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Master’s Thesis
Parametric Decay and Anomalous
Scattering from Tokamak Plasmas
Søren Kjer Hansen
s113079
Supervisors: Stefan Kragh Nielsen and Mirko Salewski
Section for Plasma Physics and Fusion Energy
Department of Physics
Technical University of Denmark
25 June 2016
Abstract
In this thesis we investigate the parametric decay instability by which a high-powered
electromagnetic (pump) wave in the microwave/millimetre wave frequency range may be
converted into two electrostatic daughter waves, one having a frequency comparable to that
of the pump wave and one having a frequency much lower than that of the pump wave,
with the sum/difference of the two daughter frequencies being equal to the pump frequency.
The above type of instability may, unsurprisingly, be important in many experiments
involving high-power microwave/millimetre wave radiation. The present work is motivated
by observation of strong scattering, dubbed anomalous, during some collective Thomson
scattering (CTS) experiments aiming at determining the parameters of high-temperature
tokamak plasmas, which are currently the leading contenders for realising fusion based
power plants, by means of a much weaker scattering process. The observed anomalous
scattering spectra have several features attributable to parametric decay instabilities, most
notably peaks close to the pump wave frequency possessing characteristic frequency shifts
and occurring when particular plasma resonances become accessible to the pump wave.
The parametric decay instability which has been investigated, both analytically and numerically, in this work is one in which the electromagnetic pump wave decays into a
high-frequency electron Bernstein wave and a low-frequency lower hybrid wave, causing
a frequency shift very similar to the one observed in anomalous scattering spectra from
CTS experiments at the ASDEX Upgrade tokamak. The above parametric decay instability has previously been invested theoretically by [Porkoláb, 1982] and observed in the
Versator II tokamak by [McDermott et al., 1982]; in this work, we have generalised the
analytical results obtained by these authors somewhat. We have further investigated the
parametric decay instability numerically in ASDEX Upgrade shot 28286, where an electromagnetic beam power threshold of 11.95 kW, which is exceeded by a large margin in
the experiments, was found, along with a frequency shift 837.5 MHz of the high frequency
daughter wave involved in the instability relative to the pump wave frequency. The above
frequency shift agrees well with the frequency shift of (880 ± 50) MHz for the anomalous
scattering peak observed in the experiment, indicating that anomalous scattering in the
CTS experiments at ASDEX Upgrade may very well be caused by the above parametric
decay instability.
i
Dansk Resumé
I denne afhandling undersøges den parametriske henfaldsinstabilitet ved hvilken en kraftig
elektromagnetisk bølge (pumpe) i mikro-/millimeterbølgefrekvensområdet omdannes til
to elektrostatiske datterbølger. Den ene af disse har en frekvens, der er sammenlignelig
med pumpens, mens den anden har en frekvens, der er langt lavere end pumpens; summen af/forskellen imellem de to datterbølgefrekvenser svarer til pumpens frekvens. Det
er ingen overraskelse, at den ovennævnte instabilitetstype muligvis er vigtig i forsøg, som
involverer kraftig mikro-/millimeterbølgestråling. Motivationen bag denne afhandling er
observationer af kraftig spredning (omtalt som anomal) i visse eksperimenter, hvor parametrene relateret til højtemperatur tokamak-plasmaer, der i øjeblikket er grundlag for de mest
lovende forsøg på at virkeliggøre fusionskraftværker, forsøges at målt vha. den langt svagere
kollektive Thomson spredningseffekt (CTS). De observerede anomale spredningsspektre
har flere karakteristika, som kan tilskrives parametriske henfaldsinstabiliteter, først og
fremmest toppe tæt på pumpens frekvens med karakteriske frekvensskift, som fremkommer, når bestemte plasmaresonanser bliver tilgængelige for pumpen.
I denne afhandling har vi undersøgt den parametriske henfaldsinstabilitet ved hvilken
den elektromagnetiske pumpe henfalder til en højfrekvent elektron-Bernstein-bølge og en
lavfrekvent nedre-hybrid-bølge, som giver et frekvenskift, der er meget tæt på det, som observeres i forbindelse med de anomale spredningssprekre fra ASDEX Upgrade-tokamaken.
Den ovennævnte parametriske henfaldsinstabilitet er tidligere blevet undersøgt teoretisk af
[Porkoláb, 1982] og observeret i Versator II-tokamaken af [McDermott et al., 1982]; i denne
afhandling er de analytiske resultater, først præsenteret af de ovennævnte forfattere, blevet
gjort en smule mere almene. Vi har yderligere undersøgt den parametriske henfaldsinstabilitet numerisk i ASDEX Upgrade skud 28286 og fundet en effektgrænse på 11,95 kW,
der som regel er langt overskredet i eksperimenterne, samt et frekvensskifte på 837,5 MHz
for den tilhørende højfrekvente datterbølge ift. pumpen. Dette frekvensskifte stemmer fint
overens med frekvensskiftet på (880 ± 50) MHz ift. pumpen, som observeres i det anomale
spredningsspektrum i forsøget, hvilket indikerer, at anomal spredning i CTS-eksperimenter
ved ASDEX Upgrade med stor sandsynlighed kan tilskrives den ovennævnte parametriske
henfaldsinstabilitet.
ii
Preface
The present thesis is submitted as fulfilment of the prerequisites for obtaining the Master
of Science and Engineering degree at the Technical University of Denmark (DTU). The
work has been carried out at the Department of Physics (DTU Physics) with the Section
for Plasma Physics and Fusion Energy (PPFE) headed by Professor Volker Naulin. The
work on this 35 ECTS point thesis was carried out from 4 January 2016 to 25 June 2016.
There are several people whom I should like to thank for their help and support during
this project. First of all, I should like thank my supervisors Stefan Kragh Nielsen and
Mirko Salewski for their continual interest in my project and details pertaining to it, and
for providing me with the experimental background often missing in theoretical studies of
parametric decay instabilities; in addition to this, I am grateful to them for introducing
me to both theoretical and experimental authorities in the tokamak community, as well as
for their comments on various drafts of this thesis which improved the presentation significantly. I also thank Professor Jens Juul Rasmussen for providing useful references on the
non-resonant parametric decay instability and Professor Evgeniy Z. Gusakov for providing useful comments on the project in its mid stage. Further, I gratefully acknowledge a
stay at the Max Planck Institute for Plasma Physics (IPP) in Garching, Germany, under
the auspices of the EUROfusion MST-1 Programme. At IPP, I should particularly like to
thank Severin S. Denk, Alf Köhn, and Professor Jörg Stober for their work in support of,
and interest in, my project.
On a less formal level, I should like to thank my officemates at various times Emma Goos,
Asger Schou Jacobsen, Michael Løiten Magnussen, Aske Anguasak Olsen, Jeppe Miki Busk
Olsen, Bernhard Schießl, and Alexander Simon Thrysøe, whose thirst for coffee always
provided a convenient excuse for taking a break, as well as the members and supporters
of Club 47, particularly my co-founders Kristoffer "Kakao" Bitsch Joanesarson and Mads
Givskov Senstius, whose thirst for beer, combined with my own, may have hampered
progress on this project slightly. Finally, I thank my family for their love and support
during this project and my life in general to this point.
Søren Kjer Hansen
Section for Plasma Physics and Fusion Energy
Department of Physics
Technical University of Denmark
25 June 2016
iii
Contents
List of Figures
vi
1 Introduction
1
1.1
Fusion Research and the Tokamak Concept . . . . . . . . . . . . . . . . . .
1
1.2
Basics of Parametric Decay . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Parametric Decay in Plasma Physics . . . . . . . . . . . . . . . . . . . . . .
7
2 Kinetic Theory of Parametric Decay
10
2.1
Motion of an Electromagnetically Driven Plasma Particle . . . . . . . . . . 10
2.2
Fundamental Kinetic Theory of Parametric Decay . . . . . . . . . . . . . . . 11
2.3
The Parametric Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . 15
2.4
Growth Rate of the Parametric Decay Instability . . . . . . . . . . . . . . . 18
3 Parametric Decay near the Upper Hybrid Resonance
21
3.1
Cold Theory of High-Frequency Electromagnetic Plasma Waves . . . . . . . 21
3.2
Electrostatic High-Frequency Daughter Waves at the Upper Hybrid Resonance 29
3.3
Electrostatic Low-Frequency Daughter Waves . . . . . . . . . . . . . . . . . 34
3.4
Parametric Decay into Electron Bernstein and Lower Hybrid Modes . . . . 38
4 Wave Propagation and Parametric Decay in Tokamak Plasmas
46
4.1
Geometric Optics and Simple Theory of Wave Amplification . . . . . . . . . 46
4.2
Advanced Wave Amplification: Full-Wave, WKB, and Hybrid Approaches . 54
4.3
Parametric Decay in Inhomogeneous Plasmas . . . . . . . . . . . . . . . . . 60
4.4
Numerical Investigations of Parametric Decay in ASDEX Upgrade . . . . . 66
5 Conclusions and Outlook
71
Bibliography
74
iv
List of Figures
1.1
Diagrammatic representation of parametric three-wave processes . . . . . .
6
1.2
Idealised parametric decay (angular) frequency power spectrum . . . . . . .
6
1.3
Semi-logarithmic contour plot of the CTS signal around ω0 /(2π) = 105 GHz
versus t and ω/(2π) in ASDEX Upgrade shot 28286 . . . . . . . . . . . . . .
9
Plot of the CTS signal versus gyrotron power P0 in ASDEX Upgrade shot
32563 around t = 4.6 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.4
3.1
CMA diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2
CMA contours for ω0 /(2π) = 105 GHz in ASDEX Upgrade shot 28286 at
t = 2.900 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3
Real part of the Fried-Conte plasma dispersion function, Z(ξ), along with
asymptotic approximations, for 0 < ξ < 5 . . . . . . . . . . . . . . . . . . . 32
3.4
2 v 2 ) from Eq. (3.43) . . . . . . . . . . . . . . . . . . 41
Contour plot of ω12 /(k⊥
Ti
√ 3 −ω2 /(k2 v2 ) 2 3
2 v 2 ) from Eq. (3.43) 41
Contour plot of 2 πω1 e 1 ⊥ T i /(k⊥ vT i ) with ω12 /(k⊥
Ti
3.5
4.1
Accessibility to the upper hybrid layer for ω0 /(2π) = 105 GHz in ASDEX
Upgrade shots 28286 and 32563 . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2
Semi-logarithmic plot of the field amplification for the reflected ω0 /(2π) =
105 GHz X-mode radiation in ASDEX Upgrade shot 28286 at t = 2.900 s . . 59
4.3
Plot of the field amplification for the reflected ω0 /(2π) = 105 GHz X-mode
radiation in ASDEX Upgrade shot 32563 at t = 4.500 s . . . . . . . . . . . . 59
4.4
Semi-logarithmic plot of the power threshold and the field amplification for
the reflected ω0 /(2π) = 105 GHz X-mode radiation in ASDEX Upgrade shot
28286 at t = 2.900 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5
Frequency of the low-frequency daughter modes in ASDEX Upgrade shot
28286 at t = 2.900 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6
Contour plot of γ(r(s2 ), k(s2 ), ω2 )/|vg (r(s2 ), k(s2 ), ω2 )| versus s2 and ω1 /(2π)
for P0 = 500 kW in ASDEX Upgrade shot 28286 at t = 2.900 s . . . . . . . . 69
v
4.7
Numerically obtained gyrotron power threshold P0,th versus ω1 /(2π) in ASDEX Upgrade shot 28286 at t = 2.900 s . . . . . . . . . . . . . . . . . . . . 69
vi
Chapter 1
Introduction
In this chapter we introduce the basic concepts and motivation behind the more formal discussions of the subsequent chapters. First, we review the current state of fusion research
with particular attention to magnetic confinement and the tokamak concept. Then, we
consider the basic properties of parametric decay, introducing the generic features of parametric processes, and, finally, we specialise to parametric decay in the context of plasma
physics and present the motivation for this work in detail.
1.1
Fusion Research and the Tokamak Concept
Fusion of the nuclei of light elements, to produce heavier ones, is one of the most important basic processes taking place in the universe. The fusion reactions that occurred in the
interior of early stars are responsible for producing all elements heavier than helium, without which Earth and life as we know it would not exist. The strongly exothermic nature
of light element fusion, due to the general increase of binding energy (smaller mass, by
Einstein’s mass-energy equivalence) per nucleon for elements with higher atomic numbers
(upto iron, 26), also accounts for the power radiated by stars; the fusion reactions taking
place in the solar interior provide the vast majority of the power necessary to sustain life
on Earth. With this in mind, it is not too difficult to envisage an earthbound fusion based
power plant. However, as indicated by the above examples, fusion reactions only occur at
a significant rate under extreme conditions resembling those of a stellar core, at least in
terms of temperature, since large particle energies are required to overcome the electrostatic repulsion between the involved nuclei before the short-ranged, but much stronger,
attractive nuclear forces cause fusion to occur. This means that the realisation of a fusion
based power plant is intimately connected with high-temperature plasma physics.
The goal of producing power by means controlled nuclear fusion has thus been driving
research in plasma physics ever since World War II. The practical reason for wanting to
realise fusion as an earthbound energy source is its many favourable aspects compared with
currently existing alternatives. First of all, the operation of a fusion power plant would not
1
be associated with the production of harmful emissions, such as carbon and sulphur dioxide.
Second, the fuel necessary to run a fusion power plant consists of light elements which are
present in, or may be bred from, compounds that are abundant on the surface of Earth,
and has an energy density (by mass) higher than or comparable to that of uranium, thus
providing resources to maintain current levels of power consumption for billions of years at
affordable prices. Third, the output of a fusion power plant would be easily regulated, and
since the fusion process is essentially a (very) high temperature combustion process, not
relying on chain reactions etc., a fusion power plant would not be at risk of a melt-down in
case of an accident. Finally, the waste generated in connection with a fusion power plant
would be low-radioactive and safe to handle/recycle within 100 years, thus eliminating the
need for long term storage facilities [Freidberg, 2007]. The main hurdle to be overcome,
in order to develop a commercially viable fusion reactor, is the problem of creating and
maintaining the very high temperature plasma, necessary for a significant number of fusion
reactions to occur, in a sufficiently effective way to extract energy from the process; so far,
the necessary conditions have only been achieved regularly in the centre of atomic bomb
explosions, slightly discouragingly, making the hydrogen bomb the "practical" earthbound
application of fusion to date.
The process which has attracted most attention in connection with fusion research is the
reaction of the hydrogen isotopes deuterium (hydrogen-2) and tritium (hydrogen-3) producing helium-4 and a free neutron, as well as liberating 17.6 MeV of energy per reaction.
The reason why this particular reaction has received so much attention is its very large
cross-section at relatively low collision energies, translating to relatively low temperatures
necessary for it to occur, compared with all other thermonuclear reactions; even so, the
optimal temperature for the deuterium-tritium reaction to occur is 15 keV or 1.7 × 108 K
[Freidberg, 2007]. There are two main schemes by which these temperatures may be
reached: inertial and magnetic confinement.
Inertial confinement is essentially similar to the scheme used in a hydrogen bomb: a pellet of frozen deuterium and tritium is compressed by means of high-powered radiation,
lasers in the case of inertial confinement fusion and X-rays generated by the atomic bomb
explosion in case of a hydrogen bomb, raising its temperature and density to the point
where fusion reactions start occurring at a significant rate. In order to ensure a high gain
from inertial confinement fusion, the pellet needs to be compressed in a very symmetric
fashion, which is achieved by placing it in a carefully engineered cavity (Hohlraum) illuminated by a large number of carefully aligned high-powered lasers (in a hydrogen bomb
the problem is solved using the Teller-Ulam design, the details of which remain classified).
The quantity determining whether or not ignition is achieved in inertial confinement fusion is directly proportional to the pellet radius, favouring large bomb-like explosions over
the small controlled explosions desired in a power plant. In spite of this, a net gain in
controlled experiments, compared with the amount of power absorbed by the pellet, has
been reported recently by researchers from the National Ignition Facility at the Lawrence
Livermore National Laboratory in the United States of America [Hurricane et al., 2014];
while an impressive technical feat and an important step, allowing the physics of ignited
fusion plasmas to be studied, we note that there are several fundamental problems which
2
remain to be solved in order for a power plant based on inertial confinement fusion to be
feasible. These problems are mainly related to the inherently pulsed nature of the inertial
confinement scheme, i.e., the need to compress a significant number of pellets per second
in order to have a steady power supply. First of all, the lasers compressing the pellets have
efficiencies on the order of a few percent, and only a fraction of the laser power is ultimately
absorbed by the pellet, requiring the energy produced by the fusion reactions relative to
the laser power absorbed by the pellet to be increased by several orders of magnitude
in order for the process to produce net power. Second, the lasers currently available are
far from capable of delivering ignition pulses at the rate necessary for steady state power
output. Third, even if the above issues are resolved, the cost and manufacturing time of
the carefully engineered cavities, required to ensure even compression of the pellet and
destroyed each time a pellet is ignited, need to be reduced by many orders of magnitude in
order for inertial confinement fusion energy to be economically viable. Thus, while inertial
confinement fusion may provide useful insight into the physics of ignited fusion plasmas,
it seems that its pulsed nature is quite undesirable for a power plant.
Magnetic confinement provides the possibility of generating steady state ignited fusion
plasmas, although pulsed schemes such as magnetised target fusion also exist. The basic
principle of magnetic confinement is that charged plasma particles will, in the first approximation, gyrate around magnetic field lines, and thus closed magnetic field lines in a
vacuum vessel may be used to confine a low-density fusion grade plasma with a relatively
small energy/particle flux to walls. However, there are some complications since particles
moving in an inhomogeneous magnetic field, which is unavoidable if closed field lines are
to exist inside a vacuum vessel, will slowly drift across the magnetic field and, unless these
drifts are handled properly, the plasma confinement will not be sufficient to maintain a
fusion grade plasma. Two basic magnetic confinement fusion concepts, using different approaches to solve the basic drift problem, exist: the stellarator concept and the tokamak
concept.
The stellarator concept was developed at the Princeton Plasma Physics Laboratory in the
United States of America in the 1950s and acts to solve the drift problem through an external magnetic field configuration which acts to average out particle particle drifts, so as to
keep plasma particles confined. Stellarators, consequently, tend to have quite complicated
geometries, with discrete symmetries at most, and early stellarators had relatively poor
confinement, suffering from suboptimal magnetic field configurations due to the limited
computing available for designing them. While this has since improved, the complicated
geometry of stellarators still leads to them being very difficult to manufacture and requires
stellarators using super conducting coils to be cooled and heated at extremely slow rates in
order to avoid straining the materials used, which would destroy the very carefully designed
external magnetic fields. While these problems may eventually be resolved, stellarator development still lacks somewhat behind that of tokamaks, which we shall consider next.
The largest stellarator experiment currently in existence is Wendelstein 7-X located at the
Greifswald branch of Max Planck Institute for Plasma Physics in Germany, which started
operations in December 2015.
The tokamak concept was developed in the Soviet Union in the 1950s, and acts to solve
3
the drift problem by averaging out the particle drifts using magnetic fields generated by
currents in the plasma [Wesson, 2004]. Tokamaks have relatively simple toroidal geometries, which are favourable compared with those of stellarators in terms of manufacturing
complexity, as well as in terms of the geometry related complications of the physics. However, the need to drive currents in tokamaks is evident from the fact that the background
magnetic field, by Ampère’s law and the toroidal geometry, will be proportional 1/R and
can only have a ϕ-component (in a cylindrical coordinate system characterised by a radial coordinate R, an azimuthal angle ϕ, and a z-coordinate, with the z-axis being the
symmetry axis of the torus), which does not allow drifts to be averaged out without an additional magnetic field component originating from a ϕ-directed plasma current. This is no
problem in short pulses, where the plasma may be heated and current driven by inductive
effects resulting from changing the external magnetic field, but for the long steady state
pulses desired in a fusion power plant it is clear that this will be effective, especially since
inductive (Ohmic) heating becomes very small due to the high plasma conductivity once
the temperature exceeds a few keV [Freidberg, 2007]. While much of the necessary current
may be provided by so-called bootstrap currents, which can be optimised by engineering
the tokamak density profiles [Freidberg, 2007], it is still necessary to drive a significant
amount of current externally and this, along with heating of the plasma to temperatures
necessary for a significant amount of fusion reactions to occur, may be accomplished using the fact that the charged plasma particles, electrons and ions, may interact strongly
with electromagnetic radiation of the right frequency [Freidberg, 2007]. In this work we
shall mostly be interested in interaction of tokamak plasmas with relatively high-frequency
∼ 100 GHz electromagnetic radiation, generated by so-called gyrotron sources, and which
tends to interact strongly with electrons in tokamaks while its frequency is too high for
the ions respond appreciably to it.
Since this work is mainly concerned with tokamak applications a few words on the tokamak
theory and nomenclature are in order. The basic magnetohydrodynamic theory of tokamaks is described in a number of books, e.g., [Wesson, 2004], [Bellan, 2006], [Freidberg,
2007], and [Mazzucato, 2014], and while we shall not consider it in detail here it will be
used implicitly in many of the tokamak applications in this work. One main result of the
theory, which we shall quote, is that plasma particles tend to stay on surfaces of constant
magnetic flux the torus, ψ; there will generally be a number such flux surfaces which are
closed, where plasma may exist, and a number which are open, where the plasma will
eventually be lost etc. The surface separating these regions is known as the last closed flux
surface, and is often considered the boundary between the (real) plasma and the region
outside the plasma known as the scrape-off layer; often one also sets ψ = 1 at this surface,
then ψ < 1 in the plasma and ψ > 1 in the scrape-off layer. Due to the approximate
1/R-dependence of the magnetic field strength, the side of the vacuum vessel/plasma with
large R-values is known as the low-field side, while the side with small values is known as
the high-field side.
The largest tokamak currently in existence is JET (Joint European Torus) located at the
Culham Centre for Fusion Energy in the United Kingdom. The next generation fusion research reactor ITER (International Thermonuclear Experimental Reactor), currently under
4
construction near Cadarache in France, is also a tokamak. In this work we are mostly concerned with results obtained from the ASDEX (Axially Symmetric Divertor Experiment)
Upgrade tokamak located at the Garching branch of the Max Planck Institute for Plasma
Physics in Germany. For reference we note that plasma discharges in ASDEX Upgrade
are characterised by a shot number, e.g., 28286 and 32563, as well as a time t, which is
calculated from the shot start. The reconstructed plasma parameter profiles (equilibria) in
ASDEX Upgrade may be calculated using a number of different diagnostics in place around
the plasma; in this work, we have used Thomson scattering (TS) equilibria for older shots
and integrated data analysis (IDA) equilibria, reconstructed by collecting data from many
diagnostics, for newer shots.
1.2
Basics of Parametric Decay
The parametric decay instability is a fundamental nonlinear (three-wave) process in which
an externally applied pump wave of large amplitude decays into two daughter waves.
Parametric decay and general parametric processes occur in many branches of physics and
engineering. The simplest example of a parametric process is the parametric resonance from
mechanics. In this problem a spring has its resonance frequency, spring parameter, hence
the term parametric, modulated periodically by a "pump". Once the pump amplitude
exceeds a certain threshold, depending on the mean resonance frequency, pump frequency,
and damping rate, oscillations at half the pump frequency become unstable [Landau and
Lifshitz, 1969]. A more familiar related problem is that of a child, of any age, propelling
itself on a swing by changing the location of its centre of mass (pumping) at twice the
frequency of the swinging motion [Swanson, 2003].
The above examples contain the essential features of all parametric instabilities: an instability threshold depending on the amplitude of the pump, the strength with which the
system parameters are modulated, and selection rules related to (angular) frequencies, ω,
and wave vectors, k, obeyed by the excited modes. The selection rules may be understood
from conservation of energy and momentum by the photons/phonons/plasmons involved
in the three-wave process [Sagdeev and Galeev, 1969], since each of these is associated
with energy ~ω and momentum ~k; ~ = 1.055 × 10−34 J · s is the reduced Planck constant.
Fig. 1.1 shows the main parametric three-wave processes: decay of the pump wave, with
parameters ω0 , k0 , into a low-frequency daughter wave, with parameters ω1 , k1 , and a
high-frequency daughter wave, with parameters ω2 , k2 , as well as, recombination of the
pump wave and the low-frequency daughter wave to excite a high-frequency
daughter wave,
P
with
P parameters ω3 , k3 ; in the two cases conservation of energy, j ~ωj , and momentum,
j ~kj , yield the selection rules
ω2 = ω0 − ω1 ,
k2 = k0 − k1 ; ω3 = ω0 + ω1 , k3 = k0 + k1 .
(1.1)
We note that processes involving recombination of the pump and high-frequency waves,
recombination of the low-frequency and high-frequency waves, as well as more than three
waves, also occur, but these are higher-order effects. For the mechanical examples above
5
Figure 1.1 – Diagrammatic representation of the
most important parametric three-wave processes;
to the left, decay of the
pump wave (ω0 , k0 ) into
a low-frequency (LF) wave
(ω1 , k1 ) and a down-shifted
high-frequency (HF) wave
(ω2 = ω0 − ω1 , k2 = k0 −
k1 ); to the right, recombination of the pump and
LF waves to excite an upshifted HF wave (ω3 = ω0 +
ω1 , k3 = k0 + k1 ).
Figure 1.2 – Idealised (angular) frequency power spectrum excited by the
parametric decay instability. Apart from
the externally applied pump, blue line
around ω0 , an LF daughter peak, orange
line around ω1 , and HF daughter peaks,
one down-shifted, gold line around ω2 ,
and one up-shifted, purple line around
ω3 , occur. Peaks are generally observed
when ω1 , ω2 , and ω3 , coincide with linear
modes, the up-shifted peak is generally
weaker than the down-shifted.
only the frequency matching rule exists and the excited "waves" both have the same
frequency; in this case the up-shifted wave is also a higher-order effect [Landau and Lifshitz, 1969]. When dealing with inhomogeneous media it is usually necessary to interpret
the selection rules within a geometric optics/WKB (Wentzel-Kramers-Brillouin) framework where waves are treated as locally plane waves satisfying a local dispersion relation, since plane waves with well-defined ω and k are generally not true eigenmodes of
such systems. However, the basic interpretation of parametric processes in terms of threephoton/phonon/plasmon interactions is valid within a more general setting. This is evident
from the fact that spontaneous parametric down-conversion of high-intensity laser generated light, corresponding to the left process in Fig. 1.1, may be used to create entangled
photons, useful for fundamental tests of quantum mechanics (Bell’s inequality) and quantum "teleportation" protocols, as a direct consequence of the simultaneous generation of
two daughter photons from a single pump photon under the fulfilment of certain selection
rules [Agarwal, 2013].
An idealised version of the (angular) frequency power spectrum generated by the parametric decay instability is shown in Fig. 1.2. The nearly monochromatic pump wave (blue
line) gives rise to a low-frequency peak (orange line), a down-shifted high-frequency peak
(gold line), and an up-shifted high-frequency peak (purple line). Although the selection
rules in Fig. 1.1 allow a continuous spectrum to be excited, the response will generally
be very weak, and the instability threshold very high, unless the modes involved coincide,
6
or nearly coincide, with linear modes of the system; for this reason a power spectrum in
the presence of parametric decay will ideally look like Fig. 1.2 with peaks corresponding
to linear modes excited by the parametric decay instability. In the mechanical examples
considered earlier, the above effect is clearly manifested, as the lowest instability threshold
is obtained when the frequency of the decay oscillations is the mean resonance frequency
of the spring/swing [Landau and Lifshitz, 1969].
1.3
Parametric Decay in Plasma Physics
The parametric decay instability in plasma physics was first described theoretically by
[Silin, 1965], for an unmagnetised plasma. For a magnetised plasma, the first theoretical
description was given by [Aliev et al., 1966]; the main result of that article is similar to
one which we shall derive in Chapter 2. In the years following these early theoretical treatments, a number of experiments confirming the existence of parametric decay instabilities
in laboratory plasmas were performed, see [Amano and Okamoto, 1969] for a description
of some of these; the path of derivation followed in that article, as well as its inclusion of
collisional effects, also seems to have influenced later treatments, e.g., [Porkoláb, 1974] and
[Porkoláb, 1978], strongly. The fact that plasma inhomogeneities usually set the strictest
limit on the power necessary to excite parametric decay instabilities was pointed out by
[Rosenbluth, 1972], who also considered the parametric decay instability in connection with
inertial confinement fusion. During the 1970s and 1980s a large number of theoretical works
considering parametric decay instabilities in tokamaks, as well as in general magnetised
plasmas, occurred, e.g., [Porkoláb, 1974], [Berger et al., 1977], [Porkoláb, 1978], [Ott et al.,
1980], [Porkoláb, 1982], [Sharma and Shukla, 1983], [Murtaza and Shukla, 1984], [Stefan
and Bers, 1984], and [Kasymov et al., 1985]. Many of the parametric decay instabilities
described in these works were subsequently observed in tokamaks, e.g., by [McDermott et
al., 1982], [Takase et al., 1984], [Van Nieuwenhove et al., 1988], and [Rost et al., 2002]. In
1981, stimulated electromagnetic emission from the ionosphere, during experiments where
high-powered radio-frequency were launched into it, was further observed at the Heating
facility near Tromsø in Norway [Leyser, 2001]; this has since been attributed to parametric
decay instabilities occurring in the ionosphere, see [Murtaza and Shukla, 1984], [Leyser et
al., 1994], and [Leyser, 2001] for discussions/reviews.
