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Suppression of runaway electron avalanches by radial diffusion P. Helander, L.-G. Eriksson, and F. Andersson Citation: Phys. Plasmas 7, 4106 (2000); doi: 10.1063/1.1289892 View online: http://dx.doi.org/10.1063/1.1289892 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v7/i10 Published by the American Institute of Physics. Related Articles Properties of convective cells generated in magnetized toroidal plasmas Phys. Plasmas 19, 082304 (2012) Two-dimensional plasma expansion in a magnetic nozzle: Separation due to electron inertia Phys. Plasmas 19, 083501 (2012) Generation of dust projectiles passing over an obstacle in the plasma sheath Phys. Plasmas 19, 083701 (2012) Three-dimensional numerical investigation of electron transport with rotating spoke in a cylindrical anode layer Hall plasma accelerator Phys. Plasmas 19, 073519 (2012) Revisited global drift fluid model for linear devices Phys. Plasmas 19, 072317 (2012) Additional information on Phys. Plasmas Journal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions PHYSICS OF PLASMAS VOLUME 7, NUMBER 10 OCTOBER 2000 Suppression of runaway electron avalanches by radial diffusion P. Helander EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, United Kingdom L.-G. Eriksson Association EURATOM-CEA sur la Fusion, CEA Cadarache, F-13108 St. Paul lez Durance, France F. Andersson Department of Electromagnetics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden 共Received 6 April 2000; accepted 29 June 2000兲 The kinetic theory of runaway electron avalanches caused by close Coulomb collisions is extended to account for radial diffusion. This is found to slow down the growth of avalanches. An approximate analytical formula for the growth rate is derived and is verified by a three-dimensional Monte Carlo code constructed for this purpose. As the poloidal magnetic flux that is available to induce an electric field in a tokamak is limited, avalanches can be prevented altogether by sufficiently strong radial diffusion. The requisite magnetic fluctuation level is sensitive to the mode structure and the speed of the disruption. It is estimated to be ␦ B/B⬃10⫺3 for parameters typical of large tokamaks. 关S1070-664X共00兲03010-X兴 I. INTRODUCTION II. SECONDARY RUNAWAY GENERATION The generation of runaway electrons in disruptions is a common feature in present-day tokamaks. The runaways are generated in the thermal quench of the plasma, when the plasma temperature drops to the region of 10 eV.1 The resistivity then becomes very large and the loop voltage therefore rises rapidly—typically to a few hundred volts in the center of the Joint European Torus 共JET兲.2 A large fraction 共typically around a half in many tokamaks兲 of the pre-disruption plasma current can then be converted to a runaway current, and the loop voltage drops to nearly zero. For future larger experiments, runaway electrons can be a severe problem since their loss to the first wall may cause severe localized surface damage. It is thus desirable to avoid runaway production in disruptions as far as possible. Magnetic fluctuations may be helpful in this respect since they increase the loss rate of runaway electrons, as has been reported from experiments in Ref. 3. In the present paper we investigate to what extent the presence of radial diffusion caused by magnetic fluctuations can be expected to prevent the production of runaways in large tokamaks. Our paper is organized as follows. In Sec. II we briefly review the physics of runaway production, focusing on so-called ‘‘secondary’’ generation, which is thought to be the dominant mechanism in tokamaks with large plasma current. In the following section the influence of radial diffusion on runaway production is calculated analytically. The result is then compared with simulations done with a three-dimensional, time-dependent Monte Carlo code constructed for this purpose, as described in Sec. IV. Our conclusions are summarized in Sec. V, where we estimate the magnetic fluctuation level required to suppress avalanches. The generation of runaway electrons has been studied theoretically for over four decades. Early work focused on the formation of fast electrons through the combined effects of a constant electric field and velocity–space diffusion as described