PHYSICS OF PLASMAS
VOLUME 7, NUMBER 2
FEBRUARY 2000
Particle pinch in collisionless drift-wave turbulence Frank Jenkoa) Max-Planck-Institut fu¨r Plasmaphysik, EURATOM Association, Boltzmannstr. 2, 85748 Garching, Germany
共Received 13 July 1999; accepted 18 October 1999兲 Nonlinear numerical simulations show that the particle flux from collisionless drift-wave turbulence is directed up-gradient for sufficiently high values of e ⫽L n /L T e . This ‘‘particle pinch’’ results from the completely different perpendicular dynamics of slow 共resonant兲 and fast 共nonresonant兲 electrons, making it a genuinely kinetic effect which cannot easily be described by fluid models. Moreover, the linearly stable system self-sustains its turbulent state through a finite-amplitude 共nonlinear兲 instability. Therefore, quasilinear estimates of turbulent transport caused by collisionless drift waves are practically useless and have to be replaced by nonlinear kinetic simulations like the present one. © 2000 American Institute of Physics. 关S1070-664X共00兲00802-8兴
I. INTRODUCTION
described in terms of a distribution function on a phase-space grid, and the basic nonlinear equations are solved numerically on a massively parallel computer using explicit finitedifference methods.17 It will be shown that such nonlinear numerical simulations of collisionless drift-wave turbulence yield a surprising result: The turbulent particle flux is directed inward for sufficiently high values of e . This anomalous particle pinch is caused by the completely different perpendicular dynamics of slow 共resonant兲 and fast 共nonresonant兲 electrons, making it a genuinely kinetic effect which is hard to capture in fluid models. The remainder of this paper is organized as follows. In Sec. II we describe the basic equations as well as the numerical methods used to solve them. The nonlinear simulation results on the particle transport from collisionless drift-wave turbulence are presented in Sec. III. The roles of linear and nonlinear processes in the turbulent state will be discussed in Sec. IV with a particular emphasis on nonlinear instability. In Sec. V we give some conclusions.
Although the particle sources of tokamak plasmas are usually located in the outer edge region, one always observes radial density profiles which are peaked in the center of the plasma.1 This basic experimental fact 共usually called ‘‘pinch effect’’兲 is the prerequisite for the successful operation of all existing tokamaks. Yet it is very counterintuitive and proves that particle transport in magnetically confined plasmas generally cannot be described as a diffusive process. A widely used phenomenological remedy is to express the particle flux ⌫ as a sum of an outward diffusive term and an inward convective term, ⌫⫽⫺D dn/dr⫺n v in , where D is the coefficient of turbulent diffusion, dn/dr is the radial density gradient, and v in is the pinch velocity.2 To put this Ansatz on a solid theoretical basis has proven to be a difficult problem. The neoclassical Ware pinch,3 which describes the collision-induced inward drift of trapped particles, is much too small to explain the experimental observations.4 Therefore, the particle pinch must be caused by turbulent fluctuations. It was shown that plasma turbulence driven by temperature gradients5–8 共with the plasma density playing only a passive role兲 or trapped particles9–13 can exhibit up-gradient particle transport. However, the temperature gradient results were too small to explain the pinch effect in the edge region of tokamaks which is stronger than that in the core by one to two orders of magnitude, and trapped particles are in general absent in the edge. Now, it is well known that the observed anomalous transport in the steep gradient zone of tokamak plasmas can in part be attributed to drift-wave turbulence.14 A careful investigation of the relevant plasma parameter regime shows that the electron motion parallel to the magnetic field is usually at most weakly collisional.15 As a consequence, driftwave turbulence 共which has been described almost exclusively by Braginskii-type fluid models16 so far兲 should really be treated within the framework of kinetic theory. Here, we use a hybrid model of drift-kinetic electrons and cold ions in three-dimensional sheared slab geometry. The electrons are
