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Plasma Physics and Controlled Fusion, Vol. 27, No.4, pp. 433 to 445, 1985Printed in Great Britain.

0741-3335/85$3.00 + .00© 1985 Institute of Physics and Pergamon Press Ltd.

SPECTRUM OF TURBULENT ION ACOUSTIC WAVESIN A MAGNETIC FIELD

APARNA BASU, A, SURJALALSHARMA* and A, C DASPhysical Research Laboratory, Ahmedabad 380 009, India

(Received 3 November 1984)

Abstract-The perturbed orbit theory and the superposition principle of dressed particles are used to obtainthe saturated spectrum of turbulent ion acoustic waves, driven by an electron current along an externalmagnetic field in a plasma. For a typical set of parameters the spectrum is numerically computed. Theshape of the spectrum in k space, in w, and the angular spectrum are compared with observations andnumerical simulations.

L INTRODUCTIONTHE ION acoustic wave is one of the most widely observed plasma modes, and hasbeen studied extensively, The linear theory of these waves is well understood, butthe nonlinear and turbulent states are not fully understood with the existing theoriesof plasma turbulence, These theories need to be verified by comparison of predictedcharacteristics of the turbulent state with observations, The electric field fluctuationspectrum is one such characteristic that can be measured in the laboratory as wellas in space plasmas, Ion acoustic spectra have been studied in the laboratory forunmagnetized plasmas (YAMADAand RAETHER,1974; MASE and TSUKISHIMA,1975;SLUSHERet al., 1976, 1980; STENZEL,1978; KAWAIet al., 1978) as well as for magnetizedplasmas (GEKELMANand STENZEL,1978), Also spectral measurements have been madein space plasmas (GURNETT et al., 1978, 1979), Computer simulation of the currentdriven ion acoustic instability have yielded spectral information in the presence ofa magnetic field (BISKAMPand CHODURA, 1971; ISHIHARAand HIROSE, 1983) as wellas in the field free case (ISHIHARAand HIROSE, 1983),

The earliest theoretical calculation's of the ion acoustic spectrum (KADOMTSEV,1965)./ suffered from certain drawbacks, such as the low-k divergence, which were rectified"'by the addition of ion-neutral collisions (GALEEVand SAGDEEV,1973). These were

essentially quasilinear calculations applicable to weakly turbulent situations. Theinclusion of higher order perturbations to the particle motion, in a renormalized quasi-linear theory, removed the low-k divergence in collisionless plasmas (HORTONand CHOI,1979), The renormalized quasilinear theory of ion acoustic turbulence has been dealtwith in detail by the above authors, who have obtained the spectrum through a modesimulation technique, and have also reviewed the literature on spectral observationsand computer simulation results, All these calculations belong to a class of initialvalue problems, where the time evolution is studied by taking explicitly into accountthe feedback action of the oscillations on the single particle distribution functions,while the wave-wave interaction is treated perturbatively,

A second class of problems arises if connection to an external energy source is main-tained, Under such conditions, the system can 'go to a new equilibrium state, where

* Present Address: Plasma Physics Programme, Physical Research Laboratory, Ahmedabad 380009, India.

433

-434 A. BASU et al.

