Can Landau-fluid models describe nonlinear Landau damping?

Can Landaufluid models describe nonlinear Landau damping?Nathan Mattor Citation: Phys. Fluids B 4, 3952 (1992); doi: 10.1063/1.860350 View online: http://dx.doi.org/10.1063/1.860350 View Table of Contents: http://pop.aip.org/resource/1/PFBPEI/v4/i12 Published by the American Institute of Physics. Additional information on Phys. Fluids BJournal 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

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Can Landau-fluid models describe nonlinear Landau damping? Nathan Mattor Lawrence Livermore National Laboratory, Livermore, Calz$ornia 94551

(Received 30 March 1992; accepted 31 July 1992)

The question of whether the “Landau-fluid” description of wave-particle resonances [e.g.., Hammett and Perkins, Phys. Rev. Lett. 64, 3019 (1990)] can describe nonlinear Landau damping is addressed. It is found that this model can provide an adequate description of the Compton scattering of electron drift waves, but fails in the case of ion-temperature- gradient-driven modes near the threshold. The general conclusion is that Landau- fluid models have difficulty describing the dynamics associated with the product of two or more deeply resonant propagators.

I. INTRODUCTION

Traditionally, theories of plasma fluctuations have been either of the “fluid” or the “kinetic” variety. A principal difference has been that fluid theories cannot describe wave-particle resonances; this has required the more complete kinetic description to follow the evolution of the particle distribution in phase space. Unfortunately, the more detailed kinetic description has borne with it greater difficulty, both in analytical solubility and in numerical time consumption. To circumvent the difficulties associated with kinetic theory, recent efforts have attempted to model Landau damping in a fluid description of a plasma. I-3 The general procedure is to add an ad hoc term to the fluid equations that correctly reproduces the more complete kinetic results. Comparisons between Landau-fluid models and l inear kinetic theories have generally met with some success. However, the utility of reproducing linear kinetic theory is somewhat questionable, since the problems typi- cally addressed can generally be solved without the need for the Landau-fluid model. The true merit of the Landau- fluid approach is addressing nonlinear issues; it is here that kinetic theory can become prohibitively complicated. Therefore, the question arises as to whether the Landau- fluid model adequately describes nonlinear kinetic effects.

For drift waves and ion-temperature-gradient-driven (ITG) turbulence, one of the more important nonlinear kinetic effects is ion Compton scattering (ICS), also known as nonlinear Landau damping4 (NLLD). Physi- cally, this effect is a resonance between a particle and the beat wave between two or more modes. Alternatively, this effect can be viewed as a wave scattering from a particle into another wave (viewed this way, the process is known as Compton scattering). In the present terminology, these two processes are basically the same, involving resonant interaction between multiple waves and a particle; the principal difference is whether the final state of the energy is in the particle distribution (nonlinear L.andau damping) or in the waves [ Compton scattering).

For drift wave and ITG turbulence, the importance of ICS is that it provides a nonlinear saturation mechanism. ICS competes with, and often dominates,5-‘1 the more commonly invoked fluid saturation (wave-wave reso-

nances). The effects of ICS can be quite significant. It can cause large reductions of the turbulent saturation level and transport (relative to mixing length levels). For ITG modes near the threshold,739 ICS becomes dominant, and the usual ,x-r/g estimate becomes augmented by a “kinetic reduction factor,” and more accurately becomes x-<‘;l”y/G, where c=ti/v%llui. Other effects of ICS can be nonlocal transfer of energy in wave number space,8’11 and inelastic scattering (nonconservation of wave energy), which can allow turbulent saturation without the need to couple to stable modes.’

This work addresses the issue of whether the Landau- fluid model correctly describes the effects of Compton scattering on drift waves and ITG turbulence. The strategy is simple. For a paradigm model, both the Landau-fluid and gyrokinetic equations are considered. With each, a weak turbulence expansion is performed on the nonlinearity, de- riving the two Compton scattering terms, which are then compared in drift wave and ITG orderings. It will be assumed that the reader is somewhat familiar with the weak turbulence theory of drift waves; details of the derivations can be found in Refs. 5 or 7.

This study will restrict consideration to the L.andau- fluid model of Hammett and Perkins,’ which has the ad- vantages of being physically transparent, algebraicly simple, and capable of producing the correct linear theory to the accuracy sought here. Because ICS generally becomes important in the regime kgi- O( 1 ), it will be necessary to add finite Larmor radius (FLR) effects to the Hammett and Perkins model. This will be done simply by adding the polarization drift and nothing more. Much more elaborate FLR Landau-fluid models existzY3 (which include separate evolution of parallel and perpendicular pressures), but the inclusion of too many terms prevents a systematic weak turbulence expansion, and also tends to obscure the underlying physics. In defense of the present approach, one should note that the discrepancies found here (discussed two paragraphs below) are a property only of resonant propagators, which have no FLR dependence. This study should not be regarded as a categorical dismissal of the Landau-fluid approach, but rather as an illustration of cer- tain effects that might require caution.

The principal results of this study are the following.

3952 Phys. Fluids B 4 (12), December 1992 0899-8213/92/123952-10$04.00 @ 1992 American Institute of Physics 3952


( 1) For drift waves, the Landau-fluid model adequately describes the damping rate from ion Compton scattering to the accuracy sought here.

(2) For ITG modes, the Landau-fluid model is unable to reproduce the magnitude of the nonlinear damping rate of the gyrokinetic theory in both, and in the elasticity of scatter. As a result, the Landau-fluid model fails to capture the “kinetic reduction factor,” and predicts x-y/# instead of the ICS corrected x-[~~/G.

