tokamak equilibria with toroidal...
TRANSCRIPT
~' ~H~!~EHflX
Tokamak Equilibria with Toroidal Flows
FURUKAWA Masaru, NAKAMURA Yuji, HAMAGUCHI Satoshi and WAKATANI Masahiro Graduate School of Energy Science, Kyoto University, Uji 611-0011, Japan
(Received 16 April 2000 / Accepted 3 1 July 2000)
Abstract
Axisymmetric magnetohydrodynamic (MHD) equilibria with toroidal flows are shown with a
numerical code based on the finite elementmethod (FEM). In the present study, reversed magnetic shear
configurations with toroidal flows which simulate the JT-60U and DIII-D discharges approximately are
examined. When the pressure profile is flat around the magnetic axis, the separation between magnetic
axis and maximum-pressure position becomes large with the effect of toroidal flows. Except this point
the toroidal flow with a Mach number less than 0.2 does not change the MHD equilibria. As for the
aspect ratio dependence, both the magnetic-axis shift and the separation between magnetic axis and
maximum-pressure position are larger for the smaller aspect-ratio equilibrium when a toroidal-flow Mach
number is fixed.
Keywords: axisymmetric MHD equilibrium, toroidal flow, MHD equilibrium code, magnetic-axis shift,
maximum-pressure position, JT-60U, DIII-D, Iow aspect-ratio tokamak
1. Introduction
Recently flows in toroidal plasmas have attracted
much attention in experimental and/or theoretical
research of magnetically confined fusion plasmas.
Experimental measurements have shown that there exist
significant plasma flows in tokamak plasmas. Toroidal
flows can be driven by Neutral Beam Injection (NBI)
[ I] and poloidal flows by radial electric fields [2]. It has
been reported that flow shear can reduce macro- and
microscopic instabilities of plasmas [3,4].
In MagnetoHydroDynamics (MHD) stability analyses, static equilibria are routinely assumed and
effects of plasma flows are often ignored. Indeed it is a
difficult task to treat general plasma flows correctly
since (i) the mathematical structure of the equilibrium
equations with steady flows may be hyperbolic in the
presence of large poloidal flows [5,6], and (ii) the
system of equations governing the plasma stability with
finite flows is not self-adjoint [7].
However the problem can be simplified if some
appropriate conditions are imposed on the plasrna flows.
For example, several numerical equilibrium codes with
only toroidal flows were developed in the past. Kerner
and Jandl solved the Grad-Shafranov (G-S) equation for
MHD equilibria with toroidal flows using the Finite
Element Method (FEM) [8]. In their paper, some
numerical results including ASDEX-Upgrade equilibria
are presented and the dependence of magnetic-axis shift
on the ellipticity of the plasma cross-section and the
plasma flows is examined. Cooper and Hirshman obtained tokamak equilibria with toroidal flows based
on a variational approach [9] similar to the method used
in the VMEC (Variational Moment Equilibrium Code)
[10]. In their work, JET (Joint European Torus)
authors ' e-mail.' furukawa @ center. iae.kyoto-u. ac.jp
937
j~ ;~7 ･ ~;~~~Al:1 ~L-~~~~L
equilibria are studied. Semenzato et al. developed a
numerical code to calculate axisymmetric equilibria with
toroidal and small poloidal flows using the FEM [1 I].
Their numerical solutions are based on the continuation
method: the solution with a finite flow is obtained in an
iterative scheme that starts from a static equilibrium. An
example of JET equilibrium with a general flow is
presented in their paper.
We have developed a numerical code to calculate
MHD equilibria with toroidal flows for axisymmetric
plasmas. The MHD equilibrium equations are reduced to
a second-order elliptic partial differential equation, i.e.,
the G-S equation, if only toroidal flows are taken into
account [6]. In our code, the G-S equation with fixed
boundary condition is solved nurnerically using the
FEM, as in the numerical code developed by Kerner and
Jandl [8]. In the Ref. [8], the equilibria are obtained by
specifying the pressure and poloidal current profiles. In
our code, one can specify the pressure and the one of the
following quantities; (i) poloidal current profile, (ii)
safety factor profile and (iii) toroidal current density
profile. As for the matrix solver, we have applied the
preconditioned (incomplete Cholesky decomposition)
conjugate gradient method, which is one of the efficient
methods for sparse matrices.
