J. Plasma Fusion Res. SERIES, Vol.2 (1999) 121-124

Reactively Driven Drift Modes in Sheared FlowvneNfeSJovo*, JOVANOvIC Duian and SHUKLA padma K.r
Institute of Physics, P.O. Box 57, Yu-l 1001 Belgrade, Yugoslavia.
llnstitut
fiir Theoretische Phlsik, Fakultiit fiiur Physik undAstronomie, Ruhr Universittit Bochum, D-44780 Bochum, Germany.
(Received: 18 December 1998 / Accepted: 3 June 1999)

Abstract
Equations describing drift waves in a strongly inhomogeneous plasma with a sheared flow and asheared magnetic field are derived. New stabilization criteria are obtained in the linear regime. In thenonlinear regime, it is found that the governing equations admit stationary solutions in the form oftripolar vortex and vortex chains that are driven by the equilibrium plasma flow and magnetic shear. The
tripolar vortex obtained in this paper is surprisingly similar to corresponding structures in twodimensional flows in ordinary fluids.

Keywords:
drift waves, sheared flow, tripolar vortices, vortex chains.

1. Introduction

a sheared magnetic field, keeping the terms which
describe the modification of the ion polarization drift
associated with the equilibrium flow velocity. New

Magnetic field shear has a strong stabilizing effecton drift waves, especially for exponential density
profiles when they are completely stabilized. Thisstabilization criteria are obtained in the linear regime. In
the nonlinear regime, it is found that the corresponding

stabilization follows as a result of the convection of thewave energy from the mode rational surface towards theion Landau damping layer. In realistic geometries,

governing equations admit stationary solutions in theform of a tripolar vortex and vortex chain.

inhomogeneities along the magnetic field lead totoroidal coupling which may destabilize drift waves by acomplete elimination of the shear effects. Standard
approach to the problem of shear stabilization of drift
waves is to treat the diamagnetic drift velocity y* as aconstant, which is hardly satisfied in the edge of atokamak plasma. In configurations of this type, with asteep density gradient and radially sheared transverse
velocity field, the magnetic shear stabilization criteria

2. Basic Equations and Derivations
We study low-frequency electrostatic perturbations,d/0t << O1, of a collisionless, spatially nonuniform
plasma, no = no(x), with a sheared flow, rio =vo(x)6,- ,p_laced in a sheared magnetic field of the form

Bo=i,Bo +aJBo(d/dx)/(.r), where l(d/dx)"f(x)l << l.
We take ions to be cold, and the electron temperature
gradient effects are neglected.

The nonlinear ion continuity and the parallel
motion equations describing the system [2] can be

are severely restricted, and the velocity profile curvaturevf,(,r) is found to play a crucial role. In this paper, wefirst derive linear equations for drift waves in astrongly
inhomogeneous plasma with a sheared plasma flow andwritten as@ I 999 by The Japan Society of Plasma*C o rre spondin g autho r' s e - mail : v ranj e s @ phy. b g. ac. y u

Science and Nuclear Fusion Researcht21

Vranjel J. et al., Reactively Driven Drift Modes in Sheared Flowfunction along the x axis for.tr J @, with the distance6(x) between its zeros going to 6o = tt/"'l t at large -r.
Thus, in this case we have an oscillatory, and possibly
stabilized solution along the x-axis, without any explicit

xVr(@*urr) Orl[#.;up2V2)(O[(l -+ BoQ)- Bo(p + Y)l+Lv, ii,=e,dependence on the magnetic shear. On the other hand,when the above inequality is reversed, the solution is(1)unstable. Assuming that Vs = V.o, we obtain thefollowing condition for the oscillatory solution ft;r >* . * d,xY ,1@* r. r) o,][6v,- Qif@))=0.

,E/(LT -i5. where obviously L3 > L? . According to
the experimental data L* = L, = I cmi consequently, fora stabilized drift wave, we must have lL2l > I cm.

