PFC/JA-79-17
EFFECT OF MAGNETIC FIELD RIPPLE ON ENERGETIC
IONS IN ALCATOR A
J. J. Schuss
November 1979
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 nonexclusive, royalty-free license in and to any copyright covering this paper.
EFFECT OF MAGNETIC FIELD RIPPLE
ON ENERGETIC
J.
J.
IONS IN ALCATOR A
Schuss
Abstract
The effect of toroidal magnetic field ripple on energetic
ion confinement in Alcator A is theoretically investigated.
A heuristic Monte Carlo particle simulation shows that for
sufficiently large v
/v the ion loss rate due to ripple trapping
is independent of the toroidal fraction of the torus contained
in the magnetic well.
The resulting power loss is estimated
to be large enough to limit the efficiency of rf heating on
Alcator A.
2
The effect of toroidal field ripple on ion confinement in
tokamaks has been previously considered,1- 5 and can significantly
deplete the ion distribution function at high energies.
would occur for vj, /v < /oB/B,
where AB is the peak to peak
change in BT due to the ripple and AB/B << 1.
occurs at small values of v
This
Since this loss
/v, it would most strongly affect the
loss rate of the nearly perpenducular ions generated by rf heating.
In Alcator A there is a ripple of 2.5% on axis and 7.5% at
the plasma edge caused by the gap in the toroidal magnet at each
of four access ports.
This ripple, as graphed in Fig. (1),
is
localized to about 20% of the machine's toroidal angle, and is too
large to be substantially reduced by the rotational transform. '3
In this letter we present calculations that show that for most
values of v
/v there is no reduction in the loss rate of initially
toroidally circulating ions due to the fact that the ripple is
localized only over a fraction of the torus.
v
Thus for small
/v large loss rates for energetic ions can be present in Alcator
A, and may have substantially lowered the efficiency of the recent
lower hybrid heating experiment on this tokamak. 6
An ion is trapped in the ripple well for v
/v
< 6
S1/2
where
2
6 = AB/B.
The ion has a vertical VB drift velocity VD
~ v /(2w ciR)
and scatters out of the loss cone in a time Tout = 2T D6, where
7
TD is the 90* scattering time.
drifts a distance d =
2
Thus an average trapped particle
6T D v 2 /(2wciR).
When d '% a/2, where a
is the plasma minor radius, the trapped ions are rapidly depleted
by the VB drift, leaving a hole in velocity space.
Since d a E 5/2
d = a/2 defines a sharp boundary in energy above which an isotropic
3
distribution would scatter into this loss cone in a time '
at which d = a/2 the loss time would vary
Below this energy E
as E. -7/2
E. < E
T D'
3/2
since the slowing down time varies as E. 3,
1,2.
,
for
the loss rate due to the VB drift would quickly diminish
below the energy deposition time and not significantly affect rf
heating efficiencies.
E
> E
at
In this letter we will consider ion energies
which the loss cone will be depleted and for E. < E
neglect this loss.
we will
We will investigate the effect of ripple
localized to a small fraction of the torus, a situation uniquely
relevant to the Alcator A tokamak.
We can write down the orbit averaged Fokker-Planck equation
for the energetic ion distribution f. as 8,9
3f.
v
+
_t3
s
+
f
3
T v
s
3
+v
2
f)
Ov
rBm
+ 1/2
V
v
W
1
c
Ts
1
2
O
1>
2i
E3
0
where
3J3/2
s
(1
m
4
e 1/2
ne knA
e
05
2
-
i
4
2E
vcc
Ec =
2 /m
(3/4)
i /M) l/3T
and where we have let Zeff = 1, let the energetic ions be of
the same mass as the background species, and ignored charge
exchange and energy diffusion.
v is the ion velocity, E = vjj /v,
EO and Bm are respectively E and B at the center of the ripple
well, and <> denotes an orbit average.
If we approximate the
(1) by a constant over most of the torus
magnetic field of Fig.
and a cosine in the well, <B /B-l> can be calculated in terms
m
of elliptical integrals,
-
However, it is easy to see that <Bm/B-l>
6 when the ripple is located only over a small fraction of
the torus.
Thus for
02 >> ', <Bm/B-l>/E 02 << 1 and Eq.
