Fundamentals of Plasma Physics

310 Chapter 10. Stability of static MHD equilibria

result with the additional stipulation that the normal component of the velocity vanishes at
the wall (i.e., the wall is impermeable).
A system consisting of a magnetofluid surrounded by a vacuum region enclosed by an
impermeable perfectly conducting wall will therefore have its total internal energy con-
served, that is

∫

Vmf

d^3 r

(

ρU^2

2

+

P

γ− 1

+

B^2

μ 0

)

+

∫

Vvac

d^3 r

B^2

μ 0

=const. (10.56)

whereVmfis the volume of the magnetofluid andVvacis the volume of the vacuum region
between the magnetofluid and the wall.
This total system energy can be split into a kinetic energy term

T=

∫

d^3 r

ρU^2

2

(10.57)

and a potential energy term

W=

∫

V

d^3 r

(

P

γ− 1

+

B^2

μ 0

)

(10.58)

so that
T+W=E (10.59)
where the total energyEis a constant. HereVincludes the volume of both the magnetofluid
and any vacuum region between the magnetofluid and the wall.
We will now consider a static equilibrium (i.e., an equilibrium withU 0 =0) so that

0=J 0 ×B 0 −∇P 0. (10.60)

Dotting with eitherB 0 orJ 0 shows that

B 0 ·∇P 0 =0, J 0 ·∇P 0 =0 (10.61)

so that∇P 0 is normal to the surface defined by theJ 0 andB 0 vectors.

10.3.2Self-adjointness of potential energy as a consequence of energy integral

From Eq.(10.57) it is seen that in static equilibriumT 0 =0.This means that all the internal
energy must be in the form of stored potential energy, i.e.,

W 0 =E. (10.62)
It is now supposed that thermal noise causes a small motion of orderǫto develop at
each point in the magnetofluid. This means there is a first-order velocity

U 1 ∼ǫ. (10.63)

The displacement of afluid element is obtained by time integration to be

ξ=

∫t

0

U 1 dt′; (10.64)