PFC/JA-82-1
QUASILINEAR CALCULATION OF ION TAILS AND NEUTRON RATES
IN A D-T PLASMA DUE TO LOWER HYBRID WAVES
J. J. Schuss
Plasma Fusion Center
Massachusetts Institute of Technology
Cambridge, MA 02139
December, 1981
This work was supported by the U.S. Department of Energy Contract No.
DE-AC02-78ET51013. Reproduction, translation, publication, use and
disposal, in whole or in part by or for the United States Government
is permitted.
By acceptance of this article, the publisher and/or recipient acknowledges
the U.S. Government's right to retain a non-exclusive, royalty-free
license in and to any copyright covering this paper.
QUASI LINEAR CA LCULATION OF [ON TAT LS
AND NEUTRON RATFS IN A D-T PLASMA
DUE TO LOWER HYBRID WAVES
J. J. Schuss
Massachusetts Institute of Technology
Plasma Fusion Center
Cambridge, Mass. 02139
ABSTRACT
Quasilinear theory is used to calculate the distribution function and neutron rate of a D-T plasma being
heated by a lower hybrid wave. It is shown that
Q values approaching
temperatures and for small deuterium concentrations.
I
1 are possible for sufficiently high bulk
Lower hybrid wave heating experiments have been carried out on several tokamaks; in several cases the
production of fast ion tails due to RF injection were reported.'- Specifically the Alcator A lower hybrid
experiment observed the formation of an energetic ion tail in the plasma center having a tail temperature
T7- > 15 keV and extending out to cnergics E > 50 kCV. 4 - 0 Such ion tails are predicted by the quasilinear
theory of Karney! It has been shown that this theory is not inconsistent with the ion tail observations of the
Wega experiment. Furthermore, a Monte-Carlo quasilinear ion heating code has shown consistency between
its results and the Alcator A RF produced ion tails.q--I These results demonstrate a reasonable agreement
between experimental results and the quasilinear theory of lower hybrid ion heatinig.
Stix12 has employed quasilinear theory to calculate the neutron rates and
Q of a D-T plasma being heated
by ion cyclotron waves. In this paper we calculate the Q of a D-T plasma being heating by lower hybrid waves
employing the quasilinear formalism of Ref. 7. Here we shall show that Q values approaching 1 are achievable
with lower hybrid heating by reducing the deuterium fraction and increasing T.
Karney7 has shown that in steady state the ion distribution function can be approximated as f(U) =
Fi(v )jfr exp(-mjvj/2Tj), where
Fi(v)= F exp
f
v dv .
1
V1 (I + D (v_ )1Ci(vs)
(1)
Here we defined'"
D
e2 E2 w 2
Di~v)
=2m
=
dkG(k )
k"va"
v 2vn~~~~t
'k
JkJAcrky
~ A
~k2
k2_ V2 _W2
(~ Iv~ilV 23 + ~VilpTdlmi)
where vi/ and v"' are defined in Ref. 13, V2 = T;/m and all species temperatures are equal. The summa-
tion is over all ion species and electrons. and f G(k±)dk± = A = kma. - kjn i,. 0 (k) is the spectral
shape of E2(k 1 ). For a D-T plasma
67rne 4 1n 12 [v
5
3i 2
.
(~(~)6~~cll f
m Ti
( -+ ) +
V,
m-r
1-6) + 2(
~
)6)+
m
v32
2
~
(2)
6 .9 9VtD
=
-o
t
tbT = 7.48V(T
6 = nD/n,
The power dissipated by the wave is 7
Pd =d=2
wn,
T
m2
dvLFD(v_)DD(vL)v3
u(vj)/Co(v )
I+ D
Jo
**c dvLF,,(vL)Dr(vL)V3 M2r
v
v)lrv
+ (1 - 6)jd
I + Dr(vL)/Cr(vL
Jo
(3)
(3
]
The neutron rate per unit volume due to tail-bulk collisions is then
RiN = n.(1
where ODT(v)
--
(4)
v_7T(v_) v_±.(FD(v-L) + Fr(vj))
2vd
r
6)
is the neutron production cross section and is a function of the relative ion velocity. We
then see that the ratio between neutron and alpha particle production power and the RF power dissipation is
Fq. (1) becomes
Q = RNENIPd, where EN = 17.6 McV. For (v_) >> /
F
(w)
'exp
FD(vL) m F
k .PLASi
dvw(vjv,)(vDaD
)pD
7QV
onal)+'IoaDVoD
~
+
v
+
a
V
1
(5)
where
W2
.e 2Eg
2
CL
I
kV1C
V
= 6irne4e In A
Thus, the shape of the ion tail is strongly characterized by -yD. and its amplitude is determined by w/k
1
,.