In the present paper, we study a relation between
the magnetic-axis shift and the maximum-pressure
position. In the presence of toroidal flows, the magnetic-
axis shift is enlarged and the constant-pressure surfaces
deviate from the magnetic surfaces due to the centrifugal forces of the flows. We use the maximum-
pressure position as a measure of the deviation between
constant-pressure surfaces and magnetic surfaces. When
the magnetic-axis shift is enhanced due to the toroidal
flow, MHD stability properties may be affected by the
change of the equilibrium properties. For example, when
the deviation of the pressure contours and magnetic
surfaces may not be neglected, the effect of the poloidal
pressure gradient on each magnetic surface becomes
important in the MHD stability analyses. Furthermore,
the modified pressure profile including the effect of the
toroidal flow, may have effects on the heat and particle
deposition profiles in the NBI and the pellet ablation
analyses. Thus equilibria reconstructed from measured
data without the inclusion of the effects of the plasma
flows may be insufficient for understanding the stability,
heating and fueling properties. Since we have used fixed
boundary conditions in the code, our results are
approximate for the experimental conditions. However
the results from our code are useful for understanding
2000 ~F 9 ~
the general properties of equilibria with toroidal flows.
The rest of the paper is organized as follows. In
Sec. 2, the G-S equation with a toroidal flow is derived
from the steady state MHD equations. The numerical
method to solve the G-S equation is briefly discussed in
Sec. 3, where comparison of our numerical results with
the analytic solutions by Maschke and Perrin [12] is also
made for the confirmation of the correctness of our
code. The code is then applied to reversed magnetic
shear equilibria with steep pressure gradients and strong
rotational shear which simulate the DIII-D and JT-60U
discharges approximately [3,13]. It is found that the
magnetic-axis shift is not significantly affected by the
experimentally observed flows in JT-60U, although the
maximum-pressure position is largely separated from
the magnetic axis when the pressure profile is flat
around the magnetic axis. In Sec. 4, the magnetic-axis
shift and the deviation of the maximum-pressure
position from the magnetic axis are obtained as
functions of the aspect ratio. Low aspect-ratio tokamaks,
for example START, have an attractive feature that the
high beta plasrnas have been obtained [.141. It is found
that for the lower aspect ratio tokamaks, the magnetic-
axis shift and the deviation of the maximum-pressure
position from the magnetic axis become larger. For the
same flow velocity, the higher pressure (or density) will
be realized in the outer region of the torus in the lower
aspect ratio configurations. Concluding remarks are
given in Sec. 5.
2. Grad-Shafranov Equation including a
Toroidal Flow In this section, we derive the G-S equation
including a toroidal flow from the MHD equilibrium
equations, which are given by
v
v. ( pv ) = O ,
Pv ' Vv :: J XB
( =, )
VP o y p
/10 J= V XB ,
Vx(v xB)=0 ,
V. B=0 ,
-Vp,
(1 )
(2)
(3)
(4)
(5)
(6)
where p is the mass density, p is the pressure, v is the
flow velocity, J is the electric current, B is the magnetic
field, Yis the specific heat ratio, and //o is the vacuum
permeability.
938
I~~1:,;t~;~i ~-~~~~~~~~
tl)1 /*~1Fil
as
From Eq.
Tokamak Equilibria with Toroidal Flows
(6), the magnetic field can be expressed
B = fVc + Vep x Vx , (7)
where f is an arbitrary function of R and Z in the usual
cylindrical coordinates, and x and ip denote the poloidal
flux function and the toroidal angle, respectively.
Similarly from Eq. (1) the momentum vector can be
expressed as pv = gVc + Vc x Vh(X) in general, where
h(X) denotes a stream function of the poloidal flow.
Here we assume that the flow is in the direction of
symmetry (i.e., the toroidal direction) and
pv =gVep , (8) where g being an arbitrary function of R and Z. By
substituting the cross product of Eqs. (7) and (8) into
Eq. (5), one can verify that glpR2 is a function of x, i.e.,
Q(X)s g (9) pR2 '
which leads to
v = R2 ~2 (;OVc . (10)
The plasma current J may be obtained from Eqs.
(4) and (7) as
/10 J = (A*X)Vc+ Vf x Vc , (1 1)
where
A*= a2 LL a + a2 (12) aR2 R aR aZ2 '
From Eqs. (7), (10) and (1 1), the three components
of Eq. (2) can be evaluated. First, taking the ep
component of Eq. (2), we obtain
Vf xVep . Vx =0 . (13) Since f may be considered as a function of x and the
poloidal angle e, Vf may be written as
Vf = af Vx+ af Ve (14) aX a e '
Substituting this expression into Eq. (13), we obtain dfl
de = O and therefore
f = f(;If) . (15) The remaining two components are taken in the
directions of VR and Vx, which lead to
pR Q 2 ap (1 6) = aR x'
T~]ll , ~~~4til
A*X=-/10 R2 ap _ f df . (17)
aX R dX
Equation (16) describes the pressure gradient on each
magnetic surface and Eq. (17) is the G-S equation
including the toroidal flow through the pressure p (X, R).
Equation (16) must be solved simultaneously with Eq.
(3) for p and p. However, if the flow is purely toroidal,
as given in Eq. (10). Eq. (3) is satisfied trivially.
