ANOMALOUS ELECTRON HEAT-TRANSPORT DRIVEN BY LOW-FREQUENCY ELECTROMAGNETIC TURBULENCE

VOLUME 59, NUMBER 13

PHYSICAL REVIEW

LETTERS

28 SEPTEMBER1987

Anomalous

Electron

Heat Transport

Driven

by

Low-Frequency

Electromagnetic

Turbulence

A. A.Thoul,

P. L.

Similon, and

R.

N. Sudan

I

aboratory

of

Plasma Studies, Cornell University, Ithaca, New York 14853 (Received 5August 1987)

We consider the anomalous electron and heat transport in a tokamak plasma. The electrons are de-scribed by the nonlinear drift-kinetic equation. We analyze transport through the averaged response function in the presence ofdrift-Alfven wave turbulence. In contrast to recent findings by Terry, Dia-mond, and Hahm, we conclude that magnetic fluctuations lead toasubstantial transport ofboth parallel and perpendicular energies. The latter, previously neglected, is found to be significant and ofthe order ofthe test-particle diflusion.

PACS numbers: 52.35.ka, 47.25.Jn, 52.35.Kt, 52.35.Qz

Observations in large tokamak machines' have shown

that electron heat transport is a major energy-loss mech-anism, and is 2 orders ofmagnitude larger than what the classical diflusion theory predicts. It is believed that magnetic turbulence is an important candidate to explain anomalous transport in atokamak plasma.

Terry, Diamond, and Hahrn have recently attempted to improve the quasilinear models and have included in their study the self-consistency of the particle evolution and the fields. Constraints appear through

quasineutral-ity and Ampere's law. They reach the conclusion that the magnetic fluctuations do not contribute to the

relaxa-tion of the average electron distribution function, and

therefore do not contribute to the radial transport ofany

velocity moment.

In this Letter we show that while electron transport is indeed determined by ion transport because of quasineu-trality, there can nevertheless be a significant electron parallel and perpendicular heat transport, in contrast to the conclusion of Ref.

3.

Electrons difluse because of resonant processes such as Compton scattering. Because of the resonance condition v~~«,

=

(Lo/k~~) &&v, , where

L,

=

T/m„

these particles have small parallel energy. They create self-consistent electromagnetic fields in response to which other electrons are exchanged to

main-tain quasineutrality. However, these other electrons may have vll of order v,, because they are not constrained to be resonant. This explains how, in spite ofambipolarity, there can be transport of electron parallel energy. The reasoning above applies equally for the transport of per-pendicular energy. It will be shown that the particle

dy-namics is independent of v&,. therefore the transport of perpendicular energy is of the order of the test-particle

diflusion.

We consider a slab geometry with density and temper-ature gradients, and without magnetic shear. The mag-netic field is along the z direction, the gradients are

along the (radial)

x

direction, and _y is the (poloidal)

ig-norable coordinate. The electrons are governed by the drift-kinetic equation in which we keep the nonlinear

E

x

_B

drift, nonlinear magnetic flutter, and nonlinear parallel acceleration, but we neglect the dift'usion in

ve-locity space. The ion dynamics is linearized and depends on the electrostatic potential p only, with a constant of proportionality v. The ion current is neglected, and

non-linear ion response is not considered here. The plasma is thus described by the drift-kinetic equation, the quasineutrality condition, Ampere's law, and the relation

between the perturbed ion density n; and the electrostat-ic potential 4t, in Fourier representation k= (k,eo):

e "'ll

t(co k~~L'p)hp

—

t(co cow

)f

p( 'L~~) Fp(vz )

—

_p &~~

=

g

wk._t, p

—

A~[ ht,-,

k'+k" =k c Jr

d'v

ht,

+

(1

—

vt,.)