II. BASIC EQUATIONS AND NUMERICAL METHODS
The equations used to treat the nonlinear dynamics of collisionless drift waves in a three-dimensional sheared slab geometry are described in detail elsewhere.15,17 The time evolution of the perturbed electron distribution function f (x,y,z,w 储 ,w⬜ ,t), electrostatic potential (x,y,z,t), and parallel ion velocity u 储 (x,y,z,t) is given in dimensionless form by d t f ⫽⫺ T f m y ⫺ ␣ e w 储 ⵜ 储 共 f ⫺ f m 兲 ,
共1兲
d t ⵜ⬜2 ⫽ⵜ 储 u 储 ⫺
共2兲
冉 冕
冊
␣ e w 储 f dw 储 ,
⑀ˆ s d t u 储 ⫽⫺ⵜ 储 ⫹ 储 ⵜ 2储 u 储 ,
共3兲
where d t ⫽ t ⫹z⫻ⵜ •ⵜ, T ⫽1⫹ e (w 2 ⫺3/2), fm ⫺3/2 ⫺w 2 ⫽ e is the background Maxwellian, ⵜ 储 ⫽ z , and ⵜ⬜2 ⫽( x ⫹sˆ z y ) 2 ⫹ 2y . The most important parameters in this model are ␣ e ⫽( v T /qR)/(c s /L n ), e ⫽L n /L T , and sˆ e
⫽(dq/dr)/(q/r). The parameter ⑀ˆ s ⫽(qR/L n ) 2 and the par-
a兲
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Phys. Plasmas, Vol. 7, No. 2, February 2000
Particle pinch in collisionless drift-wave turbulence
allel ion viscosity 储 are secondary. Here v T ⫽(2 T e /m e ) 1/2 is the electron thermal velocity, c s ⫽(T e /M i ) 1/2 is the ion sound speed, T e is the electron temperature, m e and M i are, respectively, the electron and ion mass, L n ⫽ 兩 ⵜ ln n兩⫺1 and L T e ⫽ 兩 ⵜ ln Te兩⫺1 are the profile scale lengths, and 2 qR is the field line connection length. The parallel, perpendicular, velocity, and time normalization scales are, respectively, qR, s ⫽c s /⍀ i , v T , and L n /c s , where ⍀ i is the ion cyclotron frequency. The coordinate system (x,y,z) is aligned to the background gradients and the sheared magnetic field: x ⬀⫺ⵜn⬀⫺ⵜT e , z⬀B, and ⵜx⫻ⵜy•ⵜz⫽1. Its nonorthogonality is reflected in the metric18 and in the parallel boundary condition,19–21 S(x,y,z⫹2 )⫽S(x,y⫺2 sˆ x,z) for any scalar quantity S. In the perpendicular (x,y) plane, periodic boundary conditions are applied. Numerically, we use a nonlinear Vlasov code which is based on an explicit finite-difference scheme for the electron distribution function in phase space.17 The nonlinear E⫻B convection terms are computed according to a multidimensional second order upwind method by Colella.22 The parallel electron dynamics, ␣ e w 储 ⵜ 储 ( f ⫺ f m ), uses van Leer’s second order upwind method.23 Note here that upwinding is applied to the nonadiabatic part of f, ( f ⫺ f m ), instead of the full distribution function. Thus the numerical scheme reflects the underlying physics of the parallel electron dynamics which forces ( f ⫺ f m )→0. The remaining terms are also finite differenced to second order and evaluated at the current time step. The computational domain is 32 s ⫻64 s ⫻2 qR, corresponding to about 2 cm⫻4 cm⫻30 m for typical tokamak conditions. w 储 runs from ⫺3 v T to 3 v T . 共Note that in a slab model, one velocity space dimension can be integrated out, so that the computational velocity space for f is one-dimensional.17兲 The grid is typically 32⫻64⫻16⫻80 ˆ nodes in (x,y,z,w 储 ) space; the model parameters are ⫺2 ˆ ˆ ⬅2 ␣ e ⫽10, s ⫽3/ , ⑀ s ⫽3600, and 储 ⫽60. The value ˆ corresponds to the plasma parameters q⫽3, chosen for R⫽1.65 m, and L n ⫽2.5 cm, i.e., typical high confinement mode 共H-mode兲 edge parameters in the axisymmetric divertor experiment ASDEX Upgrade.24 The initial condition for the distribution function f is a localized disturbance in real space and a Maxwellian in velocity space f 共 x,y,z,w 储 ;t⫽0 兲 ⫽p e ⫺b(x
2 ⫹y 2 ⫹z 2 )
f m共 w 储 兲 ,
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FIG. 1. Turbulent particle flux ⌫ as a function of e . For e ⬍2 the turbulent particle flux depends only weakly on e . However for e ⬎2.5 it becomes negative and is, therefore, directed inward.
