Whistlerization and anisotropy in two-dimensional electron magnetohydrodynamicturbulenceSheikh Dastgeer, Amita Das, Predhiman Kaw, and P. H. Diamond Citation: Physics of Plasmas (1994-present) 7, 571 (2000); doi: 10.1063/1.873843 View online: http://dx.doi.org/10.1063/1.873843 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/7/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reconnection events in two-dimensional Hall magnetohydrodynamic turbulence Phys. Plasmas 19, 092307 (2012); 10.1063/1.4754151 Forward and inverse cascades in decaying two-dimensional electron magnetohydrodynamic turbulence Phys. Plasmas 16, 042307 (2009); 10.1063/1.3111033 Analysis of cancellation in two-dimensional magnetohydrodynamic turbulence Phys. Plasmas 9, 89 (2002); 10.1063/1.1420738 On two-dimensional magnetohydrodynamic turbulence Phys. Plasmas 8, 3282 (2001); 10.1063/1.1377611 Hydrodynamic regime of two-dimensional electron magnetohydrodynamics Phys. Plasmas 7, 1366 (2000); 10.1063/1.873953

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Whistlerization and anisotropy in two-dimensional electronmagnetohydrodynamic turbulence

Sheikh DastgeerInstitute for Plasma Research, Bhat, Gandhinagar, 382428, India

Amita Dasa)

University of California at San Diego, La Jolla, California 92093

Predhiman KawInstitute for Plasma Research, Bhat, Gandhinagar, 382428, India

P. H. DiamondUniversity of California San Diego, La Jolla, California 92093

~Received 17 August 1999; accepted 8 November 1999!

A detailed numerical simulation to understand the turbulent state of the decaying two-dimensionalelectron magnetohydrodynamics is presented. It is observed that the evolved spectrum is comprisedof a collection of random eddies and a gas of whistler waves, the latter constituting the normaloscillatory modes of such a model. The whistlerization of the turbulent spectra has been quantifiedby novel diagnostics. In this work, results are presented only in the regime where the spatialexcitation scales are longer than the electron skin depth. Simulations suggest that spectra at shortscales are comparatively more whistlerized. The long scale field merely acts as the ambient fieldalong which whistler waves propagate. It is also observed that, in the presence of an externalmagnetic field, the power spectrum acquires a distinct directional dependence. This anisotropy isdominant at short scales. It is shown that such an anisotropy at short scales results from a cascademechanism governed by the interacting whistlers waves. ©2000 American Institute of Physics.@S1070-664X~00!04602-4#

I. INTRODUCTION

The problem of turbulence has been studied both in thecontext of ordinary as well as magnetofluids. Progress in thestudy of the problem of turbulence in the magnetofluids~e.g.,plasma! is comparatively recent and indicates interestingsimilarities and differences vis a vis ordinary fluids. One ofthe most important distinctions from the neutral, noncon-ducting fluids stems from the fact that magnetofluids, likeplasmas can support a variety of waves. Turbulence in themagnetohydrodynamic~MHD! model, which supports dis-persionless Alfve´n waves as normal modes, has been ex-plored in considerable detail. It is observed that in the case ofMHD, the existence of these waves leads to subtle effects inturbulence. There are suggestions that there may be a modi-fication in the power spectral index from that predicted onthe basis of Kolmogorov’s dimensional arguments. This istermed the Alfve´n effect.1 The interactions amongst theAlfven waves are fairly restrictive, with only oppositely trav-eling waves interacting with each other. This results instrong anisotropy of the spectrum in the presence of externalmagnetic fields. The magnetohydrodynamics~MHD! modelfor plasma applies for the description of phenomena wherethe time scales are reasonably slow and the dynamics ismainly governed by the ion species. There exists anothermodel which depicts MHD phenomena occurring at fastertime scales; in this case ions do not have time to respond

because of their heavy mass and merely provide a neutraliz-ing background, and the dynamics is entirely governed bythe lighter electron species. Such a model is known as theelectron magnetohydrodynamics~EMHD! model.2 In con-trast to MHD, this particular model supports dispersivewaves known as whistlers. This model was more or lessunexplored so far and has attracted interest lately due to itspotential applications in fast switches,Z pinches, solar phys-ics, magnetospheric reconnection, and ionosphericphenomena.2,3 A recent numerical work by Biskampet al.4,5

on decaying EMHD turbulence has shown agreement withKolmogorov’s dimensional argument6 for the scaling ofpower spectrum with wave number. On the basis of this re-sult, they indicate that the whistler effect may not play anysignificant role in EMHD turbulence.

