Ion containment in neutral injection heated tokamaks

Volume 91A, number 8

PHYSICS LETTERS

4 October 1982

ION CONTAINMENT IN NEUTRAL INJECTION HEATED TOKAMAKS R.D. GILL Cuiham Laboratory (Euratom/UKAEA Fusion Association), Abingdon, Oxon, 0X14 3DB, UK Received 26 March 1982 Revised manuscript received 24 June 1982

The results of several high-power neutral injection heating experiments are used to derive scaling laws for the ion temperature and the ion containment time valid for both ohmically heated and high-power injection heated discharges.

Although empirical scaling laws have been established [1—3]for the energy containment time and the ion temperature (T1) in ohmically heated tokamaks, similar laws have not yet been established for tokamaks heated by high-power neutral beam injection. This is partly because of the increased difficulty in making simplified plasma energy balances in these discharges. This letter uses a set of simplifying assumptions to derive scaling laws for the ion temperature and the ion energy containment time (rEj) in injection heated discharges. This approach is an alternative to the use of 1-D simulation codes or transport analysis codes to determine the plasma energy balances. In the derivation [2] of the scaling law for T1 for ohmically heated discharges, an energy balance is made between the power input (Fej) to the ions from collisions with the electrons and the ion losses (FL) which are assumed to be determined by plateau conduction losses. In high-power injection heated clischarges the power input to the plasma ions (P~1)from the beam is generally much greater than ~ei and the energy balance equation is P~1+ = Pj, where Fej can be calculated approximately. This energy balance can be used to analyse the scaling of both TEi and r1 with injection power and other parameters. This approach neglects the role of profiles and other effects in order to obtain a simple expression for the scaling laws. The relation of the ion losses determined from experiment to those predicted by the full neoclassical theory of ion conduction losses are discussed.

0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland

Ohmic heating. In order to obtain an expression for the ion power losses in ohmically heated discharges, the ion temperature scaling calculations of ref. [2] have been repeated using the data set of refs. [1,3] supplemented by further DITE data taken from refs. [4—61. Assuming that all profiles vary as [1 22 and that the ion collisionality parameter is in the plateau region, the power loss from the central region of the plasma (n/a ~ 0.5) can be written as (in mks units, T1 in eV, and assuming q(r/a = 0.5) = ~q~) —

~

—

,-

5/2

-

i—, 2

L—a~~Tin~q~vAj/B)W,

(1)

where T1 is the central ion temperature, ñe (=n~)iS the line averaged electron density, qL the safety factor at the limiter, A1 the ion atomic weight and the toroidal magnetic field strength. Equating F’L to ~ei (n/a ~ 0.5) then gives the usual Artsimovich expression [2] for T~,and a comparison of the calculated values of T1 with those of the data sets determines ci = (5.4 ±0.4) X 10—22. Neutral beam heating. In injection-heated tokamak discharges at low density, the ion temperature is usually raised sufficiently to take the plasma ions out of the plateau and into the collisionless regime of collisionality and the form of eq. (1) needs modifying. The modified equation will be determined from experiment and, guided by the theoretical expression for the coffisionless conduction coefficient [7], it is assumed that, over a restricted range of the collisionality parameter, ~t,the power loss from the ions can be written as

407

Volume 91A, number 8

PHYSICS LETTERS

4 October 1982

Q~=A~PL, (2) where A and B wifi be determined from experiment. v’ =(R/r)3/2qR/v1r1withR the plasmamajor radius

N en ~

~O L~- 00

N ~

en ~o N r—

c~C

N

en ~

~ ~.O r-~00 o\ C

~o

~

r~

v1 the ion velocity and r1 the ion collision time [7] i

enen

00

~.

NN

00 r~Ioo

~

~•C•~

drogen plasmas and which is necessary [8] to include the effects of collisions with impurities. The subtraction of 0.41 makes Z10~= 1 for Zeff= 1. ~3’ is the value of r4~evaluated at n/a = 0.5 and with q(r/a = 0.5)

c

00 o~ en

i-,

I

includingthefactorZj00~~Z~ff—0.4l forhy-

~

I N N C C C C C C C C C N N ~ ~ C C N en ‘,~ ~ en ~ ~ en en N N N N N r~ ~ I

the power input to the ions from the neutral injection and electron—ion collisions ~q~• A power balance is then made between Q1 and Qi=P~I+Pei=f~PNI+Fei, (3)

I ~

— — — — N — N N — —

— — — — —

.