with v x ⫽⫺ y , N⫽ 兰 f d 3 w, and (3/2) P⫽ 兰 w 2 f d 3 w where 具 ••• 典 denotes space and time averaging. ⌫ and Q are normalized to D GB(n/L n ) and D GB(nT e /L n ), respectively, with the gyro-Bohm transport coefficient D GB⫽cT e s /eBL n corresponding to 3.2 m2/s for the nominal plasma edge parameters given above 共and using T e ⫽200 eV and B⫽2.5 T兲. The dependence of these normalized fluxes on the dimensionless model parameters is, in general, nonlinear and needs to be computed numerically. The results of a e scan for ⌫ and Q are shown in Figs. 1 and 2. Here e has been varied between 0 and 4 with a stepsize of 0.5. For e ⬍2 the turbulent particle flux depends only weakly on e . However for e ⬎2.5 it becomes negative and is, therefore, directed inward. In this regime the ⌫ curve displays a strong drop with e . On the other hand, the heat transport rises strongly with increasing e over the whole parameter range. It is to be noted, however, that the heat flux is normalized with respect to the density scale length L n , not the electron temperature scale length L T e . To correct for that fact, one would have to divide every data point of the Q curve by its corresponding value of e . Then the increase would be less pronounced but still substantial. To get a better understanding of this up-gradient particle transport at sufficiently high values of e we compute the w 储 spectrum ⌫(w 储 ) of ⌫⫽ 兰 ⌫(w 储 )dw 储 . The result for e ⫽4 is shown in Fig. 3. It indicates clearly that the roots of this anomalous particle pinch lie in the completely different per-
共4兲
with u 储 and both set to zero. If the simulations are started with a density fluctuation amplitude well above unity the system then relaxes towards a turbulent steady state characterized by stationary transport levels 共see Sec. IV兲. Due to the periodic boundary condition in the radial direction one can maintain this state for arbitrarily long simulation times without getting quasilinear flattening effects. III. NONLINEAR SIMULATION RESULTS
The turbulent particle and heat transport in the radial direction is characterized by the fluxes ⌫⫽ 具 Nv x 典 ,
Q⫽ 共 23 ) 具 Pv x 典 ,
共5兲
FIG. 2. Turbulent heat flux Q as a function of e . Q rises strongly with increasing e over the whole parameter range. Note, however, that Q is normalized here with respect to L n , not L T e .
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FIG. 3. w 储 spectrum of the particle flux ⌫ for the case e ⫽4. Slow and fast particles show different behavior with respect to the particle transport.
pendicular dynamics of slow 共resonant兲 and fast 共nonresonant兲 electrons. In fact, the graph in Fig. 3 can be parameterized extremely well by the expression 2
⌫共 w 储 兲⫽
兺
i⫽1
2
⌫ i共 w 储 兲 ⫽
兺 ␣ i exp关 ⫺  i共 w 储 ⫺ ␥ i 兲 2 兴 ,
i⫽1
共6兲
for w 储 ⭓0 and ⌫(⫺w 储 )⫽⌫(w 储 ) with ␣ 1 ⫽⫺1.4,  1 ⫽2.0, ␥ 1 ⫽0.27, ␣ 2 ⫽0.83,  2 ⫽2.1, and ␥ 2 ⫽1.2. This means that ⌫ consists of two parts: 共i兲 A positive 共down-gradient兲 fast particle contribution ⌫ 1 and 共ii兲 a negative 共up-gradient兲 slow particle contribution ⌫ 2 . In the case shown in Fig. 3 we get ⌫ 1 ⫽⫺2.4 and ⌫ 2 ⫽2.0 yielding ⌫⫽⫺0.4. Obviously we have here a genuinely kinetic effect 共different regions of w 储 space have different dynamics兲 which is hard to capture by fluid models. In analogy to that, one can also parameterize the w 储 spectrum of the heat flux Q as computed from the simulations. For the case e ⫽4—which is shown in Fig. 4—we get ␣ 1 ⫽0.54,  1 ⫽1.8, ␥ 1 ⫽0, ␣ 2 ⫽2.1,  2 ⫽2.0, and ␥ 2 ⫽1.4. So both the radial particle and heat fluxes are composed of fast and slow particle parts. In the latter case, however, both contributions are positive 共down-gradient兲. Let us make another important observation here. In order to permit a larger time step in numerical simulations of plasma turbulence involving the motion of electrons along magnetic field lines, the so-called adiabatic cutoff technique25 was invented. It is based on the idea that in the presence of only very low-frequency modes, with parallel phase velocities /k 储 Ⰶ v T , only the very slow electrons are
FIG. 4. w 储 spectrum of the heat flux Q for the case e ⫽4. The ‘‘hole’’ 3
around w 储 ⫽0 results from the ( 2 ) ⌫ part of Q.