by the Fokker–Planck equation.4–7 This is sometimes referred to as the primary generation of runaways. More recently it has been realized that in a tokamak with sufficiently large plasma current, close Coulomb collisions 共which are not described by the usual Fokker–Planck equation兲 between thermal electrons and existing runaways can lead to exponential multiplication of the latter—a so-called runaway avalanche.8–10 The point is that a single such collision can kick the thermal electron above the critical energy for runaway acceleration by the electric field. The total number of e-foldings caused by this ‘‘secondary generation’’ mechanism in a disruption can be estimated as10 1070-664X/2000/7(10)/4106/6/$17.00 冕␥ r dt⬃ I , I A ln ⌳ 共1兲 where I is the plasma current and I A ⫽4 m e c/ 0 e⫽0.017 MA is the Alfvén current. Consequently, in small tokamaks this mechanism is not effective, but becomes the dominant mechanism in a next-step device. It has been observed on a small scale in nondisrupting TEXTOR 共Tokamak Experiment for Technology Oriented Research兲 plasmas.11 Mathematically, secondary runaway production is governed by the kinetic equation, 冉 冊 f eE 储 f 2 f ⫽C 共 f 兲 ⫹S, ⫺ ⫺ t m ec p p 共2兲 for the electron distribution f . 10 The velocity–space coordinates are the relativistic momentum p, which has been normalized to m e c 共with m e the rest mass兲, and ⫽p⬜2 / p 2 b ⫽(1⫺ 2 )/b, where ⫽ p 储 /p and b⫽B/B max is the magnetic 4106 Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions Phys. Plasmas, Vol. 7, No. 10, October 2000 Suppression of runaway electron avalanches by . . . field normalized to its maximum on the flux surface in question. Coulomb collisions are described by the right-hand side of the equation, where C( f ) denotes the Fokker–Planck collision operator. For superthermal electrons it has the form7 C共 f 兲⫽ 1 p 2 冋 册 共 1⫹ p 2 兲 f ⫹ 共 1⫹Z eff兲 冑1⫹ p ⫺2 L共 f 兲 , p 共3兲 where Z eff is the effective ion charge, ⫽4 ⑀ 20 m 2e c 3 /n e e 4 ln ⌳ is the collision time for relativistic electrons, and 1 L⫽ 共 1⫺ 2 兲 2 共4兲 is the Lorentz scattering operator. Relativistic effects associated with the thermal electron distribution have been neglected, but are retained for the fast electrons. S is a source term of fast electrons produced by close collisions between existing runaways and slow electrons. In the limit where the former are strongly relativistic and move nearly parallel to the magnetic field while the momentum imparted to the slow electrons is relatively small, S can be calculated by retaining only the leading term in the quantum-mechanical Mo” ller scattering formula.12 This approximation,10 whose accuracy was confirmed in Ref. 13, gives S 共 p, 兲 ⫽ 2⫽ nr 2 ln ⌳ 2b ⫺1 冑1⫹p 2 ⫹1 冑b ⫺1 ⫺ ␦ 共 ⫺ 2 兲 p2 冉 1 p 1⫺ 冑1⫹ p 2 冊 , 共5兲 . Note that this source is proportional to the density of existing runaways n r and to the collision frequency for close collisions 1/( ln ⌳) rather than the usual Fokker–Planck collision frequency 1/ . Rosenbluth and Putvinski10 solved the orbit-average of Eq. 共2兲 analytically in several limits and constructed an interpolation formula, ␥ r ⯝ E⫺1 ln ⌳ 冉 冑 3 共 Z eff⫹5 兲 1 4 共 Z eff⫹1 兲 2 ⫻ 1⫺ ⫹ E 3 共 Z eff⫹5 兲共 E 2 ⫹4/ 2 ⫺1 兲 冊 ⫺1/2 , 共6兲 for the runaway production rate ␥ r ⫽d ln nr /dt, whose correctness they confirmed numerically. Here E⫽ 兩 E 储 兩 /E c is the normalized electric field, ⫽1⫺1.46⑀ 1/2⫹1.72⑀ describes the effects of finite toroidicity, ⑀ ⫽r/R is the inverse aspect ratio, and E c ⫽m e c/e is the critical electric field below which no runaway is possible.7 Thus, we have approximately ␥ r ⬃ E⫺1 . 2 ln ⌳ 共7兲 The estimate 共1兲 is obtained by integrating this growth rate over time, assuming E 储 ⰇE c , using E 储 ⯝L(dI/dt)/(2 R), and approximating the plasma inductance by L⯝ 0 R. In a tokamak disruption, the electric field usually exceeds the critical one, so that EⰇ1. The energy threshold for 4107 electron runaway is then low, and electrons with pⲏE ⫺1/2 run away. Most secondary electrons are thus generated with relatively low energy (pⰆ1) and with velocity vectors nearly perpendicular to the magnetic field (⯝1); see Eq. 