Fundamentals of Plasma Physics

274 Chapter 9. MHD equilibria

9.7 Static versus dynamic equilibria

We define (i) a static equilibrium to be a time-independent solution to Eq.(9.1) having no
flow velocity and (ii) a dynamic equilibrium as a solution with steady-stateflow velocities.
ThusU=0for a static equilibrium whereasUis finite and steady-state for a dynamic
equilibrium. For a static equilibrium the MHD equation of motion reduces to

∇P=J×B. (9.22)

For a dynamic equilibrium the MHD equation of motion reduces to

ρU·∇U+∇P=J×B+νρ∇^2 U (9.23)

where the last term represents a viscous damping andνis the kinematic viscosity. By tak-
ing the curl of these last two equations we see that for static equilibria the magnetic force
must be conservative i.e.,∇×(J×B)=0whereas for dynamic equilibria the magnetic
force is typically not conservative since in general∇×(J×B)=0. Thus, the charac-
ter of the magnetic field is quite different for the two cases. Static equilibria are relevant to
plasma confinement devices such as tokamaks, stellarators, reversed field pinches, spher-
omaks while dynamic equilibria are mainly relevant to arcs, jets, magnetoplasmadynamic
thrusters, but can also be relevant to tokamaks, etc., if there areflows. Both static and
dynamic equilibria occur in space plasmas.

z

r

finite pressure

in cylinder

magnetic force pinches in

hydrodynamic pressure pushes out

cylindrical plasma w ith

axial currentdensity Jz
r

Figure 9.7: Geometry of Bennett pinch.