REFERENCE IC/66/98

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

THE EFFECT OF MAGNETIC SHEARON PLASMA INSTABILITIES

W. E. DRUMMONDA N D

K. W. GENTLE

1966

PIAZZA OBERDAN

TRIESTE

IC/66/98

International Atomic Energy Agency

UTTEEIIATIOHAL CENTRE FOR THEORETICAL'PHYSICS

THE EFFECT OP MAGNETIC SHEAR ON PLASMA IN STABILITIES

*

. DBUMMOMD*

and

K.W. GEETLE*

TRIESTE

August 1966

* ?'ermanent addresses: Department of Physics, University of Texas,Austin, Texas,'

A B S T R A C T

A new formulation of the effects of magnetic shear on a

plasma is developed. The linearized Vlasov equation is solved

in the natural co-ordinate system that shears with the magnetic

field. Instead of an algebraic dispersion relation for waves,

a differential equation in k-space results. The solutions of

this equation must be explored to determine effects of shear on

plasma waves. An approximate solution is obtained for the drift-

wave instability and the criterion for shear stabilization is

calculated. The criterion is close to previous results,

but the nature of the stable mode is quite different.

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TICS EFFECT OP MAGNETIC SH3AR OH PLASMA INSTABILITIES

Introduction

In the past several years , a considerable number of micro-

i n s t a b i l i t i e s driven by gradients or r e s i s t i v i t y have been found in

plasmas. The calculat ions have a lso indicated that a sheared magnetic

f ie ld can s t ab i l i ze or reduce these low-frequency i n s t a b i l i t i e s /~"l-4_7"-

Examining r e s i s t i ve i n s t a b i l i t i e s with f luid equations, ROBERTS and

TAYLOR Z~\_/ found unstable modes that are modified by shear but not

en t i r e ly s t a b i l i z e d . In cont ras t , HOSEITBLUTH, C05RIV an (^ others ^f~l-3_7",

working d i r e c t l y from the Vlasov equation, find that shear s t ab i l i z e s

the unstable modes quite generally (excepting the temperature gradient

i n s t a b i l i t i e s ) . However, they do not solve the Ylasov equation d i rec t ly

for waves in the presence of shear . The equation i s solved in the

absence of shear, and the solut ions are then modified in physically

plausible ways to account for the shear . Impl ic i t ly or e x p l i c i t l y ,

the approach assumes wave modes with 1c fixed in spaoe. As the f ie ld

shears, the magnitude of' k)( changes, and the effect of shear i s

deduced from the effect of the changed \ x on the wave. I t i s not

clear whether t h i s approach allows twis t ing modes where 1c ro ta tes with

_5 5 these twis t ing modes were the type found to be unstable "by Eobsrts

and Taylor. To es tab l i sh the efficacy of shear s t ab i l i z a t i on , i t must

bo shown that shear s t a b i l i z e s a l l possible unstable modes of the unwant-

ed type, not jus t a subset having an assumed form.

A straightforward approach to the question of shear s t a b i l i -

zation i s to transform the Vlasov equation to the sheared co-ordinate

system. Tlie shear enters the theory na tura l ly and accurately, and i t s

effects may be investigated without making any r e s t r i c t i v e assumptions

about the so lu t ions . We have solved the l inearized Vlasov equation in

a co-ordinate system that shears as B and obtained the equation for

waves in such a system.

A d i f f e ren t i a l equation in ic for the wave po ten t i a l replaces

the famil iar a lgebraic dispersion r e l a t i o n . As an example of the conse-

quences, we have calculated an approximate solution for the density-gradient

d r i f t wave, which has a phase veloci ty along_B between the electron and

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ion thermal velocities. The shear required to stabilise this wave

is approximately the saaic as that calculated previously /7"̂ _7j "DU"''

the physical mechanism scorns quite different.

Solution of the linearized Vlasov eauatioii

~;lo wish to transform co-ordinates from X, Y, Z to a sot of natural

co-ordinates X, *\, '» S i n ̂ i 0 ^ "^ie magnetic field is constant and

always along the "?-a.xis. The transformation is pictured in Fig. 1.

and specified "by

\=. Y c*s fcu) \ i s m §^ (i)

!=->( sm GCX) + 2. cos

Expressing all vectors and variables in these new co-ordinates, the

Vlasov equation assumes the form

;-];ere P =̂ "^ /^x — ^ ^ denotes the shear parameter. The form

of the equation has changed "because *./i.x is not the invariant form

of the gradient operator. Following the usual procedure, we write

"V- Tj+'f with the perturbation T, expressed as the

Pourior integral )) ^ ^ \ V^) ̂ \ j ^ ( \ ^ ^ v \ k\&$ . Confining

ourselves to low frequencies where the electric field -is electrostatic,

we have ^ x •*" "^ T where f̂ is also represented as a Fo-arier

i-ate^ral like "f̂ . The components of 3 must be calculated in X, Y, 7, co-

and then transformed "because the gradient operator is not simple in the

new co-ordinates. The. resulting linearized equation for -f. (Xf t<tj, k W

can. be written in the general foru

= V ( i^ k O

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T

In spite of the additional terms, an equilibrium solution

of the ordinary Vlasov equation is also a solution of the unperturbed

equation 0-̂ . ̂ Ot = O here if £»(x,0 is obtained by replao-

ing ^K\=t ky V X ^ J . The new T,, can be written

as

(4)

where t — v^n) *^A* ~ ^%-i. ', if necessary, a gravitational force

in the x direction oan be added exactly as in the shear-free case.

