9 MHD equilibria
9.1 Why use MHD?
Of the three levels of plasma description – Vlasov, two-fluid, and MHD – Vlasov is the
most accurate and MHD is the least accurate. So, why use MHD? The answer is that
because MHD is a more macroscopic point of view, it is more efficient to use MHDin
situations where the greater detail and accuracy of the Vlasov or two-fluid models are
unnecessary. MHD is particularly suitable for situations having complex geometry because
it is very difficult to model such situations using the microscopically oriented Vlasov or
2-fluid approaches and because geometrical complexities are often most important at the
MHD level of description. The equilibrium and gross stability of three-dimensional, finite-
extent plasma configurations are typically analyzed using MHD. Issues requiring a 2-fluid
or a Vlasov point of view can exist and be important, but these more subtle questions can
be addressed after an approximate understanding has first been achieved using MHD. The
MHD point of view is especially relevant to situations where magneticforces are used
to confine or accelerate plasmas. Examples of such situations include magnetic fusion
confinement plasmas, solar and astrophysical plasmas, planetary and stellar dynamos, arcs,
and magnetoplasmadynamic thrusters. Although molten metals are not plasmas, they are
described by MHD and in fact, the MHD description is actually more appropriate and more
accurate for molten metals than it is for plasmas.
We begin our discussion by examining certain general properties of magnetic fields in
order to develop an intuitive understanding of the various stresses governing MHD equilib-
rium and stability. The MHD equation of motion,
ρ[
∂U
∂t+U·∇U
]
=J×B−∇P, (9.1)
is a generalization of the equation of motion for an ordinaryfluid because it includes the
J×Bmagnetic force. Plasma viscosity is normally very small and is usually omitted
from the MHD equation of motion. However, when torques exist, a viscous damping term
needs to be included if one wishes to consider equilibria. This is because sucha damping is
required to balance any torque;otherwise the plasma will spin up without limit. Such situ-
ations will be discussed in Section 9.9;until then, viscosity will assumed to be negligible
and will be omitted from Eq.(9.1).
264