Therefore we may write p = pT(X)/mi, where p = mjni +
meme ::: m*n, n ni = ne' and T ~ Tj + T*. Here mj(m*),
nj(ne) and Tj(T*) are the ion (electron) mass, density and
temperature, respectively. The temperature is assumed
constant on each magnetic surface, T = T(X), due to the
high thermal conductivity along magnetic field lines.
We then integrate Eq. ( 16) to obtain
p=F(X)exp mjQ2 ( R2 -R:. ) . (18)
2T
Here the integration constant has been taken as p = ~(X)
along the R = Rax in the poloidal cross-section, where
Ra* denotes the major radius of the magnetic axis.
3. Numerical Results The G-S equation (17) can be solved if the source
terms dp/dxlR and fdf/dx are given with some appropriate boundary conditions. In the present study we
use fixed boundaries and impose x = O at the plasrna
edge. In our FEM code, 4-node rectangular isoparametric elements are mainly employed except for
the region around the magnetic axis, where triangular
elements are employed. Since the G-S equation is
nonlinear, we solve it iteratively. In each iteration,
elements are moved so as to follow the magnetic
surfaces. The area integration in each rectangular
element is obtained from the 2 x 2 Gaussian quadrature.
As the source terms in Eq. (17) we specify either (i)the
pressure and poloidal current profiles, (ii) the pressure
and safety factor (i.e., q) profiles or (iii) the pressure and
toroidal current-density profiles. The details of (ii) and
(iii) are given in the Appendices. As mentioned in the
Introduction, we have applied a preconditioned
(incomplete Cholesky decomposition) conjugate
gradient method for the matrix solver. For the convergence check, the following quantity is monitored
for each iteration,
1 ERROR = ~ Ni~ Ni~
l.h.s. of Eq. (17) - r.h.s. of Eq. (17)
r.h.s. of Eq. (17) ,
939
j~ ;~:7 ･ f~~~~~A~ i~:A7~~~~~~*
where Nj~ denotes the number of grid points except the
boundary points. To obtain a solution with an ERROR
less than 10-6 for 60 x 40 grids in the upper half plane,
typically it requires about 3 minutes on a Pentium 11 400
MHZ personal computer.
3.1 Comparison between analytical and
numerical solutions
In this section, we compare a numerical solution
with an analytic solution by Maschke and Perrin 1 12] for
the confirmation of the the correctness of our code.
Maschke and Perrin obtained the analytical solution of
the G-S equation given by
R2 X -X* = CP R~*
8. - I Z2 R2 R2 +
4 R~^ 4R~. R~^
(
- ) + 1 1 + yco2R2 exp yco2R2 y2 nl4
~/ 2 Rax 2 Rax
C ea~1 Ya) ) 1 ~ [ 2-2 ( +1 l] :=const ' exp 2 8 2Yco
- J ~~ [ 2= ' p PoRax~(X) yco const exp
X 2
(19)
(20)
(21)
yco2 m j Q(X)2 const (22) R~. ~ T(X) ~
where the integration constant X* is chosen such that X =
O at the plasma edge, 8a is a constant related to the
ellipticity of the plasma cross-section, and Ra* denotes
the position of the magnetic axis. We note that p = X
and fdf/dx = O are assumed in the G-S equation (17) for
this solution.
Figure I shows the flux contours of the analytical
and numerical solutions. The corresponding safety-
factor profile is plotted in Fig. 2, where the horizontal
axis denotes the normalized toroidal flux v//v/;a' Here we
have assumed that the aspect ratio is about 3 and the
Mach number (i.e., the ratio of the toroidal flow velocity
to the thermal velocity) is about 0.6. The plasma beta is
about 1.4% at the magnetic axis. The number of grid
points are 80 in the radial direction and 40 in the
poloidal direction in the upper half plane. The ERROR is
less than 10-6. Both the numerical and analytical
2000 ~~ 9 ~I
solutions are in good agreement. The slight difference in
the safety factor (in Fig. 2) near the magnetic axis (Vf/v(*
-~ -N
2.5
Numerical Analytical - - - - - - ･ -
2
1 .5
l
0.5
o
-0.5
-1
-1.5
-2
-2.5
¥ ¥1 ' ,, ¥¥¥.t,1/~ '
~l-'::~{~::'L/¥'~~~=~'~'-- -
l/7! ¥¥~
l '!
Fig.
9
8
7
6
5
4
3
2
o
1. 8 2.3 2.8
R [m]
3.3 3.8
~~~
1 Flux surface contours of both ana[ytical and numerical solutions. The analytical solution is
given by Maschke and Perrin [12].