Substituting this form of fc in the expression for C in (3) and

dropping terms of order £ ' , we can rewrite C in a very convenient

form:

(5)Eq.(3) is formally more complicated than the usual Vlasov problem,

which is integrated in a form with c^ — Q f but it possesses a

simple integral:

(6)

With expression (5) for C(t), this equation can be integrated by parts

to give

c— 8.

For the usual J£ time dependence for perturbed quantities, this

equation becomes

(8)

Tho equation for \̂ in the absence of shear can "be cast in a form

identical to this. The integrals proceed formally in the same way,but the procedure can no longer "be visualized simply as an integral

along unperturbed orbits. The k vector in (8) is also a function

of t. The new trajectories for the integration are defined by the

following aquations:

N ^ S ^ (9)

The first two equations are the familiar ones for the unperturbed

orbitSj hut the second two are new and are an explicit consequence

of the shear. In general, they present a major complication in

integrating Eq.(8).

Bqs.(9) ma;.' ha integrated to determine the time dependence of

X , V, and V :

•where V̂ * (Y***^"!.] and O" = +1 (for ions), — 1 (for electrons).

These expressions are to "be, substituted in (8) and the time integrals

performed. With k left in this form, the integrations are intraotable,

"but for most waves of interest, the wave number along the field, "^ ,is

much smaller than the wave number across the field, Nw .(In the usual

notation for the unsheared oase, +•%.<** v^y .) In this approximation,

the last two equations of (10) may be expanded to give

-5-

The integration of (8) would now be straightforward -were it not that

\ is \\Y V ^ V w ) , where both x and K.- are functions

of t . A convenient technique is to expand l in a Taylor series,

which can be written formally as an exponential;

3}

Using this, (8) can be rewritten in standard form as

- ̂ CM* + S ̂ >/% ^ X Y L * ^ ) (13)

where T-^'--t and *b = V^V^ ^ + <r£ v

Each of the five sine functions of ^ must be expanded in a Bessel

series. The general sum is formidable, but we are interested only in

tho limit of UJ <<. SL- . Eliminating terms proportional to \w/JL

links the indices of the first and third terms as p and -p. After

the time integration, the fifth term makes contributions that decrease

like ( S/k^j( V^O J in each higher order. We shall assume that

this is much less than one, an assumption less restrictive than the

snail LarmoT radius limit of K ̂ 0 <<-L . Then only the vfo term

from the fifth sine term need be kept . Performing the average over

T , which now connects the heretofore independent second and fourth

sine term, one is left with the following sums and integrals:

where 0 = -K

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The two siims are identical and can be found analytically by use of

Bessel function identities. Completing the sums and integrating

over time leaves

This is the solution of the linearized Vlasov equation in the presence

of shear.

Equation for waves in the plasma

To calculate dispersion relations or differential equations

for waves, the density must be found. This requires integrating (15)

over V x and V- . The integration over Vl̂ can be performed in the

usual manner if the dependence in the denominator is ignored. We shall

ignore it and replace Vj_ by vj_ there. The justification for this

approach is that we shall find no corrections arising from this terra;

the result is unaffected by the approximation we use for it. Completing

the v integration, we have an expression for the perturbed density n :

(16)

The z here is a function of V« , but the correction iB always quite

small and will be ignored. Therefore a may be evaluated at V^=o and

brought outside the integral. The resulting expression still differs

from the customary integral for ths W function in having a branch line

instead of a simple pole. However, the branch points are close together,

and the branch line can be chosen to connect them. The function is

specified by requiring that it behaves like the usual pole at distance

far from the branch points. Careful examination of the function has

T

shown that it is virtually equal to the 17 function; corrections to

either real or imaginary parts are of order •(S/k.j) (_ \i.

a parameter we have already assumed snail. Regardless of the magni-

tude,the corrections will only change the quantitative results, not

the qualitative "behaviour of the function. This calculation yields

an expression for the perturbed density

; (IT)

V.'X ^tn * \ "SV̂ )J wiwhere now t = ̂ V.'X ^ t n * \ SV̂ )J witli ^ as Larmor radius.