We define the concept of whistlerization of the turbu-lence viz. how closely the whistler mode relationshipbk

5kck ~physically an equipartition between the axial and po-loidal magnetic energies! is obeyed in the turbulent state. Asstated previously, the power spectral index in EMHD turbu-lence does not seem to be significantly influenced by thewhistler effect. However, whistlerization does take place, asevidenced by the simulations of Biskampet al.,4,5 who haveshown that at longer scales than the skin depth (kde,1)there is significant equipartition between the axial and poloi-dal magnetic energies. One of the objectives of the presentwork is to understand the degree of whistlerization of theturbulent state in EMHD in quantitative detail. For this pur-a!Permanent address: Institute for Plasma Research, Gandhinagar, India.

PHYSICS OF PLASMAS VOLUME 7, NUMBER 2 FEBRUARY 2000

5711070-664X/2000/7(2)/571/9/$17.00 © 2000 American Institute of Physics

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pose, we seek to address the following issues in the subse-quent sections;~i! quantification of the extent of whistleriza-tion of the spectra,~ii ! scale dependence of whistlerization,~iii ! dependence of whistlerization on the strength of externalfield, and~iv! time evolution of a random initial spectra to-ward a whistlerized state. Another objective is to study theinfluence of an external magnetic field on the spectrum, andthe reasons thereof. For this we would largely explore~i! thecharacteristic difference in the spectrum in directions paralleland perpendicular to the external field~e.g., anisotropy in thespectrum! and ~ii ! scale dependence of anisotropy, etc.

The paper is organized as follows. In Sec. II we discussbriefly the EMHD model and the numerical results on whis-tlerization. We conclude that the turbulence evolves toward astate comprised of a mixture of a gas of whistler waves~in-teracting randomly with each other! and a collection of ran-dom eddies. The tendency for whistlerization seems to bestronger at shorter scales in the inertial range. Basically, thelong scale field acts as an ambient magnetic field alongwhich the short scale whistler perturbations propagate. Nu-merical studies of the spectrum in the presence of externalmagnetic field are presented in Sec. III. In the presence of anexternal magnetic field, we observe that the spectrum ac-quires directional dependence. It is observed that the perpen-dicular scales are predominantly shorter in comparison toparallel scales. Furthermore, this disparity in parallel andperpendicular scales is stronger at shorter scale lengths. Itseems quite likely that the observed anisotropy arises as aresult of interactions amongst whistler waves, which prima-rily propagate parallel to the magnetic field. In Sec. IV weprovide a theoretical understanding of observed anisotropy inthe power spectrum on the basis of whistler wave interac-tions. Section V contains the conclusion.

II. WHISTLERIZATION IN EMHD TURBULENCE

Electron magnetohydrodynamics~EMHD! is the de-scription of electromagnetic phenomena in a magnetizedplasma at fast time scales where only electron dynamics isimportant. Ions merely provide a stationary neutralizingbackground. The EMHD model has been discussed in con-siderable detail in some of the earlier works.2,4,5,7,8Here, weconcentrate on the case when the electromagnetic perturba-tions vary only in two dimensions, e.g., thex–y plane. TheEMHD model is then represented by the following set ofcoupled equations in the two fieldsb andc.

]

]t~c2¹2c!1 z3“b•¹~c2¹2c!5h¹2c, ~1!

]

]t~b2¹2b!2 z3¹b–““

2b1 z3¹c–“¹2c5h¹2b. ~2!

The total magnetic field can be expressed in terms of thesefields asB5 z3“c1bz. The two evolution equations havebeen written in normalized variables. The length scales arenormalized by the electron skin depthde5c/vpe , magneticfield by a typical amplitudeB0 and time by the correspond-ing electron gyrofrequency. The dispersion relation for whis-

tlers, the normal mode of oscillation in the EMHD frequencyregime, can be obtained by linearizing Eqs.~1! and~2! in thepresence of uniform magnetic fieldB0y pointing in the ydirection as

v5skkyB0y

~11k2!.