—

&t,

=0,

T

f

3 &'ll - k e

Jld

v ht,

+

₂

—

A((t

=0,

kD2,

T

(2)

(3)

e-n; np vk (l)k,

where h(v~~) is the nonadiabatic part of the perturbed electron distribution function, II and

J

refer to components Jh

parallel and perpendicular to the equilibrium magnetic field Bp

=Bbp,

and p and A~~ are the electrostatic and vector

po-tentials. The nonlinear coupling coe%cient is wk,k-

=

(c/B)bp

(k'x

k");

the diamagnetic frequency is given by

toy (L((,L ~) Cog

'+

Cog(L~~

+

v~

—

_{3v, )/2v, , where co+}

= —

_(cT/eB)k'

_{bpx(Vttp/n p) and co~}

= —

_(cT/eB

_)k

_bpx(V

T/T);

the local Maxwellian distribution function is given by fp(v~~)

=(2trv,

) '

exp(

—

v~~/2v, ) and

Fp(v~) =(2trv,

)

VOLUME 59, NUMBER 13

PHYSICAL REVIEW

LETTERS

28 SEPTEMBER 1987 x exp(

—

v~/2v,

);

and k_D, is the Debye wave number.

We wish to study the transport ofboth parallel and perpendicular energy, because whereas the resonance condition

v~~„,=(Lo/k~~) (&v, may limit v~~, no such limitation exists on v~. Considerable simplification occurs because the v~~

and v& dependence in Eqs.

(I)

—

(3)

can easily be separated, ifthe nonadiabatic perturbation

of

the distribution func-tion is written as

&(L'ii,L

~)

=p(L

ii)Fo(L

~)+q(vll)Fp(vz)(v~

2L' )/2L', .

Equation

(I)

then splits naturally into two equations, one each forpk and Ltk, both independent

of

v~:

e

-

t'll kii vii

)

p k+bk(vii

)

,k k'+k" =k &' ll Pk"+ 4k (L'll)~ , k'

(5)

~' ll ktlL'll)qk +Pk(L II) ₀

+

II

g

Lvk'k" 0

+

[[ Lik"

+~k

(vJJ) ,k k

+k"=k,

,k'

(6)

where bk

= —

i[LO

—

LO+

—

COs(L~~

—

_{v, )/2v, ]fp(v~~) and} pk =iro+fp(v~~). Arbitrary external source terms gk and Xk have been introduced here in order to study how the plasma responds to external perturbations

(i.

e.,its

relax-ation); from this, the transport properties

of

the config-uration can be deduced.

Notice that qk does not appear in Eqs.

(2)

and

(3),

so that pk, pk, and A~~k form a closed system. The

poten-tials do not depend on qk, which is only passively con-vected according to

(6).

We study the transport through the mean evolution of a given perturbation in the turbulent field, via the aver-age response function computed at long wavelengths, i.e.,

at small kp (somewhat related discussion has been made

by Krommes, Oberman, and Kleva

).

The objective of the calculation is to determine the nonlinear fluxes in response to arbitrary sources _{gk, and kk,} in abackground of turbulent drift-Alfven fluctuations. To obtain the to-tal fluxes across the magnetic surfaces, we calculate

them at koll

=koJ

=0,

which corresponds to integrating

them over the magnetic surface. The average ofthe

non-linear terms in the right-hand sides

of

Eqs.

(5)

and

(6)

is denoted by

=

and A, respectively, and is the divergence

of the nonlinear fluxes. The radial fluxes

of

particle, parallel energy, and perpendicular energy are then given by

I

„=

(Lkp ) np dL'ii

=

r„„=

—

_{(Lko, )} 'no

IH

=

(Lkp

)

'no2v

& dv[~ k

+g

dv[~Ak .

(9)

The first term in Eq.