The observations of parametric decay instabilities described above all occur when the
pump waves encounter a region where their electric fields are enhanced, for reasons described later in this work, since this is also relevant to the parametric decay instability
discussed in this work. Such regions are, as we shall see, often not easily accessible in
tokamaks and may be avoided even if they are. As parametric decay instabilities are undesired for most tokamak applications, this type parametric decay instability has received
relatively little attention within the tokamak community since the 1980s; the anomalous
scattering observations, which have motivated this work, do nonetheless serve as a reminder
that unexpected (nonlinear) phenomena may occur if regions of field enhancement become
accessible to high-powered radiation.
7
The parametric decay instabilities which have received most attention within the tokamak
community in recent years require the waves involved to be localised, since this may significantly reduce the threshold of the parametric decay instability in a inhomogeneous plasma.
The localisation can either occur if the waves are trapped around a minimum/maximum
of the plasma parameters [Rosenbluth, 1972] or if the pump wave is backscattered by the
parametric decay instability [Pesme et al., 1973]. Although the parametric decay instability threshold in the presence of localised waves was discussed by, e.g., [Ott et al., 1980]
and [Sharma and Shukla, 1983], more recent theoretical investigations tend to focus exclusively on this type of waves, see, e.g., [Gusakov and Surkov, 2007], [Gusakov and Popov,
2010], [Popov and Gusakov, 2015a], and [Popov and Gusakov, 2015b]. Experimental observations of the above type of parametric decay instabilities in tokamaks are provided
by the correlation of strong scattering of microwaves with the presence magnetic islands,
rotating local minima/maxima of the plasma parameters, reported by [Westerhof et al.,
2009] and [Nielsen et al., 2013]. These observations often rely on use equipment originally
designed for collective Thomson scattering (CTS) experiments, in which weak scattering of
a strong beam of radiation with a wavelength long enough to resolve the collective plasma
oscillations, usually in the gyrotron frequency range, may be used to infer many plasma
parameters not easily determinable by other means, e.g., the ion temperatures and fast ion
distribution functions, see [Froula et al., 2011] and [Kjer Hansen, 2014] for reviews of the
basic theory.
The motivation for this work is related to the CTS experiments themselves, as these sometimes give rise to strong scattering in situations where no such effects were initially expected, or where the origin of the strong scattering is atleast not fully understood. This
strong scattering has been termed anomalous scattering and is frequently observed in
CTS experiments at ASDEX Upgrade. An example of the anomalous scattering spectrum obtained in ASDEX Upgrade shot 28286 around the CTS gyrotron frequency of
ω0 /(2π) = 105 GHz, which has been filtered out of the shown data, is seen in Fig. 1.3.
The spectrum clearly contains a lot of noise which is to be expected when looking at a
saturated spectrum where nonlinear phenomena are of importance. However, in connection with the discussion of the parametric decay instability in Section 1.2, we note the two
lines appearing quite symmetrically symmetrically around the pump wave at frequency
shifts of ±(880 ± 50) MHz for t > 2 s; the frequency shift is determined using the accurate
pump wave frequency ω0 /(2π) = 104.93 GHz along with the frequency grid spacing of 100
MHz around them in Fig. 1.3. These peaks look remarkably similar to the high-frequency
part of Fig. 1.2. This should, further, be coupled with the fact that strong anomalous
scattering has a highly nonlinear dependence on the CTS gyrotron power P0 , being essentially zero until a particular threshold is exceeded, as is evident from Fig. 1.4, which shows
the result of modulating the CTS gyrotron power, P0 , in ASDEX Upgrade shot 32563
around t = 4.6 s: the signal is very small for P0 < 200 kW, but has increased significantly
once P0 > 300kW, as might be expected for a parametric decay instability with a power
threshold around this value. Finally, although not visible from Figs. 1.3 and 1.4, the occurrence of anomalous scattering seems to be strongly correlated with the accessibility to the
so-called upper hybrid layer for CTS gyrotron radiation reflected from the high-field side
8
Figure 1.3 – Semi-logarithmic
contour plot of the CTS signal
around ω0 /(2π) = 105 GHz versus t and ω/(2π) in ASDEX Upgrade shot 28286. Strong anomalous scattering is observed in
much of this shot, but the feature of most interest in this work
are the lines with frequency shifts
around ±(880 ± 50) MHz relative
to the gyrotron visible for t > 2 s.
Figure 1.4 – Plot of the CTS
signal versus gyrotron power P0
in ASDEX Upgrade shot 32563
around t = 4.6 s. By modulating P0 , a very small scattering signal is found for P0 < 200 kW,
but once P0 > 300 kW the signal strength has increased significantly, indicating a parametric decay power threshold between these
values.
wall of the tokamak vacuum vessel; the CTS gyrotron launches always take place from the
low-field side. This last accessibility fact, along with the frequency shift observed in Fig.
1.3 for the plasma parameters of ASDEX Upgrade, leads us to suspect that one particular
parametric decay instability of relevance to anomalous scattering in ASDEX Upgrade is
one in which the electromagnetic pump wave decays to a high-frequency so-called electron
Bernstein wave and a low-frequency so-called lower-hybrid near the upper hybrid layer,
as described by [Porkoláb, 1982] and observed previously on the Versator II tokamak by
[McDermott et al., 1982]. The remainder of this work is therefore dedicated to describing
parametric decay instabilities near the upper hybrid layer, with particular emphasis on the
one considered by [Porkoláb, 1982] and on extending the results obtained in that paper.
9
Chapter 2
Kinetic Theory of Parametric Decay
In this theory chapter we review the kinetic theory of parametric decay in a magnetised
plasma, loosely following [Porkoláb, 1974] and [Porkoláb, 1978]. First, the motion of a
charged particle in a constant magnetic field, under the influence of an electromagnetic
pump wave, is investigated. Then, the detailed kinetic theory of parametric decay is presented, culminating with the so-called parametric dispersion relation and an expression for
the growth rate of the parametric decay instability in a homogeneous magnetised plasma.
2.1
Motion of an Electromagnetically Driven Plasma Particle
We consider a plasma particle of species σ, with mass mσ and charge qσ ; at time t the
particle is located at the position r(t) and has a velocity v(t) = dr(t)/dt; the particle
moves in an electric field E(r, t) and a magnetic field B(r, t). In the non-relativistic limit,
the motion of such a particle is governed by the Newton’s 2nd law,
dv(t)
qσ
=
[E(r(t), t) + v(t) × B(r(t), t)].
dt
mσ
(2.1)
In general E(r, t) and B(r, t) should be determined self-consistently through the Maxwell
equations with current and charge densities being derived from kinetic or fluid equations.
Initially we will, however, simply consider particle motion in externally prescribed fields
E(r, t) and B(r, t). As we are dealing with a magnetised plasma driven by an electromagnetic pump wave, two external contributions are considered, a harmonic electromagnetic
pump field and a steady background magnetic field.
At present, we take the pump field to be harmonic in both space and time, i.e., we assume
it to be a plane wave, giving a contribution Re(E0 eik0 ·r−iω0 t ) to the electric field and a
contribution Re(B0 eik0 ·r−iω0 t ) to the magnetic field, where k0 is the pump wave vector,
ω0 is the angular pump frequency, and E0 , B0 are the complex amplitude vectors of
the pump electric and magnetic fields, respectively. Since the pump fields are associated
with an electromagnetic plane wave, the forces due to the magnetic field will only be
10
comparable to those due to the electric field for particle speeds in the relativistic regime
[Jackson, 1999]. Our calculations are non-relativistic, and we are thus justified in neglecting
the pump magnetic field when calculating particle motion. We will further utilise that
non-relativistic particles traverse a negligible fraction of the pump wavelength (2π/k0 )
during a pump period (2π/ω0 ) to ignore any spatial variation of the pump field and set
E(r, t) = E(t) = Re(E0 e−iω0 t ), B(r, t) = B; B is the steady magnetic field and we are
ignoring any spatial variation. With these assumptions
dv(t)
qσ
[Re(E0 e−iω0 t ) + v(t) × B].
=
dt
mσ
(2.2)
This is a system of inhomogeneous linear differential equations with constant coefficients
which may be solved using standard techniques, i.e., by finding a particular solution to the
inhomogeneous problem and a general solution to the homogeneous one. The solution of
the homogeneous equations is the well-known cyclotron motion of particles in a constant
magnetic field. It is of great importance when determining the linear, and nonlinear, response of a magnetised plasma. However, it is the (particular) inhomogeneous solution that
is responsible for parametric processes. Our main focus will therefore be on obtaining such
a solution. This is done relatively simply by inserting the ansatz vdσ (t) = Re(vdσ0 e−iω0 t )
in Eq. (2.2) and solving for vdσ0 , taking B along the z-axis,


2 )
(iω0 E0x − ωcσ E0y )/(ω02 − ωcσ
qσ 
2 ) ,
(ωcσ E0x + iω0 E0y )/(ω02 − ωcσ
vdσ0 =
(2.3)
mσ
iE0z /ω0
here ωcσ = qσ B/mσ is the (angular) cyclotron frequency of species σ. Upon a further
integration with respect to time, selecting the particular solution of the form rdσ (t) =
Re(rd0 e−iω0 t ), we find
ivdσ0 −iω0 t
rdσ (t) = Re
e
.
(2.4)
ω0
2 ) are finite frequency generalIn terms of particle drifts, the terms involving ωcσ /(ω02 − ωcσ
2
2 ) are finite frequency
isations of the E(t) × B/B -drift, the terms involving ω0 /(ω02 − ωcσ
generalisations of the polarisation drift, and the z-component is simply the quiver velocity
of a charged particle, reacting to Ez (t), in the absence of B [Bellan, 2006].
2.2
Fundamental Kinetic Theory of Parametric Decay
Since we are interested in the case where the daughter waves are electrostatic in nature the
problem is governed by the Vlasov-Poisson system; to avoid unnecessary complications,
we are ignoring collisional effects in the basic theoretical sections, but the main results
will remain valid in the presence of weak collisions [Amano and Okamoto, 1969]. The
Vlasov-Poisson system is given by,
∂fσ (r, v, t)
qσ
∂φ(r, t)
∂fσ (r, v, t)
∂fσ (r, v, t)
−iω0 t
+v·
+
−
+ v × B + Re(E0 e
) ·
= 0,
∂t
∂r
mσ
∂r
∂v
(2.5)
11
∂ ∂φ(r, t)
1 X
qσ nσ (r, t),
·
=−
∂r
∂r
0 σ
(2.6)
R
where fσ (r, v, t) is the distribution function of species σ, nσ (r, t) = all v fσ (r, v, t) dv is
the density of species σ, φ(r, t) is the electrostatic potential associated with the daughter
waves, ∂/∂r and ∂/∂v denote gradients with respect to r and v, and 0 = 8.854×10−12 F/m
is the vacuum permittivity. The traditional method of characteristics [Stix, 1992] is not
applicable to solve the equations in this standard form, as the pump term results in an
explicit time dependence of the unperturbed distribution function along the characteristics.
However, the problem may be remedied by going into a frame oscillating with the velocity
induced by the pump, where inertial forces will exactly cancel the pump term. In this
frame we define position x = r − rdσ (t), velocity u R= v − vdσ (t), distribution function
Fσ (x, u, t) = fσ (r, v, t), density Nσ (x, t) = nσ (r, t) = all u Fσ (x, u, t) du, and electrostatic
potential Φ(x, t) = φ(r, t). Using these definitions, along with the chain rule, we find
∂x(r, t)/∂t = −drd (t)/dt = −vd (t), ∂u(v, t)/∂t = −dvdσ (t)/dt = −(qσ /mσ )[vdσ (t) × B +
Re(E0 e−iω0 t )], ∂/∂r = ∂/∂x, ∂/∂v = ∂/∂u, and thus
∂fσ (r, v, t)
∂Fσ (x(r, t), u(v, t), t)
∂Fσ (x, u, t)
∂Fσ (x, u, t)
=
=
− vdσ (t) ·
∂t
∂t
∂t
∂x
∂Fσ (x, u, t)
qσ
[vdσ (t) × B + Re(E0 e−iω0 t )] ·
,
−
mσ
∂u
∂fσ (r, v, t)
∂Fσ (x, u, t) ∂fσ (r, v, t)
∂Fσ (x, u, t) ∂φ(r, t)
∂Φ(x, u, t)
=
,
=
,
=
. (2.7)
∂r
∂x
∂v
∂u
∂r
∂x
Plugging the above expressions into Eqs. (2.5) and (2.6) yields
∂Fσ (x, u, t)
qσ
∂Fσ (x, u, t)
∂Fσ (x, u, t)
∂Φ(x, t)
+u·
+
+u×B ·
= 0,
(2.8)
−
∂t
∂x
mσ
∂x
∂u
∂ ∂φ(r, t)
1 X
·
=−
qσ Nσ (r − rdσ (t), t);
∂r
∂r
0 σ
(2.9)
note that different oscillating frames are used for each species, but the quantities related
to a given species should always be evaluated in its own oscillating frame, as indicated in
the Poisson equation.
Now, assuming that |∂Φ(x, t)/∂x| is not too large, i.e., that few daughter waves have
been generated, as is appropriate at the onset of instability, Eq. (2.8) may be linearised, and the linearised equation solved using the method of characteristics. When
(0)
(1)
linearising, we write Fσ (x, u, t) = Fσ (u) + Fσ (x, u, t), with corresponding density
(0)
(1)
Nσ (x, t) = Nσ + Nσ (x, t); the quantities with superscript (0) (order 0) yield a timeinvariant uniform background density, without specifying the velocity distribution, while
the quantities with superscript (1) (order 1) are perturbations of the background. The
(1)
(0)
(1)
(0)
ordering |Fσ (x, u, t)| Fσ u) and |Nσ (x, u, t)| Nσ is implicit, and the earlier assumption of a relatively small (order 1) |∂Φ(x, t)/∂x| justifies taking Φ(x, t) =
Φ(1) (x, t), with Φ(x, t) → 0 for |x| → ∞ such that Φ(x, t) is spatially localised, φ(r, t)
12
obeys similar relations. We are, further, justified in neglecting the nonlinear product
(1)
[∂Φ(x, t)/∂x] · [∂Fσ (x, u, t)/∂u] and it is generally assumed that B = B(0) . Now, separating terms of orders 0 and 1, Eqs. (2.8) and (2.9) become
(0)
∂Fσ (u)
= 0,
∂u
(1)
(1)
(0)
(1)
∂Fσ (x, u, t)
∂Fσ (x, u, t)
qσ
qσ ∂Φ(x, t) ∂Fσ (u)
∂Fσ (x, u, t)
(u×B)·
+ u·
+
=
·
,
∂t
∂x
mσ
∂u
mσ ∂x
∂u
(2.10)
(u × B) ·
X
qσ Nσ(0) = 0,
σ
1 X
∂ ∂φ(r, t)
·
=−
qσ Nσ(1) (r − rdσ (t), t).
∂r
∂r
0 σ
(2.11)
(0)
The order 0 part of Eq. (2.10) simply requires that ∂Fσ (u)/∂u lies in the plane of u
(0)
and B, which is the case if Fσ (u) is symmetric
q around the axis of B, i.e., for B along
(0)
(0)
the z-axis, Fσ (u) = Fσ (u⊥ , uz ) with u⊥ = u2x + u2y . This is generally a very good
approximation in magnetised plasmas, since velocities perpendicular to the magnetic field
are mixed on a short time scale ∼ 1/ωcσ . The order 0 part of Eq. (2.11) requires the
unperturbed plasma to be charge neutral. This is a necessary requirement for a small
|∂Φ(x, t)/∂x| and is also generally a very good approximation in fusion relevant plasmas,
due to the large electrostatic forces associated with the plasma having a net charge.
The order 1 part of Eq. (2.10) may be solved using the method of characteristics, i.e.,
by going into a frame moving with the particles along the unperturbed (cyclotron) orbits
and performing a temporal integral in this frame; this is done in many standard texts,
e.g., [Krall and Trivelpiece, 1973], [Stix, 1992], [Swanson, 2003], [Bellan, 2006], and [Mazzucato, 2014]. We shall not repeat such an analysis here, but simply note that it results
(1)
in a linear, but generally nonlocal, relation between Nσ (x, t) and Φ(x, t). The linearity
of the problem, along with the assumed homogeneity of the background plasma and B,
means that the relation takes on a particularly simple local Rform Rin Fourier-Laplace space
∞
[Grosso and Pastori Parravicini, 2000]. If we let g̃(k, ω) = all x ( 0 g(x, t) eiωt−ik·x dt) dx
define the Fourier-Laplace transform of g(x, t), the relation between density and potential
perturbations may be written as
Ñσ(1) (k, ω) = −
0 k 2
χσ (k, ω)Φ̃(k, ω),
qσ
(2.12)
where χσ (k, ω) is the linear susceptibility of species σ and k = |k|; a similar relation holds
in the presence of collisions if χσ (k, ω) is modified appropriately [Amano and Okamoto,
1969]. On a formal note, we are neglecting any initial deviations from the background
distributions in the above expression. A spatial Fourier transform is used, which means
(1)
that the mode wave vector, k, is real, and requires Nσ (x, t) and Φ(x, t) to be spatially
localised, consistent with earlier assumptions. A temporal Laplace transform is employed,
meaning that ω is generally a complex quantity. This allows us to account for Landau
13
damping as well as exponentially growing temporal (absolute) instabilities; Re(ω) is the
angular mode frequency and Im(ω) is the amplitude e-folding rate.
The main problem now, apart from finding a tractable form of χσ (k, ω) in the cases of
interest, is that Φ̃(k, ω) is the Fourier-Laplace transform of Φ(x, t) which is defined in
different oscillating frames for each species, i.e., there are as many different versions of
(1)
Φ̃(k, ω) as of Nσ (k, ω). This issue is resolved by expressing Φ̃(k, ω) in terms of the
Fourier-Laplace transform of φ(r, t) (potential in the stationary frame). To do this, we
first use the transformations r = x + rdσ (t) and Φ(x, t) = φ(r, t) to write
Z Z ∞
Z Z ∞
φ(r, t) eiωt+ik·rdσ (t) dt e−ik·r dr.
Φ(x, t) eiωt−ik·x dt dx =
Φ̃(k, ω) =
all x
all r
0
Next, we evaluate
expressed as
eik·rdσ (t) .
0
(2.13)
Insertion of rdσ (t) from Eq. (2.4) allows k · rdσ (t) to be
k · rdσ (t) = Re
ik · vdσ0 −iω0 t
e
ω0
= µσ sin(ω0 t − βσ ),
(2.14)
with µσ = |k · vdσ0 |/ω0 and βσ = arg(k · vdσ0 /ω0 ) (arg is the phase angle). The parameter
µσ is, as we shall see, a measure of the parametric coupling strength to the Fourier mode
specified by k and is of great importance to the theory of parametric decay; plugging in
vdσ0 from Eq. (2.3) gives an explicit form of µσ , with B along the z-axis,
v
u u ω0 Im(kx E0x + ky E0y ) + ωcσ Re(kx E0y − ky E0x ) kz Im(E0z ) 2
u
+
+
2
|qσ | u
ω0
ω02 − ωcσ
u µσ =
,
mσ ω0 u
ω0 Re(kx E0x + ky E0y ) − ωcσ Im(kx E0y − ky E0x ) kz Re(E0z ) 2
t
+
2
ω0
ω02 − ωcσ
(2.15)
showing that µσ is proportional to the pump electric field strength, though also dependent
on the polarisation. The above form agrees with the ones given by [Porkláb, 1974] and
[Porkoláb, 1978] in the special cases considered there. The phase angle βσ has no particular
physical significance. It may be changed by shifting the point at which t = 0. Now, using
a Fourier series identity, also used when analysing the linearised Vlasov equation [Bellan,
2006], eik·rdσ (t) is determined as
eik·rdσ (t) = eiµσ sin(ω0 t−βσ ) =
∞
X
Jn (µσ ) ein(ω0 t−βσ ) ,
(2.16)
n=−∞
where Jn is a Bessel function of the 1st kind of order n. Inserting the above expression
in Eq. (2.13), and identifying the Fourier-Laplace argument of each term in the infinite
series, gives
∞
X
Φ̃(k, ω) =
Jn (µσ ) e−inβσ φ̃(k, ω + nω0 ),
(2.17)
n=−∞
14
and thus by Eq. (2.12),
Ñσ(1) (k, ω)
∞
X
0 k 2
=−
χσ (k, ω)
Jn (µσ ) e−inβσ φ̃(k, ω + nω0 ).
qσ
n=−∞
(2.18)
Eq. (2.18) gives an explicit relation between Ñ (1) (k, ω) and φ̃(k, ω + nω0 ), with n ∈ Z. By
taking the Fourier-Laplace transform of the order 1 part of Eq. (2.11), and performing an
analysis completely analogous to the one above, a relation between φ̃(k, ω) and Ñ (1) (k, ω +
nω0 ), with n ∈ Z, may be found:
X qσ Z Z ∞
(1)
iωt−ik·r
φ̃(k, ω) =
N (r − rdσ (t), t) e
dt dr
0 k 2 all r 0 σ
σ
(2.19)
∞
X qσ X
(1)
−inβσ
=
Ñσ (k, ω + nω0 ).
Jn (−µσ ) e
0 k 2 n=−∞
σ
Now, Eqs. (2.18) and (2.19) constitute a set of coupled algebraic equations which, when
solved, reveals the coupling between different Fourier-Laplace modes due to the pump term
if the amplitude of the daughter waves is not too large. Note that only Fourier-Laplace
modes differing by an integer multiple of ω0 are coupled. This may be understood in terms
of the selection rules for parametric processes (Fig. 1.1). No coupling between modes with
different k exists due to the dipole approximation k0 ≈ 0. However, modes with ±k and
±Re(ω) are coupled through the requirement of real Nσ (x, t)(1) and φ(r, t) [Grosso and
Pastori Parravicini, 2000], leaving the selection rules valid.
2.3
The Parametric Dispersion Relation
While Eqs. (2.18) and (2.19) do in principle contain a complete solution to the problem
of finding the homogeneous parametric decay instability threshold/growth rates, with the
dipole and linearised daughter wave assumptions, which may be found approximately by
solving the linear system truncated at |n| 1, such an approach will generally yield
results which are difficult to analyse and interpret. Thus, it is desirable to make some
further physically motivated assumptions in order to obtain a more tractable parametric
dispersion relation; this is done in the present section to some extent following the approach
of [Porkoláb, 1974].
First of all, we are considering a fully ionised plasma consisting of electrons with mass
me = 9.110 × 10−31 kg and charge qe = −e = −1.602 × 10−19 C, and an arbitrary number
ionic species each characterised by a mass mi and a charge qi = Zi e, all satisfying mi /Zi ≥
1836me , Zi ∈ N. For the cases of interest in this work, ω0 ∼ ωce ωci , and thus
the parametric coupling constants from Eq. (2.15) obey the ordering µe µi ; this is
ordinarily true, unless ω0 ≈ ωci , and reflects that the electron displacement due to the
pump field is generally much larger than the ion displacement, since mi /Zi me . Now,
assuming µe . 1, which generally holds in the cases of interest, we conclude that the
15
coupling between different Fourier-Laplace modes is only of importance for the electrons
and thus set µi ≈ 0, Jn (µi ) ≈ Jn (−µi ) ≈ δn0 , where δnm is the Kronecker δ, i.e., the ion
response is taken to be linear and the reference frame of the ions is just the stationary
frame. With these assumptions Eqs. (2.18) and (2.19) may be rearranged, respectively, as
follows
X
0 k 2 X
χi (k, ω)φ̃(k, ω),
e
(1)
Zi Ñi (k, ω) = −
i
Ñe(1) (k, ω)
i
∞
X
k2
0
χe (k, ω)
Jn (µe ) e−inβe φ̃(k, ω + nω0 ),
=
e
n=−∞
"
#
∞
X
X
e
(1)
φ̃(k, ω) =
Zi Ñi (k, ω) −
Jn (−µe ) e−inβe Ñe(1) (k, ω + nω0 ) .
0 k 2
n=−∞
(2.20)
(2.21)
i
Note that the linear ion response has allowed us to deal with all ionic species through
P
P
(1)
i Zi Ni (k, ω) and
i χi (k, ω), meaning that multiple ionic species do no complicate
the problem significantly; however, this does not apply to the problem of finding χi (k, ω)
in the presence of collisions. Now, substituting φ̃(k, ω) from Eq. (2.21) into Eq. (2.20),
P
(1)
upon moving terms containing i Zi Ñi (k, ω) to the left hand side in the first expression,
yields
"
#
∞
X
X
X
X
(1)
1+
χi (k, ω)
Zi Ñi (k, ω) =
Jn (−µe ) e−inβe Ñe(1) (k, ω + nω0 ),
χi (k, ω)
i
i
n=−∞
i
(2.22)
"
Ñe(1) (k, ω) = χe (k, ω)
∞
X
#
Jn (µe ) e−inβe
n=−∞
X
Zi Ñi (k, ω + nω0 ) − S ,
(2.23)
i
P∞
P∞
−i(n+m)βe Ñ (1) (k, ω + (n + m)ω )]. The
where S =
e
0
m=−∞ [ n=−∞ Jn (µe )Jm (−µe ) e
seemingly complicated double sum S turns out to have a very simple form. By changing summation over m to be over l = n + m, and using Neumann’s addition theorem
P
∞
n=−∞ Jn (µe )Jl−n (−µe ) = Jl (0) = δl0 [Abramowitz and Stegun, 1964], we find
S=
∞
X
l=−∞
"
∞
X
#
Jn (µe )Jl−n (−µe ) e−ilβe Ñe(1) (k, ω + lω0 ) = Ñe(1) (k, ω),
(2.24)
n=−∞
and thus Eq. (2.23) may be rewritten as
[1 + χe (k, ω)]Ñe(1) (k, ω) = χe (k, ω)
∞
X
Jn (µe ) e−inβe
n=−∞
X
(1)
Zi Ñi (k, ω + nω0 ).
(2.25)
i
The problem of solving Eqs. (2.22) and (2.25) is already much simpler than that of solving
Eqs. (2.18) and (2.19), and the only additional restrictions are that we are dealing with
an electron-ion plasma, which is essentially always the case in tokamaks, and that the
16
ion response to the pump wave is linear, which is almost always satisfied for ω0 ωci .
However, Eqs. (2.22) and (2.25) are still quite complicated and generally only amenable
to approximate numerical solutions. We shall therefore make further assumptions, which
are more restrictive, but still reasonable for the cases of interest in this work.
q
2 (0)
In addition to ω0 ωci , our pump wave satisfies ω0 ωpi ; ωpi = (Z
i e) Ni /(0 mi )
P
is the angular plasma frequency of any ionic species. This means that | i χi (k, ω)| 1
for Re(ω) & ω0 [Bellan, 2006], i.e., ions do virtually not respond respond at frequencies comparable to, or larger than, that of the pump wave. Now, associating ω with a
low-frequency mode, e.g., the lower hybrid or an ion Bernstein mode, the above assumption allows us neglect the ion response for all but this mode and, consequently, we set
P
P
(1)
(1)
i Zi Ñi (k, ω)δn0 . Inserting this in Eq. (2.25) yields
i Zi Ñi (k, ω + nω0 ) ≈
Ñe(1) (k, ω + nω0 ) =
X
χe (k, ω + nω0 )
(1)
J−n (µe ) einβe
Zi Ñi (k, ω),
1 + χe (k, ω + nω0 )
(2.26)
i
and plugging the above expression into Eq. (2.22), using J−n (µe ) = (−1)n Jn (µe ) and
Jn (−µe ) = (−1)n Jn (µe ), both valid for n ∈ Z, we obtain a parametric dispersion relation,
#
" ∞
X
X
X
χ
(k,
ω
+
nω
)
e
0
1+
χi (k, ω) =
,
(2.27)
χi (k, ω)
Jn2 (µe )
1
+
χ
(k,
ω
+
nω
e
0)
n=−∞
i
i
which is similar to Eq. (3.5) from [Aliev et al., 1966]. The above equation is valid for
arbitrary µe , so long as the ion response remains linear. However, we are interested in
the case of µe 1, where the Taylor expansions, J02 (µe ) = 1 − µ2e /2 + O(µ4e ), J12 (µe ) =
2 (µ ) = µ2 /4 + O(µ4 ), and J 2 (µ ) = O(µ2|n| ) may be used to rewrite Eq. (2.27),
J−1
e
e
e
e
n e
ignoring terms of order µ4e or higher, as follows
X
1 + χe (k, ω)
1 + χe (k, ω)
µ2e X
1+
χi (k, ω) + χe (k, ω) =
χi (k, ω) 2 −
−
;
4
1 + χe (k, ω − ω0 ) 1 + χe (k, ω + ω0 )
i
i
(2.28)
the terms of order µ2e have been re-expressed using χe (k, ω ± ω0 )/[1 + χe (k, ω ± ω0 )] −
χe (k, ω)/[1 + χe (k, ω)] = 1/[1 + χe (k, ω)]
P − 1/[1 + χe (k, ω ± ω0 )]. Now, introducing the
linear dielectric function (k, ω) = 1 + i χi (k, ω) + χe (k, ω), noting that (k, ω ± ω0 ) =
1+χe (k, ω ±ω0 ) since the high-frequency ion response has been neglected, and ignoring the
2 in the bracket on the right hand side of Eq. (2.28), we arrive at a parametric dispersion
relation similar to the one used by [Kasymov et al., 1985] to study parametric decay near
the upper hybrid resonance,
µ2e X
1
1
(k, ω) = −
χi (k, ω)[1 + χe (k, ω)]
+
.
(2.29)
4
(k, ω − ω0 ) (k, ω + ω0 )
i
[Porkoláb, 1974] has a similar expression; it is, however, derived from quite different assumptions since a lower hybrid pump wave is used in that article.
17
Eq. (2.29) shows the basic coupling between different frequency components implied by Fig.
