Fundamentals of Plasma Physics

286 Chapter 9. MHD equilibria

9.9 Dynamic equilibria:flows

The basic mechanism driving MHDflows will first be discussed using the simplified as-
sumptions of incompressibility and self-field only. This situation isrelevant to simple arcs,
basic concepts of magnetoplasmadynamic thrusters, and to currents driven inmolten met-
als. The more general situation where the plasma is compressible and where there are
external, applied magnetic fields in addition to the self-field will then be addressed.

9.9.1 Incompressible plasma with self-field only

MHD-drivenflows involve situations where∇×(J×B)is non-zero. This means the MHD
forceJ×Bis non-conservative in which case its finite curl acts as torque or equivalently as
a source for hydrodynamic vorticity. When the MHD force is non-conservative a closed line
integral

∮

dl·J×Bwill be finite, whereas in contrast a closed line integral of a pressure
gradient

∮

dl·∇Pis always zero. Since a pressure gradient cannot balance the torque
produced by a non-conservativeJ×B, it is necessary to include a vorticity-damping term
(viscosity term) to allow for the possibility of balancing this torque. Plasma viscosity is
mainly due to ion-ion collisions and is typically very small. With theaddition of viscosity,
the MHD equation of motion becomes

ρ

(

∂U

∂t

+U·∇U

)

=J×B−∇P+ρυ∇^2 U (9.64)

whereυis the kinematic viscosity. To focus attention on vorticity generation,transport,
and decay the following simplifying assumptions are made:

1. The motion is incompressible so thatρ=const.
   This is an excellent assumption for a molten metal, but is questionable for aplasma.
   However, this assumption could be at least locally reasonable for aplasma if the region
   of vorticity generation is smaller in scale than regions of compressionor rarefaction
   or is spatially distinct from these regions. The consequences of compressibility will
   be discussed later.


2. The system is cylindrically symmetric.
   A cylindrical coordinate system r,φ,zcan then be used andφis ignorable. This
   geometry corresponds to magnetohydrodynamic thrusters and arcs.


3. Theflow velocitiesUand the current densityJare in the poloidal direction (randz
   plane). Restricting the current to be poloidal means that the magnetic fieldis purely
   toroidal (see Fig.9.10).

The most general poloidal velocity for a constant-density, incompressiblefluid has
the form

U=

1

2 π

∇ψ×∇φ (9.65)

whereψ(r,z)is theflux offluid through a circle of radiusrat locationz.Note the analogy
to the poloidal magneticflux function used in Eqs.(3.146) and (9.34). SinceUlies entirely
in ther,zplane and since the system is axisymmetric, the curl ofUwill be in theφ
direction. It is thus useful to define a scalar cylindrical “vorticity”χ=rφˆ·∇×U;this