PLASMA
DYNAMICS
VIII.
PLASMAS AND CONTROLLED NUCLEAR FUSION
E.
High-Temperature Toroidal Plasmas
Academic and Research Staff
Prof.
Prof.
Prof.
Prof.
Prof. B. Coppi
Dr. D. B. Montgomery]
Prof. G. Bekefi
Prof. A. Bers
1.
R. A. Blanken
R. J. Briggs
L. M. Lidsky
W. M. Manheimer
Prof. R. R. Parker
Prof. K. I. Thomassen
Dr. R. Gajewski
Dr. P. A. Politzer
MODEL FOR THE TURBULENT ION HEATING IN
TOKAMAK TM-3
Recently reported experiments on the Tokamak TM-3 machine have shown that
the ion energy distribution contains a low-energy part at a temperature of approxil
The striking feature of this tail is that it
mately 50 eV and a high-energy tail.
continuously disappears as the applied magnetic field is increased from 10 kG to
We propose that the enhanced heating of the tail is due to an electrostatic
26 kG.
ion-cyclotron instability driven by the electron drift current parallel to the applied
We summarize the pertinent results of linear and quasi-linear themagnetic field.
ories in support of this model.
Linear Instability Theory
The
instability under
has been
consideration
past
studied in the
by several
authors.2, 3 The regime of plasma parameters pertinent to TM-3, where Te >> T.
Di = ion Debye length),
1, k£Di> Ti/Te (a = ion Larmor radius,
and, for klai
was not covered by these previous studies,
must be rederived.
The
ll)
almost across B o (k>>k
v
dispersion
however,
relation for
T.
i
VT(kv-W)
1(k
1 eO)
kllve
Te
e
vi
S= -
where f(w) is
< (/k)
waves
<v
e,
propagating
where vi and
The growth rate
has been solved.
0)
imaginary part of
is given by (y
electrostatic
and with phase velocities v i
are the ion and electron thermal velocities,
and the instability conditions
-klaa2
i
no
I (ka
22
Z)
expcI(
- nwci2
ci
8f
given in Eq. 4.
This work
AT(30-1)-3980).
was
It is positive as long as
supported
by the
U.S.
Atomic
Energy
Commission
(Contract
tDr. D. Bruce Montgomery is at the Francis Bitter National Magnet Laboratory.
QPR No. 99
77
(VIII.
PLASMAS AND CONTROLLED NUCLEAR FUSION)
ue
(k /k
-e
(
V. >
) (W/W
1
ci
a.
+
FT3
1 e
+
m
oc
I
2
n
/k2a2
(2)
,e(k a.)2
mT
1L
2 2
-ke22
me
i
n=-j
where
Ci
kv-c
kllVv i
n
'(3)
and w, k , and kll satisfy the real part of the dispersion relation:
T.
1 +
-k2a2
1 + (k2Di2 =
0
ok
2a
202 n ka2)a
+
2
2 2
ci J
n=1
f
-
, kla
.
(4)
Equation 4 may be written
T.
1
2
2
"c i
2
+ (klai
e
f
= f.
(5)
pi
Consider first the limit (k<Di)2 << Ti/Te, which has been solved numerically. 3
this limit we note that Eqs.
for a constant ratio of Ti/T
1,
e
2, and 4 depend only on w/woci,
, as Bo is
changed,
k ai,
In
and k1 /kl.
Hence,
both the growth rate -y/wci, and
the instability onset condition, Eq. 2, remain unchanged if ki and kll are changed
proportionately to Bo. Also, in this limit, the calculation by Lominadze and Stepanov
shows that for the onset of instability, kla i remains close to unity for a wide range
of Ti/Te (see Fig. VIII-1 and Table VIII-1).
For the case of interest in TM-3, where
k _Di > Ti/T e ,
moves w closer
rate
it can
to nci
be seen from Fig. VIII-1 and Eq. 5 that increasing Bo
and thus cyclotron damping will decrease the growth
y/wci.
Using TM-3 data
60 v i
ve/3, ne
tail at B
in the operating regime of strong anomalous resistance
10 kG,
and it
Te
1 keV, T ~ 50 eV), we observe the hot-ion
Se
11
disappears at B 0
26 kG. If the density is taken as 1011
( ci/p .)2 varies from 1/25 to 1/4 for this variation in B
computations
well
satisfied
QPR No. 99
(ue
1011 - 10 i2/cm3
to these parameters shows that at
and the growth
rate
should be
.