Here s56 and indicates the direction of propagation. Thewhistler wave excitation leads to a coupling of the formbk

5skck between the two perturbed fields. It is interesting tonote that the above-mentioned relation leads to an equiparti-tion between the poloidal and the axial components of themagnetic/kinetic energies. This should be contrasted withMHD where the Alfven wave excitation leads to an equipar-tition between magnetic and kinetic energies. There can beno equipartition between magnetic and kinetic energies as aresult of whistlerization of the spectrum. The dominance ofmagnetic or kinetic energy is dependent on whether the typi-cal turbulent scales of excitation are longer or shorter thanthe electron skin depth. Also, unlike Alfve´n waves, thesewaves are dispersive. They propagate obliquely to the exter-nal magnetic field with a group velocity

Vg5sB0ykykx~12k2!

k~11k2!2x

1sB0y@k2~11k2!1ky

2~12k2!#

k~11k2!2y.

We now carry out numerical investigations to under-stand the turbulent state in EMHD. For this purpose weevolve the two fieldsb andc numerically using Eqs.~1! and~2!, respectively, with the help of a fully de-aliased pseu-dospectral scheme. In this scheme the fieldsb and c areFourier decomposed. Each of the Fourier modes are thenevolved, the linear part exactly, whereas the nonlinear termsare calculated in real space and then Fourier transformed inkspace. This requires going back and forth in real andk spaceat each time step. The fast Fourier transform~FFT! routinesare used to go back and forth in the real andk space at eachtime integration. The time stepping is done using a predictorcorrector with the midpoint leap frog scheme. Some testsimulations were carried out at a lower resolution of 1283128. However, most results that we present here corre-spond to the resolution of 2563256 and 5123512 modes.For numerical studies the dissipation operator of Eqs.~1! and~2! were replaced by higher powers of Laplacian~cubic! inorder to restrict dissipative effects at shorter scales. The typi-cal values of the dissipation parameterh were chosen to liebetween 531021 to 531022 in the various runs. The initialspectrum of the two fieldsb andc is chosen to be concen-trated on a band of scales and their phases are taken to berandom. The Fourier modes of the two fields are chosen tobe entirely uncorrelated to begin with. Furthermore, in thepresent work we have chosen to restrict our studies to scaleskde&1 where wave propagation effects are comparativelystronger.

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The initial energy is distributed unequally in the poloidalc and axialb fields. In Figs. 1 and 2 we show the evolutionof total energy~dashed line!, and the energy in the fieldsbandc, which has been termed as axial and poloidal energies,respectively. The plots in Fig. 1 are for the case where theinitial axial energy is chosen to be greater than the poloidalenergy. We observe that the axial energy decays and thepoloidal energy grows so as to achieve equal distribution ofenergies, in the two fields, ultimately. Figure 2 depicts theother scenario wherein the initial poloidal energy is largerthan that in the axial field. In this case, the poloidal energyreduces and the axial energy increases with time, so that

finally both acquire similar values. This shows that the ener-gies associated with the two fields strive to achieve equipar-tition.

A complete whistlerization of the turbulent spectra, i.e.,when the spectrum is comprised of a collection of randomlyphased whistlers interacting with each other, will automati-cally lead to an equal distribution of energy in the axial andpoloidal fields. This is so because the relationship,bk

56kck , is satisfied for whistler modes. However, one can-not infer the opposite, i.e., from the observed equality ofpoloidal and axial energies it cannot be concluded with cer-tainty that the spectrum is completely whistlerized. This is sobecause the axial and poloidal energies represent integralquantities, where contribution from each mode has beensummed. Thus to ensure the extent to which the relationshipbk56kck is satisfied for each of the individual modes, weintroduce a set of new diagnostics, which are monitored dur-ing the course of numerical simulations. We define

V1~ t !51

(kx ,ky

(kx ,ky

abs~ ubku22k2ucku2!

~ ubku21k2ucku2!

to represent the extent of whistlerization of the spectrum. Itis clear that for a completely whistlerized stateV1 will van-ish. The more the state is whistlerized, the lower the value ofV1(t). In the definition ofV1 we have summed suitablynormalized nonwhistlerized contribution of power from eachmode. The contribution from each mode is normalized by itsown spectral content. This treats all the modes on an equalfooting. However, an alternative definition of the kind

V2~ t !5(kx ,ky

abs~ ubku22k2ucku2!