FIG. 5. k y spectra of the particle flux ⌫ for the cases e ⫽1 共solid line兲 and e ⫽4 共dashed line兲.
subject to wave-particle interactions 共electron Landau damping兲. Consequently, one might be able to choose a cutoff velocity above which all electrons can be assumed to be adiabatic, i.e., their response to potential fluctuations is described by the 共dimensionless兲 linearized Boltzmann relation f ⫽ f m . This allows for a significant increase of the time step in the simulations. However, in our present model the relevant mode is the electrostatic version of the shear Alfve´n wave26 for which the above relation reads /k 储 ⫽ ␣ e /k⬜ Ⰶ ␣ e or k⬜ Ⰷ1 in dimensionless units. Obviously the necessary condition for the use of the adiabatic cutoff technique is not fulfilled as can also be seen directly from Figs. 3 and 4 where the entire w 储 space—not just a narrow region around w 储 ⫽0—contributes significantly to the particle and heat transport, indicating non-adiabatic electron dynamics.15 In conclusion one can say that the notion of an adiabatic cutoff velocity is not helpful in drift-wave turbulence studies. Instead the entire velocity space dynamics must be treated. Let us now turn to the k y spectra of the turbulent particle and heat fluxes, ⌫ and Q. The results for e ⫽1 and e ⫽4 are shown in Figs. 5 and 6. The Q spectra in both cases differ only in the low k y region where there is enhanced activity for higher e . In contrast to that, the two ⌫ spectra are qualitatively different: For k y ⭐k crit y the ⌫ spectrum changes sign, and for k y ⬎k crit y it is reduced by some two orders of magnitude for e ⫽4 as compared to e ⫽1. (k crit y was observed to rise with increasing e .) Therefore, it is clear that the larger
FIG. 6. k y spectra of the heat flux Q for the cases e ⫽1 共solid line兲 and e ⫽4 共dashed line兲.
Phys. Plasmas, Vol. 7, No. 2, February 2000
Particle pinch in collisionless drift-wave turbulence
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FIG. 9. Spatially averaged radial particle flux as a function of time for e ⫽1. The mean value is shown as a dashed line. FIG. 7. Probability distribution P( ␣ ,k y ) for the k y components of density and potential fluctuations having a given relative phase ␣ in the case e ⫽1. A positive phase shift corresponds to an outward particle flux and vice versa.
perpendicular spatial scales are mainly responsible for both the enhanced heat flux and the particle pinch observed in the simulations for sufficiently high e . The important role of the larger scales is confirmed by the probability distribution P( ␣ ,k y ) of the phase shifts ␣ 苸 关 ⫺ , 兴 between the k y components of density and potential fluctuations. A positive ␣ corresponds to a positive 共i.e., outward兲 contribution to the particle flux and vice versa. Figures 7 and 8 show that while for e ⫽1 all k y components lead to an outward flux, for e ⫽4 one can distinguish three regions in k y space: 共1兲 For k y ⭐0.4 the phase shift is clearly negative resulting in the observed particle pinch, 共2兲 for 0.4 ⭐k y ⭐1.3 ␣ is 共on average兲 slightly positive, 共3兲 and for 1.3⭐k y the phase shift is distributed over a wide range of mainly negative values. These results are consistent with the observations made earlier in Fig. 4 and stress the fact that drift-wave induced transport is nondiffusive at larger perpendicular scales. This is reflected by the strong self-consistent coupling between N and at small k y which stems from the relatively fast parallel electron dynamics as compared to the perpendicular dynamics on larger scales.
FIG. 8. Phase shift distribution function P( ␣ ,k y ) for the case e ⫽4. For k y ⭐0.4 the phase shift is clearly negative resulting in the observed particle pinch.
