Radio frequency control of impurity transport

Volume 91A, number 2

PHYSICS LETTERS

23 August 1982

RADIO FREQUENCY CONTROL OF IMPURITY TRANSPORT H. SUGAI Department of Electrical Engineering, Nagoya University, Nagoya 464, Japan Received 5 February 1982 Revised manuscript received 18 May 1982

New control methods of impurity-ion transport are proposed: one is the impurity-flux drive by radio frequency (RF) waves in a plasma, and the other is the RF control of the diffusion coefficient by impurity-ion heating.

Uncontrolled inward diffusion of impurities is one of the major problems for nuclear fusion along with plasma confinement in a magnetic field. Much attention has been focused on methods of minimizing the transport of contaminants into a plasma. Recently, some novel ideas [1] using neutral-beam injections

the collisional diffusion flux F’, and the second term to the wave driven flux rw. For visualizing the underlying physical processes of r”, let us consider a single particle of mass m and charge q in a RF travelling wave of frequency w, wavenumber k, and amplitude E0. The motion is governed

have emerged, and they have been tested in tokamak experiments [2]. In this letter I propose two new methods of radio frequency (RF) control of impurity transport. One is the impurity-flux drive by externally launched waves in the plasma, the other is the control of the diffusion coefficient by RF impurity-ion heating. Owing to the radial outward flow generated by RF, the impurity diffusion or even reverses the direction of the inward diminishes collisional diffusion, Oscillating electromagnetic fields can cause steady flow (i.e., time averages are nonzero) in plasmas. Welliciiown examples tms are tne ponueromotive iorce or other mechanisms which can make RF power (J’ E) nonzero, such as Landau damping or collisions. Another mechanism, which makes no use of wave dissipation, is the steady particle flow carried by waves. If, for example, a fluid density n 0 and velocity v0 supports a wave with the perturbed density ~ and yelocity then we have the total flux density F(t, x) (n0 + n~i)(v0+ ii). Averaging in space and time provides the steady flow

by a nonlinear equation, d x(t)/dz’

(q/m)E0sin[kx(t)

=

—

wt]

In the case of infinite wavelength (k 0), the particle exhibits a simple sinusoidal oscillation around the fixed point. For the travelling wave with finite k, however, the particle slowly drifts in the direction of ~.,/k 2, as ifiustrated with in fig.the 1. flow This isvelocity because(k/2w)(qE0/mw) the particle in the moving -~

E

.~CCELERATION —‘

-

01

,~‘

- -

t=~=~w/k

C

X /

-- -

--

DECELERATION

<~~> t

~,

/r’(~t,

\\

X1,

—

flçjVç~ + /_‘_~~ \fl V I

,

(1\ ~I)

where ( ) denotes an average over the wave period. The first term at the right-hand side corresponds to 0 031-9163/82/0000-—0000/$02.75 © 1982 North-Holland

C’

x

Fig. 1. Physical picture of the origin of the wave driven flux. Single particle trajectory in x—t diagram, together with the electric field E of travelling wave.

73

Volume 91A, number 2

PHYSICS LETTERS

wave sees the accelerating phase (E> 0) longer than the decelerating phase (E < 0). In order to control the impurity transport by F”, I choose an electrostatic ion cyclotron wave (ICW) among numerous kinds of waves in plasmas. The primary reason to controlflux theunchanged. impurity-ion flux only and retain theis main-ion The electrostatic ICW is the best for this purpose, as we shall see below. Let us model the waves and plasma in a slab geom-

etry with the uniform magnetic field B in the z direction andconsists the density x direction. The plasma of an changing electron in andthe multi-ion species fluids. The relevant hydrodynamic equations are

where

23 August 1982 =

w + iv0,

and ~ is the wave potential. According to eqs. (3) and (5), one finds the density perturbation 0i ~(k~ + ky~)+ K ~ = 0(_ik~+~ 2)w Wi kYt~ o 0G~G 1 a m (w~ ~

)

L

—

(6)

+

m 0ww1 where = (dn00/dx)/n00. On the other hand, the strongly magnetized electrons obey the Boltzmann 1e and equilibrium, i.e., ~eequation, = e~/Te.Substituting into Poisson’s one obtains the~ dispersion equation of electrostatic ICW and drift waves, 2

(a/at + u —

an

0 ‘V)u0 = (q0/m0)(E + (T0/m0n0)Vn0 v,~u0,

U0

X B)

1

—

0/at + V~(n0o0) = 0,

+~

~

2

(2)

t w(w

(3)