Numerical Analytical
Fig.
o 0.2 0.4
V~/v,:~
0.6 0,8 1
2 Safety factor profiles of both analytical and numerical solutions corresponding to Fig. 1. The
horizontal axis denotes the normalized toroidal f]ux vrfv,;~-
940
~~~e~~~)~ 7L n~I Tokamak Equilibria with Toroidal Flows ~~iJll ~~~t4~
= O) is caused by insufficient numerical resolution
around the magnetic axis. The position of the magnetic
axis is R** = 3.0000 for the analytical solutions and R**
= 2.9993 for the nurnerical solutions, i.e., the difference
is 0.7 mm when the major radius is 3 m. Other numerically obtained equilibrium quantities are also
found to agree well with those given by the analytical
solution. We have also confirmed that our code can
reproduce the Solov'ev equilibrium [15] in the absence
of plasma flows.
3.2 Reversed magnetic shear equilibria
We first present numerical results of reversed
magnetic shear equilibria which simulate the JT-60U
discharges approximately [13]. The equilibrium
parameters employed here are; major radius R~*j = 3'37
m, minor radius a = 0.84 m, ellipticity K = 1.54,
triangularity 6 = 0.2, and vacuum toroidal field BTO = 3'4
T at R = R~*j' The temperature, density, and toroidal
rotation frequency at the magnetic axis are given by To
= 10 keV, no = 1.0 x 1020 m~3, and Qo = 5.7 x 104 rad/s,
respectively. The safety factors are assumed to be qo =
4.4 at the magnetic axis, q~i* = 2.1 at its minimum, and
qa = 3'8 at the plasma edge. The q~i~ surface is located
around the normalized average minor radius <r>/<r>a =
0.65, where <r> = ~~:1lc, S is the area inside the poloidal
cross-section of a magnetic surface and <r>a denotes the
value at the plasma edge. Unlike actual experiments of
JT-60U, we consider a plasma of deuterium (50~;~o) and
tritium (50%) and therefore the ion mass is assumed to
be 2.5 times the proton mass.
The temperature and the rotation frequency have
steep gradients inside the q~j~ surface. Under the
conditions employed here, the Mach numbers are 0.2 at
the magnetic axis and -0.1 at the edge. Note that the
flow direction at the axis is opposite to that at the edge.
In Fig. 3 we present the magnetic surfaces, pressure
l.5
l
0.5
~_ O
-0.5
~~
5
4
3
2
l
o o
10
8
~6 ~"
~4 2
o
0.2 0.4
<r)1<r>.
0.6 0.8 l
-l
- I .5
2.4 2.9 3.4
R [m]
3.9 4.4
o
6.0e+04
0.2 0.8 0.4 0.6
<r>/<r>~
l
4.0e+04
;~"~ 2 Oe+04
c~
o.Oe+00
-2.0e+04
o 0.2 0.8 0.6 0.4
<r>!<r>~
Fig.3 A model equilibrium of JT-60U reversed magnetic shear discharge with a steep pressure gradient and strong rotational shear at the norma]ized minor radius <r>/<r>. - 0.6. An aspect ratio A = 4.01, an ellipticity lc= 1.54, and a
triangularity 6 = 0.2. In the left figure, the solid and dashed contours correspond to magnetic surfaces and
pressure contours, respectively. The magnetic axis and the maximum-pressure position are denoted by + and x, respective]y. The horizontal axes for the figures of q, T, and (2 denote the average minor radius.
941
~~ ;~7 ･ ~~~~~~~A~S'~･:~-/~.-:r~~~~d~L'
contours, q, T, and Q profiles obtained from numerical
calculations. We used 60 radial grids and 40 poloidal
grids for the calculation.
The position of the magnetic axis is Rax = 3'5 128
whereas, in the case of the static equilibrium (i.e., zero
flow) with the same other parameters, we obtain R~~ -
3.5082 from our numerical code. The difference is only
0.55% of the minor radius, i.e., the magnetic axis
position is hardly affected by the toroidal flow in this
case, which is expected from the analytic estimate
discussed in Sec. 4. The maximum-pressure position
Rpmax deviates from the position of the magnetic axis by
6,8 cm outwards, which is 8.1% of the minor radius. It
should be noted that the pressure is no longer constant
on each magnetic surface. Standard MHD stability
analyses based on the equilibrium condition that the
pressure is constant on each magnetic surface, therefore,
need to be reexamined in the presence of plasma flows.