(kx ,ky~ ubku21k2ucku2!

can also be employed, where contribution from each mode isnormalized by the average power. This definition selectivelygives importance to modes having higher spectral power.

The evolutions ofV1 andV2 are shown in Figs. 3 and 4,respectively, for various values of the external fieldB0 . Thefollowing points are clearly evident from the plot ofV1 . ~i!V1 rapidly decreases from its initial value and attains a sta-tistically steady value, which is relatively small. This con-firms the tendency for whistlerization of the spectrum, whichhappens even when the external field is absent.~ii ! Theasymptotic value ofV1 decreases with increasing magneticfield. Thus, the extent up to which a spectrum is whistlerizedis dependent on the strength of magnetic field.~iii ! The rateof whistlerization increases with magnetic field, the initialdrop to the asymptotic value can be seen to be steepest forB054, and gets shallower with decreasing value ofB0 . It isalso apparent that the asymptotic value ofV1(t) shows fluc-tuations in time. These fluctuations are more rapid with in-creasing external fields. It appears that the time scale for thespectrum to get whistlerized, as well as its statistical behav-ior, are linked with a typical whistler time scale defined bythe magnetic field strength and a suitably averaged spatialscale@e.g.,twhistler;1/(kavkav•B)].

A comparison of the two plots ofV1 and V2 drawsattention to another interesting feature. It is clear thatV2 is

FIG. 1. The evolution of total energy~dashed line!, axial ~b! field energy~dotted line!, and the poloidal (c) field energy~solid line! when the initialaxial energy was greater than the poloidal energy.

FIG. 2. Similar to Fig. 1, however, for this case the initial poloidal energywas chosen to be larger than the axial energy.

573Phys. Plasmas, Vol. 7, No. 2, February 2000 Whistlerization and anisotropy in two-dimensional . . .

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always larger thanV1 . SinceV2 represents modes havinghigher spectral power, a comparative higher value ofV2

indicates that modes having higher spectral powers are lesswhistlerized. Since the power spectrum decays ask27/3, thisalso suggests that the large scale regime of the spectrum isless whistlerized. This clearly suggests that whistlerizationoccurs predominantly at short scales. We provide further evi-dence of the short scale part of the spectrum being morewhistlerized in Fig. 5. The various plots of Fig. 5 show theevolution of a function similar toV1 in which only certainscales in the summation have been included. Results havebeen shown both at the resolution of 2563256 @Figs. 5~a!and 5~b!# and at 5123512 @Figs. 5~c! and 5~d!#. In Figs. 5~a!and 5~c! and Figs. 5~b! and 5~d! we have chosen to plot the

evolution ofV1 containing contributions typically from longand short scales alone, respectively. As can be seen the typi-cal values in Figs. 5~a! and 5~c! are closer to unity whencompared with Figs. 5~b! and 5~d!. It is also evident from theplots that as one successively incorporates contributionsfrom shorter scales the value of the functionV1 keeps reduc-ing. Furthermore, this inference holds consistently even aswe increase the resolution from 2562 to 5122. Thus it can bestated that the extent of whistlerization is greater at shortscales. This is reasonable as the large scale fields merely actas an effective ambient magnetic field along which the whis-tlers propagate. We would, however, like to point out that theshort scales for which we are drawing inferences here aretypically longer than the electron skin depth.

It should be noted here that the linear whistler frequencyv lin5kk–B is larger at short scales. Hence, the conditions forapplicability of weak turbulence theory viz.v lin@DvNL

~where DvNL represents the nonlinear broadening of themodes! are more likely to be satisfied at short scales. Thus atshort scales linear wave physics dominate and the small scalemodes may be viewed as whistler wavepackets, whereaslong scales are effectively magnetic eddies.

III. SPECTRAL ANISOTROPY IN THE PRESENCE OFEXTERNAL MAGNETIC FIELD

In this section we study the evolution of the spectralproperties of decaying EMHD turbulence in the presence ofan external magnetic field. It is well known that the presence

FIG. 3. Evolution ofV1 for B050 ~dotted line!, B051 ~dashed line!, andB054 ~continuous!.

