Revista Brasileira de Aplicações
de Vácuo, Vol. 11, n. 2, 1992.
BDarSTRAP
CURRENT FOR SMALL ASPECT
RATIO
D.L.G.
P. H. Sakanaka and
TOKAMAK EQUILIBRIA
Andreatta
Instituto
de Física,
Universidade Estadual de Campinas
13081 Campinas, SP, Brasil
C.P. 6165
-
S. Tokuda
Japan Atomic Energy Research Institute
Naka Establishment, Tokai-mura, Ibaraki, Japan
ABSTRACT :
We
A joint project, TBR-2E, has been proposed
by three Institutions in Brazil: University
of são Paulo (USP), State University of
Campinas (UNICAMP), and National Institute
of Space Research (INPE), with the
cooperation of Oak Ridge National
present equilibrium features of the
very small aspect ratio tokamak, TBR-2E,
with the aspect ratio of 1.6, which is
being designed in Brazil - a joint project
with the participation of the University
of são Paulo, the State University of
Campinas, and the National Institute for
Space Research.
Laboratory, USA 181.
The design values of TBR-2E is given in
Table I, the expected plasma parameters
are given in Table 11 and the main cotes
for the project is shown in figure 1.
The equilibria have been studied by using
the SELENE-J cede developed at JAERI,.
Japan, by Tokuda et aI. We have
concentrated our study on thedetermination
of the stability limit by using the
critical pressure criterium for ballooning
stability and Mercier criterium for other
MHD medes. The B-limit values were
calculated for the case of the noninductive current and found that its
maximullllies at elongation of 1.7.
Increasing the triangularity, the B-limit
values increase, but the maximum continues
to stay at the garoevalue of elongation.
Major radius
Mirrar radius
Ro = 0.39 m
a = 0.25 m
Maximum elongation
Toraidal field at Ro
Plasma current
h/a = 1.7
=
0.50
Qf lindrical safety factor
Plasma pulse duration
Table
We have algo studied the effect of the neo
classical transport properties by changing
the plasma temperature (ar B values). In
particular, we have studied the trapped
particles and bootstrap current. We have
found that at temperatures as low as 600 eV
the transport is already in banana regime
and that the bootstrap current may account
for a significant part of the total plasma
current.
I:
Main
= 0.63 T
Ip
=
260 kA
qc = 4.5
T
= 100 ms
p
design values for TBR-2E
Maximum average density
Average electran
temperature
One-tUTI1 loop voltage
(Z ff
Maximum B (Troyon)
Table
B<j>
lI: Expected
e
<ne>=1.0xlO20nr3
Te = 700 eV
= 2) Vloop = 2 V
8 = 10 %
plasma values for TBR-2E
1. INTRODUCTION
In section 2 the physical model is briefly
described. In section 3 we report
studies
Triggered by Troyon scaling 111, there
have been many studies of very small
aspect ratio tokamaks 12, 31 which show
that, from the stability point of view, as
one lowers the aspect ratio the plasma
beta value increases. There are aIso
several proposals and projects for the
construction of a very small aspect ratio
tokamak 15, 4, 6, 71.
of equilibrium stability limit (mercier
and ballooning) using the equilibrium cod~
SELENE-J, developed by S. Tokuda et aI 191.
In section 4 we present
the study of the
bootstrap current generated in TBR-2E
tokamak as a function of plasma current.
100
I
Revista Brasileira de Aplicações
Z
PHYS:t:ÇN.:::)!QDEJH
o
de Vácuo, Vol. 11, n. 2, 1992.
'jV\
c':"'õC?L
'o"!
"\"
~oo"
000.
where F
=
:13
',," ,',
,o,,~;..; U'L(~\;
, ,
RB
fie~d)"ot.!:H:'!"Graçd~Sha.fri'\:o.ov,
" o..,
tI
,
=
a2
a2
lI:~:--aR2°
,+,a z 2:
1n equation,
0(:2)1J<I>'
i5
+
t
= -R2~
RJ
r
"
CO"",:,'Co'"
i ',i:>
= O,.
RJ.
'I'
fi
;,'0, J,..'
.' ")j
+
<J
"",0
:0
0",0,
;'--;:,
~ r
B>
'
10
00
,
'
'j ,)"
'
,
"
0'
0
-
0.0
,-",
-- ..