1.1: a low-frequency daughter wave, characterised by ω, Re(ω) = ω1 > 0, exists along with
a down-shifted high-frequency daughter wave, characterised by ω − ω0 , Re(ω − ω0 ) = −ω2 ,
remember the coupling between ±ω2 [Grosso and Pastori Parravicini, 2000], and an upshifted high-frequency daughter wave, characterised by ω+ω0 , Re(ω+ω0 ) = ω3 , to order µ2e .
If the right hand side is negligible, Eq. (2.29) is nothing but the linear dispersion relation
for electrostatic plasma waves, (k, ω) = 0, and no discernible coupling between different
frequency components, i.e., no parametric decay instability, exists. In general, µ2e 1 so
this is true for most Fourier-Laplace modes. The important exception occurs when the
high-frequency daughter modes coincide with a linear plasma mode, i.e., (k, ω ± ω0 ) ≈ 0.
In this case, the right hand side of Eq. (2.29) may acquire a non-negligible value even
though µ2e 1. If the low-frequency daughter waves are also close to a linear mode,
(k, ω) ≈ 0, it may be very easy to satisfy Eq. (2.29), and thus the considerations around
Fig. 1.2 have been validated. The requirement (k, ω ± ω0 ) = 1 + χe (k, ω ± ω0 ) ≈ 0 justifies
neglecting the 2 in the bracket on the right hand side of Eq. (2.28) and we further note
that it has been tacitly assumed that higher order terms in Eq. (2.27) are off-resonance,
1 + χe (k, ω ± nω0 ) 6≈ 0 for |n| > 1, when using the Taylor expanded Eq. (2.28) to represent
Eq. (2.27); the higher order resonances will generally lead to higher instability thresholds
than the fundamental one, so we need not consider them under normal circumstances.
2.4
Growth Rate of the Parametric Decay Instability
We shall now use Eq. (2.29), along with the considerations above, to derive an expression
for the growth rate (amplitude e-folding rate) of the parametric decay instability, γ =
Im(ω). To do this, we first assume that only the down-shifted high-frequency daughter
wave is on-resonance and, consequently, neglect the
P up-shifted high-frequency daughter
wave, reducing Eq. (2.29) to (k, ω) = −(µ2e /4) i χi (k, ω)[1 + χe (k, ω)]/(k, ω − ω0 ).
Next, we need to express (k, ω − ω0 ) = (k, −ω2 + iγ) approximately near the linear
mode. This is done by assuming weak growth/damping such that ω2 |γ| and, except
exactly at the linear mode, |Re()| |Im()|. Then, for the linear mode Re[(k, −ω2 )] = 0,
and (k, −ω2 + iγ) may be found by Taylor expanding Re[(k, −ω2 + iγ)] to order γ 1
and Im[(k, −ω2 + iγ)] to order γ 0 around −ω2 [Bellan, 2006]. Doing this, also using
that Re[(k, ω)] is an even function of ω, while Im[(k, ω)] and ∂ Re[(k, ω)]/∂ω are odd
(1)
functions of ω, to ensure that Nσ (x, t) and Φ(x, t) are real valued [Grosso and Pastori
Parravicini, 2000], yields
∂ Re[(k, ω)] ∂ Re[(k, ω)] (k, −ω2 + iγ) ≈ iγ
,
+ i Im[(k, −ω2 )] = −i[γ + Γ(k, ω2 )]
∂ω
∂ω
ω=−ω2
ω=ω2
(2.30)
where Γ(k, ω2 ) = Im[(k, ω2 )]/(∂ Re[(k, ω)]/∂ω)|ω=ω2 is the mode damping rate (negative
growth rate) of linear theoryP
[Bellan, 2006]. Plugging this expression into the dispersion
2
relation, (k, ω) = −(µe /4) i χi (k, ω)[1 + χe (k, ω)]/(k, −ω2 + iγ), and isolating γ +
18
Γ(k, ω2 ), now gives
P
i i χi (k, ω)[1 + χe (k, ω)]
µ2e
γ + Γ(k, ω2 ) = − Re
4
(k, ω)(∂ Re[(k, ω)]/∂ω)|ω=ω2
P
P
2
µe | i χi (k, ω)|2 Im[χe (k, ω)] + |1 + χe (k, ω)|2 i Im[χi (k, ω)]
.
=
4
|(k, ω)|2 (∂ Re[(k, ω)]/∂ω)|ω=ω2
(2.31)
The above equation does still not explicitly determine γ, since ω = ω1 + iγ; different
approximations are appropriate depending on the conditions fulfilled by (k, ω) = (k, ω1 +
iγ). If the low-frequency mode corresponds to an exact linear plasma mode, known as
resonant parametric decay, it is necessary to expand (k, ω1 + iγ) in a manner similar to
(k, −ω2 + iγ), leading to (k, ω1 + iγ) ≈ i(∂ Re[(k, ω)]/∂ω)|ω=ω1 [γ + Γ(k, ω1 )], assuming
|γ| ω1 . Further noting that finite γ effects in the numerator on the right hand side of
Eq. (2.31) will only lead to higher order corrections, we arrive at the following equation
for γ in the case of resonant parametric decay
P
Re { i χi (k, ω1 )[1 + χe (k, ω1 )]}
µ2e
[γ +Γ(k, ω1 )][γ +Γ(k, ω2 )] = −
. (2.32)
4 (∂ Re[(k, ω)]/∂ω)|ω=ω1 (∂ Re[(k, ω)]/∂ω)|ω=ω2
This quadratic equation is easily solved to give γ, showing only the possibly positive root,
"s
P
µ2e Re { i χi (k, ω1 )[1 + χe (k, ω1 )]}
1
2
[Γ(k, ω1 ) − Γ(k, ω2 )] −
γ=
2
(∂ Re[(k, ω)]/∂ω)|ω=ω1 (∂ Re[(k, ω)]/∂ω)|ω=ω2
(2.33)
− Γ(k, ω1 ) − Γ(k, ω2 ) ,
and the threshold of the resonant parametric decay instability in a homogeneous plasma
(condition for γ > 0),
P
Re { i χi (k, ω1 )[1 + χe (k, ω1 )]}
µ2e
−
> Γ(k, ω1 )Γ(k, ω2 ).
(2.34)
4 (∂ Re[(k, ω)]/∂ω)|ω=ω1 (∂ Re[(k, ω)]/∂ω)|ω=ω2
In the case where the low-frequency mode does not correspond to an exact linear plasma
mode, known as non-resonant parametric decay, for instance in the presence of significant
Landau damping, it is unnecessary to consider finite γ effects on the right hand side of Eq.
(2.31), and the following linear expression for γ is found,
P
P
µ2e | i χi (k, ω1 )|2 Im[χe (k, ω1 )] + |1 + χe (k, ω1 )|2 i Im[χi (k, ω1 )]
−Γ(k, ω2 ). (2.35)
γ=
4
|(k, ω1 )|2 (∂ Re[(k, ω)]/∂ω)|ω=ω2
This expression is similar to the one used by [Porkoláb, 1982] to study parametric decay
near the upper hybrid resonance in the Versator II tokamak, and the threshold of the
non-resonant parametric decay instability in a homogeneous plasma, trivially, becomes
P
P
µ2e | i χi (k, ω1 )|2 Im[χe (k, ω1 )] + |1 + χe (k, ω1 )|2 i Im[χi (k, ω1 )]
> Γ(k, ω2 ). (2.36)
4
|(k, ω1 )|2 (∂ Re[(k, ω)]/∂ω)|ω=ω2
19
The non-resonant parametric decay instability is seen to vanish in the limit of vanishing
Im[χe (k, ω1 )] and Im[χi (k, ω1 )], i.e., its existence actually depends on damping of the
low-frequency mode. As noted by, e.g., [Porkoláb, 1974], the above observation makes it
possible to interpret non-resonant parametric decay in terms of nonlinear Landau damping
of the low-frequency mode, which is often referred to as a quasi-mode in this case. While
this may be understood on the basis of Eq. (III-9) from [Sagdeev and Galeev, 1969], a
more intuitive physical picture of the process is provided by [Weiland and Wilhelmsson,
1977]. Essentially, the low-frequency quasi-mode acts to couple the high-frequency pump
and daughter modes, through interaction with the plasma particles causing it to be Landau
damped, and thus transfers energy from the pump wave to the high-frequency daughter
wave. As we shall see, parametric decay instabilities in tokamaks may often be of the
non-resonant type, and, further, the fact that non-resonant parametric decay may be seen
as simple amplification of the high-frequency daughter wave, at the expense of the pump
wave, has important consequences for its threshold in inhomogeneous plasmas.
20
Chapter 3
Parametric Decay near the Upper
Hybrid Resonance
In this chapter we use the theory of Chapter 2 to investigate the parametric decay instability
near the upper hybrid resonance frequency. We first review the cold plasma theory of highfrequency electromagnetic waves, which leads to the existence of the upper hybrid resonance
and determines its accessibility in tokamak plasmas, in order to set the scene for this and
later chapters. Then, we discuss the high-frequency electrostatic daughter waves likely to
be involved in parametric decay near the upper hybrid resonance; these are found to be
a limiting case of electron Bernstein waves, whose dispersion relation and linear damping
rate are derived. After dealing with the high-frequency electrostatic daughter waves we
turn our attention to the low-frequency electrostatic daughter waves involved in parametric
decay; the most likely candidates are found to be (pure) ion Bernstein and lower hybrid
waves, and these are consequently discussed in some detail. Finally, we derive the growth
rate and (homogeneous) instability threshold of parametric decay into electron Bernstein
modes and lower hybrid (quasi-)modes, as studied by [Porkoláb, 1982], generalising the
results of that paper somewhat.
3.1
Cold Theory of High-Frequency Electromagnetic Plasma
Waves
We are considering parametric decay originating from a high-frequency electromagnetic
(microwave) pump wave. We therefore initiate this chapter with a discussion of such waves
in homogeneous magnetised plasmas which clearly demonstrates the special importance of
the upper hybrid resonance in connection with parametric decay in this frequency range.
The present treatment is based on cold plasma theory in which kinetic/thermal effects are
neglected; this is justified when the group velocity of the pump wave along the magnetic
field is large compared with the characteristic particle velocities, an assumption that has
already been invoked when treating the pump as purely time varying in Chapter 2, [Bellan,
21
2006]. In addition to the assumption of a cold plasma response, the high frequency of the
pump wave, ω0 ∼ ωce ωpi , ωci , allows us to neglect the ion response to the pump wave
altogether, also in keeping with the assumptions of Chapter 2. With these assumptions, and
if the pump-induced velocity is small compared with the characteristic electron velocities,
the response of the plasma to the pump wave is governed by the cold electron momentum
equation [Braginskii, 1965],
∂Ve (r, t)
∂Ve (r, t)
e
[E(r, t)+Ve (r, t)×B(r, t)]−νei (r, t)Ve (r, t), (3.1)
+Ve (r, t)·
=−
∂t
∂r
me
R
where Ve (r, t) = [ all v vfe (r, v, t) dv]/ne (r, t) is the electron fluid velocity, νei (r, t) is the
electron-ion collision frequency, multiple ionic species may be included, and we have taken
Vi (r, t) = 0. The basic collisional drag term −νei (r, t)Ve (r, t) has been retained for later
convenience, although it is, strictly speaking, not a part of cold plasma theory; we note
that a similar collision term was also included in the original treatment [Appleton, 1932],
but that a proper account of dissipation of high-frequency electromagnetic waves in fusion
plasmas requires inclusion of relativistic and kinetic effects [Mazzucato, 2014], as well as
the inherent anisotropy of electron-ion collision dynamics in a magnetic field [Swanson,
2003], which are beyond the scope of the present discussion.
In the model, electrical current is only carried by the electrons and the electrical current
density is J(r, t) = −ene (r, t)Ve (r, t). With this Faraday’s and Ampère’s laws, which are
the only Maxwell equations necessary in this discussion, become,
∂
∂B(r, t)
× E(r, t) = −
,
∂r
∂t
c2
∂
ene (r, t)Ve (r, t) ∂E(r, t)
× B(r, t) = −
+
,
∂r
0
∂t
(3.2)
where c = 2.998 × 108 m/s is the vacuum speed of light. Taking the time derivative of
Ampère’s law, and inserting ∂B(r, t)/∂t from Faraday’s law, yields an equation for E(r, t)
in terms of ne (r, t) and Ve (r, t),
∂
∂
e ∂[ne (r, t)Ve (r, t)] ∂ 2 E(r, t)
−c2
×
× E(r, t) = −
+
.
(3.3)
∂r
∂r
0
∂t
∂t2
Now, treating E(r, t) and Ve (r, t) as small (order 1) quantities, Eqs. (3.1) and (3.3) may be
linearised around the time-invariant homogeneous magnetised (order 0) plasma equilibrium
used in Chapter 2. The procedure results in trivial order 0 parts of Eqs. (3.1) and (3.3),
while the order 1 parts are, respectively,
∂Ve (r, t)
e
=−
[E(r, t) + Ve (r, t) × B] − νei Ve (r, t),
∂t
me
(3.4)
(0)
∂
∂
∂ 2 E(r, t)
eNe ∂Ve (r, t)
c
×
× E(r, t) +
=
;
∂r
∂r
∂t2
0
∂t
(3.5)
2
B is the background magnetic field, νei is the electron-ion collision frequency given by
(0)
the background quantities, and Ne is the background electron density. By plugging in
22
the assumed plane wave variation of the pump, E(r, t) = Re(E0 ei(k0 +iκ0 )·r−iω0 t ), we have
added an imaginary part κ0 to the wave vector to account for spatial wave damping, with
the assumption |κ0 | |k0 |, along with a similar ansatz for the electron fluid velocity,
Ve (r, t) = Re(Ve0 ei(k0 +iκ0 )·r−iω0 t ), Eqs. (3.4) and (3.5) are recast as algebraic equations
for E0 and Ve0 ,
e
−iω0 Ve0 = −
(E0 + Ve0 × B) − νei Ve0 ,
(3.6)
me
(0)
c2 (k0 + iκ0 ) × [(k0 + iκ0 ) × E0 ] = i
eNe
ω0 Ve0 − ω02 E0 .
0
(3.7)
If we let ω0 → ω0 − iνei , Eq. (3.6) is exactly the same as the algebraic equation obtained
when calculating single particle motion in the presence of the pump wave (Section 2.1).
The link between Ve0 and E0 can therefore immediately be determined from Eq. (2.3).
However, in the present case, it is more practical to write the result as a matrix-vector
equation,


iωce
ω0 + iνei
0
2
2
 (ω0 + iνei )2 − ωce

(ω0 + iνei )2 − ωce


e
−iωce
ω0 + iνei


Ve0 = −i
M·E0 , M = 
0
 , (3.8)
2
2
2
2
 (ω0 + iνei ) − ωce (ω0 + iνei ) − ωce

me


1
0
0
ω0 + iνei
where the mobility tensor, M, in the right hand expression, is given for B along the z-axis.
Plugging the above expression into Eq. (3.7), and using the identity (k0 + iκ0 ) × [(k0 +
iκ0 ) × E0 ] = [(k0 + iκ0 )(k0 + iκ0 ) − (k0 + iκ0 ) · (k0 + iκ0 )1] · E0 , where the dyadic and
scalar products are linear in both arguments, and 1 is a unit dyad, allows Eq. (3.7) to be
rewritten as,
(k0 + iκ0 , ω0 ) · E0 = 0,
(3.9)
with the dielectric tensor, (k0 + iκ0 , ω0 ), defined in a manner similar to the ones from
[Bernstein, 1975] and [Bravo-Ortega and Glasser, 1991],
2
ωpe
c2 (k0 + iκ0 )(k0 + iκ0 )
c2 (k0 + iκ0 ) · (k0 + iκ0 )
(k0 + iκ0 , ω0 ) =
M;
+
1
−
1
−
ω0
ω02
ω02
(3.10)
q
(0)
ωpe = e2 Ne /(0 me ) is the angular electron plasma frequency. Since Eq. (3.9) is a
homogeneous linear equation a necessary requirement for it to have a non-trivial solution
(E0 6= 0) is that (k0 + iκ0 , ω0 ) be a singular matrix [Arfken and Weber, 2005]. This
requirement yields a (linear) dispersion relation similar to the ones considered earlier for
electrostatic waves,
(k0 + iκ0 , ω0 ) = det[(k0 + iκ0 , ω0 )] = 0,
(3.11)
and from which connections between k0 , κ0 , and ω0 for the linear plasma modes may be
determined. Just as in the electrostatic case, the assumptions |κ0 | |k0 | and |Re()| |Im()|, except exactly at a linear mode, allow Eq. (3.11) to be expanded around k0 (Re()
23
to order 1 and Im() to order 0). Doing this, the real and imaginary parts of the dispersion
relation yield, respectively,
∂ Re[(k, ω0 )] Re[(k0 , ω0 )] = 0, Im[(k0 , ω0 )] + κ0 ·
= 0.
(3.12)
∂k
k=k0
We shall return to the imaginary part of the dispersion relation in the next chapter when
dealing with wave propagation (and absorption) in inhomogeneous plasmas. However, for
the remainder of this section our focus will be on the real part. In general, Re[(k0 , ω0 )] may
be obtained by taking the determinant of the hermitian part of (k0 , ω0 ), H[(k0 , ω0 )] =
[(k0 , ω0 ) + (T )∗ (k0 , ω0 )]/2 [Arfken and Weber, 2005]. With from Eq. (3.10), and M
from Eq. (3.8), we have (for B along the z-axis)
c2 k0 k0
H[(k0 , ω0 )] =
+ 1−
ω02

S − n2y − n2z
=  iD + nx ny
nx nz
c2 k02
ω02
1−
2
ωpe
H(M)
ω0

−iD + nx ny
nx nz
,
ny nz
S − n2x − n2z
2
2
ny nz
P − nx − ny
(3.13)
where
n=
ck0
,
ω0
2 (ω 2 − ω 2 + ν 2 )
2
ωpe
ωpe
ce
0
ei
,
P
=
1
−
2 ,
2 − ν 2 )2 + 4ω 2 ν 2
(ω02 − ωce
ω02 + νei
0 ei
ei
2 (ω 2 − ω 2 − ν 2 )
ωpe
ωce
ce
0
ei
D=
.
2 − ν 2 )2 + 4ω 2 ν 2
ω0 (ω02 − ωce
0 ei
ei
S =1−
(3.14)
Taking, without loss of generality, n = n[sin(θ), 0, cos(θ)], with θ ∈ [0, π[ being the angle
between k0 and B and where the refractive index, n, is allowed to take on both positive
and negative values, and evaluating Re[(k0 , ω0 )] = det{H[(k0 , ω0 )]} = 0, the following
dispersion relation is obtained,
[S sin2 (θ)+P cos2 (θ)]n4 −{(S 2 −D2 ) sin2 (θ)+P S[1+cos2 (θ)]}n2 +P (S 2 −D2 ) = 0. (3.15)
This bi-quadratic equation is easily solved, yielding, upon some further simplifications of
the discriminant,
p
(S 2 − D2 ) sin2 (θ) + P S[1 + cos2 (θ)] ± (S 2 − D2 − P S)2 sin4 (θ) + 4P 2 D2 cos2 (θ)
2
n =
.
2[S sin2 (θ) + P cos2 (θ)]
(3.16)
While the above expression represents a general dispersion relation for electromagnetic
waves in a cold plasma, its interpretation is not immediately obvious. We can, however,
already say a number of things from the basic structure of Eq. (3.16). First of all, the
terms in the square root are non-negative, so n is either real or purely imaginary. The
present treatment is only valid for real n, which is also the only case in which propagating
cold electromagnetic plasma waves exist. Second, there are generally two different plane
24
wave modes which are allowed to propagate in a magnetised plasma, one corresponding to
each sign in front of the square root term. This is not unique to a magnetised plasma, but
a general feature of anisotropic optical media [Born and Wolf, 2002]. Since the dispersion
relation has the form of a generalised eigenvalue problem, the polarisations (directions of
E0 ) for the two modes will be orthogonal when they are non-degenerate [Born and Wolf,
2002].
The two modes may, in one scheme, be termed ordinary (O) and extraordinary (X); for
S 2 − D2 < P S the plus corresponds to the O-mode and the minus to the X-mode, the
situation is inverted for S 2 −D2 > P S, the motivation behind these definitions will become
clear. In order to describe the features of the O-mode and the X-mode we consider the
cases of propagation perpendicular and to B, θ = π/2, and parallel to B, θ = 0. For
perpendicular propagation, the dispersion relation reduces to
n2 = P (for O-mode),
n2 =
S 2 − D2
(S + D)(S − D)
=
(for X-mode).
S
S
(3.17)
For parallel propagation, the plasma also supports two modes, which we shall call the Rmode and the L-mode; for P > 0 the R-mode corresponds to the X-mode and the L-mode
to the O-mode, the situation is inverted for P < 0. These modes satisfy the dispersion
relations
n2 = S + D (R-mode), n2 = S − D (L-mode).
(3.18)
In the case of perpendicular propagation, the O-mode dispersion relation is independent
2 in the absence of collisions, just as in an unmagnetised plasma
of B, n2 = 1 − ω02 /ωpe
[Bellan, 2006], while the X-mode dispersion relation depends strongly on B. In the case
of parallel propagation, both modes depend on the magnetic field, seen from the fact that
2 /[ω (ω ± ω )] in the absence of collisions. The R-mode corresponds to
S ± D = 1 − ωpe
0 0
ce
a right hand polarised wave and the L-mode to a left hand polarised wave, which is the
motivation behind the names [Bellan, 2006]. The fundamental properties of the dispersion
relations, e.g., whether or not the plasma supports propagating waves and the topology
of wave normal surfaces (plots of 1/n), change at points where n2 → 0 (cut-offs) and
n2 → ±∞ (resonances) for θ = π/2 and θ = 0; this point is discussed in many standard
texts, e.g., [Stix, 1992], [Swanson, 2003], and [Bellan, 2006]. We therefore proceed to discuss
these principal cut-offs and resonances, as well as their relation to parametric decay.
At cut-offs the pump wavelength becomes infinite, k0 → 0. Propagating waves will, in
general, exist only on one side of the cut-off in plasma parameter space, and propagating
waves encountering a cut-off, at a plasma-vacuum interface or in a slightly inhomogeneous
plasma, will be reflected or, if their amplitude is sufficiently large, decay parametrically;
linear absorption at cut-offs is normally very small, since k0 → 0 and, generally, κ0 /k0 1.
The reason that parametric decay is likely to occur near a cut-off is the field enhancement
associated with waves "piling up" before being reflected [White and Chen, 1974]. We defer
a detailed discussion of such field enhancement to Chapter 4.
In the absence of collisions, the O-mode has a cut-off for θ = π/2 when ω0 → ωpe , with
2 /ω 2 < 1. This is the well known cut-off at the
propagating waves (n2 > 0) existing for ωpe
0
25
plasma frequency which is also present in unmagnetised plasmas and metals. The X-mode
has cut-offs for θ = π/2 when S ± D → 0, which are also cut-offs of the R-mode and the
L-mode for θ = 0, respectively. In the absence of collisions, the R- and L-cut-offs occur for
2 /ω 2 → 1 ∓ |ω |/ω , since ω
ωpe
ce
0
ce = −eB/me < 0. The condition for propagating waves
0
2 /ω 2 < 1 ∓ |ω |/ω . The occurrence of a resonance between
to exist is in all cases ωpe
ce
0
0
2 /ω 2 > 0, means that the L-cut-off cannot, in
the cut-offs, along with the requirement ωpe
0
general, be ignored for the X-mode at θ = π/2, although it provides a seemingly less strict
limit than the R-cut-off, the situation will be visualised later. We note that both the R2 /ω 2 → (1 − ω 2 /ω 2 )2 .
and L-cut-offs are characterised by ωce
pe
0
0
Parametric decay near cut-offs is frequently observed in ionospheric modification experiments, where strong electromagnetic, usually O-mode, radiation is used to pump the ionospheric plasma, whose response, in the form of stimulated electromagnetic emission, may
be recorded by ground based receiving stations, and show many features attributable to
parametric decay. The parametric decay processes in these experiments usually take place
2 /ω 2 1,
near the ω0 = ωpe cut-off. However, the ionospheric plasma is very overdense, ωpe
ce
and thus the R- and L-cut-offs, as well as the upper hybrid resonance, to be discussed later,
are located quite close to one another, allowing a large number of parametric processes to
occur for the incident O-mode radiation, and the reflected O-mode and X-mode radiation.
A review is given by [Leyser, 2001]. In tokamaks and other laboratory plasmas, reflectom(0)
etry diagnostics probe Ne and other plasma parameters by collecting electromagnetic
radiation reflected by the various cut-offs [Mazzucato, 2014]. Since there is no need to use
high-power sources for these measurements, the possibility of parametric decay is virtually
non-existent, and we shall, consequently, not devote any more effort to the discussion of
parametric decay near cut-offs in tokamak plasmas.
At resonances the pump wavelength becomes infinitesimal, k0 → ∞. In general, propagating waves will also exist only on one side of a resonance. There are, however, a larger
number of possible outcomes of a propagating wave encountering a resonance than a cutoff: the wave may be absorbed linearly, converted to a different type of wave linearly or,
again if its amplitude is sufficiently large, decay parametrically. Linear absorption is likely
to occur at a resonance since k0 → ∞, while κ0 /k0 1 usually remains relatively constant; linear mode conversion is likely to occur if linear absorption or parametric decay is
insignificant, and parametric decay may, just as near a cut-off, be likely to occur due to
field enhancement [White and Chen, 1974].
From Eqs. (3.17) and (3.18), along with the fact that P and S − D never diverge, since ω0
is positive and comparable to ωpe , it is clear that resonances only occur at θ = 0 or θ = π/2
for S + D → ∞ or S → 0. We therefore examine these cases separately. In the absence
2 /(ω − |ω |), and consequently, S + D → ∞, which is an
of collisions, S + D = 1 − ωpe
0
ce
R-mode resonance at θ = 0, occurs for ω0 → |ωce |. This is the so-called electron cyclotron
2 /ω 2 > 1; no cyclotron resonance occurs for the
resonance; propagating waves exist for ωce
0
X-mode at θ = π/2, since S → ∞ for ω0 → |ωce |, in fact (S + D)/S → 0 here. Hot plasma
theory reveals strong collisionless linear (cyclotron) damping near the electron cyclotron
resonance and its harmonics, which is, unsurprisingly, the mechanism exploited in electron
cyclotron resonance heating of tokamak, as well as other magnetised, plasmas [Freidberg,
26
Figure 3.1 – CMA diagram illustrating
regions with qualitatively similar propagation characteristics for cold electro2
magnetic waves in terms of ωpe
/ω02 (nor(0)
2
/ω02 (normalised
malised Ne ) and ωce
2
B ), neglecting collisional effects. The
qualitative behaviour changes at principal cut-offs/resonances, i.e., the O-cut-off
ω0 = ωpe (blue line), the R-cut-off (orange line), the L-cut-off (gold line), the
R-resonance ω0 = |ωce | (cyclotron resonance, purple line), and the upper hybrid (UH) resonance (green line). Areas shaded yellow support no propagating X-mode, the area shaded orange supports no propagating O-mode, and the
area shaded red supports no propagating
cold electromagnetic waves at all.
Figure
3.2 – CMA
contours for ω0 /(2π) =
105 GHz in ASDEX Upgrade
shot
32563
at
t = 2.900 s (equilibrium reconstructed using TS). The
colour coding of principal
cut-offs/resonances is the
same as in Fig. 3.1; only
the R-cut-off, R-resonance,
and
UH-resonance
are
present in the plasma for
ω0 /(2π) = 105 GHz. Background contours indicate
plasma and vessel locations.
(R, z) are cylindrical coordinates, the z-axis is the
torus symmetry axis.
2 /(ω 2 − ω 2 ) and S → 0, the X-mode resonance at
2007]. Without collisions, S = 1 − ωpe
ce
0
2
2
2
θ = π/2, occurs for ω0 → ωpe + ωce = ωU2 H . This is the long advertised upper hybrid
q
2 + ω 2 is the (angular) upper hybrid frequency; propagating
resonance and ωU H = ωpe
ce
2 /ω 2 + ω 2 /ω 2 > 1. Linear damping at the upper hybrid
waves exist for ωU2 H /ω02 = ωpe
ce
0
0
resonance is usually quite small unless it (nearly) coincides with the cyclotron resonance
or one of its harmonics, and thus linear mode conversion or parametric decay are very
likely to happen near it.
The propagation characteristics resulting from the above cut-offs/resonances may, in the
absence of collisions, be visualised using the CMA (Clemmow-Mullaly-Allis) diagram, seen
in Fig. 3.1. The diagram shows the lines representing cut-offs/resonances at θ = 0 and
2 /ω 2 (normalised N (0) ) along the x-axis
θ = π/2 in a plasma parameter plane with ωpe
e
0
27
2 /ω 2 (normalised B 2 ) along the y-axis. These lines separate the plasma parameter
and ωce
0
plane into 8 areas which permit qualitatively different propagation of cold electromagnetic
plasma waves, when only electron dynamics are included. The propagation characteristics
may be visualised by plotting the wave normal surfaces in each area, as is done in many
standard texts, e.g., [Stix, 1992], [Swanson, 2003], and [Bellan, 2006], but are essentially
determined by whether or not propagation of the O-mode or the X-mode is allowed at
θ = 0 and/or θ = π/2 around the cut-offs/resonances. In Fig. 3.1 we only indicate the
areas in which no propagating X-mode (yellow areas), no propagating O-mode (orange
area) or no propagating cold electromagnetic waves at all (red area) exist, which is the
case if no propagating waves exist at both θ = 0 and θ = π/2 for the mode(s) in question.