FIG. 4. Similar to Fig. 3, but showing evolution ofV2 .

FIG. 5. Evolution ofV1 with contributions in the summation only fromcertain regions of scales.~a! and~b! Results for a resolution of 2562. ~c! and~d! Results at 5122. For the dotted, dashed, and solid lines of~a! the scalelengthsuku included in the sum lie in the range 0<uku,2k1 , 0<uku,4k1

and 0<uku,8k1 respectively, wherek150.0125. The scale lengths in thethick solid, dotted, and dashed lines of~c! lie in the range 0<uku,2k2 , 0<uku,4k2 and 0<uku,8k2 , respectively, wherek250.006 25. The solidlines of ~b! and ~d! represent the case when all the scales have beensummed, however, the dashed lines of both~c! and ~d! are when the scalelengths lie in the regime 1/3kmax<uku,2/3kmax. Herekmax51.6 and 2/3kmax

basically is the shortest scale available after aliasing.

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of a uniform magnetic field in the context of MHD gives riseto an anisotropic spectrum.9,10 The power spectrum of thevarious variables in the case of MHD are known to showdistinct directional dependence. This is clear from the MHDsimulations carried out by Shebalinet al.9 and most recentlyby Oughtonet al.10 It is believed that this happens primarilydue to the excitation of Alfve´n waves which preferentiallypropagate parallel to the external magnetic field, and hinderthe process of spectral cascade parallel to the external field.This results in the anisotropy of the spectrum. Thus the ori-gin of anisotropy is basically the restrictive interactions ofthe Alfven waves for which only oppositely traveling wavesinteract. In the absence of external magnetic field, Alfve´nicexcitations are generated along the long scale turbulent field~which has a random orientation, and hence is isotropic!.These Alfvenic excitations restrict the cascade mechanismand are also responsible for the deviation of power spectrumindex from the Kolmogorov value of25/3 to a value23/2.1,11

In the context of EMHD turbulence, it has been notedthat the whistler effect on spectral index is negligible.4,5 Thenumerically observed spectral index can be explained purelyon the basis of Kolmogorov’s dimensional arguments, with-out invoking any role, whatsoever, for the whistler waves. Anatural question then arises whether in EMHD turbulence thespectrum is maintained isotropic even in the presence of astrong external magnetic field. This is what we intend tostudy in this section. We have seen in Sec. II that in theturbulent EMHD state both whistlers and random eddies co-exist. We have also seen indications that in the presence ofexternal magnetic field the spectrum becomes more whistler-ized, and that the short scale part of the spectrum showsstronger tendencies toward whistlerization. The intriguingquestion in this regard is whether these whistlers have any

role to play in the cascade mechanism. For this purpose wewould like to investigate the spectral properties of the twofields b andc in the presence of varying external magneticfield. We would also like to investigate a possible distinctionbetween short and long scales with regard to the anisotro-pization of the spectrum.

The numerical scheme has already been outlined in Sec.II. We have chosen the external magnetic fieldB0 to bealong they axis in our numerical studies. We then define thefollowing ratio:

RQ~ t !5(kx

(kyky

2uQ~kx ,ky ,t !u2

(kx(ky

kx2uQ~kx ,ky ,t !u2

. ~3!

Here,Q stands for any one of the following variablesb, c,¹2b, and¹2c. It is clear that if the spectrum ofQ is isotro-

FIG. 6. Evolution ofR, the anisotropy factor for the variablesb, c, ¹2b,and¹2c. The upper curve hovers around unity for the case when there is noimposed external fieldB050; the lower curve represents the case whenB057.5.

FIG. 7. Contour plots for the variableb in the evolved state forB050 andB057.5.

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pic then the ratioRQ will be close to unity. Any deviationfrom unity is an indication of anisotropy in the spectrum.