Figure 4 shows the enhancement of the magnetic-
axis shift due to the flow (Rax ~ R~~)/a and separation of
the maximum-pressure position from the magnetic axis
Rax)/a as functions of the Mach numbers M at (R -pmax
the magnetic axis. In these calculations, only the
magnitude of the rotation frequency is varied and the
rotation frequency profile Q(X)/Qo and the other
parameters are the same as those used in Fig. 3. It is
seen in Fig. 4 that the magnetic-axis shift changes only
slightly due to the toroidal flow. For example, the
increase of the shift from that in the static equilibrium is
only 2. 1% of the minor radius even for M = 0.4, which
is two times larger than the currently observed
experirnental value. In contrast with this, the separation
of the maximum-pressure position from the magnetic
axis notably increases as a function of M. For M = 0.4, it
is 14.7% of the minor radius. In Fig. 5, the magnetic
surfaces and pressure contours are shown for the M =
0.4 equilibrium. The deviation of the pressure contours
from the magnetic surfaces is much larger in this case,
compared with the case of M :: 0.2 shown in Fig. 3.
For comparison, we have also calculated a sequence of model equilibria with typical parameters
corresponding to the DIII-D experiments [3]. The
magnetic and pressure contours, q, T, and Q profiles are
shown in Fig. 6. The Mach number at the magnetic axis
is 0.38. In this case, the enhancement of the magnetic-
axis shift due to the flow R - Rst is 2 9% of the minor ' ax ax'
radius, which is similar to that of the JT-60U case with
the Mach number 0.4. On the other hand, the difference
of the maximum-pressure position from the magnetic
axis R - R is O 78% of the minor radius which is
pmax ax ' '
~i~76 ~~~i~ 9 ~f~ 2000 ~ 9 ~
quite small compared with that in the JT-60U case. As
in the JT-60U case, (Ra^ ~ R~~)/a and (R -pmax
R*x)/a are
plotted in Fig. 7. It is seen that R~~ax does not deviate
from the magnetic axis significantly, which is different
16
- 14 ~~~)
~ 12 ~ ~ Q~1 10
~
Q~~~8
~ ~6 ~ 1~~
Q~1 4 ~
Q~
~2 o
~1~ --~..
(Rax~R~)/a
(R pmax~Rax)/a
. Ja"
..er"
)eF"
Fig.
o O. 1 0.2
M
0.3 0.4
4 Mach number dependence of (R,* - R~~)/a and R,*)/a for JT-60U model equilibria with (R -p~""
reversed magnetic shear. The horizontal axis denotes the Mach number M at the magnetic axis.
~ ~ -N
Fig.
l .5
1
0.5
o
-0.5
-1
1 .5
2.4 2.9 3.4
R [m]
3.9 4.4
5 Pressure contours and magnetic surfaces for a JT-
60U model reversed magnetic shear equilibrium with the Mach number 0.4 at the magnetic axis.
942
~ Jf ~ju ~^m~ ~C Tokamak Equilibria with Toroidal Flows ~f]II , H~ ~~- ~~
l
0.8
0.6
0.4
ty
lO
8
6
4
2
o
10
0.4 0.8 O 0.2 0.6 <r)1<r).
1
0.2
~_ o
-0.2
-0.4
-0.6
-0.8
-1
l 1 .5
R [m]
2 2.5
8
~6 ~~s)
~4 2
o
O O.2 0.4 0.6 0.8 <r>/(r>.
2.5e+05
2.0e+05
l .5e+05
1 .Oe+05
5.0e~(~
0.0e+00
l
I~
(::
o 0.2 0.6 0.8 l 0.4
<r>/(r>.
Fig. 6 A model equilibrium of Dlll-D discharge with negative centra] shear. In the left figure, the solid and dashed
contours correspond to magnetic surfaces and pressure contours, respective[y. The magnetic axis and the maximum-pressure position are denoted by + and x, respectively. The horizontal axes for the figures of q, T, and
Q denote the average minor radius.
3
~ 2.5 ~S
~ ~ ~ 2 Q~l
~ Q<~c~ 1.5
~ ~~
~ - 1 ~:x*
Qf*
~ 0,5
o
--~- (Rax~Rsatx)/a
- ---- (Rpmax~Rax)/a ~
ldl'~iI"
""e'l'P'
"d"'~""(3'~"
__Jo
o O, 1 0.2
M
0.3 0.4
from the JT=60U case.
This result may be explained as follows. From Eq.
(18) the pressure profile can be expressed as
p = ~(X) exp L M2 (X) (R2 - 1)] . (23)
where R R/R*^ and M2 ~ miQ2R~^12T. On the midplane,
x is uniquely determined by R and therefore the pressure
and the Mach number may be considered as functions of
R. Thus we assume the pressure p and the Mach number
profiles as
2a p=1-(R -1) P (24) M= Mo [1 - (R - 1)2aM] (25)
Fig. 7 Mach number dependence of (R.* - R*t)/a and "*
R**)/a for Dlll-D model equilibria with (R -p~** negative central shear. The horizonta[ axis denotes the Mach number M at the magnetic axis.
