Fundamentals of Plasma Physics

278 Chapter 9. MHD equilibria

Consider the virial expression

∇·(T·r) =

∑

jk

∂

∂xj

(Tjkxk)

= (∇·T)·r+

∑

jk

Tjk

∂

∂xj

xk

= TraceT

(9.32)

whereTraceT=

∑

jkTjkδjk. From Eq.(9.31) (or, equivalently from the matrix form in
Eq.(9.11)) it is seen that
TraceT=3P+B^2 / 2 μ 0
ispositive definite.
We now integrate both sides of Eq.(9.32) over all space. Since the right hand side of
Eq.(9.32) is positive definite, the integral of the right hand side over all space is finite and
positive. The integral of the left hand side can be transformed to a surface integral at infinity
using Gauss’ theorem
∫
d^3 r∇·(T·r) =

∫

S∞

ds·T·r

=

∫

S∞

ds·

[(

P+

B^2

2 μ 0

)

I−

1

μ 0

BB

]

·r.

(9.33)

Since the plasma is assumed to have finite extent,P is zero on the surface at infinity.
The magnetic field can be expanded in multipoles with the lowest order multipole being
a dipole. The magnetic field of a dipole scales as∫ r−^3 for largerwhile the surface area
dsscales asr^2 .Thus the left hand side term scales as

∫

dsB^2 r∼r−^3 and so vanishes
asr→∞.This is in contradiction to the right hand side being positive definite and so the
set of initial assumptions must be erroneous. Thus, any finite extent, three-dimensional,
static plasma equilibrium must involve at least some currents external to the plasma. A
finite extent, three-dimensional static plasma can therefore only be in equilibrium if at least
some of the magnetic field is produced by currents in coils that are external to the plasma
and that are held in place by some mechanical structure. If the coils werenot supported
by a mechanical structure, then the current in the coils could be considered as part of the
MHD plasma and the virial theorem would be violated. In summary, a finite extent three-
dimensional plasma in static equilibrium with a finite internal hydrodynamic pressureP
must ultimately have some tangible exterior object to “push against”. The buttressing is
provided by magnetic forces acting between currents in external coils and currents in the
plasma.

9.8.3 Three-dimensional static equilibria: the Grad-Shafranov equation

Despite the simple appearance of Eq.(9.22), its three-dimensional solution is far from triv-
ial. Before even attempting to find a solution, it is important to decide the appropriate way
to pose the problem, i.e., it must be decided which quantities are prescribed and which are
to be solved for. For example, one might imagine prescribing a pressure profileP(r)and