11. Since the Rosenbluth potentials are defined through integrals of the distribution, the field-particle term introduces a full block for each Legendre mode into the normally sparse matrix describing the system. However, the integral dependence on F also implies that significantly fewer modes are required to accurately describe the potentials (compared to F ), and the additional computational cost 5 A. Stahl et al Nucl. Fusion 56 (2016) 112009 Figure 4. (a) Conductivity (normalized to the Spitzer value) and (b) normalized runaway density, as functions of time for different collision operators (non-relativistic full linearized: solid; relativistic test-particle: dashed) and E-field strengths (E /ED = 1%: yellow; E /ED = 5%: red; E /ED = 6%: blue), considering only Dreicer runaway generation. The parameters T = 1 keV, n = 5 ⋅ 1019 m−3 and Z eff = 1 were used. where n r is the number density of runaway electrons, n̄ is the density normalized to its reference value, and δ D is the Dirac δ-function. In the derivation, the momentum of the incoming particle is assumed to be very large (simplifying the scattering cross section) and its pitch-angle vanishing (ξ = 1). It is also assumed that the incoming particle is unaffected by the interaction. These conditions imply that the generated secondary particles are all created on the curve ξ = ξ2 = δy /(1 + 1 + δ 2y 2 ) ne 2 σS = L (Z eff ) , (11) Z eff mνee where L (Z eff ) is a transport coefficient which takes the value L 2 in a pure hydrogen plasma [10]. Figure 4 demonstrates that the conductivity calculated with code reproduces the Spitzer value for moderate electric-field strengths, if the conservative collision operator is used, and the initial Maxwellian adapts to the applied electric field on a time scale of roughly 10 collision times. For field strengths significantly larger than Ec, the conductivity starts to deviate from σS, as a runaway tail begins to form (figure 4(b)); in this regime, the calculation in [10] is no longer valid. Using the collision operator in equation (2) consistently leads to a conductivity which is lower by about a factor of 2, as expected (see for instance [28]). The runaway growth is also affected, with the conserving operator leading to a larger runaway growth rate. (which is a parabola in [ y∥, y⊥] -space), and that all runaways (from the point of view of the avalanche source) are assumed to have momentum p = γv /c = δy 1 (since SˆRP ∝ n r). They can therefore contribute equally strongly to the avalanche process. This has the peculiar and non-physical consequence that particles can be created with an energy higher than that of any of the existing runaways. The δ-function in ξ is numerically ill-behaved, as it produces significant oscillations (Gibbs phenomenon) when discretized using the Legendre-mode decomposition employed in code (see figure 5(a)). An operator that relaxes the assumption of very large runaway momentum has been presented by Chiu et al [11]. It has the form 4. Improved operator for knock-on collision The Fokker–Planck collision operators discussed in section 3 accurately describe grazing collisions—small-angle deflections which make up the absolute majority of particle interactions in the plasmas under consideration. Large-angle collisions are usually neglected as their cross section is significantly smaller, but in the presence of runaway electrons they can play an important role in the momentum space dynamics, as an existing runaway can transfer enough momentum to a thermal electron in one collision to render it a runaway, while still remaining in the runaway region itself. Such knock-on collisions can therefore lead to an exponential growth of the runaway density—an avalanche [13, 29]. In the absence of a complete solution to the Boltzmann equation, we model avalanche runaway generation using an additional source term in the kinetic equation (1), evaluated for y > yc. A commonly used operator was derived by Rosenbluth and Putvinski [13] and takes the form ⎡ ⎛ 3πδ 3 n 1 ∂ 1 SˆRP = r n¯ 2 ⎢ δ D(ξ − ξ2 ) 2 ⎜⎜ ˜ ⎢ n y y ∂ 16 ln Λ − + δ 2y 2 1 1 ⎝ ⎣ ˆCh( y, ξ ) = n¯ 2πe n˜δ x ( y )4F ( y ) Σ(γ , γ ), S(13) in in in m2c 3 ν˜ee y 2ξ 4 where Σ(γ , γin ) = − 3 ⎡ γin 2 ⎢(γin − 1)2 (γin 2 − 1)(γ − 1)2(γin − γ )2 ⎣ ⎤ (γ − 1)(γin − γ ) (2γin 2 + 2γin − 1 − (γ − 1)(γin − γ ))⎥ ⎦ γin 2 (14) is the Møller scattering cross section [12] and F is the pitch-angle-averaged distribution of incoming runaways with properties yin and γin. All incoming particles are thus still assumed to have zero pitch angle (ξ = 1), but their energy distribution is properly taken into account. In code, F is computed from the 0th Legendre mode of F; F = 2F0. From the conservation of 4-momentum in a collision, the momentum-space coordinates are related through ⎞⎤ ⎟⎥ , ⎟⎥ ⎠⎦ (12) 6 A. Stahl et al Nucl. Fusion 56 (2016) 112009 Figure 5. Contour plots of the magnitude of the source in (a) equation (12) and (b) equation (13) in (y∥,y⊥) momentum space, given the same electron distribution. The plotted quantity is log10 Ŝ and yc defines the lower bound of the runaway region. The angle-averaged source magnitudes are shown in (c). The parameters T = 1 keV, n = 5 ⋅ 1019 m−3, Z eff = 1 and E = 1 V m−1, with max( y ) = 70, were used to obtain the distribution, and the simulation was run for 300 collision times with primary generation only. (γ − 1)(γin + 1) ξ= , (15) (γ + 1)(γin − 1) (as expected). This reduces the sensitivity of the avalanche growth rate to the choice of momentum cut-off (as long as ycut − off ⩽ yc), and reaffirms our choice ycut − off = yc. Figure 5(c) shows the source terms integrated over pitchangle, and as expected, the source in equation (13) extends only up to y ymax / 2, whereas the source in equation (12) is non-vanishing also for larger momenta. The amount of secondary runaways generated by the two sources agrees well at low energies, but less so further away from the bulk. In this particular case, the total source magnitude ∫ Sˆ y 2 dy dξ agrees to within 25%, as most of the secondaries are created close to the boundary of the runaway region. which restricts the region where the source is non-vanishing (this relation is analogous to the parabola ξ2 in the case of the operator in equation (12)). Since the electrons participating in a collision are indistinguishable, it is sufficient to consider only the cases where the energy of the created secondary runaway is less than half of the primary energy, (γ − 1) ⩽ (γin − 1)/2, which with the above equation leads to the condition ξ ⩽ ξmax = γ /(γ + 1) . By the same argument, the maximum attainable runaway energy in the simulation (the maximum of the momentum grid) leads to the condition ξ ⩾ ξmin = (γ − 1)(γmax + 1)/(γ + 1)(γmax − 1) . The magnitudes of the two sources (12) and (13) are computed from a given typical runaway distribution function, and shown in figures 5(a) and (b). Curves corresponding to the parabola ξ2, as well as the limits ξmin and ξmax are also included. Note that the amount of numerical noise is significantly reduced for the source in equation (13). In order to avoid double-counting the small-angle collisions described by the Fokker–Planck–Landau collision operator C, the knockon source must be cut off at some value of momentum sufficiently far from the thermal bulk. As can be seen from the figure, however, the magnitude of both sources increases with decreasing momenta, and the avalanche growth rate is therefore sensitive to the specific choice of momentum cutoff. Since our particular interest is the generation of runaway electrons, we choose to place the cut-off at y = yc, so that the sources are non-vanishing only in the runaway region [5, 15]. Secondary particles deposited just below the threshold—although not technically runaways—could eventually diffuse into the runaway region, thereby potentially increasing the Dreicer growth rate. In [30], such effects were however shown to be negligible for the operator in equation (12), indicating that the vast majority of particles deposited at y < yc are slowed down rather than accelerated 4.1. Avalanche growth rates for the different operators In general, the avalanche growth rate produced by the two sources can differ substantially. We will illustrate this point by considering the Møller cross section in more detail. We choose to quantify the source magnitude for an arbitrary distribution by computing the cross section, integrated over the energy of the outgoing (secondary) particle and normalized to r 20, with r0 the classical electron radius. In other words, we look at the total normalized cross section for an incoming particle with γin to participate in a knock-on collision resulting in avalanche [31]: KCh(γin ) = ∫γ (γin − 1) /2 + 1 c Σ(γ , γin )dγ ⎡ γ 2 γin 2 = (γin 2 − 1)−1 ⎢ in + ⎣ γc − 1 γc − γin + ⎤ 2γin − 1 ⎛ γc − 1 ⎞ γin + 1 ln⎜ − γc⎥ , ⎟+ ⎥⎦ γin − 1 ⎝ γin − γc ⎠ 2 (16) where γc = (E /Ec )/(E /Ec − 1) corresponds to the critical momentum for runaway generation and the upper integration 7 A. Stahl et al Nucl. Fusion 56 (2016) 112009 Figure 6. (a) Contours (black, white) of the ratio of total cross-sections (KCh /KRP) for an electron with pin to contribute to the avalanche process, as a function of pin = γinvin /c = γin 2 − 1 and E /Ec. (b) Ratio of avalanche growth rates ( [ΓCh − ΓD]/[ΓRP − ΓD] ) in code simulations. The parameters T ∈ [0.1 eV, 5 keV], E /Ec ∈ [1.1, 1000] , n = 5 ⋅ 1019 m−3 and Z eff = 1 were used. was set to either tmax, the first time step for which n r > 5%, or the first time step in which the growth rate started to become affected by the proximity of the runaway tail to the end of the simulation grid, whichever occurred first. The parameters of the scan were chosen to focus on the most interesting region of figure 6(a)—by performing longer simulations on larger momentum grids, the upper part of the figure could also be studied. Exact agreement between figures 6(a) and (b) can not be expected, since the source, in addition to the cross section, depends on the details of the runaway distribution. Figure 6(a) should thus be viewed as a simplified analytical estimate for figure 6(b). The different regions identified in figure 6(a) are still apparent in figure 6(b), however they are somewhat shifted in parameter space. In particular, the region where the Rosenbluth–Putvinski operator produces a higher growth rate is larger, whereas the opposite region—where the operator in equation (13) dominates—is smaller, or at least shifted to higher values of E /Ec. Figure 7 shows all the data points in figure 6(b), as a function of temperature. The figure confirms that the region where the more accurate operator produces a significantly higher growth rate is only accessible at temperatures T < 100 eV (in the domain of validity of a linearized treatment). As is evident in the figure, however, regions where the Rosenbluth– Putvinski operator significantly overestimates the avalanche growth rate (points below 1 on the vertical axis) are present at all temperatures. The operator in equation (13) is thus of general interest. Since the electric field spike responsible for the acceleration of runaways during a tokamak disruption is induced by the temperature drop, and therefore occurs slightly later than the drop itself, the temperature is low during the majority of the acceleration process. For significant runaway acceleration, E /Ec 1 is therefore required, and during the initial part of the acceleration process, parameters are likely those corresponding to the blue region of figure 6(b), where the improved avalanche source produces a significantly higher growth rate than the Rosenbluth–Putvinski operator. Postthermal-quench temperatures in ITER are expected be as low as 10 eV and peak electric fields in disruptions can reach 80 V m−1 or more [32]. Towards the end of the thermal quench, the boundary stems from the condition leading to ξmax. This expression is relevant to the source in equation (13), which uses the complete cross section (14), whereas for the more simple source in equation (12), only the leading-order term in γin in the scattering cross section is taken into account. This corresponds to taking the high-energy limit of the above equation, so that 1 KRP = (17) γc − 1 becomes a simple constant. To systematically explore the relative magnitude of the two sources, the ratio KCh /KRP is plotted in figure 6(a). As expected, the two expressions agree very well at high primary momenta. At somewhat lower momenta, of the order γ ≈ p 5, two distinct regions are discernible. For E /Ec 10 (the orange region), the simplified cross section is larger than the full expression, and the Rosenbluth–Putvinski operator (12) is likely to overestimate the avalanche generation. For E /Ec 10, the opposite is true, and the operator in equation (13) has a significantly larger cross section for E /Ec 30 (the blue region). The more accurate operator (13) should thus be expected to produce more runaways when the runaway population is at predominantly low energies, and E /Ec is large. For both of these conditions to be fulfilled simultaneously (and at the same time avoid a slide-away scenario), the temperature must be low so that E /ED 1 even for large E /Ec. The effect is also likely to be most apparent at relatively early times, before the runaway tail has extended to multi-MeV energies. code simulations support the above conclusions and show excellent qualitative agreement, as shown in figure 6(b). The figure shows the ratio of final avalanche growth rates (ΓCh − ΓD)/(ΓRP − ΓD), with Γi = n−r 1(dn r /dtˆ ) the growth rate obtained in a code run using source i (here the subscript D denotes pure Dreicer generation). Each marker in the figure is thus computed from three separate code runs. As a proxy for pin, the average runaway momentum pr,av in the final time step tf of the simulation without a source was used, and for a given E /Ec, different pr,av were obtained from simulations with varying values of T (and corresponding values of E /ED). The simulations were run for tmax = 5000 collision times, and tf 8 A. Stahl et al Nucl. Fusion 56 (2016) 112009 implementation described. The operator was found to reproduce the expected Spitzer conductivity in the relevant parameter regime and showed excellent conservation properties. The use of such an operator is essential for the correct current evolution in self-consistent modelling, and in particular when studying the interplay between current and electric-field evolution and runaway-electron generation during a disruption. The process of avalanche multiplication of the runaway population via close Coulomb collisions was also considered, and an improved operator, relaxing some of the approximations of the commonly used Rosenbluth–Putvinski operator, was discussed. It was found that the avalanche growth rate can be significantly affected—increased for low temperatures and high E /Ec and decreased for low E /Ec—by the use of the new operator. The change to the growth rate can be especially large during the early stages of the runaway acceleration process, thus potentially affecting the likelihood of a given runaway seed transforming into a serious runaway beam, and use of the improved operator is of particular relevance in disruption scenarios. The work presented in this paper paves the way for a better understanding of runaway-electron dynamics in rapidly changing scenarios, for instance during tokamak disruptions. It enables more accurate assessment of the risks posed by runaway electrons in situations of experimental interest, par ticularly in view of future tokamaks such as ITER. Figure 7. Ratio of avalanche growth rates ( [ΓCh − ΓD] / [ΓRP − ΓD] ) in code simulations, as a function of temperature. The same parameters as in figure 6 were used. normalized electric field is then E /Ec ≈ 1300 (with E = 80 V m−1, T = 50 eV and n = 1 ⋅ 10 20 m−3). A typical ITER disruption would thus (at least initially) be firmly in the blue region of figure 6(b), and the avalanche growth should be significantly higher than what the Rosenbluth–Putvinski source predicts. As the temperature is low, the runaways will also spend a comparatively long time at low momenta ( p 1), where the disagreement between the operators is most pronounced. Note that, according to the figure, an average runaway energy of several MeV (p > 5–10) is needed for the difference between the growth rates to become small for all E /Ec, at which point the most energetic electrons will have reached energies of several tens of MeV or more. However, since the electric field changes rapidly, the runaways may experience parameters corresponding to both the orange and blue regions in figure 6(b) before reaching such energies. Further work is therefore needed to assess the overall impact on the avalanche growth of using the improved operator (13), although it is clear that its use is essential for accurate analysis. Acknowledgments The authors are grateful to I. Pusztai and E. Hirvijoki for fruitful discussions. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. The authors also acknowledge support from Vetenskapsrådet, the Knut and Alice Wallenberg Foundation and the European Research Council (ERC2014-CoG grant 647121). 5. Summary Runaway electrons are intimately linked to dynamic scenarios, as they predominantly occur during disruptions and sawtooth events in tokamaks. An accurate description of their dynamics in such scenarios requires kinetic modelling of rapidly changing plasma conditions, and mechanisms such as hot-tail runaway generation add to the already interesting set of phenomena of importance to the evolution of the runaway population. In this paper we have described the modelling of several such processes, using the numerical tool code to calculate the momentum-space distribution of runaway electrons. In particular, we have investigated rapid-cooling scenarios where hot-tail runaway-electron generation is dominant. 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Fülöp, Effective critical electric field for runaway electron generation, Physical Review Letters 114, 115002 (2015). http://dx.doi.org/10.1103/PhysRevLett.114.115002 http://arxiv.org/abs/1412.4608 week ending 20 MARCH 2015 PHYSICAL REVIEW LETTERS PRL 114, 115002 (2015) Effective Critical Electric Field for Runaway-Electron Generation 1 A. Stahl,1,* E. Hirvijoki,1 J. Decker,2,1 O. Embréus,1 and T. Fülöp1 Department of Applied Physics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden 2 École Polytechnique Fédérale de Lausanne (EPFL), Centre de Recherches en Physique des Plasmas (CRPP), CH-1015 Lausanne, Switzerland (Received 26 August 2014; published 17 March 2015) In this Letter we investigate factors that influence the effective critical electric field for runaway-electron generation in plasmas. We present numerical solutions of the kinetic equation and discuss the implications for the threshold electric field. We show that the effective electric field necessary for significant runawayelectron formation often is higher than previously calculated due to both (1) extremely strong dependence of primary generation on temperature and (2) synchrotron radiation losses. We also address the effective critical field in the context of a transition from runaway growth to decay. We find agreement with recent experiments, but show that the observation of an elevated effective critical field can mainly be attributed to changes in the momentum-space distribution of runaways, and only to a lesser extent to a de facto change in the critical field. DOI: 10.1103/PhysRevLett.114.115002 PACS numbers: 52.25.Xz, 52.55.Fa, 52.55.Pi, 52.65.Ff Introduction.—In a plasma, an electron beam accelerated by an electric field is damped by collisional friction against the bulk plasma and by emission of electromagnetic radiation. Since the collisional friction decreases with increasing velocity of the electrons, a large enough electric field may overcome the collisional damping and accelerate electrons to relativistic speeds, leading to the formation of a runaway-electron (RE) beam. In laboratory plasmas, much attention has been given to the potentially dangerous, highly relativistic RE beams that can be generated in tokamak disruptions [1]. Runaway acceleration can also occur in nondisruptive plasmas due to the Ohmic electric field, if the plasma density is low. In addition, runaway electrons are ubiquitous in atmospheric and space plasmas, e.g., as a source of red sprites in the mesosphere [2] and in lightning discharges in thunderstorms [3], and their occurrence in solar flares has been suggested [4]. The critical (threshold) electric field for runaway generation, Ec ¼ ne e3 ln Λ=4πϵ20 me c2 , is a classic result in plasma physics [5]. Because of relativistic effects, it is the weakest field at which electron runaway is possible. Here, ne and me are the number density and rest mass of the electrons, respectively, ln Λ is the Coulomb logarithm, c is the speed of light, and e is the magnitude of the elementary charge. Recent experimental evidence from several tokamaks indicates that the electric field strength necessary for RE generation could in fact be several times larger than the critical electric field [6–8]. As the classic expression for the critical field only considers the balance between electric field and Coulomb collisions, many potential mechanisms affecting the RE generation are left out. In this Letter we investigate the role of the background plasma temperature and synchrotron radiation reaction as possible explanations for these observations. 0031-9007=15=114(11)=115002(5) An appreciation for the importance of temperature and synchrotron effects can be gained by considering the energy balance for an electron experiencing electric field acceleration, collisional damping, and the AbrahamLorentz radiation reaction force [9]: 3γ 2 γ2 3γ 2 Frad ¼ k v̈ þ 2 ðv · v_ Þ_v þ 2 v · v̈ þ 2 ðv · v_ Þ2 v ; c c c ð1Þ where k ¼ e2 γ 2 =ð6πε0 c3 Þ, v is the electron velocity, and γ ¼ ½1 − ðv=cÞ2 −1=2 is the relativistic mass factor. At the critical electric field, acceleration due to the electric field balances the friction due to collisions and radiation, so the particle energy is constant. This means that γ_ ¼ 0 and v· v_ ¼0, implying that qE·vþFc ·v−ðe2 γ 4 =6πϵ0 c2 Þ_v · v_ ¼0. At constant energy, v_ ¼ ðe=γme Þv × B, so that v_ · v_ ¼ ð1 − ξ2 Þv2 ω2c . Here B is the magnetic field, ξ ¼ p∥ =p is the cosine of the particle pitch angle, p ¼ jpj ¼ γv=c is the magnitude of the normalized momentum, and ωc ¼ eB=γme is the Larmor frequency. This means that for the electric field to accelerate the electron, it has to be larger than E 1 c2 2ε0 B2 1 − ξ2 v 2 γ ≡ hðv; ξÞ: > 2þ Ec ξ v 3ne me ln Λ ξ c ð2Þ Note that hðv; ξÞ > 1, always. In Fig. 1, h is illustrated for typical ITER [10] plasma parameters as a function of ξ and the electron kinetic energy Ekin. From this simple estimate of the single electron energy balance, we see that the value of E=Ec needed to accelerate electrons can be significantly larger than unity and increases with Ekin in the MeV range. This warrants a more thorough investigation of the RE dynamics close to the critical electric field. The first term 115002-1 © 2015 American Physical Society 102 function describing the particle speeds is determined by the temperature. This introduces a temperature dependence to the effective critical field, since the number of particles with speed above any threshold speed is temperature dependent. Mathematically, this can be understood from the primary runaway growth rate [5]: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnr ∼ ne νee E −3ð1þZeff Þ=16 exp ½−1=ð4EÞ − ð1 þ Zeff Þ=E dt 101 0.6 E/Ec ξ=v /v 0.8 0.4 100 0.2 −1 0 10 1 2 10 2 10 Ekin / (mec ) FIG. 1 (color online). Magnitude of the normalized electric field (E=Ec ) necessary to compensate for collisional friction and synchrotron damping for a single electron in an ITER-like plasma with ne ¼ 1020 m−3 , T e ¼ 10 keV, and B ¼ 5 T. 10000 0 3000 E/E D =1 −4 1000 −8 300 0.1 100 0.0 2 0.0 1 30 10 1.2 2 3 − 12 − 16 −1 log10 (n−1 e dnr /dt) [s ] in Eq. (2) is related to the collisional damping, which is temperature dependent, while the second term—which vanishes for purely parallel motion (ξ ¼ 1)—is due to the radiation reaction force. We will consider both these effects in more detail. The single electron estimate depends strongly on ξ, v, and various plasma parameters, but neglects the collisional Coulomb diffusion that spreads the electrons in velocity space. An accurate estimate for the threshold electric field can thus only be obtained using kinetic calculations that take into account the details of the electron distribution function. Here we make use of COllisional Distribution of Electrons (CODE) [11], an efficient finite-difference–spectral-method tool that solves the two-dimensional momentum space kinetic equation in a homogeneous plasma. The Coulomb collision operator in CODE is valid for arbitrary electron energies [12]. Often secondary (or avalanche) generation of REs, resulting from knockon collisions between REs and thermal electrons, is the dominant RE generation mechanism, and CODE has also been equipped with an operator describing this process [13]. For the present work, an operator for synchrotron emission backreaction based on the analysis in Refs. [14,15] was implemented. With the numerical electron distribution function from CODE, we may investigate the runaway generation dynamics for a wide range of plasma parameters, to calculate, e.g., the synchrotron radiation spectra of the REs [16], or to study wave-particle interactions [17,18]. The parameters used in this Letter reflect those common to magnetic fusion experiments, but the arguments are generally applicable. In particular, no effects specific to fusion plasmas (such as a toroidal field configuration) have been assumed. Temperature dependence of the critical electric field.— At E ≳ Ec , only electrons already moving with approximately the speed of light may run away. Since the number of plasma particles is finite (especially in a laboratory context), the actual highest speed achieved by the background electrons may be significantly less than c. Thus, if the critical speed for RE generation at a given E field is larger than this maximum speed, no electrons will be able to run away. The width (in velocity space) of the distribution (where νee ¼ ne e4 ln Λ=4πε20 m2e v3th is the collision frepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi quency, vth ¼ 2T e =me is the electron thermal velocity, and Zeff is the effective charge number of the plasma), which is exponentially small in E ¼ E=ED ¼ ðT e =me c2 ÞðE=Ec Þ, where ED is the Dreicer field [19]. There is thus an inherent temperature dependence in the primary runaway growth rate at a given value of E=Ec , and for significant RE production on a short time scale it is not enough to only require E > Ec [1,20]. We define a runaway electron as any electron with p > pc ¼ ðE=Ec − 1Þ−1=2 , where pc is the critical momentum for electron runaway. In the absence of avalanche generation, a quasisteady state for the RE distribution can be calculated using CODE. In Fig. 2, the RE growth rate for this primary distribution is displayed as a function of the electron temperature and E=Ec . The figure indicates that, for all temperatures T e ≲ 5 keV, the fraction of the electron population that runs away in 1 s is less than 10−20 for all E=Ec < 1.5. In a plasma with ne ≲ 1020 m−3 and a volume of a few tens of m3 (typical of fusion experiments), essentially no runaway production (let alone detection) is thus to be expected. It is also clear from the figure that, for lower temperatures, a stronger normalized electric field would be required for significant RE production (note that Fig. 2 essentially covers the whole temperature range of magnetic fusion plasma operation). We note that this temperature dependence may increase the sensitivity of future hotter tokamaks (like ITER) to the problem of deleterious RE formation. The white and black contours in Fig. 2 show the corresponding values of E=ED , and we may conclude that E=ED must be at least larger than 1%–2% for significant runaway formation to occur, in Te [eV] 1 10 week ending 20 MARCH 2015 PHYSICAL REVIEW LETTERS PRL 114, 115002 (2015) − 20 5 10 30 E/Ec 100 300 FIG. 2 (color online). Primary runaway growth rate (particle fraction per second) as a function of temperature and electric field, in the absence of synchrotron effects. White and black contours refer to E=ED . The plasma parameters ne ¼ 5 × 1019 m−3 and Zeff ¼ 1.5 were used. 115002-2 agreement with previous analytical findings [1,20]. In practice, there are thus two conditions that must be fulfilled: E=Ec > 1 and E=ED > k, for some k ∼ 1%–2%. The second criterion is more restrictive for temperatures below 5.1 and 10.2 keV (for k ¼ 1% and 2%), respectively. Momentum loss due to synchrotron emission.