The pressure profiles from the above formula are plotted
for (ab, aM) = (1, 1), (1, 2), (2, 1) and (2, 2) in the case
of Mo = 0.2 in Fig. 8. Note that the larger ap (or aM)
represents the flatter pressure (or flow) profile around
the magnetic axis. It is shown that the profiles with aM
= I and 2 are similar for a fixed a~ whereas the profiles
with different c~ differ much for a fixed aM. For a~ = 2,
943
~~ ;~7 ･ f^~~~i~*~~:~--･-"~* 2000 ~~ 9 ~
~~
l ,4
1 .2
1
0.8
0.6
0.4
0.2
o
a;p= I ,aM= 1
------ ap=2,aM:=1
- ' aj'=1,aM=2 -'-'-'-'-'-'- aj'=2,aM=2
(flc"/~
ll / / / /
/
/
/
*~~~'.
~~'~.
~~
¥ ¥
o 0.5 1
R
l.5 2
8 Model-pressure profiles p(R) defined by Eqs. (23)-
(25) for (ap, aM) = (1, 1),(1, 2),(2, 1) and (2, 2). The
lines for (ap, aM) = (1, 1) and (1, 2) nearly overlap.
The maximum-pressure positions are ~ = I and R
:: 1.3 for ap = I and 2, respective]y. The magnetic axis corresponds to ~ = 1. The seperation
between the magnetic axis and the maximum-pressure position is larger for the flatter-pressure
profile (ap = 2) around the magnetic axis.
~ s ( ~ ( =r ) o ~ d rB dA 21'1rdP B
dr dr dr Rmaj
+ poRmaj r ( d~'2 2 ) dP Q Q +2p , dr dr
With the boundary conditions
As(a ) = O ,
dA drS (O) = O
Fig.
~ is flat around the magnetic axis R = I . In this case the
small modification of the pressure profile due to the
centrifugal force of the toroidal flow has a relatively
large effect on R p~*^ .
For a:p = 1, however, p is peaked
at R = I even if M is finite. Therefore the maximum-
pressure position Rp~*^ hardly changes from the position
of the magnetic axis in the case of a~ = I .
Thus it can be concluded that the effects of the
toroidal flows on the magnetic configurations are
negligibly small for the experimentally typical flows.
For the equilibrium with flat pressure profile around the
magnetic axis, which is often observed in high
performance discharges, the flow effect may not be
negligible.
4. Aspect Ratio Dependence of Shafranov S h ift
In this section, the aspect ratio dependence of the
Shafranov shift and the maximum-pressure position is
discussed.
4.1 Analytic Shafranov shift
The equation for the Shafranov shift As(r) of a
tokamak equilibrium with a toroidal flow was obtained
by Green and Zehrfeld [5] in the large aspect-ratio limit,
which is given by
)
(26)
(27)
(28)
Here r is the minor radius, r = a denotes the plasma
edge, Be(r) is the poloidal magnetic field. The pressure
p, the mass density p, and the plasma rotation frequency
Q are assumed to be functions of r. The terms including
Q in Eq. (26) stem from the plasma flow.
The second-order ordinary differential equation
(26) can be solved for given equilibrium profiles. Here
we consider the following profiles:
p=po (1 - , (29) )
r a2
r Be Bea a '
p= po = const .,
Q =QO (1 - a 2 r 2
with which we obtain from Eq.
),
(26),
(30)
(31)
(32)
As(O) I ( 5 2 ) a = ~?+1+-Mp . . (33) 2A 4 6
Here A ~~ R~*j/a is the aspect ratio, pp is the poloidal
beta, and Mp is the poloidal Mach number I e
~?E 2 Po( P ) (34) B~
Mp2 = kto Po R~.J ~20 ~: M2 ~?, (35)
B~
with <･･･> denoting the volume average. In Eq. (33), 1/4
reflects the current density profile.
It follows from Eq. (33) that As (hereafter As
denotes the magnetic-axis shift As(O)) is inversely
proportional to the aspect ratio in the large aspect-ratio
limit when pp is fixed. As for the flow effect in Eq. (33),
it is proportional to the square of the Mach number M2.
When the Mach number is several 10 percent, which is
typical in the present day experiments, the flow has a
contribution of a few percent to As, which is small. It is
944
,t~ ~~ ~~ ~r~~~'RE)~
0.16
Tokamak Equilibria with Toroidal Flows
0.14
0.12 ~~~'
~~ 0.1 ~~ "~l~ 0.08
Q::_ O 06
0.04
0.02
d
6
~----~>---------
.;e---~ (Rax~RSt )/a --'--ar
p"' A(Rax~R~)/a ----<)----
l 2 3 4 5 6 7 8 9 10
0.29
0.28
0.27 -~
026 ~ . ~~ "Ca~~
0.25 i~
Q~_
0.24 ~:
0.23
0.22
10
9
-8 ~:~o
~ ~:7 ~ ~ Q~j6 ~
,~
Q~5 ~
4
Fig.
