Magnetohydrodynamic (MHD) confinement theory has to a large extent
revolved around the stability of ideal MHD equilibria without flow, as
described by the Grad-Shafranov equation. Such ideal equilibria are very
numerous: as numerous as the solutions of the Euler equations for
inviscid pipe flow, for example. Investigating the stability of such
ideal equilibria is a task that could literally go on forever. However,
if small but non-zero resistivity and viscosity and appropriate boundary
conditions are introduced, the class of MHD steady states collapses to a
very small one. In a periodic, straight cylinder, zero-flow,
current-carrying steady states exist, and their stability boundaries and
slightly non-linear behavior can be investigated. However, for toroidal
geometry, the situation appears very different, and there seem to be no
current-carrying, steady-state, MHD configurations without flow, or at
least none with physically realizable resistivity profiles. The
necessity of including flow in stability analyses, together with the
loss of self-adjointness in the dissipative, linearized MHD equations,
appears as a setback for MHD theory; but it may also open the door to
the kind of realism that is taken for granted in Navier-Stokes fluid
theory.

Reference: J.W. Bates and D. Montgomery,
Phys. Plasmas 5, 2649 (1998).