Vol.2, No.1, 54-55 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.21008
Relationship between normalized thermal energy and
conductivity for cylindrical tokamak geometry
M. Asif
Department of Physics, COMSATS Institute of Information Technology, Lahore, Pakistan
Received 23 November 2009; revised 4 December 2009; accepted 7 December 2009.
ABSTRACT
The relation between normalized thermal en-
ergy and conductivity, with no particle sources
in cylindrical tokamak geometry has been de-
rived. It is obtained, p






. This relationship
is of importance for tokamak confinement and
heating.
Keywords: Thermal Energy; Cylindrical Geometry;
Confinement
1. INTRODUCTION
It was previously shown [1], using a numerical analysis,
that from the poloidal and toroidal components of Ohm’s
law for a tokamak, there follows a relation between the
normalized thermal-energy content, β, and the normal-
ized plasma current, 1/q, which only depends on the
plasma refueling and not on thermal-energy transport.
Previously, the relation was examined only numerically
and it is the purpose of [2] to provide an approximate
analytical expression for this relation valid for tokamaks,
showing explicitly the dependence on the fueling pa-
rameters. The relation [3] implies a constraint on the
thermal-energy content which cannot be violated, what-
ever the energy transport present and the plasma heating
used. It should be noted that a different relation [4,5] has
been derived. Analogous to that of above models, can be
obtained by following the same procedure but using
Ohm’s law [6]. The main focus of this paper is to show
the relation between normalized thermal energy and
conductivity, with no particle sources in cylindrical
tokamak geometry. It is obtained, p






. This rela-
tionship is of importance for tokamak confinement and
heating, since it provides a direct constraint on the ther-
mal-energy content which cannot in any case be violated,
whatever the energy transport presents.
2. RELATION BETWEEN NORMALIZED
THERMAL ENERGY AND
CONDUCTIVITY
We consider the problem of determining the steady state
in cylindrical tokamak geometry with no particle sources.
First of all, we recall the expressions for the toroidal and
poloidal components of Ohm’s law, which can be put in
the following form [6], with no particle sources in the
limit of large aspect ratio and circular magnetic surfaces:
E
j




 (1)
dp EB
dr



 (2)
where all quantities are surface-averaged and depend on
the minor radius r,
J

is the toroidal component of the
current density, E

is the electric field, 

and


is the parallel and perpendicular resistivities, B

and
B

are the toroidal and poloidal components of the
magnetic field, p is the plasma pressure. The safety fac-
tor is q=rB

/R B

and R is the major radius. The minor
radius of the last closed magnetic surface will be indi-
cated by a, and B

and
J

are related by Ampere’s
law

dr
rBd
r
j



0
1
 (3)
Substituting Eq.1 into Eq.2, one obtains the following
equation:






Bj
dr
dp (4)
It should be stressed that, in comparing tokamak ex-
periments with theory, one can certainly use the cylin-
drical approximation where all quantities depend only on
minor radius r; however, one must then consider the
relations between surface-averaged quantities. In this

M. Asif / Natural Science 2 (2010) 54-55 55
case, the perpendicular component of Ohm’s law con-
tains the toroidal corrections (Pfirsch-Schlüter and vis-
cosity or trapping terms) which are proportional to the
square of the safety factor and which are absent for the
exactly cylindrical case. In this paper these terms were
not considered.
The normalized thermal-energy content
p

can be
defined by

p
aB
p2
0
2



 (5)
where .. indicates the volume average and, in the
cylindrical approximation,
20
2()
a
pprrdr
a
 (6)
So that, if p(a)=0, we also have, by integrating by parts,
2
20
1adp
prdr
dra
  (7)
Substituting Eq.7 into Eq.5, one obtains the following
equation:

drr
dr
dp
aBa
a
p
2
0
22
0
2




(8)
Substituting Eqs.3 and 4 into Eq.8, one obtains the
following equation:
p






(9)
We have obtained relation between normalized ther-
mal energy (poloidal beta) and conductivity, with no
particle sources in cylindrical tokamak geometry. This
relationship is a generalization of the usual Bennett rela-
tion and it is of importance for tokamak confinement and
heating. It should be noted that different relations,
analogous to that given above, can be obtained by fol-
lowing the same procedure using Ohm’s law [6]. The
consideration of the generalized Bennett relation is of
importance for the problem of plasma confinement and
heating, since it provides a direct constraint on the ther-
mal-energy content which cannot in any case be violated,
whatever the energy trans-port present and the heating
used.
3. CONCLUSIONS
We have obtained relation between normalized thermal
energy and conductivity, with no particle sources in cy-
lindrical tokamak geometry. It is obtained, p






.
This relationship is of importance for tokamak confine-
ment and heating, since it provides a direct constraint on
the thermal-energy content which cannot in any case be
violated, whatever the energy transport present and the
heating used.
REFERENCES
[1] Segre, S.E. and Zanza, V. (1992) A profile constraint and
the value of beta in tokamak experiments. Plasma Phys-
ics and Controlled Fusion, 34(11), 1619-1637.
[2] Segre, S.E. and Zanza, V. (1993) A relation between the
normalized thermal energy and the safety factor for to-
kamaks. Nuovo Cimento, 15D(6), 897-901.
[3] Bennett, W.H. (1934) Magnetically self-focusing streams.
Physical Review, 45, 890-897.
[4] Asif, M. (2008) Relationship between current density and
mass density for ohmic tokamak plasmas. International
Journal of Modern Physics B, 22(30), 5329-5333.
[5] Asif, M. (2009) Relationship between viscosity and con-
ductivity for tokamak plasmas. Brazilian Journal of
Physics, 39(1), 59-61.
[6] Segre, S.E. and Zanza, V. (1992) Is the perpendicular
resistivity in tokamaks really much greater than the
Spitzer value? Physics of Fluid, 4(10), 3457-3459.