—The importance of synchrotron backreaction as a limiting factor for the maximum energy achieved by REs has been discussed before [21], and its importance in RE dynamics has been investigated [14], also in the context of the critical field for RE generation [8]. An accurate description of the RE dynamics close to the critical field based on first principles does, however, require kinetic modeling, and we will investigate the effect of the synchrotron emission on the effective critical field using CODE. In a homogeneous plasma, the distribution function f for electrons experiencing an electric field, Coulomb collisions, and synchrotron radiation backreaction is determined by the gyro-averaged Fokker-Planck equation, ∂f eE∥ ∂f 1 − ξ2 ∂f ∂ þ þ · ðFrad fÞ ¼ Cffg þ Sava ; ξ þ ∂t me c ∂p ∂p p ∂ξ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} radiation where E∥ is the parallel (to −B) electric field, Cf·g is the collision operator, Sava is the source of secondary (avalanche) runaways, and Frad is given by Eq. (1). Note that the operator ð∂=∂pÞ · ðFrad fÞ conserves the number of particles, unlike the corresponding operator used in Ref. [14]. (The simplified operator used in Ref. [14] was justified due to the focus on the high-energy tail of the distribution function.) The magnetic force jFm j ≃ ωc me v, characterized by the Larmor frequency ωc ¼ eB=γme, typically dominates both the electric force and the radiation reaction force. This implies that v · v_ ≃ 0, and that in the coordinates p and ξ, the term accounting for the effects of synchrotron radiation backreaction can be written as ∂ 1 ∂ γp3 ð1 − ξ2 Þ · ðFrad fÞ ¼ − 2 f ∂p τr p ∂p 2 ∂ ξð1 − ξ Þ f ; þ ∂ξ γτr 0.7 1000 0.5 300 0.3 100 0.1 30 2 3 5 10 30 0.9 m−3] (a) 3000 19 10000 where τr ¼ 6πε0 ðme cÞ3 =ðe4 B2 Þ [14] is the radiation damping time scale. What ultimately determines the relative importance of the synchrotron effects is the ratio of the collision time 1=νee to the radiation time scale τr . For a given magnetic field, we therefore expect the largest effect on the distribution for high temperatures and low densities, 2 since 1=ðνee τr Þ ∼ T 3=2 e B =ne . The radiation reaction force acts as an additional drag, which increases with particle momentum. Therefore, it ultimately prevents the REs from reaching arbitrary energies [21], and given enough time, the system will reach a steady state where the RE growth rate vanishes. This occurs only once the REs have reached very high energies; in the initial phase (which is our interest in this section), the RE growth rate is well defined. The rate is calculated as the flux through a sphere of constant p, located well inside the RE region. The change in this RE growth rate in CODE as a result of the synchrotron radiation reaction is presented in Fig. 3. From the figure, we conclude that the synchrotron losses can reduce the RE rate substantially for weak E fields—by several orders of magnitude at high temperatures and low densities—and it is therefore essential to include these effects when considering near-critical RE dynamics. The sharp cutoff for weak fields is in line with the change in effective critical field associated with the inclusion of the synchrotron drag [see Eq. (2)]. The full kinetic simulation thus agrees qualitatively with the singleparticle estimate in the Introduction. For stronger electric fields, the effects are less pronounced. We note that in the post-thermal-quench conditions associated with disruptions in tokamaks (low T e, high ne ), the effects of synchrotron radiation reaction on the RE growth rate are likely to be negligible, whereas in the case of RE generation during the plasma current ramp-up phase (high T e , low ne ), they can be substantial. There is thus a qualitative difference in the momentum space dynamics in these two cases—at least for near-critical electric fields— and conclusions from ramp-up (or flattop) scenarios do not necessarily apply under postdisruption conditions. Critical field under experimental conditions.—Until now we have only considered primary (Dreicer [19]) generation, i.e., when the electrons gradually diffuse through velocity space due to small-angle collisions and run away as they ne [10 electric field Te [eV] week ending 20 MARCH 2015 PHYSICAL REVIEW LETTERS PRL 114, 115002 (2015) 0.7 3 0.5 1 0.3 0.3 0.1 30 0.9 (b) 10 0.1 2 3 4 5 6 7 8 9 10 E/Ec E/Ec FIG. 3 (color online). Contour plots of the (a) temperature and (b) density dependence of the ratio between the primary RE growth rate in CODE with and without synchrotron effects included. The parameters B ¼ 4 T, Zeff ¼ 1.5, and (a) ne ¼ 1 × 1019 m−3 , (b) T e ¼ 2 keV were used. To ensure reliable results in (a), the parameter region has been restricted as the growth rates are negligible for low T e and E fields, and E=ED approaches unity for high T e and E fields (cf. Fig. 2). 115002-3 PRL 114, 115002 (2015) week ending 20 MARCH 2015 PHYSICAL REVIEW LETTERS reach the critical velocity. An electron can enter the runaway region also through a sudden collision at close range, which throws it above the critical speed in a single event. This leads to avalanche multiplication of the REs, which is the dominant mechanism in many cases. The avalanche growth rate is γ ava ∝ nRE ðE=Ec − 1Þ [13] (where nRE is the runaway density), and the total runaway growth rate is then the sum of primary and avalanche processes. Above the critical field, the RE population will be growing and below it will be decaying; according to the estimate in Ref. [13] (taking only losses due to Coulomb collisions into account), the transition should occur at E=Ec ¼ 1. However, in an experiment designed to test this at the DIII-D tokamak [7], the measured transition from RE growth to decay occurred at E=Ec ¼ 3–5. In the experiments, after first generating a substantial RE population, the plasma density was rapidly increased so as to raise Ec . The value of E=Ec for the transition was determined from the change from growth to decay in the hard-x-ray (HXR) signal, as well as from the synchrotron emission (in the visual range). These signals were thus treated as a straightforward representation of the number of REs. This assumption is not necessarily valid in the present context, as it neglects the influence of the distribution of RE energies on the emitted radiation. We illustrate this by calculating the synchrotron emission from RE distributions obtained with CODE. The synchrotron radiation emitted by an electron is highly dependent on its energy and the curvature of its trajectory, with highly energetic particles with large pitch angles emitting most strongly and at the shortest wavelengths. The total emission is therefore very sensitive to the shape of the electron distribution function [16]. A change in the distribution shape may in fact lead to a reduction in the emitted synchrotron power, even though the size of the RE population is constant, or even increasing. Here we use the code SYRUP [16] to calculate the synchrotron spectrum emitted by CODE distributions, in order to investigate the role of this effect in the experiment in Ref. [7]. Accessing the physics underlying the evolution of the RE distribution in the experiment may by done in CODE by ramping down the electric field strength in the presence of a significant RE tail, dominated by the avalanche mechanism. The purpose here is to give a qualitative explanation for the observed discrepancy. The CODE calculation was started with a constant E=Ec ¼ 12, to produce a significant RE tail, and after 1.2 s the electric field was gradually ramped down during 1 s—a time scale consistent with the experiments described in Ref. [7]. The synchrotron spectrum at each time step was calculated using SYRUP. In the calculations, the maximum particle energy was restricted to 22 MeV, in agreement with experimental observations [22]. (This maximum energy limit cannot be attributed to the effects of the synchrotron losses; other mechanisms, such as radial transport, must (a) (b) FIG. 4 (color online). (a) Emitted synchrotron power in the visual range and (b) kinetic energy contained in the RE population during electric field ramp-down. In (a), each curve is normalized to its peak value. The parameters T e ¼ 1.1 keV, ne ¼ 2.5 × 1019 m−3 , and Zeff ¼ 1.2 were used. The black dotted lines denote the beginning of the E-field ramp-down phase. be invoked to explain it, but this is outside the scope of this Letter.) Figure 4(a) shows the total emitted power in the visual spectral range (400–700 nm) during the simulation, for various B-field strengths. The figure shows that as E=Ec decreases, the emitted power transitions from growth to decay in all cases, even though E=Ec is still well above unity. There is a clear dependence on the magnetic field strength, as the transition occurs at E=Ec ≃ 7.4 and 10.6 for B ¼ 2.5 and 3.5 T, respectively. This suggests that the origin of the effect is indeed the influence of the synchrotron reaction force on the distribution. In the experiments in Ref. [7], the field was 1.5 T; in this case, we observe an apparent effective critical field E=Ec ≃ 4.2, in agreement with the experimental value of 3–5. Although the RE growth rate decreases with E=Ec , in the calculations it remains positive for all E=Ec > 1.1, 1.3, and 1.4 for B ¼ 1.5, 2.5, and 3.5 T, respectively. The effective critical field is thus close to unity in all cases considered, and is B dependent, as expected from the previous section. The kinetic energy content of the RE population also continues to increase well after the emitted synchrotron power has started to decrease, as shown in Fig. 4(b). We therefore conclude that what is observed in Fig. 4(a) is not a fundamental change in the critical electric field but a reduction in synchrotron emission. The changing electric field leads to a reduced accelerating force, which modifies the force balance, causing a redistribution of electrons in velocity space towards lower energies. The density of highly energetic particles with large pitch angles thereby decreases, leading to a substantially reduced synchrotron emission in the visual range. Trends similar to those shown in Fig. 4(a) are seen also in the infrared spectral range. Since several tokamaks are equipped with fast visual or IR cameras dedicated to observing the synchrotron emission from REs, 115002-4 PRL 114, 115002 (2015) PHYSICAL REVIEW LETTERS experimental study of the effect we describe should be within reach. In particular, confirming the B dependence of the apparent elevated critical field would be of interest. In the experiments, E=Ec was decreased via a density ramp-up. Increasing the density also indirectly modifies T e , Zeff , and the loop voltage [7] and reduces the magnitude of the synchrotron effects. In the calculation in Fig. 4 these changes were not taken into account; however, the plasma parameters where chosen to reflect those observed at the time of the transition from growth to decay in DIII-D shot 153545, considered in Ref. [7]. Although the trend shown in Fig. 4 appears consistently and the results are largely independent of the details of the ramp in electric field strength and the energy cutoff, the specific value of E=Ec for which the transition from growth to decay occurs is sensitive to the plasma parameters. Similarly, the details of the avalanche source used [13] are expected to affect only the specific value of the transition, not the qualitative behavior in Fig. 4 (indeed, the same trend is seen even when the avalanche process is not included at all). It was suggested in Ref. [7] that a significant part of the detected HXR signal was due to RE bremsstrahlung emission. Like the synchrotron emission, bremsstrahlung is sensitive to the RE distribution function—the general argument made above may be invoked to explain the elevated growth-to-decay transition also in the case of the HXR signal. The effects considered in this section thus offer a plausible explanation for the mechanism behind the experimentally detected elevated electric field for transition from RE growth to decay. Our simulations show that the observed increase is mainly an artifact of the methods used to determine it, and only to a lesser extent the result of a fundamental change to the critical field itself. The impact on runaway mitigation schemes in future tokamaks is likely to be negligible, especially considering that in a postdisruption scenario the impact on the distribution function from synchrotron backreaction is small (due to the low T e and high ne ). Conclusions.—We have shown that several factors can influence the effective critical electric field for both generation and decay of runaway electrons. The temperature dependence of the RE growth rate means that in practice E=ED > 1%–2% is required for substantial RE generation. In addition, the drag due to synchrotron emission backreaction increases the critical field; for weak E fields, the runaway growth rate can be reduced by orders of magnitude. The synchrotron effects on the distribution are most prominent at high temperature and low density, however, and their practical impact is likely negligible in postdisruption tokamak plasmas. By the same token, the effects can be substantial during ramp-up and flattop. Deducing changes to the size of the runaway population using radiation can be misleading, as the emission is very sensitive to the momentum-space distribution of the week ending 20 MARCH 2015 runaways. This can lead to a perceived elevated critical field in electric field ramp-down scenarios, despite a continuing increase in the energy carried by the runaways and their density. Our results are consistent with recent experimental observations, giving a possible explanation for the observed elevated critical field. The authors are grateful to M. Landreman, G. Papp, G. Pokol, and I. Pusztai for fruitful discussions. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training program 2014-2018 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. * [email protected] [1] P. Helander, L.-G. Eriksson, and F. Andersson, Plasma Phys. Controlled Fusion 44, B247 (2002). [2] T. F. Bell, V. P. Pasko, and U. S. Inan, Geophys. Res. Lett. 22, 2127 (1995). [3] A. V. Gurevich, G. M. Milikh, and R. A. Roussel-Dupre, Phys. Lett. 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Fülöp, NORSE: A solver for the relativistic non-linear Fokker-Planck equation for electrons in a homogeneous plasma, Computer Physics Communications 212, 269-279 (2017). http://dx.doi.org/10.1016/j.cpc.2016.10.024 http://arxiv.org/abs/1608.02742 Computer Physics Communications 212 (2017) 269–279 Contents lists available at ScienceDirect Computer Physics Communications journal homepage: www.elsevier.com/locate/cpc NORSE: A solver for the relativistic non-linear Fokker–Planck equation for electrons in a homogeneous plasma✩ A. Stahl a, *, M. Landreman b , O. Embréus a , T. Fülöp a a b Chalmers University of Technology, Göteborg, Sweden University of Maryland, College Park, MD, USA article info Article history: Received 9 August 2016 Received in revised form 20 October 2016 Accepted 31 October 2016 Available online 18 November 2016 Keywords: Non-linear relativistic Fokker–Planck equation Kinetic plasma theory Energetic electrons Runaway electrons a b s t r a c t Energetic electrons are of interest in many types of plasmas, however previous modeling of their properties has been restricted to the use of linear Fokker–Planck collision operators or non-relativistic formulations. Here, we describe a fully non-linear kinetic-equation solver, capable of handling large electricfield strengths (compared to the Dreicer field) and relativistic temperatures. This tool allows modeling of the momentum-space dynamics of the electrons in cases where strong departures from Maxwellian distributions may arise. As an example, we consider electron runaway in magnetic-confinement fusion plasmas and describe a transition to electron slide-away at field strengths significantly lower than previously predicted. Program summary Program title: NORSE Program Files doi: http://dx.doi.org/10.17632/86wmgj758w.1 Licensing provisions: GPLv3 Programming language: Matlab Nature of problem: Solves the Fokker–Planck equation for electrons in 2D momentum space in a homogeneous plasma (allowing for magnetization), using a relativistic non-linear electron–electron collision operator. Electric-field acceleration, synchrotron-radiation-reaction losses, as well as heat and particle sources are included. Scenarios with time-dependent plasma parameters can be studied. Solution method: The kinetic equation is represented on a non-uniform 2D finite-difference grid and is evolved using a linearly implicit time-advancement scheme. A mixed finite-difference-Legendre-mode representation is used to obtain the relativistic potentials (analogous to the non-relativistic Rosenbluth potentials) from the distribution. © 2016 Elsevier B.V. All rights reserved. 