This is by no means an exhaustive characterisation of the propagation characteristics in
2 /ω 2 > 1
each area. For instance, a propagating O-mode exists, with our definitions, for ωpe
0
2
2
and ωce /ω0 > 1, but this is a so-called helicon mode, which does not propagate at θ = π/2,
and thus very different from the "normal" O-mode [Bellan, 2006]. However, it is sufficient
for our present purposes.
In Fig. 3.2, we show the application of CMA contours to an inhomogeneous tokamak
plasma. Specifically, that of ASDEX Upgrade in shot 28286 at t = 2.900 s (reconstructed
using TS), with an electromagnetic pump wave frequency ω0 /(2π) = 105 GHz, corresponding to the frequency of the CTS gyrotron used in this shot. Due to the assumed time
invariance of the equilibrium, which is good given the short wave-plasma interaction time,
and linear plasma response, ω0 remains constant throughout the plasma; this point is dis2 /ω 2 and ω 2 /ω 2 are just
cussed in more detail in Chapter 4. Thus, the CMA variables ωpe
ce
0
0
(0)
normalised versions of Ne
plasma equilibrium.
and B 2 , whose spatial variation are given by the reconstructed
Fig. 3.2 illustrates the basic accessibility problems for electromagnetic waves in tokamak
plasmas, and especially the ones encountered in CTS at ASDEX Upgrade. It is easy,
at least in theory, to choose a pump frequency that is sufficiently high (low) to allow
the electromagnetic O-mode (X-mode) access to the whole plasma, which is desired in
scattering experiments; in ASDEX Upgrade high-frequency, above |ωce |/(2π), O-mode is
used for CTS. However, access to the electron cyclotron and upper hybrid resonances
is limited for X-mode radiation launched from the low-field side, due to their location
(0)
behind the R-cut-off, resulting from the peaked nature of Ne around the plasma centre
and the fact that B ∝ 1/R from Ampère’s law applied to the axis-symmetric tokamak,
neglecting plasma currents. The electron cyclotron resonance is of course accessible to
X-mode radiation launched from the high-field side, as implemented on the Versator II
tokamak by [McDermott et al., 1982]; if strong absorption occurs around this resonance,
the upper hybrid resonance will, however, still remain inaccessible. In addition, we note
that high-field side launches are rarely used in tokamaks, neither for electron cyclotron
resonance heating nor for scattering diagnostics. This is due to the limited space available
for launch ports/mirror systems, as well as the possibility of heating in O-mode or at higher
harmonics of the electron cyclotron frequency in both modes from the low-field side for
electron cyclotron resonance heating [Freidberg, 2007], and the easy accessibility to the
entire plasma, for high-frequency O-mode and (moderately) low-frequency X-mode, for
28
scattering diagnostics.
This lack of accessibility and necessity to heat/probe tokamak plasmas by means of a
parametric decay instability are the likely explanations for the relatively limited interest
in parametric decay near the upper hybrid resonance within the tokamak community since
the 1980s. As seen in Fig. 3.2, and as will be demonstrated numerically in Chapter 4, the
upper hybrid resonance in ASDEX Upgrade may, nevertheless, be available to 105 GHz
O-mode launched from the low-field side via a reflection from the high-field side vessel
wall. This proposal of course requires a significant fraction of the reflected power to be
coupled into the plasma in X-mode, and that the power is not absorbed at the electron
cyclotron resonance, to fulfil the latter assumption it is generally necessary for the electron
cyclotron resonance layer to be outside the bulk plasma. While the correctness of the
above assumptions depends quite delicately on the launch geometry, and precise spatial
(0)
variation of Ne and B, we shall see that they may often be satisfied for the CTS launch
geometry in ASDEX Upgrade, and that anomalous scattering is strongly correlated with
their fulfilment. The previously mentioned likelihood of parametric decay of strong Xmode radiation near the upper hybrid resonance, coupled with the correlation between
accessibility to the upper hybrid resonance layer and anomalous scattering, showing many
spectral features attributable to a parametric decay instability, leads us to suspect that
parametric decay near the upper hybrid layer plays an important role in this phenomenon.
This clearly warrants a more detailed investigation.
3.2
Electrostatic High-Frequency Daughter Waves at the Upper Hybrid Resonance
Having discussed the pump wave dispersion relation and established the significance of the
upper hybrid resonance, we proceed to discuss the high-frequency electrostatic daughter
waves which may be generated near the upper hybrid resonance. While cold plasma theory
does describe essentially electrostatic waves with propagation at θ ≈ π/2 near the upper
hybrid resonance [Bellan, 2006], these are basically continuations of the electromagnetic
modes described earlier, and thus do not satisfy the requirements of small wavelength
compared with the pump wave. In addition, their group velocities are ordinarily quite
large, leading to large convective losses for the inhomogeneous cases in Chapter 4. The
modes, in which we are interested, require a kinetic treatment and are (a limiting case
of) so-called electron Bernstein waves, first described by [Bernstein, 1958]. They still
satisfy θ ≈ π/2, since they are subject to heavy Landau/cyclotron damping for θ 6≈ π/2
[Rasmussen, 1994], but generally have much smaller wavelengths than the cold waves, as
well as much smaller group velocities and consequently smaller convective losses in the
inhomogeneous case [Ott et al., 1980].
As all linear electrostatic waves, the electron Bernstein waves are characterised by a linear dispersion relation Re[(k, ω2 )] = 0, as well as a linear
P damping rate Γ(k, ω2 ) =
Im[(k, ω2 )]/(∂ Re[(k, ω2 )]/∂ω)|ω=ω2 , where (k, ω) = 1 + i χi (k, ω) + χe (k, ω). Neglecting, as always, the high-frequency ion response, we find that the electron Bernstein
29
waves are characterised by
1 + Re[χe (k, ω2 )] = 0,
Γ(k, ω2 ) =
Im[χe (k, ω2 )]
.
(∂ Re[χe (k, ω)]/∂ω)|ω=ω2
(3.19)
In order to determine these characteristics, we clearly need at model for χe (k, ω). As was
mentioned in Chapter 2, such a model may in general be obtained by finding a relation
(1)
between Ñe (k, ω) and Φ̃(k, ω) from the order 1 kinetic equation within a perturbation
scheme, e.g., the one given by Eq. (2.10). Different χe (k, ω) will be obtained for different
(0)
order 0 distribution functions Fe (u); we shall from now on only consider cases in which
(0)
Fe (u) is a Maxwellian distribution,
(0)
Fe(0) (u) =
Ne
2
2
e−u /vT e ,
3
3/2
π vT e
(3.20)
p
where vT e = 2Te /me is the electron thermal speed, Te is the electron temperature in
R
(0)
(0)
energy units, the distribution function is normalised such that all u Fe (u) du = Ne , and
the background electron fluid velocity has, as for the cold electromagnetic waves, been set
to zero. The Maxwellian distribution is characteristic of thermal equilibrium, hence the
inclusion of temperature in its definition; it maximises entropy for a given plasma density
and (centre of mass frame) energy [Bellan, 2006]. In the presence of collisions, and the
absence of sources, e.g., fusion generated fast particles, all (non-relativistic and classical)
distribution functions will eventually relax to a Maxwellian distribution [Helander and
Sigmar, 2001]. However, the low collisionality of fusion plasmas, along with the injection
and generation of fast particles by various processes, means that a Maxwellian is by no
means the only reasonable background distribution.
Because parametric decay (in homogeneous plasmas) is often limited by collisional damping we are going to add a simple collision operator to the right hand sides of Eqs. (2.8)
and (2.10) when determining χe (k, ω). The collision operator, which we shall employ, is
a particle conserving Bhatnagar-Gross-Krook (BGK) operator, introduced for single component systems by [Bhatnagar et al., 1954] and for two component plasmas by [Gross and
(0)
(0)
(1)
Krook, 1956]. It is given by −νe [Fe (x, u, t) − Ne (x, t)Fe (u)/Ne ] = −νe [Fe (x, u, t) −
(1)
(0)
(0)
Ne (x, t)Fe (u)/Ne ] [Pécseli, 2013], where νe is an empirical electron collision frequency,
which we shall usually identify with the electron Coulomb collision frequency. Unlike the
Fokker-Planck collision operator, it is not derived from first principles, but should rather
be viewed as a simple way of describing relaxation towards the Maxwellian distribution
(0)
Fe (u) taking place on a time scale 1/νe . The above operator ensures conservation of the
number of electrons at all times, and the susceptibilities obtained using it should generally
agree well with more sophisticated BGK treatments which also include energy conservation
at all times [Opher et al., 2002]. However, the basic 1/u3 dependence of the Coulomb collision frequency and the inherent anisotropy of the collision dynamics of charged particles in
a magnetic field is not included, so the results relying strongly on collisional effects should
be considered semi-quantitative at best.
30
With the above assumptions of a Maxwellian background distribution and a particle conserving BGK collision operator, the electron susceptibility may be determined by the
method of characteristics [Amano and Okamoto, 1969], and is (for B along the ±z-axis
and kz > 0),
∞
ω + iνe P
ω + iνe − nωce
2 r2
2
2
−k
In (k⊥ rLe ) e ⊥ Le Z
2 1+
2ωpe
kz vT e n=−∞
k z vT e
,
χe (k, ω) = 2 2
(3.21)
∞
P
iνe
ω + iνe − nωce
k vT e
2 r2
−k⊥
2
2
Le Z
1+
In (k⊥ rLe ) e
kz vT e n=−∞
kz vT e
√
√
where rLe = vT e /( 2|ωce |) = me Te /(eB) is the electron Larmor radius, In is a modified
Bessel function of the 1st kind of order n, and Z is the Fried-Conte plasma dispersion
function, the
√ real part of which is seen, along with some asymptotic approximations, in Fig.
3.3; vT e /( 2ωpe ) is the electron Debye length. This general χe (k, ω) can be used to describe
a large number of electrostatic high-frequency modes, but is generally too complicated to be
treated analytically. Fortunately, for the limiting case of electron Bernstein waves, in which
we are interested, a number of approximations are appropriate. First of all, propagation is
2 1. Further, the collision frequency is
taking place almost perpendicular to B so kz2 /k⊥
assumed to be small compared to the wave frequency νe /ω 1. Finally, the wavelength
2 r 2 1;
of the electrostatic daughter waves is large compared to rLe such that be = k⊥
Le
generally, rLe is a very small length scale in a tokamak plasma, so it is not too difficult
2 1/r 2 . Additionally, it will
to have electrostatic daughter waves with k k0 and k⊥
Le
be implicitly assumed that ω 6≈ nωce (for n ∈ Z), i.e., that the wave frequency, which
turns out to be close to the upper hybrid frequency, does not coincide with a harmonic of
the electron cyclotron frequency, such that we can take all terms to be non-resonant when
Taylor expanding.
In order to allow easier manipulation of Eq. (3.21), when obtaining the simplified dispersion
relation, we introduce the dimensionless quantities ξe = (ω+iνe )/(kz vT e ), ξνe = νe /(kz vT e ),
2 v 2 (1 + k 2 /k 2 ) = 2ω 2 b (1 + k 2 /k 2 ),
and ξce = ωce /(kz vT e ). Also noting that k 2 vT2 e = k⊥
ce e
z
z
Te
⊥
⊥
Eq. (3.21) is rewritten as,
∞
P
In (be ) e−be Z(ξe − nξce )
2
ωpe
n=−∞
χe (k, ω) =
.
∞
P
kz2
−be Z(ξ − nξ )
2
1
+
iξ
I
(b
)
e
ωce be 1 + 2
νe
n e
e
ce
k⊥
n=−∞
1 + ξe
(3.22)
Further,
using the identity In (be ) = I−n (be ) (for n ∈ Z) along with 1 = e−be [I0 (be ) +
P∞
2 n=1 In (be )] (in the numerator) [Abramowitz and Stegun, 1964], this becomes
∞
P
2
−b
e
ωpe e
I0 (be )[1+ξe Z(ξe )]+
In (be ){2+ξe [Z(ξe −nξce )+Z(ξe +nξce )]}
n=1
.
χe (k, ω) =
∞
P
kz2
2
−b
e
ωce be 1+ 2 1+iξνe e
I0 (be )Z(ξe )+
In (be )[Z(ξe −nξce )+Z(ξe +nξce )]
k⊥
n=1
(3.23)
31
Figure 3.3 – Real part of the Fried-Conte plasma dispersion function, Re[Z(ξ)] =
R∞
√
2
(1/ π)P{ −∞ [e−x /(x − ξ)] dx}, along with asymptotic approximations, for 0 < ξ <
5. The ’exact’ Z-function (blue line) is calculated using the method described by
[Froula et al., 2011], and is plotted along with the asymptotic approximations used
in this work: −1/ξ (orange line, generally accurate for |ξ| > 3), −1/ξ − 1/(2ξ 3 )
(gold line, generally accurate√for |ξ| > 2), and −1/ξ − 1/(2ξ 3 ) − 3/(4ξ 5 ) (purple line,
generally accurate for |ξ| > 2); see, e.g., [Bellan, 2006] for a derivation of these.
2 k 2 r 2 = b 1, we have that |ξ | ∼ ξ
Because kz2 rLe
e
e
ce ∼ 1/(kz rLe ) 1, and we
⊥ Le
are thus justified in using an asymptotic approximation for Z(ξe ∓ nξce ). Since In (be ) =
√
2
|n|
O(be ) for be 1, we use the approximation Z(ξe ) ≈ −1/ξe − 1/(2ξe3 ) + i π e−ξe for the
(n = 0)-term in the numerator. This is accurate for |ξe | > 2, from Fig. 3.3; the Landau
√
2
damping term, i π e−ξe , additionally requires |νe ω/(kz2 vT2 e )| . π/4, which may be satisfied
since νe /ω 1, [Stix, 1992]. For the (n 6= 0)-terms, the less accurate approximation
Z(ξe ∓nξce ) ≈ −1/(ξe ∓nξce ) (see Fig. 3.3) is sufficient in the numerator. We note that the
terms involving Z the denominator are proportional to the small factors ξνe /(ξe ∓ nξce ) =
νe /(ω+iνe ∓nωce ) ≈ νe /(ω∓nωce ), such that we can use Z(ξe ) ≈ −1/ξe for the (n = 0)-term
and Z(ξe ∓nξce ) ≈ 0 for the (n 6= 0)-terms in the denominator. With these approximations
2
∞
P
√
1
n2 ξce
2
2
−b
−ξ
e
e
ωpe e
I0 (be ) − 2 + i πξe e
−2
In (be ) 2
2
2ξe
ξe − n2 ξce
n=1
χe (k, ω) ≈
.
(3.24)
kz2
ξνe
2
−b
e
ωce be 1 + 2
1 − i I0 (be ) e
ξe
k⊥
2 /(ω + iν )2 ](k 2 /k 2 ) ≈ (ω 2 /ω 2 )(k 2 /k 2 ) Now, ξνe /ξe ≈ νe /ω 1, 1/(2be ξe2 ) = [ωce
e
z
ce
z
⊥
⊥
2
2
2
2 2
−ξ
3
−ξ
2
3
3
3
2 /ω 2 )(k 2 /k 2 ) 1 (since
1, ξe e e /be = ξe e e /(ξe be ) ≈ [ω /(kz vT e )] e−ω /(kz vT e ) (2ωce
z
⊥
2
2 2
2 /(ξ 2 − n2 ξ 2 ) = n2 ω 2 /[(ω + iν )2 − n2 ω 2 ] ≈
[ω 3 /(kz3 vT3 e )] e−ω /(kz vT e ) ≤ 0.4099), and n2 ξce
e
e
ce
ce
ce
2 /(ω 2 − n2 ω 2 ) − i[2n2 ω 2 ω 2 /(ω 2 − n2 ω 2 )2 ](ν /ω), approximations are order 1 in
n2 ωce
e
ce
ce
ce
32
2 1 and ν /ω 1, so to this accuracy (remembering that 1/(1 + x) ≈ 1 − x),
kz2 /k⊥
e
2
ωpe
√ ω 3 −ω2 /(k2 v2 )
kz2
νei
kz2
−be
−be
z
T
e
−1 + i2 π 3 3 e
− 1 − 2 + i I0 (be ) e
χe (k, ω) ≈ 2 2 I0 (be ) e
ω k⊥
ω
k z vT e
k⊥
∞
∞
2
2
2 ω2
−b
−b
n ωpe
n2 ωpe
νe 4e e X
2e e X
.
In (be ) 2
+
i
I
(b
)
×
n e
2
2 )2
be
ω − n2 ωce
ω be
(ω 2 − n2 ωce
n=1
n=1
(3.25)
The approximate version of χe (k, ω), which we have sought, is finally obtained by expand2 , and ν /ω, using that ebe = 1 − b + O(b2 ),
ing the above equation to order 1 in be , kz2 /k⊥
e
e
e
|n|
2
3
2
4
I0 (be ) = 1 + O(be ), I1 (be ) = be /2 + O(be ), I2 (be ) = be /8 + O(be ), and In (be ) = O(be ),
2
2 ω2 b
ωpe
3ωpe
ce e
−
2
2
2
2
2 )
ω − ωce (ω − ωce )(ω 2 − 4ωce
(3.26)
2
2 (ω 2 + ω 2 )
2
ωpe
ωpe
√ ω 3 −ω2 /(k2 v2 ) kz2
ωce
ce νe
z
T
e
+ 2
+ i2 π 3 3 e
2 + i (ω 2 − ω 2 )2 ω .
2
ω
ω 2 − ωce
k z vT e
k⊥
ce
χe (k, ω) ≈ −
If we set νe = νei , the collisional part of this susceptibility is the same as that used by [Stefan
and Bers, 1984] to study parametric processes during electron cyclotron resonance heating
of tokamaks, when θ ≈ π/2, which has been assumed. To obtain the dispersion relation for
the electron Bernstein waves described by this equation we compute 1 + Re[χe (k, ω2 )] = 0,
1−
2
2 ω2 b
2 ω2
ωpe
3ωpe
ωpe
kz2
ce e
ce
−
+
= 0.
2 ) k2
2
2 )(ω 2 − 4ω 2 )
ω22 − ωce
(ω22 − ωce
ω22 (ω22 − ωce
ce
2
⊥
(3.27)
2 . While it is possible to
This is equivalent to a cubic equation for ω22 , given be and kz2 /k⊥
solve such an equation, we can get a more tractable form by making a few approximations,
in keeping with what we have already assumed. The starting point is to note that, to order 0
2 , ω 2 ≈ ω 2 ; this also shows that the dispersion relation does indeed support
in be and kz2 /k⊥
2
UH
waves with (angular) frequencies near ωU H , which are important for parametric decay near
2 ∆
the upper hybrid resonance. Now, inserting ω22 = ωU2 H + ωpe
U H , where |∆U H | 1 is
2 . Doing this, we
assumed, Eq. (3.27) may be expanded to order 1 in ∆U H , be , and kz2 /k⊥
find
2 k2
be
ωce
z
∆U H ≈ −
−
(3.28)
2 ,
2 /(3ω 2 )
1 − ωpe
ωU2 H k⊥
ce
and thus
ω22
≈
ωU2 H
−
2
ωpe
2 k2
be
ωce
z
+ 2
2 ,
2 /(3ω 2 )
1 − ωpe
ωU H k ⊥
ce
(3.29)
which is the dispersion relation used by [Sharma and Shukla, 1983] and [Murtaza and
Shukla, 1984] to discuss electron Bernstein waves generated near the upper hybrid layer;
we note that it is also similar to the dispersion relation used by [Porkoláb, 1982] in the limit
2 /ω 2 , as assumed in that paper, if we let k 2 /k 2 → k 2 /k 2 which only introduces
of small ωpe
ce
z
z
⊥
4 . The above dispersion relation has a cut-off at ω = ω
deviations of order kz4 /k⊥
2
U H ; at
33
this point the cold upper hybrid waves and the hot electron Bernstein waves coalesce and
linear mode conversion may occur. Note that mode conversion occurs near a resonance for
the cold upper hybrid waves and near a cut-off for the electron Bernstein waves, showing
that the electron Bernstein waves usually have k k0 , as previously claimed. At θ√= π/2,
i.e., kz = 0, propagating waves, with
√ ω2 ∈ R, exist for ω2 < ωU H when ωpe < 3|ωce |,
and for ω2 > ωU H when ωpe > 3|ω
√ce |. Additionally, the waves are backward propagating, ∂ω2√
(k)/∂k ∝ −k, for ωpe < 3|ωce |, and forward propagating, ∂ω2 (k)/∂k ∝ k,
for ωpe > 3|ωce |. Both the above observations have important consequences for the
parametric
hybrid resonance, separating the underdense
√ decay instability near the upper
√
(ωpe < 3|ωce |) and overdense (ωpe > 3|ωce |) cases [Sharma and Shukla, 1983]. Conventional tokamak plasmas
√ are underdense, generally satisfying ωpe . |ωce |. We are further
assuming that ωpe 6≈ 3|ωce |, since the upper hybrid resonance is assumed to be removed
from the 2nd harmonic electron cyclotron resonance; if the latter assumption is not fulfilled,
parametric decay may be inhibited due to bandgap effects [Leyser et al., 1994]. The fact
that underdense plasmas support ω2 < ωU H means that propagating electron Bernstein
waves exist on the same side of the upper hybrid resonance as the propagating X-mode,
which satisfies ω0 < ωU H , see Fig. 3.1, and allows parametric decay to occur without
an initial linear mode conversion of the X-mode radiation, as is usually assumed in the
overdense case, e.g., by [Gusakov and Surkov, 2007].
After having discussed the dispersion relation of the electron Bernstein waves, we turn
our attention to their damping rate, Γ(k, ω2 ) = Im[χe (k, ω2 )]/(∂ Re[χe (k, ω)]/∂ω)|ω=ω2 .
This is easily obtained from Eq. (3.26). However, since the numerator is already a small
quantity, we can use an order 0 approximation for the denominator, ∂ Re[χe (k, ω)]/∂ω ≈
2 /(ω 2 − ω 2 )2 , with which
2ωωpe
ce
Γ(k, ω2 ) ≈
2 )2
2
2
√ (ω22 − ωce
ω22 + ωce
ω22 + ωce
2 )
−ω22 /(kz2 vT
e +
π
e
ν
≈
νe ;
e
2 v3
kz k⊥
2ω22
2ω22
Te
(3.30)
the last approximation stems from the fact that ω2 /(kz vT e ) 1, which makes the Gaussian
in the Landau damping term extremely small; the motivation for including Landau damping at all is its possible significance in damping the low-frequency daughter waves. This
damping rate is important for the threshold of parametric decay into electron Bernstein
modes and low-frequency (quasi-)modes near the upper hybrid resonance in a homogeneous
plasma.
3.3
Electrostatic Low-Frequency Daughter Waves
We have already discussed and described the electromagnetic pump wave and the highfrequency daughter wave generated near the upper hybrid resonance. So now, all that
remains, before discussing the parametric decay instability near the upper hybrid resonance, is a discussion of the low-frequency daughter waves involved in the instability. In
order to determine these, we shall make the assumption that the low-frequency electron
response is also governed by Eq. (3.26); this imposes a much stricter limit on the smallness
34
2 than for the high-frequency waves, due to the assumption ω /(k v ) 1, reof kz2 /k⊥
1
z Te
lated to use of the asymptotic expansions of Z; however, the approximation Z(ξ) ≈ −1/ξ
is already accurate for |ξ| > 3, from Fig. 3.3, so this requirement is not quite as strict as
2 in the
it may seem. With these assumptions, and neglecting ω12 in comparison with ωce
sum/difference terms, we find the following low-frequency electron susceptibility,
2
2
2 2
ωpe
3ωpe
ωpe
ωpe
√ ω13 −ω2 /(k2 v2 ) kz2
νe
z
1
Te
χe (k, ω1 ) ≈ 2 −
b
+
π
.
−1
+
i2
e
+
i
e
2
3
2
2
2
3
ωce
4ωce
ωce ω1
ω1
kz vT e
k⊥
(3.31)
2 is a large number ω 2 /ω 2 & m /(Z m ) ≥ 1836, which again
The pre-factor of kz2 /k⊥
i
i e
pe
1
2 when using the this version of
shows the strict requirement on the smallness of kz2 /k⊥
χe (k, ω1 ) to describe the electron response. The low-frequency waves described, when using the above expression, are, as we shall see, pure ion Bernstein waves and lower hybrid
waves [Rasmussen, 1994]. These modes differ from most low-frequency modes in having
|χe (k, ω1 )| ∼ |χe (k, ω2 )| ∼ 1, rather than |χe (k, ω1 )| |χe (k, ω2 )|, which is assumed to be
generally true in some reviews describing parametric decay, e.g., [Porkoláb, 1978]. This is
related to them having a nearly adiabatic electron response along the magnetic field lines,
and not the usual isothermal low-frequency electron response, as a consequence of k being
almost perpendicular to B. Low-frequency modes having an electron response different
from the one above, particularly the ion-acoustic mode, may be important for parametric
decay of electron Bernstein waves generated by linear mode conversion or parametric decay in tokamaks [Sharma and Shukla, 1983]. However, as we are considering parametric
decay of the non-converted X-mode radiation, and following the experimental evidence of
[McDermott et al., 1982] and more recent observations from ASDEX Upgrade seen in Fig.
4.2, we shall not consider these modes analytically. They will, however, be included in the
numerical work of Chapter 4.
Apart from χe (k, ω1 ), we also need a model for χi (k, ω1 ) in order to describe the lowfrequency modes. In general, χi (k, ω) is given by an expression similar to Eq. (3.21),
assuming Maxwellian ion background distributions, with all quantities modified to describes those of the ionic species in question. To obtain a tractable expressions, we shall,
from now on, consider a simple plasma having only one ionic species. This is sufficient
for most current day tokamak experiments, including the ones from ASDEX Upgrade to
be analysed in Chapter 4. However, to model a realistic fusion plasma, would require us
to include at least one additional ionic species. In a simple plasma, background charge
p
(0)
(0)
neutrality requires Ni = Ne /Zi , such thatpωpi =
Zi me /mi ωpe , and, in addition
to this, ωci = (Zi me /mi )ωce ; thus ωpi /ωci = mi /(Zi me )ωpe /ωce 1, for a tokamak
plasma with ωpe ∼ ωce . The Coulomb collision frequencies for p
electrons and ions (with
Maxwellian background distributions) further satisfy νi /νe =
2Zi me /mi (Zi Te /Ti )3/2
[Braginskii, 1965], and, as all ionic species species satisfy, Zi me /mi 1 and Zi ∼ 1,
while tokamaks have Te /Ti ∼ 1, it holds that νi /νe 1, which justifies neglecting collisional effects on the ionic susceptibility. Collisions do, however, have some subtle effects
on the ion response.
These effects stem from the fact that ωci /|ωce | = Zi me /mi 1 such
p
that νi /ωci = mi /(Zi me )(Zi Te /Ti )3/2 νe /|ωce | νe /|ωce |, and therefore, while in general
νe /|ωce | 1, ions may satisfy νi /ωci & 1, in which case they experience a significant de35
flection due to collisions before executing a single Larmor orbit. In this case, even though
collisional damping is negligible for the ions, it is appropriate to treat the ions using a
susceptibility neglecting the effect of B for ω1 ωci , as ion orbits are virtually straight
lines in this case. An important modification of the ion response occurring as a result of
this is the appearance of Landau damping perpendicular to B, due to the loss of gyrophase
information caused by collisions [Swanson, 2003].
The above considerations yield two basic models for χi (k, ω1 ). In the case of νi /ω1 & 1 and
ω1 ∼ ωpi ωci , we may use χi (k, ω1 ) from an unmagnetised hot plasma with a Maxwellian
background ion distribution (neglecting collisions) [Bellan, 2006],
χi (k, ω1 ) =
2
2ωpi
k 2 vT2 i
ω
Z
1+
kvT i
ω1
kvT i
.
(3.32)
The waves described by this dispersion relation are (warm) lower hybrid waves, and will
be discussed shortly. We note that, while the condition νi /ωci & 1 may be satisfied for the
relatively low B and Ti in small tokamaks like Versator II, or in ionospheric plasmas when
collision frequencies are modified to include neutrals, it is at most marginally satisfied near
the edge of larger tokamaks like ASDEX Upgrade. When νi /ω1 1, the above model
is not always appropriate, and ions should be considered magnetised. It is convenient to
use the susceptibility from Eq. (3.23) as our starting point, modifying all quantities to
describe the ionic ones, and neglecting collisions, with which
∞
P
2
−b
i
ωpi e
I0 (bi )[1+ξi Z(ξi )]+
In (bi ){2+ξi [Z(ξi −nξci )+Z(ξi +nξci )]}
n=1
χi (k, ω1 ) =
;
2 b (1 + k 2 /k 2 )
ωci
i
z
⊥
(3.33)
2 r 2 = k 2 v 2 /(2ω 2 ) =
for reference, ξi = ω1 /(kz vT i ), ξci = ωci /(kz vT i ), and bi = k⊥
ci
Li
⊥ Ti
[Ti /(Zi Te )][mi /(Zi me )]be be . Since vT i vT e , we have ξi |ξe | 1, which justifies
using the asymptotic approximation Re[Z(ξi ∓ nξci )] ≈ −1/(ξi ∓ ξci ) for all n, assuming
ξi 6= nξci for any integer n, only excluding a very narrow band around each harmonic of
2 Z m /m 1, so we can also neglect any effects
ωci . It is further assumed that kz2 /k⊥
i e
i
2
2
of finite kz /k⊥ , and inserting the definitions of ξi and ξci , we thus have,
∞
2
n2 ωpi
2e−bi X
Re[χi (k, ω1 )] ≈ −
In (bi ) 2
2 .
bi
ω1 − n2 ωci
(3.34)
n=1
Because bi be , we do not have any immediately obvious way of simplifying the above
expression, as a large number of terms may contribute; the best we can do is to obtain
a dispersion relation by combining it with Re[χe (k, ω1 )] from Eq. (3.31), which results
in a system describing an infinite number of modes, one between each harmonic of ωci
[Bernstein, 1958]. These are the so-called pure ion Bernstein modes. They are "pure",
since they closely resemble their electron counterpart, due to the virtually adiabatic electron response along B, in contrast to the neutralised ion Bernstein waves, where electrons
respond virtually isothermally along B [Rasmussen, 1994]. For high-order ion Bernstein
36
modes, with ω1 ωci and/or bi 1, it is possible to invoke some approximations simplifying the the problem considerably, see, e.g., [Piliya, 1994] and [Swanson, 2003]. The
most important conclusion drawn from this procedure, as well as numerical investigations
of the resulting dispersion relation, is that the results obtained using the unmagnetised
χi (k, ω1 ) are reproduced, but that the warm lower hybrid waves couple to ion Bernstein
waves around each harmonic of ωci [Verdon et al., 2009]; in this case, the perpendicular
"Landau" damping occurring, in a weakly inhomogeneous magnetic field, is really linear
mode conversion of the lower hybrid mode energy to ion Bernstein mode energy near each
harmonic of ωci [Swanson, 2003].