(3) It is argued that the fundamental difficulty can be stated as follows: the Landau-fluid model does not accurately describe the dynamics associated with the product of two or more deeply resonant propagators. [Here “deeply resonant propagator” refers to a resonance producing denominator of the form l/(w- ku), with w<kv.]

To see this third point more concretely, consider two modes k and k’ with frequencies o and w’, which need not be normal modes. Kinetically, the product of the two propagators can be evaluated invoking the Plemelj formula

1 (a-kku)(ti’-k’v) = (PV) + &:opk [+(v-;)

)

where (PV) stands for the principal value, and the usual analytical continuation formula has been applied12 to provide continuity in w and w’. Of course, the Dirac S functions describe Landau damping terms. On the other hand, the Landau-fluid model replaces these propagators with

1 (a-kv)(&-k’v)

Landau-fluid 1 -

(w-ilklvth)(W’-ilk’Ivth) *

In the case where none or one of these propagators is fluid, then the Landau-fluid model is a fair approximation. For example, if u>k+, then the above two prescriptions give the same result, both when m’)k’vth (both being dominated by the principal value parts, which vary as I/am’), and when ti’(k’vth (both varying as a’/ak’2v2 in the principal value and i/uk’vth in the resonant contribution). However, when two of the propagators are deeply resonant, that is o/kv,,,(l and ti’/k’vth(l, then the Landau- fluid approximation, varying as l/k%&,, becomes an inac- curate approximation of the kinetics, which vary as l/kov,,, after the velocity integral is carried out. Mathe- matically, in the kinetic theory resonances combine addi- tively, while in the Landau-fluid theory they combine mul- tiplicatively. Physically, the reason is that the Landau-fluid model approximates all the particles as twice resonant, while in reality there are only two narrow bands of resonance: one at VII =w/k and one at vII =w’/k’. From this heuristic argument, one can conclude that the Landau- fluid model describes nonlinear Landau resonances when the basic waves are fluid, but not when they are deeply resonant. This is the underlying reason why the Landau-

fluid model fails to capture the kinetic reduction factor for threshold ITG turbulence.

The remainder of this paper is organized as follows. In Sec. II, the physical model is presented. In Sec. II A we present the gyrokinetic derivation of ICS, and in Sec. II B we present the Landau-fluid version. In Sec. III we present a comparison and discussion of the Landau-fluid and gyrokinetic results, and Sec. IV is a summary. The Appendix is provided to show that the results of the heuristic derivation of Sec. II B persist under a more complete analysis.

II. MODEL AND WEAK TURBULENCE EXPANSIONS

The physical model here is chosen to be as simple as possible, while still retaining enough physics to describe ion Compton scattering of drift waves and ITG modes. The equilibrium consists of an infinite plasma with straight, parallel magnetic field lines, B= B& density no(x), ion temperature T&x), and electron temperature T , homogeneous in the y and z^directions, with “radial” (3 gradients of density [scale length L, = (d In n,/dx) -‘I, and ion temperature [scale length L,= (d In Tddx)-‘I. There are three-dimensional electrostatic fluctuations of electric potential $(x,f_), the nonadiabatic part of the ion distribution function, h (x,v,t), density n( x,t), fluid velocity 7( x,t), and ion temperature Ti( x,t), all of the form

S(x,t) = c f,,, expbk-x-id, W

where f represents a generic fluctuation. Fluctuations are assumed to obey the “usual” drift wave/ITG ordering, with perpendicular wavelengths in the range Pi -‘Z k,)L,‘, L,‘, frequencies in the range Ri,w, and the parallel phase velocity in the range dkll(va where ~a= TJm, so that electrons are adiabatic. Ions are allowed to be either fluid (Ui< w/k,,) or resonant (Vim w/k,,), in keeping with the Landau-fluid/gyrokinetic ion description.

A. Gyrokinetic description

First, a brief review is given of the gyrokinetic description of ion Compton scattering. This has been addressed in a number of studies, so here only a brief derivation is given, for the point of comparison with the Landau-fluid model. Details are described in Refs. 5- 11.

Ion dynamics are described by th_e phase space distribution function p( x,v,t) =Fc(x,v) + f (x,v,t), where F, iz a local Maxwellian [i.e., with no(x) and T,(X)], and f is the rapidly varying part of the distribution function. The nonadiabatic part off evolves according to the nonlinear gyrokinetic equation13 in the electrostatic limit:

e4k -i(ti-kllv~I)~~+i(w-ti~i)FoJo <

1

=&, c :(k’xk”)J’,&&, , I 1 k’+V’=k 0’ 0’1

0’+d’=0

where

(1)

3953 Phys. Fluids B, Vol. 4, No. 12, December 1992 Nathan Mattor 3953


kyCTi d In 110 d In Ti o*i=-&- dx 9 Vi= dInno’

3;2 e-m,u”/2Ti, ai=-f&,

1

Jo is the zeroth Bessel function with argument k,vl/~, JA has argument&‘ivt/S1, and II and,1 refer to the magnetic field direction, b=B/ I B I (here, b =z^ 9. The right-hand side of Eq. ( 19 describes mode coupling due to_ the EX B nonlinearity. Electrons are adiabatic ( &=noe+/T,), and Eq. (1) is closed with the quasineutrality equation, Ze=Kiy or

(l+~)no$=~d3vJo~)~k, (29

where r= TdTP Equations ( 1) and (2) describe drift waves and ITG modes in both the fluid (rid 1) and kinetic (qi- 1) regimes. The fluid equations can be obtained from the velocity moments of Eq. ( 19, in the w/k,, > Vi limit.