'
'eoef:t;icieiifs:, of
~,;:
','
,
',,~:
.
êlectroris
CÇ>ulomb°
°j
..'
,0,.,',"-,
"'-"'--"c':
07
<,c
",cor
that the
'
~quil,ibrium i.s.,oc;leteqtined"by
~pe,cifying the
density.
an<t--t;~mpej::.p.tur~
prof:L1;es,
ofo th,e
o
el,-ec:tr:91\and
',~d~s~~t~t~:",the, j'IJJp.soi?riate
boundary
anç1,co:nstr<:~i.n-t~.con.qitipns
o .In
tl;l~~'Ly.rP,Fk thêytotbaL
°t?ip~qa.li::urr!:'!n.tI:I,p,o
is
employeq
'i
P
lI"
,C}~ tr~~
o'
=
1opS-'
(5)'
'000 '
_1
=:-
IpS
,'5
;.
'
'.
4
:
,,
,
'
,
p.O)
D "'- I o,
~
--."
,>
"~
"27f'd'w..<,B,>,oo"
1
,
I BS' ,;:
.occ
, 9~:"r..~n,
,"' ~: 'L
<
"
-;
con,s:t,r;,a,i'1t;.
cond:itipn:
,I
'
+"
'o
OHM,~
"",
1r?,':"ere;.o
o
r-'
' "'0"-
!JniytJ~~ough
'.""
,
,~,;
,
,
'--~,
'êc,
,:-."c;
forces 6í'the impU'rity ar'e
:o,1rnpqfit,y"t;!f,-fe,ctE!
enters in,
",-,
+
o
c.
(C;:irbon),i-S:"
'" ".
,
,j
-:,'~,àTli,~i.9ns;op::!
(4)
F~Fi =ó -.]1O i F20' r. ~o +,
F'<')~2°>~,:.'~o
dlj!°
'<B'2>"ç11Jt-",
f
]
,.
:"'-0;""'..'
~ and°lnaiI;l;iôn~>'
r r'
dF
'
o>. .
.r
.o::;,~, o'
The quantityd
c:,!.ll be express,ed in terms
oí the f1ux s~rÍ:ace_aver'aged para11el
current dens,itX1'<;:t,oB>
"
,~<j
negl,eçteo(L
'
'
:
assumptions
i'lrec;
", ~" ?"
';""'i
Equation 6 demonstràt~s
{,O,
A:!i.'itP ;l;'i"",;nS:9)
,
,,'...
",
thermodynamic
o
with p = p(lj!)being the equi1ibrium pressure prpfi~le Equ,ation (3} is for
,t!:1'e 0';"0
:-e.gion cii1§.id~et,:t:he.p1a.~ma :p.nd-eq.uatiçm-,
(4.),: t
t
';',
" "
'o ""-"~~ '~;v',,;:::,;":t.,
o only;bne.~mpurity,,'ion-,consider-ed;
j
current
',.'
",
Q,nT i
o, p+~smà Li~, i~1 ã;') staf:Lo~1ry
s,tale,::~.<E',~>' 'f
= <B~;V:~?Vi.P~~~ , wi~h"' ~h 1?~J:-PgtÍ1~'~S)l1?:'J:urp
°,0
1n, vacuU!JI,o,
\:', ',' 2'
F
ion-speC;:L7~{, pl<;lS~~~
,,'the :neoclassJ.caL
"
c;cY'
00
muHip1e
used:
(3)
'r:{
5E'dlj!"lf;o"
~j]o
',-, ".. . "-',
vo1ta907'-~'O
- J:.pdF
dlj!
d
"':)
anel'1aa ~;[êo.'tq~-B~~g~n~~i;i,
ç:6)iíOnlb <;:ÓY~~$iÓn
t~me, R~f~~§~o i~h[101~,C Tq,é i~o§~q..é;i~!ü:s
ANt~
,:R'íiRO
dlj!]10
,-'i,oo':",'
',1'\
1n this work, the" following
(2)
,"
thetQJ:oj}da1
density given by
,
and
a
1
,
:eCO,';;,';'
for.finite4aspect ratio, arb~trar~ocrosssecttoní ail col1isio~ality regime~ and'
oegu..at-ion,!'cS
2",J~:,
:'1"":,'
',L
'Ó)i '..-1:, o"
R]10J<I>'e'o' ~,
po1oida1
f1ux fhnction
lI*1j!