(1) reduces
to the velocity diffusion equation in a homogeneous plasma.
would therefore expect that for large E
unaffected by the well.
f(001
< 61/2) = 0;
Equation
We
the diffusion would be
(1) could be solved by requiring
the resulting solution for a source at E
> 0
would not allow any ions to cross the hole in velocity space and
f
i
(0
0
< 0) = 0.
sufficiently small ions
However, in reality for
can diffuse in velocity space as they travel between ripple wells
so that they can acquire a AE = /2DAt =
where D1 = v3T /v c3,
between ripple wells.
E
c between ripples,
Lt = RA$/vjj , and L$ is the toroidal angle
We then obtain
5
/ 2DRL#
For E <
c
1/3
ubstantial diffusion, including a v
occur between ripple wells and Eq.
reversal can
(1) breaks down.
This is
because the orbit averaged Fokker-Planck equation assumes that
an orbit time is much less than a characteristic diffusion time,
which is not true for E < Ec.
14
ne = 2.4 x 10
10 keV
-3
cm
For Alcator A parameters, A4
during lower hybrid heating, and for E
= v/2,
=
(an energy characterizing many rf produced energetic ions),
nu 0.07.
In order to obtain the solution to the ripple loss for
small E, a simplified Monte Carlo scheme was employed.
A particle
on the magnetic axis was followed around the torus which had a
ripple well at each of the four access ports;
several times between
ripple wells and within the wells the particle was scattered in
velocity space by a Gaussian weighted scattering operator based
on the local Fokker-Planck equation.9
in a ripple well it scattered into
lost if E. > E .
1
W(j
If when the particle was
< 61/2, it was treated as
In order to speed execution time each magnetic
well was approximated as
B
B
(0
BTB
The wells occupy
outside well
(3)
=3
(l-6)
inside well
a fraction of the torus = 1/N.
This procedure
does not take into account the possible effects of banana orbits
6
on this diffusion from ions located off the magnetic axis, or
from highly energetic ions on axis with large radial orbit
excursions.
Figure
(2a) shows the result of following 500 particles
as they are lost while circulating around the torus; here
= 2.4 x 10
n
14
cm
-3
, E = 10 keV, initial ( = 0.05, N = 5, and
each ion starts out between two wells in the ripple free region.
In this case the velocity drag term is omitted and the number of
ions remaining are graphed vs. time.
ions left
decays to l/e of its
loss cone.
Figure
In 127 yisec the number of
initial value due to scattering into the
(2b) shows f(E) 19 yvsec after the ions are
Substantial spreading of ( has occurred
injected into the torus.
and a sizeable fraction of the ions now have E < 0 by virtue of
having a reversed v11 between ripple wells.
These particles would
not be treated properly by the orbit-averaged Fokker-Planck
equation.
Figure
(3) shows the l/e loss time : T
for the same plasma parameters as Fig.
vs. initial (
(2) but for varying N.
We see that for E = 0.3 changing N from 2 to 10 produces little
change in T ; furthermore, while
not go to zero as E
-+
T
k c2
for E > 0.1 it does
0, but asymptotes a finite value.
can be explained heuristically as follows.
For E >>
This
E , the
c
loss time will be the time for an ion with initial E to diffuse
toE c plus the additional time for the ion with E = Ec to be lost.
For
>>
= Ec
c 2, this tire will essentially be
ion is small. For
<
TZ
42TD,
since the loss tine of a
c the ion distribution function will
spread by pitch angle scattering to Lr ^
c between wells;
loss time would then saturate according to a loss rate
the
7
RA$
D
NE 0
(
k %
-
61/2) 2
c
RL0
NE-01/2
c 2) (1 -
(
02
ripple well when E =
E < E
2
0
c
CV
where
(4rD -
6) is the value of E inside the
C outside.
Equation (4) states that for
the total loss rate will be the loss rate of ions inside
the magnetic well multiplied by the fraction of time they spend
there.
c>> 6 1/2 , Eq.
For
c<< 61/2 we obtain T
the variation of -
.
(4)
4Nc2 TD c/(46
saturates.
Figure
).
4N c
Figure
2
T
For
(4a) shows
from the Monte Carlo code vs. 6 for v
6- 1/2 as expected unless 6 /2<<
T
-1
shows T
(4b) shows T
proportionality to N.