Anomalous scattering spectra contain features which are attributable to both lower hybrid
and ion Bernstein modes, as expected from the above discussion: an isolated peak at a
relatively large frequency shift, and a number of peaks at a smaller frequency shifts differing
roughly by ωci /(2π), respectively. While the lower hybrid feature has ω1 ωci , the ion
Bernstein features observed often have ω1 6 ωci , and it is therefore not from the outset
evident if the high-order ion Bernstein approximations are appropriate or not; however,
if be is not very small they most likely will be. In order to obtain tractable analytical
and numerical results, which may be benchmarked against previous treatments, for both
homogeneous and inhomogeneous plasmas, our focus will be on the lower hybrid waves
for the remainder of this work. We note that a proper treatment of decay into electron
Bernstein and ion Bernstein waves in inhomogeneous plasmas may require inclusion of
wave localisation effects setting up an absolute parametric decay instability [Gusakov and
Popov, 2010], which is outside the scope of the present work, but further investigations of
this type of decay are clearly warranted.
2 /(k 2 v 2 ){1 +
To describe the lower hybrid modes, we proceed to simplify χi (k, ω) = 2ωpi
Ti
[ω/(kvT i )]Z[ω/(kvT i )]}, in order to obtain their dispersion relation. We assume that
ω/(kvT i ) is large enough to justify an asymptotic expansion Z[ω/(kvT i )] ≈ −kvT i /ω −
√
2
2 v2 )
T i ; this assumption is somewhat restrictive,
k 3 vT3 i /(2ω 3 ) − 3k 5 vT5 i /(4ω 5 ) + i π e−ω /(k
√
but generally acceptable for ω/(kvT i ) > 2, see Fig. 3.3. In this case,
χi (k, ω1 ) ≈
2
ωpi
ω12
√
3 k 2 vT2 i
ω13 −ω12 /(k2 v2 )
Ti
−1 −
+ i2 π 3 3 e
,
2 ω12
k vT i
(3.35)
2 k2 r2 ≈
and further noting that k 2 vT2 i = (Ti /Te )(me /mi )k 2 vT2 e /ω12 = 2(Ti /Te )(me /mi )ωce
Le
2 = b (1 + k 2 /k 2 ), and the approximation is order 1 in b
2[Ti /(Zi Te )]ωci |ωce |be , since k 2 rLe
e
e
z
⊥
2 (this is equivalent to setting k ≈ k in the ion susceptibility),
and kz2 /k⊥
⊥
2
ωpi
√
3Ti ωci |ωce |
ω13 −ω12 /(k2 v2 )
⊥ Ti
χi (k, ω1 ) ≈ 2 −1 −
be + i2 π 3 3 e
.
Zi Te ω12
ω1
k ⊥ vT i
(3.36)
We remember that for ω1 ∼ ωpi , ωpe ∼ ωce , ωci |ωce | ∼ ω12 , and the term involving
be 1 is indeed a small perturbation of χi (k, ω1 ); while the Landau damping argu2 ) does contain a term of order
ment ω12 /(k 2 vT2 i ) ≈ [Zi Te /(2Ti )][ω12 /(ωci |ωce |be )](1 − kz2 /k⊥
2
2
2
2
(1/be )kz /k⊥ , it is assumed that kz /k⊥ Zi me /mi 1, which justifies the order 1 approximation of k 2 vT2 i .
37
From Eqs. (3.31) and (3.36) we may now find a linear dispersion relation, Re[(k, ω1 )] =
1 + Re[χe (k, ω1 )] + Re[χi (k, ω1 )] = 0, for the low-frequency daughter waves,
!
2
2 ω |ω |
2
2
2
ωpi
ωpe
ωpe
ωpe
Ti ωpi
kz2
ci ce
1+ 2 − 2 −3
+
b
−
(3.37)
e
2 = 0.
2
ωce
4ωce
Zi Te
ω1
ω14
ω12 k⊥
2 , may be solved using standard
This bi-quadratic equation for ω12 , in terms of be and kz2 /k⊥
techniques. However, just as for the high-frequency electron Bernstein waves, we note
2 , ω 2 ≈ ω 2 /(1 + ω 2 /ω 2 ) = ω 2 ω 2 /ω 2
2
that, to order 0 in be and kz2 /k⊥
pe
ce
ce pi
1
pi
U H = ωLH , where
ωLH = |ωce |ωpi /ωU H is the (angular) lower hybrid frequency, which also corresponds to an
X-mode resonance at θ = π/2 in cold plasma theory, neglecting collisions, if ion dynamics
2 (1 + ∆
is included and ωci ωpi , [Bellan, 2006]. We may thus insert ω12 = ωLH
LH ), with
|∆LH | 1, in Eq. (3.37), and expand the resulting equation to order 1 in ∆LH , be , and
2 . This yields,
kz2 /k⊥
!
2
ωpe
Ti ωci |ωce |
mi kz2
(3.38)
+
b
+
∆LH ≈ 3
e
2
2 ,
Zi Te ωLH
Zi me k⊥
4ωU2 H
2
2 , valid in a simple plasma, we
such that, utilising the identity ωci |ωce |/ωLH
= ωU2 H /ωpe
arrive at the dispersion relation for (electrostatic warm) lower hybrid waves,
"
#
!
2
ωpe
Ti ωU2 H
mi kz2
2
2
ω1 ≈ ωLH 1 + 3
(3.39)
+
be +
2 .
2
Zi Te ωpe
Zi me k⊥
4ωU2 H
The above expression is equivalent to the electrostatic dispersion relation quoted by [Verdon
2 /ω 2 and Z = 1 (as assumed in that article) if we let k → k which
et al., 2009] for large ωpe
i
⊥
ce
2 ) and k 4 /k 4 , as well as the one given by [Porkoláb,
only introduces errors of orders be (kz2 /k⊥
z
⊥
2 /ω 2 and Z = 1 (as assumed in that article) if we let k 2 /k 2 → k 2 /k 2
1982] for small ωpe
i
z
ce
z
⊥
4 . This dispersion relation always has a cut-off
which only introduces errors of order kz4 /k⊥
for ω1 → ωLH , and only supports propagating waves with ω1 > ωLH .
The decay rate of lower hybrid waves is Γ(k, ω1 ) = Im[(k, ω1 )]/(∂Re[(k, ω)]/∂ω)|ω=ω1 ,
where (k, ω) = 1 + χi (k, ω) + χe (k, ω). Using Im[(k, ω1 )] = Im[χi (k, ω1 )] + Im[χe (k, ω1 )],
2 /ω 3 (to order 0 in b and k 2 /k 2 ),
along with ∂ Re[(k, ω)]/∂ω ≈ ∂ Re[χi (k, ω)]/∂ω ≈ 2ωpi
e
z
1
⊥
we find
√ 2
2
ωpe
ω1
πω1
ω2
Ti ω1 −ω12 /(kz2 v2 )
2 v2 )
−ω12 /(k⊥
Ti +
Te
Γ(k, ω1 ) ≈ 2 2
e
e
ω1 + 2 21 νe , (3.40)
Zi Te kz vT e
2ωce ωpi
k⊥ vT i k⊥ vT i
which shows that lower hybrid waves may be subject to significant ion Landau damping
for all permissible directions of k, as well as electron Landau damping if kz is finite.
3.4
Parametric Decay into Electron Bernstein and Lower Hybrid Modes
In this section we shall, at long last, derive the temporal growth rate and homogeneous
threshold of a parametric decay instability near the upper hybrid resonance. Particularly,
38
we consider decay into the previously discussed (high-frequency) electron Bernstein modes
and (low-frequency) lower hybrid modes. In order to ascertain whether the parametric
decay instability is resonant or non-resonant, we first restate the dispersion relations from
Eqs. (3.29) and (3.39), remembering that ω2 = ω0 − ω1 ,
2 k2
ωce
z
2
2
2
(ω0 − ω1 ) = ωU H − ωpe A2 be + 2
,
(3.41)
2
ωU H k⊥
mi kz2
2
2
ω1 = ωLH 1 + A1 be +
;
(3.42)
2
Zi me k⊥
2 /(4ω 2 ) + [T /(Z T )](ω 2 /ω 2 )} and A = 1/[1−ω 2 /(3ω 2 )] have been introA1 = 3{ωpe
i
i e
2
pe
pe
ce
UH
UH
2 v2 ) =
duced to allow easier manipulation of the equations. Our goal is to obtain ω12 /(k⊥
Ti
2
2
2 , since
[Zi Te /(2Ti )]ω1 /(ωci |ωce |be ) in terms of the plasma and pump parameters, and kz /k⊥
this quantity, by Eq. (3.40), determines whether the lower hybrid mode is subject to
strong Landau damping or not, i.e., whether the parametric decay instability is resonant
or non-resonant. In order to accomplish this, we first insert ω12 from Eq. (3.42),
2 2
1
mi kz2
mi kz2
Zi Te ωLH
Zi Te ωpe
1
ω12
+ A1 +
+ A1 +
=
,
2 v 2 = 2T ω |ω |
2
2
be
Zi me be k⊥
2Ti ωU2 H be
Zi me be k⊥
k⊥
i
ci ce
Ti
(3.43)
and note that all that is now needed is to find an expression for 1/be in terms of the
2 . To find this expression, we once again use
plasma and pump parameters, q
and kz2 /k⊥
Eq. (3.42), plugging ω1 = ωLH
2) ≈ ω
1 + A1 be + [mi /(Zi me )](kz2 /k⊥
LH {1 + (A1 /2)be +
2 ) into Eq. (3.41), and isolating b , retaining, as always, only terms of
[mi /(2Zi me )](kz2 /k⊥
e
2,
order 1 in be and kz2 /k⊥
be ≈
2 ω 2 /ω 2 ]k 2 /k 2
ωU2 H − (ω0 − ωLH )2 + [(ω0 − ωLH )ωLH mi /(Zi me ) − ωpe
ce
UH z
⊥
2
A2 ωpe − A1 (ω0 − ωLH )ωLH
2
ω 2 − (ω0 − ωLH )2 + (ω0 − 2ωLH )ωLH [mi /(Zi me )]kz2 /k⊥
= UH
.
2 − A (ω − ω
A2 ωpe
1 0
LH )ωLH
(3.44)
From the above equation it is clear that the assumption be 1 is only valid near the
upper hybrid layer, where ω0 ≈ ωU H , since it is generally assumed that ω0 ∼ ωU H ∼
ωpe ωLH , while A1 ∼ A2 ∼ 1, such that the leading terms in the numerator should
nearly cancel, i.e., ωU2 H − ω02 ≈ 0, in order for the denominator to be significantly larger
than the numerator. The propagating X-mode near the upper hybrid layer further
√ has
ω0 < ωU H , which, along with A2 > 0, since we are interested in plasmas with ωpe < 3|ωce |,
shows that be will generally be positive, as is appropriate for propagating waves, and that
ωU2 H − (ω0 − ωLH )2 > (2ω0 − ωLH )ωLH . Keeping in mind that ω0 ωLH , it is thus clear
2 1 will always represent a small (fractional)
that the term involving [mi /(Zi me )]kz2 /k⊥
perturbation of be . This justifies expanding 1/be to order 1 in {(ω0 − 2ωLH )ωLH /[ωU2 H −
2 , with which
(ω0 − ωLH )2 ]}[mi /(Zi me )]kz2 /k⊥
2 − A (ω − ω
A2 ωpe
1
(ω0 − 2ωLH )ωLH mi kz2
1 0
LH )ωLH
≈
1− 2
(3.45)
2 .
be
ωU2 H − (ω0 − ωLH )2
ωU H − (ω0 − ωLH )2 Zi me k⊥
39
2,
Plugging the above expression into Eq. (3.43), and only keeping terms of order 1 in kz2 /k⊥
we get
(
2
2 − A (ω − ω
A2 ωpe
Zi Te ωpe
ω12
1 0
LH )ωLH
≈
2
2
2
2
2Ti ωU H
k ⊥ vT i
ωU H − (ω0 − ωLH )2
(3.46)
ωU2 H − (ω0 − ωLH )2 − (ω0 − 2ωLH )ωLH mi kz2
× 1+
2 + A1 .
Zi me k⊥
ωU2 H − (ω0 − ωLH )2
This expression does contain the information we have sought. However, as it stands, its
2 -terms
consequences are not apparent. We therefore note that, by neglecting the ωLH
relative to the other terms in the numerators and denominators, and using the fact that
ωU2 H − ω02 ≈ 0 at the upper hybrid layer, we have,
"
#
2 2
A2 ωpe
ω12
mi kz2
Zi Te ωpe
mi kz2
1+
+ A1 1 −
,
(3.47)
2 v 2 ≈ 4T ω 2
2
2
2Zi me k⊥
2Zi me k⊥
k⊥
i
Ti
U H ω0 ωLH
2 /(ω ω
which also maximises the leading term A2 ωpe
0 LH ) 1 in the bracket. Now, plug2
2
2 )}, A = 1/[1 − ω 2 /(3ω 2 )], ω
ging in A1 = 3{ωpe /(4ωU H ) + [Ti /(Zi Te )](ωU2 H /ωpe
2
LH =
pe
ce
p
2
2
2
Zi me /mi |ωce |ωpe /ωU H , ω0 ≈ ωU H , and ωU H = ωpe + ωce , we find
3 /|ω |3 r
2
3ωpe
Zi Te ωpe
ω12
mi 1 + [mi /(2Zi me )]kz2 /k⊥
ce
≈
+
2
2
2
2
2
2
4Ti 1 + ωpe /ωce
Zi m e
1 − ωpe /(3ωce )
4|ωce |
k⊥ vT i
(3.48)
2
2
1 − [mi /(2Zi me )]kz /k⊥
3
mi kz2
×
+
1−
.
2
2 /ω 2
1 + ωpe
4
2Zi me k⊥
ce
2 v 2 ) from Eq. (3.48), with ω /|ω | ∈ [0, 1], the parameter range
A contour plot of ω12 /(k⊥
pe
ce
Ti
2 ∈ [0, 0.1], the range in
relevant to conventional tokamak plasmas, and [mi /(Zi me )]kz2 /k⊥
2
2
which [mi /(Zi me )]kz /k⊥ 1 is to some extent valid, for Zi Te /Ti = 1 and mi /(Zi me ) =
3672, parameters of a deuterium (Zi = 1 and mi /me = 3672) plasma with Ti = Te ,
representing a canonical tokamak plasma, is seen in Fig. 3.4. From Eq. (3.48) it is evident
2
2 2
that the principal contribution to ω√
1 /(k⊥ vT i ), the first term in the equation, is rapidly
2 1 is
increasing with ωpe /|ωce |, for ωpe < 3ωce , while the influence [mi /(Zi me )]kz2 /k⊥
small due to its assumed smallness; both these points are also clearly seen in Fig. 3.4.
The present theory is not valid for very small ωpe /|ωce |, since the asymptotic approximation
2 v2 ) > 2
of Z for χi (k, ω1 ), leading to the lower hybrid dispersion relation, requires ω12 /(k⊥
Ti
in order to be reasonable (see Fig. 3.3), and Fig. 3.4 clearly demonstrates that for a typical
tokamak plasma this is only satisfied for ωpe /|ωce | > 0.4; note that by Eq. (3.48) the limit
depends linearly on Ti /(Zi Te ), so 0.4 is by no means a universal limit of validity. The
requirement on ωpe /|ωce | can also be understood from the lower hybrid dispersion relation,
2 1, and
Eq. (3.42), where A1 be 1 is assumed: for small ωpe /|ωce | we have ωU2 H /ωpe
2
2
2
2
thus A1 = 3{ωpe /ωU H + [Ti /(Zi Te )]ωU H /ωpe } 1, imposing a very strict condition on
the smallness of be , which may be difficult to satisfy together with the selection rules
for parametric decay. The linear proportionality to Ti /(Zi Te ) is also evident from this
argument, but the limit imposed by it may differ from 0.4, as we shall see.
40
Figure 3.4 – Contour plot
2 2
from
Eq.
of
ω12 /(k⊥
vT i )
(3.48) (for Zi Te /Ti = 1 and
mi /(Zi me ) = 3672). From the
plot, the strong dependence
2 2
of ω12 /(k⊥
vT i ) on ωpe /|ωce |, as
well as the very small influence
2
1, is
of [mi /(Zi me )]kz2 /k⊥
clearly seen.
Note that the
asymptotic expansion of Z in the
low-frequency ion susceptibility
2 2
requires ω12 /(k⊥
vT i ) > 2, imposing the condition ωpe /|ωce | > 0.4
for accurate results in this figure.
Figure
3.5 – Contour plot
√
2
2 2
2 3
of 2 π[ω12 /(k⊥
vT i )] e−ω1 /(k⊥ vT i ) ,
the term determining the size
of Im[χi (k, ω1 )] relative to the
order 0 |Re[χi (k, ω1 )]| in Eq.
2 2
(3.36), with ω12 /(k⊥
vT i ) from Eq.
(3.48) (for Zi Te /Ti = 1 and
mi /(Zi me ) = 3672). From the
plot, and Eq. (3.40), it is evident that significant ion Landau damping of the lower hybrid
waves is present if ωpe /|ωce | <
0.7, which is usually satisfied in
tokamaks.
2 v 2 ) from Eq. (3.48) has been discussed in detail, we
Now that the expression for ω12 /(k⊥
Ti
proceed to determine the strength of ion Landau damping expected for the lower hybrid
waves. From Eq. (3.36), the relative size of Im[χi (k, ω1 )] and the order 0 |Re[χi (k, ω1 )]|
√
2 v2 )
3 v 3 )] e−ω12 /(k⊥
T i , and from Eq.
is given by 2 π[ω13 /(k⊥
(3.40) we see that the size of
Ti
ion Landau damping is also essentially determined by this quantity. Consequently, we
√
2 v2 )
3 v 3 )] e−ω12 /(k⊥
T i with ω 2 /(k 2 v 2 ) from Eq. (3.48), for Zi Te /Ti = 1 and
plot 2 π[ω13 /(k⊥
1
Ti
⊥ Ti
mi /(Zi me ) = 3672, in Fig. 3.5. This figure clearly illustrates that the lower hybrid
waves are subject to significant ion Landau damping for typical tokamak parameters, i.e.,
Zi Te /Ti = 1 and ωpe /|ωce | < 0.7 at the upper hybrid layer. We shall therefore consider
the parametric decay instability to be non-resonant, and use Eq. (2.35) to determine its
growth rate, γ. We keep in mind that this theory has a rather limited range of validity,
0.4 < ωpe /|ωce | < 0.7 for Zi Te /Ti = 1 and mi /(Zi me ) = 3672, and that, due to the linear
2 v 2 ) with Z T /T , non-resonant parametric decay is only expected to
increase of ω12 /(k⊥
i e
i
Ti
occur for Zi Te /Ti . 1, a similar point is made by [Porkoláb, 1982]. Fortunately, these
conditions are usually satisfied in tokamaks.
For reference, we repeat γ for the non-resonant decay instability from Eq. (2.35), in the
41
case of a simple plasma, and noting that, since we are considering decay into waves satisfying the lower hybrid dispersion relation, Re[(k, ω1 )] = 0 and (k, ω1 ) = Im[χi (k, ω1 )] +
Im[χe (k, ω1 )] by design, along with (∂ Re[(k, ω)])/∂ω)|ω=ω2 = (∂ Re[χe (k, ω)]/∂ω)|ω=ω2 ,
since we are neglecting the high-frequency ion response,
γ=
µ2e |χi (k, ω1 )|2 Im[χe (k, ω1 )] + |1 + χe (k, ω1 )|2 Im[χi (k, ω1 )]
− Γ(k, ω2 ).
4 {Im[χi (k, ω1 )] + Im[χe (k, ω1 )]}2 (∂ Re[χe (k, ω)])/∂ω)|ω=ω2
(3.49)
All the quantities going into this expression are essentially determined by Eqs. (2.15),
(3.30), (3.31), and (3.36), along with the dispersion relations Eqs. (3.41) and (3.42),
remembering the selection rule ω2 = ω0 − ω1 . In order to obtain a tractable form of γ, we
consider only the leading terms, as has already been done for Γ(k, ω2 ), and, furthermore
2 v2 )
due to the fact that µ2e 1. For the of the Landau damping arguments, ω12 /(k⊥
Ti
2 /k 2 )ω 2 /(k 2 v 2 ), we use ω 2 /(k 2 v 2 ) from Eq.
and ω12 /(kz2 vT2 e ) = [Ti /(Zi Te )](Zi me /mi )(k⊥
z
1
1
⊥ Ti
⊥ Ti
(3.46) or an approximate version of this. Thus, the terms we are going to use, taking
2 ≈ ω 2 − ω 2 , and saving µ2 for later, are given as: (∂ Re[χ (k, ω)]/∂ω)|
ω22 − ωce
e
ω=ω2 ≈
e
ce
0
2
2 )2 , Γ(k, ω ) ≈ [(ω 2 + ω 2 )/(2ω 2 )]ν , and
2ω2 ωpe /(ω02 − ωce
2
e
ce
2
2
2
ωpi
√ ω13 −ω2 /(k2 v2 )
√ ω13 −ω2 /(k2 v2 )
ωU2 H
1
1
T
i
T
i
⊥
⊥
χi (k, ω1 ) ≈ 2 −1 + i2 π 3 3 e
= 2 −1 + i2 π 3 3 e
,
ωce
ωLH
k⊥ vT i
k⊥ vT i
(3.50)
"
#
2
2
2
3
ω
ωpe
√ ωpe
ω1
2 )
pe νe
−ω12 /(kz2 vT
e +
χe (k, ω1 ) ≈ 2 + i 2 π 2
2 v3 e
2
ωce
ω
ωLH kz k⊥
ce ωLH
Te
"
#
(3.51)
2
2
ωpe
ωpe
ωU2 H √ Ti
ω12
νe
ω1 −ω12 /(kz2 v2 )
Te +
e
.
= 2 +i 2 2 π
2 v2 k v
ωce
ωce
Zi Te k⊥
|ωce |ωU H ωpi
Ti z Te
For µ2e , we note that the pump wave is essentially electrostatic near the upper resonance
[Bellan, 2006], such that we can take E0 to be linearly polarised (and parallel to k0 ),
meaning that the ratio between the real and imaginary parts is the same for all (non-zero)
components. Further taking
k = [k⊥ , 0, kz ] and neglecting the terms parallel to B in Eq.
p
(2.15), since kz /k⊥ Zi me /mi 1, and, in addition, µ2e 1 is assumed,
e2 k 2 |E0x |2
≈ 2 ⊥2
2 )2
me (ω0 − ωce
2 |E |2
4
2 |E |2
ωce
|E0x |2 /B 2
ωce
ωce
0y
0y
= 2be
,
1+ 2
1+ 2
2 )2
ω0 |E0x |2
vT2 e
(ω02 − ωce
ω0 |E0x |2
(3.52)
which is similar to the expression used by [Porkoláb, 1982]. From the last equality it is also
clear that µ2e is indeed a small quantity (since be 1), provided that the characteristic
velocity induced by the pump |E0x |/B is not much larger than vT e , assuming that ω0 6≈ ωce ,
as was done when analysing the electron Bernstein waves; the violation of this criterion
would, in many cases, also require the inclusion of relativistic effects, which were neglected
at a much earlier stage. Now, plugging all of the above expressions into Eq. (3.49), first
µ2e
42
4 ), we arrive at
noting that {Re[χi (k, ω1 )]}2 ≈ {1 + Re[χe (k, ω1 )]}2 ≈ ωU4 H /ωce
4 + Im[χ (k, ω )] Im[χ (k, ω )]
ωU4 H /ωce
µ2e
e
1
e
1
− Γ(k, ω2 )
4 {Im[χi (k, ω1 )] + Im[χe (k, ω1 )]}(∂ Re[χ(k, ω1 )]/∂ω)|ω=ω2
2 2
2
2 |E |2
ωce ωU H 1 + Le Li ω02 + ωce
be |E0x |2 /B 2
ωce
0y
−
≈
1
+
νe
(3.53)
2
4
ω2 ωpe
Li + Le
vT2 e
ω02 |E0x |2
2ω22
2 |E |2
2
ωU2 H 1 + Le Li ω22 + ωce
be 0 |E0x |2
ωce
0y
=
1
+
−
νe ,
8 Ne(0) Te
ω2 Li + Le
ω02 |E0x |2
2ω22
2 v2 )
2 /ω 2 ) Im[χ (k, ω )] = 2√πω 3 e−ω12 /(k⊥
T i /(k 3 v 3 ) and Le =
where the factors Li = (ωce
i
1
1
U
H
⊥ Ti
√
2
2
2
2 /ω 2 ) Im[χ (k, ω )] = [2 πT /(Z T )]ω 3 e−ω1 /(kz vT e ) /(k 2 v 2 k v )+[ω 2 /(|ω |ω
(ωce
i
i e
z
ce U H )]
e
1
T
e
pe
1
T
i
UH
⊥
(νe /ωpi ) determine ion Landau and electron Landau (plus collisional) damping of the lower
hybrid quasi-mode, respectively. The last expression is essentially similar to Eq. (7) from
[Porkoláb, 1982], if we associate θ from that paper with (1 + Le Li )/(Li + Le ), since the
2 /ω 2 (used in that paper),
fact that ω2 ≈ ωU H , along with the assumption of small ωpe
ce
makes be ωU2 H /ω2 ≈ be ωU H ≈ be |ωce |. One also has to account for differences due to the
use of CGS-units in that paper, and SI-units in this work; in the present case, this is done
through the substitution 0 → 1/(4π) [Jackson, 1999].
γ≈
Our interest is now to maximise γ, since this will yield the fastest growing mode, with the
smallest threshold of the parametric decay instability, which is expected to dominate the
saturated anomalous scattering spectrum. The parameters, which we can vary, are essentially the ones determining the direction of k, namely angle between k and B, through
2 , and the angle between k and E , through |E |2 /|E |2 . The opti[mi /(Zi me )]kz2 /k⊥
0y
0x
⊥
0⊥
mum for the latter case is very easy to determine: since we have |E0x |2 + |E0y |2 = |E0⊥ |2 ,
2 /ω 2 < 1, at the upper hybrid layer, the largest value is
which is a constant, and ωce
0
clearly obtained by having |E0x |2 = |E0⊥ |2 and E0y = 0, meaning that k⊥ is parallel
2 is slightly less ob(or anti-parallel) to E0⊥ . The optimal value of [mi /(Zi me )]kz2 /k⊥
2
2
2
vious. While be from Eq. (3.44), ω1 /(k⊥ vT i ) from Eq. (3.46) (Li ), and ω2 , depend
2 , we note that these dependences are higher-order effects,
nominally on [mi /(Zi me )]kz2 /k⊥
which is also evident from Figs. 3.4 and 3.5. Thus, in order to be consistent with the
approximations used in deriving γ, the above dependences should be neglected and be ≈
2
2
2
2 −A ω ω
2
2 2
(ωU2 H −ω02 +2ω0 ωLH )/(A2 ωpe
1 0 LH ), ω1 /(k⊥ vT i ) ≈ [Zi Te /(2Ti )](ωpe /ωU H ){[(A2 ωpe −
2
2
2
A1 ω0 ωLH )/(ωU H − ω0 + 2ω0 ωLH )] + A1 }, where we have neglected ωLH relative to ω02 ,
2 and ω ω
ωpe
0 LH , and ω2 ≈ ωU H , should be used. The only term left carrying a (non2 , through ω 2 /(k 2 v 2 ) = [T /(Z T )](Z m /m )
negligible) dependence on [mi /(Zi me )]kz2 /k⊥
i
i e
i e
i
z Te
1
2 /k 2 )ω 2 /(k 2 v 2 ), is L . However, since T /(Z T ) ∼ 1, (Z m /m )k 2 /k 2 1, and
(k⊥
e
i
i e
i e
i ⊥ z
z
1
⊥ Ti
2 v 2 ) > 2 is assumed, it is already clear that ω 2 /(k 2 v 2 ) 1, meaning that electron
ω12 /(k⊥
z Te
1
Ti
Landau damping is extremely small. This allows us to neglect it as well, and set Le ≈
2 /(|ω |ω
2
2
[ωpe
ce U H )](νe /ωpi ), making the final result independent of [mi /(Zi me )]kz /k⊥ 1
to lowest order, and equivalent to letting kz → 0. Using the above considerations, the
43
following max(γ) is found,
2
ωpe
√
νe ω13 −ω2 /(k2 v2 )
1
⊥ Ti
1
+
2
π
3 v3 e
|ωce |ωU H ωpi k⊥
be 0 |E0⊥ |2 ωU2 H
ω2 + ω2
T
i
# − 2 2 ce νe ,
"
max(γ) ≈
2
8 Ne(0) Te ω2
2ω2
ωpe
√ ω13 −ω2 /(k2 v2 )
νe
1
T
i
⊥
+
2 π 3 3 e
|ωce |ωU H ωpi
k⊥ vT i
(3.54)
2
2
2
2
2
where the approximations be ≈ (ωU H − ω0 + 2ω0 ωLH )/(A2 ωpe − A1 ω0 ωLH ), ω1 /(k⊥ vT2 i ) ≈
2 /ω 2 ){[(A ω 2 − A ω ω
2
2
[Zi Te /(2Ti )](ωpe
2 pe
1 0 LH )/(ωU H − ω0 + 2ω0 ωLH )] + A1 }, and ω2 ≈ ω0 ≈
UH
ωU H , should be used/satisfied. Note that, if we neglect collisions, the factor including
2 v2 )
3 v 3 (2√πω 3 )] eω12 /(k⊥
T i , which is the reciprocal of
effects of Landau damping becomes, [k⊥
1
Ti
the function plotted in Fig. 3.5; [Porkoláb, 1982] and [McDermott et al., 1982] simply approximate its value by 1, when estimating the threshold of the parametric decay instability.