where Ij!is the
the operator
'o"
1:~: ~~!g~~~ft§~a~~~~'
~~~i~~~~~~:~r~;~:i~::~
(Bt' toroida1 magnetic
derl.-ve.d":"'h' o)
Api.zIT;
,
,t;,C-G.aJ,ot,o:;
V;$~"~~:"':~o-,.+~~:f;~,~
f=
,\,
d Q,nPi
. "
"'0'
By expressing the equi1ibrium magnetic
'o
~:','T"
'1'
'r
o;
~,jr")'
':'e"":
fietç1as0000
.'
C'o."
:;aJ-:,
,
"
'-
OJ
surfaceaveraged para11el current,
(5) obeys the
<Jo~>, 4nequation
The
+
generalized Ohm' s law
+
+
<JoB>
+
=
+
110,
+
<q>oB'>OHM
+
lhl o" Ó .
:t
+
<JoJ::S>BS
+
,;
I~~j~Y..f!~d~~~r;~r
~Y.
:t
<JoJ::S>NOHM
o'Ie
--. "
L
(6)
+ +.
where
<JoB>OHM is the
<Jo)3~BS }:Jle
,;PRptos,tr~w
{
IBS=i',,:
ohmic current,
,o
.-F,urr.ent!p.I'}~,c'"
T .
O"'
,
'"
-,2-:'1,;0
O'r
<llB'> ÉS::'O;r 1:: ,;
l
'"
v :J fP F [<p ,>c~V~
-
:
f
,,,o
":'1
C
"
,o-
'
,
,,
<,133:l':
",
,
,
,d o ,;,'
'"
.
;'
j'.
"
i,"";;";:):-'"
Ite::r;:ejps
.J
-1: -
-'!O'~,."
,:it~oth~~R~a~~g'r1'etic
",'
:,c;urre,nt
<,JoJLt'!OH,MJ,h~
the
extern'al
flows to maintain charge neutrality, 70
conditions on each flux surface, IOHM is
The, ohniic
c
non-O-p"mli J-E 4FFen t Ç!!,~;,~~n' by.
o o
,
spurcj=!!3",.êu,cho
"i'lS"o N,B1, canel. R.F-oYJa~e.
'andthe'
J:Joots"'tràpcurr'ents
are'
Pfir,sch-:;'gchlute,1i::,,;retu{n
,.curr,ent
~'-+
and
,is,?1;:he
tb~OIl11!tc,cur.:r.i~nt"'0a!1d'o.IB:S'
boo1:;!?t,I(ap
:;p;: :::~ /~~~'<~,i,.'
",'3
",i
<1"13>
.
°l'~~::;i;,~/,~:
.L"
..'~E.
",--,.L
i
",,-,
b
such
"C)H
'B~
APe
t
:;::r~;p:':'::~~
:~'!if~~
=
d~1
'
ATe =
cI\jJ
c
' o.
!10f
,the
-
"
ti'
,)
c, C'i
"s.atoisf,ied,t
.cnon~
",f"--C"
~
,>8:1
c~lQ),ois
,}:'1E(.r.eg,ar,dE2,d",as
eigenval,~e..,,<,,;;:o'T
~
: -
tag,e L :L,si'u,p'd<'lt~dat.~aqh o~it~F9tiqn
thàt,,:~8.uatAon
VL -,ca,n
'r~r~
"~C 1, -' So,
"
'"""..:
":',,
"de
:,;"
,,:::
'" :1'.1['-'
'::'~
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So,tvÍf?:.(Lo~0:H1.1p.tJ.gn ",.;1.2)...:ct;ogej;.,I:1,er wi,-,t,h
, J'. q
equa1::.i:9ns",(5)..éj1)d
(,9+ iter,,!-t,ive,ly,
~L '.(OnE~
;7~
tur;no:ypl
=-F(LeA,,+.LeA,)
L,C?pr;reDk
whiph
inear;
'!o'P'f_l
00,
:i;-i~';
~
,f"
'"
,,"-
,.,
<
'
=
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~:iss -i'i!~e];i,c.a
l-ly,)i;n,Y~:rt~g
r!;>y
us in,9, o)t;.h~ -(~óu.b.1;e ,)Ç;y.c,Hi,c 8-+:eg1:1ct i, <P;rl I1)E;'!tfQ-odr,
in - ~1:1_EhC'3~tEE~i",!-gf,Ç9q§ç!i~atl?;AJ1ii
Z.) l~itl1,
'
.ti.9,?sEtmb'3fi,~1?4--bot!pqFY
1J]~1:hod::.f.'1.-,l2-1
~', J ~
~
Revista Brasileira de Aplicações
3.
