Stability of vortex flows in magnetized plasmas. I. NonBoltzmann vorticesNikhil Chakrabarti, Amita Das, Predhiman Kaw, and Raghvendra Singh Citation: Physics of Plasmas (1994-present) 2, 3296 (1995); doi: 10.1063/1.871164 View online: http://dx.doi.org/10.1063/1.871164 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/2/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A general framework for non-Boltzmann Monte Carlo sampling J. Chem. Phys. 124, 054116 (2006); 10.1063/1.2165188 Experimental detection of rotational non-Boltzmann distribution in supersonic free molecular nitrogen flows Phys. Fluids 17, 117103 (2005); 10.1063/1.2130752 Generalized simulated tempering realized on expanded ensembles of non-Boltzmann weights J. Chem. Phys. 121, 5590 (2004); 10.1063/1.1786578 Two-dimensional coherent structures of drift waves exhibiting non-Boltzmann relationships in toroidalplasmas Phys. Plasmas 5, 381 (1998); 10.1063/1.872720 Stability of vortex flows in magnetized plasmas. II. Boltzmann vortices Phys. Plasmas 3, 4360 (1996); 10.1063/1.872052

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Stability of vortex flows in magnetized plasmas. I. Non-Boltzmann vortices Nikhil Chakrabarti. Amita Das, Predhiman Kaw, and Raghvendra Singh Institute for Plasma Research, Bhat, Gandhinagar-382 424, India

(Received 3 January 1995; accepted 26 April 1995)

In this paper the stability of long scale vortex flows to secondary perturbations with short parallel wavelength is analyzed. It is demonstrated that flow ellipticity can drive secondary instabilities of ion-acoustic and shear Alfven waves provided certain resonance conditions between the rotation speed and the Doppler shifted secondary wave frequency are satisfied. Such secondary processes can act as a major sink of energy and may help in our understanding of nonlinear saturation of low frequency instabilities. © 1995 American Institute of Physics.

I. INTRODUCTION

It is widely believed that low-frequency instabilities (w~ wei) in a magnetically confined plasma are responsible for the observed anomalous transport. Recent theoretical work on drift, Rayleigh-Taylor, Kelvin-Helmholtz and other such instabilities indicate that often polarization-drift nonlinearities dominate and lead to an accumulation of en­ergy at long scales.' The nonlinear state may thus be highly turbulent or may contain a number of coherent, long-lived and nearly two dimensional vortex structures.2,3 In order to investigate a new mechanism of energy dissipation in large amplitude, long scale vortices, we consider in this paper, the possible excitation of secondary instabilities with a short par­allel wavelength. The secondary wave may be ion-acoustic or a shear Alfven wave, which on leaving the vortex region may lose its energy by parallel Landau damping to ions or to electrons. As a specific example, we study a model vortex equilibrium with finite ellipticity at the core and study its stability to dispersive ion-acoustic and shear-Alfven pertur­bations. Our calculations demonstrate that the ellipticity of the vortex acts as a free energy source for instabilities and attempts to parametrically drive secondary waves when the Doppler shifted secondary wave frequency matches the mean rotation frequency of the vortex or one of its multiples. Physically, this free energy source is related to velocity shear in the elliptical vortex flow. In the case of ion-acoustic and Alfven waves, instabilitiy is explicitly demonstrated and the estimates of the growth rates are obtained.

The stability of two dimensional hydrodynamic vortices to three dimensional perturbations was numerically investi­gated for the first time by Pierrehumbert.4 He demonstrated instability of a broad class of two dimensional vortices and showed that the eigenmode is concentrated near the centre of the vortex. An elegant analytic method, using the techniques of Floquet theory, was developed by Bayly5 to give a math­ematical interpretation of the above results. In plasma phys­ics, the stability of ion-temperature gradient vortices was dis­cussed by Cowley et at.;6 however, they considered only the long, thin eddy approximation, which reduces the calculation essentially to a one-dimensional eigenvalue problem. The two-dimensional plasma vortices and their stability were dis­cussed in a preliminary manner in some earlier reports.7

•S

The present paper gives a detailed discussion of the stability

of two-dimensional vortices or pseudo-two-dimensional vor­tices in a plasma.