1. Introduction Energetic electrons, having speeds significantly larger than the average speed of the thermal population, are ubiquitous in plasmas. Examples are found for instance in the solar corona [1] and wind [2], and in solar flares [3,4]; in the ionosphere of the Earth [5] and lightning discharges [6]; as well as in laboratory laser-plasma accelerators [7] and inertial [8] and magnetic-confinement [9] fusion plasmas. In the latter case, understanding the dynamics of the energetic electrons is of particular concern, as so-called runaway electrons [10,11] generated during disruptions – events where the plasma rapidly cools and strong electric fields are induced – have ✩ This paper and its associated computer program are available via the Computer Physics Communication homepage on ScienceDirect (http://www.sciencedirect. com/science/journal/00104655). Corresponding author. E-mail address: [email protected] (A. Stahl). * http://dx.doi.org/10.1016/j.cpc.2016.10.024 0010-4655/© 2016 Elsevier B.V. All rights reserved. the potential to cause severe damage to a tokamak fusion reactor. This problem is only expected to become more severe in future devices since the runaway generation is exponentially sensitive to the available plasma current [12,13]. In a spatially homogeneous plasma, the main processes influencing energetic-electron dynamics are: the presence of an accelerating electric field; magnetization (causing directed motion); Coulomb collisions; dynamic changes in plasma parameters such as the temperature; radiative losses (associated with synchrotron and fast-electron bremsstrahlung emission); and waveparticle interaction. The combined influence of these processes has been shown to lead to phenomena such as bump-on-tail formation [14,15] and local isotropization [16,17] in the high-energy tail of strongly anisotropic electron populations. Since analytic treatment is possible only in special cases, the evolution of the electron distribution function f must in general be studied using kinetic simulations. 270 A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 Many numerical tools solve the kinetic equation for f , taking some subset of the processes mentioned above into account. In collisional plasmas, the Fokker–Planck operator describing the Coulomb collisions is the main source of complexity in the problem, which in general is described by a stiff integro-differential diffusion equation, and numerical treatments can be broadly categorized based on the level of sophistication of the collision operator employed. A number of continuum tools have been developed that use linearized (around a Maxwellian) collision operators, especially in the case of fully non-relativistic problems, but also in scenarios where the electrons are allowed to reach relativistic energies [18–20]. In addition, several fully non-linear tools are available for non-relativistic scenarios [21–28]; however, to our knowledge, no tool treats the relativistic non-linear collision operator in its entirety. Both the integrated-tokamak-modeling tool TASK and CQL3D (which is focused on heating and current drive in tokamaks) successfully implement the first few Legendre modes of the relativistic non-linear collision operator [29,30]. While this approach guarantees the conservation of density, momentum and energy, it cannot resolve fine structures in the momentum-space distribution, making it unsuitable for accurate study of the fastelectron dynamics. In CQL3D, the implementation is general and therefore in principle supports the use of any number of modes, however in practice, the maximum number of modes cannot exceed 3 to 5 because of numerical problems [30]. In the magnetic-fusion community in particular, there is a pressing need for a tool with the ability to handle situations where relativistic particles comprise a significant part of the overall electron distribution, as these are the situations of greatest danger to the integrity of the fusion device [13]. Such scenarios arise primarily when the electric field magnitude is (at least) a significant fraction of the so-called Dreicer field [10], ED = ne3 ln Λ/4πϵ02 T ; where n, T and −e are the electron number density, temperature and charge, ln Λ is the Coulomb logarithm, and ϵ0 is the vacuum permittivity. For such field strengths, the electric field overcomes the maximum collisional friction force affecting the electrons. However, also for E < ED , the distortion of the distribution can become substantial, leading to the break-down of linearized codes. In contrast, the so-called critical field Ec = Θ ED , with Θ = T /me c 2 the bulk temperature normalized to the electron rest mass, is equivalent to the minimum collisional friction force experienced by highly relativistic electrons. It therefore describes the weakest field at which runaway-electron generation can occur [31], since the accelerating force must overcome the friction (and the latter decreases with increasing particle energy). For non-relativistic bulk-electron temperatures, Ec is much less than ED and runawayelectron generation can in general be studied using a linearized treatment since the electric field can fulfill both E > Ec and E ≪ ED simultaneously. A fully non-linear relativistic tool is however needed in the scenarios of highest importance, where the runaway population becomes comparable to the thermal population or the electric field is of order ED . In this paper, we describe such a tool: the new finite-difference code NORSE (NOn-linear Relativistic Solver for Electrons), which efficiently solves the kinetic equation in 2D momentum space. NORSE includes a fully relativistic non-linear Fokker–Planck operator for electron–electron collisions [32–34] and synchrotronradiation-reaction effects [14,35]. Time-dependent plasma parameters make the investigation of dynamic scenarios possible. With a non-linear treatment, the distribution is not restricted to being approximately Maxwellian, and strong electric fields (compared to ED ) can therefore be applied. However, if the distribution departs strongly from a Maxwellian, concepts such as temperature and the thermal collision time are not well defined. In many scenarios of practical interest, the distribution will nevertheless stay close to collisional equilibrium, and we will make use of familiar concepts where appropriate. The kinetic equation and the operators for the various mechanisms mentioned above are discussed in Section 2. The numerical implementation is then outlined in Section 3 and validated in Section 4 through a comparison to previous work in several limits. Finally, in Section 5 we use NORSE to investigate the properties of strongly distorted electron distributions and the condition for electron slide-away. 2. Kinetic equation To study the momentum-space dynamics of energetic electrons, we will solve the kinetic equation neglecting any spatial dependence. A point in the 3D momentum space is represented in spherical coordinates by p = (p, ξ , ϕ ), with p = γ v/c the magnitude of the (normalized) momentum, ξ = p∥ /p the cosine of the polar angle, and ϕ the azimuthal angle. Here v is the speed of the particle, c is the speed of light, and γ is the relativistic mass factor. The spherical symmetry of our problem is broken by the presence of the electric field E, and we therefore let the electric field define the parallel direction. If the plasma is magnetized, only the electricfield component parallel to B contributes to the acceleration (i.e. B∥E), in which case ξ is the cosine of the pitch–angle and ϕ is the gyro angle. We will assume the electron distribution function f = f (t ; p, ξ ) to be independent of ϕ , reducing the problem to a two-dimensional one. The kinetic equation describing the evolution of f can be written as ∂f eE ∂f ∂ − · + · (Fs f ) = C {f } + S , (1) ∂t me c ∂ p ∂p where Fs is the synchrotron-radiation-reaction force (in the presence of a magnetic field), C {f } is the Fokker–Planck collision operator describing microscopic Coulomb interactions between the plasma particles, and S denotes sources and sinks (of ∫ for instance heat or particles). The distribution f satisfies n = d3 p f , with n the number density of electrons. The parallel component of the momentum-space gradient, appearing in the term describing the Hamiltonian motion of the electrons due to the electric field, becomes E E · ( ) ∂f ∂f 1 − ξ 2 ∂f = ξ + . ∂p ∂p p ∂ξ (2) In what follows, we will detail the synchrotron-radiation-reaction and collision terms of Eq. (1), as well as the various source terms. 2.1. Synchrotron-radiation reaction The reaction force experienced by electrons emitting synchrotron radiation can be derived from the Lorentz–AbrahamDirac force. In a homogeneous plasma, it can be written as (see for instance [14] and references therein) ( 3 ) ( ) 1 ∂ γ p (1 − ξ 2 ) ∂ ξ (1 − ξ 2 ) ∂ · (FS f ) = − 2 f + f ∂p p ∂p τr ∂ξ γ τr [ ∂ f ∂ f 1 1 − ξ2 γ 2p =− −ξ τ γ ∂p ∂ξ ) ] (r 2 2 + 4p + f , 1 − ξ2 (3) where τr = 6π ϵ0 (me c)3 (4) e4 B2 is the radiation time-scale. Here B is the magnetic field strength. The total synchrotron power emitted by a relativistic particle is proportional to p2⊥ = p2 (1 − ξ 2 ), and the back-reaction experienced by the electrons therefore increases with perpendicular A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 271 momentum. The efficacy of the synchrotron-radiation reaction is thus closely linked to collisional pitch–angle scattering, which can redistribute parallel momentum gained from the accelerating field. through 2.2. Electron-ion collision operator The five potentials are analogous to the two Rosenbluth potentials g and h in the non-relativistic case [37], and reduce to these in the appropriate limit. The notation (which differs from that in Ref. [34], see Appendix B) has been chosen to highlight the existence of two ‘‘branches’’ of potentials, distinguished by the application of different La operators. Crucially, in the non-relativistic limit (as La reduces to the Laplace operator), Υ0 = Π0 → h and Υ1 = Π1 → g. A sketch of the derivation of the explicit expressions obtained in our coordinate system is given in Appendix A; here we list only the final result, with the terms grouped according to the derivative of f . The collision operator can be written as In a fully ionized plasma, the collision operator C contains contributions from collisions with electrons (Cee ) and ions (Cei ): C {f } = Cee {f } + Cei {f }. (5) The electron–electron collision operator is the main source of complexity in our problem, and will be discussed in Section 2.3. In contrast, we will assume a stationary, Maxwellian ion population. This, together with the mass difference between the species involved in the collision, significantly simplifies the operator for electron-ion collisions (unless the ratio between ion and electron temperatures is comparable to their mass ratio). In the ion restframe, the operator is [36]: Cei {f } ≃ Zeff νγ p3 L{f } = Zeff νγ [ p3 ] ∂f (1 − ξ ) , 2 ∂ξ ∂ξ 1 ∂ 2 2 where Zeff = n j nj Zj is the effective charge (with the sum taken over all ion species j), L is the Lorentz scattering operator, and ν= L2 Υ 1 = Υ 0 , L1 Π0 = f , L1 Π1 = Π0 . Cee {f } α (6) ∑ −1 ne4 ln Λ L0 Υ0 = f , C (7) 4π ϵ02 m2e c 3 is the collision frequency for relativistic electrons. The operator Cei describes pitch–angle scattering, but no change to the magnitude of the electron momentum. This is because the ions are assumed to be much heavier than the electrons (i.e. mi ≫ γ me ), so that the energy lost by the electrons through collisions can be neglected. C (p) 2.3. Electron–electron collision operator To describe electron–electron collisions, we will use the fully relativistic non-linear collision operator of Beliaev and Budker [32], in the form developed by Braams and Karney [33,34]. The operator is valid for collisions between arbitrary species of arbitrary energy (i.e. the bulk population is not required to be non-relativistic). For electron–electron collisions, it takes the form [34] Cee {f } = α ( ) ∂f ∂ · D· − Ff ∂p ∂p C (ξ 2) (8) where α = 4πν/n, D is the diffusion tensor and F is the friction vector. These are given by D = γ −1 [LΥ− − (I + pp)Υ+ ] , C (ξ ) (9) F = γ −1 KΠ , (10) where I is the unit tensor and L and K are defined by ( ) ∂ 2 Υ− ∂ Υ− , · (I + pp) + (I + pp) p · ∂ p∂ p ∂p ∂Π KΠ = (I + pp) · . ∂p LΥ− = (I + pp) · (11) C (pξ ) (12) Here, Υ− , Υ+ and Π are linear combinations of potential functions, given by Υ− = 4Υ2 − Υ1 , Υ+ = 4Υ2 + Υ1 , Π = 2Π1 − Π0 , (13) where we denote the five potentials introduced by Braams and Karney as Υ0 , Υ1 , Υ2 , Π0 and Π1 . These are defined using the differential operator La Ψ = (I + pp) : ( ) ∂ 2Ψ ∂Ψ + 3p · + 1 − a2 Ψ , ∂ p∂ p ∂p (14) 2 ∂ 2f ∂f 2 ∂ f + C (p) + C (ξ ) 2 2 ∂p ∂p ∂ξ 2 (ξ ) ∂ f (pξ ) ∂ f +C +C + C (f ) f , ∂ξ ∂ p∂ξ C (f ) (15) 2) = C (p with pre-factors C (p2 ) L2 Υ2 = Υ1 , (i) (16) given by γ 3 ∂ Υ− p ∂p γ (1 − ξ 2 ) ∂ 2 Υ− γ ξ ∂ Υ− − +2 2 , p2 ∂ξ 2 p ∂ξ ∂ Υ2 1 (2 + 3p2 )(8Υ2 − Υ0 ) − 16γ = γp ∂p ( ) ∂ Υ1 ∂ Υ0 γ 3 ∂ 2 Υ− 1 ∂ Υ− + 6γ −γ −2 + ∂p ∂p p ∂ p2 p ∂p ( )( ) 2 1 1 ∂ Υ− 2 ∂ Υ− + 2+ 2 2ξ − (1 − ξ ) 2 γp p ∂ξ ∂ξ ∂Π −γ , ∂p ( 1 − ξ 2 γ 2 ∂ Υ− = 2 γp p ∂p [ ] ) ∂ 2 Υ− ∂ Υ− 1 + 2 (1 − ξ 2 ) − ξ − Υ , + p ∂ξ 2 ∂ξ 2 2 2 2 ξ (1 − ξ ) ∂ Υ− γ (1 − ξ ) ∂ Υ− γ ξ ∂ Υ− =− −2 −2 3 γ p4 ∂ξ 2 p3 ∂ p∂ξ p ∂p ( ) 2 1 − ξ 2 ∂ Υ− + +3 γ p4 γ p2 ∂ξ ( ) ∂ Υ2 ∂ Υ1 ∂ Υ0 ∂Π 1 − ξ2 4 −3 + + − 2 γp ∂ξ ∂ξ ∂ξ ∂ξ ξ + 2 2 Υ+ , γp [ ] γ (1 − ξ 2 ) ∂ 2 Υ− ∂ Υ− =2 , p − p3 ∂ p∂ξ ∂ξ 2 ) ∂Π ∂ Π 1 ( = −γ − 2 + 3p2 ∂ p2 γp ∂p 1 − ξ 2 ∂ 2Π ξ ∂Π − +2 2 . γ p2 ∂ξ 2 γ p ∂ξ = γ (8Υ2 − Υ0 ) − 2 (17) (18) (19) (20) (21) (22) 2.4. Heat and particle sources A strong electric field is a source of energy that quickly heats the distribution function. In contrast to a linearized treatment (where this heat must be removed to ensure the validity of the 272 A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 linearization), this energy source is automatically accounted for in the non-linear solution. Sometimes, it is however of interest to remove the excess heat from the bulk as it is applied. In reality, the bulk temperature is not always increasing during fast-particle generation, for instance because of energy loss due to radiation emission or heat conduction. A heat sink also serves as a way to vary the temperature of the thermal population, which makes it possible to model dynamic scenarios where the plasma parameters change on a time scale similar to that of the acceleration dynamics. To be able to model density changes, a particle source must also be included. An advantageous way to formulate a heat sink is to write it in divergence form ) ∂ ( · kh Sh f = kh ∂p ( 2 p ∂ Sh (p) ∂ + Sh (p) f, ∂p ∂p dW dt = me c 2 Sh (p) + ( d3 p (γ − 1) − Ω + eE me c ∂f ∂p · ) ∂ ∂ · (Fs f ) − C {f } + kh · (Sh f ) , ∂p ∂p (24) ( ) γ (p) ) exp − , Θ 4π Θ K2 1/Θ n (25) ( where Kν (x) is the modified Bessel function of the second kind (and order ν ), has the energy moment W (Θ ) = me c 2 n Θ K2 (1/Θ ) pmax,Ω ∫ ( γ) . Θ dp p2 (γ − 1) exp − 0 (26) The magnitude kh can be determined from the requirement that the energy supplied by the heat sink should equal W (Θ2 ) − W (Θ1 ) for two temperatures Θ1 and Θ2 at subsequent time steps. Here pmax,Ω denotes the upper boundary of Ω in p; if pmax,Ω → ∞, the above integral can be evaluated analytically, yielding W (Θ ) = m e c 2 n ( K3 (1/Θ ) K2 (1/Θ ) −1−Θ ) ≡ me c 2 nW (Θ ). (27) ] γ −1 + ap ( Θ ) f M ; Θ (28) a linear combination of the energy and density moments of a Maxwellian (with an overall scaling factor kp analogous to kh ). The quantity ap can be determined from the constraint