LIMITING
de Vácuo, Vol. 11, n. 2, 1992.
BETA VALUES
4.
FOR TBR-2E
To solve the set of equations (2, 3,4 and
5), we start by giving a set of vertical
field and elongation coil positions and
the profiles of ne(~)' Te(~)' Ti(~),
Zeff(~), and <j.Ê> in a general forro:
f(~)
=
(fo - fa ) (1 - ~)CL(~) + fa , (11)
a(~)
=
ao + (aE
-
ao)~m
.
(12)
Besides that, we have taken other
parameters of TBR-2E and have chosen the
triangularity, O, to be 0.3. By varying
its elongation, K, from 1.0 to 2.2 we have
checked the 8-limit, i.e., the maximum
plasma 8 value before plasma becomes
either ballooning ar Mercier unstable. We
have increased the plasma temperature to
insure the increase of 8.
Figures 2 through 9 show a typical ruo for
this study. In particular, we have taken a
non-inductive current profile which brings
a safety factor profile with qo being near
unity and moderate magnetic shear.
"
BOOTSTRAP
CURRENT
COLLISIONALITY
AND NEOCLASSICAL
FOR TBR-2E
We have ruo equilibrium cases for TBR-2E
taking a close look at the neoclassical
collisionality parameter and the bootstrap
current.
The ratio between the average fraction of
trapped particles to that of circulating
particles is given by
ft
(13)
9 =t
fc
where
1
3 < B2>
fc
4(B
=
ft
1
max
-
ÀdÀ
)2
i
o
(14)
</l-ÀB/B
max
>
(15)
fc
Now the average transit frequency of the
species a is defined by
=v
w
ta
Figure 2 plots the <J.B> profile; figure 3
shows the pressure and the q profiles;
figure 4 showsthe pressure derivative and
the critical ballooning pressure
derivative. As the temperature is increased
the critical ballooning pressure
derivative (dashed liDe) are raised. When
this curve goes above the pressure
derivative at some points, the ballooning
instability sets in. In particular, figure
4 is the limiting case when plasma tas
just become unstable. Figure 5 is the
Mercier creterion (thick solid liDe) and
the three different terms which the
Mercier criterion is composed of: shear
term (dotted line), well term (thin solid
line), and the destabilizing term
(dashed
line). Figures 6 and 7 show the pressure,
q, De' and Te profiles in R-space. Figure
8 shows the current density term: noninductive current (thin solid line),
pfirsch-Schlüter current (dashed), and the
total current profile (thick solid). Figure
9 shows the contour plot of ~(R, Z)function.
where
Ta
1
(-)
- vea
-
I~B
(16)
Roq
v'g= I (V1jJX ve) .vel-\
vTa
=
12Ta/ma'
RO
is the position of the magnetic axis, and
q is the safety factor. The collisionality
parameter v*is given by 1121
v*
=
~
2.92
(~)
~B
The three collisionality
species can be written.
Pfirsch-Schlüter
(17)
<B2>
«B.VB/B)2>
regime:
regimes for each
!
1«
WtaTaa
Plateau regime:
1 <
--<
1
< <1
,
v* W
Banana Regime:
T
ta aa
1
«- 1 .
v
W
T
ta aa
Now, when there are no trapped particles,
neoclassical conductivity aNC reduces to
the classical Spitzer conductivity a sp1' tzer
.
As we increase the temperature the stable
equilibrium configuration becomes either
ballooning ar Mercier unstable. We have
plotted the 8-limit versus the elongation,
K, in figure 10, for two values of the
triangularity, 0= 0.1 and o = 0.3. We see
that the maximum 8-limit lies about
elongation of K = 1.7, and the 8-limit is
larger for larger triangularity. For K =
= 1.7 and ó = 0.3 we have 8 = 0.075.
This
calculations is the most stringent 8 limit
calculation.