The organization of this paper is as follows: In section II, a formulation of the equilibrium problem is presented con­sidering an elliptically rotating vortex flow. In the next sec­tion, the basic equations governing the electrostatic perturba­tions are written and a complete stability analysis is presented for the case when the eqUilibrium vortex is of the non-Boltzmann type. In section IV, we have analyzed the stability of vortex solutions to Alfven wave perturbations. In the stability analyses we have employed a mathematical technique similar to that of Bayly.5 Our equations can be cast into the form of a Hill's equation. The stability has been analytically discussed in the limits of small and large ellip­ticity. We have also obtained the estimate of growth rates for small ellipticity using multiple scale analysis. An extensive numerical study has been done to obtain the instability do­main for arbitrary values of eccentricity (€) of the vortex. Section V provides a brief discussion of the results and their relevence to nonlinear theories of low frequency turbulence in magnetized plasmas.

II. BASIC EQUILIBRIUM

In this section we formulate the eqUilibrium vortex solu­tions which are formed from processes involving non­Boltzmann electrons. For this study we considered a colli­sionless inhomogeneous plasma in an external magnetic field (B = B oz) with warm electrons and cold ions in a slab geom­etry. We also make a simplifying assumption that time scale involved is larger than ion cyclotron time-which allows us to use the guiding centre or the drift approximation of ve­locities. The basic equations governing an electrostatic 2-D equilibrium vortex are the ion continuity equation

d:rO -V1.'[; V1.$o+(ZxVcI>o· V)V1.cI>o] =0,

and the vorticity equation

iJ 2 A

at v 1. cI>0+ V 1. ·[(zX V$o' V)V 1. cI>0]=0,

(1)

(2)

where dldt=(iJlat+ Vi.' V) and No. cI>0 are eqUilibrium density and potential respectively. The perpendicular veloc­ity can be written as V1. =zxV1>o+(alat+zXV1>o'V) (- V 1. 1>0)'

3296 Phys. Plasmas 2 (9), September 1995 1070-664Xi95J2(9)/3296f6f$6.00 © 1995 American Institute of Physics

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Several authors I ,2.9 have obtained vortex like solutions for the nonlinear equations describing Kelvin-Helmholtz waves,· Rayleigh-Taylor waves, etc. where the electrons do not satisfy a Boltzmann distribution. We regard these vorti­ces as the basic equilibrium and study the excitation of sec­ondary instabilities in a localized region around the core. Around the elliptic vortex core, we may approximate the dimensionless potential1O

-12 ¢o=e<PoITe as

n (x2 ) ¢o= 2-;+ Ey2 , (3)

where n=c<pol BWci'=c(<Pol BcsLl.)(asILl.) = (VEXB/cs) X (as ILL), Ll. is the perpendicular scale length of the vor­tex. Note that for ExB flow velocities much less than c and s LJJas'l> 1, the parameter n is a small parameter, n~l. The velocity flow generated by this potential may be written

V1.=n(~Y-Cxx)-n2(xY-YX), (4)

where n(~l) has been used as an expansion parameter. The first term represents the ExB flow and the second term the polarization drift flow. Since E~ I and n~l, the polarization drift teml produces negligible effects as compared to ExB flow terms; in our subsequent analysis, contribution due to polarization drift flows in the equilibrium are therefore ne­glected. It may readily be verified that the vortex core solu­tion Eq. (3) exactly satisfied the eqUilibrium Eqs. (1) and (2) for homogeneous magnetized plasmas (VNo=O).

The following points may be emphasized for this equi­librium model: (a) It is the simplest distorted eqUilibrium around the o-point of a vortex and ignores the higher order distortions like triangularity of constant potential surfaces [X3,y3 terms in Eq. (3)]; (b) the eqUilibrium may, in prin­ciple, have a variation along the field line; however, as long as the secondary -waves are much shorter in parallel wave­length, we may ignore the parallel variation of equilibrium; (c) the eqUilibrium vortex may in general be propagating with a finite uniform velocity in the y-z plane; however for our calculation, we go to the frame of this vortex where this velocity may be taken as zero.