that the energy moment of Sp should vanish (so that the source supplies particles, but no heat), giving ap ( Θ ) = − = (23) from which kh can be determined in each time step by demanding that dW /dt = 0 (for an ideal heat sink). (Note that this approach does not automatically enforce energy conservation after discretization; only physical sources of heat are taken into account. If desirable, numerical heating caused by the discretization can be eliminated by instead requiring the numerically calculated energy moment of f to be constant.) The same heat source can be used to induce changes to the bulk temperature, but in this case the magnitude kh is calculated differently. We note that the relativistic equilibrium Maxwell– Jüttner distribution fM (p) = [ Sp = kp ) since it will then automatically conserve particles (before discretization). Here Sh (p) is an isotropic function of momentum, with Sh its p-component. kh is the magnitude, to be determined. In practice, the exact momentum-space shape of the sink depends on the processes responsible for the heat loss. A detailed investigation of this is left for future work; here we let Sh have the shape of a Maxwellian for simplicity. Apart from the electric-field term, synchrotron-radiation reaction also changes the total heat content of the distribution by removing energy, primarily at large particle momenta. However, the momentum-space region Ω of interest need not necessarily encompass the entire computational domain. For instance, it is sometimes desirable to maintain a fixed energy content in the thermal population, while simultaneously allowing the energetic particles to gain energy. Physically, this corresponds to heat sinks that only affect slow particles. In such cases, collisions may also transfer energy into or out of Ω. The total energy change in Ω can thus be written as ∫ Changes to the density can be introduced using a particle source of the form ∫∞ 1 Θ 2 Θ − ∫0 ∞ 0 dp p2 (γ − 1)2 exp(−γ /Θ ) dp p2 (γ − 1) exp(−γ /Θ ) 3(1 + Θ ) W (Θ ) − 3. (29) As Θ → 0, the non-relativistic limit ap = −5/2 is recovered, whereas in the ultra-relativistic case (Θ ≫ 1), ap → −4. The density moment np of the source is given by np n [ = kp Θ [ = kp W (Θ ) ] + a(p) W (Θ ) + 2 Θ −3 W (Θ ) + 1 + Θ W (Θ ) ] , (30) from which the magnitude kp that gives a desired density change can be determined. The bracket takes the asymptotic value −1 at both Θ → 0 and Θ ≫ 1, but reaches a minimum of −1.18 for intermediate temperatures. 3. Numerical method 3.1. Discretization We choose to represent the distribution f on a two-dimensional finite-difference grid in p and ξ , and use a 5-point stencil to discretize the momentum-space derivatives. Moments of the distribution and other integrals are calculated using a composite Simpson’s rule. The grid points can be chosen non-uniformly in both p and ξ , making it possible to efficiently resolve both a Maxwellian bulk (assuming there is one) and a high-energy tail. Specifically, the p grid should preferably be densely spaced for small p to resolve the bulk, but since the tail generally varies over larger momentum scales, coarser spacing can be used at larger momenta to reduce the computational expense. Similarly, the ξ grid should be densely spaced close to ξ = 1 (the parallel direction) to resolve the tail drawn out by the electric field. Alternatively, in scenarios without a preferred direction of acceleration, a grid which gives a uniform spacing in the polar angle (arccos ξ ) is often appropriate. Due to the polar nature of the coordinate system, the point at p = 0 is special; the value of the distribution at p = 0 should be independent of ξ . The total number of grid points is thus Nξ × (Np − 1) + 1, with Np and Nξ the number of grid points in the respective coordinate, and a single (rather than Nξ ) grid point appropriately describes the system at p = 0. For the calculation of the potentials Υi and Πi (here collectively denoted by Ψ ), it is advantageous to decompose the ξ coordinate in Legendre modes (rather than use a finite difference grid), since these are eigenfunctions of the collision operator. The distribution and potentials are then written as f (p, ξ ) = Nl ∑ l=0 fl (p)Pl (ξ ), Ψ (p, ξ ) = Nl ∑ Ψl (p)Pl (ξ ), (31) l=0 where Pl is the lth Legendre polynomial. The potentials are integral moments of the distribution function, and their calculation is a A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 smoothing operation. Therefore, a small number Nl of Legendre modes typically suffices to accurately describe the potentials, unless the bulk of the distribution deviates significantly from the origin of the coordinate system. Thus, it is usually reasonable to choose Nl to be much smaller than Nξ , the number of points in the ξ grid. The mapping between the 2D-finite-difference-grid and finitedifference-Legendre-mode representations can be formulated as a single matrix operation, where a mapping matrix ML can be constructed to represent the summation in Eq. (31). In general, ML is not square, but the inverse mapping can be performed by taking the Moore–Penrose pseudo-inverse of ML to find the inverse in a least-squares sense. (This only needs to be done once in each NORSE run.) The solution is exact in the sense that norm[ML fl (p) − f (p, ξ )] is of order the round-off error, and the mapping between the two representations can thus be performed to machine precision at very small computational cost. The parallel axis is the symmetry axis of the problem. Therefore, we require that the derivative of the distribution with respect to p⊥ at a fixed p∥ must vanish as p⊥ → 0. This condition must be imposed as a boundary condition at p∥ = 0, but is automatically satisfied for all non-vanishing p∥ . At p = pmax , we impose the Dirichlet condition f (pmax ) = 0 for all ξ . 3.2. Calculation of potentials The Legendre modes of the potentials Ψ can be calculated from the distribution using Eq. (15), which becomes L0,l Υ0,l = fl , L2,l Υ1,l = Υ0,l , L1,l Π0,l = fl , L1,l Π1,l = Π0,l , L2,l Υ2,l = Υ1,l , (32) where ∂ 2Ψ La,l Ψ = γ 2 + ∂ p2 ( ) ( ∂Ψ l(l + 1) Ψ (33) + 3p + 1 − a2 − 2 p ∂p p 2 ) is obtained by decomposing the differential operator La (described in our coordinate system by Eq. (A.8)) into Legendre modes. Inverting Eqs. (32) results in operators which can determine the potentials from an arbitrary f , and a block-diagonal sparse matrix for each potential Ψ can be constructed to efficiently calculate Ψl from fl for all l using a single matrix multiplication, in accordance with the discussion in Section 3.4. The above calculation requires that boundary conditions for the potentials be specified. In general, for a function φ (p, ξ ) to be continuous at p = 0, its Legendre modes φl (p) must satisfy ∂φl (0)/∂ p = 0 for l = 0 and φl (0) = 0 for l > 0. Boundary conditions at p = pmax can be determined from Eq. (31) in Ref. [34], which gives explicit expressions for the potentials Ψl in terms of weighted integrals over fl . The calculation of these boundary conditions is discussed in Appendix B. 3.3. Time advance To advance the system in time, we employ a linearly implicit time-advancement scheme based on the first-order backwardEuler method. The scheme avoids the restriction on the time step imposed on explicit methods by the CFL condition, and is straight-forward to implement, as it only requires building and inverting a single matrix in each time step. Compared to fully implicit methods, however; the time step has to be kept relatively short, and the overall computation time can still be considerable when simulating a long time span. As long as the time step is short enough, good accuracy is achieved, and this simple scheme is sufficient for our purposes. The method is formulated as follows. The entire kinetic equation, excluding the time derivative, can in general be written as 273 an operator O{Ψ {f }, f }, where Ψ represents the five potentials Υi and Πi , which depend on the distribution f . In a fully implicit time-advancement scheme, this operator should be evaluated at the next time step (k + 1): O{Ψ {f k+1 }, f k+1 }. If the potentials are instead evaluated based on the distribution at the current time step, f k , O can be written as a regular matrix operation O{Ψ {f k }, f k+1 } = k k Mmn f k+1 , where the matrix Mmn = M k (pm , ξn ) describes a set of linear equations. This makes the time-advancement scheme linearly implicit, and M k can be explicitly evaluated in each time step and the system solved using standard matrix-inversion techniques. The Backward-Euler method for our problem can then be written as f k+1 = f k + ∆tM k f k+1 , (34) where ∆t is the time step. 3.4. Performance NORSE is written in Matlab, using an object-oriented structure. To make efficient use of the Matlab language, care has been taken to formulate the problem in terms of matrix multiplications and avoid loops where they are detrimental to performance. To this end, many parts of the operators of the kinetic equation are precalculated to speed up the matrix building in each time step. As an example, the first term of the electron–electron collision operator Eq. (16) at time step k (together with Eq. (17)) can be written as 2) C (p ( ) ∂ 2 f k+1 = C0 Υ0k + C2 Υ2k + C− Υ−k D2pp f k+1 , ∂ p2 (35) where the various operators, defined as C0 = −γ , C− = −2 D2pp = γ3 p ∂2 , ∂ p2 C2 = 8γ , γ (1 − ξ 2 ) Dp − Dp = p2 ∂ , ∂p D2ξ ξ + 2 γξ D2ξ ξ = p2 (36) Dξ ∂2 , ∂ξ 2 Dξ = ∂ , ∂ξ (37) are all independent of f , and can thus be pre-calculated. Constructing this part of the linear system in each time step is thus reduced 2 to determining the potentials Υik from f k and constructing C (p ) D2pp in accordance with Eq. (35), using just a few matrix operations. The above algorithm is efficient, making the matrix inversion associated with the solution of the resulting linear system the most costly part of each time step. The overall computational cost can be reduced by approximately a factor of 2 by employing an iterative scheme using the generalized minimal residual method (gmres [38]), which is available in Matlab as a standard subroutine. By periodically (every nLU time steps) solving the system exactly using LU-factorization, and supplying the L and U factors as preconditioners for the next nLU − 1 steps, gmres converges in just a few iterations if nLU is sufficiently small. In certain scenarios – such as where an initial transient requires high temporal resolution, but the subsequent relaxation happens on a significantly longer time scale – adaptive time-step schemes can be very effective in reducing the computational expense. Such a scenario is for example considered in Section 4.1. Here, we use a simple adaptive-time-step scheme based on information about the number of iterations needed for convergence of the gmres algorithm. If few gmres iterations are needed for convergence (ngmres < nopt , where nopt is some desired optimal number), the change in the distribution in each time step is small, indicating that the step length can be increased. Conversely, the step length should be reduced if ngmres > nopt . Employing the techniques discussed above makes the implementation efficient, and moderately sized test cases usually run on a standard laptop in less than a minute. 274 A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 Fig. 1. Collisional relaxation from a starting distribution consisting of two shifted Maxwellians. Panels (a)–(d) show the 2D distribution at various times, and panel (g) shows corresponding cuts along the positive parallel axis. Panel (e) shows the conservation of density and energy and panel (f) the time step used, as functions of time. The numerical parameters Np = 250, Nξ = 65, Nl = 25, pmax = 10, and the initial time step dτ0 = 0.001 were used. A uniform grid was used in p, whereas a non-uniform grid giving uniform spacing in the polar angle (arccos ξ ) was used for the ξ coordinate. 4. Tests and benchmarks The kinetic equation solved by NORSE is valid for strongly non-Maxwellian electron distributions, as well as relativistic temperatures and particle energies. In this section, we validate the implementation by comparing to the two limits of arbitrary temperature but weakly distorted distribution (Section 4.2), and nonrelativistic but fully non-linear distribution (Section 4.3). However, let us first look at a proof-of-principle scenario, demonstrating both the non-linearity as well as the high-temperature validity of NORSE . In this section, we will repeatedly make use of the normalized time τ = ν t (i.e. the time in units of relativistic-electron collision times) and the normalized distribution F = f /fM (p = 0) (so that if the initial distribution is a Maxwellian, F initially takes the value unity at p = 0). We will also use the normalized electricfield magnitude Ê = eE /me c ν = E /Ec . Throughout the rest of the paper, we will always apply fields with an implicit minus sign, so that electrons will be accelerated towards positive p∥ . 4.1. Proof-of-principle non-linear scenario: collisional relaxation of a two-maxwellian initial state In this section we demonstrate the validity of the NORSE implementation by considering a basic non-linear test case: the collisional relaxation of two initially shifted Maxwellians. A shifted Maxwell–Jüttner distribution (e.g. the equilibrium distribution with temperature Θb in a frame boosted by pb in the parallel direction, as seen from the stationary frame) takes the form ) γb γ − pb p∥ ( ) exp − fM,b (Θb , pb ) = , (38) Θb 4π Θb K2 1/Θb √ with γb = 1 + p2b and p∥ = pξ . We consider two initial Maxwellians, each with a temperature of 10 keV (Θb = 0.0196), and each shifted the equivalent of three thermal speeds (pb = 0.59) along the symmetry axis (in opposite directions). The initial n ( state is depicted in Fig. 1(a), with panels (b)–(d) showing the subsequent evolution of the distribution function. Panel (g) shows a cut of the distribution along the positive parallel axis at the same time steps. The parameter values E = 0 and B = 0 were used, and to isolate the behavior of the non-linear electron self-collision operator, a pure electron plasma was assumed (Zeff = 0). The number density of each Maxwellian in its rest frame was set to n = 1019 m−3 , resulting in a total initial number density of ntot = 2γb n = 2.326 · 1019 m−3 . The expected final-state Maxwellian (cyan, thin dashed line in panel (g)) has a temperature of 61.3 keV, which can be calculated by equating Eq. (27) (with n → ntot ) with the combined energy content in the two shifted Maxwellians: Wtot = 2Wb , Wb [ ] = γb2 W + me c 2 n γb (γb − 1) + (γb2 − 1)Θb (39) (with W given by (27)), and solving for Θ . The final equilibrium state shows excellent agreement with the theoretical prediction. The relative error (compared to the initial value) in the density and energy contents of the NORSE solution are shown in Fig. 1(e) as functions of time. For the numerical parameters used, the density is conserved to within 0.05%, whereas the relative error in energy saturates at the 0.5% level. Fig. 1(f) shows the time step used by the adaptive-time-step scheme, normalized to the initial time step. In this particular case, the scheme is very effective since the time evolution involves an initial transient followed by a comparatively slow asymptotic relaxation. The final time step was approximately 104 times longer than the initial time step, and a total of 312 time steps were used (as opposed to ∼ 4 · 105 had the initial time step been used throughout the entire calculation). 