The damping rate from ion Compton scattering can be derived in the weak turbulence limit, in which the nonlinearity on the right-hand side of Eq. ( 1) is treated pertur- batively and the modes have random phases. For brevity, Eq. ( 1) is represented as

-iL~,k(o)~~+~LZ,k(o)~k= s Ck’,k”&kthk” . 0 us’ cd”

First, hk = @) + 12 .--2’ + . . * is expanded in powers of the

amplitude of &, which is assumed small. Thus, to first order,

Ll,kb)h, , -“I =L2 ,(&iik . (39 0 ‘IJ

Equation (3) and the quasineutraJity equ_ation yield a spectrum of linear normal modes, +k = &?(o - Ok). To second order, the linear modeswform beat fluctuations, given by

Note that g$ is not itself a normal mode; it is a beat wave. Accordingly, c/=ok - tikl, not the linear frequency wk”. Iteratively, substituting L$? into the nonlinearity of Eq. ( 1) produces the third-order fluctuation

Ll,k(Ok)~~3" c i~k;ir,k,$kr~$j. k’

(59

Substituting li3) into Eq. (2) yields the nonlinear damping rate from ion Compton scattering, valid in the weak turbulence regime,

NL,kin Ok $y%g,l, (69

where the superscript “kin” denotes the kinetic theory version (as opposed to fluid). In Eq. (69, ek is the linear dielectric function,

(79

where

%-* Z;+~(l-$j L. \ A o/ is the linear stability threshold of the ITG mode in the shearless slab limit,t4 &&v’Zkll, and ec3) is the dielectric term following from the third order of the expansion,

(3),kin= ‘k,kr

d3t JO(klvl)L2~k(Ok)Ck’,k”C-k~,k

’ L&+)2L,,-k8t( --0”)

+ s

d3v Jo(klv19L2,-k,(O-k,)Ck,,k,,ck,-k,

Ll,k(Wk)Ll,-k’(W-k’)L*,-k”( -d’) ’ (89

O”zEOk - ok, (nor o of the beat wave!), k”=k-k’, rn=eVEl,(b), I, is the nth modified Bessel function, and b= k: p”. The velocity integrals in the ec3) terms can be carried out with a lot of algebra but no fundamental difficulty, but produce complicated functions with no obvious physical interpretation. Instead, only the limits of drift waves and threshold 17i modes will be considered.

Drift waves have the ordering o-o*, and jrw/~‘~kIIUdl. F or simplicity rli’0 is taken. The linear dielectric then reduces to

199

where the superscript “DW” denotes the drift wave limit (as opposed to ITG). For ec3), the velocity integrals can be carried out by noting that in each of the triply resonant denominators, the terms Lt&(mk) and Lt,-kP(m-k,) are in the fluid regime; only Lt,-k” (ti”) can produce an appre- ciable resonant res onse. grals to evaluate 6”

Carrying out the velocity inte- k,k,, and inserting these into the expres-

sion for @FL, after some algebra produces the damping rate from ion Compton scattering, AC’ = Im @FL,

; (kxk’$G(b,b’)

&lk{l (W”--Ozi) e-s,,z x .-

t2(cSkk+k,k,,)2 (109

and also a nonlinear frequency shift,15 retained here for the sake of comparison with the fluid theory,

(119

where



G( b,b’) E s

m @‘:)J;( $&)J;( @~)e-~:. 0

Ref. 1, which form is used here. The heat flow is most easily expressed in k space, as

For ITG modes, Refs. 7 and 9 have indicated that ion Compton scattering is important near the instability threshold. In this regime, the ordering is y ( w, - k$~f/o,~ (allowing a weak turbulence expansion), and {< 1, so that ions are “deeply resonant.” All three propagators L, in the denominators of e&), are in the resonant regime. Each of these makes a resonant contribution, which can be evaluated using the Plemelj formula for each of the resonances, with analytical continuation in w. After straightforward algebra, the contribution to the growth rate from ion Compton scattering can be shown to be

.,fS,kin,ITG = fi@k@, (3Ti-2) r0 p:fii

vq 1+ l/T)

k$+j&]

(12)

where the contribution from the second velocity integral in Eq. (8) vanishes to this order by asymmetry of the summand in kJ,, and it has been noted that Im( w) (Re(w) for validity of the weak turbulence expansion. The superscript “ITG” in Eq. ( 12) denotes the qrmode limit (as opposed to the drift wave). Discussion of Eq. ( 12), including symmetry properties and their implications for scattering, can be found in Ref. 9.

To summarize this section, Eqs. (10) and (12) represent the damping rate of drift waves and ITG modes (respectively) from ion Compton scattering. They will be used to assess the validity the Landau-fluid results.