III. INSTABILITY OF ELLIPTIC VORTICES WITH ELECTROSTATIC PERTURBATIONS

In this section we study the stability of elliptic vortices described above to electrostatic perturbations. In the stability analysis we assume that the perturbations to have a suffi­ciently low frequency W and short parallel wavelength kill such that wi kllv the ~ 1; this allows us to use the Boltzmann relation for electrons viz. iie=Noexp(e¢ITe)' where the tilde denotes the perturbations. Physically this means that for such perturbations electron inertia is negligible and the high ther­mal conductivity of electron suppress the temperature fluc­tuations justify the Boltzmann relation. A closed system of two equations, describing the evolutions of electrostatic po­tential and parallel ion velocity can be obtained by combin­ing the ion continuity equation with the quasineutrality con­dition and writing the parallel component of ion velocity. The linearized equations are given by

Phys. Plasmas, Vol. 2, No.9, September 1995

( :t +~ Z X V ¢o' V ) ¢ I - V 1. • [ ( :t + Z X V ¢o . V ) V 1. ¢l

+(ZXV¢I),VVl.¢O]+VIIVll=O, (Sa)

(:t +ZXV¢O,V)VII=-Vll¢l' (5b)

To derive the above equations the following normalizations are made: All co-ordinate variables i.e. x,Y,z(r) are scaled b~ 'as', the ion larmor radius at electron temperature (T,,) [glVen by the ratio of sound speed (c s ) to ion cyclotron frequency (wcJ]. Time variables are scaled by cyclotron . -1

tIme w· and all velocities are scaled by sound speed csC= ~T>mi)' Total potential (equilibrium plus perturbed) is scaled by Te Ie. The notations used are as follows:

¢1 =e¢ITe' ¢o=e<PoITe, n(=iiINo, vII=vll/cs> t=twci, r=rlas and the operator V1. =ialax+yalay, VII= al az. Eqs. (5) are linear partial differential equations with space dependent coefficients entering through ¢o(x,y) and its derivatives. We now use Bayly's5 method to solve this set of equations. In this method one eliminates the space dependent terms coming from the equilibrium flows by as­suming the wave vector to be an explicit function of time. Assuming perturbations of the form 11 =11 (t)exp(ik· r) (i.e. fourier analyzing in space but not in time) and taking the wave vector k=k(t), one may eliminate the eqUilibrium flow terms by adjusting the explicit time dependence of k1. . The above equations may then be replaced by

2 . • . (1 +k1.)¢1 + 2k1. ·k1.¢1 +zkllvll=O,

vII+ ikll¢1 =0,

where the time dependence of k x' ky is given as:

. n kx + -;- ky=O,

(6a)

(6b)

(7a)

(7b)

Rescaling the time by the vortex rotation time n -1, Eqs. (7) may be solved to obtain

kx=kocos(t+ 80),

ky = ckosin( t + 80 ),

(8a)

(8b)

Here ko=kassina, a is the angle between wave vector and z axis and 80 is the initial phase. We are now left with wave l~e equation for the perturbation VII and ¢1 in which perpen­dIcular wave vectors are periodic functions of time. We may now combine Eqs. (6) and the above solutions to derive a single equation for the perturbation variable 'It=(l+ki)¢1

d2'1t ( 1 )

--;--'J" + 'It - ° dr a+bcos2r· - , (9)

where r=t+ 80 , a=(nlkIICs )2[1 +k6(l +(:2)/2] and b=(nlkIICs )2[k&C1- c2)/2]. Note that the basic equations above have been transformed into a second order ordinary

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14.0

1.29 r....:::~..:---

O.O~ ______________ ~ ____________ ~ 1.0 5.0

€ 9.0

FIG. J. Plot of klTc;fil2 versus € (ellipticity of the vortex) for ko=O.3. The shaded region is the unstable domain. The slope of the lower curve ap­proaches a constant value 4hr2 for high €.

differential equation in one variable qr, which has the gen­eral character of the standard Hill-equation and can be solved by the method of Floquet theoryP

We first consider the case with 15= I Le. the case of a rigidly rotating circular vortex flow. In this case b == 0 so that the 'cos27' term disappears from Eq. (9), giving the disper­sion relation w 2 =k[c;/(l+k6) of the standard dispersive ion-acoustic waves. It is thus clear that, instabilities, if any, will be driven by (I - (2)cos27 term in Eq. (9), i.e. due to the terms which arise from the finite ellipticity of the basic flow.