IV. THE CRUCIAL ROLE OF THE NONLINEAR INSTABILITY
It is well known that linear collisionless drift waves in two-dimensional sheared slab geometry are 共marginally兲 stable.27–29 However, Hirshman and Molvig30 argued that the linear mode structure might be too delicate to have relevance to turbulence. This conjecture was confirmed in linear computations by adding a cross-field diffusion term which yielded a positive growth rate.31 It was also shown in fluid simulations that this nonlinear instability carries over into the strongly collisional regime.32,33 Here resistivity replaces electron Landau damping as the main free energy sink in the parallel dynamics of the turbulence. Most notably, it was shown that resistive drift wave turbulence in twodimensional sheared slab geometry is self-sustaining if initialized at sufficiently high amplitudes whereas it dies out otherwise.34 Three-dimensional collisionless drift-wave turbulence is driven by the same inherently nonlinear mechanism and its collisional counterpart.35 Therefore, quasilinear estimates of turbulent transport from collisionless drift waves are practically useless and have to be replaced by fully nonlinear kinetic simulations. This aspect sets the present investigation apart from most previous work on inward particle transport by plasma turbulence. Now, linear collisionless drift waves in an unsheared slab are also known to yield a 共quasilinear兲 particle pinch for e ⬎2. At the same time, however, the growth rates of all modes are negative, such that turbulence must decay in such a regime in absence of some other mechanism to sustain it. 关The most widely invoked sustaining mechanism is the ion
FIG. 10. Spatially averaged radial particle flux as a function of time for e ⫽4. Again, the mean value is shown as a dashed line.
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mixing mode5,6 in which destabilizing ion dynamics in the form of the ion temperature gradient 共ITG兲 mode is retained. However, ITG modes are absent in the present model due to the cold ion approximation.兴 The relationship between the flux and the growth rate 共outward flux for growing modes, inward flux for damped modes兲 is usually stated for the quasilinear flux and the linear growth rate. However, this relationship also applies to the nonlinear regime, where the rate of input of free energy into the density fluctuations, t E n ⫽ t 具 N 2 /2典 , is given by ⌫.15 Therefore when ⌫ is positive, the flux is outward and there is a free energy input into the density fluctuations. When it is negative, there is a pinch, but there is now no energy input into the density fluctuations. If collisionless drift-wave turbulence was driven entirely by density gradients, it would thus be hard to envision how one could have a finite turbulence level and a pinch at the same time. However, one must not forget that the nonlinear driftwave instability is driven equally well by electron temperature gradients.36 In the high e regime, free energy tapped from ⵜT e is put into the electron temperature fluctuations and drives the system. As density fluctuations play only a passive role, the particle transport can therefore be upgradient (⌫⬍0). One observes nondecaying, self-sustained turbulence exhibiting a particle pinch 共see Figs. 9 and 10兲. V. CONCLUSIONS
To summarize, we have presented nonlinear numerical simulations of collisionless drift-wave turbulence which show that the turbulent particle flux is directed up-gradient 共inward兲 for sufficiently high values of e . This particle pinch can be attributed to the completely different perpendicular dynamics of slow 共resonant兲 and fast 共nonresonant兲 electrons, making it a genuinely kinetic effect 共different regions of w 储 space have different dynamics兲 which is hard to capture by fluid models. The w 储 spectra of ⌫ and Q can be parameterized extremely well in terms of superpositions of two Maxwellian distributions. Thus it can be seen that ⌫ consists of a positive 共down-gradient兲 fast particle part and a negative 共up-gradient兲 slow particle part. In the case of Q, however, both contributions are positive 共down-gradient兲. From these results we also concluded that it is not permissible in our case to use adiabatic cutoff techniques which would allow for larger time steps in the numerical simulations. This is because a narrow region around w 储 ⫽0 does not exist, outside of which the electrons are adiabatic. So there is no shortcut to treating the entire velocity space dynamics in studies of collisionless drift-wave turbulence. Finally we observed that both the enhanced heat flux and the particle pinch with increasing e are primarily due to changes in the low k y regions of the ⌫ and Q spectra. This fact highlights the role of larger perpendicular spatial scales with respect to turbulent radial transport.