While this approximation certainly gives a reasonable order of magnitude of the growth
rate/threshold, it does not seem to be entirely appropriate for obtaining quantitatively
correct results.
The threshold of the above parametric decay instability in a homogeneous plasma (condition for max(γ) > 0) is,
(0)
|E0⊥ |2 >
2 ν
4Ne Te ω22 + ωce
e
"
0 be
ωU2 H ω2
2
ωpe
√ ω3
νe
2
2 2
2 π 3 13 e−ω1 /(k⊥ vT i ) +
|ωce |ωU H ωpi
k⊥ vT i
2
ωpe
νe ω13 −ω2 /(k2 v2 )
1
⊥ Ti
1+2 π
3 v3 e
|ωce |ωU H ωpi k⊥
Ti
√
#,
(3.55)
which is similar to Eq. (3) from [McDermott et al., 1982], when taking into account the
2 /ω 2 leading to b ω 2
2
assumptions of small ωpe
e U H ≈ be ωce , (Li + Le )/(1 + Le Li ) ≈ 1, and
ce
2
2
ω2 ≈ ω0 , as well as the use of CGS units, 0 → 1/(4π), and the subscript 1 referring to the
high-frequency daughter waves, in that article.
By plugging in values representative of the upper hybrid layer for ω0 /(2π) = 105 GHz, near
the edge and in the bulk plasma of ASDEX Upgrade, for shot 28286 at t = 2.900 s, seen
in Fig. 3.2), we may determine the (homogeneous plasma) threshold of |E0⊥ |, and further
ascertain to which extent the current theory applies. We use the Coulomb
collision rate,
q
√
(0)
3/2
(0)
4
3/2
2
3
3
3
νe = νei = ln(Λ)e Zi Ne /[3(2π)
me 0 Te ] with Λ = (12π/e ) 0 Te /Ne [Swanson,
2003], set Ti = Te , since only Te is available from the equilibrium, and take the plasma
ions to be deuterium, for which Zi = 1 and mi /me = 3672. Then, near the edge we
(0)
have: Ne = 1.83 × 1019 m−3 , Te = Ti = 157 eV, and B = 3.49 T, where B is chosen
to make ω0 = ωU H , the reasonableness of this value has been checked, with which the
parametric decay instability occurs for |E0⊥ | > 21.8 kV/m. Similarly, well inside the bulk
(0)
of the plasma: Ne = 2.53 × 1019 m−3 , Te = Ti = 351 eV, and B = 3.39 T, which yields a
parametric decay instability for |E0⊥ | > 56.2 kV/m.
These electric field thresholds have the same order of magnitude as the homogeneous value
obtained by [McDermott et al., 1982] for the Versator II tokamak; their relation to the
44
beam power will be considered in Chapter 4. We further note that ωpe /|ωce | = 0.393 (near
the edge) and ωpe /|ωce | = 0.476 (in the bulk), showing that the asymptotic expansion of Z
for the lower hybrid (quasi-)mode and the assumption of non-resonant parametric decay
are atleast approximately valid (by Figs. 3.4 and 3.5). However, we note that A1 be is
significant in both cases, 17.3 near the edge and 2.20 in the bulk, making use of the simple
lower hybrid dispersion relation, Eq. (3.42), questionable; we shall, for now, consider the
results to be qualitatively reasonable, but this clearly demonstrates the need for the more
accurate (numerical) treatment of the low-frequency modes to be carried out in Chapter
4.
The homogeneous threshold in the bulk plasma, |E0⊥ | > 56.2 kV/m, is not significantly
larger than that near the edge, |E0⊥ | > 21.8 V/m; this is largely due to the fact that
the parametric decay instability (in a homogeneous plasma) is only limited by Coulomb
−3/2
collisions, which become less important at higher temperatures (νe ∝ Te
). In Chapter
4, we shall see that inhomogeneities generally provide a significantly higher threshold than
the Coulomb collisions, and the limit imposed by these will usually be smaller near the edge
than in the bulk. However, near the upper hybrid resonance, |E0⊥ | will still be limited only
by Coulomb collisions in the model which we shall apply, making a relatively low-power
threshold parametric decay instability possible both near the edge and in the bulk plasma,
although this may not be the case if Landau damping of the pump wave is included. As
a final remark on collisions, we note that the collisional term Le is negligible relative to
Li in both cases (Le /Li ∼ 10−6 ), such that ion Landau damping is essentially the only
mechanism behind driving the non-resonant parametric decay instability, consistent with
the interpretation of this type of instability in terms of nonlinear Landau damping at the
end of Chapter 2.
45
Chapter 4
Wave Propagation and Parametric
Decay in Tokamak Plasmas
The preceding chapters contain a theory of parametric decay of high-frequency waves in
homogeneous plasmas. However, real tokamak plasmas always have some degree of inhomogeneity which must be taken into account. First, we wish to review the theory of wave
propagation in inhomogeneous plasmas, in order to demonstrate the accessibility of the upper hybrid resonance in CTS experiments at ASDEX Upgrade and to evaluate E0 near the
upper hybrid layer which, as demonstrated, is the critical factor determining whether or not
the parametric decay instability occurs. Second, we wish to extend the theory of parametric
decay to inhomogeneous plasmas, especially for the case of non-resonant parametric decay,
as this is the most important for decay into electron Bernstein modes and lower hybrid
(quasi-)modes as shown in Section 3.4. Finally, we wish to investigate parametric decay
near upper hybrid layer in ASDEX Upgrade numerically in order to investigate validity
of the analytical theory and to obtain results taking the precise plasma/beam conditions
more directly into account.
4.1
Geometric Optics and Simple Theory of Wave Amplification
The standard approach to determining the propagation characteristics of high-frequency
(ω0 ∼ ωce ) waves in tokamak plasmas is geometric optics. The geometric optics approximation is based on an expansion of the (linearised) Maxwell and kinetic/fluid plasma equations in the parameters 1/(k0 `) and 1/(ω0 T ), where ` and T are characteristic length/time
scales over which the background plasma parameters vary, and 1/(k0 `), 1/(ω0 T ) 1, i.e.,
weak spatial and temporal inhomogeneity, is assumed [Bernstein, 1975]. While k0 k,
the plasma parameters generally vary on a much larger length scale, making the earlier
assumption valid for high-frequency electromagnetic radiation in most cases and virtually
always for the electrostatic waves. The order 0 term of the expansion is essentially an ex46
tension of the dispersion relation of homogeneous theory to each point space, when waves
are taken to be locally plane. Its main result, concerning the propagation of waves, may
be derived by considering the motion of the individual photons/phonons/plasmons, whose
canonical momentum, p, and energy, H, from the basic quantum model are p(t) = ~k(t)
and H(t) = ~ω(t), respectively; k(t) and ω(t) represent the wave vector and angular
frequency of the locally plane wave at the particle position, r(t). Now, the local dispersion relation may be written in the form ω(t) = ω(r(t), k(t), t), or H(t) = H(r(t), p(t), t),
where H(r, p, t) is the Hamiltonian governing the (classical) motion of the photon/phonon/
plasmon through the canonical equations, dr(t)/dt = (∂H(r, p, t)/∂p)|r=r(t),p=p(t) and
dp(t)/dt = −(∂H(r, p, t)/∂r)|r=r(t),p=p(t) [Landau and Lifshitz, 1969]. Upon a simple
change of variables, these equations may be written as
dr(t)
∂ω(r, k, t) ∂ω(r, k, t) dk(t)
=
=−
,
,
(4.1)
dt
∂k
dt
∂r
r=r(t),k=k(t)
r=r(t),k=k(t)
which are similar to the ray equations obtained from the geometric optics analysis by
[Bernstein, 1975]. The above equations show that the photons/phonons/plasmons propagate at the local group velocity of the wave, vg (r, k, t) = ∂ω(r, k, t)/∂k, which is, as a
logical consequence, also the velocity with which the wave transports energy/momentum,
a well-known result [Swanson, 2003]. The spatial variation of ω(r, k, t) defines an effective
potential in which the photons/phonons/plasmons move. However, using Eq. (4.1) and
the chain rule, it is easily shown that
∂ω(r(t), k(t), t)
∂ω(r, k, t) dω(t)
,
(4.2)
=
=
dt
∂t
∂t
r=r(t),k=k(t)
and thus ω(t) is constant along the particle trajectories, unless the dispersion relation
ω(r, k, t) depends explicitly on t, just as expected for a Hamiltonian system. While the
above formulation is not covariant, it is capable of describing waves with an arbitrary dispersion relations, e.g., electromagnetic O-mode and X-mode, electrostatic electron Bernstein waves and electrostatic lower hybrid waves, and may easily be recast in a covariant
form [Bravo-Ortega and Glasser, 1991]. It is often convenient to work with a more general
dispersion relation D(r, k, ω, t) = 0, since ω(r, k, t) can often not be expressed as a simple
function; D, which is often referred to as the Hamiltonian within geometric optics, may be
a local version of Re() from the earlier chapters, but this is not necessary. The equations
of motion resulting from the above kind of dispersion relation are easily found, using the
chain rule and the fact that D is constant when the dispersion relation is satisfied, to write
∂D(r, k, ω(r, k, t), t)
∂D(r, k, ω, t) ∂ω(r, k, t) ∂D(r, k, ω, t) =
+
= 0, (4.3)
∂k
∂k
∂k
∂ω
ω=ω(r,k,t)
ω=ω(r,k,t)
∂D(r, k, ω(r, k, t), t)
∂D(r, k, ω, t) ∂ω(r, k, t) ∂D(r, k, ω, t) =
+
= 0, (4.4)
∂r
∂r
∂r
∂ω
ω=ω(r,k,t)
ω=ω(r,k,t)
47
from which Eq. (4.1) becomes,
dr(t)
∂D(r, k, ω, t)/∂k ∂D(r, k, ω, t)/∂r dk(t)
=−
=
,
dt
∂D(r, k, ω, t)/∂ω r = r(t), k = k(t)
dt
∂D(r, k, ω, t)/∂ω ω = ω(t)
r = r(t), k = k(t)
ω = ω(t)
,
(4.5)
as well as the above equations, along with dD(t)/dt = ∂D(r(t), k(t), ω(t), t)/∂t = 0 and,
once again, the chain rule to write
dω(t)
∂D(r, k, ω, t)/∂t .
(4.6)
=−
dt
∂D(r, k, ω, t)/∂ω r = r(t), k = k(t)
ω = ω(t)
In the case where the dispersion relation does not depend explicitly on t, which is the only
one that we shall consider in the following, we have, as previously mentioned, dω(t)/dt = 0,
i.e., ω(t) = ω is constant along the trajectories, and we may further arc-length parametrise
the trajectories through the parameter s, with ds = |vg (r, k)| dt = [|∂D(r, k, ω)/∂k|/
|∂D(r, k, ω)/∂ω|] dt, giving
dr(s)
∂D(r, k, ω) ∂D(r, k, ω)/∂k = −sign
,
ds
∂ω
|∂D(r, k, ω)/∂k| r=r(s),k=k(s)
(4.7)
∂D(r, k, ω) ∂D(r, k, ω)/∂r dk(s)
= sign
,
ds
∂ω
|∂D(r, k, ω)/∂k| r=r(s),k=k(s)
which are similar to the ray equations used to study wave propagation in tokamaks by
[Mazzucato, 2014].
Since Eqs. (4.1) and (4.7) are ray equations from geometric optics, the above derivation
indicates that geometric optics can be interpreted as a theory treating photons/phonons/
plasmons as classical particles, with the rays representing the classical trajectories of the
(quasi-)particles. This interpretation of geometric optics is, to the author’s knowledge,
rarely seen in the literature, perhaps due to the generally macroscopic applications of geometric optics and microscopic applications of photons/phonons/plasmons; it does, however,
seem sensible, as classical particle theories and geometric optics are both generally valid
when the background parameters vary on scale which is very large compared to the characteristic length/time scales of the underlying wave phenomena. We also remind that the
inverse treatment of classical particles as waves represents the well-known Hamiltonian
analogy, which has been applied very successfully in the development of quantum mechanics and electron microscopy [Born and Wolf, 2002]. As a final remark on this subject, we
note that the particle interpretation of geometric optics allows, e.g., Snell’s law and the law
of reflection to be understood in terms of conservation of energy and momentum along the
invariant directions for the individual photons/phonons/plasmons, providing a very neat
physical picture of these laws.
In the remainder of this report the trajectories (rays) described by the above equations
of motion will be interpreted within the conventional geometric optics framework, i.e., as
describing the propagation path of a narrow (Gaussian) beam through the plasma. We
48
also note that absorption of the (electromagnetic) beam along the rays may be evaluated
by extending the imaginary part of the dispersion relation in Eq. (3.12) to apply at
each point in space, just
R was done for the real part. This is done by evaluating the
optical thickness, τ = 2 ray κ0 · ds, where ds = vg (r(s), k0 (s), ω0 )ds/|vg (r(s), k0 (s), ω0 )|
is a line element along the ray, assuming a time invariant dispersion relation; e−τ is the
intensity, which is proportional to |E0 |2 , at the end point of the ray relative to that at the
start point [Mazzucato, 2014]. Using the above definition of ds, along with vg (r, k, ω) =
−(∂ Re[(r, k, ω)]/∂k)/(∂ Re[(r, k, ω)]/∂ω) and Eq. (3.12), we find
Z sf
Z
∂ Re[(r(s), k, ω)]
Im[(r(s), k, ω)]
sign
κ0 ·ds = 2
τ =2
ds, (4.8)
∂ω
|∂ Re[(r(s), k, ω)]/∂k| k=k0 (s),ω=ω0
ss
ray
where ss and sf characterise the start and end points of the ray, respectively. Now, multiplying and dividing the above expression by (∂ Re[(r(s), k0 (s), ω)]/∂ω)|ω=ω0 , remembering that sign(x)x = |x|, |vg (r, k, ω)| = |∂ Re[(r, k, ω)]/∂k|/|∂Re[(r, k, ω)]/∂ω|, and
Γ(r, k, ω) = Im[(r, k, ω)]/(∂Re[(r, k, ω)]/∂ω), where Γ a linear temporal damping rate
similar to the one used for the electrostatic waves,
Z sf
Z tf
Γ(r(s), k0 (s), ω0 )
τ =2
ds = 2
Γ(r(t), k0 (t), ω0 ) dt;
(4.9)
ss |vg (r(s), k0 (s), ω0 )|
ts
the last equality comes from the fact that dt = ds/|vg (r(s), k0 (s), ω0 )|, ts and tf correspond temporal equivalents of ss and sf , respectively. This equation indicates that spatial
and temporal damping/growth can be related by introducing an imaginary wave number
κ(r, k, ω) = Γ(r, k, ω)/|vg (r, k, ω)|, a fact that will be useful when considering parametric
decay in inhomogeneous plasmas. We do, however, remark that the distinction between
temporal and spatial growth/damping is generally more complicated, depending on the
details of the problem under consideration, for instance only spatial damping is present for
the pump wave, since it is continuously being excited by the gyrotron, as well as the frame
of reference in which observations are being made, see [Stix, 1992] and [Swanson, 2003] for
detailed discussions; the imaginary wave vector κ0 may also be used to describe undamped
spatially localised waves, e.g., surface plasmons, in the case where κ0 ⊥ vg [Balanis, 2012],
but these shall not be of further concern to us in this work.
We have assumed the imaginary part of the dispersion relation to be small, both when
deriving the ray equations with no creation/annihilation of the photons/phonons/plasmons
(purely real ω and k) and when including the effects of absorption through Eq. (3.12),
following the development of [Bernstein, 1975]. This assumption is often questionable
in regions of strong linear absorption, e.g., near the electron cyclotron resonance and its
harmonics. However, geometric optics may still be valid if the above analysis is repeated
for the full complex dispersion relation, provided that absorption only occurs in a narrow
region [Bravo-Ortega and Glasser, 1991]. In spite of this, the validity of geometric optics is
generally limited near cut-offs and resonances, including the upper hybrid resonance, where
reflection/linear mode conversion/field enhancement may occur on length scale comparable
to the wavelength; this is the motivation for the non-geometric optics based investigations
in the next section. For the remainder of this section our focus will be on the questions
49
of accessibility to the upper hybrid layer in CTS experiments at ASDEX Upgrade, for
which geometric optics should be ideal, and a simple geometric optics based theory of field
enhancement, which will atleast indicate field enhancement near the upper hybrid layer.
The upper hybrid layer accessibility in CTS experiments at ASDEX Upgrade is studied by
tracing the central beam ray of the ω0 /(2π) = 105 GHz O-mode radiation injected from the
low-field side to the high-field side wall and the corresponding ray of the ω0 /(2π) = 105 GHz
X-mode radiation reflected by the high-field side wall back into the plasma; examples of the
results obtained using this procedure for ASDEX Upgrade shots 28286 and 32563 are seen
in Fig. 4.1. The ray tracing is done using the wr-code, developed by Henrik Bindslev at
DTU Physics, which finds rays by numerical integration of Eq. (4.7) and evaluates τ along
the rays from an expression similar to Eq. (4.8), for given ω0 , initial r and k0 -direction (k0
is fixed by ω0 along with the dispersion relation), and equilibrium plasma profiles of B,
(0)
Ne , and Te (only electron dynamics are included). The dispersion relation used by the
wr-code is quite complicated, including kinetic and relativistic effects, and found in, e.g.,
[Mazzucato, 2014]. However, its essential features, apart from Landau/cyclotron damping,
are captured by the cold plasma theory described in Section 3.1, as is evident from Fig.
4.1. The reflections from high-field side wall are calculated using the law of reflection for
geometric optics, i.e., by determining the local normal vector of the surface and switching
the sign of the component of k parallel to it. When implementing this in the code, we
further assume that dispersion relations of the O-mode and X-mode are not too different
at the high-field side wall, since it is only possible to specify the direction of k; in reality,
this should be well satisfied due to the very low plasma density at the wall, but the actual
wall is also covered with tiles, the geometry of which has not been taken into account.
Fig. 4.1 shows the accessibility to the upper hybrid layer for ω0 /(2π) = 105 GHz in
ASDEX Upgrade shot 28286 at t = 2.900 s (left pane, TS equilibrium), and shot 32563 at
t = 4.000 s and t = 4.500 s (middle and right panes, respectively, IDA equilibria). For all
three panes, the fat blue line is the ray trajectory of the incident O-mode radiation and the
fat red line is the ray trajectory of the reflected X-mode radiation; the orange line is the
R-cut-off, the purple line is the electron cyclotron resonance, the green line is the upper
hybrid resonance, the yellow shaded area indicates that no propagating X-mode exists, the
background contours indicate the plasma and vessel locations, and (R, z) are cylindrical
coordinates with the z-axis being the torus symmetry axis (conventions similar to the ones
from Figs. 3.1 and 3.2).
In shot 28286, the O-mode radiation is launched from gyrotron 1, located above the plasma
centre in the z-direction on the low-field side (R1 = 2.364 m, z1 = 0.320 m), and the upper
hybrid resonance is accessible to the reflected X-mode radiation with a sizeable, though not
excessive, absorption of power at the electron cyclotron resonance in most of the shot; this
is illustrated by the optical thickness τ = 1.12 of the X-mode ray traced for t = 2.900 s.
The above observation is consistent with the fact that the anomalous scattering features
are observed throughout shot 28286.
In shot 32563, the situation is more complicated. Here two gyrotrons, labelled 3 and 4
and both located at the same (R, z)-value below the plasma centre on the low-field side
50
Figure 4.1 – Accessibility to the upper hybrid layer for ω0 /(2π) = 105 GHz in ASDEX Upgrade shots 28286 and 32563. In all three panes, the fat blue line is the ray
trajectory of the incident O-mode radiation and the fat red line is the ray trajectory
of the reflected X-mode radiation; the orange line is the R-cut-off, the purple line
is the electron cyclotron resonance, the green line is the upper hybrid resonance,
the yellow shaded area indicates that no propagating X-mode exists, the background
contours indicate the plasma and vessel locations, and (R, z) are cylindrical coordinates with the z-axis being the torus symmetry axis (conventions similar to the ones
from Figs. 3.1 and 3.2). The left pane shows the ray trajectories from gyrotron 1
in shot 28286 at t = 2.900 s (TS equilibrium), while the middle and left panes show
the ray trajectories from gyrotron 3 in shot 32563 at t = 4.000 s and t = 4.500 s,
respectively (IDA equilibria). It is clear that the upper hybrid resonance, in the
bulk plasma, is accessible to gyrotron 1 in shot 28286 at t = 2.900 s, although some
absorption, τ = 1.12, occurs at the electron cyclotron resonance. For shot 32563, the
upper hybrid resonance is inaccessible to gyrotron 3 at t = 4.000 s, as virtually all
power is absorbed at the electron cyclotron resonance; however, at t = 4.500 s the
external magnetic field has been reduced, moving the resonances/cut-off closer to the
z-axis and making the upper hybrid resonance accessible to gyrotron 3 in the edge,
although absorption at the electron cyclotron resonance is still significant, τ = 2.99.
(R3,4 = 2.364 m, z3,4 = −0.320 m), but at different toroidal angles ϕ, are active; the ray
traces shown in the middle and right panes of Fig. 4.1 are both started from gyrotron 3.
At t = 4.000 s, the upper hybrid layer is clearly inaccessible to radiation from gyrotron
3, with essentially all power being absorbed at the electron cyclotron resonance; however,
at t = 4.500 s the external magnetic field has been reduced, moving the resonances/cutoff closer to the z-axis and making the upper hybrid resonance accessible to gyrotron 3,
although absorption at the electron cyclotron resonance is still significant, τ = 2.99, for
the traced X-mode ray. In spite of the significant absorption, we note that parametric
decay may still occur for gyrotron 3 at t = 4.500 s, since the upper hybrid resonance is
located near the plasma edge. Gyrotron 4 is launches its O-mode radiation at a steeper
angle, which results in the upper hybrid layer being accessible to it, with virtually no
absorption, already at t = 4.000 s. The above observations are also consistent with the
51
fact that anomalous scattering features are start occurring when gyrotron 3 is on around
t = 4.500 s, while they are present when gyrotron 4 is on as early as t = 4.000 s.
Having discussed the accessibility of the upper hybrid layer to the X-mode radiation reflected from the high-field side wall, we turn our attention to the important question of
relation between the gyrotron input power, P0 , and the electric field amplitude within the
geometric optics and Gaussian beam approximations. Our treatment of the beam is very
rough: we simply assume its width to evolve as that of a Gaussian beam in free space
along the central beam rays traced in Fig. 4.1, with the focus point and beam waist being
determined by the gyrotron. In the following we merely summarise the results of this procedure and refer to standard texts on Gaussian beam optics, e.g., [Saleh and Teich, 2007],
for the derivation. The electric field amplitude of a Gaussian beam in free space is,
|E0 (ρ, s)| = |E0 (0, s)| e−ρ
2 /W 2 (s)
,
(4.10)
with ρ being the shortest distance from the central beam ray to the point in question and
the beam half-width, W (s), given by
s
4c2 (s − s0 )2
W (s) = W0 1 +
,
(4.11)
ω02 W04
where W0 is the beam waist (smallest beam half-width) and s0 is the s-point of the beam
waist; for the gyrotrons at ASDEX Upgrade, W0 = 2.29 cm and s0 = 85.4 cm are used
(setting s = 0 at the gyrotron launch point). We note that the evolution of the beam
width is not changed at reflection from the high-field side wall, since we are treating it as
a locally plane mirror whose radii of curvature are much larger than W [Saleh and Teich,
2007]. In order to determine the desired relation between P0 and |E0 (0, s)|, we first assume
the amount of power in the beam, P, to be a function of s alone, which may be determined
by evaluating the optical thickness and other losses along the central beam ray, such that
P(s) = P0 F e−τ (s) ,
(4.12)
where F is the fraction of power coupled back into the plasma in X-mode, after reflection
from the high-field side wall, and τ (s) is the total optical thickness of the ray, including
the (generally very small) optical thickness of the incident O-mode ray, at point s. The
above assumptions obviously requires the beam to be relatively narrow and collimated,
since a Gaussian profile, with a power found by evaluating the optical thickness along
the central ray alone, is not realistic if the plasma parameters, etc., vary significantly
across the width of the beam. The above assumption may also be utilised to estimate
F. This is done by calculating first calculating the polarisation vector along the central
beam ray of the incident O-mode radiation, e0 (r(s), k0 (s), ω0 ), i.e., the (non-trivial) unit
norm eigenvector solving H[(k0 , ω0 )] · e0 (k0 , ω0 ) = 0, with H[(k0 , ω0 )] from Eq. (3.13),
n = n[sin(θ), 0, cos(θ)], and n given by the O-mode root of Eq. (3.16), assuming that
this polarisation is characteristic of all beam points with the same s-value. Then, the
change of polarisation upon reflection using the standard rules [Saleh and Teich, 2007] and
the components of this polarisation vector along the (orthogonal) O-mode and X-mode
52
polarisation vectors is calculated upon to reflected beam re-entering the plasma; the of reentry point is not completely well-defined, so, to have a definite problem in which re-entry
occurs prior to the reflected beam encountering the upper hybrid resonance, we define it
to be either the electron cyclotron resonance or the last closed flux surface, the one chosen
being the one first encountered. The square norm of the X-mode component is an estimate
for F. This value can vary quite strongly depending on the gyrotron setting, for instance,
F = 0.0678 for gyrotron 1 in ASDEX Upgrade shot 28286 at t = 2.900 s (left pane of Fig.
4.1) while F = 0.5892 for gyrotron 3 in ASDEX Upgrade shot 32563 at t = 4.500 s (right
pane of Fig. 4.1), and should not be considered more than a rough estimate.
The connection between P(s) and |E0 (0, s)| can now be established using the fact that
energy propagates at the local vg within the geometric optics approximation, from which
the radiation intensity, I, may be written as the product of the wave energy density, U ,
and |vg |. We additionally assume that the energy at a given s all propagates with the
vg of the central beam ray, which is approximately true for the narrow collimated beams
under consideration, such that I(ρ, s) ≈ U (ρ, s)|vg (r(s), k0 (s), ω0 )|. This result is of course
only useful if an expression for U is known in terms of E0 , which is, fortunately, the case
for weakly absorbing plasmas [Swanson, 2003]; in this case the result has further been
showed to carry over to inhomogeneous plasmas within the geometric optics framework
[Bernstein, 1975]. Considering, as always, only the variation of |E0 (ρ, s)| with ρ, it holds
that E0 (ρ, s) ≈ |E0 (ρ, s)|e0 (r(s), k0 (s), ω0 ) (with the polarisation appropriate to X-mode),
and thus the expression for U , from [Swanson, 2003], becomes
0 |E0 (ρ, s)|2 ∗
∂{ωH[(r(s), k0 (s), ω)]} U (ρ, s) ≈
e0 (r(s), k0 (s), ω0 ) ·
· e0 (r(s), k0 (s), ω0 ),
4
∂ω
ω=ω0
(4.13)
where the hermitian part of the dielectric tensor, H(), from Eq. (3.13), extended to hold
at each point in space (and time), is used in this work, and we remind of our slightly nonstandard definition of , which follows [Bernstein, 1975] and [Bravo-Ortega and Glasser,
1991], while (0 /4)ω0 H[(k0 , ω0 )] is referred to as the Maxwell operator by [Swanson,
2003]. Now, the desired relation can be established by noting that P(s) may be found by
integrating I(ρ, s) over the whole plane related
R ∞ to a given s (assuming the central beam
ray to be nearly straight), i.e., P(s) = 2π 0 ρI(ρ, s) dρ, which is easily evaluated using
the above results and substituting the integration variable, ρ, with 2ρ2 /W 2 (s), yielding
s
8P0 F e−τ (s)
|E0 (0, s)| =
;
(4.14)
2
π0 W (s)|vg (r(s), k0 (s), ω0 )|E(r(s), k0 (s), ω0 )
E(r, k, ω) = e∗0 (r, k, ω) · (∂{ωH[(r, k, ω)]}/∂ω) · e0 (r, k, ω) has been introduced in order to
simplify the notation. We note that in free space |vg (r, k0 , ω0 )| = c and E(r, k0 , ω0 ) = 2,
since the dispersion relation reads ω0 = ck0 and waves are transverse (k0 · e(r, k0 , ω0 ) = 0),
such that the above expression may be written in the form P(s) = (π/4)c0 |E0 (0, s)|2 W (s)2 ,
which is the well-known result for a Gaussian beam in free space [Saleh and Teich, 2007],
as it should be. The above expression also shows that |E0 (0, s)| increases for larger P0 ,
smaller W , and smaller |vg |, which is consistent with geometric optics based intuition and,
53
further, indicates field enhancement at points where |vg | → 0, e.g., at the upper hybrid
resonance. Unfortunately, according to the above expression, |E0 (0, s)| diverges at the upper hybrid resonance layer, which is not physically plausible and, additionally, the above
expression does also not indicate field enhancement near cut-offs, which is a well-known
phenomenon. It thus seems that the above expression does not adequately describe field
enhancement very close to cut-offs and resonances, which are often the places where it is
of most interest in connection with parametric decay. This was to be expected, since geometric optics generally breaks down at these points; the above expression is, however, still
useful, as it allows the field amplitude to be fixed for a given P0 in the more sophisticated
field enhancement results of the next section.