共5兲. However, as in Ref. 10, we assume that they are superthermal, thus taking (T e /m e c 2 ) 1/2ⰆE ⫺1/2Ⰶ1, where T e is the bulk electron temperature. III. RADIAL DIFFUSION In the analysis reviewed above it was assumed that there is no loss of runaway electrons. In practice, these particles undergo radial transport due to magnetic fluctuations and are thus imperfectly confined. This reduces the growth rate of runaway avalanches since the runaways leaving the plasma do not produce any new ones in close collisions with thermal electrons. In this section, we calculate the reduction of the avalanche growth rate caused by radial diffusion described by the addition of a term, f 1 rD , r r r 共8兲 on the right-hand side of Eq. 共2兲, where D is the diffusion coefficient. The effect of diffusion on primary runaway generation has previously been investigated by Catto et al.,14 but this problem does not appear to have been analyzed in the context of secondary runaway production other than in the numerical study of Ref. 15. The addition of the diffusion term 共8兲 adds another dimension to the kinetic equation, which then becomes very difficult to solve analytically. 关Already without diffusion the kinetic equation 共2兲 poses a formidable mathematical problem, which was, however, successfully solved in Ref. 10.兴 However, an approximate analytical solution can nevertheless be found by noting that there is a separation of time scales in the problem. The acceleration to relativistic speed of an electron that has recently been knocked above the runaway threshold by a close collision occurs on a time scale acc⫽ /E, but the time scale for growth of the runaway population is much longer: as we have already noted, it is ␥ r⫺1 ⬃(2 ln ⌳)acc . Therefore we expect that finite radial diffusion can interrupt the avalanche even if the diffusion is not strong enough to alter the velocity–space kinetics of the runaway generation process. The momentum space can thus be divided into a lowenergy region (p⬍ p ) where the avalanche mechanism op* erates undisturbed, and a high-energy region (p⬎p ) where * the accelerating runaways undergo radial diffusion.16 The flux of runaways from the former to the latter region is given by the Rosenbluth–Putvinski formula 共6兲. In the simple, but unrealistic, case where the diffusion coefficient is independent of energy, integrating the kinetic equation 共2兲 over the high-energy region then gives the following simple equation for the runaway density: nr nr 1 ⫽ ␥ rn r⫹ rD , t r r r where the growth rate ␥ r is given by Eq. 共6兲. 共The source term S can be ignored for p⬎ p .) If, for simplicity, the * Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions 4108 Phys. Plasmas, Vol. 7, No. 10, October 2000 Helander, Eriksson, and Andersson plasma has a circular cross section, this equation is easily solved by separation of variables. Keeping only the 共Bessel兲 eigenfunction corresponding to the lowest eigenvalue, n r 共 r,t 兲 ⬀J 0 共 kr 兲 , 冉 冊 a 2 . 2.4 共9兲 The diffusion coefficient D of fast electrons is thought to be governed by magnetic turbulence in the plasma,17 but its precise value is not known. In the simplest picture, the electrons follow stochastic magnetic field lines and thus diffuse radially out of the plasma with a diffusion coefficient of the order D RR ⫽ q v 储 R 冉 冊 ␦B B L共 f 兲 ⯝ 冕 ⬁ 共10兲 which was first derived by Rechester and Rosenbluth.18 Here ␦ B is the magnetic fluctuation level, and the magnetic field is assumed to be fully stochastic. However, runaway confinement is known to be much better than that predicted by Rechester–Rosenbluth diffusion.17 It was first pointed out by Mynick and Krommes19 that the finite Larmor radius and magnetic drift velocity of the fast electrons can considerably reduce the diffusion coefficient. Some of their results were later disputed by Myra and Catto20 共and subsequently corrected in a joint paper21兲, who found that the poloidal component of the drift does not, in fact, reduce the diffusion coefficient in broad-band magnetic turbulence. Their analysis shows that the effect of magnetic drift on diffusion is generally very complicated and depends on details of the turbulence such as its poloidal localization and radial correlation length. In general, drifts tend to be important when the orbit width ⌬ e exceeds the mode width ⌬ m . The details can vary substantially, however, and the diffusion coefficient may decrease slowly 共algebraically兲 or rapidly 共exponentially兲 with ⌬ e /⌬ m depending on the poloidal structure