4.2. Weak-electric-field limit: conductivity for relativistic temperatures In Ref. [34], Braams and Karney use the relativistic electron– electron collision operator to calculate the plasma conductivity for a wide range of temperatures. The operator is linearized around a stationary Maxwellian, and the zeroth and first Legendre modes are calculated numerically as an initial-value problem. The results are compiled in their Table 1, which contains normalized conductivities for Θ ∈ [0, 100] (recall that Θ = 1 corresponds to T = me c 2 ≃ 511 keV) and Zeff ∈ [0, ∞]. The unit used is √ σ̄ = eme ln ΛZeff j 4πϵ02 T 3/2 E , (40) where j is the current density. To demonstrate that our implementation reproduces the above results, we similarly calculate the conductivity of a quasi-steadystate distribution found by evolving the system from a Maxwellian A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 275 eventual loss of fast-electron confinement constitute a serious threat to the plasma-facing components of fusion reactors. The energy and current carried by the runaway electrons, and thus the potential for damage, increase the larger the distortion of the electron distribution. Unlike existing tools such as LUKE [19] and CODE [20,41], NORSE can be used to study the cases of highest runaway-electron growth rate. 5.1. Runaway region of momentum space Fig. 2. Normalized conductivity in NORSE (lines) for various temperatures and plasma compositions. Data points from Table 1 of Ref. [34] are also shown (squares). The electric field corresponded to E = 10−3 ED , and n = 5·1019 m−3 and B = 0 were used. initial state using a constant electric field corresponding to E = 10−3 ED . However, we make no simplification to the collision operator and retain adequate resolution in ξ to accurately resolve the distribution function in 2D momentum space. Fig. 2 overlays NORSE results with the data in Ref. [34] for Zeff = 1, 2, 5 and 10, and all tabulated temperatures. Excellent agreement is seen for all parameters. In the figure, the data points at Θ = 0 in Ref. [34] are compared to NORSE runs with Θ = 10−5 , however all temperatures Θ < 10−3 give good agreement as the obtained values of σ̄ are essentially independent of the temperature in this range. 4.3. Non-relativistic limit: highly anisotropic distributions Several codes exist that solve the non-relativistic kinetic equation using non-linear collision operators. To validate the non-linear aspect of NORSE, we will compare to conductivities reported by Weng et al. in Ref. [39]. In their Fig. 3, conductivities as functions of time are presented for electric fields as strong as the Dreicer field ED , leading to highly distorted distributions. Results are shown for E /ED = 0.01, 0.1 and 1, with Zeff = 1 and B = 0. Fig. 3(a) reproduces the results in Fig. 3 of Ref. [39] using NORSE. The units used are those of the original figure: conductivities are given as j̄/Ê, with j̄ = jZeff /nec√ Θ 3/2 a normalized current density, and the time unit used is Ê τ / Θ . The parameters Θ = 1 · 10−4 (corresponding to a temperature of T = 51eV) and n = 5·1019 m−3 were used in NORSE. Data points extracted from the figure in Ref. [39] are included for comparison. The agreement is very good in general, demonstrating that NORSE behaves as expected also for highly non-linear distributions. The error is somewhat larger (and systematic) for the weakest electric field, however in this case, numerical heating in the results of Ref. [39] cannot be ruled out [40]. The final distributions for the three field strengths are shown in 3(b). As can be seen, the distributions deviate strongly from the initial Maxwellian for the two higher field strengths, and even at the weakest field, a substantial tail of runaway electrons is produced. For the strongest field, a small ‘‘bump’’ in the distribution is seen at p∥ = 0, which indicates that electron-ion collisions are strong enough to ‘‘capture’’ a sub-population of the electrons, despite the strong accelerating electric field. 5. Application: Runaway electrons in fusion plasmas As an example of how NORSE can be used to provide new physical insight, we consider the case of runaway-electron generation in magnetically confined fusion plasmas. Under the influence of strong electric fields, electrons quickly accelerate to relativistic speeds since the friction force they experience decreases with increasing velocity. The localized heat loads associated with the What constitutes a runaway particle can be defined in several ways. The definitions usually employed in theoretical works, as well as many numerical tools, assume the distribution to be close to Maxwellian and are therefore not directly applicable in our context [10,42]. In addition, it has been pointed out that synchrotron radiation reaction may have a significant impact on the runaway region [43–46], however the commonly used definitions only account for collisional friction. We will define a runaway region based on particle trajectories in momentum space [42], neglecting the effect of diffusion but allowing for arbitrary electron distributions as well as synchrotron radiation reaction. For an arbitrary electron distribution, the lower boundary of the runaway region (the separatrix) can be obtained by considering the forces that affect a test particle: ∂Π γ p(1 − ξ 2 ) + , (41) ∂p τr 2 2 2 eE 1 − ξ 1 − ξ ∂Π ξ (1 − ξ ) dξ ξ ξ ξ = FE − FC − FS = +α − , dt me c p γ p2 ∂ξ γ τr dp dt = FEp − FCp − FSp = eE me c ξ + αγ (42) where the expressions for the force associated with the electric field, FEi , the collisional electron–electron friction FCi and the synchrotron radiation-reaction force FSi are taken from Eqs. (1), (A.3) and (3), respectively. Asymptotically, particles on the separatrix neither end up in the bulk population nor reach arbitrarily high energies, but instead settle at the point pc of parallel force balance at ξ = 1 (in the absence of diffusion). pc can be determined from dp/dt = 0 at ξ = 1, since the separatrix becomes purely perpendicular to the parallel axis as ξ → 1. The separatrix is then traced out by numerically integrating the above equations from ξ = 1 to ξ = −1. In the limit of non-relativistic temperature (Θ ≪ 1), small departure from a Maxwellian, and B = 0, the result agrees with the standard expression [42]. To ensure consistency with the distribution, the separatrix in NORSE is calculated in each time step. 5.2. Distortion-induced transition to electron slide-away For electric fields stronger than approximately Esa = 0.215ED , all electrons in a Maxwellian distribution experience net acceleration, since the field overcomes the maximum of the collisional friction force. This is known as electron slide-away [10,47]. However, in a non-linear treatment, the condition for slide-away can in principle be modified since the collisional friction depends on the shape of the electron distribution. The distortion of the distribution associated with a moderately strong electric field turns out to have a large effect on the effective Dreicer field at which the transition to the slide-away regime occurs. This is illustrated in Fig. 4, which shows the distribution at several time steps, as well as the separatrix and the force balance (neglecting diffusion) as a function of p at ξ = 1. In the figure, the distribution is evolved under a constant electric field which initially corresponds to 5% of the Dreicer field (E = 0.05ED,0 ≈ 0.23Esa,0 , with ED,0 and Esa,0 the Dreicer and slide-away fields at the initial temperature). The distribution quickly becomes distorted, and soon after t = 0.15τ 276 A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 Fig. 3. (a) Normalized conductivity in NORSE (lines) as a function of time for various E-field strengths. Data points extracted from Fig. 3 of Ref. [39] are also shown (squares). (b) Cuts in the parallel direction through the final distributions in (a). The numerical parameters were Np = 300, Nξ = 55, Nl = 15, pmax = 0.3, using 2000, 400 and 300 time steps for E /ED = 0.01, 0.1 and 1, respectively. Fig. 4. (a)–(d) Contour plots and (e) cuts along the parallel axis of the distribution at different times during a NORSE run. In (a)–(c), the white dashed lines are the separatrices defining the lower boundary of the runaway region. (f) Sum of forces (neglecting diffusion) on the parallel axis. The physical parameters were Θ = 0.01 (T = 5.11 keV), n = 5 · 1019 m−3 , Zeff = 1, Ê = 5 (corresponding to E /ED,0 = 0.05 and E = 0.22 V/m), B = 0, and Np = 100, Nξ = 35, Nl = 5, pmax = 3.54, and dτ = 2.8 · 10−4 were used. the slide-away regime is reached. This can be seen in Fig. 4(f), where the sum of forces (Eq. (41)) becomes positive everywhere on the parallel axis, indicating that the electric field at that time corresponds to the instantaneous slide-away field, E = Esa (t). No separatrix therefore exists for later times (see Fig. 4(d)). An important effect of the electric field is to quickly heat the bulk of the distribution, and this turns out to be the main cause of the induced transition to slide-away. Since (neglecting the weak dependence on ln Λ) the Dreicer field ED ∼ 1/T , an increase in temperature lowers the effective Dreicer field and thus the threshold for electron slide-away. An approximate effective temperature Teff can be estimated from the energy moment W of the NORSE distribution by solving Eq. (27) for Θ . The effective Dreicer field can then easily be evaluated. Fig. 5 highlights the importance of the heating effect by showing the time to a transition to slide-away under constant electric fields of various strengths (starting from an equilibrium distribution). Lines denote when the effective temperature becomes such that E > Esa (Teff ), whereas squares denote the actual transition in NORSE, calculated from the force balance. The agreement between these two values is very good in the entire range of electric-field values, demonstrating that the bulk heating is the dominant effect in the modification of the slide-away threshold. Only at fields very close to Esa,0 does the values obtained using the effective temperature noticeably overestimate the time to transition, indicating that here, other effects start to become important as well. The figure also shows that the process leading to a transition to slide-away is quick, also for relatively weak fields. At E /Esa,0 = 0.3 (E /ED,0 ≈ 0.065), the transition happens around 30 thermal collision times for Zeff = 1, and at E /Esa,0 = 0.1 (E /ED,0 ≈ 0.022), the corresponding figure is 500. In practice, various processes can lead to heat losses, as previously noted. This can partially or entirely offset the heating caused by the electric field, and in many situations the modification to the slide-away threshold may not be as dramatic as demonstrated here. In addition, a feedback mechanism commonly exists between the accelerating electric field and the distribution (through changes in the plasma current). In such scenarios, a reduction in the electric field may be induced due to the changes in the distribution before these have become too extensive, thus limiting the distortion and potentially avoiding a transition to slide-away altogether. 6. Conclusions The study of energetic-electron populations in plasmas has long been of interest, but when considering relativistic particles in kinetic simulations, the work has so far been restricted to linearized treatments of the Fokker–Planck collision operator. In this paper, we remove that limitation by introducing a new efficient computational tool (NORSE) which includes the fully non-linear relativistic collision operator in the differential form developed by Braams and Karney, as well as electric-field acceleration and synchrotronradiation reaction. A 2D non-uniform finite-difference grid is used to represent momentum space, however when evaluating the five relativistic potentials (analogous to the two Rosenbluth potentials in the non-relativistic case), a mixed finite-difference–Legendremode representation is used since the potentials are given by simple 1D integrals in a Legendre-mode decomposition. The system is A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 [ ∂f + γ Dpξ (Υ− ) ∂p ( ) ] 1 − ξ2 ∂f ∂ Υ− g D ( Υ ) + p + − Υ eξ ξ ξ ξ ξ − + γ p2 ∂p ∂ξ 277 (A.2) and Ff = γ 1 − ξ 2 ∂Π ∂Π f ep + f eξ , ∂p γ p2 ∂ξ (A.3) where ] ∂ 2 Υ− ∂ Υ− ∂ Υ− −ξ +p , 2 ∂ξ ∂ξ ∂p [ ] 1 ∂ Υ− ∂ Υ− Dϕϕ (Υ− ) = 4 p −ξ , p (1 − ξ 2 ) ∂p ∂ξ [ ] 2 2 1−ξ ∂ Υ− 2 ∂ Υ− Dpξ (Υ− ) = Dξ p (Υ− ) = p − p p4 ∂ p∂ξ ∂ξ Dξ ξ (Υ− ) = Fig. 5. Time to transition to slide-away as a function of electric-field strength for various values of Zeff . Times calculated from the effective temperature of NORSE distributions are indicated by lines, while the actual times obtained in NORSE (based on force balance) are indicated by squares. Here, a subscript 0 indicates an initial value, and τth,0 = τ /(2Θ0 )3/2 denotes the initial thermal-electron collision time. The physical parameters were Θ0 = 1 · 10−4 , n = 1019 m−3 , B = 0. evolved using a linearly implicit time-advancement scheme, and a simple method for adapting the time step during runtime has been implemented. NORSE has been successfully benchmarked in both the relativistic-weak-field and non-relativistic–non-linear limits. As an application, we have used NORSE to investigate scenarios relevant to the study of runaway electrons in magneticconfinement-fusion plasmas. We find that the quick heating of the bulk associated with the application of medium to high-strength electric fields (compared to the Dreicer field) leads to a transition to the electron slide-away regime, despite the E field being weaker than the threshold value Esa = 0.215ED for the initial distribution. The time scale for this transition is relatively short, ranging from a few to a few hundred thermal collision times for E fields in the range E /Esa > 0.1. These effects cannot be consistently captured in a linearized treatment, and this example thus illustrates that NORSE opens new avenues of investigation into the dynamics of relativistic electrons in plasmas. Acknowledgments The authors would like to thank E. Hirvijoki for his initial work on the Braams and Karney collision operator, S.