large fluctuating fields co-exist with linearly unstable particle distribution functionsand enhanced levels of particle diffusion. It is to this class of problems that the presentwork is addressed. Formal expressions for such stationary fluctuation spectra, forguiding-centre plasmas, have been obtained by COOK and TAYLOR(1973), while ionacoustic spectra in the absence of a magnetic field have been obtained in the hydro-dynamic limit by ICHIMARU(1973). Another calculation (SLEEPERet al., 1973) deter-mines the angular extent of the spectrum in the k plane. Since most experiments inturbulent heating, etc. are conducted in the present of a confining magnetic field, wehave examined the ion acoustic instability in a magnetic field in the low frequencylimit. The stationary turbulent spectrum of COOK and TAYLOR(1973) was obtainedby using the superposition of dressed particles. The turbulent dielectric function usedby them for "dressing" the particles was later modified by CATTO(1978). The formalresult of COOK and TAYLOR has not been used extensively to calculate turbulencespectra. We have examined in this paper, conditions under which the spectrum canbe explicitly calculated, and obtained an analytical form for the ion-acoustic spectrumin the presence of strong spatial diffusion. To obtain the dielectric function for theturbulent plasma, we have used the Dupree-Weinstock perturbed orbit formalism(DUPREE, 1966, 1967; WEINSTOCK,1968, 1969). The paper is organized as follows. InSection 2 we obtain the non-linear ion acoustic dispersion relation, and also solveit for the nonlinear frequency and growth rate. In Section 3, the turbulent diffusiontensor is derived and the relative importance of its different components for electronsand ions is examined. In Section 4, we obtain an analytic expression for the 3-dimensional ion acoustic spectrum S(k) in a magnetic field, and evaluate it for a typicalset of parameters. The frequency spectrum S(w), and the angular spectrum S(8) arealso obtained. The results are compared with earlier theoretical work, observations,and computer simulation results. The conclusions of the paper are summarized inSection 5.

2. ION ACOUSTIC WAVES IN A MAGNETIC FIELD

In a magnetic field B, the ion acoustic mode is split into the "fast" mode, whosefrequency w is greater than the ion cyclotron frequency Qi, and the "slow" mode withw ~ Qi. We consider here the slow mode in a fully ionized plasma, where both theelectrons and ions are hot. As usual, the thermal velocities VTj = (Tjjmj)1/2 tT, beingthe temperature in energy units and m, the mass of the ph species; j = e (electrons)or i (ions)) are such that VTi < c, < VTe, c, = (TejmY/2 being the ion acoustic speed.The electrons have an average drift Vo along the magnetic field, but the ions haveno directed velocity. The space averaged particle distribution function foiv) is athermally isotropic Maxwellian:

r.()_ 1 [_(V,,-voY+viJJO) v - (2 )3/2 exp 2 2

7WTj VTj(1)

where VOe = Do, VOi = 0 and II and ..1. refer to the directions parallel and perpendicularto B.

For perturbations of the type exp ( - iwt + ik ·x) the Vlasov-Poisson equations yield

-Spectrum of turbulent ion acoustic waves in a magnetic field 435

the linear dielectric function for electrostatic modes in a magnetic field:

where Wpj = (4nnjq;/mY/2 is the plasma frequency, In{A.j) the modified Bessel functionof order n with argument Aj = kip;; pj = VTj/0.j is the gyroradius, 0.j = IqjlB/mjc thegyrofrequency of the ph species, and Z(O the plasma dispersion function (FRIED andCONTE, 1961). In the limit W ~ 0.j, equation (2) reduces to

(k2 -2)

ReBdk, W) = A 1 _ ~~s (3)

and

I (k) _ (n)1/2[w - kllvo 1 {(W - kIlVO)2}mBL ,W - - 2 2 exp - 2 2

2 kllVTe k ADe k VTe

(4)

where ADj = (Ti/4nnje2)1/2 is the Debye length, A = (1 + k2 A&e + kip;)/k2 A&e, Cs =

c.(l - kipf)1/2/(1 + eA&e + kip;)1/2 and ps = cs/0.j; Cs is the speed of the slow mode.Equations (3H4) yield the linear frequency WL and growth rate YL:

(5)

Y.L =

where r = T'J'T; When YL is positive, the wave with frequency WL is driven unstableby the electron drift along Ba.

The linear dispersion relation was obtained, above, by using the usual method ofintegration along unperturbed trajectories, assuming the wave fields to be infinitesimal.In order to consider the effect of finite fields, as in the case of a turbulent plasma,the perturbation of particle orbits by these fields must be taken into account. Thiseffect becomes important close to resonance even for small values of the electric fieldand can lead to significant modification of the particle orbit. If the turbulence hasa short autocorrelation time rac = Ikil(w/k)I-1 compared to the particle trapping timerrr = (mi/ekE)1/2 the motion of a test particle is governed by a series of random orbitdeflections. The motion, averaged over a statistical set of background waves, is

-436 A. Bxsu et al.