Frank Jenko
Let us conclude by emphasizing that our results have been obtained in the framework of a rather simple model— neglecting, e.g., collisions, magnetic curvature, and finite ion temperature effects. Also, equilibrium sheared E⫻B flows have not been taken into account despite their important role in the edge region of tokamak plasmas where the present computations are applicable. Although the physical mechanisms described in this paper are interesting in themselves, the possible importance of any of these additional effects remains a topic for future investigation in nonlinear kinetic simulations. Comparisons with experiment should be qualified in the light of these considerations. ACKNOWLEDGMENT
I greatfully acknowledge useful discussions with Bruce D. Scott. J. Hugill, Nucl. Fusion 23, 331 共1983兲. B. Coppi and N. Sharky, Nucl. Fusion 21, 1363 共1981兲. 3 A. A. Ware, Phys. Rev. Lett. 25, 15 共1970兲. 4 F. Wagner and U. Stroth, Plasma Phys. Controlled Fusion 35, 1321 共1993兲. 5 B. Coppi and C. Spight, Phys. Rev. Lett. 41, 551 共1978兲. 6 T. Antonsen, B. Coppi, and R. Englade, Nucl. Fusion 19, 641 共1979兲. 7 W. W. Lee and W. M. Tang, Phys. Fluids 31, 612 共1988兲. 8 B. D. Scott, Plasma Phys. Controlled Fusion 34, 1977 共1992兲. 9 V. V. Yankov, JETP Lett. 60, 171 共1994兲. 10 J. Nycander and V. V. Yankov, Phys. Plasmas 2, 2874 共1995兲. 11 M. B. Isichenko, A. V. Gruzinov, and P. H. Diamond, Phys. Rev. Lett. 74, 4436 共1995兲. 12 M. B. Isichenko, A. V. Gruzinov, P. H. Diamond, and P. N. Yushmanov, Phys. Plasmas 3, 1916 共1996兲. 13 P. W. Terry, Phys. Fluids B 1, 1932 共1989兲. 14 P. C. Liewer, Nucl. Fusion 25, 543 共1985兲. 15 F. Jenko and B. D. Scott, Phys. Plasmas 6, 2418 共1999兲. 16 S. I. Braginskii, in Reviews of Plasma Physics 共Consultants Bureau, New York, 1965兲, Vol. 1, p. 205. 17 F. Jenko, Ph.D. thesis, Technische Universita¨t Mu¨nchen, 1998. 18 K. V. Roberts and J. B. Taylor, Phys. Fluids 8, 315 共1965兲. 19 R. L. Dewar and A. H. Glasser, Phys. Fluids 26, 3038 共1983兲. 20 M. A. Beer, S. C. Cowley, and G. W. Hammett, Phys. Plasmas 2, 2687 共1995兲. 21 B. D. Scott, Plasma Phys. Controlled Fusion 39, 471 共1997兲. 22 P. Colella, J. Comput. Phys. 87, 171 共1990兲. 23 B. van Leer, J. Comput. Phys. 32, 101 共1979兲. 24 W. Suttrop et al., Plasma Phys. Controlled Fusion 39, 2051 共1997兲. 25 A. M. Dimits and W. W. Lee, J. Comput. Phys. 107, 309 共1993兲. 26 W. W. Lee, J. Comput. Phys. 72, 243 共1987兲. 27 D. W. Ross and S. M. Mahajan, Phys. Rev. Lett. 40, 324 共1978兲. 28 K. T. Tsang, P. J. Catto, J. C. Whitson, and J. Smith, Phys. Rev. Lett. 40, 327 共1978兲. 29 T. M. Antonsen, Phys. Rev. Lett. 41, 33 共1978兲. 30 S. P. Hirshman and K. Molvig, Phys. Rev. Lett. 42, 648 共1979兲. 31 B. D. Scott, S. Camargo, and F. Jenko, in Fusion Energy 1996 共IAEA, Vienna, 1997兲, Vol. 2, p. 649. 32 B. D. Scott, Phys. Rev. Lett. 65, 3289 共1990兲. 33 D. Biskamp and A. Zeiler, Phys. Rev. Lett. 74, 706 共1995兲. 34 B. D. Scott, Phys. Fluids B 4, 2468 共1992兲. 35 F. Jenko and B. D. Scott, Phys. Rev. Lett. 80, 4883 共1998兲. 36 B. D. Scott, Plasma Phys. Controlled Fusion 39, 1635 共1997兲. 1
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