B. Landau-fluid description

The Landau-fluid description represents a large simpli- fication over the kinetic theory. The idea is simply to use the fluid equations, and add a heat flow term to model the influence of Landau damping. For the present model described above, the basic equations are the continuity equation,

the

if ?c!i+V*(niVi) =O;

parallel momentum equation,

g u[I i+ (Vi l V)uII i= -A VII Pi-;, v,,+; I 1

(13)

(14)

and the ion pressure equation,

a a zPi+V**VPi+ rpiV*Vi= -z qi,

where uIl i is the ion velocity parallel to the magnetic field, pi=niTi is the ion pressure, and I is the ratio of specific heats. The term qi in the ion pressure equation represents the effects of Landau damping, and has been discussed in

where x1 is a dimensionless constant of order unity. The perpendicular ion velocity, vii, is the sum of the ExB, diamagnetic, and polarization drifts, where

VDi=& b^XVPi 9

As before, electrons are quasineutral and adiabatic, ni=n,=noe$/T,. As discussed in the Introduction, FLR terms are added to the Hammett and Perkins model through the inclusion of Vpi, and nothing more. Separate evolution equations for parallel and perpendicular pressures2T3 are not considered.

As is standard, these equations can be converted into a more workable form by substituting the perpendicular drifts into the fluid equations, taking ni=na(x) +Ki(X,t), pi=po(X) +pi(x,t), normalizing time and distance to l/n, and ps=cs/ai (where c;=TJm,>, taking $=e+/T, %‘= Z/no, ?I= 51 i/C,, p=~i/Tdto, retaining only the nonlinearities coming from perpendicular convection, and noting the usual Hinton and Horton cancellation.16 This yields

l+rli & u-v:)~+b( l+- T v: v$ 1

-iTxv(~+~4T:~+v,,~,=o, (17)

& ;;I,+b^xv&+ -v,y-v,,& (18)

where uDe= -d In n/dx (in normalized units), and CkGflXluiJ$ /I$ I is the Landau-fluid damping coeffi- cient. Equation (19) is written informally, with the left side in configuration space, and the right side in Fourier transform space. The proper configuration space version of the right-hand side can be obtained by a Fourier transform, as discussed in Ref. 1.

Equations (17)-( 19) are fairly standard; however, they contain two terms that are often neglected, but are important for describing nonlinear Landau damping. The first is the Landau-fluid heat flow term in Eq. ( 19), which models Landau damping. The second is the diamagnetic vorticity convection” in Eq. ( 17), which is the only term in Eq. (17) that directly couples to pressure fluctuations.



Here, Eqs. ( 17)-( 19) need not be solved completely; it is only necessary to derive the nonlinear damping rate from ion Compton scattering. As in the kinetic theory, a weak turbulence expansion is the most expedient means of de- riving this. Expanding all the nonlinearities in Eqs. (17j- (19) to third order requires a considerable amount of algebra, which tends to obscure physical clarity. For the present purposes, it is adequate simply to identify the chain of terms responsible for Compton scattering, and pertur- batively expand them to t.hird order. Of course, this leaves open the question of whether any important contributions have been neglected. To answer this, a more systematic analysis is presented in the Appendix; while this is more complicated, the results do not differ substantially from those inferred from the following heuristic analysis.

The perturbation expansion parallels the kinetic weak turbulence theory, and proceeds as follows. Equations (17)-( 19) are Fourier transformed in x and t. Equation ( 19) is used to calculate the nonlinear response of F to the electrostatic potential by iterating in successive powers of 4, which is assumed to be small. The first-order response is

(-io+cIJp k --(l)= ( -i~~+~)$k,

where 0:: = (1 + 77i)w*,J7 and ti*,=kyD= -rw,,.? AS in the kinetic theory, this-linear-response yields a spectrum of linear normal modes, $k = &S(o - wk), and similarly for 3, and E To second &-der, the linear modes form beat fluctuations, given by

( -id~+ck”)&yJ= (b^xik’)-lk~~ktp:‘)

+(b^xik)&‘&g~’ k’

@* “-icmk#/r - - -

akj-ic-k# #k$--kl * (21)

The term FE? is then substituted into the E nonlinearity in the continuity equation, EQ ( 17), resulting in the following nonlinear correction to co:

NL,LF Ok l , 5 @~-~;;S,,

=(1+&l

X CO: +i+/r w;p-ic-kl/r

‘ok-tick - mkl-ic_.kr 1 &kr 1’3 (22’)

where the superscript “LF” denotes the Landau-fluid prediction (as opposed to kinetic). Equation (22) is the Landau-fluid analog to Eq. (6) from kinetic theory. Note that no ordering has been applied that would distinguish between drift waves and vi modes, and so, subject to the validity of the Landau-fluid model, Eq. (22) should apply to both.

Physically, this Landau-fluid procedure mirrors the steps in th_e gyrokinetic derivation. The term F (‘) is analogous to h . (2)* both are virtual modes, both describe a per-

turbation in the ion distribution driven by the beating of two linear normal modes, and through both of these Landau damping enters the Compton scattering rate. The substitution of F Q’ into the continuity equation (producing a third-order density response) is analogous to the gyrokinetic step in which L(““! is substituted into the nonlinearity of the gyrokinetic equation, (giving gc3’ and a third-order density response upon substitution into the quasineutrality equation).

III. DlSCUSSlON

Having obtained both the gyrokinetic and t.he Landau- fluid versions of the Compton scattering rate, these results are compared next.