diffusive. Consequently the sharp well defined trajectories in the linear case are replacedby average trajectories in the turbulent case. In other words the free particle propa-gator exp(iwt - ik .x) in the linear theory is replaced by exp(iwt - ik II v II t - kiD 1-t -k1-kIID1-lIt2 - kfrDt3/3) where D1-, DII and D1-11 are the components of the diffusiontensor D of the guiding centre in phase space. The use of this propagator in the calcula-tion of the dielectric function leads to a nonlinear dispersion relation. The same relationmay be obtained by replacing co by t» + i'1j where '1j = k· D· k, in the resonant termof the linear dispersion relation (CATTO, 1978):

Separating CNL into real and imaginary parts, and proceeding exactly as before weobtain, on assuming '1j ~ WNL, the nonlinear frequency WNL and growth rate YNL,

(8)

(9)

where Y~ is the linear growth rate YL, as given by equation (6), modified by the introduc-tion of the electron and ion diffusion terms, viz.,

(10)

The coupled equations (8) and (9) may be decoupled and solved for YNL ~ n, to giveWNL :::::kllcs(l - '1NkTIv}y/2. The wave frequency shifts as a result of scatteringoff the particles, while the resonance broadening term has a stabilizing effect. Thecontribution to YNL from n, is enhanced by the factor w~dkTIvL indicating that theions play the main role in the saturation mechanism where primary resonance withthe background spectrum is considered for both species. The ion temperature T, hasbeen assumed to be sufficiently high so that enough ions resonate with the backgroundwaves to make resonance broadening an effective saturation mechanism.

3. TURBULENT DIFFUSIONIn a background oflow freq uency turbulence, the guiding centre of a particle diffuses

in phase space. Assuming isotropy in the plane perpendicular to B, the threeindependent components of the diffusion tensor are the (E x B) spatial diffusion D 1-

across B, velocity space diffusion D II along B and a cross diffusion term D 1-11 in mixedvelocity and configuration space.

Spectrum of turbulent ion acoustic waves in a magnetic field 437

The correlation functions Jdt<xlr)xit + r» define the components of D in phasespace, where x can be the velocity or the acceleration; and

x(r) = LXkexP( -iwr + ik-Rtrj).k

(11)

The correlation must be evaluated along the orbit R(r) of the particle. Under stationaryconditions,

00

o; = f f dt<lxjxjlexp(ik·~R(t»)exp(iwt)

o

and its components are

and

(12)

where it has been assumed that the ensemble average exp(iwt)< IXkl2exp( - ik· ~R(t»)may be written as <lxkI2)I with the orbit integral I given by

00

1= f dtexp(i(W-kIlVII)t-kiD.lt-k.lkIlD.lllt2-~kTIDllt3). (13)

o

Also, use has been made of the expansion of the perturbed orbit <exp( - ik· ~R(t»)in terms of the cumulants of the electric field distribution, and cumulants greater than2 have been neglected (WEINSTOCK, 1968).

Before evaluating the components of the diffusion tensor, it is necessary to examinetheir relative importance, both for ions and electrons. In general, the diffusion proceedsalong different directions with different time scales, the dominant diffusion processamong them having the shortest time scale. The exponents in the perturbed orbit satisfythe ordering:

(14)

where t is some large multiple of the correlation time and e is the angle betweenk and B. If we take t ~ OJ-1, we find that for electrons, parallel diffusion dominatesfor waves propagating in the cone 0-88° about B. For ions, parallel diffusion dominatesup to ~ 37° and perpendicular diffusion between 37° and 90°. There is no regime

-438 A. BASU et al.

in which D .lll dominates over D IIand D.l. The level of turbulence also determines whichdiffusion process will be dominant. This is considered later in this section. We evaluatethe diffusion coefficients from equation (12) by neglecting the coupling terms. Eachcoefficient is determined in the region in which it is dominant.

