Fundamentals of Plasma Physics

9.8 Static equilibria 277

a resounding ‘no’, is provided by a virial theorem due to Shafranov (1966). A virial is
a suitably weighted integral over the entirety of a system and contains information about
how an extensive property such as energy is partitioned in the system. For example, in
mechanics the virial can be the time average of the potential or of the kinetic energy.

ball of plasma

surrounded by vacuum

x

y

z

Figure 9.9: Plasma sphere with finite pressure surrounded by vacuum

The MHD virial theorem is obtained by supposing that a self-confining configuration
exists and then showing this leads to a contradiction.
We therefore postulate the existence of a spherical plasma with the following properties:

1. the plasma has finite radiusaand is surrounded by vacuum,


2. the plasma is in static MHD equilibrium,


3. the plasma has finite internal pressure and the pressure gradient is entirely balanced
   by magnetic forces due to currents circulating in the plasma;i.e. there are no currents
   in the surrounding vacuum region.
   Using Eqs.(9.10) and (9.11) the static MHD equilibrium can be expressed as

∇·T=0 (9.30)

where the tensorTis defined as

T=

(

P+

B^2

2 μ 0

)

I−

1

μ 0

BB. (9.31)

Letr=xxˆ+yyˆ+zˆzbe the vector from the center of the plasma to the point of observation.