If we choose the critical
ballooning pressure derivatives at every
flux s~rface, this 8 limit might be largeL
Figure 11 shows the partition between
trapped and circulating particles forthe
case of Te = 600 eV. The trapped particles
occupy between 60% to 80% of the total
particles, which is very high.
ln figure 12 we see the ratio between the
neo-classical and Spitzer conductivy (1),
which reaches about the value 0.4, and the
collisionality curve (2), which means that
when it is above 1 it is in banana regime.
As one can see, the major part of electrons
is in banana regime.
102
I
i
Revista Brasileira de Aplicações
de Vácuo, Vol. 11, n. 2, 1992.
5Ct'
.)0'
,:>
t"
v
~."
'"nG
'"
2720
"
1030
~650
";;f',.
"""""'::"C::':;,,~
4,0
1050
'..
6.0
-"...""".,.
"i
5}O
(\J
E' 3.0
'"',M'
"
~i
~:
8
4:b
W
C1>
0:::
~i.2.0
3.0 tT
(f)i",
W
0:::
.-"
"'0'
'
a.i
81'
;1
2,0
:J,Q
"(\J
i_o, ~-,~..-,-,~
f',
o
1.0
~
O
O
Q
,--
,.
1.0
Figure 3: p and a versus
Figure 1: Main cotes (in millimeters) for
the design of TBR-2E
'"
:0r--
~
- .j
c
,o
...'
l'
I
~
19
4.0
"t",
1:
c.
\
-2;0
I
\,
a.
3.0
<
73;9.
,
+
/\
i
-,o _.,,-~,
.
lho 2,0
J-?
'V
"'O"
-
,
'0
0.5
O
"",
0.5
1.0
~
'o,.
Figure 4: p' and P'cr
T
;...
-4.0,
I~O
o
,'..""'
'~
1.0
..,~'...,
.,<'
.:;.'~
r-.
Figure ~: <J.B>
versus v~
;",
103
.\.
"'...
versus ~
~
Revista Brasileira de Aplicações
de Vácuo, Vol. 11, n. 2, 1992.
2.0
1.0
>(,!)
a:
w
z
w
O
Mer
Well
1.0
1.6
1.4
-1.0
Dest
""""-----------
0.8
,,/
-2.0
1.2
->
0.8 I1.0
,;;- 0.6
l
0.5
O
-';0.4
G>
te
u
1.0
0f
Figure 5: Mercier Criterium
..
0.6
0.4
0.2
0.2
o
1.0 2.0
3.0
4.0
R
5.0
6.0
Figure 7: Ne and Te versus
o
7.0
R
6.0
4.0
5.0
N
"ze 3.0
q
p
4.0
~
w
a:
~
VJ
3.0
2.0
w
a::
a.
O"
2.0
3.0
1.0
1.0
o
1.0 2.0
3.0
4.0
R
5.0
6.0
o
7.0
2.0
n 1.0
Figure 6: p and q versus radius
o
"
'" ,;
,
-1.0
1.0
Figure
,
",'" J' PS
2.0
3.0
.8:
4.0
R
6.0
Current densities
versus
104
5.0
radius
7.0
Revista Brasileira de Aplicações de Vácuo, Vol. 11, n. 2, 1992,
5.0
4.0
3,0
N
1.0
5.0
2.0
4.0
1.0
>-
o
;iz
~
O
1.0
2.0
3.0
4.0
R
5.0
6.0
7.0
..
G.I
N
3.0
'õ,
o
0.5
b
......
Figure 9:
b
contours plot
8
o
7
0.5
O
1.0
~
-- 6
'I-
Figure 12: Ratio between Neo-classical
and Spitzer conductivity
(curve a) and collisionality
I:' 5
E
><
(curve
b) versus
~.
C!l..4
3
-o-
l::.. 0.3
~
l::. a 0.1
2
0.5
1.0
1.5
2.0
2.5
K
0.6
Figure 10: S-lirnit versus elongation for
different triangularities
0.5
0.4
1.0
- Q.
......
(J)
0.3
0.2
0.1
"
0.5
0.0
"""',
""""~
0.0
fe----------....
o
O
0.5
0.2
0.4
0.6
0.8
1.0
1.2
IE:~J"
Figure 13: IBS/lp versus IESJ
different I .