Q)

On the other hand, for a quadratic magnetic shear

Here, we have introduced the stream function rp of thezero-order shear flow vs(;) = dEldx, and also v*e= dV

determined by f'(x) = x2lL2s, we have the following limit

dx = - n6c?lnoQi.

for F(.r):

In the linear limit, combining Eqs. (l)-(2), weobtain:.t'

14. potlo=14-ti+---l' a+bx+cx'
ld*'

Ild*'n, llaoJ'(x)-lf
t=0,lo,*ur*+c,x2+'+bx+cx'll|

ao=k2y# a=l-rr, U =-?
4r=-r,L', ,2i i+ uo*=>ki+l.
Lorv] C t 'Lo, an(3), ll L.\ a" L:k"'>ll. l-*+'-ll' l\Lsl
Yn L'.
tlLi|L:h')vo.kyH-r.,.

p'and, in order to simulate conditions typical for a Hmode plasma [3,4], we modeled the unperturbed profilesof the radial and diamagnetic velocities in the form"\L' Lil-.//I\,andIL,A)

Integrating Eq. (2) we obtain

6v, = Qlf(x)+ F(@ + BoE - uBox).

@)

\(8)

Here, .f is an arbitrary function of the given argument,
which we adopt it in the linear form I = F ' (@ + tp a.r), where Fis an arbitrary constant. From the conditionof vanishing perturbations for x -+ 6, we have

For the linear profile of the magnetic shear,/'(x) - xlLt,
we obtain
I\zcr-c'''Ithe unperturbed gradients, i.e., 0l0t = -u0l3y.

and/ -' \%o(x)=Y.'|1-a;1.
L:-JI|

However, for typical experimental values of the above
quantities [3,41, i.e., P = O.l cm, l,5 = 70 cm, and Ln =L* = | cm, the shear term in Eq. (7) is much smaller
than the others, and the stability condition is practicallythe same as for the linear magnetic shear.
In the nonlinear regime, we look for travelling
solutions of Eqs. (1), (2), which are stationary in thereference frame moving with a constant velocity u
almost perpendicularly to the magnetic field lines andL r.r.*un(x)=%lr* +.{1.

(6)l-lVo' Li\Lsl =+=>1.

Here we used the following normalization

k,\4IL,\ ci

bt=-b, ,r=\-+o*-*,V.owhich for Vo = V*o = p ctlLn yields
I| / \4l-uz
1. t'zT+v.o+vo,L;-l11-obtained ifvo
c --(s)An oscillatory, and thus stabilized solution for / > 0, istt

I=4lgo-v(y-1')--l\l,:,:,\ ti- t'.I ' L;I,=

II"tl'5v, - F ' iD, and Atf@) =- FQ@)+ Fux. (9)For / > 0, which is equivalent to LllL? > (kj + DVoIV-o

any nontrivial solution of Eq. (3) is an oscillatory

By using Eq. (9), we may integrate Eq. (1), yieldingt22

VranjeS J. et aI., Reactively Driven Drift Modes in Sheared Flowf

Vl- tl @ + 9"(x) + Y(x) + f(x)= g IA + e@)- uxl,,.(**,.fr,t",)(10)

where we have set p'Y' -+ Y2, Boe --> tp, Bsu -) u,
BsY - Y, and T.FB6 f / e -+ f. Here, Q(() is also an arbitrary function of the given argument, and we take it inthe form 9G) = G6 + G1 . f. Such simple expressions
for the functions .7 and f are possible if the unperturbed=^[(f -h)",.',fr*,]Ii r<,rt Kl^
11 2"

quantities cp(x), Y(x) and/(x) satisfy\n;land, consequently, Eq. (10) can be rewritten in the form:l,t -h)"-',,,](V1- l) Qqrcrx2 =Go + G, ((D+rcrxz). (12)^[;Equation (12) will be solved separately inside andoutside of a circle with an arbitrary radius 16, allowing
for different values of Gfl"f and G[n1 outside and insideof the vortex core, on condition of localized solutions
for r --r -. In the cylindrical co-ordinates r,O,Eq. (12)

t2K,.

Ktri

A,=-L
K22

in the vicinity of the first zero of the Bessel function "/2,giving L2 = 2.68. This yields the following values forthe above constants: fo = -53.85, Fz = -25.87, ao =5.75, ar= -7.46, A = 1.28, and B = -0.6. A typicalprofile of the tripolar vortex, given by Eqs. (13), (14),corresponding to these numerical values, is presented inFigs l. and 2.
Equation (10) admits also several types of vortex
chain solutions. Here we shall present only one. For theshear flow in the form of an asymmetric function aroundthe velocity a, given by v6("r) = u * AoKtanh(rx), and in
the case Y(x) + f(x) - 0, taking the function Q in the- B+2-rl
laotr(12)- Ailcos2e. (14)|'lHere, Gfiot = 0, Giut = KzlKr and Ke.2, Js.2arc modified

Bessel, and Bessel functions of the given order,
respectively, and

11=t +ci"'=l+*,(r7).