The evolution ofRQ for the variablesb, c, ¹2b, and¹2c is shown in the four plots of Fig. 6. The upper curve inall these plots, which hovers around unity, is forB050. Asexpected, it testifies to the fact that the spectrum of all thevariables remains isotropic in the absence of any externalmagnetic field. The lower curve is for the case when theexternal magnetic field has been chosen to beB057.5. It canbe seen that the value ofR corresponding to all the fourvariables drops off from unity and saturates at a low value.The plots also show thatR¹2b andR¹2c saturate at a muchlower value than compared withRb and Rc , respectively.This clearly implies that the spectral anisotropy in the vari-ablesb andc is dominant at shorter scales. Furthermore, it isalso evident that finallyRc.Rb , suggesting that the spec-trum of b is more anisotropic compared to that ofc. Since

the magnetic scalar potentialc is known to cascade towardlonger scales, the weak anisotropy observed in its spectrumcan be attributed to the dominance of long scales in it.

The definition of R involves a numerator which isweighted byky

2 , corresponding to the wave vector parallel tothe direction of external magnetic field, whereas the denomi-nator containskx

2 , the perpendicular wave vector, as the fac-tor inside the summation. SinceR asymptotes toward a valuelower than unity, it is obvious that the spectrum is evolvingso as to have comparatively shorter scales perpendicular tothe magnetic field. The contour plot of the variableb in thex–y plane depicted in Fig. 7 clearly exemplifies the domi-nance of short scales perpendicular to the external magneticfield in the spectra ofb. For, c, which exhibits compara-tively weak anisotropy, the dominance of short scales per-pendicular to the magnetic field cannot be gauged directlyfrom its contour plot~e.g., Fig. 8!.

We also observe that by reducing the value ofh, thedissipation parameter results in an increased anisotropy ofthe spectrum. Figure 9 shows the evolution ofR for the vari-able b for varying values ofh. This can be understood byrealizing that decreasingh is tantamount to pushing the dis-sipation scale toward shorter scales. Thus the short perpen-dicular scales generated in the cascade continue to survive ina larger domain of scales, ash is reduced, therebyR reducesfurther.

We define the average perpendicular and parallel scalesas

^ki2&5

(kx ,kyky

2Q~kx ,ky ,t !2

(kx ,kyQ~kx ,ky ,t !2

,

^k'2 &5

(kx ,kykx

2Q~kx ,ky ,t !2

(kx ,kyQ~kx ,ky ,t !2

.

FIG. 8. Contour plots for the variablec in the evolved state forB050 andB057.5.

FIG. 9. Evolution ofRb for varying h, the dissipation parameter,h50.05for the dotted line,h50.005 for the solid line, andh50.0005 for the dottedsolid line.

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Figures 10 and 11 show the evolution of the average perpen-dicular and the parallel scales forb andc, respectively. It isclear from the plots that average scale lengths forb aresmaller in comparison to those ofc. WhenB0 is finite thereis a greater disparity between the perpendicular and parallelscales ofb in comparison toc, with perpendicular scalelengths being shorter compared to parallel scales. Figure 10shows that in the presence of finite external magnetic fieldB0, ^k'

2 & for b shows steady rise initially. This tendency,however, does not continue forever. Rather, the curve subse-quently flattens out and later exhibits the same slowly decay-ing tendency as that of the parallel scales. However, a dis-tinct disparity between the perpendicular and parallel scalespersists.

IV. THEORETICAL DISCUSSION

We would now like to understand what causes the an-isotropy in the spectrum in the presence of external magneticfield, and why it is that it leads to a generation of short scalespreferentially in the perpendicular direction. It should bementioned at the outset, that unlike MHD, where the expla-nation for anisotropy is premised on an outright absence ofcertain kind of interactions amongst the Alfve´n waves, hereone can at best hope to provide an argument based on no-tions of more/less likelihood of certain interactions amongstwhistlers. This is primarily because the whistlers, unlikeAlfven waves, are dispersive in nature.

It is clearly evident from the numerical results of Secs. IIand III that the observed anisotropy and the whistlerizationof the spectrum are related issues. Both occur preferentiallyat short scales. In this section we describe how whistleriza-tion could possibly be responsible for the ensuing anisotropyin the spectrum in the presence of an external magnetic field.

For this purpose we first try to understand the nonlinear

interaction amongst the whistler waves. The EMHD equa-tions in thekde!1 limit can be written as

]B

]t1ve–¹B5B–¹ve . ~4!

