Fundamentals of Plasma Physics

10.3 The MHD energy principle 307

perfectly

conducting

w all

Gaussian

pillbox

equilibrium

surface

x

x

perturbed

surface

vacuum

origin

ds

magnetofluid

vacuum

Figure 10.5: Two-dimensional cut of three-dimensional plasma equilibrium and perturba-
tion;Gaussian pillbox is used to relate quantities in vacuum to quantitiesin plasma. Each
volume element at a positionxis displaced by an amountξ(x). The displacement of a
volume element at the surface is illustrated.

10.3.1Energy equation for a magnetofluid

The energy content of a magnetofluid can be obtained from the ideal MHD equations if it is

assumed that all motions are sufficiently fast to be adiabatic but slowenough for collisions

to keep the pressure isotropic. The ideal MHD equations are

ρ

(

∂U

∂t

+U·∇U

)

=J×B−∇P (10.36)

E+U×B=0 (10.37)

∇×E=−

∂B

∂t

(10.38)

∇×B=μoJ (10.39)