To look for growing solutions, we now express Eq. (9) in the standard Hill form l3

:; +[ eo+2 X,I emcos2m7}qr=0, (10)

where

eo k and em=eo( l~;oeor The general solution of Eq. (10) has the Floquet form l3

qr = exp(A 7)Lnanexp(2in7), where A is characteristic expo­nent and determines the growth rate. A can be obtained from the equation

Sin2e~A) ==Dx(0)Sin2( 1T~); (11)

where Dx(O) is the determinant of the infinite dimensional Hill's matrix evaluated at A=O. The elements of the matrix are given by the expression C mm = 1 and Cmn = - em _ n /(4m 2 - eo), where both m and n take the val­ues from -00 to 00.

For arbitrary values of 15 and k~C;/02 the characteristic exponent equation [Eq. (11)] has to be solved numerically. Fig. I gives a plot showing the unstable domain (shaded

3298 Phys. Plasmas, Vol. 2, No.9, September 1995

region) in the parameter space defined by [kUc;/02,E]. From Fig. 1 it is clear that for 15= 1, the mode is unstable only near the resonant matching condition kllcs=O. As 15 is in­creased, the unstable domain widens. For very large value of 15(15;;;;.10), the lower stability boundary goes asymptotically to a straight line. In our numerical investigation we have also explored the possiblity of exciting instabilities of higher or­der resonances for ion-acoustic wave viz. (k11csfO)2/( I +k6( 1 + ( 2»/2=n(n=2,3, ... ). However it is found that eo and em are not independent and that there is hardly any regime of parameter space where a significant instability of this type may exist.

In realistic situations this growth rate of the secondary instability will have to overcome the natural Landau damp­ing of the ion-acoustic wave due to wave paricle interaction. However if its wavelength is few times the ion Debye length, the Landau damping rate becomes exponentially small and can be overcome easily. Thus we find that an elliptic vortex flow can drive ion-acoustic waves unstable in a broad region of parameter space. These secondary waves can leave the vortex region and get damped thereby acting as sinks of en­ergy for the equilibrium vortex. We now propose some ana­lytic calculations to demonstrate the growth of ion-acoustic waves explicitly for the cases when (1 - (52 ) is small or large.

Let us first examine the behaviour of the solution of Eq. (9) in the limit of weak ellipticity when the parameter /3= (1- (2)/[ (2Ik6) + (1 + (52)], can be treated as an expan­sion parameter. In this limit we solve Eq. (9) by the mUltiple time scale method,14 introducing a new variable 71 = /3r. The formal procedure consist of assuming a perturbation expan­sion of the form

Substituting for qr in Eq. (9) and equating equal powers of /3 gives:

a2qto (kllc s IO)2 aT2 + 1 +k6(l + (52)/2 qro=O, (l2a)

Now when k[c;=02[1 +k6(l + (52)/2] which is the condi­

tion for the resonant matching of the vortex frequency and the secondary wave frequency, we have the zeroth order so­lution of Eq. (12a) as, qro(r,TI)=A(Tt)COS7+B(7r)sinT, where the constants of integration A and B are taken as func­tions of 71' This dependence is determined by substituting for qr 0 in (12b) and requiring that the reSUlting equation have secularity free solutions. This condition leads to the equa­tions

dA B dB A ---=0 d7t 2

and ---=0 d7t 2

(13)

which imply A.B~exp(7112). Thus we conclude that for small /3. the amplitude of the secondary instability grows exponentially with growth rate - /30. The vortex ellipticity

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is therefore shown to be responsible for the growth of ion­acoustic waves at a rate determined by the vortex frequency o and the ellipticity parameter f3.