(9)

represents the heat transport as if it were simply given by the particle flux times the thermal energy, and the second term represents the correction due to the non-Maxwellian nature

of

the per-turbed distribution function. The fluxes I calculated here are the flux increments coming from an increase in temperature and density gradients. The equilibrium

fluxes are obtained by integration of I as the gradients are raised from zero to their equilibrium values. To ob-tain the response to a perturbation in density Snk, (for both the ions and the electrons) or temperature 6T~~~k, or

6T~k„we

choose as our source term a low-frequency

(cop) perturbation

of

a Maxwellian distribution:

and

(kLa' )(

)

lrop(&lk/no)fp(L'((

)

—

Lrop(DTLkJT)fp(v(t)(L')( L' _)/2eL'e,

(IO)

~ko(L'll) Lcoo(~Tako/T)fo(L'ii).

The average response functions (SP/8g) and (Sq/&,)

(where _p and

j

are the responses to the perturbations g and X, distinct from the Iluctuating quantities _p and

_q)

are then given to the linear approximation by _pk,

=

(kg

(

—

igloo)

—

(e/T)

pk

Joand

qk,

=

Xk'J(

—

icoo)

The.

corrections due to the nonlinear relaxation are evaluated here with the direct-interaction approximation

(DIA)

in the renormalized weak-turbulence limit. As a result, we find that =can be written as a linear function

of

pk, pk,

and A~~k, and A as a linear function

of

qk, pk, pk, and

Note that the following conservation laws are

verified by the exact equation:

_gk

[&

—

(vL/c)A~~lk =k

=0

and

_gk[p

—

(v~~/c)&~t]k&k

=0,

&v~~, which means that in a reference frame moving with the parallel velocity of the particle the drift is perpendicular to both the local electric field and the magnetic field. This property is

preserved by the DIA statistical closure.

When n; is not in phase with P, e.g.,when the ions are

dissipative, the particle Aux is ambipolar and regulated

by the ion response. What we wish to emphasize here is

more apparent if we take n; in phase with _p, i.e., if

Imv=0.

In this case the ions cannot diff'use, and

be-cause

of

ambipolarity the electrons do not diftuse either. We find that although I„vanishes for ko small, generally

r„and

IH do not vanish. We can illustrate the

cancel-lations occurring in the calculation ofthe particle flux by

VOLUME 59, NUMBER 13

PHYSICAL REVIEW

LETTERS

28 SEPTEMBER 1987 choosing a few terms in

=.

Define ak

=1

—

vk, dk

=k

/ko„

the propagator g»

=(

—

iro+ikiivii)

',

and the dispersion function

Gk '

=

a»

—

_J

dviig»b» ₎

dk+

_J

dii~g»b»vii/c

+ J

dviig»bkvii/c

Then the approximate expression

of:"

provided by the DIA closure is

2 k'+k"=k v pk gk' _{Gk'gk '~k}' C2 C v dk

+

J"dviiigk b» C dvllgkiPk C Jtd~_iig kb ,_k C II

Jt

dviig» pk C 2 Gk'gk 'bk ' _ak

J

dv((g» bk

_J

dvi(gk,p» C 2

T

I+k'k"I

(+"

p)k" k'+k" =k e

+

J"dviig» bk, C "'ll

J

dviig» p» C vII

x'Gk,

C vII r 2 a»,

Jrdv iong»,bk, Jl dviigk 'pk

+

Jrdv iigk'bk'

J _{C C} v

Jt

dv_{iong» pk} i

+.

.. .

(12)

C _j

If

we integrate this expression over ~II at k

=kp

to get the particle flux, it is easy to see how the cancellations occur _by

multiplying the first term in the curly brace by Gk Gk

'.

With use of Ampere's law, all _{these terms cancel out,} in the

limit of small ko . More explicitly, the range of ko is determined by the inequalities IVno/nol, IVT/TI

«ko„«k„",

where

k"

represents a typical wave number

of

the turbulence. In other ~ords, ambipolarity is achieved when the flux is

averaged over the spatial extent ofthe modes. The dots in Eq.

(12)

represent other similar terms which cancel out in

the same manner.

When Eq.