First the drift wave limit is discussed. The gyrokinetic prediction is given by Eqs. ( 10) and ( 11). The Landau- fluid prediction, taken in the drift wave limit ( @k/ck - 0 */~~X1Uikil)l and 7ji-+ 0 for simplicity) yields

&JL,LF.DW k =7( 1;~: j ; af!;z$z,,

(231

Clearly the nonlinear frequency shift? Re(wFL), is in close agreement with Eq. (11). There are some minor discrepancies in the kL dependence (owing to the fact that the gyrokinetic theory retains terms higher order in k,), but these are of order unity and not of concern here. However, while in agreement, this frequency shift has nothing to do with the Landau-fluid approximation or ICS; here it is given simply to support the claim that the correct term in the Landau-fluid expansion has been retained. A test of the Landau-fluid model is the comparison of Im CIJ:~, and this also turns out to be in agreement with the gyrokiietic predict.ion. To see this, note that the drift wave ordering Wk/ktlZ’i)l means that bot.h drift wave ICS terms are dominated by modes with [Ok* N (ok. In this case, the gyrokinetic drift wave damping rate, Eq. ( lo), reduces to

YJkCs,kmw= _ *! o c~ ;;p;bj ; (kxk’)%G(b,b’)

(24)

and the analogous Landau-fluid term, from Im c$~ in Eq. (231, is

3ps,LF,Dw = _ okT( :+ni, 3 (kxk’$k?

Up to details of kl and T dependence, these two are in reasonable enough agreement to conclude that the Landau- fluid model provides a good description of the Compton scattering of drift waves to the level of accuracy sought here.



The accuracy of the Landau-fluid model for threshold ITG modes is considered next. Before addressing nonlinear

Another important aspect of nonlinear Landau damping concerns the conservation of wave action, also known

dynamics, it is useful to consider the linear Landau-fluid as the wave occupation number. In order to discuss wave model, which turns out to compare well with the more complete gyrokinetic theory.7 To examine the threshold

action conservation properly, a wave kinetic equation is necessary. This can be derived through standard weak tur-

regime, the small parameter e<l is used, where bulence procedures,‘* and takes the form

E_Ok-kliUi<l klpi a* ’

verifiable a posteriori. Equations ( 17)-( 19) are linearized and Fourier transformed. With the above ordering, then the continuity equation, Eq. ( 17), reduces to

io,,( 1 -Kk:)&+ik&=O, (26)

where K= ( 1 + qi)/T* Eliminating Ek from Eqs. ( 18) and ( 19) produces

_ vk=

i$ck( 1 + l/T) + k[& - iq& - r$/T 4k * (27)

Substituting & into Eq. (26) yields the linear ITG mode frequency and growth rate in the zero Larmor radius limit,

LFJTG wk =-($) $+i(,&-;) z. (28)

Equation (28) verifies the ordering scheme, and agrees well enough with the kinetic theory prediction,7 as long as J$ is not too far above 1. This supports the claim that the Landau-fluid model adequately describes the linear theory.

For the Compton scattering rate of threshold ITG modes, the prediction of the Landau-fluid model is not so good. To see this, first note from Eq. (28) that when 1 vi- qc 1 <ck/ 1 we 1 [where here qc= r/( 1 + r&7) - 11, then y(o and the weak turbulence expansion can be performed, leading to Eq. (22). Taking the imaginary component of Eq. (22) and applying the above ordering yields the Landau-fluid prediction for the ICS damping rate of threshold ITG modes,

(29) Comparing Eq. (29) with the gyrokinetic prediction, Eq. ( 12), it can be seen that the magnitudes are quite different: neglecting k, pi terms (all of which tend to be of order 1) the Landau-fluid damping rate varies as ww,/kif, while the gyrokinetic rate varies (relatively) as w&/ukll. The ratio of the former to the latter is then c*-o*/,$vf; this is much less than unity for deeply resonant ions, meaning that the Landau-fluid model greatly underestimates the magnitude of nonlinear Landau damping. In fact, this ratio is just the magnitude by which ion Compton scattering is expected to reduce the level of the spectrum and transport below the mixing length leve1,7P9 so the Landau-fluid predictions will completely wipe out this “kinetic reduction factor,” pre- dicting instead the mixing length level of turbulence.

x(~k,@k’) I&k’~*~~k~*=o. (30)

Equation (30) can be applied to any of the limits considered in this paper (gyrokinetic or Landau fluid; drift wave or ITG) by inserting the appropriate Ek and $$, or equiv- alently by Eq. (6)) by inserting the appropriate qcs. Con- servation of wave action follows from summing Eq. (30) over k. If the symmetries of the nonlinear exchange term are such that the sum vanishes, then a conserved quantity can be identified.

For drift waves, wave action is conserved by ion Compton scattering. In the kinetic theory, action can be defined as

From Eqs. (30), (6), and (IO), the total NpDW evolves by

& ; N~DW= (Ffzi) + F ; “rp ,$LkhDW

x l+kl*k#‘k’l*- (31)

The nonlinear term can easily be seen to be zero, since from the form of $LPki”lDW in Eq. (IO), the sum reverses sign when the indices k and k’ are exchanged. The Landau fluid theory correctly predicts action conservation for drift waves under ICS exchanges, with a slight adjustment of the wave action definition,

LF DW _ %$k: Nk ’ =w: b#)kl**

Again employing Eq. (30), and this time Eq. (25), the evolution of the Landau-fluid drift wave action obeys

& ; NkFvDW= (;f;;) - ; ; (kXk’)i

(32) Noting that tik=@*b( 1 + kf ), and that wk N wk, for

Compton scattering exchanges, it can be seen that the summand again reverses sign under the exchange of k and k’, and so vanishes.