A
9 Aspect ratio dependence of the quantities (R** -
R~~)/a and A(R - Rst)/a The Mach number is held ax '* constant (0.2) at the magnetic axis. As the aspect
ratio A is decreased, (R** - R~~)/a increases.
confirmed with the numerical results presented in Sec.
3.
Figure 9 shows the numerical results of (R** - R~~)/
a, which corresponds to the last term in Eq. (36), and
A(R.x ~ R~~)/a for D-shaped plasrnas with pTO = 2,10Pol
p~o = 10%, and the Mach number 0.2 at the magnetic
axis. The equilibrium profiles are given as functions of
the normalized poloidal flux x = (Xo ~ X)/xo Where xo
denotes the poloidal flux at the magnetic axis, such as ~
= (1 - ~2)2 mjQ2R~x/2T = I - ~ and dlT/d~ = I - ~2.
Here IT denotes the toroidal current. As A is decreased,
(Rax ~ R~~)/a increases. In addition, A(Rax ~ R~~)/a seems
to approach a constant as A is increased. They are
consistent with Eq. (33).
4.2 Maximum-pressure position The pressure given by Eq. (18) may be rewritten as
mi !22r2(2A + 1)
2T (36)
3
~rJII , ~ft4~
p - ~(;O exp
= ~(;O exp * ) ( 2 mQ2R:. 2 + 1
AA 2T
(37)
Here we have used R = R** + r for R > Ra* and A ~ Ra*/r,
where r is a minor radius. From Eq. (36), Rp~"* is an
increasing function ofA, since the exponent is positive,
if r, Q and T are held constant. However, if the Mach
number is held constant, i.e, mi(~~R~x/2T is fixed in Eq.
is a decreasing function of A. Numerical (37), R pmax
1 2 3 4 5 6 7 8 9 lO
Fig. 10
A
Aspect ratio dependence of (Rp"," - R**)/a. The
Mach nvmber is held constant (0.2) at the magnetic axis. As A is decreased with the Mach
number being fixed, separation of the maximum-pressure position from the magnetic axis increases.
results also show this behavior, as presented in Fig. 10.
The profiles of~, mi(~R~x/2T and dlT/d~ are the same as
those in Sec. 4.1. As A is decreased from 10 tol.5,
(R -pmax Rax)/a increases by a factor of about 2.5. In Fig.
1 1, the magnetic surfaces and pressure contours are
shown for A = 1.5 (1eft) and A = 10 (right). The solid
and dashed contours correspond to the magnetic
surfaces and the pressure contours, respectively. The
deviation of the maximum-pressure position (R -pmax
Rax)/a is greater for A = I .5 than that for A = 10. In these
calculations, D-shaped plasmas with pTO = 10% and the
Mach number 0.2 at the magnetic axis are assumed.
As mentioned above, the pressure amplification
factor, exp[miQ2(R2 - R~x)12T], is larger for the smaller
A when the Mach number is fixed. Thus if the same
velocity of flow could be realized in tokamaks with
various aspect ratios, the higher pressure (or density) is
obtained in the outer region of the torus for the smaller
A configurations.
5. Concluding Remarks The nurnerical code to calculate general
axisymmetric MHD equilibria with arbitrary toroidal
flows has been developed. This code is flexible and
efficient compared to that in Ref. [8]. As an input to this
code, one may choose any of the following pairs of
profiles; (i) the pressure and the poloidal current, or (ii)
the pressure and the safety factor q, or (iii) the pressure
945
j~ ;~7 ･ ~~~~~~Ar~ ~~A~~-.._"*~1 2000 ~~ 9 ~l
1 .5
1
0.5
~o N
-0.5
-1
- I .5
,, *, i +xh l '*
0.5 1 1.5 2 R [m]
2.5
-~ ~ N
1 .5
1
0.5
o
-0.5
-l
- I .5
9 9.5 10 10.5 R [m]
11
Fig. 11 Magnetic surfaces and pressure contours for A = 1.5 (left) and A = 10 (right) equilibria corresponding to Figs. 9
and 10. The solid and dashed contours correspond to the magnetic surfaces and pressure contours, respectively.
The magnetic axis and maximum-pressure position are marked by + and x, respectively. The magnetic-axis shift
and separation of the maximum-pressure position from the magnetic axis for A = 1.5 are larger than those for A
= 10.
and the toroidal current density. The code can handle
arbitrary shape of plasma cross-sections. We have
confirmed that this code is consistent with the analytical
solutions of the G-S equation by Maschke and Perrin
with sufficiently high accuracy. Since we have used
fixed boundary conditions in the code, it is not fully
sufficient for experimental data analyses. However, our
code will be usable for studying the toroidal flow effects
on MHD stability theoretically, which is our future
work.