of the turbulence. In any case, since the orbit width for a relativistic electron increases linearly with p, 共11兲 the confinement improves progressively for fast electrons as they are accelerated by the electric field.22 Here q is the tokamak safety factor and B T the magnetic field strength in Teslas. The situation is thus considerably more complicated than indicated by the simple criterion 共9兲, and we need to calculate how a velocity-dependent diffusion process affects a runaway avalanche. To this end, we again integrate the kinetic equation 共2兲 in the high-energy region, p⬎p , but this * time only take the integral over perpendicular momentum. Since most runaways have nearly parallel momenta, p⯝p 储 , the pitch-angle scattering operator 共4兲, p 2储 f p⬜ ⫹O 共 f 兲 , 2 p⬜ p⬜ p⬜ 0 C 共 f 兲 d 2 p⬜ ⫽ 冉 冊 1⫹Z eff F 1 , 共 1⫹ p ⫺2 兲 F⫹O p p2 for p⯝ p 储 Ⰷ p⬜ , where F is the distribution function in parallel momentum, F⫽ 冕 ⬁ 0 qm e c qp ⌬ e⯝ ⫽ 1.7 mm, eB BT 冊 if f / p 储 ⬃ f /p 储 and f / p⬜ ⬃ f /p⬜ with p⬜ Ⰶ p 储 . The first term, which is the largest, is therefore annihilated by integration over perpendicular momentum, and the integral of the collision operator 共3兲 becomes 2 , 冊 冉 is approximately k⫽2.4/a, where a is the minor radius, one obtains the confinement time e ⫽1/k 2 D. If this is shorter than the inverse avalanche growth rate ␥ r⫺1 no avalanche occurs. Thus, in this simple case the avalanche is expected to be interrupted if D⬎D crit⫽ ␥ r 冉 p p 储 p⬜2 f p储 f L共 f 兲 ⫽ ⫺ ⫺ 2 p 储 p⬜ p⬜ p p 储 p⬜ p⬜ f 2 p⬜ d p⬜ . Thus we obtain the following equation for strongly relativistic runaway electrons: F F F ⫹ 共 E⫺1 兲 ⫽ rD 共 p 兲 . t p r r r 共12兲 The second term in Eq. 共12兲 contains the difference between acceleration by the electric field and collisional drag. As before, the population of high-energy electrons governed by Eq. 共12兲 is fed from below 共in p兲 by the avalanche mechanism, which operates in the region p⬍p according to Eq. * 共6兲 and thus provides a boundary condition on F(p ), 共 E⫺1 兲 F 共 p ,r,t 兲 ⫽ ␥ r n r ⫽ ␥ r * 冕 * ⬁ p 共13兲 F 共 p 兲 d p. * Note the unconventional form of this condition: the boundary data depend on the integral of the solution. In applying the strongly relativistic approximation 共12兲 to all electrons in the high-energy region p⬎p , we have assumed that p * * Ⰷ1. It is not difficult to find separable solutions to Eq. 共12兲, 冋 F 共 p,r,t 兲 ⫽J 0 共 kr 兲 exp ␥ t⫺ E⫺1 冕 p p 册 共 ␥ ⫹k 2 D 共 p ⬘ 兲兲 dp ⬘ . * 共14兲 In general, the full solution must be sought as a superposition of several of these eigensolutions, but in the present case the boundary condition 共13兲 is satisfied by a single eigensolution if the growth rate ␥ is chosen so that E⫺1⫽ ␥ r 冕 ⬁ p * 冋 d p exp ⫺ E⫺1 冕 p p 册 „␥ ⫹k 2 D 共 p ⬘ 兲 …dp ⬘ . * 共15兲 More generally, Eqs. 共12兲 and 共13兲 are satisfied by any sum of such solutions corresponding to different Bessel eigenvalues k. Again, for our purposes it is sufficient to keep only the term corresponding to the lowest k since this term has the highest growth rate. Therefore, the relevant solution is given by Eq. 共14兲 with k⫽2.4/a. Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions Phys. Plasmas, Vol. 7, No. 10, October 2000 Suppression of runaway electron avalanches by . . . As already mentioned,16 the parameter p should satisfy * k D p /(E⫺1)Ⰶ1. In this case, the range of both integrals * in Eq. 共15兲 may be extended down to p⫽0 since the contribution from the interval 0⬍ p⬍ p only makes a small con* tribution. Thus, although we earlier assumed p Ⰷ1, we may * still take p ⫽0 in Eq. 共15兲, which determines the avalanche * growth rate ␥ if the runaway diffusion coefficient D(p) is known. If the latter is independent of p, we simply obtain 4109 IV. MONTE CARLO SIMULATION 2 ␥ ⫽ ␥ r ⫺k 2 D, 共16兲 which reproduces the criterion 共9兲. As already remarked, in practice D(p) decreases with increasing p; in fact, the integral 冕 ⬁ 0 共17兲 D 共 p 兲 dp In order to verify the ideas described above and to fully simulate the avalanche process, we have constructed a Monte Carlo code, ARENA 共Avalanche of Runaway Electrons, Numerical Analysis code兲. This code numerically solves the orbit-average of Eq. 