Vol.2, No.2, 95-97 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.22015
Magnetohydrodynamic equations for toroidal plasmas
M. Asif
Department of Physics, COMSATS Institute of Information Technology, Lahore, Pakistan;
Received 23 November 2009; revised 4 December 2009; accepted 30 December 2009.
ABSTRACT
A set of reduced MHD equations is derived us-
ing the standard energy balance equation. By
applying assumption of internal energy, i.e.
constuR 
2, a set of reduced magnetohydro-
dynamic equations are obtained for large aspect
ratio, high

tokamaks. These equations in-
clude all terms of the same or der as the toroidal
effect and only involve three variables, namely
the flux, stream function and internal energy.
Keywords: Magnetohydrodynamics; Aspect Ratio;
Viscosity
1. INTRODUCTION
Magnetohydrodynamic (MHD) instabilities [1], such as
tearing modes, play an important role in plasma behav-
iour of tokamaks. They may influence both the particle
and the energy confinement and are considered to be one
of the main reasons for disruptions. Hence they have
been studied by many authors [2-4]. The method of re-
duced magnetohydrodynamic based on a large-aspect
ratio expansion has provided a powerful method for lin-
ear and nonlinear numerical computations.
In previous studies [2-4], the density of plasma is as-
sumed to be constant. This assumption is valid in cylin-
drical geometry because the divergence of velocity is of
the order 3

with

being the ratio of the minor ra-
dius to the length of cylinder. Thus we can neglect the
divergence of velocity in the continuity equation. By
applying assumption of density, i.e. 2
R const  , Ren
et al. [5] has derived the reduced MHD Equations in
toroidal geometry for large aspect ratio, low

toka-
maks. Since the divergence of velocity is of the order
2

in toroidal geometry the same order as the toroidal
effect, and cannot be neglected. Hence the assumption of
constant density does not agree satisfactorily with the
continuity equation in toroidal geometry. The present
paper extends this work to high

tokamaks.
In this paper, by using a new assumption about inter-
nal energy that is, 2 Ru const a set of reduced MHD
Equations is derived. Where R is major radius and u is
an internal energy. The reduced MHD Equations involve
three variables: the flux, stream function and internal
energy. The Equations can be used to calculate the
nonlinear evolution of tearing modes for toroidal plas-
mas.
2. REDUCED MHD EQUATIONS
The basic MHD Equations are of the form
35
.0
22
dP P
dt


 (1)
PJB
t




 
 (2)
J
B

 (3)
BE
t



 (4)
EBJ



 (5)
In MHD, if 1
P
u



, denote internal energy than
standard energy Balance Equation has the form
.0
du u
dt


 (6)
For the sake of simplicity, we adopt the quasi-cylin-
drical coordinate


,,r

 with

being the toroidal
angle. By assuming the inverse aspect ratio

to be a
small quantity, the ordering of the high

tokamaks [4].

1
1,,OB
rr





1
,,,,,OBJ u
tR







2,,OBJ







M. Asif / Natural Science 2 (2010) 95-97
96
where 00
BR
BR

 is the externally applied magnetic
field with 0 R being the coordinate of the geometric
center of plasma and
0
0
R
R
BB


. B

 is the toroidal
magnetic field produced by plasma current.
J

is the
toroidal current and


the toroidal velocity. The sub-
script  denotes perpendicular to

. The u is the
internal energy assumed to be of the order

, for high

tokamaks. For simplicity we use B

to express
RB

, B

 to RB

,
J

to RJ

,


to R


. Thus
B

is of the order 1

,
J

is of zeroth order while
B

 and


are of the order

. Furthermore we assume

to be of the order 2

, for high

tokamaks [4].
Introducing the vector potential
A
for magnetic fiel-
d
B
, that is,


 AeAeAA rr (7)
Then B can be expressed as



































 r
A
R
A
r
R
Be
A
e
A
R
Arr
r
1
(8)
From Eq.(8) one can see that
A
is of zeroth order
and r
A
and
A

are of the order 2

. Compared with
equilibrium magnetic filed expressed as B


BB

 
  
, where
A
is the flux function.
Later it will be shown that the effect of r
A
and
A

is
of the order of 2

and can be neglected so that within
our approximation B can be determined by
A
only.
Substituting Eqs.(8 ) and (5) into Eq.(4) we obtain
 ,JB
t
A


 

 