(12)

is multiplied by vti and then integrated over vii to get the parallel energy flux, these cancellations do

not occur, showing that the transport ofparallel energy does not vanish in general. The result for A is similar tothe

re-sult for

=,

and the corresponding terms are k'+k" =k

C C

2

dk

+

_J

dviig»

bk,

J"di

iig»pk

C C J"dvi~gk b» 12 vll Jl dviig» pk C 2 ₁₂ vll

—

_{Gk g»,p»,}

2 a»

—

J

dvi~gk.b», dv~igk,p»

C C vll

f

„vll

v Ilg» bk

_J

dviig» pk C C k'+k" =k r 2 vll

f

f

vll

f

vII

&'_{Gk ak}~

—

J' dviig»,bk. Ji dvi&g»,

p» z

+

Ji dviig»,b»,

C C

Jrd.

,

g,

,

p,

'

C

If

we integrate this equation over vll at k

=kp,

we see that the flux of perpendicular energy I

₀

vanishes only when

there is neither average temperature gradient,

AT=0,

nor perpendicular temperature perturbation, 6T~k

=0.

The or-der ofmagnitude

of

this energy flux is obtained by considering the special case where AT

=0,

which implies P»,

=0

and qk.

=0.

Then the only term which survives in Eq.

(13)

is the first one, which corresponds to the quasilinear estimate. 1450

VOLUME 59, NUMBER 13

PHYSICAL

REVIEW

LETTERS

28 SEPTEMBER 1987

In conclusion, we have shown that in spite of ambipo-larity, there can be transport of both electron parallel

and perpendicular energies, the latter being ofthe order of the test-particle diff'usion. The transport of parallel energy is due to the contribution ofthe nonresonant elec-trons. The transport of perpendicular energy is due to a portion ofthe distribution function which is evolved only passively in the electromagnetic fields. The results ob-tained here have deep implications for transport studies in tokamak machines. Contrary to Terry, Diamond, and

Hahm's assertion, the electron heat transport due to electromagnetic fluctuations is large, and is of the order of the test-particle diff'usion in the turbulent field. It is clear from the expressions

(12)

and

(13)

that the

equi-librium fluxes, obtained by integration of the I's, Eqs.

(7)-(9),

as the gradients are raised from zero to their

equilibrium values, will be nonlinear functions of

T

and ofAT.

We wish to thank Dr.

J.

A. Krommes for reporting to

us his own work on this subject in advance of publica-tion. This work was supported by U.

S.

Department of Energy Contract No.

DE-F602-84ER53174,

and by the Cornell College ofEngineering.

'See, for example, P.H. Rebut eI al., in Proceedings ofthe

Eleventh International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Kyoto, Japan, 1986 (to be published), paper IAEA-CN-47/A-I-2.

P. H. Rebut and P.P. Lallia, in Proceedings ofthe Second International Conference on Plasma Physics, Kiev, USSR, 1987 (to be published).

P. W. Terry, P.H. Diamond, and T. S. Hahm, Phys. Rev.

Lett. 57, 1899(1986).

4A.B.Rechester and M.N. Rosenbluth, Phys. Rev. Lett. 40,

38 (1978).

5J. A. Krommes and C. B.Kim, Princeton Plasma Physics Laboratory Report No. PPPL-2473, 1987 (to be published);

C. W. Horton, in Handbook

of

Plasma Physics, edited by A. A. Galeev and R. N. Sudan, Basic Plasma Physics Vol. 2

(North-Holland, Amsterdam, 1984),p.413.

63. A. Krommes, C.Oberman, and R.G. Kleva, 3.Plasma Phys. 30, 11

(1983).

7B. B. Kadomtsev, Plasma Turbulence (Academic, New York, 1965).

sJ.A. Krommes, in Handbook

of

Plasma Physics, edited by A. A. Galeev and R. N. Sudan, Basic Plasma Physics Vol. 2 (North-Holland, Amsterdam, 1984),p. 183.

9R.E.Waltz, Phys. Fluids 25, 1269(1982).