/v = 0.05.
, in which case T
vs. N and obtains the expected
We note the Eq.
(4) is only an estimate
of the absolute magnitude of Tk due to its heuristic derivation.
We note that as 1/N
-+
0 and the fraction of the machine
containing ripple goes to zero, v
should vanish.
From Eq.
(4)
we see that for E < Ec' \i a 1/N and satisfies this criteria.
S>>
2 T -1
c'
and is seemingly independent of N.
ever, as stated above,this expression for v
for
valid if it is much smaller than the v, of Eq.
2
>
N(o/
c
be that of Eq.
for N
+
(0
-
61/2) 2;
(4) even for
(4).
>
For
How-
c is only
This requires
for N sufficiently large vk will
> Ec and will vanish as expected
-.
All of the previous results ignore ion slowing, which will
8
eventually cause E
to this loss.
of Eq.
to drop below E
and no longer be subject
The first and second terms on the right hand side
(1) describe this slowing;
it has been shown that the
ion velocity will vary as 8 ,9
v =
is
where v
[(v
3
3
+ vc) e
the initial
3t/T
velocity.
taken into account in Fig.
as in Eq.
3 1/3
S - VC]
(5) ; when E. < E
I
(5).
This slowing down has been
Here each ion velocity varies
the ion is no longer lost and its
remaining energy is dumped into the plasma.
that for E
(5)
Figure
(5) shows
= 10 keV, and for the Alcator A lower hybrid heating
parameters almost half the ion power is lost through the ripple
wells;
for higher E
the loss fraction is even greater.
Here
1/
we let v11 /v initial = 0.3 -, (T /Ei) 1 / , as we assume that the
rf creates energetic ions in times short compared to
T D.
perpendicular acceleration would result in larger v
/v due to
collisional scattering.
Slower
This example illustrates that when the
fraction of the torus containing ripple is only 0.2, the loss rate
is still large enough to seriously limit RF heating efficiencies.
Again, it should be noted that particles orbiting off the magnetic
axis may be affected differently due to their banana orbits.
How-
ever, it has been measured that the energetic ion and neutron
10
production is localized to r/a < 0.2 during rf heating in Alcator A.
Finally, the rotational transform due to the plasma current
will reduce the effective ripple. 1 ' 3
It can be shown that an ion
vertically above the magnetic axis will see an effective ripple 6eff
9
6
0
=
-
(2
2
_sin-
(
))
(6)
where 60 is the ripple in the absence of plasma current and
a = 2rARN'q(r)],
cos N').
6
0.1% for r/a
60 keV.
and where we model BT = B0 (1 -
r/R cos
e
-
60(r)
has been estimated for Alcator C11 at less than
'\
This E
0.3, which results in E
being of the order of
is much higher than that of Alcator A; we would
therefore expect that the ripple loss process would be greatly
reduced in Alcator C from Alcator A.
-10-
ACKNOWLEDGEMENTS
The author is happy to acknowledge the valuable suggestions of Dr.
Martin Greenwald and Dr. Robert J. Goldston concerning this work, and
Wayne Hamburger, Magnet Design Group, M.I.T., for the calculation of
BT c1()
of Fig. 1. This work was: supported by U.S. Department of Energy Contract
No. DE-AC02-78ET51013.A002.
-11-
REFERENCES
1. T.E. Stringer, Nucl. Fusion 12, 689 (1972).
2. D.L. Jassby, H.H. Towner, and R.J. Goldston, Nucl. Fusion 18,
825, (1978).
3. J.W. Connor and R.J. Hastie, Nucl. Fusion 13, 221 (1973).
4. J.N. Davidson, Nucl. Fusion 16, 731 (1976).
5.
Fusion 17, 557 (1977).
K.T. Tsang, Nucl.
6. J.J. Schuss, et al, Phys. Rev. Lett. 43, 24 (1979).
7. L. Spitzer, Jr., in Physics of Fully Ionized Gases,
(John Wiley & Sons, Inc., New York) 1962.
8. J.A. Rome, D.G. McAless,
Nucl.
J.D. Callen,
and R.H.
Fowler,
Fusion 16, 55 (1976).