(a) Parallel diffusion in velocity spaceFrom equations (12) and (13), we obtain,

(15)

where Ei3(rx) = J~ dtexp( _t3 - 3rxt) is Hardy's generalization of the Airy integral(WATSON,1966). Far from resonance, where (w - kIIVII)/(knDII/3)1/3 ~ 1,the asymptoticexpansion of Ei3 gives no orbit perturbation effects. For particles within the broadenedresonance (w - kIIVII)/(knDII/3)1/2 ~ 1, the series expansion of Ei3 may be taken.Assuming that the wave spectrum is peaked about the wavenumber k, we get

[(w/k V)2 J3/4D ~ 276 k (W/ )3/2 1 - 02 II - IIII ~. II m . (W/m) (16)

where W ~ <ecpk) is the turbulent wave energy. In this case of broadened resonancemost of the particles will fall within the width of the resonance and will be subjectedto turbulent diffusion. Taking D II ::::::2.76 k II (W/m)3/2, the level of turbulent energyrequired for resonance with the bulk of the ions is W ~ Te(l - ,-1/2)2 and for theelectrons W ~ T'; With, ~ 10, the first condition becomes W ~0.5 Ts, so that the ionsare affected more than the electrons by the resonance broadening effect. ForW ~ 0.1 Ti, under consideration in this paper, the parallel diffusion is not significantfor either electrons or ions.

(b) Turbulent diffusion across BEquations (12) and (13) yield (DUPREE, 1967),

D _ c2

~ E 2 k~P.l.l-2B2L.<1 HI)( -k )2+k4D2

k co IIVII .l.l(17)

In the limit of D.l ~ 0, equation (17) reduces to the quasilinear diffusion coefficientD.lq = (nc2 /B2)Lk< IEHI2) x b(w - klivll) which has the classical B-2 dependence andis peaked about the resonant velocity. In the limit of strong diffusion when Iw - k IIv III ~kl D.l, D.l ~ c# fiB, which is Bohm like (B - 1) and the independent of the particlemass and velocity. This non-resonant character of D has been used in the calculation ofthe spectrum in Section 4.

The bulk of the ions are within the broadened resonance when

(18)

-Spectrum of turbulent ion acoustic waves in a magnetic field 439

and for T ~ 10, this gives W ~ O.3k II Te/kipi. For the electrons a similar conditiongives

W ~ J?-kll (~)1/2 t:kl.Pi mer

(19)

Thus the perpendicular diffusion of the electron is not important. From equations(16), (18) and (19) we conclude that the diffusion of the ions across the magnetic fieldis dominant among the different diffusion processes.

4. STATIONARY SPECTRUM

The spectrum of fluctuations in a quiescent plasma may be expressed in terms ofthe plasma dielectric function by using a superposition principle of dressed particles(ROSTOKER, 1961; HUBBARD, 1961; COOK and TAYLOR, 1973). Consequently theproblem of determination of turbulent spectra is effectively reduced to one of findingthe correct dielectric function in the turbulent state. With the nonlinear dielectric func-tion CNL obtained in Section 2, the turbulent spectrum is given by

(20)

A similar expression for the spectrum, obtained by ICHIMARU(1973), reduces toequation (20), if mode coupling terms are neglected. It may be noted that CNL is ingeneral a function of the particle velocity since it contains the resonant diffusion D.However, in the limit of strong diffusion, D becomes insensitive to the particle velocityas discussed in Section 3, and the integration can be performed treating D as a constant.From equations (7), (8) and (9), we can write, taking W = co, + iw;,

(21)

where c.c. stands for complex conjugate. With Wo = WNL - iYNLas a solution of thenonlinear dispersion relation, we get,

1 = _1_[1 + 2 Real{ N - _N_*_}]ICNdk,wW A2 Wr-WO Wr+Wo

(22)

where

2( 2 2 2) 4N = Wo Wo + YNL + YNL

2Wo(W6 + W62)

and an asterisk indicates the complex conjugate of that quantity.Substituting equation (22) in equation (20) and assuming that the space averaged

r

-440 A. BASU et al.

distributionji., is given by equation (1), the ion contribution to the fluctuation spectrumis obtained as

(23)

Similarly the contribution of the drifting electrons is

= 1 + 2 Real {NeZ(fi" kIIVO) + NeZ(Ji kIIVO)} (24)2kllvTe 2kllVTe

where u., = N/fikllvTe,i.Combining equations (23) and (24) for a charge neutral plasma we obtain the fluctua-

tion spectrum,

The spectrum, normalized to the electron thermal energy is

Equation (26) gives the spectrum of electric field fluctuations of turbulent ionacoustic waves in equilibrium with a steady electron drift Vo. This is the main resultof our paper.

