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DE84 002364
EXCITATION
OF
INTERNAL KINK MODES
BY
TRAPPED ENERGETIC BEAM IONS
By
L. Chen,
R.B.
White,
and
M.N.
Rosenbluth
OCTOBER
33
PLASMA
PHYSICS
LABORATORY
PRINCETON UNIVERSITY
PRINCETON,
NEW
JERSEY
rat nt tt.s.
2-7«-a»-3073.
.
DISTRiaUTIOH OF THIS DOOiaEHT tS EffltHWFffl
EXCITATION OP INTSflNAL KINK MODES BY
TRAPPED ENERGETIC BEAM IONS
Liu Chen and R. B. White
Plasma Physics Laboratory, Princeton University
Princeton, New Jersey 08544
and
11.
N. Rosenbluth
Institute of Fusion Studies, University of Texas
Austin,
Texas 78712
ABSTRACT
Energetic trapped particles are shown to have a destabilizing effect on
the internal kink mode in tokamaks. The plasma pressure threshold for the
mode is lowered by the particles. The growth rate is near the ideal
magnetohydrodynamic value, but the frequency is comparable to the trapped
particle precession frequency. A model for the instability cycle gives
stability properties, associated particle losses, and neutron emissivity
consistent with the "fishbone" events observed in PDX.
DISCLAIMER
This report was prepared as on account of work sponsored by an agency of the United Stales
Government. Neither the United States Government nor any agency thereof, nor Bny of their
employees, makes any warranty, express or implied, or assumes any legal liability or responsi-
bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or
process disclosed, or represents that its use would not infringe privately owned rights. Refer-
ence herein to any specific commercial product, process, or service by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-
mendation, u favoring by the United States Government or any agency thereof. The views
and opinions of authors expressed herein do not necessarily state or reflect those of the
United States Government or any agency thereof.
-2-
In recent poloidal divertor experiments (PDX) with high-power nearly
perpendicular beam injection, bursts of large-amplitude magnetohydrodynamic
(MHD) fluctuations, dubbed "fishbones" from the characteristic signature on
1 2
the Mirnov coils, have been observed. ' These "fishbone" bursts are found to
be correlated with significant losses of energetic beam ions and thus have
serious implications for the beam-heating efficiencies and the achievable g
values in tokamaks.
Detailed experimental measurements have identified the mode structure of
the "fishbone" as an m = 1, n = 1 mode with additional m > 2 components,
supposedly due to the finite e&
D
toroidal-coupling effects. (Here m and n
are,
respectively, poloidal and toroidal mode numbers, e = a/R is the inverse
aspect ratio, and g is the poloidal beta.) The plasma pressure threshold for
the mode is consistent with that of the internal kink mode. The most crucial
feature is that all components rotate toroidally with a frequency comparable
to the precession frequency of the trapped beam
ions.
This resonance feature
indicates that proper
understand!*,
j
of both the stability and the beam loss
mechanisms require a kinetic treatment of the plasma dynamics.
In this Letter, we employ a gyrokinetic description ' for the trapped
beam ions and demonstrate that the internal kink mode can be excited at a
lower threshold than that of the ideal HHD prediction, with a frequency given
by the precession frequency. A model for the "fishbone" cycle gives MHD
amplitudes,
particle losses, and neutron emissivity in close agreement with
those observed in PDX.
We consider a large-aspect-ratio tokamak plasma consisting of core (c)
and hot (h) components. For the purpose of formal orderings, we use
e = a/R << 1 as the small parameter. Since we are interested in the parameter
range of the first stability boundary of the internal kink mode, we order
-3-
B » 0(1) and, for simplicity, Li, ~ 0(e). Temperatures are ordered as
T
c
<-1 keV)/T
h
(~50 keV) -
0(
e
2
),
which implies t>
h
/n
c
<- 0(e
3
) and, hence,
overall charge neutrality may be assumed. We also have, for PDX parameters,
u)
~ u
dh
- 6 x 10
4
>>
<,j,
c
,
m
dc
,w
A
= v
ft
/qR ~ 2 x 10
6
and thus
— 2 5
I(J]/OI
A
I
~
1
u>
(ih
/u)
A
I - Die >J similar to the usual internal kink ordering. Here
u
dh
is the toroidal precession frequency of the trapped hot particles, and u»
and
U>J
denote, respectively, diamagnetic and magnetic drift frequencies.