-M. Weng for his helpful assistance, and I. Pusztai, S. Newton, T. DuBois and G. Wilkie for constructive discussions. This work was supported by the Swedish Research Council (Dnr. 2014-5510), the European Research Council (ERC-2014-CoG grant 647121), and the Knut and Alice Wallenberg Foundation (Dnr. KAW 2013.0078) . A.S. would also like to acknowledge travel support from Adlerbertska Forskningsstiftelsen. Appendix A. Derivation of electron–electron collision operator in (p, ξ ) coordinates In our coordinate system, the non-zero components of the metric are gpp = 1, gξ ξ = p2 1 − ξ2 , gϕϕ = p (1 − ξ ), 2 2 (A.1) with g = 1/gii . Note also that the position vector is just p = p ep (with ep the unit vector along p), since the coordinate system is spherical. Using this and some algebra, we can write the terms in the parenthesis of Eq. (8) as ii D· ∂f = ∂p [( ) ∂ 2 Υ− ∂ Υ− ∂f γ3 + γ p − γ Υ + ∂ p2 ∂p ∂p ] γ (1 − ξ 2 ) ∂f + g D ( Υ ) ep ξ ξ p ξ − p2 ∂ξ 1 − ξ2 p4 [ (1 − ξ 2 ) (A.4) come from the expression for ∂ 2 Υ− /∂ p∂ p in the operator L. Writing out Eq. (8) in components, we get ]p ) ∂f − Ff ∂p ∂p ([ ]ξ )] ( ) ∂ ∂f + D· − Ff ≡ α Ap + Aξ , ∂ξ ∂p Cee {f } = α [ 1 ∂ p2 ( [ p2 D · (A.5) where the superscripts p and ξ denote the corresponding vector components. Carrying out the differentiations, we find the p-term to be p2 A p = ∂ Υ− ∂p ) 2 ∂ Υ ∂ Υ− ∂ 2 f − − γ (1 − ξ 2 ) + 2γ ξ 2 ∂ξ ∂ξ ∂ p2 ([ 3 ] p + + 2γ p (8Υ2 − Υ0 ) γ [ ] ∂ Υ2 ∂ Υ1 ∂ Υ0 ∂ Υ− + γ p2 −16 +6 − − 2γ 3 ∂p ∂p ∂p ∂p 2 p(1 − ξ 2 ) ∂ 2 Υ− 3 ∂ Υ− − 2γ p − ∂ p2 γ ∂ξ 2 ) 3 ∂ Υ p ξ ∂ Υ ∂ 2 Υ− ∂ f − − − γ (1 − ξ 2 ) +2 + 2γ ξ 2 ∂ p∂ξ γ ∂ξ ∂ p∂ξ ∂ p ∂ 2f 2 + γ p Dpξ (Υ− ) ∂ p∂ξ [( 3 ) ] p ∂ Dpξ (Υ− ) ∂ f + + 2γ p Dpξ (Υ− ) + γ p2 γ ∂p ∂ξ [( 3 ) ] 2 ∂ Π ∂ f p ∂ Π ∂ Π − γ p2 − + 2γ p + γ p2 2 f , (A.6) ∂p ∂p γ ∂p ∂p ( γ p2 (8Υ2 − Υ0 ) − 2γ 3 p whereas the ξ -term becomes ∂ 2f ∂ Dpξ (Υ− ) ∂ f +γ ∂ p∂ξ ∂ξ ∂p ( ) 2 1 − ξ2 ∂ Υ− ∂ f + g D ( Υ ) + p − Υ ξ ξ ξ ξ − + γ p2 ∂p ∂ξ 2 [ ( ) ξ ∂ Υ− + − 2 2 g ξ ξ D ξ ξ (Υ − ) + p − Υ+ γp ∂p ( ( )] ) ∂ 2 Υ− ∂ Υ+ ∂f 1 − ξ 2 ∂ gξ ξ Dξ ξ (Υ− ) + + p − γ p2 ∂ξ ∂ p∂ξ ∂ξ ∂ξ [ ] 1 − ξ 2 ∂Π ∂f ξ ∂Π 1 − ξ 2 ∂ 2Π − + 2 − f. γ p2 ∂ξ ∂ξ γ p2 ∂ξ γ p2 ∂ξ 2 Aξ = γ Dpξ (Υ− ) (A.7) 278 A. Stahl et al. / Computer Physics Communications 212 (2017) 269–279 Table B.1 Some analytical expressions for yl,a useful for validating the recursive calculation. l\a 0 1 2 2 −(3 + 2p2 )/p3 −(15γ /p4 + 6γ /p2 ) −105γ 2 /p5 − 42γ 2 /p3 + 27/p3 + 18/p −945γ 3 /p6 − 378γ 3 /p4 + 483γ /p4 + 258γ /p2 −3γ /p3 3/p2 − 15γ 2 /p4 −5(12γ /p3 + 21γ /p5 ) −45(7γ 2 /p4 + 21γ 2 /p6 + 1/p2 ) −3/p3 −(5/2)γ /p2 × (15 − 15γ 2 /p2 + 21/p2 ) −(35/2)γ 2 /p3 × (15 − 15γ 2 /p2 + 21/p2 ) + 15/p3 −(315/2)γ 3 /p4 × (15 − 15γ 2 /p2 + 21/p2 ) + 135γ /p4 + 30γ /p2 × (15 − 15γ 2 /p2 + 21/p2 ) 3 4 5 Combining and re-grouping the terms according to the derivative of f , we arrive at the final expressions (17)–(22). In obtaining some of these results, we have used the differential operator La , which can be written as ( ) ∂ 2Ψ 2 ∂Ψ + + 3p 2 ∂p p ∂p ( ) 2ξ ∂ Ψ (1 − ξ 2 ) ∂ 2 Ψ − 2 + 1 − a2 Ψ , (A.8) + p2 ∂ξ 2 p ∂ξ together with L2 Υ± = 4Υ1 ± Υ0 , to remove third order derivatives. La Ψ = γ 2 Appendix B. Boundary conditions for the potentials Υi and Πi at p = pmax To calculate the potentials Υi,l and Πi,l (or collectively Ψl ) from fl , boundary conditions at p = pmax must be specified. These can be determined from Eq. (31) in Ref. [34], which for the final grid point becomes Ψl (pmax ) = pmax ∫ Nl,∗ (pmax , p′ ) 0 p ′2 γ′ fl (p′ )dp′ , (B.1) where the place holder ∗ denotes a set of indices, distinct for each potential. These indices – which in the notation of Ref. [34] specify the order of differential operators La to apply to obtain a given potential from f (cf. Eq. (15)) – are Υ0 : 0, Υ1 : 02, Υ2 : 022, Π0 : 1, Π1 : 11. (B.2) The quantity Nl,∗ is defined as Nl,∗ (p, p′ ) ⎧ y (p)j (p′ ), ⎪ ⎨ l,a l,a ′ if ∗ = a, yl,a (p)jl,aa′ (p ) + yl,aa′ (p)jl,a′ (p′ ), if ∗ = aa′ , = (B.3) ′ ′ ′ ⎪ ⎩yl,a (p)jl,aa′ a′′′ (p′′ ) + yl,aa′ (p)jl,a′ a′′ (p ) + yl,aa′ a′′ (p)jl,a′′ (p ), if ∗ = aa a , where yl,a (p) and jl,a (p) are two independent solutions to the homogeneous equation La,l Ψl,a = 0 (here Ψl,a represents one of the one-index potentials; either Υ0 and Π0 , depending on the value of a), and the other yl,∗ and jl,∗ can be calculated from these using relations given in [34]. The problem of finding Nl,∗ can be reduced to recursively calculating jl,a for a = 0, 1, 2, and all l of interest, however the recursive calculation is numerically non-trivial. A method for achieving accurate results is outlined in Appendix 7 of [34]. Validation of the obtained jl,a can be done using the equation after Eq. 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Journal of Physics: Conference Series 775, 012013 (2016). http://dx.doi.org/10.1088/1742-6596/775/1/012013 http://arxiv.org/abs/1610.03249 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 Runaway-electron formation and electron slide-away in an ITER post-disruption scenario A Stahl1 , O Embréus1 , M Landreman2 , G Papp3 and T Fülöp1 1 Department of Physics, Chalmers University of Technology, Göteborg, Sweden Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA 3 Max Planck Institute for Plasma Physics, Garching, Germany 2 E-mail: [email protected] Abstract. Mitigation of runaway electrons is one of the outstanding issues for the reliable operation of ITER and other large tokamaks, and accurate estimates for the expected runawayelectron energies and current are needed. Previously, linearized tools (which assume the runaway population to be small) have been used to study the runaway dynamics, but these tools are not valid in the cases of most interest, i.e. when the runaway population becomes substantial. We study runaway-electron formation in a post-disruption ITER plasma using the newly developed non-linear code NORSE, and describe a feedback mechanism by which a transition to electron slide-away can be induced at field strengths significantly lower than previously expected. If the electric field is actively imposed using the control system, the entire electron population is quickly converted to runaways in the scenario considered. We find the time until the feedback mechanism sets in to be highly dependent on the details of the mechanisms removing heat from the thermal electron population. 1. Introduction Runaway electrons pose a severe threat to the safety and reliability of ITER and other highplasma-current fusion devices [1]. The larger the runaway-electron population, the larger the threat to the integrity of the device. However, if the electron momentum-space distribution function becomes highly non-Maxwellian due to the presence of a high-energy tail of runaways, existing numerical tools employing linearized collision operators are no longer valid. The same is true if the electric field (even momentarily) becomes comparable to the Dreicer field [2]. We recently presented NORSE [3] – an efficient solver of the kinetic equation in a homogeneous plasma – which includes the full relativistic non-linear collision operator of Braams & Karney [4, 5]. NORSE – which will be discussed in Section 2 – is able to model Dreicer and hot-tail runaway generation in the presence of electric fields of arbitrary strength and synchrotronradiation reaction: one of the most important energy-loss channels for runaways [6]. Since NORSE is able to treat highly distorted distributions, a range of new questions may be addressed. One issue of particular interest is: will non-linear phenomena accelerate or dampen the growth of runaways? Naturally, this is of great importance in view of ITER and other large tokamaks, as it potentially impacts the requirements on the disruption mitigation system (the design of which is currently being finalized) [1, 7]. In addition, the electric field is expected to reach values as high as 80-100 V/m during the current quench in ITER [8], and runaway Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 generation is likely to be strong enough for 60% or more of the plasma current to be converted to runaway current. In Section 3, we use NORSE to model the evolution of the electron population in a typical ITER post-disruption scenario. If the electric field is strong enough, the net parallel force experienced by electrons due to the electric field and collisions becomes positive in the entire momentum space, leading to a phenomenon known as electron slide-away. This is expected to happen when E > 0.215ED ≡ ESA , where ED = ne3 ln Λ/4π20 T is the so-called Dreicer field [2]; n, T and −e are the electron number density, temperature and charge; ln Λ is the Coulomb logarithm; and 0 is the vacuum permittivity. The associated surge in runaway current can have a large impact on the potential for material damage, as well as the subsequent evolution of the parallel electric field. The slideaway process cannot be consistently modelled using linear tools such as CODE [9, 10] or LUKE [11], which assume a Maxwellian background plasma and therefore require E ESA , as well as that the runaway fraction is small nr /n 1. A strong electric field represents a source of energy that quickly heats up the electron distribution. This heating can induce a transition to the slide-away regime – even under a fixed applied electric field which is initially below the threshold E < ESA – since the collisional friction is lower in a hotter distribution. As a consequence, the Dreicer field is also lower, making the effective normalized field E/ED,eff higher for a given field strength. If the temperature increase is large enough, the slide-away regime is reached, which happens at a field of E/ED,eff = 0.215 in the case of a constant applied electric field, i.e. it coincides with the standard slide-away field at the effective temperature Teff [3]. In practice, many processes act to remove heat from the plasma. In a cold post-disruption plasma, line radiation and bremsstrahlung from interactions with partially ionized impurities are important loss channels, as is radial heat transport. Including a heat sink in numerical simulation of such scenarios is therefore desirable, and the sink effectively acts to delay or prevent the transition to slide-away. In this paper, we demonstrate that the evolution of the runaway electron population – including the time to reach slide-away – is highly sensitive to the properties of the applied heat sink, making a detailed investigation of the various loss channels an area of interest for future work. 2. NORSE We will use the newly developed fully relativistic non-linear tool NORSE [3] to study the dynamics of the electron population. NORSE, which is valid in spatially uniform plasmas, solves the kinetic equation eE ∂f ∂ ∂f − · + · (Fs f ) = Cee {f } + Cei {f } + S, (1) ∂t me c ∂p ∂p where f is the electron distribution function, t is the time, me and p are the electron rest mass and momentum, E is the electric field, c is the speed of light, Fs is the synchrotronradiation-reaction force, Cee is the relativistic non-linear electron-electron collision operator, Cei is the electron-ion collision operator, and S represents heat and particle sources or sinks. For a detailed description of the various terms and operators, see Ref. [3]. For the remainder of this paper, we define the electric field such that electrons are accelerated in the positive pk direction. In NORSE, the particle momentum p is represented in terms of the magnitude of the normalized momentum p = γv/c (where v is the velocity of the particle and γ is the relativistic mass factor) and the cosine of the pitch angle ξ = pk /p. The kinetic equation is discretized using finite differences in both p and ξ. A linearly implicit time-advancement scheme is used, where the five relativistic Braams-Karney potentials [4] – analogous to the Rosenbluth potentials in the non-relativistic case – are calculated explicitly from the known distribution. These are then used to construct the electron-electron collision operator Cee , and the remainder of the kinetic equation is solved implicitly. 2 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 For the results presented in this paper, a non-uniform finite-difference grid was used to improve the computational efficiency. In the pitch-angle coordinate, the grid points were chosen with a dense spacing close to ξ = ±1 – in particular at ξ = 1 where the runaway tail forms – but with a sparser grid at intermediate ξ. In the p-direction, a grid mapping with a tanh step in spacing was used in order to produce a grid with dense spacing at low momenta (to accurately resolve the bulk dynamics), and larger spacing in the high-energy tail, where the scale length of variations in f is larger. The runaway region in NORSE was determined by studying particle trajectories in phase space, neglecting momentum-space diffusion but including self-consistent collisional friction and synchrotron-radiation reaction [3]. The trajectory that terminates at ξ = 1 and p = pc marks the lower boundary of the runaway region, since particles that follow it neither end up in the bulk nor reach arbitrarily high energies. The critical momentum pc is the momentum at which the balance of forces in the parallel direction (ξ = 1) becomes positive (i.e. the lowest momentum at which the accelerating force of the electric field overcomes the collisional and synchrotronradiation-reaction drag). If the balance of forces is positive for all p at ξ = 1, however, all electrons experience a net acceleration, and the population is in the slide-away regime. 2.1. Heat sink Including a heat sink (HS) in the numerical simulations is of great importance for accurate modelling of the distribution evolution during a disruption. The heat sink used in NORSE to remove heat from the thermal population has the form S = ∂/∂p · kh Sh f , where Sh (p) is an isotropic function of momentum (i.e. Sh k p) and kh (t) is the magnitude of the source. The terms in the kinetic equation that affect the total energy content are the electric-field and synchrotronradiation-reaction terms, however; when considering a subset Ω of momentum space, collisions can also transfer energy in or out of Ω, and a corresponding term must be included. The total energy change dW/dt in Ω can thus be written as Z eE ∂f ∂ ∂ dW = me c2 d3 p (γ − 1) − · + · (Fs f ) − C{f } + kh · (Sh f ) . (2) dt me c ∂p ∂p ∂p Ω The magnitude kh of the sink in each time step can be determined by requiring dW/dt = 0. In this work, we take Ω to represent the pthermal bulk of the distribution, which we define as all particles with v < 4vth,0 , with vth,0 = 2T0 /me the thermal speed at the initial temperature T0 . In this study, the p-component of Sh was chosen to have the shape of a Maxwellian at the desired temperature T . In practice, the momentum dependence of the sink will be more complicated and subject to the details of the particular physical processes at work. It will also likely have a limited energy-removal rate, dictated by for instance spatial gradients or impurity content, which could limit its efficiency in maintaining a given temperature. A detailed investigation of the characteristics of the sink is left for future work; the aim of this paper is to highlight the sensitivity of the runaway-electron evolution to the particulars of the sink, and for that purpose we will impose a limit on the energy-removal rate, as will be discussed in the next section. 