We now present the details of analysis for large E. When ~ 1, all the Om's are of the order liE and Eq. (10) takes the form

d2

'1J1' [00. ] d'?+ 7T80lo 8( r-m7T)'l"=O. (14)

This equation describes an oscillatory motion with decayingl growing amplitude for stable/unstable solutions. The critical value of 00 representing the boundary of the stable and un­stable regions can be determined by demanding the ampli­tude of the oscillations to be constant. This can be sought simply by expressing the above equation as two dimensional map in terms of the variable Xn and Yn which represent d'lJl'ldr and 'IJI' at r=n7T as follows:

-7T80 ) (xn-t) (xn-l) 2 ==A ,

1-7T 00 Yn-t .... Yn-l and then seeking the second order fixed point of the map. This implies that the required condition is obtained by setting the determinant of the matrix A 2 as unity. The critical value of 00 is then obtained as, Oc=4/7T2.

The condition 00= Oc=4/7T2 may be rewritten as kHc;lko(k5+ 1) 1!202=4 E/7T2, which is indeed the asymp­totic (large E) slope observed for the lower stability bound­ary in the numerical result (Fig. 1). This result may also be seen directly by applying the Floquet theory to Eq. (14). The Hill matrix now takes the form such that the diagonal ele­ments obey, C mm = I, as before, but the non-diagonal ele­ments obey, Cmn = Oo/(Oo-4n2

). The analytic expression for the determinarit of such a matrix is 15

.. 00 ( 4n2 )2 7T200 .'( 7T ffo) D",(O)= 11 Oo-4n2 =-4- cosec - -2- .

The marginal stability boundary can be found by solvingJ~q. (11) for AR=O,A[= I glVlng the condition D",,(0)sin2( 7Tffo/2) = 1. Substituting the expression for Doo(O) in this equation we get 00=4/7T2 in conformity with our earlier derivation. Therefore we conclude that elliptic vortex flow is responsible for exciting secondary ion­acoustic waves for both small and large ellipticity. ,

IV. INSTABILITY OF ELLIPTIC VORTICES WITH ELECTROMAGNETIC PERTURBATIONS

We now go on to study the stability of elliptic vortices to shear Alfven perturbations. In this case also, we consider collisionless low-beta (f3T==c;IV~<l) plasma in a slab ge­ometry where the magnetic field has a fluctuation over its equilibrium value viz. B=Bo+ B l . The field perturbations Bl may be calculated from the vector potential A from the relation Bl = V xA. In this case also time scales are much larger than ion cyclotron time, therefore we use the drift approximation of ion and electron velocities (ion velocities are given by ExB and polarization drift whereas electrons are moving with ExB velocity only). Note also, in this case

Phys. Plasmas, Vol. 2, No.9, September 1995

electrons will not follow the Boltzmann distribu!ion for per- . turbations because of the finite vector potential fluctuation (All #= 0) in parallel direction. To get the modified nl - cPl relationship we solve the parallel Ohm's law given by

ne[ EII+ ~(VeXB)u] = - VUPe' (15)

where EII=-VUcPl-(lIc)BAII/Bt, Ve , Pe are the parallel electric field, velocity and pressure for electrons. The above equation now gives a relation between ii, ¢,AII' To eliminate A II we use quasineutrality condition (V· J = 0) and parallel component of Ampere's law (V XB)U=(47T/C)JII which may again be written as [V (V· A-Vi A) ]11 = (47T/ c )JII' It may be mentioned that, we have neglected the term containing perpendicular vector potential A.L' Since we are considering the low-beta plasma, field compression effects are ignorable. Now out of two variables nl ,cPl, one of them can be elimi­nated using continuity equation for ions and neglecting the parallel motion of ions. In neglecting parallel ion motion we have assumed (kllcs<kIIVA), VA is Alfven wave speed. This assumption is again justified since we are considering low­beta plasma.