共2兲 along the lines described in Ref. 10, but with the additional diffusion term 共8兲. The Monte Carlo solution of this equation involves following a large number of Monte Carlo particles in phase space (r,p,); the phase space positions of these particles are changed periodically, at time intervals ⌬t, by applying Monte Carlo operators representing collisions, the accelerating electric field, and the diffusion term 共8兲. The Monte Carlo operator for the latter is given by ⌬r⫽ usually converges fairly rapidly. In this case, Eq. 共15兲 can be simplified further by noting that, because of Eq. 共7兲, ␥ 1 ⱗ Ⰶ1, E⫺1 2 ln ⌳ so that the first term of the integral within the exponent in Eq. 共15兲 is small over the convergence region of the term involving D(p ⬘ ). The integration of the latter can thus be extended to infinity, which gives us a completely explicit formula for the growth rate, 冉 ␥ ⫽ ␥ r exp ⫺ k 2 E⫺1 冕 ⬁ 0 冊 D 共 p 兲 dp . 共18兲 This simple formula is the main result of the present paper. Note that although the avalanche growth rate is reduced by the presence of radial diffusion, it cannot become negative if the integral 共17兲 converges. The physical reason for this is that the confinement time of the fast electrons approaches infinity sufficiently quickly as they are accelerated by the electric field. To understand this in a simple way, consider the simple time-dependent diffusion equation, nr 1 nr ⫽ D共 t 兲 , t r r r which models the transport of electrons whose diffusion coefficient changes with time 共because of acceleration in our problem兲. By introducing the new time variable, t ⬘⫽ 冕 t 0 D 共 t 兲 dt, where is a random number with zero mean and unit variance. Monte Carlo operators for the orbit-averaged effects of Coulomb collisions and the electric field are given in Refs. 10 and 23. It should be noted here that a Monte Carlo particle given by (r,p,) represents a number of real particles uniformly distributed in time along the unperturbed guidingcenter orbit given by the three invariants (r,p,). Thus, the procedure for obtaining the numerical solution is as follows: at the beginning of the calculation a large number of Monte Carlo particles are initialized with a Maxwellian velocity distribution and with a radial density profile which can be freely specified; after each time step, ⌬t, the particle positions in phase space are updated with the aid of the Monte Carlo operators; in addition, new runaway particles are added in accordance with the source term 共5兲; diagnostic output 共runaway growth rate, averaged energy, etc.兲 are produced after suitably spaced time intervals 共i.e., some fraction of a collision time兲. The ARENA code has been benchmarked carefully 共without diffusion兲 against the Rosenbluth–Putvinski formula 共6兲, which has been found to be accurate in all physically relevant cases. Adding radial diffusion, we have used the code to verify the approximate analysis given in Sec. III. If the diffusion coefficient is taken to be independent of momentum, the growth rate is found to drop linearly with increasing diffusion, in practically exact agreement with Eq. 共16兲; see Fig. 1. In the more realistic case where the diffusion coefficient does depend on momentum, we have verified Eq. 共18兲 by running the code with a radial diffusion coefficient given by 2 D 共 p 兲 ⫽D 0 e ⫺(p/⌬p) , this equation is transformed into an ordinary diffusion equation. It is immediately clear that if the integral defining t ⬘ converges as t→⬁, a fraction 冉 冕 exp ⫺k 2 D ⌬t⫹ 冑2D⌬t, r ⬁ 0 D 共 t 兲 dt 冊 of the electrons will always stay in the system, regardless of the absolute value of the diffusion coefficient. The avalanche growth rate 共18兲 contains exactly this reduction factor. for various values of the width ⌬ p. The runaway electrons were initialized with the 共almost flat兲 radial density profile n r (r)⫽n r0 关 1⫺0.9(r/a) 2 兴 0.1, and the ratio between the accelerating field and the critical one was E⫽ 兩 E 储 /E c 兩 ⫽100. The result of the simulations is shown in Fig. 2, where the growth rate obtained from the simulations and the one from Eq. 共18兲 are plotted as functions of ⌬p. As can be seen, the agreement is good, which indicates that Eq. 共18兲 is a convenient and accurate estimate of the reduction of the runaway Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions 4110 Phys. Plasmas, Vol. 7, No. 10, October 2000 FIG. 1. The normalized avalanche growth rate ␥ versus the diffusion coefficient in the case when the latter is assumed to be independent of momentum. The growth rate decreases linearly with D and vanishes when D ⫽D crit , as predicted by Eq. 