—

— N N

~I~NNNNNNNNNN_NNNNN_

wherethefraction(f1)ofthetotalinjectionpower (P~J)transferredtotheionscanbeevaluatedusingthe curves presented in ref. [9] at the average electron temperature (~e= ~T~(r = 0)) and average injection energy. Fei is given by (with ln A = 16) 2. (4) ~ei 1.6analysis X 1O31ñ~(Te This is applied toTj)a2R/AjT~’ the data set of table 1 which includes the main parameters of well-diagnosed dischargesinavarietyofmachines.Infig.ltheplot of(fiF~ 1+P~j)/F~against~determines,by1inear regression,A = 0.46 ±0.18 andB = 0.65 ±0.15 with a correlation coefficient n = 0.70. The points are labelledaccordingtotable 1 butitshouldbenotedthat

C C C C C C C C C C C C “)

C C C C C C C

C I~ 00 1) O~en C C ~0 C 00 C 0~ ~.O 0~ C ‘n C ‘-~ C ~-

I

~ON

—

O~ en

~o .i~ ~

0~ ~n C C

C ~ ~Oen

— C C C C C C C C C C C C C C C C C C C C C C C C C C N C C en 00 C C N C en C C ‘r, r- C ~ .—~ C ~ “) C r- ~) ~fl N 00 t- 00 C f~ — , — — — N — N —

—

‘~

I

Cl ~

~.

E-

I~ N I

~

E I~ E

0.46~°

This expression for Q

N — —

en r- O~ ~

—

=

~

> 3.3 andf(i~t)= 0.46 i3*O.65

3.49X 1016(R/a)3/2(n~q~R/r~)zj0fl .

(7)

ScalinglawsfonT1.Eqs.(3)and(6)canbecombjn-

408

a

— —

~

I

i-I~

N r00 C N N 00

— — — N

en

00 N N N N N ~O N ri en N N N N N N ~

~

I ~

‘f) ~ .‘f~ —~

N

1 equals eq. (1) at i3j~’ = 3.3 and it is therefore reasonable to determine the ion losses from (1) when i~>3.3 and from (2) when i~1”<3.3. This leads to a single expression for the ion power losses = 5.4 X ~ (6)

~-

C C ~~ N C CI C C C C C C C C O~C -~ C — N — ~— — 0000 t~ C C ~o C — ~ C C — — en *— — — en en C — 00 — — — — — Lfl C en I

An F-test applied to this data set shows that the fitted line is significant at the 1.0% confidence level. Hence 65PL. (5)

withf(i~)= 1 for for i~~3.3and

O~t~~NN

C C

B points are probably due to the overestimate [18] thethe large deviation from theofstraight line of the ISXof experimental values T~by 20—40%.

=

I

—

~ ~

~-

N N — — N N

~O ~.O ~.O ~C \O ~ N N N N N N N

~

~.

en en en en en N N r— r- N N en en 0~’ 0~ C, 0~ O\ — en en N e- — — — — — ~- ~

—~ ~

— C N

~

-~

N N

‘~

-~ ~

en ‘I-

_~ ,-~

en en

~

a

a a

_

!~.

~.

LL~

~

~

‘I~~en ~fl N en

E~E

—

~.

I

Volume 91A, number 8

PHYSICS LETTERS I

I

<~fiPNIand

I~

-

4 October 1982

—

eq. (8) can then be rearranged to give the more convenient expression 98

10833

a° 046

p,~p ei i NI

~

T~= 2.35 X 10261 L(nq~R)1.65A~. fiFNIB~ 5Z0.65j ion

065

7;

=

001

(10)

1B~/ñ~q~sjA)04.

Equating Pci and Q1 gives the usual scaling law for T1 in ohmically heated discharges which is valid for

—

I

I 0.1

001

Fig. 1. Plot of (Pei of table 1.

+

I

1

data

fiPNi)/Ft~oND versus ~i” for the

3.2 X 108 [(er

+fiPNI)B~]0.4 L ñeq~\/A~f(f~)

(8)

.

T~ = 2.57 X 106. j~ (11) The constant 2.57 x 10_6 is just 6% lower than that of ref. [2]. In the region L <3.3 this scaling law becomes modified because f(~)~ 1. However, very .