3. Runaway generation in an ITER disruption 3.1. Post-disruption scenario In this section, we use NORSE to model the evolution of the electron distribution during a typical ITER disruption. The electric field evolution (which is shown in Fig. 1a) and other parameters are taken from the ITER inductive scenario no. 2 in Ref. [8], but the temperature evolution has been simplified to facilitate the numerical calculation. We assume the electron population to be completely thermalized and use the final temperature T = T0 = 10 eV throughout our 3 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 Figure 1. a) Electric field in V/m (left vertical axis) and normalized to the Dreicer field ED at the temperature T0 and density n0 (right vertical axis), as a function of time after the thermal quench. b) Tail of the parallel electron distribution. Thin lines show f at tN (no HS), tW (weak HS) and tS (strong HS), and thick lines show f immediately before the transition to slide-away. simulation, together with the density n = n0 = 7.1·1019 m−3 . This is likely to underestimate the runaway-electron generation in the early phase, in which the temperature is still dropping, however the chosen set-up is sufficient for our purposes. We use the magnetic field on axis (B = 5.3 T) and Zeff = 1. The initial current density in the scenario is j0 = 0.62 MA/m2 , however our calculations start from a Maxwellian distribution and make no attempt to maintain the experimental current evolution explicitly. To highlight the importance of the temperature evolution of the bulk, in Section 3.2 we will consider three scenarios: no heat sink (subscript N), weak heat sink (W) and strong heat sink (S). In the no-heat-sink scenario, all the energy supplied by the electric field will remain in the simulation, leading to rapid bulk heating; with the strong heat sink, a bulk temperature of T0 = 10 eV will be enforced in accordance with Eq. 2, i.e. any excess heat in the bulk region will be removed using a heat sink. In the intermediate case of a weak heat sink, the energy-removal rate of the heat sink will be restricted to 0.5 MW/m3 . This particlar value has been chosen at will, but is meant to represent some inherent limitation in the physical processes responsible for the energy loss. In both the weak and strong cases, the heat sink will affect only the thermal population, allowing the supra-thermal tail to gain energy from the electric field. Physically, this corresponds to processes not included in the simulation (such as radial transport or radiative losses) which primarily affect the thermal population. NORSE simulations of the evolution of the electron distribution function in the presence of the electric field in Fig. 1a were performed for the three different scenarios. The simulations were aborted when the runaway population reached nr /n = 1; i.e. a transition to the slide-away regime was observed. The simulation results can however only be considered characteristic of a natural ITER disruption for current densities comparable to, or somewhat larger than, the initial value j0 , since after that point the strong response of the inductive electric field to the increased local current would invalidate the E-field evolution used. We will therefore mark the time where the current density reaches j/j0 > 5 in all plots, and denote it with tN , tW and tS , respectively, for the no-sink, weak-sink and strong-sink scenarios. The distribution evolution at later times can only be considered accessible in scenarios where the loop voltage is actively sustained using the control system. Nevertheless, this regime will turn out to be of interest, since a non-linear feedback mechanism leading to a rapid transition to slide-away is observed. 4 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 Figure 2. a) Runaway fraction and b) current density normalized to its initial value j0 , as a function of time after the thermal quench in the different heat-sink scenarios. The times tN , tW and tS (vertical thin dashed lines), mark the time where the current density reaches j/j0 > 5 for the no-heat-sink, weak-heat-sink and strong-heat-sink scenarios, respectively. 3.2. Evolution of the runaway-electron population In Fig. 1b, the tails of the distributions in the parallel direction are shown at tN , tW and tS (thin lines), as well as at the final times (thick lines) in each scenario (just before the transition to slide-away is reached). In the figure, the distribution is normalized such that F = f /fM (t = 0, p = 0), where fM is a Maxwell-Jüttner distribution, so that F initially takes the value unity at p = 0. The maximum achieved particle energies are highly dependent on whether a heat sink was applied or not; in the no-heat-sink and weak-heat-sink scenarios, the particles did not have time to reach relativistic energies, whereas in the strong heat sink case, p ≈ 19 and p ≈ 44 (corresponding to energies of roughly 9 and 22 MeV), were obtained at tS and just before reaching slide-away, respectively. The reason for this is that, as we shall see, the current density growth and subsequent transition to slide-away in the latter case occur at much later times, and the runaways have time to gain more energy. Figure 2a shows the evolution of the runaway fraction during the course of the simulation. In the no-heat-sink case, the runaway fraction increases sharply, but as shown in Fig. 1b, the runaways are all at low energy. The transition to slide-away happens already at t= 5.7 ms; early on in the electric-field evolution (cf. Fig. 1). In the two scenarios employing a heat sink, the growth in runaway fraction occurs later, but in the weak-heat-sink case the growth rate is comparable to the case when a sink is absent once the process is initiated. In this case, the transition to slide-away happens at t = 6.7 ms. With the strong heat sink, the runaway population grows steadily, eventually dominating the entire distribution at t = 8.4 ms, however in this case the transition is gradual, rather than rapid. In all three scenarios, including the one with an ideal strong heat sink, the slide-away regime is thus reached even before the electric field (calculated assuming a linear treatment) has reached its peak. Figure 2b shows the evolution of the current density. It indicates that the rapid increase in the runaway fraction is correlated with a similar increase in the current density, although in the no-sink case, the growth rate is somewhat smaller. Again, the growth in the strong-heat-sink case is gradual, rather than explosive. Note that in the no and weak heat-sink cases, the runaway fraction is still negligibly small at the start of the rapid transition to slide-away. The transition is thus not a non-linear phenomenon triggered by the size of the runaway population; it starts in a regime where linearized tools are normally expected to be valid, and before the current density becomes significantly larger than its initial value. 5 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 Figure 3. a) Effective temperature of the bulk population and b) corresponding effective normalized electric-field strength (also taking changes to the bulk density into account) as functions of time, in the different scenarios. The black solid line in panel b) corresponds to the normalized field with T = T0 and n = n0 (cf. Fig. 1). The explanation can be found by examining the thermal population. By comparing the energy moment of the bulk of the distribution (WΩ ) with that of a relativistic Maxwellian (WM (T )), WΩ = me c2 Z Ω d3 p (γ − 1)f = WM = me c2 n Θeff K2 (1/Θeff ) Z 0 γ dp p2 (γ − 1) exp − , Θeff pmax,Ω (3) an effective temperature Teff for a given distribution f can be determined by solving for Θeff = Teff /me c2 . In the above equation, pmax,Ω is the upper boundary in p of the bulk region in momentum space, and K2 (x) is the modified Bessel function of the second kind (and order two). The effective temperature Teff is plotted in Fig. 3a as a function of time for all three scenarios. In the no-heat sink case, it increases by roughly two orders of magnitude during the simulation, i.e. when energy is not actively removed from the system, the Ohmic heating is sufficient to heat the plasma to a temperature of about 700 eV before the onset of slide-away, or 55 eV if the electric field is not artificially sustained. A similar (albeit weaker) tendency is seen in the weak-heat-sink case, where the temperature increases to Teff ≈ 210 eV in the phase leading up to the slide-away transition, or 25 eV in a non-driven case. This heating is a consequence of the imposed limited maximum energy-removal rate of the heat sink in the weak case, since the temperature is efficiently kept constant in the beginning of the simulation, where (dW/dt)HS < 0.5 MW/m3 . The strong heat sink manages to keep Teff − T0 to within a few tenths of an eV during the entire simulation, corresponding to a source with unlimited (or at least higher than required) maximal energy-removal rate. The significance of the observed bulk heating is its influence on the Dreicer field ED , which (apart from the weak dependence on ln Λ) is inversely proportional to T . For a given electricfield strength, the normalized field E/ED thus increases as the bulk heats up. The effective normalized electric field is shown in Fig. 3b, indicating that the rapid growth in the runaway fraction in Fig. 2a is correlated with a sudden increase in the normalized electric field in the noheat-sink and weak-heat-sink cases. In the strong-heat-sink case, the increase in normalized field is not caused by the temperature, which is kept constant during the entire simulation, but by the decrease in the bulk density as the runaway population becomes substantial. As can be seen from the yellow dash-dotted line in the figure, this has a similar effect as a temperature increase, since ED ∼ nbulk . The effective E/ED starts to deviate from the baseline value (black solid line) 6 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 Figure 4. a) Evolution of the bulk of the electron distribution and b) balance of forces, in the direction parallel to the electric field, just before the transition to slide-away in the weak-heatsink scenario. t̂ is the time relative to the transition to slide-away in µs. already when nbulk /n0 ≈ 0.97. The feedback process is thus initiated when the runaway fraction is just 3%; a regime where linear tools are expected to be valid. The two effects of increasing Teff and decreasing nbulk lead to a positive feedback mechanism which is responsible for the rapid growth in runaway fraction and current density seen in the no-heat-sink and weak-heat-sink cases. Once the bulk temperature has increased enough (or a high enough runaway tail is produced, as seen in the strong-heat-sink case), the normalized electric field becomes strong enough to cause a depletion of the bulk through primary runaway generation. The reduced bulk density in turn leads to a more efficient heating of the remaining bulk particles. Both of these effects contribute to a reduction in the Drecier field ED , and a corresponding reduced collisional friction on the bulk electrons, which makes runaway acceleration easier. This further increases the rate of bulk depletion, and so on. Eventually the friction becomes low enough that the parallel balance of forces becomes positive everywhere, marking the transition to the slide-away regime. At this point, the bulk of the distribution can no longer be well described by a Maxwellian and the positive feedback mechanism makes the transition possible even though E/ED,eff < 0.215 (in this case at around E/ED,eff ≈ 0.15). The bulk depletion and associated change in the parallel force balance is shown in Fig. 4a and Fig. 4b, respectively, in the phase leading up to the transition to slide-away in the weakheat-sink case. Initially, the force balance is positive in the tail – i.e. particles there experience a net acceleration – while it is negative in the bulk, meaning particles are slowed down by collisions. As the feedback process starts, the minimum in the sum of forces becomes gradually less pronounced, in tandem with the depletion of the bulk population, up until the point where the sum of forces becomes positive everywhere, and the slide-away regime is reached. This highly non-linear process cannot be accurately captured by a linear model. 4. Discussion and conclusions In this paper we have examined the evolution of the electron distribution and runaway generation in an ITER-like post-disruption scenario where the electric fields reach values as high as 90 V/m. With the help of the newly developed tool NORSE, which is a relativistic non-linear solver for the electron momentum-space distribution function, we have shown that the slide-away regime, i.e. a net parallel acceleration of electrons in all of momentum space, is reached in this scenario, provided the electric field evolution used is artificially enforced by the control system. In the stage leading up to the transition, a positive feedback mechanism sets in by means of which the bulk quickly gets depleted by primary runaway generation which reduces the friction on the 7 Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas 2016 IOP Publishing Journal of Physics: Conference Series 775 (2016) 012013 doi:10.1088/1742-6596/775/1/012013 thermal population, leading to further bulk depletion, until the point where the slide-away is reached. This process can be initiated at significantly weaker fields than the slide-away field ESA expected from linear theory. The time to a transition to slide-away is highly dependent on the ability of loss processes to remove heat from the thermal electron population, but even with an ideal sink (the strongheat-sink case in Section 3), complete runaway generation was seen 8.4 ms after the thermal quench. These results were obtained without taking avalanche or hot-tail runaway generation into account, which would only lead to more prominent runaway growth. Also in the case of a disruption where the electric field is not artificially sustained, strong bulk heating leads to a rapid growth in the runaway fraction because of the increase in the normalized electric field E/ED , but the current density becomes large enough to significantly affect the electric field evolution (supressing the growth of E) before the slide-away regime is reached. This is observed in the absence of a heat sink, as well as with a heat sink with a limited maximum energy-removal rate (in this case 0.5 MW/m3 ). If the efficiency of the heat sink is not limited, the runaway fraction grows more slowly and the runaways have time to reach significantly higher energies before the electric field becomes affected by the growing current density. The severity of disruptions in ITER could thus be greatly affected by the properties of the heat sinks present in the plasma. The feedback mechanism described in this paper has important consequences for the understanding of runaway-electron dynamics. With the entire electron population experiencing a net accelerating force at much weaker electric fields than previously expected, very large runaway-electron current generation is likely. This would impact the subsequent electricfield evolution, leading to a reduction in field strength and duration which could occur at realatively early times if the heat sink has a limited energy removal rate. Therefore, it is difficult to determine the magnitude of the effect on the current evolution and post-quench dynamics without a self-consistent calculation of the electron distribution and the electric field. Nevertheless, this paper shows that feedback effects play an important role in post-disruption runaway dynamics, and that the details of the heat-loss channels may have a big impact on what strength and duration of electric field can be tolerated before the positive feedback, and possible subsequent transition to slide-away, is induced. Acknowledgments This work was supported by the Swedish Research Council (Dnr. 2014-5510), the European Research Council (ERC-2014-CoG grant 647121), and the Knut and Alice Wallenberg Foundation. M.L. was supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Science, under Award Numbers DE-FG02-93ER54197 and DE-FC02-08ER54964. 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