共16兲. growth rate caused by momentum-dependent diffusion. If ⌬p becomes very large the agreement is worse since the integral 共17兲 then converges slowly. In such cases the more accurate expression 共15兲 should be used to calculate the growth rate. V. CONCLUSIONS Our analytical calculation and numerical simulations demonstrate that radial diffusion slows down runaway avalanches, so that the growth rate ␥ r calculated in Ref. 10 is reduced according to Eq. 共15兲. If the integral 共17兲 converges quickly enough, this equation is reduced to the more explicit equation 共18兲. At first sight, this expression suggests that introducing a large diffusive loss of fast electrons does not Helander, Eriksson, and Andersson prevent an avalanche from occurring, since the growth rate 共18兲 cannot become negative. However, in practice there are at least two reasons why sufficiently rapid diffusion should still be able to effectively prevent avalanches. First, the runaway energy is not unlimited, as assumed in Sec. III. Several physical processes 共like synchrotron radiation or cyclotron instability兲 limit the energy and hence the confinement time of runaways, so that avalanches may be completely stabilized. This may, however, require a very large diffusion coefficient. Second, and perhaps more importantly, for practical purposes it is sufficient to reduce the growth rate to a level where there are not enough volt-seconds of induced electric field to achieve significant avalanche growth. Thus, although complete avalanche stabilization by radial diffusion may be difficult, it is nevertheless possible to reduce the avalanche size to a manageable, or even negligible, level. In order to estimate the magnetic fluctuation level required for this, we use the approximate expression 共1兲 for the total number of avalanche e-foldings, modified by the reduction factor from Eq. 共18兲, 冕␥ dt⬃ 冉 I k 2 exp ⫺ I A ln ⌳ E⫺1 冕 ⬁ 0 冊 D共 p 兲d p . 共19兲 We assume that the diffusion coefficient D(p) is equal to the Rechester–Rosenbluth value 共10兲 for small p and falls off at higher energies when the orbit width 共11兲 becomes comparable to the mode width of the magnetic turbulence.20 The mode width is, of course, quite uncertain; it has recently been estimated to be 0.5–4 cm in TEXTOR-94.24 This determines the value of the D(p) integral, which we write as 冕 ⬁ 0 D 共 p 兲 d p⫽p crit qcR 冉 冊 ␦B B 2 , where p crit characterizes the speed of convergence. Requiring that the number of e-foldings 共19兲 should not be larger than some number N defining the maximum tolerable avalanche size implies the following lower limit on the magnetic fluctuation level: ␦B B ⲏ 冑 a 2.4 冉 冊 E⫺1 I ln . qcR p crit NI A ln ⌳ This limit is practically independent of the argument within the logarithm, and becomes for typical large-tokamak parameters about 10⫺3 . 共For JET, E 储 ⱗ104 V/m, R⫽3 m, a⫽1 m, n e ⫽2⫻1019m3 , and p crit⬎1.) This greatly exceeds the fluctuation level in quiescent plasmas but is not unrealistic in a disruption, where ␦ B r / ␦ B ⯝0.04 has been reported.25 Thus, it appears possible that the naturally occurring, or any externally induced, magnetic fluctuations could significantly reduce the size of secondary runaway avalanches. ACKNOWLEDGMENTS FIG. 2. The normalized avalanche growth rate ␥ vs ⌬p for the diffusion 2 coefficient D(p)⫽D 0 e ⫺(p/⌬p) , with D 0 ⫽R 2 / , a/R⫽0.31, and Z eff⫽1.5. The solid line shows the result from the numerical simulation, and the dotted line the analytical approximation 共18兲. Two of the authors 共F.A. and P.H.兲 are grateful for the hospitality of CEA Cadarache, where some of this work was carried out. It was funded by the U.K. Department of Trade and Industry, and by Euratom under association contracts with Sweden, France, and United Kingdom. Downloaded 08 Aug 2012 to 194.81.223.66. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissions Phys. Plasmas, Vol. 7, No. 10, October 2000 J. A. Wesson, R. D. Gill, H. Hugon et al., Nucl. Fusion 29, 641 共1989兲. R. D. Gill, Nucl. Fusion 33, 1613 共1993兲. 3 R. Yoshino, T. Kondoh, Y. Neyatani, K. Itami, Y. Kawano, and N. Isei, Plasma Phys. Controlled Fusion 39, 313 共1997兲; Y. Kawano, R. Yoshino, Y. 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