A rough scaling of the
10 kG the instability conditions are
large,
while
for 26 kG a strong
3
(VIII.
I+ Ti/T
e
)
+ (k
Oi
PLASMAS AND CONTROLLED NUCLEAR FUSION)
2
pi
f ( /c
c= ,
k
0i
)
I+Ti/Te+(ka
)2 2
i
2
ci/pi
\tW/Wci
W/ci
Ti/, Te
(b)
=
O. I
I
(b)
Fig. VIII-1.
Dispersion characteristics for the electrostatic ion
cyclotron wave (Lominadze and Stepanov3). (a) Eq. 4
for kla i constant. (b) w(kl) at Ti/Te = 0. 1, 1.
Table VIII- 1.
Conditions for onset of instability for
me/Mi = 1/3680.
2
2
ue/Vi
k ai
cos
1. 00
23
1. 1
.05
.33
15
.8
.12
.10
12
.8
.15
Ti/T
reduction in growth rate may be expected.
rate have been initiated for
0
Computer calculations
obtaining quantitative
estimates
for the
growth
reduction
of the
in
growth rate with increasing magnetic field.
Quasi-linear Theory
We shall now examine the way in which this instability heats the ions.
linear diffusion equation for ions in the presence of weak turbulence is
and can be written
QPR No. 99
The quasi-
well known, 4
(VIII.
PLASMAS AND CONTROLLED NUCLEAR FUSION)
af.
v ' D - V f..
v
v i
=
at
In order to estimate the ion heating rate, we shall assume that fi is isotropic.
ical coordinates in velocity space, integrating Eq. 6 over the spherical angles
we obtain
af.
1
i
at
v
a
2 av
f.
1
2~
vD
In spher0 and 4,
VV
')V
where
~1
Dvv
4rr
vv
D
vv
4Tr
sin 0 dOdo
eE(k) 2
'-
2
2
S
n2 (kciVn
\'\ci
,
ckv
m
forv
2 f
>
2k
n,k
-nwc
ki
/
2
where V
v2(
S
n
nwci
Ak
'
and
2
0
for v 2 <
W
- nw)ci
ki
For TM-3 parameters and the instability conditions discussed above, we find that all
waves are linearly stable unless (w-n ci)/kl > 2. 5 v i . This, together with Eq. 8, shows
that only the tail (c ' 6 Ei) of the ion energy distribution function will be heated, which
is in agreement with experimental observations.1 For the high-velocity ions (v> 2. 5v.)
an order-of-magnitude estimate of D wv from Eq. 8 is
2
eE 2 v.2
m
3
k11v
D
vv
where IE
is an average field strength.
approximately by
hh
pi
QPR No. 99
kO~
11i
2
Ei
A heating rate vh
h
"2
Dvv/v
vv i
is then
given
(10)
(VIII.
PLASMAS AND CONTROLLED NUCLEAR FUSION)
and Ei the ion thermal
where EF is the average electric field energy,
can compare
2(m /mi)
this
heating
pe/n3De.
rate
with the
electron-ion
Thus even if the total electric field energy is
energy,
rate,
5
We
vE
106 (
small compared with the ion thermal
Furthermore, if the nonlinear process that
turbulent heating may be dominant.
stabilizes the wave is
energy transfer
10 wci/u , we find vh/
Assuming v = 5 v. and k
energy.
such that the final electric field energy scales with the linear
growth rate, then EF will decrease as the magnetic field increases.
Therefore the
heating rate of the tail (and hence its temperature) will also decrease with increasing
magnetic field strength.
A. Bers, W. M.
Manheimer,
B. Coppi
References
1.
G. A. Bobrovskii, A. I. Kislyakov, M. P. Petrov, K. A.
Shcheglov, Kurchatov Report IAE-1905, Moscow, 1969.
2.
W. E. Drummond and M. N. Rosenbluth,
3.
D. G. Lominadze and K. N. Stepanov,
4.
C.
5.
S. I. Braginskii, in Reviews of Plasma Physics, Vol.
(Consultants Bureau, New York, 1965).
Kennel and F.
QPR No. 99
Engelmann, Phys.
Phys. Fluids 5,
Razumova,
and D.
A.
1507 (1962).
Soviet Phys. - Tech. Phys.
9,
1408 (1965).
Fluids 12, 2377 (1966).
1, edited by M. A. Leontovich