For a quiescent plasma without an electron drift or magnetic field, we recover thethermal acoustic fluctuation spectrum from equation (26),

(27)

In equation (26), as YNL tends to zero, i.e. the pole of 8NL approaches the real axisfrom below, the spectrum diverges, which indicates that the state can no longer bestationary. This situation is the exact nonlinear analogue of the onset of instability

-Spectrum of turbulent ion acoustic waves in a magnetic field 441

in the linear regime described by COOK and TAYLOR(1973). 'YNL gives the growth rateof perturbations about the turbulent state, and if 'YNL = 0 all perturbations remainindefinitely in that state. It is important to note, therefore, that 'YNL = 0 does not givea condition for stationarity. For the state to be stationary, 1mB must be sufficientlyfar below the real axis to ensure that perturbations about the turbulent state decayback to the same state faster than a certain rate, which must be determined fromthe self-consistent solution of equations (20) and (17).

The spectrum S(k) has been computed from equation (26) over a range of k valuesas a function of angle fJ between k and the magnetic field B. For a typical set ofion-acoustic wave parameters, viz., r = 10, Vo = 100 Cs, Wpi = nJ2, turbulence levelW/Te ~ 0.1 and diffusion coefficient D/CsAD = 0.035, the computed spectrum normal-ized to the electron thermal energy is shown in Fig. 1. For these parameters, thespectrum peaks at kAD = 0.4 at fJ = 0°. For larger angles, the peak value falls whilethe peak shifts to lower values of k. It may be noted that the spectrum goes smoothlyto zero at k = 0 unlike earlier theoretical spectra (KADOMTSEV,1965; NISHIKAWAandWu, 1969). The spurious divergence at low k is avoided in all renormalized theorieswhere the screening of the particles is properly taken into account (HORTONand CHOI,1979). S(kll) follows a power law in kll with the index ~3.54 that is the same for allfJ as shown in Fig. 2.

The angular spectrum S(fJ) = S '; k2dkS(k fJ) and the frequency spectrum S(w) =S '; edk S sin fJdfJS(k)b(w - j(k)), where w = f(k) is the nonlinear dispersion relation, are

S 6-2

XIO

5

4

3

2

Uci /,:",pi = 2

Te I Tj = 10

W/nTe =0·[

U / C S = ice

B-

FIG. I.-The electric field fluctuation spectrum S (normalized to the electron thermalenergy) in the (k, e) plane, where e is the angle between k and B. (Dle,AD = 0.035.

!leo/wp; = 2, ToIT; = 10, WlnTe = 0.1, voles = 100)

-442 A. BASU et al.

la' • ar- 3'~5A ar-3·57)( ar" 3·50El ar=3·54

x '{0 .I a \ Is

F

102 '\

FIG. 2.~Log S YS log kll for 8 = 5° (0), 25° (6), 45° (x), 65° (0). The falling partof the spectrum follows a power law S ~ k~a with o.~ 3.54 ± 0.03 (!1,,/wp; = 2, T,/T; = 10,

W/nT, = 0.1, Vo/c, = 100).

1·0

I, 90 70 50 30 10 30 50 70 90

FIG. 3.~Angular spectrum S(8) = S k2dkS(k, 8) (!1,;/wp; = 2, T,/T; = 10, W/nT, = 0.1,Vo/c, = 100).

10

shown in Figs. (3) and (4). As shown in Fig. 5, Sew) follows a power law in w withexponent - 2.61. In the angular spectrum shown in Fig. 3, the intensity falls to halfits peak value between e = 40 and 45°. The shape of S(k) is consistent with the modesimulation results of HORTON and CHOI (1979), and See) with the calculations ofSLEEPERet al. (1973) based on the perturbed orbit theory.