This is formally identical to the result derived previously for p = O

"but the actual equations will differ significantly "because we now

have a differential operator instead of an algebraic terra in K̂ .

Functions of operators are to be interpreted as their power series

expansions.

The equation for the wave modes is obtained by substituting

the density perturbations of the two species in Pcisson's equation;

a differential equation for the potential ^f results. The solutions

of this equation must "be examined in each regime in which instabilities

have been found to determine if sufficient shear will remove the

unstable solutions. ;Je shall examine the equation for the drift-wave

(universal) . instability to determine the criterion for shear s tabi l i -

sation and the physical nature of the stable wave.

Tiro drift-wave

The drift-wave is characterised by a phase velocity intermediate

between electron and ion thermal velocities. For algebraic convenience,

we take ">^ — \ "N . The Debye length is assumed quite short, and

hence poisson's equation reduces to the requirement of charge neutrality.

The equation for the potential is

where \[ K =

-8

This is a complex differential equation in X and *"\* However, the

equation in x is similar to the equation in the absence of shear

the solutions will be either waves propagating slowly along x or non-localized standing waves in x , depending upon boundary conditions. In

either case, 1 can be Fourier analyzed in x to replace ^A% by Vk.y ,

which will be neglected compared with V.̂ . We are primarily interest-

ed in the V.«* term, for i t carries the shear effects. An equation in

k-space appears rather unphysical, but this equation can be interpreted

as the Fourier transform of a wave equation in 5 -space that has a TJICB

approximate solution. If ^ l ^ ) is taken as the Fourier transform

of a WKB solution and Bq.(l8) is then transformed back to 3 , one is

led to an equation for VC- (. ̂ ^ J of the >IKB solution:

(19)

lie have assumed a small LarmoT radius case to obtain a simple equation.

In this limit, iL4 - ^ and only the lowest order term in ^ . ' ^ ^P'U+fl t J

is retained. For the drift-wave, ^ i . can be replaced by i t s large

argument form and \AJ^ by i t s small argument expansion. The solution for

^ 1 is a wave that grows as i t propagates toward larger ^ . However,

as ^ and hence z increase, V.»e increases, causing the phase velocity

to drop. After a sufficient distance, the phase velocity approaches the

ion thermal velocity and the wave Landau-damps on the ions. The criterion

for shear stabilization in that the wave grow no more than some tolerable

number of e-folds, N, , before damping. The criterion is evaluated by

solving (19) for the imaginary part of ^ and integrating i t from the

origin to the point where damping begins. Values of **> and ^ v l - ^ J

consistent with our approximations are chosen to give the largest growth.

The requirement on the shear length for stability is

(The solution of (19) must Ibe approximated to evaluate the integral

for total growth. An alternate approximation, probably less accurate,

replaces \M/m) 5 by v* ^*\) V LH C^V*,) , giving a result near-

ly identical to that of COPPi/~3_7". The numerical results are not

significantly different.)

The physical picture of these waves is closer to the picture

given by EGBERTS and TAILOR /~4_7f or the resistive instability modes

than to the usual picture of the drift-wave. The drift-wave is general-

ly unbounded in 2 , but Landau-damps in x away from its origin.

The mode discussed here is bounded in z f but is rather arbitrary in

x ; it twists with the magnetic field along the x axis and never damps

for any x . It is quite remarkable that the shear criterion (20)

derived for this mode shall match the one derived for a physically

different wave mode that damps along x .

Conclusion

The equations through (17) and (18) are quite general and

can be used to determine the nature of any low-frequency wave in the

presence of moderate shear. This is an accurate and natural way to

introduce shear in the equations, and the treatment requires no ad hoc

terms or incomplete co-ordinate transformations to add the effects of

shear. The solution of the equations has been illustrated for one wave,

and the same general methods may be applied to determine the effects

of shear on other unstable waves.

ACKNOWLEDC&1EHTS

We wish to acknowledge the kind hospitality of Professor

Abdus Salam and the IAEA, aA the International Centre for Theoretical

Physics, Trieste. This work was supported in part by the United

States Atomio Energy Commission.

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REFERENCES

M JT .ROSENBIOTH, " M i c r o i n s t a b i l i t i e e " , p .485 , Plasma

Physios , IAEA, Vienna, 1965.

£~Z_J B.COPPI, ana M JT.ROSSHBLUTH, Plasma Physics and Control-

led Nuclear Fusion Research, Vol. I , p.6l7» IAEA, Vienna,

1966.

/ ~ 3 _ / B.COPPI, G.LAVAL, RJ?ELLAT, and M JI.ROSEHBLUTH, "Qonvective

Density Gradient Drift I n s t ab i l i t i e s " , Phys.Pluida, ( to be

published),

/~4_7 K.V.ROBERTS, and J.B.TATL0R, Phys.Fluide, ,8 , 315 (1965).

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ig. 1

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