For ITG modes, the Landau-fluid description of wave action dynamics is not entirely faithful to the kinetic theory, but possibly adequate. The gyrokinetic theory predicts that in resonances between two waves and a particle, the



action of one of the waves is dissipated, while the action of the other is conserved. This can be seen from the nonlinear damping rate in Eq. ( 12), and noting that

kll k[ kii ---=-@Gkllk;) lkiil 7 IkIll I”[1 where 0 is the Heaviside step function. The presence of the step function means that any nonzero summand element becomes zero when k and k’ are interchanged: The proper exchange matrix is lower triangular. Thus, for an interaction between k and k’ with, say, ki > kll > 0, the amplitude of k decreases, while mode k’ emerges unchanged. The lost energy is assumedly absorbed by ion heat. This “inelastic” nature of ITG Compton scattering is discussed more fully in Ref. 9, Now consider scattering elasticity in the Landau- fluid prediction. This can be addressed from applying the arguments of the previous paragraph to $cspLF,lTG, as given by Eq. (A13) [the heuristic equation (29) is not derived in enough detail to resolve this]. From Eqs. (A13) and (30), it can be seen that there is an effective “Landau- fluid ITG action,” given by

which evolves by

f ; NkFtrTG= (;z) - ; ; Uock’)i;

(33)

The key point about Eq. (33) is that when k and k’ are interchanged, the summand does not change sign: The exchange matrix is symmetric. There seems to be no way to define an action that is conserved by the nonlinear damping of Eq. (A 13 ) . Loosely, this may be interpreted to mean that fluctuation amplitude is passed not between the modes, but from the modes to the ions, as in the kinetic prediction. However, unlike the kinetic prediction, both k and k’ change amplitude during the resonant interaction, since the exchange matrix is symmetric, not lower triangular. Thus, Landau-fluid theory apparently predicts that for resonances between two threshold ITG waves and a particle, both the modes give energy to the ions. In reality, it is only the mode with lower kll that gives up energy, but possibly this makes no significant diierence for a spectrum of interacting modes. By contrast, the drift wave act.ion scattering matrix is antisymmetric, so that action passes from one wave to the other, not to particles.

Where does the Landau-fluid model go awry? This was discussed generally in the Introducbon, but can be discussed here with more specifics. The difficulty with the Landau-fluid model is the inability to capture the essence of multiply resonant propagators, such as those that appear in nonlinear theory. Specifically, in the gyrokinetic theory, Eq. (4) states that the nonlinear beat fluctuation gk? varies as

$$,ki~ - FM

(Ok-0k~--kiiVII)(Wk-kliVII) ’

where Fhl is a Maxwellian velocity distribution. The fluid pressure fluctuation associated with ?$) is obtained from the L? moment. Summing the residues from the usual Plemelj formula, this is readily seen to have the form

-I2 j,kin dv v%;; I %- ‘d-

&I - - (pv) + uk,k,, - COkki kii 1 k; 1

where (PV) stands for the principal value portion of the integral (not involved with Compton scattering). On the other hand, the second-order pressure fluctuation from the Landau-fluid model, from Eq. (21), varies as

- (1j.LF 1 P k” -i~,kk-Ol;,-ijkli/Vj)(U)k-ilk,,jvi) ’

where ikp emulates the effect of resonances in cases where o 5 kpp The accuracy of the Landau-fluid model can be understood by comparing these two denominators. Con- sider first the case where the basic linear waves are in the fluid regime, u>kllvi, as for drift waves. In this case, it is easily seen that both the Landau-fluid and gyrokinetic pressures are dominated by terms with Cr)k”Wk’r for which the two resonant contributions to PC! are equal. If, however, the basic linear waves are in a regime where ogkllvi (as for ITG modes near the threshold), then the agreement is not so good; the gyrokinetic pz? varies as I/wkll, while the Landau-tluid prediction has relative magnitude l/$vi, which is smaller by a factor of w/kllvi( 1. The resonant contribution to PC,) is smaller still, with an anomaly factor of (o/kllvi)2, which accounts for the discrepancy in .p”.

From these arguments, two general conclusions can be formulated. First, the Landau-fluid model probably does not provide an accurate nonlinear description of any waves in a deeply resonant regime, with w/ok’“<1 (where cdkin is any resonance producing drift, such as a magnetic drift frequency). Second, although this study was carried out for weak turbulence, the general conclusion might well apply to strong turbulence as well. A major change in strong turbulence is that the weak turbulence expansion ceases to converge, meaning that terms with multiple propagators become more important; thus there is no reason to believe that the associated problems go away. (The Plemelj formula and analytical continuations used in the heuristic arguments above formally require the limit Im w-+0, but realistically these do not much change I(z,I/kL’~~4.)

as long as

IV. SUMMARY

This paper has examined the ability of the Landau- fluid model to account for the ion Compton scattering (also known as nonlinear Landau damping) of drift waves and ITG modes. A weak turbulence approximation has been used to attain the Landau&id prediction for the



damping rate from ion Compton scattering, which is compared with previously known results from gyrokinetic theory. The following results have been attained.

( 1) For drift waves, the Landau-fluid model provides a reasonable approximation to ion Compton scattering, to the accuracy sought here. This pertains both to the magnitude of scattering and to conservation of wave action.

(2) For ITG modes near the threshold, the Landau- fluid and gyrokinetic predictions depart in at least two ways.

(a) The nonlinear damping rate from the Landau-fluid model is smaller by a factor of about ( o/kllui) ‘. From this, one can expect that the Landau-fluid approach will over- look the “kinetic reduction factor,” by which nonlinear ion resonances reduce the ITG turbulence leve17’9 (when present). The predictions of the Landau-fluid model will be closer to the predictions of mixing length theory.