4.2
Advanced Wave Amplification: Full-Wave, WKB, and
Hybrid Approaches
Since geometric optics is inappropriate for determining field enhancement very close to the
upper hybrid layer, this section is dedicated to different approaches by which this may be
estimated. The most accurate approach would be to return to the Maxwell equations with
current and charge densities given by the kinetic (or fluid) plasma equations; this system
of partial differential equations should then be solved, taking the full geometry of the
beam and tokamak plasma into account. The above programme is known as the full-wave
approach. We note that it is even possible to describe parametric processes directly using
full-wave approach, provided that nonlinear effects are retained in the governing equations.
However, within our present framework, it seems more appropriate to use a linearised
scheme to evaluate E0 , and then determine whether or not |E0⊥ | exceeds the threshold
necessary for the parametric decay instability to occur. While full-wave codes of this type,
specialised to handle ASDEX Upgrade plasma conditions, do exist, it has not been possible,
at the time of writing, to apply these successfully to the question of wave amplification near
the upper hybrid layer in ASDEX Upgrade. The main problem, when applying such codes,
seems to be related to the optical thickness of the electron cyclotron resonance layer in
the scrape-off layer, the modelling of which has generally been assigned a low priority due
to its previously mentioned limited accessibility and experimental importance in standard
scenarios. Until this issue is resolved, and in order to obtain less computationally expensive
and more easily interpretable results, an alternative approach is desired.
One such frequently used alternative is the previously mentioned WKB approach; it was
originally introduced in connection with quantum mechanics, but has found substantial use
within plasma physics as well, see, e.g., [Stix, 1992], [Swanson, 2003], and [Bellan, 2006] for
detailed accounts. Away from cut-offs/resonances the WKB approach is essentially similar
to geometric optics, since it describes the wave as having a fast, locally plane wave-like,
spatial variation, on a length scale of order 1/k, and a slow spatial variation of its amplitude
and k, on a length scale of order `. Near a cut-off/resonance the standard WKB solution
(0)
may be joined with a full-wave solution for B, Ne , and Te profiles expanded to lowest
non-trivial order (linear in the cases of interest in this work). The combination of these
54
techniques is what we understand as the WKB approach, which allows determination of
the amplification near a cut-off/resonance, as well as calculation of reflection/linear mode
conversion/transmission/absorption coefficients around these. Unfortunately, the WKB
approach is only applicable to problems in which the plasma parameters vary along a
single direction, i.e., slab geometries. While such a geometry is definitely not capable of
describing the global properties of tokamak plasmas, it can atleast be applied within a
limited region with some accuracy, and since we are, as previously mentioned, interested
in obtaining relatively simple estimates of the field amplification near the upper hybrid
layer we shall make the assumption of slab geometry in the following and use a full-wave,
WKB inspired, hybrid approach, taking input from the ray tracing results of Section 4.1.
The hybrid approach, which we shall employ, is largely inspired by treatment wave amplification given by [White and Chen, 1974]. It is implemented by returning to Eqs. (3.4) and
(3.5), this time assuming only harmonic time dependence of the pump wave and electron
fluid velocities, i.e., E(r, t) = Re[E0 (r) e−iω0 t ] and Ve (r, t) = Re[Ve0 (r) e−iω0 t ]. If we additionally assume the electron displacement to small compared with the scale over which B
and νei vary significantly, we can reuse Eq. (3.8) and write Ve0 (r) = −i(e/me )M(r)·E0 (r).
Plugging all of this into Eq. (3.5), the following wave equation governing the evolution of
E0 (r) is found,
ω2
∇ × [∇ × E0 (r)] − 20 E0 (r) = −F(r) · E0 (r),
(4.15)
c
2 (r)ω M(r)/c2 . We now consider, as previously advertised, a plasma
where F(r) = ωpe
0
slab varying only along one direction (perpendicular to B, taken along the z-direction)
which we shall call the x-direction, resulting in F(r) = F(x). The wave is also taken to
be essentially plane perpendicular to the x-direction, since plane waves are eigenmodes
along these homogeneous directions, such that E0 (r) = E0 (x) eik0y y+ik0z z . Using these
assumptions, along with the identity ∇ × [∇ × E0 (r)] = ∇[∇ · E0 (r)] − ∇2 E0 (r), Eq.
(4.15) is rewritten as


 

d/dx  d/dx
 d2
2
ω
2
2
0
 ik0y   ik0y  · E0 (x) −
− k0y − k0z + 2 E0 (x) = −F(x) · E0 (x).


dx2
c
ik0z
ik0z
(4.16)
For tokamak applications, there is no canonical way of choosing the direction of inhomogeneity, so in order to have a definite problem we take k0y = 0. This means that the
x-direction is parallel (or anti-parallel) to the initial k0⊥ -direction and the y-direction is
parallel (or anti-parallel) to the initial k0 ×B-direction, with k0 obtained from the geometric optics approach, which seems somewhat reasonable since the reflected X-mode radiation
generally moves quasi-perpendicular to the flux-surfaces in the (R, z)-plane of the torus
(see Fig. 4.1), and further acts to simplify the above expression. If we additionally recall
55
that the tensor elements of M (and F) from Eq. (3.8) are,
2 (x)
ωpe
ω0 [ω0 + iνei (x)]
Fxx (x) = Fyy (x) =
,
2 (x)
c2 [ω0 + iνei (x)]2 − ωce
2 (x)
2 (x)
ωpe
ωpe
ω0 ωce (x)
ω0
Fxy (x) = −Fyx (x) =
,
F
(x)
=
,
zz
2 (x)
c2 [ω0 + iνei (x)]2 − ωce
c2 ω0 + iνei (x)
(4.17)
with all other elements vanishing, the x-, y-, and z-components of Eq. (4.16) may, respectively, be written in the forms
2
dE0z (x)
ω0
2
ik0z
−
− k0z E0x (x) = −Fxx (x)E0x (x) − Fxy (x)E0y (x),
(4.18)
dx
c2
2
d2 E0y (x)
ω0
2
−
− k0z E0y (x) = Fxy (x)E0x (x) − Fxx (x)E0y (x),
(4.19)
−
dx2
c2
dE0x (x) d2 E0z (x) ω02
−
− 2 E0z (x) = −Fzz (x)E0z (x).
(4.20)
dx
dx2
c
In the case of propagation perpendicular to B, k0z = 0, Eqs. (4.18) and (4.19) form
a system for determining E0x (x) and E0y (x), independent of E0z (x) which is determined
Eq. (4.20); this shows the very clear distinction between O-mode, for which E0z (x) is finite,
and X-mode, for which E0x (x) and E0y (x) are finite, in the case of strictly perpendicular
propagation [White and Chen, 1974]. While it is possible to gain some insight from studying
such simplified cases, we shall in the following mainly be concerned implementing Eqs.
(4.18), (4.19), and (4.20) numerically for the ASDEX Upgrade shots being investigated, as
this will provide an estimate of the field enhancement which is most connected to the actual
experimental realisations. Two things are required in order to achieve the above: one, the
system of equation should be put in a form suitable for numerical computations, in this case
the form used by the ode45-command in Matlab, and, two, a suitable scheme for obtaining
the one dimensional plasma profiles to be specified. The first part requires rewriting Eqs.
(4.18), (4.19), and (4.20) as system of 1st order ordinary differential equations with the
(x-)derivative of each depend variable on the left hand side. This may be done in a
straightforward manner by introducing the depend variable ∆0y (x) = dE0y (x)/dx and
substituting dE0z (x)/dx from Eq. (4.18) into Eq. (4.20), with which
2
dFxy (x)
dFxx (x)
ω0
E0x (x) +
E0y (x) + Fxy (x)∆0y (x) − ik0z 2 − Fzz (x) E0z (x)
dE0x (x)
dx
dx
c
=
,
2
dx
ω0
− Fxx (x)
c2
(4.21)
dE0y (x)
= ∆0y (x),
(4.22)
dx
d∆0y (x)
ω2
2
= −Fxy (x)E0x (x) + Fxx (x) − 20 + k0z
E0y (x),
(4.23)
dx
c
ik0z
56
dE0z (x)
i
=
dx
k0z
ω02
2
Fxx (x) − 2 + k0z E0x (x) + Fxy (x)E0y (x) .
c
(4.24)
The terms involving dFxx (x)/dx and dFxy (x)/dx are ordinarily quite small since the above
quantities generally vary on the length scale of the plasma parameters, ` 1/k0 , however,
near a resonance their variation may be rather rapid and we therefore retain them. Since
E0x (x), E0y (x), ∆0y (x), and E0z (x) are complex variables it is necessary to plug both
the real and imaginary parts of the above equations into the system solved by ode45 in
Matlab.
In the above scheme, the profiles of Fxx (x), Fxy (x), and Fzz (x) used are simply determined
by evaluating their variation along a straight line parallel to k0 (sr ) and starting at r(sr ),
obtained from ray tracing; sr characterises a ray point relatively close to the upper hybrid
resonance. Currently, sr is determined by trial and error for each equilibrium and beam
setting being investigated, in order to ensure that points on either side the upper hybrid
resonance, which is moved slightly by the presence of collisions, are included; a less ad hoc
method would be desirable, but, given that we are interested in a rough estimate of the field
enhancement, the above method is considered sufficient. The values of dFxx (x)/dx and
dFxy (x)/dx are estimated using the centred difference scheme of the gradient-command in
Matlab, with x = (s−sr ) sin(θ) where θ is the angle between k0 (sr ) and B at the last ray
tracing point. As a final remark on the numerically implementation of the one dimensional
plasma profile, we note that the inherent anisotropy of electron-ion collision dynamics in
a magnetic field has been included by introducing a different collision frequency, νeiz (x),
2 (x)ω /[ω + iν (x)]; for a simple
along the magnetic field lines, such that Fzz (x) = ωpe
0
0
eiz
hydrogen-like (Zi = 1) plasma, νeiz (x) = 0.51νei (x) [Braginskii, 1965].
To find a unique solution of Eqs. (4.21), (4.22), (4.23), and (4.24), it is necessary to
specify the (initial) values of E0x (x), E0y (x), ∆0y (x), and E0z (x) at some point. As we
are interested in finding the field right before and just after the upper hybrid resonance,
we are essentially dealing with a one dimensional scattering/transmission problem with
a propagating wave incident on the upper hybrid resonance from the high-field side, an
evanescent "wave" existing after the upper hybrid resonance, and a transmitted (strongly
attenuated) propagating wave existing on the other side of the evanescence region. The
initial (final) condition appropriate in this type of problem is that only a transmitted
(plane) wave transporting energy (with the direction of its group velocity) away from
the upper hybrid resonance layer exists. This is implemented by first calculating the
polarisation vector e0 (r(s), k0 (s), ω0 ) of an X-mode plane wave on the transmitted side of
the upper hybrid resonance, using H() from Eq. (3.13) and the root of n appropriate
to X-mode from Eq. (3.16), which determines E0x (x), E0y (x), and E0z (x), to within a
multiplication factor giving the field amplitude. Note that the point where this is done
is taken to be a wave crest of E0 (x), since no assumption of a locally plane along the
x-direction and hence no factoring out of eik0x (x)x from E0 (x), has been made in this
case, while it has been done in the geometric optics type approximations implicit in using
e0 (r(s), k0 (s), ω0 ); this does not seem to have a strong influence on the results in the region
of interest. The initial value of ∆0y (x) is calculated by isolating it in Eq. (4.21) and using
Gauss’s law, along with the fact that electromagnetic radiation is (almost) not associated
57
with any charge density perturbation away from resonances, dE0x (x)/dx ≈ −ik0z E0z (x).
It is noted that the above procedure works even if the initial point lies within the evanescent
region.
The field amplification of the X-mode radiation is fixed by connecting the above solution,
which assumed to be valid around the upper hybrid resonance, to the geometric optics one
from Section 4.1, which is assumed to be valid away from the upper hybrid resonance. To
do this, E0 (x) is taken represent the field at the centre of a Gaussian beam and |E0 (x)|,
for the wave crest closest to the point r(sr ) at which ray tracing is stopped, is set equal
to |E0 (0, sr )| of the assumed narrow Gaussian beam corresponding to the ray, using Eq.
(4.14). The above procedure obviously requires the regions of validity of the two approximations to overlap, which we assume to be the case, and is reminiscent of the joining of
solutions made when using the WKB approach. A slightly unattractive feature of the procedure is that the rapid plane wave-like variation along the x-direction has been factored
out of E0 (0, s) in the geometric optics approximation, while it has been retained in E0 (x);
this seems to be unavoidable since it is not necessarily possible to assume E0 (x) to have a
local plane wave form near the upper hybrid resonance.
The results obtained by applying the hybrid approach described above ASDEX Upgrade
to the ray from gyrotron 1 in shot 28286 at t = 2.900 s and the ray from gyrotron 3 in shot
32563 at t = 4.500 s, seen in the left and right panes of Fig. 4.1, are shown in Figs. 4.2 and
4.3, respectively. The quantity plotted is |E0⊥ (0, sX )|/|E0 (0, 0)| versus sX , where sX is the
arc-length along the X-mode ray from the reflection point, |E0 (0, 0)| is calculated using
Eq. (4.14) for sX = 0, and |E0⊥ (0, sX )| is used for calculating field enhancement since
this is the quantity of interest for the parametric decay instability under consideration.
In both cases, the transition from use of the ray tracing result, Eq. (4.14), to the results
obtained using the full-wave slab model is clearly visible due to occurrence of the previously
mentioned rapid oscillatory amplitude variations. It is also clear that the waves experience
field enhancement around the upper hybrid resonance before and become evanescent when
passing it, as expected for X-mode radiation.
In shot 28286 at t = 2.900 s, Fig. 4.2, the transition from geometric optics to the slab
model seems to be relatively smooth when looking at the wave crests; this also seems to be
the case when looking at the straight line continuation of the geometric optics ray. Some
relatively rapid variations of the field amplitude occur in connection with the ray crossing
the electron cyclotron resonance, around sX = 7 cm, but these are quite small compared
with the very large field enhancement, of 2.10 × 103 , occurring near the upper hybrid
resonance, around sX = 20 cm (note the semi-logarithmic nature of Fig. 4.2). There are
two main reasons for this large field enhancement: one, the upper hybrid is encountered
in the bulk plasma where the electron temperature is quite high, Te ≈ 350 eV, making
νei , which is the only quantity limiting |E0⊥ | at the upper hybrid resonance, very small,
νei /ω0 ≈ 6 × 10−6 near the upper hybrid resonance; two, the incidence of the straight line
used in the slab model is almost perpendicular to B, ck0z /ω0 = 0.0032, which means that
our model takes the incidence to be almost parallel to the gradients of the slab, where
the most extreme field enhancement is expected. We note that while the observed field
enhancement is not unexpected, when considering the results obtained by [White and Chen,
58
Figure 4.2 – Semi-logarithmic
plot of the field amplification,
|E0⊥ (0, sX )|/|E0 (0, 0)| versus sX ,
for the reflected ω0 /(2π) =
105 GHz X-mode radiation in ASDEX Upgrade shot 28286 at t =
2.900 s. A relatively smooth transition from the geometric optics to
the full-wave slab solution, along
with a very large field enhancement around 2.10 × 103 near the
upper hybrid layer, is observed.
Figure 4.3 – Plot of the field amplification, |E0⊥ (0, sX )|/|E0 (0, 0)|
versus sX , for the reflected
ω0 /(2π) = 105 GHz X-mode radiation in ASDEX Upgrade shot
32563 at t = 4.500 s. A relatively abrupt transition from the
geometric optics to the full-wave
slab solution, along with a moderate field enhancement around 2.5
near the upper hybrid layer, is observed.
1974] for much lower temperatures, it is highly unlikely that a field enhancement of this
order of magnitude would occur in a model taking Landau damping and the full tokamak
geometry into account. For the present purposes, the field in Fig. 4.2 will atleast allow
us investigate parametric decay near the upper hybrid resonance in ASDEX Upgrade shot
28286 at t = 2.900 s, from which the frequency shift expected due to the parametric decay
instability may be compared with the one observed in the anomalous scattering spectrum
(Fig. 1.3); however, the P0 threshold should not be taken too seriously.
In shot 32563 at t = 4.500 s, Fig. 4.3, the transition from geometric optics to the slab
model is quite abrupt. However, the same kind of abrupt change also occurs for the
geometric optics ray, which is completely straight until encountering the electron cyclotron
resonance, around sX = 9.0 cm, after which it is deflected very strongly in the direction of
the magnetic field and starts to be damped, as seen in Fig. 4.3; the straight line parallel
to k0 (sr ), used in the slab model, is almost parallel to the initial straight ray trajectory
and accounts for the abrupt change occurring when changing from geometric optics to the
slab model in Fig. 4.3. These abrupt changes are most likely an artefact of the unreliable
high-field side scrape-off layer profiles available for studying parametric decay in ASDEX
Upgrade shot 32563 at t = 4.500 s; in spite of the abrupt changes, the field amplification is
only about 2.5 near the upper hybrid layer, around sX = 9.2 cm, which is quite close to the
ad hoc value of 5 used by [McDermott et al., 1982] and [Porkoláb, 1982]. We also note that
59
it has been necessary to switch from geometric optics to the slab model at a relatively early
point in order to encounter the upper hybrid resonance, meaning that τ = 0.28, rather
than 2.99, at the point where the ray tracing is stopped which should significantly lower
the P0 threshold of the parametric decay instability. Although the electron temperature is
substantially lower near the upper hybrid resonance in the scrape-off layer in shot 32563 at
t = 4.500 s, Te ≈ 8 eV, than the corresponding bulk value in shot 28286 at t = 2.900 s, the
electron density is also substantially reduced resulting in comparable electron-ion collision
frequencies: νei /ω0 ≈ 8 × 10−6 near the upper hybrid layer in shot 32563 at t = 4.500 s.
The likely explanation for the much smaller field amplification in shot 32563 at t = 4.500 s,
than in shot 28286 at t = 2.900 s, is therefore to be found in the substantially larger
ck0z /ω0 = 1.377 in the present case.
4.3
Parametric Decay in Inhomogeneous Plasmas
Now that we have a method by which the electric field near the upper hybrid resonance
may be estimated for given plasma and beam parameters, we turn our attention to the
modifications of the threshold of the parametric decay instability due to the inhomogeneity
of the plasma.
Before proceeding to the matter of calculating the threshold of the parametric decay instability in a truly inhomogeneous plasma, we consider the homogeneous plasma threshold
at the point of maximum electric field in ASDEX Upgrade shot 28286 at t = 2.900 s.
(0)
The plasma parameters extracted at this point are: B = 3.36 T, Ne = 2.55 × 1019 m−3 ,
Te = 378 eV, and Ti is set equal to Te since only Te is available from the equilibrium
used; by use of Eq. (3.55), as always, with νe = νei , these parameters give the threshold
|E0⊥ | > 63.5 kV/m for a deuterium plasma (Zi = 1 and mi /me = 3672). From Eq. (4.14)
and the maximum field amplification of 2.10 × 103 , shown in Fig. 4.2, the power threshold
for gyrotron 1, necessary to exceed this |E0⊥ |, is found to be a very low P0,th = 3.13 W.
Obviously, this value is usually exceeded by many orders of magnitude in CTS experiments, but it is quite irrelevant since plasma inhomogeneities will be shown to provide
a threshold about three orders of magnitude above this value. We note that, in spite of
the very large field amplification, our homogeneous plasma threshold is only around an
order of magnitude lower than the value of 75 W cited for the Versator II experiments
by [McDermott et al., 1982]; the main reason for this is of course that only 6.78 % of the
incident O-mode power is coupled into the reflected X-mode according to our model. The
very low homogeneous plasma threshold, provided by collisions, also justifies the neglect
of collisions when considering the threshold of the parametric decay instability in inhomogeneous plasmas. No results for ASDEX Upgrade shot 32563 at t = 4.500 s has been
included since the analytical theory of Section 3.4 yields nonsensical results, most notably
a negative be , due to the very low ωpe /|ωce | resulting from the upper hybrid layer being
encountered in the scrape-off layer.
While the above conclusions are important, the most interesting information provided
by the considerations related to the homogeneous plasma threshold are related to the
60
frequency shift due to the parametric decay instability and the validity of the approximations used in the derivations of Chapter 3 in ASDEX Upgrade shot 28286 at t =
2.900 s. The
1 ) is found, using the results of Section 3.4:
√ (angular) frequency shift (ω
2
2 )} and b ≈
ω1 ≈ ωLH 1 + A1 be , with A1 = 3{[ωpe /(4ωU2 H )] + [Ti /(Zi Te )](ωU2 H /ωpe
e
2
2
2
2
2
(ωU H − ω0 + 2ω0 ωLH )/{ωpe /[1 − ωpe /(3ωce )] − A1 ω0 ωLH }, to be ω1 /(2π) = 832 MHz.
This is only 5.45 % smaller than, and within the uncertainty of, the frequency shift of
(880 ± 50) MHz measured in the anomalous scattering spectrum from ASDEX Upgrade
shot 28286 at t = 2.900 s, Fig. 1.3. Given the large number of assumptions, as well as
the reconstructions of equilibrium and electric field profiles, involved in arriving at the
above result, an agreement to this accuracy is quite impressive and an important indication that anomalous scattering is attributable to parametric decay near the upper hybrid
resonance; the above estimate of ω1 /(2π) will be further refined later in this section. Based
on the above coincidence of frequency shifts, we expect the assumptions involved in deriving the lower hybrid dispersion relation, which are the most restrictive in the theory of
parametric decay from Section 3.4, not to be violated. Since kz ≈ 0 by assumption, this
is confirmed by the fact that A1 be ≈ 0.567 which, while not much smaller than 1, is small
enough to justify the Taylor expansions used in the derivations of Section 3.4; for reference
2 /(3ω 2 )] ≈ 0.0373 1, so the approximations used for the high-frequency elecbe /[1 − ωpe
ce
tron Bernstein waves are satisfied with a considerably larger margin, as previously claimed.
The fact that the assumptions made in connection with the derivation of the lower hybrid
dispersion relation are actually significantly more reasonable, than indicated by the discussion at the end of Chapter 3, is connected with the fact that ω0 is actually slightly larger
than ωU H just before the upper hybrid resonance layer, which was reduces the value of
be below the minimum value at ω0 = ωU H considered in Chapter 3. This is most likely
attributable to the inclusion of collisions in the full-wave slab model applied near the upper
hybrid resonance, along with the near cancellation of the main contributions to be in this
region; we note that the important conclusion of a non-resonant parametric decay instability, drawn in Section 3.4 and necessary for the above results to be valid, is still satisfied at
the points of significant electric field in ASDEX Upgrade shot 28286 at t = 2.900 s. In fact,
the most questionable assumption close to the upper hybrid resonance is the early made
dipole approximation, k0 k, which was already expected from the fact that linear mode
conversion of X-mode radiation to electron Bernstein waves and vice versa, may occur in
this region, and noted by, e.g., [Porkoláb, 1978]; we shall return to this matter later in this
section, but not until after the fundamental theory of parametric decay in inhomogeneous
plasmas has been discussed.
The fact that the threshold of the parametric decay instability is raised, generally significantly, for an inhomogeneous plasma was noted by [Rosenbluth, 1972], who considered
the resonant parametric decay instability in a weakly inhomogeneous medium within a
WKB framework. The main conclusion drawn in the above article is that the parametric
decay instability in an inhomogeneous plasma is usually convective (spatial), rather than
absolute (temporal), since the wave vector selection rule for the linear plasma modes can
only be approximately fulfilled within a finite region of the inhomogeneous plasma, determined by the scale length of the plasma parameters. An exception to this rule of thumb
61
occurs when parametric decay takes place at a local minimum/maximum of the plasma
parameters, where the scale length becomes infinite and waves may be trapped, e.g., at
the O-point of a magnetic island in a tokamak plasma; the occurrence of such structures
is, as previously mentioned, strongly correlated with anomalous scattering during electron
cyclotron resonance heating in tokamaks, [Westerhof et al., 2009] and [Nielsen et al., 2013],
and a theoretical model describing this in terms of parametric decay into two trapped
upper hybrid waves has been published recently by [Popov and Gusakov, 2015a]. The
parametric decay instability may also be limited by an inhomogeneous pump wave, i.e., if
the amplitude of E0 is only significant within a limited region of space, as was noted by
[Pesme et al., 1973], who also showed that an absolute parametric decay instability is possible in the case where the pump wave is backscattered parametrically by a plasma wave,
which has been used to explain low-threshold parametric decay instabilities in (overdense)
spherical tokamaks more recently [Gusakov and Surkov, 2007].
The above considerations are all valid for resonant parametric decay. However, based on
the results of Chapter 3, we expect the parametric decay instability considered in this work
to be of the non-resonant type. This case is relatively simple compared with the resonant
one above, chiefly due to the fact that the non-resonant parametric decay instability can
be viewed as simple amplification of the high-frequency daughter waves by the pump wave,
a point discussed in Section 2.4, but see also [Weiland and Wilhelmsson, 1977]. In the case
where the high-frequency daughter waves are not trapped, which is the one of relevance to
the cases considered in this work, we can simply determine the spatial gain (negative optical
thickness), G = −τ , along the rays of the high-frequency daughter waves; the geometric
optics approximations are used due to the short wavelength of these waves. This is done
using an expression similar to Eq. (4.9), replacing Γ by γ for the non-resonant parametric
decay instability,
Z sf
γ(r(s), k(s), ω2 )
γ(r(s), k(s), ω2 )
G=2
ds = 2
∆s;
(4.25)
|vg (r(s), k(s), ω2 )|
ss |vg (r(s), k(s), ω2 )|
in the last equality hγ/|vg |i represents the average value of γ/|vg | along the ray, and
∆s = sf − ss is the length of the ray in question. The above expression is similar to one
used by [Berger et al., 1977] to study non-resonant parametric decay of lower hybrid waves,
as well as the one used by [Ott et al., 1980] to study non-resonant parametric decay into
electron Bernstein waves during electron cyclotron resonance heating in tokamaks. We shall
generally assume that amplification of the high-frequency daughter waves only takes place
in a narrow region, e.g., in the region of the enhanced field near the upper hybrid layer, such
that we can replace the average of γ/|vg | along the ray with its value at a representative
point sc , hγ(r(s), k(s), ω2 )/|vg (r(s), k(s), ω2 )|i ≈ γ(r(sc ), k(sc ), ω2 )/|vg (r(sc ), k(sc ), ω2 )|
and interpret ∆s as the ray length over which amplification takes place; the pump electric
field strength used in γ should still represent an average in the amplification region. With
this approximation, we are ready to determine the inhomogeneous plasma threshold of
the non-resonant parametric decay instability of electromagnetic pump waves into highfrequency electron Bernstein modes and low-frequency lower hybrid quasi-modes, for which
the homogeneous plasma threshold was determined in Section 3.4.
62
The first thing to note about the inhomogeneous plasma threshold is that it is not quite as
well-defined as the one in a homogeneous plasma (max(γ) > 0), since the total amplification
of the power in the high-frequency daughter waves along a ray is eG and the input power
is essentially provided by thermal fluctuations, which are not necessarily large enough for
the parametric decay instability to be observed for all max(G) > 0; rather max(G) should
have a sufficiently large positive value. This value is somewhat arbitrary, so here we
follow the convention of [Rosenbluth, 1972] and define the threshold to be max(G) > 2π,
corresponding to the thermal fluctuation input power being amplified by a factor of 535.5,
keeping in mind that the parametric decay instability may be observable for values close
to, but slightly below, this. Inserting the above results, for now suppressing the implicit
spatial dependence, yields the inhomogeneous plasma threshold,
γ ∆s
max
> π.
(4.26)
|vg (k, ω2 )|
For the electron Bernstein waves, vg (k, ω2 ) = ∂ω2 (k)/∂k may be determined using Eq.