JB
t
A
(9)
where

is the gauge potential. Eq.(9) can be deduced
as

4
2
BO
R





Hence we can introduce stream function .U for

0
,UB

 (10)
where U is the order

and

can be expressed as

2
3
0
RUO
R


   (11)
where the relation 00
BBR

 is used. The divergence
of

is obtained as

23
0
1
... RU O
R

   (12)
This Eq.(12) shows that the divergence of

is of
the order 2

. Taking the

component of Eq.(9) and
substituting Eqs.(10) and (12) into Eq.(9), we obtain the
flux evolution equation

2
2
0
..
AR
BU O
tR


 (13)
Taking the curl of
B
, we get the expression of cur-
rent

 BRJ  .
2






222 .
 OARR   (14)
where the

-derivatives in operator are of the higher
order and can be dropped. From the internal energy
evolution Eq.(6) by substituting Eq.(11), we obtain


2
dRu O
dt

 (15)
The term on the right-hand side of Eq.(15) can be ne-
glected because the term on the left-hand side is of ze-
roth order. This indicates that if we assume the internal
energy to be constant, Eq.(15) cannot be satisfied. We
assume
constuRuR  0
2
0
2 (16)
This assumption satisfies Eq.(1 5) and includes the tor-
oidal effect.
Taking the curl of momentum Eq.(2) after multiplying
by 2 R we can eliminate
J
 and B

. Then we get


22
22
0
00
.1 .
dW RR
BJR uO
dt RR


 (17)
where
22,WR U
2
0
..
dR
U
dtt R





The

-derivatives in the Laplacian are also of the
higher order and can be neglected. Using the Assump-
tion of Eq.(16) we can directly obtain the parallel vis-
cosity equation. Then a set of reduced MHD equations
can be written as
2
0
.
AR
BU
tR



 (18)
 

 .1. 2
0
2
0
2
0uR
R
R
JB
R
R
dt
dW (19)

20
dRu
dt  (20)


ARRJ  22.

(21)
22,WR U (22)

M. Asif / Natural Science 2 (2010) 95-97
97
The Assumption 16 that the product of the square of
the major radius and the internal energy is a constant, is
quite stringent as taking this to be a constant, the first
driving term in the Grad-Shafranov equation becomes
just proportional to the flux derivative of the logarithm
of the major radius, which shows a rather weak depend-
ence. Therefore, this assumption seems to drastically
narrow down the range of equilibrium configurations to
which it is applicable. On the other hand, it has been
observed [6] that a high density region appears near the
inside limiter, which means that the density profile at the
inside and outside of plasma along a flux surface is
asymmetric. The pressure is calculated as the product of
experimental temperature and density. Since the internal
energy is related to the pressure as 1
P
u

, we can say
that the pressure distribution is nonuniform poloidally
and the pressure is higher at the inside of plasma than at
the outside. The result is, however, consistent with our
expectation. On the other hand, density and pressure
profile widths are clearly correlated [7].
3. CONCLUSIONS
In summary, we derived the reduced MHD Equations
(18-22) by using the Assumption 16 about the internal
energy in a large aspect ratio limit. These equations in
clude all terms of the same order as the toroidal effect
and only involve three variables, namely the flux, stream
function and internal energy. These equations can be
used to investigate the time evolution of tearing mode
for the high

, large aspect ratio limit for tokamak
Plasmas.
REFERENCES
[1] Wesson, J.A. (1978) Hydromagnetic stability of toka-
maks. Nuclear Fusion, 18(1), 87-132.
[2] Fitzpatrick, R., Hastie, R.J., Martin, T.J. and Roach, C.M.
(1993) Stability of coupled tearing modes in tokamaks.
Nuclear Fusion, 33(10), 1533-1556.
[3] Strauss, H.R. (1983) Finite-aspect-ratio MHD equations
for tokamaks. Nuclear Fusion, 23(5), 649-655.
[4] Strauss, H.R. (1977) Dynamics of high

tokamaks.
Physics Fluids, 20(8), 1354-1360.
[5] Ren, S.M. and Yu, G.Y. (2001) Reduced equations in
toroidal geometry. Chinese Physics Letters, 18(4),
556-558.
[6] Frigione, D. et al. (1996) High density operation on
frascati tokamak upgrade. Nuclear Fusion, 36(11), 1489-
1499.
[7] Minardi, E. et al. (2001) Stationary magnetic entropy in
ohmic tokamak plasmas: Experimental evidence from the
TCV device. Nuclear Fusion, 41(1), 113-130.