Figure 1 shows graphs of typical deuterium and tritium ion tails versus EL. the perpendicular ion energy
for -[ = 7.05 and y = 113. Here paramctcrs similar to those anticipated in the Alcator C lower hybrid
Thfaing cxperiment 14 are used. k_1 ,,, and k. ,,,,are determined from the dispcrsion relation
kie,
22
+k2
+ kL.
0
=0
(G)
where
= 1 426,W2o
mjw
W2Ce
2
3 T wp
3Z
4 =2
mw W2
6i
4n 26 i
T
MW2 mw
2
ni/ne
and where the electric field spectrum is chosen to extend from k?,in = 2.75w/c to krn.. = 3.25w/c. We see
that as the ID is increased, the extent in energy of the ion tail increases and in this case
Q increases from 0.049
to 0.246. We also note that the deuterium tail is generally of greater amplitude than the tritium tail. This ratio of
the tritium tail amplitude to the deuterium tail at v_ just greater than w/km.
2k2
RATIO = exp
MT
Nevertheless, the tritium tail will be flatter since its -rr is larger (j'i From this expression, we can calculate
Q versus -y
MD)
is
(7)
mi).
for fixed plasma parameters and fixed wave kn,, kma,
and w. In doing this it is useful to also graph 6A, where
6
A =AVPd
AS
(8)
where AV is the volume being heated, A is its external area and S, the power flow into this volume per unit
area, is
S Edw(fro + 2E.2k)
87r=
8w kL
(9
(9)
For 64 ~ I the RF power would be absorbed in the order of one pass. For 6.1 < 1, many passeC would be
required to absorb the RF power, (which might result in edge plasma heating) whereas for 6. > 1, the RF
power would be abosrbed at the edge of AV. For 64 ~ 1 the following calculation of Q is meaningful. (A ray
tracing calculation would be necessary to calculate heating details.) For a tokamak 64 = (P/S)Ar/2 where
Ar is the minor radius of the heating volume.
Figure 2 shows graphs of Q: 6A and R. the ratio between the RF tail produced neutron rate and the
thermal plasma neutroti rate ror 6 = 0.5 and 0.1. We see that higher %alticsorQ arc achieable w ith lower 6.
Thi, is easily explainbihlc by noting that both the RF powker dissiplition and the neutron rate arc doinaiited by
-1
the deuterium util. From Eqs. (1-4), we can then deduce for yL > 1
(1
--
6M)mDE,
f
dv
V_
67re 4 czvInfu
(10)
-vjF(vI)v±sir(v±)
dv
(v3)(1+ ,,)
V
From Eq. (10) we see that Q is not directly dependent on ne, except that as n, increases. k±0 m0. increases (from
Eq. (6)), which then increases Pd and lowers Q. Q is proportional to (1 - 6) and will increase as T increases,
which increases tOD and lowers the electron drag on the deuterium ions. i.owering n, sufliciently will raise Q
by lowering k
however, as this is done 6A also rapidly decreases and the absorption becomes too weak
for the resulting high Q values to have any real meaning. Also, as -ID continues to increase Q will decrease, as
ODT(v±) decreases at sufficiently large energy while the electron drag term grows. Finally, these results arc only
useful when R > 1, as only then is the deuterium tail sufficiently large compared to the thermal tail to make a
consideration of an RF Q meaningful.
Figure 3 graphs Q versus 6D for T = 5 keV. We thus see that for larger T, Q can approach 1. While in this
case 6A < 1. it is proportional to Ar: picking a larger heating minor radius would increase it linearly. Figure
3 employs 2.75 < kIc/w < 3.25; these values of k. are not strictly appropriate, as they would subject the
lower hybrid wave to strong electron Landau damping when T, = 5 kcV. Figure 4 shows a similar graph with
2.0 < kc/w < 2.5 and with n, now increased to 1.0 x 101 5 cm-3. The previous Q values are approximately
recovered; the higher density is only necessary to restore k_
e
-
160/cm.
In conclusion, we have calculated the lower hybrid wave damping, neutron rates and
Q of a D-T plasma.
For properly selected nc, T > 5 keV, and small deuterium fraction, values of Q ~1 can be achieved.