10

ed to derive a scaling law for the ion temperature (in eV) in low-density, high-power injection heated discharges: =

(9)

3.2 X 108(fjP~

~ > 3.33, 0.1

‘

which is valid for i~<3.3. When ~ > 3.3

rzO7O

The comparison of this scaling law with experiment, again for the points of table 1, is shown in fig. 2. There is good agreement. In high-temperature low-density discharges Fei

nearly all ohmic discharges in the data sets considered had i~> 3.3 and this modified form is therefore of limited applicability. Ion energy containment time. A value for the ion energy containment time (rEj) can be calculated from the plasma energy content and eq. (6) TEi

= 3.5 X ~ (12) Comparison with neoclassical condition losses. The

plateau neoclassical ion conduction loss [19] Q1(plateau) is evaluated at n/a = 0.5 assuming q (n/a = 0.5) _1

~~ This is compared with the empirical power loss in fig. 3 where Q1/Q~(plateau)is plotted as a func—

10

1EXP

/

-

(keV) I

I

I

10— —

1—

01

—

001

—

—

—

EmpIricQt~H_H

~

/ /

_~~‘

0 (plotecu)

/

Theory, ö

~HR

0.13

/

—

/

/

Kpi~teou

1 0.1

0.01 I

0.1

1

II

0.1

1

1

1.0

100

10

(keV) Fig. 2. Ion temperature from the scaling law [eq. (8)] for injection heated discharges compared with experiment.

Fig. 3. Comparison between empirical ion losses and theoretical ion conduction coefficients calculated from neoclassical theory (H—H [7]; H—R [20]).

409

Volume 91A, number 8

PHYSICS LETTERS

tion of ~ Also shown is the neoclassical heat conduction coefficient divided by the plateau value. If the ion heat loss is entirely due to neoclassical processes then these two curves should coincide. However, there is a considerable divergence between the curves, which increases with decreasing Within the present approach the difference cannot be adequately understood. The empirical curve is in agreement with theory at high collisionalities but exceeds theory by about a factor of five at low collisionality. In some experiments (e.g. PLT [14,15] Q1 has alarge contribution from ~.

charge exchange (Qex) and convection losses (QD), but in other experiments (e.g. DITE [4J)this contribution is quite small. Because the plasma neutral density or equivalently the particle confinement time is a poorly measured tokamak parameter it is difficult to assess in a general way the magnitude of the contribution of Qex and QD to Q1. It is therefore probably reasonable to regard Q1 as an upper limit for the ion conduction loss, Summary. An analysis of high-power neutral injection experiments is used to derive an expression for the ion power loss and ion energy containment time in a tokamak over a wide range of collisionality. In the coffisionless regime the ion power loss exceeds the calculated neoclassical conduction loss by up to a factor of five. Scaling laws are derived for the ion temperature in both ohmically and injection heated discharges.

References [1] W. Pfeiffer and R.E. Waltz, Nuci. Fusion 19(1979)51.

[2] L.A. Artsimovich, NucI. Fusion 12 (1972) 215. [3] J. Hugili and J. Sheffield, Nucl. Fusion 18 (1978) 15. [4] R.D. Gifi et al., Proc. 9th Europ. Conf. on Controlled fusion and plasma physics, Vol. 2 (Oxford, 1979) p. 151. [5] R.D. Gill et al., Pioc. 8th Europ. Conf. on Controlled fusion and Plasma Physics, Vol. 1 (Prague, 1977) p. 171. [6] R.D. Gifi, G.D. Kerbel and A.A. Mirin, to be published.

[71 F.L. Hinton (1976) 239. and

R.D. Hazeltine, Rev. Mod. Phys. 48

[8] P.E. Stott, Plasma Phys. 18(1976) 251. [9] L.A. Berry, Symp. on Plasma heating in toroidal devices, Varenna, 1974, (Editrice Compositori, Bologna) p. 151. [10] V.M. Leonov et al., Plasma physics and controlled nuclear fusion research, Brussels, 1980, Vol. 1 (IAEA, Vienna, 1981)etp.al., 393, [11] M. Murakami Plasma physics and controlled nuclear fusion research, Brussels, 1980, Vol. 1 (IAEA, Vienna,, 1981) ~. 377. [12] D. Meade et al., Plasma physics and controlled nuclear fusion research, Brussel, 1980, Vol. 1 (IAEA, Vienna, 1981) p. 665. [131 D.W. Swain et al., Proc. 9th Europ. Conf. on Controlled fusion and plasma physics, Oxford, Contrib. paper B2.2 (1979) p. 44. [14] H. Eubank et al., Plasma physics and controlled nuclear fusion research, Innsbruck, 1978, Vol. 1 (IAEA, Vienna, 1979) p. 167.

[15] H. Eubank et al., Phys. Rev. Lett. 43 (1979) 270.

[16] V.S. Vlasenkov et al., Plasma physics and controlled nuclear fusion research, Innsbruck, 1978, Vol. 1 (IAEA, Vienna, 1979) p. 211. [17] K. Bol, private communication. [181 M. Murakami, private communication. [19] A.A. Galeev and R.Z. Sagdeev, JETP 26(1968) 233. [20] F.L. Hinton and M.N. Rosenbiuth, Phys. Fluids 16 (1973) 836.

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4 October 1982