~2

where a denotes the species of ion, and T0 and v~are the temperature and the collision frequency, respectively.ofHere, we consider low-frequency wave frequency w andawavenumber k =electrostati.c (kr, ky, ks). Suppose that a quantity Q(r, t) such as the density, the velocity and potential, is slightly perturbed from its equilibrium value:

~l~PG —

~l2)k2 a

1

+

[x ~

w~0k~= 0,

(7)

+ (_ik~+ ~ 2koTe, k~)j w~= n2 where 00q~/e0m0,and 2 = k~e + = noee + k~. k Now, one can calculate the u-species-ion flux in K0

the x direction by the relation F~= Re(~ 0~Y~’) ~

(8)

,

Q(r,

t)

Q0(x)

+

Q

exp(ik r

—

iwt).

In absence of steady electric field, the stationary solutions of eqs. (2) and (3) give the collisional diffusion flux in the x direction, =

—D0 dn00/dx,

(4)

where the diffusion coefficient D

=

v,~p~/(l + 2~2~

= q0B /m and p~= T0/m0~L~. The ion thermal motion is not important if the wave phase velocity is much greater than the diffusion velocity, with the density-gradient scale length much greater than the wavelength. For simplicity, the cold-ion model (T0 = 0) is used in the following wave analysis. Linearizing eq.(2) allows each component of ~ to be found; ~,

where Re and * denote real part and complex conjugate, respectively. In the case of drift waves (k~ 0), the driven gradient in they vanishes direction.forThe fluxprincipal F~’alongflux the isdensity v = 0 —~

since the phase difference between ~i’~and ~ is 90°. Thus, the drift wave is not appropriate for controlling the impurity transport. Wave-particle interactions may alter the phase difference and cause anomalous impurity diffusion [31. by electrostatic ICW, we To estimate F~’driven considerSubstituting a simple case where ~ (6)v~, and ~ k~, i~, eqs. (5) and into (8)k~ yields ~“

K~.

the wave driven flux scaled by the collisional diffusion flux F~, 2P~ ~a Te 21, Z~e~2 W

—

kr~kxI

‘(

2)+ikywiFZqI(w? (q~/m0w1) X [k~w?/(W? ç~ = (q 0~/m0w1) =

—

~l~)]

0 IF~’/F~1_w2/cl2)2~aPo(TJ

,

p,~

2iTJ()

—

X [_ikxW =

74

where kr = Re(k~),q0

,/(w~ ~l~) + k~w~/(w? —

(q0~/m0w1)k~ ,

—

(5)

=

Z0e, and L0 denotes the

gradient scale length introduced by I(dn00/dx)/n001 = ilL0. For a rough estimation of F~’, we substitute k~p0 1 and w/~l,, 1.1 into eq. (9), for ~l0/~0

Volume 91A, number 2

PHYSICS LETTERS

23 August 1982

2,andZ 0=l,L0/p010 0=8.These parameters yield I F~’/F~I 1 for the wave amplitude Je~/TeI= i0~. A more precise evaluation has been done under =~o~T/T ‘

e

conditions with main-ion when (ZM 5%= impurity-ion i) of mass ratio (Z1 mI/mM = 8) is mixed = 16 and temperature TM = T1 = Te, with the same gradient scale length LM = L1 (L1/p1 = 102). The wave dispersion given by eq. (7) is shown in fig. 2a for W~M/&l~ = 1, ~‘2~/v~ = l0~,and ky = k~= 0. The electrostatic ICW has a cutoff(k~= 0) at w = ~ and a resonance (k~ oo) slightly below the ion—ion1/2. hybrid frequency For these con[4], WihI have f~1(1 +mInOI/mMnOM) ditions, calculated the wave driven flux F”’ defined by eq. (8), and the results are shown in fig. 2b for the wave amplitude Ie~/TeJ= 10~.It should be noted in this figure that the impurity-ion flux is at least two orders of magnitude larger than the mainion flux. This indicates the excellent selectivity of the RF impurity control. The control efficiency increases near the resonance frequency, and there the very small RF power might be sufficient to reverse the direction of the inward collisional diffusion. Strictly speaking, the collision term in eq. (2) expresses ion—neutral collisions. So, the calculations are applicable to a weakly ionized plasma in a scrape-off layer or a divertor region where the neutral gas pressure reaches [5] 10— 4_iO_2 Torr. For the fullyionized plasma core of a tokamak, more complicated calculations are required: the toroidal geometry gives rise to the neoclassical Pfirsch—SchlUter radial trans2/Z)Fc where Z = ZI/ZM, q is a safety factor, fC is evaluated in terms of port flux [6], F~and(2q Coulomb-collision frequency. Therefore, the RF con-