Therefore the basic governing linearized equations for the evolutions are the ion continuity equation, parallel Ohm's law, and the combination of quasi neutrality condition and Ampere's law in a nondimensional·form viz.

anI [a at + i x V cPo' V n 1 - V.L· at V 1- cP 1 + (i x V cPo' V)

V 1. cPt + (ix V CPI)' vv .L <Po] = 0, (16a)

BAil A at +zxV 4>0' VAII= V II (nl - 4>t>, (16b)

B V 1-' at V.L cPl + V.L· [(iXV cPo"V)V ,lcPI +(£xV cPl

(16c)

where the symbols have their usual meanings as in the pre­vious section and AII""'(eAllITe)(cslc), V'i,=B21 (47Tm inO)'

We now use Bayly's method to. solve these equation to see whether the Alfven perturbations grow or not. Proceed­ing exactly in the same manner as the previous case we get the equation for perturbed potential as

d2 1/1 [. 2Ccos2r 3 C2(1-cos4r) H - (I+Ccos27) +2: (I+Ccos2r)2

-w'i,{ 1+ ~5(1+E2)(1+f,'COS27)} ]1/1=0, (17)

where

'J (Of,'sin27 ) ( 1- E2) 1/I=kiexp 1+ f,'cos2r cP, C= 1 +) ,

7=t+OO, wA=kIIVA/O.

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12.0 r---------,------=~~

9.0

FIG. 2. Plot of k[ V~ID. 2 versus E (ellipticity of the vortex) for ko =0.2. The shaded region is the unstable domain.

For ~ = 0, this gives us the standard Alfven wave dispersion relation w2=kffV~(l +k6). Note that ~=O corresponds to E= 1 viz. a circularly symmetric vortex. Thus the possiblity of an instability exists only when E is not equal to one, i.e. an elliptic vortex.

For arbitrary ~ this equation may be analyzed by stan­dard method of Floquet theory as discussed earlier. We have solved the equation by above method numerically for fixed values of ko and arbitrary values of E. The plot of domains of instability in the w~ - E space is given in Fig. 2. Note that bands of instability originate (when E"'" I) near the resonant frequencies wi = 1,4,9 ... , etc.; the deviations from exact inte­ger relationships arise due to the finite ko corrections. When E is large, the bands of instability broaden significantly. We have also plotted the dotted line corresponding to the condi­tion 00=0 where 00=w~-EI2-1/2E+ 1 is the constant term of the Floquet form [similar to Eq. (10)] of the Alfven wave equation (Eq. (I 7». This is because the domain corre­sponding to 00 <0 is always unstable irrespective of the value of Om; in the figure this corresponds to the shaded region below the dotted line. We note from the figure that for high values of E, this unstable domain smoothly merges with the unstable domain of higher resonances.

We next analyze Eq. (I7) in the limit when ~ is small using the method of multiple time scales. 14 The basic result is that rfJ grows exponentially with a growth rate =0.5~n( I + kffV~), when the zeroth order frequency resonantly matches the rotation frequency 0 (Le. 0 2 = kIT V~[ 1 + k6(l + E2)/2 J). Thus the vortex ellipticity also is responsible for the growth of shear Alfven wave per­turbations at a rate determined by the vortex frequency 0 and the ellipticity parameter ,.

V. DISCUSSION AND CONCLUSION

We have investigated the stability of an elliptical vortex core to secondary perturbations. The basic physical mecha-

3300 Phys. Plasmas, Vol. 2, No.9, September 1995

nism for the instability presented in this paper may be under­stood in terms of a parametrically driven oscillator. In the frame rotating with the mean rigid .angular velocity of the fluid, the waves have a time dependent wave vector kJ. (t) whose dependence arises through the ellipticity parameter (1 - E2); this can resonantly drive the ion-acoustic and AI­fven waves provided certain resonance conditions are satis­fied. For large deviation from circularity, off resonance driv­ing is also possible as seen from the stability diagram (Figs. 1 and 2). It may be noted that the flow is stable for E= 1 which corresponds to a rigidly rotating circular flow. Thus the free energy for this instability is essentially coming from the ellipticity parameter associated with the flow. Once the instability is excited, we expect the vortex to copiously radi­ate secondary waves, which on leaving the vortex region get damped by natural damping mechanisms. In this manner, long scale vortices can shed energy and there need be no accumulation of energy at the long wavelength end.