Here ve52¹3B is the electron velocity. Linearization ofEq. ~4! in the presence of an external magnetic field in theydirection leads to the usual dispersion relation for whistlers,e.g., vk5skkkyB0 , along with the relationshipvek

52skkBk . Here sk561 and refers to the direction ofpropagation of thekth mode. Now, Fourier analyzing Eq.~4!leads to

]Bk

]t1(

k1

~vek2k1–ik1Bk1

2Bk2k1–ik1vek1

!50.

Substituting the linearized expression forvek2k1we obtain

]Bk

]t5(

k1

ik1–Bk2k1Bk1

~sk2k1uk2k1u2sk1

uk1u!.

According to Kolmogorov’s theory for fluid turbulence thetriad of wave vectors forming the sides of an equilateral tri-angle are best suited for efficient power cascade. Assumingthat this holds even for the case of EMHD turbulence, thecascade mechanism would entail interactions between neigh-boring scales. Thus the spectrum will be altered only bythose interactions which haveuk2k1u'uk1u. When uk2k1u'uk1u holds, the strength of the nonlinear interaction is weakif both sk2k1

andsk1are of the same sign, i.e., either both

are positive or both are negative. Thus, for copropagatingwaves, the nonlinearity is weak. It should be noted here, thatunlike Alfven waves, for whistlers the interaction betweencopropagating waves does not vanish identically, it is merelyweak, in comparison to counterpropagating waves~i.e., whenthe twos ’s correspond to opposite signs!. Thus the phenom-

FIG. 11. Evolution of the average perpendicular~solid line! and parallelscales~dotted line! for the variablec.

FIG. 10. Evolution of the average perpendicular~solid line! and parallelscales~dotted line! for the variableb.

577Phys. Plasmas, Vol. 7, No. 2, February 2000 Whistlerization and anisotropy in two-dimensional . . .

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enon of spectral wave cascade would be governed primarilyby nonlinear interactions amongst counterpropagating whis-tlers.

The matching conditions for the frequency and the wavevectors for such kind of interaction can be written as

6v35v12v2 ,

k35k11k2 .

Here the suffix 1 and 2 refer to the two interacting whistlerwaves which generates a third whistler wave referred by in-dex 3. Selecting positive sign beforev3 and using the dis-persion relation for whistlers we obtain

k3k3y5k1k1y2k2k2y .

Now, we eliminatek3y by usingk3y5k1y1k2y , which gives

k1y

k2y5

k31k2

k12k3.

Now since k1'k2'k3 , the above-mentioned equationshows thatk1y@k2y ~choosing a negative sign beforev3

would merely have resulted in the opposite inequality. Thisdoes not matter since 1 and 2 are merely dummy suffixes!.Thus, instead of havingk2y→0 as in Alfven turbulence wehave herek2y!k1y , which implies that there is very littlecascade to small scales in they direction. Cascade to smallscales thus can occur only along the perpendicular direction.This explains the initial increase of^kperp

2 &.The disparity in the average parallel and perpendicular

scales in the asymptotic state can be understood by the fol-lowing reasoning. Squaring the frequency matching condi-tion and elimination of the vectork3 leads to

k22k1y

2 1k12k2y

2 14k1y2 k2y

2

522k1yk2y@~k11k2!21~k1y2 1k2y

2 !#

22k1xk2x~k11k2!2. ~5!

The positivity of the left-hand side of the Eq.~5! requiresthat the two wave vectorsk1 andk2 have either both or oneof the components of opposite sign. However, the naturaltendency of power cascade in 2d EMHD turbulence is to-ward short scales. Furthermore, both whistlerization andanisotropization of the spectrum are short scale phenomena.This, thus excludes the possibility of having both compo-nents of the interacting wave vectors of opposite signs. Thecomparison of the coefficients of 2k1yk2y and 2k1xk2x ap-pearing in Eq.~5!, then suggests that during the initial stages,when the spectrum is isotropic, the permissible interactionswill predominantly havey components of the two interactingwave vectors as antialigned andx component as aligned.This results in preferential shortening of scales along thexdirection ~direction perpendicular to the applied magneticfield!, and hence explains the dynamic development of an-isotropy in the spectra. However, clearly this process is selfdefeating. Once a certain disparity between the average par-allel and perpendicular scales are developed, Eq.~5! can besatisfied by interactions amidst the wave vectors which leadto either the generation of short scales along parallel or per-pendicular directions with equal likelihood. It can then be

expected that there will be no preferential shortening of per-pendicular scales. The initial increase of^kx

2&, which laterstops growing, bears testimony to this scenario.