5 % I M pu R IT ‘

I

~

krF~

WAVENUMBER

/~

k1

kr 0.1

DAMPING RATE / / RESONANCE

.~

00’ /UTOFF — — — I .05 LI FREQUENCY

1.15

will

.2

1

I0~

I0~

0’

IMPURITY ION

x z

(b)

I e ‘~/Te I I 0

—

~

I

00

0. 0

02

I

-

CUTOFF .05

1.1

FREQUENCY

RESONANCE 1.15

1.2’

will 1

trol efficiency may be the reduced approximately 2 /Z. Also, temperature gradientbyef-a factor of 2qbe taken into account in the tokamak fect should plasma. When the RF power is increased, selective heating of the impurity-ion is expected in the processes of coliisionless or collisional wave damping. Then, the diffusion flux itself changes since the diffusion coefficientD 1 i.’1p~is a function of ion temperature T1. In a scrape-off layer or a divertor region, impurityion collisions with particles areperiphery, frequent, the and 2 Inneutral a plasma core or hence D1 ~with T?’main-ions or other impurity-ions are collisions dominant, and soD 12. Also, the diffusion 1 cr Tf’ .

Fig. 2. (a) The solid line shows the wavenumber kr Re(kx) normalized to the impurity-ion gyroradius pj. The dashed line shows the damiing constant ki = Im(kx) normalized to kr. (b) Wave driven flux pW normalized by the collisional diffusion flux rC as a function of the frequency ~ normalized by the impurity-ion gyrofrequency f~.

caused by toroidal field ripple is characterized as D1 T?[2. Therefore, the RF impurity-ion heating diminishes theneutral-collision Coulomb-collision diffusion, while it enhancesinthe diffusion and the ripple diffusion. Thus, the RF heating provides additional means of controlling the flux of impurities. 75

Volume 91A, number 2

.PHYSICS LETTERS

Excitation and accessibility of the waves in question may be accomplished in the toroidal plasma just the same way as in two-ion hybrid resonance (minority ion) heating [7]: fast Alfvén waves are launched outside of the plasma, and near the ion— ion hybrid layer they convert into ion Bernstein waves which are the electrostatic ICW modified by the finite-Larmore-radius effect. In the hot plasma core, the cold-ion fluid model breaks down and the kinetic approach is required; this gives another restriction for practical applications of the present calculations. In principle, however, the RF control of impurity transport would be possible even in the hot plasma core. Recently, a tokamak experiment closely related to this proposal has been reported by the TFR group [8], a heavy impurity(argon) has been efficiently pumped out of a two-ion component plasma during ICRF heating. At the present moment, the mechanisms of the impurity pump-out cannot be identified definitely, but the possibilities of a change of the diffusion coefficient or of a wave driven flux are mentioned in this paper. To summarize, two new methods of RF impuritytransport control are proposed. Strong impurity-ion flow is generated by an electrostatic ion cyclotron wave which is externally launched at a frequency

76

23 August 1982

above the impurity-ion gyrofrequency. A simple fluid model provides a useful expression for the wave driven flux in an inhomogeneous multi-ion species plasma. With a relatively small amplitude of the wave, one can already control the direction as well as the magnitude of the impurity flux of the selected species. Also, RF heating of the impurity-ion makes additional control of the diffusion coefficient possible. The author wishes to acknowledge helpful discussions with Professor T. Okuda and Dr. S. Takamura. References Li] K.H. Burrell, T. Ohkawa and S.K. Wong, Phys. Rev. Lett. 47 (1981) 511. [2] R.C. Isler et al., Phys. Rev. Lett, 47 (1981) 649. [3] B. Coppi, G. Rewoldt and T. Schep, Phys. Fluids 19 (1976) 1144. [4] S.J. Buchsbaum, Phys. Fluids 3 (1960) 418. [5] M. Petravic, D. Post, D. Heifetz and J. Schmidt, Phys. Rev. Lett. 48 (1982) 326. [6] See, for example, W.M. Stacey, Fusion plasma analysis (Wiley, New York, 1981) p. 124. [7] F.W. Perkins, Nuci. Fusion 176 (1977) 1197. [8] TFR Group, Association Euratom-CEA sur la Fusion, Fontenay-aux-Roses, Internal Report EURCEA-FC

1131 (1981).