9. R.J. Goldston, Ph.D. Thesis, Princeton University (1977).
10.
D. Gwinn, J.J. Schuss, and W. Fisher, Bull. Am. Phys. Soc.
24, 998 (1979).
11.
M.
.
Greenwald, J.J. Schuss, and D.Cope (to be published).
-12-
FIGURE CAPTIONS
Fig. 1. Normalized BT vs.
for the Alcator A tokamak; a=10 cm,
R=54 cm and all values are calculated on the torus midplane.
A-R=44 cm and 6=2.9%; B-R=49 cm and 6=2.3%; C-R=54 cm and
6=2.5%; D-R=59 cm and 6=3.5%; and E-R=64 cm and 6=7.5%.
Fig. 2. (a) Number of ions vs. time from the Monte Carlo code,
ignoring ion slowirig.
ne= 2 .4x101 4 cm~1, E =10keV, v,./v
initial=0.05, deuterium, N=5 and 6=0.02. (b) f( ) at
t=19 lisec after launching.
of AE=0.1.
The
axis is divided into blocks
The ions having E<0 would not be correctly
described by the bounce averaged Fokker-Planck equation.
Fig. 3. T vs. E initial from the Monte Carlo code ignoring ion
slowing; here E.=10keV, deuterium, ne=24x101 4cm-3 and 6=2%.
L-N=5, X-N=10, and 0-N=2.
T
We see that for E initial=0.3,
is nearly independent of N.
Fig. 4. (a)
Uc
vs. 6 from the code ignoring ion slowing; n =2.4xl0'4cm~s,
E =10keV, deuterium, and N=5.
to 6-1/2.
6=0.02.
The dotted line is proportional
(b) T t vs. N for the same parameters as (a) except
The dotted line is proportional to N.
Fig. 5. Number of ions vs. time from the code including ion slowing.
Here ne=2.4x101 4 cm-1, E1 =10keV, v/v initial = 0.3, N=5,
6=2%, Te=lkeV and E =5keV.
At t>0.79 msec the number of ions
is constant since now E <E
and the ripple drift loss subsides.
(b) Fraction of ion energy deposited in plasmas vs. time.
A total of 46% of the initial ion energy is lost in the ripples.
-
I
I,0
0
ro)
0
C\J
0
uLJ
g I]ZI]lV\JON
01
0
0
00
00
LC)
fO
SNOI iO
0
0
Cu)
0
0
OJflN
0
i
I
I
Go
0
.......-
m4v
(\'
I
Ln
I
0
ro
0
Qa
N~
C\i
SNOI
jo
I
C\J
co
j3-L fN
I
0
I
7.0
6.0-
5.0
4.0
E
3.0
2.0x
1.0-
0.2
0.4
0.6
0.8
1.0
o
o
0
0
0
0
c\J
0
01
0
0 0
C~I.
2/4
k. -J
j
Cj
00
0
0
0
SNOI jO
0
0
j-V\JfN
0
0
0
(Nj
I
(NJ
I
7
0
q
I
I
0
Cd
co
D
0
d
d
VV\SV-ld.NI G]-iiSOdiG(
,k!dI N] NOI iO NOIIDVdi
0
E
I
0
LC)
0
0
(\j
0
00
0
o
98
0C\j
0
GIZ 1]V1VN JON
0
0
0
0
0
0
0
0
0
0
SNOI iO di~VJfN
0
0
co
-o~
04
Lo
C~C\J
C(\J
.SNOI~~-e
n
.o
7.0
I
I
iI
I
I
I
I
I
IL
6.0 --
5. 0 -
A
0
4.OF-
U,
E
I~1
3.0k-
2.0
X
1.0
41
I
i
0.2
I
I
II
0.4
I
I
II
0.6
II
I
0.8
1.0
LOZ
o
o
0
0
(OaS
TI)2'
00
0N
ro CI.
(Ds
H) /-
0
0
b E
0
000
0
0
SNOI
jo
0
0
djioAN
0
0
0
-I
I
C\J
(\J
I
-Q
C-)
Cl)
0
q
I
co
I
II
O;
o
(\J
b
o
d
v V l SV~1d NI GIISOdIG
L\Od]N] NOI AO NOIIDV di
0
0
E