290 Chapter 9. MHD equilibria
Thus, the system of equations can be summarized as (Bellan 1992)∂
∂t(χ
r^2)
+U·∇
(χ
r^2)
= υ∇·1
r^2∇χ−μ 0
4 π^2 ρr^4∂I^2
∂z(9.85)
∂
∂t(
I
r^2)
+U·∇
(
I
r^2)
=
η
μ 0∇·
(
1
r^2∇I
)
(9.86)
r^2 ∇·(
1
r^2∇ψ)
= − 2 πχ (9.87)U =
1
2 π∇ψ×∇φ. (9.88)This system cannot be solved analytically, but its general behavior can be described in a
qualitative manner. The system involves three scalar variables,χ,ψ,andIall of which are
functions ofrandz.Boundary conditions must be specified in order to have a well-posed
problem. If a recirculatingflow is driven, thefluidfluxψwill be zero on the boundary
whereasI will be finite on electrodes at the boundary. In particular, the currentIwill
typicallyflow from the anode into the plasma and then from the plasma into the cathode.
The vorticityχwill usually have the boundary condition of vanishing on the bounding
surface.
Initially, there is noflow soψis zero everywhere. An initial solution of Eq.(9.86) in
this no-flow situation will establish a current profile between the two electrodes. For any
reasonable situation, thisI(r,z)profile will have∂I^2 /∂z=0so that a vorticity source
will be created. The source will be quite localized because of ther−^4 coefficient. Once
vorticityχis created as determined by Eq.(9.85), the vorticity acts as a source term for the
fluidfluxψin Eq.(9.87), and so a finiteψwill be developed. Thus aflowU(r,z)will be
created as specified by Eq.(9.88), and thisflow will convect bothχ/r^2 andI/r^2.
A good analogy is to think of ther−^4 ∂I^2 /∂zterm as constituting a toroidally sym-
metric centrifugal pump which acceleratesfluid radially from large to smallr.This radial
acceleration takes place atzlocations where the current channel radius is constricted. The
pump then accelerates the ingestedfluid up or down thezaxis, in a direction away from the
current constriction. This vortex generation can also be seen by drawing vectors showing
the magnitude and orientation of theJ×Bforce in the vicinity of a current constriction.
It is seen that theJ×Bforce is non-conservative and provides a centrifugal pumping as
described above.
An arc or magnetoplasmadynamic thruster can thus be construed as a pump which
sucksfluid radially inward towards the smaller radius electrode and then shoots thefluid
axially away from the smaller electrode towards the larger radius electrode. The sign of the
current does not matter, since the pumping action depends only on thezderivative ofI^2 .If
the equations are put in dimensionless form, it is seen that the characteristicflow velocity
is of the order of the Alfvén velocity.