(b) Both gyrokinetic theory and the Landau-fluid model predict deeply inelastic ion Compton scattering, in which a large portion of wave energy is transferred to ion heat. However, gyrokinetic theory predicts that the energy is lost entirely by the mode with the lower 1 k,, I, while the Landau fluid model predicts that both modes lose energy.

(3) It is concluded generally that the Landau-fluid model has a problem in describing effects dependent on the product of two or more resonant propagators. Possibly the ion Compton scattering considered here is only a subclass of the totality of such effects.

What the Landau-fluid model has previously shown is that it is possible to add a term to the fluid equations, which gives the correct linear response to perturbations. The present results call into question whether this term is as accurate nonlinearly.

ACKNOWLEDGMENTS

Useful conversations with Dr. L. D. Pearlstein, Dr. A. M. Dimits, and Dr. T. S. Hahm are gratefully acknowl- edged. I am indebted to Dr. B. I. Cohen for a stimulating and useful critique, and for a reading of the manuscript.

This work was supported at Lawrence Livermore Na- tional Laboratory by U.S. Department of Energy Contract No. W-7405ENG-48.

APPENDIX: DETAILED PERTURBATION EXPANSION OF THE LANDAU-FLUID EQUATIONS

In Sec. II B we presented a heuristic weak turbulence expansion on the Landau-fluid equations. Nonlinear cor- rections to the parallel velocity fluctuation, rh” and F/l”, and other terms were ignored. This saved a good deal of algebra, but leaves open the question of whether any terms were neglected that would alter the conclusions. In this appendix we present a more systematic weak turbulence expansion on the Landau-fluid equations, Eqs. (17)-( 19>, with no such omission. The conclusions remain intact.

For economy, calculations will be performed with Eqs. (17)-( 19) placed in a matrix form. Fourier transforming Eqs. (17)-( 19) in x and t, these can be written as

3959 Phys. Fluids El, Vol. 4, No. 12, December 1992

where the matrix propagator is given by

w -kll

0 w

i r --c -- k k

7 7

-kll

3 w+iq

the fluid fields are given by

+ i&(w) =

I- \ nk

0

‘k f 0

,Fk, Co

the linear source term is given by

zks&+~;Kk3),

where K = (1 + vi)/r,wz = (1 + ~)w,,Jr, and the nonlinear coupling matrix is given by

-k;’ 0 -k;’

Zk,E 0 1 0 .

t 1 0 0 1

In order not to confuse spatial vectors with field vectors, the latter have been represented with “arrow” notation. Left arrows represent column vectors, right arrows represent row vectors, and double arrows represent matrices. Equation (Al > is written without yet applying the relation K=$, in order to represent moments of the gyrokinetic equation. This allows a weak turbulence expansion paral- leling that of Sec. II B, with quasineutrality applied only after representing fluid moments in successive powers of 4. Most of thezol@wing dzrivation follows directly from Sec. II A, with M, L, and w, replacing L,, L,, and h, respectively.

A weak turbulence expansion can now be performed on, 1 ,Eq. (Al ) . The lowest-order (linear) response, wk (w), is given by

z(l) wk (w) =i?&& . (A21

As in the kinetic calculation, the first-order responses (which will be seen to form a spectrum of linear normal modes) form second-order beat waves, given by

i&,(d) ;,$)(d) =i(kXk’),,~k~-k,[~_k,li;j,-l(w)~k

0 -aI

-h$$?;,( -d)&], (A31

Nathan Mattor 3959


where o”=ti--w’, and only the phase coherent terms have been retained. The third-order response is obtained from substituting %p’(o) into the nonlinearity in Eq. (Al ), and is given by

~k(~):&3’(o)= c (kxk’);zk,%;‘l(W”) k’o’

(A4)

With the first-, second-, and third-order fluid responses now given, a closed equation can be obtained from quasineutrality with adiabatic electrons,

?I) z(zl t-(3) dk=n=<,[ wk (a> + wk (0) + wk (a)], (A%

where zn is a unit row vector in the “n direction,”

z+(l 0 0).

As in the kinetic calculation, the appropriate regime is k,2 1, where the waves are sufficiently dispersive that three-wave frequency matching, tik = I&I + @k”r does

“(2) * not occur, and so the contribution from w k 1s negli- gible. Equation (A5) then becomes

l ,&d&+ c E,$@,d I&;,2$~=o, 0 k’,d (A61

where the linear dielectric is given by

+(“)) = l-i$&-‘(&$‘,

and the nonlinear dielectric is given by

E&(w,w’)= - (kXk’)?i~~;‘(w)i,~~~(ol~)

Written, thus, there is a direct correspondence between the fluidt+dielectr&s and the kinetic versions in Sec. II A, with the M,znd Lk and analogous to L,,k and L2,k, respectively. Thus, MC ’ may be regarded as sort of a “Landau-fluid plasma dispersion function,” up to a factor of 1211. For ref- erence, it is useful to write this out,

1 .&y(w) =g

I- s’+icp-; ki -kp + i&ll 4

iiclk,l a2 + ickti 0

1 i;Cp F kp+ifckkil W’

where the determinant is given by

r 1 Dk=w3+iC,&02-7 k+O-i;C&fj.

As usual, Eq. (A6) yields the linear dispersion relation from the solution of

Ek(@k) =o, and the nonlinear correction

(A7)

NL wk =-

The matrices in ec3) can be multiplied out with no fundamental difficulty, but produce complicated functions with no obvious physical interpretation. Instead, only the limits of drift waves and threshold ,qi modes will be considered.