P
1.0
~
Figure 11: Trapped and circulating
particle ratio versus ~.
105
and
--
Revista Brasileira de Aplicações
de Vácuo, Vol. 11, n. 2, 1992.
121 B.A. Carreras, L.A. Charlton, J.T.
Figure 13 shows lhe dependence of lhe
ratio IBS/Ip with IESJ,where IBS is lhe
bootstrap current, Ip is lhe total plasma
current,
E
Hogan, J.A. Holmes, E.A. Lazarus, W.
A. Cooper, and T.C. Hender, in Proc.
of lhe "11th InternationalConference
on Plasma Physics and Controlled
Nuclear Fusion Research", IAEA, Kyoto
(1986) paper IAEA-CN-47/E-1-2-4.
= a/Ro is lhe inverse aspect
ratio, and lhe current-beta
SJ =
(20)
PdV.
1
Ro\.loIZ
given by
P
J
The empty square points are values
calculated for lhe total plasma current of
Ip
=
400 kA,
and
lhe dots
are
for
Ip
=
U.-K. M. peng and D.J. Strick1er,
Nuclear Fusion 26, 769 (1986).
\4\
E.A. Lazarus et aI., "A Feasibility
Study for lhe Spherica1 Torus
Experiment". Oak Ridge National
Laboratory, Report ORNL/TM-9786.
151
T. Todd et alo "START Project",
Culham.
16\
R.M.O. Galvão, L.C.S. Goes, G.O.
Ludwig, A. Montes, and M. Veda, in
Proc. of "Fusion Energy and Plasma
Physics", Rio de Janeiro, pago 471,
260
kA. These two curves are almost
superimposed. Further verification shows
that these curves depend very little with
lhe profiles. As a result we can say that
for very small aspect ratio tokamak lhe
bootstrap current can occupy a large
portion of lhe total plasma current.
5.
13\
CONCLUSION
(1987)
1n lhe present work we computed equilibria
for lhe parameters of lhe very small
aspect ratio tokamak, TBR-2E, which is
being designed in Brasil. We have paid
close attention, first, to lhe S-limit by
varying lhe elongation with fixed
triangularity. We have found that lhe
maximum bela is obtained for values of
bela near lhe natural elongation of lhe
machine, particularly for lhe aspect ratio
of 1.6 lhe maximum lies around K = 1.7.
Two values of triangularity we examined,
0= 0.1 and 0.3. For larger o we have
larger bela maximum.
jl
181
"Genesis and Scientific Objectives of
TBR-2E - Tokamak Brasileiro 2E",
Instituto de Flsica, USP, são Paulo,
.
T. Takeda and S. Tokuda, J. Campo
Phys., 2l, 1-107 (1991).
1101
S.P. Hirshman and D.J. Sigmar, Nucl.
Fusion 21, 1079 (1981).
1111
Y.B. Kim, "Moment Approach to
Neoc1assical Flows, Currents and
Transport in Auxiliary Heated
Tokamaks", University of Wisconsin
Plasma Report UWPR 88-2 (1988).
112\
S. Tokuda, T. Takeda, and M. Okamoto,
J. phys. Soe. Japan, 58, 871 (1989).
This work was supported by JICA - Japan
Intern~tional Cooperation Agency,
JAERIJapan Atomic Energy Research Institute,
Fundação de Amparo à Pesquisa do Estado de
são Paulo and Conselho Nacional de
Desenvolvimento Cientlfico e Tecnológico.
REFERENCES
111 F. Troyon. R. Gruber, H. Saurenman,
S. Semenzato, and S. Succi, Plasma
Phys. and Controlled Fusion 26, 209
(1974)
-
106
........
G.O. Ludwig, A. Montes, P.H. Sakanaka
in Proc. of IAEA-TCN on Research
Using Sma11 Tokamaks. IAEA-TECDOC-519
page 111 (1988).
191
ACKNOWLEDGEMENTS
.
171
(1991)
The second point which we have
concentrated our study is lhe examination
of lhe neo-classical properties of TBR-2E.
We have found that for a temperature as
low as 600 eV, this tokamak operates at
banana regime. We have a1so obtained a
very general curve, lhe dependence of
1BS/112
wi th IE: SJ. This curve does not
depena much on profiles Dor on total
plasma current.
.