Magnetohydrodynamic equations for toroidal plasmas

doi:10.4236/ns.2010.22015

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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; [email protected]

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



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

dP P





 (1)

PJB









 

 (2)



 (3)







 (4)

EBJ







 (5)

In MHD, if 1







, denote internal energy than

standard energy Balance Equation has the form

du u





 (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,,OB









,,,,,OBJ u















2,,OBJ











M. Asif / Natural Science 2 (2010) 95-97

where 00



 is the externally applied magnetic

ﬁeld with 0

R being the coordinate of the geometric

center of plasma and





. B



 is the toroidal

magnetic ﬁeld produced by plasma current.



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



, B



 to RB



,



to RJ





to R





. Thus



is of the order 1



,



is of zeroth order while



 and





are of the order



. Furthermore we assume



to be of the order 2



, for high



tokamaks [4].

Introducing the vector potential

for magnetic ﬁel-

, that is,





 AeAeAA rr (7)

Then B can be expressed as





















































 r

Arr

(8)

From Eq.(8) one can see that

is of zeroth order

and r

and



are of the order 2



. Compared with

equilibrium magnetic ﬁled expressed as B







 

  

, where

is the ﬂux function.

Later it will be shown that the effect of r

and



of the order of 2



and can be neglected so that within

our approximation B can be determined by

only.

Substituting Eqs.(8 ) and (5) into Eq.(4) we obtain



,JB

A















JB

(9)

where



is the gauge potential. Eq.(9) can be deduced













Hence we can introduce stream function .U for



,UB



 (10)

where U is the order



and



can be expressed as



RUO





 

 (11)

where the relation 00

BBR



 is used. The divergence



is obtained as



...

RU O





  (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



BU O





 (13)

Taking the curl of

, we get the expression of cur-

rent





BRJ  .











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



dRu O



 (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 (16)

This assumption satisﬁes Eq.(1 5) and includes the tor-

oidal effect.

Taking the curl of momentum Eq.(2) after multiplying

by 2

we can eliminate

 and B



. Then we get



.1 .

dW RR

BJR uO

dt RR







(17)

where

22,WR 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







 (18)







 .1. 2

0uR

dW (19)



dRu

dt  (20)





ARRJ  22.



(21)

22,WR U (22)

M. Asif / Natural Science 2 (2010) 95-97

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



, 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.