Consistent with the above orderings,. we adopt the ideal MHD description
for the core plasma. For the hot component, however, we employ the
gyrokinetic description, neglecting the finite Iarraor radius correction. Tu
derive the corresponding normal mode equation, we first sum up the
collisionless equations of motion for each species and obtain
fl2
pǤ
=
c
1*3
x
5
+
i *
5
3J
"
?
sp
c
" 1 •
«&,
-
(1
>
where £ is the usual fluid displacement vector. In Bq. (1), noting that
n
n
/
n
0
~ 0(e ), wo have p =
n-jm..
In addition, the following ideal MHD
relations hold: SP„ = -[£ . VP„ + YP„(V • ?)], $E, = iujF x B/c, 5E„ = 0,
5
5
=
!?
x
'£
x
5*'
,lld
&J
= c
^ *
fif/'iir-
The perturbed distribution of the hot
component, 6F
ft
, is given by '
e 3 u SB 8
and
l
V
B
fe"
l
l--«dhJJ«
H
h
=i
m
fi
«* '
(3)
-4-
2 2
where E = v /2, u = v/2B, w
0
is the cyclotron frequency,
%/%%
s e • y,
Si|> = 5* - V^/c + v^Bj^^c, a = (ua/3E + u,
h
) F
Qh
; u.
h
5 -(i/injl^
x VHnF . )«V,
u),.
= -iv,.»V, v_ is the magnetic drift velocity, and 54 and
~ on » an ^ an
**
^ an
T
6A|I are related to J by cy5ij> = - iwg x B and
<D6A./C
= - i35(|>/ai. Noting that
the frequencies are nuch smaller than the hot-particle transit and bounce
frequencies,
Eq. (3) can be solved readily for both trapped (t> and untrapped
(u) particles. We find that 6H. = -ey^/ram and fiH. = -e25<j>/mu) + 6G
n t
,
where 6^^ =
2QBJ/(OJ
-
u
dh
).
As <$Adi/|v
|
)/<^di/|v
|
)
denotes bounce
averaging, and J = (oB/2)V • E, - (1 - 3aB/2) r. • te# with a = u/E and
(J = 3ej|/3£. Substituting SH into Eq. (2), we have 6g
n
given by
«S
h
=
"5i ' *l
p
i
5
+
l
p
,
"
p
J
e
, S,J
h
+
5P
i
I
+
t«*i
"
5P
i)S,
e
„'
where
5E -1 —S5
1
7/2
B . » 5/2 _ 2(1-
a
B)
{ | = 2 ' Tim
h
B J dad - aB) ' J dE —^ J | ( (4)
SP,
B"
1
° """dh 1
H max
correspond to kinetic contributions due to the trapped energetic particles.
Substituting 6P. into Eq. (1), we have a complete normal mode equation in
terms of jr.
To analyze the stability properties, we shall derive a dispersion
relation variationally. First, we obtain the following dispersion functional
3 *
by performing Jd x £• on Eq. 11) and assuming a fixed conducting boundary,
D
liJ
=
^MHO *
5w
k
+ 5I
' "here, with P = P
c
+ CP
i
+
P„)
h
/2,
MHD
6W„..„ = g jd x
|—j*
[^
x
x
Sj
j .
S
B
i
- 2[
&l
. VPJIC^ • K)
+
s2
i?
* h
+ 2
Si •
Jfl
2
+
VP
C
IV
- £l
2
} -
B"
1
7/2 2 nun » _, _
6W
fc
= -2 ' it n^jRHrdrJ daj dE E ' K^J —=— j
f
B*
1
6
"""dh
max
(5)
(6>