The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families
of topological invariants: the magnetic helicity invariants and the cross-helicity
invariants. It is first shown that these invariants define a natural foliation (described
as isomagnetovortical, or imv for short) in the function space in which solutions
{u(x, t), h(x, t)}
of the MHD equations reside. A relaxation process is constructed
whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic
and cross-helicity invariants being thus conserved). The energy has a positive lower
bound determined by the global cross-helicity, and it is thus shown that a steady
state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity
invariants.The stability of such steady states is considered by an appropriate generalization
of (Arnold) energy techniques. The first variation of energy on the imv folium is
shown to vanish, and the second variation δ2E
is constructed. It is shown that δ2E is
a quadratic functional of the first-order variations
δ1u, δ1h of u and h (from a steady
state U(x), H(x)), and that δ2E is an invariant of the linearized MHD equations.
Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently
obtained through consideration of the frozen-in ‘modified’ vorticity field introduced
in Part 1 of this series.Finally, the general stability criterion is applied to a variety of classes of steady
states {U(x), H(x)}, and new sufficient conditions for stability to three-dimensional
imv perturbations are obtained.
The stability of bifurcating steady states for a spatially heterogeneous cooperative system with cross-diffusion