As an illustration, we take 11 = l, Kz = I and ro= 2, andsolve the nonlinear dispersion equation (17) numerically

(13)

|J,"

K,rni
"K2 =(K, -A) r":" J2 +2A.

BoKo(lt) + BrKrQ"rr) cos20,

0S- aoJoQrr)- ll

(16)

"2

separates variables, and its solution can be written in the<Din 1r,ry*,,,B =doJn- loK^-A+,

formr)ts,(15)dt=--B,(,,*;*{ r; *.j=QG)- ya = K1x2, e"(x)+ V(x)+ f(x)= Krx2, (11)@ou' (t, 0) =t<,-A jrA,-J22

L'z, +

73=-l _ c'i.

K,+K'A-rr+-+,
L;

form:

I, (":",*",\ l.
B=_i:l--+Kr+1412"\^;' l9 @ + e - ux)= AoK2 exp l- ? r* + e - ux)|,l^oIlEq. (10) can be rewritten in the following way:

We use the following physically justified continuity
conditions at r = ro, which allows for determining theunknown constants in Eqs. (13) and (14): continuity of(vl- t @ - /(x) ("-'- r) =0,y(18)where /(x) = 8K2l(e2*' * ,-2rx + 2). An equation of thetype of Eq. (18) has been derived and solved earlier inour paper [5], where we studied the problem of self-generation of the magnetic field in a plasma with a shear
flow, thus for a completely different physical problem
with different spatial and time scales. Following thesame procedure we may write @(r, y) = @r(r) + d@(*)

the potential @, yielding finiteness of the electric field atr - 16, continuity of (dldr)<D, yielding the absence of
surface charges at r = ro, continuity of the function f({)
to avoid singularities of the starting equation. The argument of the function 0G) rs constant on the separatrix r- re. The above continuity conditions yield:t23

Vranjei J. et al., Reactively Driven Drift Modes in Sheared Flow(a)350-3-4

40x02

Fig. 1 Contour plots of the tripolar vortex describing thex

potential O(r, 0l for r, = 1, r, -- 1, ro = 2, Bo =-53.85, gz = -25.87 , do = 5.75, dz = -7 .46, A = 1.28,
B = -0.6.

(b)

>r00X

*Fig. 2 Three dimensional view of the potential from Fig.

Fig. 3 The sketch of the single (a) and double (b) vortex

1.

chain solution.

cos(t"y), where l6@(x)l << l@'(x)1, and we find two
types of localized solutions, in the form of single anddouble vortex chains presented in Fig. 3, which are

we have solved the appropriate equations using thestructures localized in the x-direction, and periodic in y-structure consists of the monopolar, and quadrupolar

direction.

part (resulting from the x2 terms by which thestandard vortex scenario, and the formation of a tripolar
vortex and vortex chain is presented. The tripolar vortex

unperturbed state is modeled). For some specific profilesof the functions E(x), ty(x), and /(;) describing the

3. Summary and Conclusions
We have studied the behavior of linear andunperturbed state, the vortex chain solutions can also befound [2]. Finally, it should be emphasized that thenonlinear solutions presented here are analogous to thewell-known structures in ordinary fluids, and we believethat they may arise as the final stage in the developmentof instability of drift waves in both laboratory, and spaceplasma configurations of this type.

nonlinear drift waves in strongly inhomogeneous
systems with sheared plasma flows. By a careful
inclusion of the shear flow effects in corresponding

equations describing drift waves in such plasma
systems, the stabilization criteria are substantiallychanged. For the given density profile and sheared
plasma flows, the linear magnetic field shear has noeffects on the behavior of drift waves; it is the sheared
flow whose influence is essential, while the role of themagnetic shear is much smaller. It enters into thestability conditions provided that its profile can beapproximated by a quadratic function. Yet, for typical

References

lllc.J.F. van Heijst et al., L Fluid Mech. 225, 3Ol(199r).

[2] J. Vranje5, Phys. Scripta 59,23O (1999).
t3l K. lda et al., Phys. Rev. Lett. 65, 1364 (1990).
[4] S. Sen e/ a/., Nucl. Fusion 34,87 (1994).
[5] J. VranjeS and D. Jovanovi6, Phys. Plasmas 3,2275

parameters in the H-mode of a tokamak plasma, theinfluence of the magnetic shear on the stability of drift
waves is insignificant. In the strongly nonlinear regime,(1996).

t24