(3.29), along with the product rule and the expression for gradients in cylindrical coordi2 k2 ,
nates, remembering that be = rLe
⊥
2
ωpe
1 ∂ω22 (k)
vg (k, ω2 ) =
=−
2ω2 ∂k
ω2
2 k B
2 k2 k
ωce
ωce
be
z
⊥
z
− 2
,
2
2 + ω2
2
2 /(3ω 2 )
1 − ωpe
ωU H k⊥
k⊥
ce
U H k⊥ B
(4.27)
where kz is the component of k parallel to B. From the above equation,
s
2
2 k2 2
4 k2
ωpe
be
ωce
ωce
z
z
|vg (k, ω2 )| =
−
+
2
2
2 /(3ω 2 )
k ⊥ ω2
1 − ωpe
ωU2 H k⊥
ωU4 H k⊥
ce
(4.28)
s
2
4 k2
ωpe
b2e
ωce
z
≈
+ 4
2 ,
2 /(3ω 2 )]2
k⊥ ω2 [1 − ωpe
ωU H k⊥
ce
2 and k 4 /k 4 in comparison with b2 and k 2 /k 2 when apwhere we have neglected be kz2 /k⊥
e
z
z
⊥
⊥
2 Z m /m 1 and b 1 made when
proximating, due to the assumptions of kz2 /k⊥
i e
i
e
deriving threshold of the parametric decay instability in Section 3.4. From Section 3.4, we
2 , meaning that the minialso have that be (and k⊥ ) are essentially independent of kz2 /k⊥
mum of |vg |, which leads to the lowest inhomogeneous plasma threshold by Eq. (4.26), is
2 /(k ω )]b /[1 − ω 2 /(3ω 2 )]
achieved for kz → 0 and has the value min[|vg (k, ω2 )|] ≈ [ωpe
e
⊥ 2
pe
ce
√
for ωpe < 3|ωce |; we further note that this is consistent with the approximations made
when determining max(γ) in Eq. (3.54). Thus, if we neglect any dependence of ∆s on
the direction of k, which is difficult to determine accurately in any case, we find the
threshold condition max(γ) ∆s/ min[|vg (k, ω2 )|] > π with max(γ) from Eq. (3.54) and
2 /(k ω )]b /[1 − ω 2 /(3ω 2 )]. By additionally neglecting the influmin[|vg (k, ω2 )|] ≈ [ωpe
e
⊥ 2
pe
ce
ence of collisions on max(γ), since the inhomogeneous threshold will be found to be significantly larger than the homogeneous collision-limited one, the inhomogeneous instability
threshold of non-resonant parametric decay into high-frequency electron Bernstein modes
63
and low-frequency lower hybrid quasi-modes is found to be,
(0)
2 /ω 2
√ ω13 −ω2 /(k2 v2 )
8Ne Te ωpe
π
UH
(4.29)
2
π 3 3 e 1 ⊥ Ti ,
2 /(3ω 2 ) k ∆s
0
1 − ωpe
k⊥ vT i
⊥
ce
q
−1
2 /[1 − ω 2 /(3ω 2 )] − A ω ω
≈ rLe
(ωU2 H − ω02 + 2ω0 ωLH )/{ωpe
1 0 LH },
pe
ce
|E0⊥ |2 >
where conditions k⊥
2 v 2 ) ≈ [Z T /(2T )](ω 2 /ω 2 ){[{ω 2 /[1 − ω 2 /(3ω 2 )] − A ω ω
2
2
ω12 /(k⊥
i e
i
1 0 LH }/(ωU H − ω0 +
pe
pe
pe
ce
Ti
UH
2
2
2
2
2ω0 ωLH )] + A1 }, and ω0 ≈ ωU H , with A1 = 3{ωpe /(4ωU H ) + [Ti /(Zi Te )](ωU H /ωpe )},
should be used/satisfied.
2 /ω 2
This threshold is similar to the one given by [Porkoláb, 1982] in the limit of small ωpe
ce
2 /ω 2 )/[1 − ω 2 /(3ω 2 )] ≈ ω 2 /ω 2 , if account is
(as assumed in that paper) where (ωpe
pe
ce
pe
ce
UH
taken of the use of CGS units in that paper, 0 → 1/(4π), and the previously mentioned
√
2 v2 )
3 v 3 )] e−ω12 /(k⊥
T i is made; setting
identification of 1/θ from that paper with 2 π[ω13 /(k⊥
Ti
√
2
2
2
3
3
3
−ω
/(k
v
)
2 π[ω1 /(k⊥ vT i )] e 1 ⊥ T i ≈ 1, the above threshold is also similar to one from [McDer2 /ω 2 (as assumed in that article).
mott et al., 1982] for small ωpe
ce
With the above inhomogeneous threshold of the parametric decay instability, we are ready
to properly investigate the parametric decay instability in ASDEX Upgrade shot 28286 at
t = 2.900 s. The electric field profile, which we shall consider, is simply that on the central
beam ray shown in Fig. 4.2, since the largest electric fields are expected to lie on this,
and the plasma parameters used for calculating the threshold are also extracted along this
ray. Before continuing we must, however, settle on a method for determining ∆s, since
"the ray length over which significant amplification of the high-frequency daughter waves
occurs" is a somewhat vague definition. This vagueness is clearly illustrated by the fact that
[Porkoláb, 1982] cites ∆s as being the free space wavelength 2πc/ω0 , while [McDermott et
al., 1982] has ∆s being related to the plasma parameter scale lengths, `, which seems more
plausible, although the precise relation is not specified. The method, which we shall apply,
is to first calculate the characteristic quantities related to the daughter modes with the
2 /[1−ω 2 /(3ω 2 )]−
lowest parametric decay thresholds, i.e., be ≈ (ωU2 H −ω02 +2ω0 ωLH )/{ωpe
pe
ce
√
A1 ω0 ωLH } and ω1 = ωLH 1 + A1 be . Then, at a point sX , ∆s = sX −sX0 , where sX0 < sX is
the closest point for which |ω1 (sX0 )−ω1 (sX )|/(2π) ≥ 100 MHz. This definition is somewhat
arbitrary, but atleast rooted in the fact that inhomogeneities will cause waves to fulfil the
selection rules, necessary for the parametric decay instability to occur, within a limited
region of space, and the frequency shift of 100 MHz is further chosen to coincide with
the width of the suspected lower hybrid/electron Bernstein feature in Fig. 1.3; different
criteria, e.g., that the frequency shift be below 10 % of ω1 (sX )/(2π), may be appropriate
within a more general setting. Now, averaging the electric field on the left hand side of Eq.
(4.29) over the region, using the field enhancement from Fig. 4.2 and Eq. (4.14), as well as
the right hand side, using the appropriate plasma and daughter wave parameters extracted
along the ray, a power threshold related to each sX , P0,th (sX ), may be calculated.
A semi-logarithmic plot of P0,th (sX ), along with the field enhancement |E0⊥ (0, sX )|/|E0 (0, 0)|
for reference, near the upper hybrid resonance is seen in Fig. 4.4. In Fig. 4.5, we further show the average value of ω1 /(2π) related to each P0,th (sX ), hω1 i(sX )/(2π), to get an
idea about the frequency shift of the parametric decay peak for instabilities occurring in
64
different parts of the upper hybrid layer. We first note that ∆s = (330 ± 30) µm for the
range of sX -values shown in Figs. 4.4 and 4.5, so the scheme used does atleast produce
relatively consistent values of ∆s in the upper hybrid layer; our ∆s is considerably smaller
than the values quoted by [Porkoláb, 1982] and [McDermott et al., 1982], but these values
also seem to be chosen slightly ad hoc. Unsurprisingly, Fig. 4.4 shows that low values
of P0,th (sX ) are strongly correlated with large values of |E0⊥ (0, |/|E0 (0, 0)|, although a
slight shift of low P0,th (sX ) relative to high |E0⊥ (0, |/|E0 (0, 0)| exists when entering the
evanescence region. The shift of P0,th (sX ) relative to |E0⊥ (0, |/|E0 (0, 0)| may be understood from the fact that sX represents the upper limit of the range used to calculate
P0,th (sX ), such that sX -values less than ∆s above a large value of |E0⊥ (0, |/|E0 (0, 0)| will
also experience a lowered P0,th (sX ); the sX -range of low P0,th (sX ) in the evanescence region thus gives an idea about the typical value of ∆s. The smallest threshold from Fig.
4.4, min[P0,th (sX )] = 2.08 kW, is attained for sX = 20.00 cm and is mainly dependent
on the very large value of |E0⊥ (0, |/|E0 (0, 0)| in last lobe before the evanescence region,
depending relatively weakly on ∆s except in extreme cases. While its value is still much
lower than the P0 used in the shot, it is as previously mentioned around three orders of
magnitude above the corresponding homogeneous plasma threshold (3.13 W), justifying
the neglect of collisions etc. From Fig. 4.5, the mean frequency shift corresponding to
min[P0,th (sX )] is hω1 i(20.00 cm) = 847 MHz, which is again very close to (3.75 % smaller
than) the frequency shift of 880 MHz observed in the anomalous scattering spectrum from
ASDEX Upgrade shot 28286, Fig. 1.3, at t = 2.900 s.
We note that the hω1 i(sX )/(2π) corresponding to the lowest threshold may often not be
the one observed, since the pump wave energy may be lost to the electron Bernstein
waves rather quickly once the threshold corresponding to the P0 used in the experiment is
exceeded. The possible implications of the above proposal may be investigated by looking
at hω1 i(sX )/(2π) at the point where P0,th (sX ) = 500 kW, which is the nominal power used
in the CTS experiments at ASDEX Upgrade. This point occurs at sX = 19.92 cm and, from
Fig. 4.5, hω1 i(19.92 cm) = 883 MHz, which is extremely close to the frequency shift of 880
MHz observed in ASDEX Upgrade shot 28286 at t = 2.900 s. Of course, the above value
depends on the somewhat arbitrary value ∆s, as well as the very uncertain quantities
F and |E0⊥ (0, sX )|/|E0 (0, 0)|, and the reconstructed equilibria, so the extremely close
agreement should probably be considered a happy accident; it does, however, indicate the
interesting possibility of a P0 -dependent frequency shift of the parametric decay instability
in inhomogeneous plasmas.
We finally revisit the validity of the approximations made in connection with parametric
decay into lower hybrid quasi-modes and electron Bernstein modes, as well as the dipole approximation, in ASDEX Upgrade shot 28286 at t = 2.900 s. The approximations, related to
the specific parametric decay instability in question, i.e., significant Landau damping and a
relatively small A1 be , seem to be well-satisfied in the region relatively low P0,th (sX ), as was
indicated in the discussion of the homogeneous plasma threshold. The validity of the dipole
approximation is assessed by comparing the value of k⊥ to that of k0 (sr ) = 1.02 × 104 m−1 ,
i.e., the electromagnetic wave number at the point where the ray tracing is stopped; the
spatial variation of the full-wave slab solution does not appear to be significantly faster
65
Figure 4.4 – Semi-logarithmic
plot of the power threshold,
P0,th (sX ), and field amplification,
|E0⊥ (0, sX )|/|E0 (0, 0)|,
versus sX for the reflected
ω0 /(2π) = 105 GHz X-mode
radiation in ASDEX Upgrade
shot 28286 at t = 2.900 s.
The minimum power threshold
min[P0,th (sX )] = 2.08 kW is
obtained for sX = 20.00 cm.
P0,th (sX ) is slightly shifted relative to |E0⊥ (0, sX )|/|E0 (0, 0)|.
Figure 4.5 – Plot of the mean
frequency of the low-frequency
modes, hω1 i(sX )/(2π), related to
the power threshold P0,th (sX ),
versus sX , in ASDEX Upgrade
shot 28286 at t = 2.900 s. At sX =
20.00 cm, where min[P0,th (sX )] occurs, the mean frequency shift is
hω1 i(sX )/(2π) = 847 MHz; for
larger values of P0 , smaller sX are
necessary for the parametric decay instability to occur, and larger
hω1 i(sX )/(2π) are expected.
than what is implied by this value, see Fig. 4.4, and note that the period of the absolute values plotted is half that of the wave itself. The value of k⊥ is generally found to
only slightly exceed k0 (sr ), being roughly 1.3 larger around sX = 20 s and increasing with
decreasing sX . This makes the dipole approximation quite questionable, and an obvious
extension of the present work would therefore be to make do without this approximation
and see whether or not substantial changes result. The fact that k⊥ and k0 (sr ) do not
exactly coincide before the evanescence region does at least indicate that the parametric
decay instability may occur before linear mode conversion, but a proper investigation of
this point would require a warm description of the X-mode radiation, which is beyond the
scope of the present work.
4.4
Numerical Investigations of Parametric Decay in ASDEX
Upgrade
Since the analytical theory, used for deriving the threshold of the parametric decay instability near the upper hybrid layer so far, has a rather limited range of validity which often
makes its predictions questionable at the points where such a parametric decay instability
66
may occur, this section is devoted to numerical investigations of parametric decay near the
upper hybrid layer for the ASDEX Upgrade. We still assume the high-frequency upper
hybrid daughter waves to obey the approximate dispersion relation from Eq. (3.29), as the
approximations involved in deriving this dispersion relation were significantly less restrictive than the ones involved in deriving the low-frequency lower hybrid dispersion relation.
We also assume the parametric decay instability to be non-resonant and convective, such
that the total gain, G, may be obtained by integrating along the rays of the high-frequency
(electron Bernstein) daughter waves using Eq. (4.25). However, we treat ω1 as a free variable and estimate G for each specific value of ω1 by integrating along a high-frequency ray
with a complementary frequency, ω2 = ω0 − ω1 , from the selection rules. The most difficult
problem in the programme mapped out above is to find the ray having the greatest G
for a given ω1 , due to the many degrees of freedom involved in determining both the ray
trajectory and amplification factor in a tokamak plasma. While optimisation is of course
possible, we shall in the following simply consider the rays starting out with k parallel to
E0⊥ , the direction of the real part is used, at the point of maximum |E0⊥ (0, sX )|, see Figs.
4.2 and 4.3, available to that particular high-frequency daughter ray. This choice should
tend to maximise µe and minimise |vg (r(s), k(s), ω2 )| in the region around the point where
the largest γ is expected, and thus tend to maximise G. However, the value of γ may
2 ∼ Z m /m since electron Landau
be significantly affected by a small, but finite, kz2 /k⊥
i e
i
damping of the low-frequency quasi-mode will be maximised around this value; this has
not been taken into account in the above optimisation.
The rays of the electron Bernstein waves are calculated by numerical implementation of
Eqs. 4.7 in Matlab, using the dispersion relation
D(r, k, ω2 ) =
2 r 2 (r)
2 (r) k 2
k⊥
ωU2 H (r) − ω22
ωce
z
Le
−
−
2
2 = 0,
2 (r)
2 (r)/[3ω 2 (r)]
ωpe
1 − ωpe
ω
(r)
k
ce
UH
⊥
(4.30)
which is a rewritten version of Eq. (3.29). Gradients in k-space are calculated using the
fact that the above expression is cylindrically symmetric in this space, while gradients in
(0)
r-space are calculated by extracting B(r), Ne (r) and Te (r), from the ASDEX Upgrade
equilibrium being investigated and using the centred difference scheme employed in the
gradient-function from Matlab; interpolation between grid points is simply linear and
done using the interp2-function from Matlab. The system of ordinary differential equations given by Eq. (4.7) is solved using the ode45-function from Matlab. We note that,
while D(r(s), k(s), ω2 ) does remain close to zero in the cases presented here, the above
scheme is lacking in both computational speed and accuracy compared with ray tracing
codes, such as wr, used to calculate the trajectories of electromagnetic rays. However, for
obtaining the basic estimates for G in which we are interested, it seems sufficient.
The point of largest |E0⊥ (0, sX )| available to the electron Bernstein waves can formally
be determined by noting that real k, as is appropriate
for propagating electron Bernstein
√
waves, requires ωU H (r) > ω2 for ωpe (r) < 3|ωce (r)| by Eq. (4.30). As previously mentioned, the propagating reflected X-mode radiation also generally has ω0 < ωU H , so there
will be some maximum value sX above which this condition may no longer be satisfied.
However, the point of maximum |E0⊥ (0, sX )|, only satisfying this condition, may be very
67
close to the cut-off at ω2 = ωU H (r) where ray tracing breaks down for the electron Bernstein waves. It is hence often necessary to choose a slightly smaller value. The numerical
scheme employed in this work is to solve to the ray tracing problem for a given maximum
|E0⊥ (0, sX )|, check whether k is real at all times, and reduce the value of sX ; obviously, this
method may fail if there are no sX capable of satisfying the requirement of real k, which
is the case for the frequencies investigated in ASDEX Upgrade shot 32563 at t = 4.500 s
due to the extreme plasma conditions in the scrape-off layer compared with the regions
in which the theory laid out this work is applicable. This means that we have to leave
a theoretical investigation of parametric decay instabilities taking place when the upper
hybrid layer is encountered in the scrape-off layer for later work and focus solely on the
bulk case in ASDEX Upgrade shot 28286 at t = 2.900 s for the remainder of this work.
To determine γ we use Eq. (2.35) for a simple plasma, containing only one ionic species,
with χe (k, ω1 ) and χi (k, ω1 ) obtained from collisionless models, in accordance with earlier
results indicating the collisional contribution to be negligible which further justifies setting
Γ(k, ω2 ) ≈ 0, and also considering ions to be unmagnetised, i.e.,
"
#
∞
2
X
2ωpe
ω1
ω
−
nω
2
2
1
ce
2 2
χe (k, ω1 ) = 2 2 1 +
,
(4.31)
rLe ) e−k⊥ rLe Z
In (k⊥
|kz |vT e n=−∞
|kz |vT e
k vT e
"
r
2
Zi Te 2ωpe
mi Te ω1
χi (k, ω1 ) =
1+
Z
Ti k 2 vT2 e
me Ti kvT e
r
mi Te ω1
me Ti kvT e
!#
.
(4.32)
As always, we set Te = Ti and consider a deuterium plasma, Zi = 1 and mi /me = 3672 in
the numerical implementation; quantities related to the electrons are extracted from the
ASDEX Upgrade equilibrium under consideration along the rays of the electron Bernstein
wave with ω2 = ω0 − ω1 . The sum in χe (k, ω1 ) is truncated at for |n| larger than a value
2 r 2 which small, by assumption, for the electron Bernstein
mainly determined be = k⊥
Le
waves, so the sum is truncated for the somewhat arbitrary, relatively large, value of |n| >
20. We also use the approximations (∂ Re[(k, ω)]/∂ω)|ω=ω2 ≈ (∂ Re[χe (k, ω)]/∂ω)|ω=ω2 ≈
2 /(ω 2 − ω 2 )2 for the electron Bernstein waves in Eq. (2.35), and the value G is
2ω2 ωpe
ce
2
calculated in accordance with Eq. (4.25) by integrating γ/|vg | along the rays of the electron
Bernstein waves with |vg | from Eq. (4.28). Since the amplification of the electron Bernstein
waves, by Section 4.3, is expected to take place within a very narrow region, which is often
significantly smaller than the step size in the traced rays, we introduced a grid containing
more points for evaluating G, interpolating linearly between the computed ray point using
the interp1-function in Matlab. The Fried-Conte plasma dispersion function (Z) is
implemented numerically in a way similar to the one described by [Froula et al., 2011], but
some corrections/modifications of the precise numerical implementation suggested in that
book, discussed by [Kjer Hansen, 2014], are employed. Finally, µe is calculated from Eq.
(2.15), using the E0 (ρ, sX ) = |E0 (ρ, sX )|e0 (r(sX ), k0 (sX , ω0 ), discussed in Sections 4.1 and
4.2, along with our knowledge of ρ, sX , k, and B along the rays of the electron Bernstein
waves based on the plasma/beam parameters.
By employing the procedure described above to ASDEX Upgrade shot 28286 at t = 2.900 s,
we arrive at the results summarised in Figs. 4.6 and 4.7. Fig. 4.6 shows a contour
68
Figure 4.7 – Numerically obtained gyrotron power threshold
P0,th versus ω1 /(2π) in ASDEX
Upgrade shot 28286 at t = 2.900 s.
The minimum power threshold
min(P0,th ) = 11.95 kW is obtained for ω1 /(2π) = 837.5 MHz,
which agrees relatively well with
the results of Section 4.3 and
the frequency shift observed in
the anomalous scattering spectrum, Fig. 1.3. We also note
that the region of low P0,th is quite
wide, as may be expected for the
non-resonant parametric decay instability.
Figure 4.6 – Contour plot
of the spatial amplification
factor
γ(r(s2 ), k(s2 ), ω2 )/
|vg (r(s2 ), k(s2 ), ω2 )| along the
electron Bernstein waves versus
ray length s2 and ω1 /(2π) for
P0 = 500 kW in ASDEX Upgrade shot 28286 at t = 2.900 s.
While the total ray lengths
considered are 6 cm in all cases,
significant spatial amplification
only occurs a narrow region of
width ∆s ≈ 200 µm around the
initial point shown in the figure.
This is in reasonable agreement
with the results of Section 4.3.
plot of γ/|vg | along the rays of the electron Bernstein waves corresponding to various
ω1 /(2π) ∈ [600 MHz; 1.225 GHz], s2 is the arc-length along the electron Bernstein wave
and s2 = 0 at the point of maximum |E0⊥ (0, sX )| described above; the gyrotron power
used in the figure, P0 = 500 kW, is the nominal in CTS experiments at ASDEX Upgrade.
To put the Fig. 4.6 (as well as Fig. 4.7) into context, all rays, corresponding to the
ω1 /(2π)-values shown, have s2 = 0 at sX = 19.99 cm, which is the global maximum of
|E0⊥ (0, sX )| from Fig. 4.2, i.e., the point where the largest field enhancement is generally
expected; rays are started both parallel and anti-parallel to E0⊥ , which is permissible
since the Hamiltonian system in Eq. (4.7) is reversible in s, and traced 3 cm in both
directions, i.e., to s2 = ±3 cm. The reason for only showing a narrow region between
s2 = ±500 µm is that significant amplification of the electron Bernstein waves only occurs
in this region, as is also indicated by Fig. 4.6. The width of the region over which
significant amplification of the electron Bernstein waves occur is ∆s ≈ 200 µm, which
is in reasonable agreement with the value of ∆s = (330 ± 30) µm obtained from quite
different arguments in Section 4.3. By integrating the spatial amplification in Fig. 4.6
69
over s2 , an estimate for G corresponding to P0 = 500 kW is obtained, and, remembering
that G ∝ P0 through γ from Eq. (2.35), µ2e , and E0 , the gyrotron power threshold of
the parametric decay instability, P0,th , defined as the smallest value for which G > 2π,
may easily be determined. The obtained value of P0,th is plotted versus ω1 /(2π) in Fig.
4.7. The minimum gyrotron power threshold, min(P0,th ) = 11.95 kW, is obtained for
ω1 /(2π) = 837.5 MHz. While min(P0,th ) = 11.95 kW is almost an order of magnitude
above value of 2.08 kW from Section 4.3, it is still far below the gyrotron power used
in the CTS experiments at ASDEX Upgrade so the main conclusion is unchanged. It is
also noted that both P0,th above should be considered no than estimates of the actual
P0,th in this problem, and the relation of this quantity to the one which would actually
be observed experimentally is of course itself rather uncertain. Keeping this in mind,
and further remembering that the above value has only been optimised in a very crude
manner, the agreement between to two results is reasonable; based on the physics alone,
the result obtained in this section is certainly more credible than that of Section 4.3. The
more interesting question is of course related to the identification of the parametric decay
instability with lower hybrid quasi-modes, which is the only case that has been considered
in detail in this work. Based on the fact that min(P0,th ) occurs for ω1 /(2π) = 837.5 MHz,
which is the grid point closest to 832 MHz, shown to be the frequency of the electron
Bernstein/lower hybrid parametric decay instability with the lowest homogeneous plasma
threshold, for the point with s2 = 0, at the beginning of Section 4.3. This, along with
the relatively broad range of ω1 /(2π) for which P0,th is quite small rather than a sharp
drop near a resonance, is a strong indication that the full χe (k, ω1 ) and χi (k, ω1 ), near the
upper hybrid resonance in ASDEX Upgrade shot 28286, are indeed well-represented by the
approximate ones used for the lower hybrid waves in Section 3.3 and that the parametric
decay instability is indeed of the non-resonant type. We finally note that the obtained
value once again lies within the uncertainty of the anomalous scattering frequency shift
of (880 ± 50) MHz observed in ASDEX Fig. 1.3, and that the slightly higher measured
frequency shift measured may be related to the parametric decay instability occurring
before the point of maximum |E0⊥ (0, sX )| due to P0 being well above P0,th in the shot, as
discussed in Section 4.3.
70
Chapter 5
Conclusions and Outlook
In this thesis, we have investigated parametric decay instabilities in tokamak plasmas
and shown that these may explain the anomalous scattering features sometimes observed
during CTS experiments in ASDEX Upgrade, particularly for shot 28286 at t = 2.900 s.
We first considered a general theory of parametric decay for electromagnetic waves in
plasmas, which assumed daughter waves to be electrostatic in nature and utilised a dipole
approximation, valid for k0 k, as done by a number of earlier authors, e.g., [Aliev et
al., 1966], [Amano and Okamoto, 1969], [Porkoláb, 1974], and [Porkoláb, 1978]. We then
proceeded to apply the theory to the case thought to be of most interest in connection with
anomalous scattering in the CTS experiments, namely parametric decay of high-frequency
electromagnetic radiation near the upper hybrid layer. Here we found parametric decay
into high-frequency (electrostatic) electron Bernstein waves and low-frequency electrostatic
waves, particularly ion Bernstein waves and lower hybrid waves, to be likely. Since the
anomalous scattering spectrum in Fig. 1.3 contains a relatively well-defined line, differing
roughly from the gyrotron frequency by the lower hybrid frequency, and also following
earlier theoretical work [Porkoláb, 1982], as well as experimental evidence from the Versator
II tokamak [McDermott et al., 1982], most emphasis was put on studying the case where
the low-frequency daughter waves are lower hybrid waves. In connection with these studies,
the results of [Porkoláb, 1982] and [McDermott et al., 1982] were generalised include effects
of finite ωpe /|ωce | on the perturbation terms, but the main conclusions of the earlier work
by these authors were confirmed for tokamak plasma condition, e.g., the fact that the
parametric decay instability is of the non-resonant type and the leading terms of the
analytical expressions determining the electric field thresholds necessary for the parametric
decay instability to occur in homogeneous/inhomogeneous plasmas.
Apart from direct studies of parametric decay instabilities, studies of wave propagation in
tokamak plasmas were also undertaken. These demonstrated the accessibility to the upper
hybrid layer for X-mode radiation reflected from the high-field side wall during the times
where anomalous scattering was observed in the CTS experiments, as well as the lack of
such accessibility when it was not. A formalism for determining the X-mode electric field
near the upper hybrid layer, where it may be strongly enhanced which is important for
71
the occurrence of the parametric decay instability in this region, based on the beam and
plasma parameters, was further developed. This included a calculation of the amount of
incident O-mode power which is coupled back to the plasma in X-mode after being reflected
by the high-field side wall, as well as an estimate of the field enhancement near the upper
hybrid layer by connecting the field amplitude calculated by geometric optics/Gaussian
beam approximations to that obtained from a full-wave solution, assuming slab geometry
and a cold plasma response (including collisions), near the upper hybrid layer, inspired by
[White and Chen, 1974]; both procedures result, at best, in rough estimates of the true
field near the upper hybrid layer in a real tokamak CTS experiment.
When combining the above results, a power threshold and a frequency shift due to the
parametric decay instability may be calculated for an ASDEX Upgrade equilibrium and
beam setting under consideration. In this thesis, the equilibrium and beam settings from
ASDEX Upgrade shot 28286 at t = 2.900 s has been investigated in detail; an attempt
has also been made at applying the theory to ASDEX Upgrade shot 32563 at t = 4.500 s,
but in this case the upper hybrid layer is encountered in the scrape-off layer where the
analytical, as well as numerical, treatments of the parametric decay instability described
above seem incapable of providing sensible results in their present forms. For ASDEX
Upgrade shot 28286 at t = 2.900 s, the quasi-analytical treatment of Section 4.3 yields
the power threshold P0,th = 2.08 kW and a mean frequency of the low-frequency daughter
wave of hω1 i/(2π) = 847 MHz, while the numerical treatment of Section 4.4 yields, P0,th =
11.95 kW and ω1 /(2π) = 837.5 MHz, where the frequency shift is now an exact value
relating to a particular high-frequency daughter ray. We note that the power thresholds
are in both cases far below the gyrotron power, P0 ≈ 500 kW, used in the shot, so a
parametric decay instability decay instability is definitely expected to occur, consistent
with the observation of anomalous scattering. However, the very low thresholds are the
result of a very strong field enhancement, upto 2.1 × 103 relative to vacuum value, which is
only slightly off-set by the fact that only 6.78 % of the incident gyrotron power is coupled
to the reflected X-mode, according to the model used, and the physicality of which is
questionable. If we are to trust the field enhancement model, a possible way of avoiding
parametric decay instabilities would be to launch radiation with a greater component
parallel to the background magnetic field, since this tends to reduce the field enhancement
predicted by it significantly. All things considered, the thresholds of 2.08 kW and 11.95 kW
do not seem beyond reason given the circumstances, and if one of the above thresholds is to
be chosen, the one of 11.95 kW seems to provide the answer having the strongest physical
foundation. The power thresholds aside, the most interesting result of the above discussion
is undoubtedly the closeness of the above frequency shifts to the one of 880 ± 50 MHz
observed in the anomalous scattering spectrum from ASDEX Upgrade shot 28286 at t =
2.900 s, seen in Fig. 1.3. This result provides strong evidence that anomalous scattering
in CTS experiments at ASDEX Upgrade are caused by a parametric decay instability of
the type described above, as was expected.
There are several questions raised in this work, the resolution of which may of sufficient
interest to warrant further investigations. First of all, it would be highly desirable to have
a credible way of calculating field enhancement near the upper hybrid layer, which might
72
also answer whether linear mode conversion occurs before the parametric decay instability
or if the opposite is the case; investigations of this type are ongoing. Second, it would
be interesting to see whether the numerical implementation of the current theory may
be extended to provide power thresholds and frequency shifts related to the parametric
decay instability when the upper hybrid resonance is encountered in the scrape-off layer,
as was, e.g., the case for ASDEX Upgrade shot 32563 at t = 4.500s. Third, the dipole
approximation was shown to be questionable near the upper hybrid resonance, so it would
be interesting to consider modifications caused not invoking this approximation; a large
amount of literature dealing with this subject already exists, though not for the precise
problem considered in this work to the author’s knowledge. Finally, it would be desirable
to investigate more equilibria, as well as high-resolution anomalous scattering frequency
spectra, in order to ascertain the degree to which the theory presented is capable of describing the frequency shifts observed in these; such investigations should be straightforward
although they would require the some of the numerical routines employed in this work to
be improved.
73
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