It is thus clear that the turbulent spectrum develops an-isotropy in the presence of an external magnetic field basi-cally due to a cascade mechanism mediated by whistler waveinteractions.

V. CONCLUSION

The role of normal modes of the system in determiningthe turbulent state and influencing the turbulent transportproperties are now increasingly recognized. For this purpose,we have taken the specific case of the electron magnetohy-drodynamics~EMHD! turbulence and investigated variousaspects of whistlerization~whistlers being the normal modeof EMHD! of its spectrum.

We have introduced novel numerical diagnostics toquantify the extent of the whistlerization of the spectrum,and conclude that whistlerization is only partial. This has adirect bearing on the turbulent diffusivity of long scale mag-netic fields. Recent works on quasilinear estimates of diffu-sivity have shown that the turbulent diffusivity would iden-tically vanish if turbulent state is comprised solely ofrandomly interacting normal modes.8,12 Our studies suggestthat the short scale part of the spectrum tends to be morewhistlerized, the long scales merely serving the purpose ofproviding an ambient magnetic field along which these nor-mal modes propagate. It will indeed be a worthwhile exer-cise to carry out such studies at higher resolutions than whathas been done here to draw conclusive and quantitative evi-dence of the phenomena of whistlerization at short scales, itsReynold’s number dependence, etc. We also observe that thespectrum gets quantitatively more whistlerized with increas-ing strength of the external magnetic field. This is expectedand is largely a consequence of reduced nonlinearity of thesystem in the presence of higherB0 .

There have been some recent attempts to invoke EMHDtime scales to provide an explanation of the fast reconnectionprocesses in the solar and astrophysical context.3 Observa-tional data on MHD scale fluctuations in solar wind indicatessignificant anisotropy.13 Spectral anisotropy in the presenceof long scale magnetic field in MHD has also been observednumerically and is being understood on the basis of a cas-cade mechanism governed by Alfve´n wave interactions.9,10 Itis thus of interest to see whether, in the context of EMHDalso, there is any dynamically generated spectral anisotropyin the presence of a mean magnetic field. Our numericalsimulation clearly shows that the spectrum develops anisot-ropy in the presence of a uniform field along a particulardirection. This can happen only if there is an asymmetry inthe nonlinear spectral transfer process relative to the meanmagnetic field. The only way nonlinear spectral transfer pro-cess could be asymmetric is if it is mediated by whistlers~asthe dispersion relation, phase, and group velocities of thesewaves depend on the direction of the mean magnetic field!.We have provided a theoretical explanation of the observedanisotropy on the basis of whistler wave interactions. This

578 Phys. Plasmas, Vol. 7, No. 2, February 2000 Dastgeer et al.

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confirms that whistler waves, the normal modes of EMHDmodel, play a crucial role in power cascade mechanism in aturbulent state.

Thus, to conclude we feel that ‘‘whistler effect’’~whis-tler wave interactions influencing the power cascade mecha-nism! is present in EMHD turbulence. However, the mani-festations of this effect is reflected largely in the subtleproperties of cascade like anisotropy, etc., rather than ongross property like spectral index~where the effect seems tobe either small or negligible!. In this context it is worthwhileto point out that a recent numerical work on MHD turbu-lence also seems to rule out any modification of the spectralindex due to the ‘‘Alfven effect,’’ which was erstwhilewidely believed to exist.14 However, the dynamical develop-ment of spectral anisotropy, mediated by Alfve´n waves in-teraction, in the presence of external magnetic field is wellestablished. Thus, it is apparent that though the normalmodes do indeed play a crucial role in power cascade mecha-nism, the ramification of this is not seen on gross propertieslike the power spectral index.

ACKNOWLEDGMENTS

We would like to thank the San Diego Supercomputercenter, a National Science Foundation~NSF! funded site ofthe National Partnership for Advanced Computational Infra-

structure~NPACI! for providing computing time on the T90supercomputer for this work. This research was partially sup-ported by Department of Energy~DOE! Grant No. DE-FG03-88ER-53275.

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579Phys. Plasmas, Vol. 7, No. 2, February 2000 Whistlerization and anisotropy in two-dimensional . . .

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