We now speculate on the possible relevance of the sec­ondary instability excitation mechanism to describe the sta­tionary turbulent state in a driven magnetized plasma. As is well known, one of the puzzles of modern nonlinear theories of low-frequency magnetized plasma turbulence is that con­ventional cascade mechanisms transfer energy towards long scales where the natural linearized damping mechanisms are negligible. In a driven turbulent plasma where energy is pumped into a band of k's either by an instability mechanism (or equivalently by external stirring), one may thus not get a stationary state because the energy is transferred by cascade mechanisms to the longest scale and accumulates indefinitely in them. It seems clear that when the energy in long scale becomes large, it must find a nonlinear mechanism of dissi­pation to give stationary state (note that the usual linear vis­cous damping is negligible at these scales). We may adopt our point of view that beyond a threshold amplitude, the long scale vortex begins to radiate energy in the form of short scale secondary waves (a process we illustrate by a linear­ized instability calculation) which propagate out of the vor­tex and are then damped in the plasma by conventional linear mechanisms (like ion viscosity, parallel ion Landau damping etc.). Thus it is very difficult to accumulate more energy than the threshold amplitude in the long scale vortex. Similar 'modulational instability' actuated transfer of energy from long to short scales and resulting 'nonlinear saturation' of long scales has indeed been observed in strong electron plasma wave turbulence. 16 One could argue that the second­ary instability mechanism discussed in this paper may only cause the vortices to shed their ellipticity and become non­linear circular vortices. However, it must be emphasized that the well-known and typical long scale solutions (like the dipole vortex solution, etc.) have elliptic vortex cores. So the circular vortices will be driven towards elliptic fonn by non­linear cascade effects pumping energy into them. We thus believe that there is a competition between cascade processes pumping energy and ellipticity into long scale vortex solu­tions and secondary instability processes extracting elliptic­ity and energy by wave radiation. This competition can in­deed lead to a saturated spectrum of vortices at the long scales, even in a driven turbulent plasma.

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We now make an estimate of the vortex amplitude re­quired for the phenomenon of secondary instability excita­tion discussed in this paper. A rough estimate can be obtained by using the resonance condition, e.g. for sound wave exci­tation, the condition O,=kllcs ' A more realistic estimate is obtained by ensuring that the secondary wave growth rate exceeds the natural damping rate of the waves due to viscous and/or ion Landau damping effects. For heavily damped shortwave length modes the damping rate 'Yd=kllc s and the threshold condition is again n;c;kllcs (because the growth rate for E:$ 1 is of order 0,). We may now re-express the condition in terms of threshold vortex amplitude given by e¢t1JTe-(mlqR)(LxLy las) where L."Ly are the radial and poloidal scale lengths of the primary vortex, m is the poloi­dal mode number of the secondary wave, 'as' is the Larmor radius, R is the major radius of the tokamak and q is the safety factor. Similarly, in the case of shear Alfven wave excitation, the threshold vortex amplitude requirement is ecf>tlJTe-(mlqR)(LxLyias)f3T -112, where f3T is the toroi­dal f3 of the plasma; Taking the typical parameters at the tokamak edge, we find that fluctuation levels observed are sufficiently large to generate secondary instabilities (espe­cially of ion-acoustic type) discussed above. We may thus argue that one possible mechanism for the observed nonlin­ear saturation of fluctuations which are dominated by inverse cascade could be the excitation of secondary instabilities, when certain critical amplitudes are exceeded. It is also con­ceivable that the vortices are getting formed by conventional inverse cascade mechanism at random locations and times and their destruction by secondary instability excita-

Phys. Plasmas, Vol. 2, No.9, September 1995

tion can account for intermittency in low-frequency magne­tized plasma turbulence observed in experiments17 and com­puter simulations. 18

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