In most experiments and simulations (GEKELMANand STENZEL,1978; BISKAMPandCHODURA,1971; ISHIHARAand HIROSE,1983), turbulence levels are lower and the ionsare unmagnetized (wpj ~ OJ), in contrast to our assumption OJ = 2wpj. In additionsome of the features observed by them may be due to cyclotron modes with n > 1,which have not been considered here. The anomalous k spectrum, dominated by waves

-Spectrum of turbulent ion acoustic waves in a magnetic field 443

0·2

0·6

0·5

f 0-4

~0·3(/)

0·1

O·O~~~ __ -L __ ~ __ ~ __ ~ __ ~L- __ ~ __ -L~

0·0 0·1 0·3 0-4 0·5 0·6 0·7 o-a

FIG. 4---Frequency spectrum S(w) = J k2dk JsinOdOS(k, O)li(w - f(k)) where w = f(k) is thenonlinear dispersion relation. (Qc;fWpi = 2, TelTi = 10, WlnTe = 0.1, Vole = 100).

FIG. 5.-The spectrum S(w) (Fig. 4), follows a power law with exponent -2.62. (Q,JWpi = 2,TelTi = 10, WlnTe = 0.1, Vo/Cs = 100).

propagating at oblique angles to B, appears to be one such feature. This will be dis-cussed in a forthcoming paper.

In the low frequency limit, w ~ ne;, the slow acoustic wave is essentially similarto the ion acoustic wave in an unmagnetized plasma, except for a change in velocityand increased dispersion. However, in the nonlinear regime the magnetic field playsan important role. As discussed in Section 3, it is the turbulent diffusion across themagnetic field that dominates over the parallel diffusion.

Since the spectra presented in this paper are for a specific set of plasma parameters,a direct comparison with experimental results cannot be made. However, it may benoted that the shape of the ion acoustic frequency spectra measured by STENZEL(1978)and MOSTOVYCHand DESILVA(1984) in the laboratory, and by GURNETTand FRANK(1978) in the solar wind, are consistent with the one shown in Fig. 4. The index ofthe power law in S(w) obtained is - 2.6 (Fig. 5).PP 27:4-F

-444 A. BASU et al.

The angular spectrum obtained by us (Fig. 3), has the same form as measured angularspectra (YAMADAand RAETHER,1974; MASE and TSUKISHIMA,1975). The energy ispeaked along the direction of the electron drift. This also agrees with the short timebehaviour of See) obtained by computer simulation (BISKAMPand CHODURA, 1973).

5. CONCLUSIONS

We have studied the spectrum of turbulent ion-acoustic waves in a plasma, in amagnetic field, consisting of drifting electrons and background ions. In the linear phase,the drifting electrons feed energy into the waves and the ions take away energy fromit by Landau damping. There is, however, a net gain of energy by the waves, thusleading to an instability. As the waves grow, they nonlinearly perturb the particleorbits, leading to energy loss to the particles, while they continue to gain energy fromthe electron drift and be Landau damped by the ions. We have considered here anonlinearly stable state as a net result of these physical processes and the spectrumis obtained by using the principle of superposition of dressed particles. The nonlineareffect considered here is the perturbation of the particle orbits by the turbulent electricfields and is incorporated in the process of "dressing" the particles. This orbit perturba-tion is represented by a diffusion process and the appropriate diffusion coefficientsand the nonlinear dispersion function are derived from renormalized plasma tur-bulence theory. The spectrum can be obtained explicitly when the resonant natureof the wave particle interaction is smeared out by resonance broadening. This requiresa sufficiently high level of turbulence. In this limit the spectrum is expressed in termsof the plasma dispersion function and is a complicated function of the wave frequencyand wavenumber. For a set of plasma parameters the spectrum has been computed.

The earlier works on ion-acoustic turbulence (HORTON and CHOI, 1979;RUDAKOVand TSYTOVICH,1971) take the ion temperature to be low, so that the ionsresonate dominantly with the low phase velocity beat waves. In this paper we takethe ions to be hot and consider their nonlinear interaction with the primary waves.Moreover, we have considered here the ion-acoustic turbulence in the presence ofa magnetic field, whereas the earlier works pertain to the magnetic field-free case.

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