We have used this code to study reversed magnetic
shear equilibria with steep pressure gradients and strong
rotational shear, which simulate the JT-60U and DIII-D
discharges approximately. It is found that, when the
Mach number is less than 0.2, toroidal flow effects on
MHD equilibria are relatively small. For example, under
the JT-60U plasma conditions, even if the toroidal flow
rotation frequency is increased to a large value (twice as
large as currently observed), the magnetic-axis shift is
hardly affected by the flow, which is expected from the
analytic expression in the large-aspect-ratio limit.
Similar results are obtained for the DIII-D equilibria. As
for the maximum-pressure position, however, its
deviation from the magnetic axis in the JT-60U is
greater than that of the DIII-D case. This may be
explained by the flatter pressure profile around the
magnetic axis in the JT-60U discharges with an Internal
Transport Barrier (ITB). This suggests that the equilibria
reconstructed from the measured data without the effect
of the plasma flows may be insufficient for understanding the high performance plasmas with ITB.
As for the aspect-ratio dependence, our numerical
results have demonstrated that the magnetic-axis shift
due to the flow is larger for the smaller A when the
Mach number is fixed, although the flow effect is weak
for the Mach number of the order 0.1-0.2. Similarly, the
deviation of the maximum-pressure position from the
magnetic axis is larger for the smaller A.
If the same velocity of flow is assumed in tokamaks with various aspect ratios, it is expected that
the higher pressure (or density) in the outer region of the
torus may be obtained for the smaller A. The presence of
poloidal pressure gradient on each magnetic surface due
to the toroidal flow may affect the MHD stability. It is
shown that the pressure variation on each magnetic
surface may be larger for the smaller A equilibria.
Therefore. MHD stability of low aspect-ratio tokamaks
may be sensitive to toroidal plasma flow.
946
~~~~t~~~'i~-~'~~*BE^~~
Tokamak Equilibria with Toroidal Flows
Finally, it should be discussed whether the results
obtained here are changed or not by imposing the free-
boundary condition. When free-boundary equilibria are
calculated by changing a plasma parameter, for example
a Mach number, the positions of the plasma columns
and the shapes of the plasma boundaries may be controlled by changing the poloidal-coil currents. If the
equilibria with nearly same plasma boundaries can be
generated by the coil systems, it is expected that the
results obtained here may not change drastically.
However, for more detailed discussions, free-boundary
calculations may be necessary and it is our future
subject.
~~Jll , H~~~'4ti~
must be satisfied on each magnetic surface at equilibrium.
Therefore we solve the following two ordinary
differential equations for x and f in each iteration:
f , dX_ I d6V~ f dv ~ 2lcq R2
2dX df2 dv dv f~r f de
R2
d dX deV~lVvl2 f dv dv R2
o l f R, a p
+ /1 dev~r aX
(A5)
Appendix A: Specification of Safety Factor Profiles
To specify q profiles in equilibrium calculations,
one needs to express the source term fdf/dx in the G-S
equation in terms of q [16]. One expression may be
obtained by the definition of q, i.e.,
~dVl q dX ,
Where vf is the toroidal flux function. Since
= f Vf I dRdZt 2lc R = J , 1 dv dev~r f
2jT R2 we obtain
= f , q I dev~r f dX R 2 27T dv
(A1)
(A2)
(A3)
where v E V/Va' V is the volume within a flux surface,
aR az aR az Va iS the whole volume, ~r~g ~ R (~"~~~~~ ~~~) is the
Jacobian.
The other expression may be obtained from the
surface average of the G-S equation,
f ( ) dO a V~r Vv 2dX av R2 1 1 dv
=-po~y dev~r ap
aX
- f fiLL de V~~ (A4) d;C R2
This gives an additional relation betweenf and q, which
(A6)
which give the source term fdf/dx in the G-S equation.
Appendix B: Specification of Current
Density Profiles
Specifying the current density profile is more
straightforward than specifying the q profile. The total
toroidal current IT may be expressed as
= f o IR ap f df /101T - dRdZ /1 R ax +- (B1) R dX
Defining I = polT127c yields
f xo dl __ I ap dX 2lT dev~r p aX
( ) V~rx ~ R aR aZ _aR aZ aX ae ae ax '
f + R R2
d f
dX' (B2)
(B3)
These equations give the following expression forfdfl
dX, i.e.,
d f 1
f ::- V~r f dX de ;f R2
o l d I a p f R 27c + /1 deV~rx aX (B4) dX
Therefore the source term fdf/dX in the G-S equation
may be replaced with this expression.
947
[1]
[2]
[31
[4]
[5]
[6]
[7]
[8]
j~ ;~'7 ･ ~~~^~:~~~~:~--･~'~~*
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~i~76 ~~~i~ 9 ~= 2000 ~# 9 ~
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948