5
Acknowledgcments
The author is happy to thank Prof. M. Porkolab for his many helpful suggcstions pertaining to this work,
and
Prof. R. Parkcr for suggesting this calculation.
'Ibis work was supported by the US Deparunent of Energy Contract No. DE-AC02-78ET51013.
(1
References
1.
S. Bernabei. C. Daughsey. W. Hooke, et al in Plasma Heating in Toroidal Devices, (Proc. 3rd Symp.
Varenna, 1974) (E. Sindoni, Ed.), Editrice Compositori, Bologna (1976) 68.
2.
T. Nagashima, 11. Fujisawa in Heating in Toroidal Plasma (Proc. Joint Varenna-Grenoble Int. Symp.,
Grenoble 1978)ff. Consoli, P. Caldirola, Eds.) Vol.2, Pergamon, Elmsford, New York (1979) 281.
3.
C. Gormezano, P. Blanc, M. Durvaux, et al, Proc. 3rd Topical Conf. on RF Plasma Heating, Pasadena
(1978) paper A3.
4.
J. J. Schuss, S. Fairfax, B. Kusse, R. R. Parker, M. Porkolab, D. Gwinn, I. Hutchinson, E. S. Marmar, D.
Overskei, D. Pappas, L. S. Scaturro and S. Wolfe, Phys. Rev. Lett. 4, 274 (1979).
5.
J. J. Schuss, M. Porkolab and Y. Takase, Bull. Am. Phys. Soc. 24, 1020 (1979).
6.
i. J. Schuss, M. Porkolab, Y. Takase, D. Cope, S. Fairfax, M. Greenwald, D. Gwinn, I. H. Hutchinson, B.
Kusse, E. Marmar, D. Overskei, D. Pappas, R. R. Parker, L Scaturro, J. West and S. Wolfe, Nucl. Fusion
21,427 (1981)
7.
J8.
Charles F. F. Karney, Phys. Fluids 22,2188 (1979).
C. Gormezano, W. Hess, G. Ichtchenko, et al, Nucl. Fusion 21.1047 (1981).
9.
J. J. Schuss, T. M. Antonscn, R. Englade and M. Porkolab, Bull. Am. Phys. Soc. 25. 1002(1980).
10.
J. J. Schuss, M. Porkolab and T. M. Antonsen, Proc. 4th Topical Conf. on RF Plasma Heating, Austin,
Texas (1981) paper C5.
11.
J. J. Schuss, T. M. Antonsen and M. Porkolab (to be published).
/12.
T. H. Stix, Nucl. Fusion 15. 737 (1975).
13.
B. A. Trubnikov in Reviews of P/asna Physics (M. A. Leontovich, Ed.) (Consultants Bureau. Ncw York
1965), Vol.], 105.
14.
M. Porkolab, I. J. Schuss, Y. Takase, et al, 8th Int. Conf. on Plasma Physics and Controlled Nucl. Fusion
Research, Brussels, Belgium, 1980, Vol. 2, 507.
7
Figure Captions
Fig.1.
F(v_) vs. EL for (a) E0 = 1 kV/cm (ID = 7.05) and (b) Eo = 4 kV/cm (yD = 113). Here
f = 4.6 GHz, k
= 233/cm.
m = 190/cm, ne = 1.0 X 1015 cm-3, T = 2 keV,
and Br = 12 T. In (a) Q = 0.049 and in (b)
deuterium and tritium.
Fig. 2.
Q=
0.246. Here the plasma has equal fractions of
Q, 6A,
and log10 (R) vs. yo for f = 4.6 GHz, T = 2 keV, Ar = 5 cm, kamn, = 3.25w/c, kmin =
2.75w/c, and B-r = 12 T. In (a) 6 = 0.5 and ne = 9 X 10" cm- 3 while in (b) 6 = 0.1 and
nc = 1.2 X 10'7 cm- 3 . In both cases ne has been optimized for the best Q consistent with 6A
Fig. 3.
Q, 6A and logjo(R) vs. YD for f = 4.6 GHz, T = 5 keV. Ar = 5 cm,
2.75w/c,Dr = 12 T, 6 = 0.1 and n, = 6 X 10" cm- 3 .
Fig. 4.
Q, 6A
1.
. = 3.25w/c, kzmin =
and log10 (R) vs. YL for f = 4.6 GHz, T = 5 keV, Ar = 5 cm k..,
2w/c, B- = 12 T,6 = 0.1 and n, = 1.0 X 10'5 cm- 3 .
8
-
= 2.5w/c, kzmin =
-n
c
co
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