First, drift waves are considered, which have the ordering ok-~*,kllui [although (wk - wp)/kllUj can be small], k;” 2 1, and 1+ vi-+0 is again taken for simplicity. The linear dispersion relation from Eq. (A7), to lowest order, is the standard result

~;~~n~=~.J/(l+k;). (-49)

The nonlinear c.orrection to the frequency is calculated from Eq. (A8). There are many terms in the calculation of et3j but this is simplified from noting that the product of the ‘1,l elements of the matrices give the dominant contribution, given drift wave ordering. The result is

a Re gFpDW o*, =7 a+ ok

and

(3),LF,DW k;” r ‘k,kl = (kxk’)i okDk,, d’2+ickd’-; kri”

With these, then the nonlinear frequency correction is NL,LF,DW

@k

=z z (kxk’)+;’

x Cd” +ic&o”- ( T/T) k/’ ~N3+ick~#d”- (r/~)k~2w”-i (l/T)CkttkF’

(A101

Equation (AlO) represents a more complete version of Eq. (23 ), which can be recovered exactly by neglecting the final two terms in each of the numerator and denominator of the polynomial quotient. From inspection, it is easily seen t.hat Eqs. (23) and (AIO) are not different with re- spect to magnitudes of the real and imaginary parts, widt.h of scatter in k’, and the action conserving symmetry of Im(aNL) under interchange of k and k’. Thus, the conclusion that the L.andau-fluid model adequately describes drift wave ICS is supported by this more detailed analysis.

For ITG modes near the instability threshold, the ordering is again ak<k~~-&B,, y(w, and k: X 1. With this, the linear dispersion relation, calculated from Eq. (A7), is



LFJTG = *It

1+1/r k;; -+iz (j&-y;), (All) - 1 - Kkf a+..

with no difference from Eq. (28). Carrying out the calculation of wrL, it is found that, to lowest order,

d Re ekFPrTG l+r a& =-Wk'

and after extensive calculation the nonlinear dielectric is, to lowest order

(3),LF,ITG %,k’

(A12)

From Eqs. (All), (A12), and the imaginary part of Eq. (A8), it is then found that

dalLF*lTG= -2 ; (kxk’)fk;2; (t-2). (A13)

Equation (A13) represents the more complete version of Eq. (29), and it is easily seen that these two expressions have the same magnitude, both of which are too small to agree with the kinetic calculation. There is a difference in the magnitude of the ITG nonlinear frequency shift (not shown here) between the heuristic calculation and that which follows from the detailed analysis of this appendix. The detailed analysis reveals some cancellations, and the resulting frequency is a factor of (w/$,)~ less in the detailed version. It turns out that neither of these are adequate to agree with the kinetic theory prediction of the

nonlinear frequency shift, which is smaller still by another factor of (w/kll)2; however, this is not of primary concern here.

‘G. W. Hammett and F. W. Perkins, Phys Rev. Lett. 64, 3019 (1990). *Z. Chang and J. D. Callen, Phys. Fluids B 4, 1167 (1992). 3R. E. Waltz, R. R. Dominguez, and G. W. Hammett, Phys. Fluids B 4, 3138 (1992).

4Unfortunately the term “nonlinear Landau damping” is somewhat am- biguous. It has been used to describe finite amplitude effects of a single wave resonant with particles, as in T. M. O ’Neil, Phys. Fluids 17, 609 (1974). In the present paper, the term is used to describe the interaction between multiple (basically linear) waves and a particle, and the effect this process has on drift waves or ITG modes.

‘R. Z. Sagdeev and A. A. Galeev, Nonlinear Plasma Theory, edited by T. M. O ’Neil and D. L. Book (Benjamin, New York, 1969), pp. 89-l 17.

6P. L. Similon and P. H. Diamond, Phys. Fluids 27, 916 ( 1984). ‘N. Mattor and P. H. Diamond, Phys. Fluids B 1, 1980 (1989). *F. Y. Gang, P. H. Diamond, and M. N. Rosenbluth, Phys. Fluids B 3, 68 (1991).

9N. Mattor, Phys. Fluids B 3, 1913 (1991). “T. S. Hahm and W. M. Tang, Phys. Fluids B 3, 989 ( 1991). “T. S. Hahm, Phys. Fluids B 3, 1445 (1991). ‘*The analytical continuation is a subtle but important component of the

evaluation of the ICS rate for deeply resonant propagators. Without the factors of k/ 1 k I, the Dirac S functions would always provide resonant contributions of opposite sign, and upon evaluating the velocity integral would cancel to lowest order. With the analytical continuation factors, the lowest-order resonant contributions do not cancel when k and k’ have opposite sign, and so the 0( 1) contribution of the Dirac S functions persists. The more complete result of this is displayed by Eq. (12).

13E. A. Frieman and L. Chen, Phys. Fluids 25, 502 (1982). 14B B Kadomtsev and 0. P. Pogutse, in Reviews of Plasma Physics, . .

edited by M. A. Leontovitch (Consultants Bureau, New York, 1970), Vol. 5, p. 249.

“N. Mattor and P. W. Terry, Phys. Fluids B 4, 1126 (1992). 16F. L. Hinton and C. W. Horton, Jr., Phys. Fluids 14, 116 (1971). “W. Horton, R. D. Estes, and D. Biskamp, Plasma Phys. 22,663 ( 1980). ‘*R. Z. Sagdeev and A. A. Galeev, in Ref. 5, pp. 90-91.



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Can Landau-fluid models describe nonlinear Landau damping?

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