Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations

Abstract

The stability (i) of fully three-dimensional magnetostatic equilibria of arbitrarily complex topology, and (ii) of the analogous steady solutions of the Euler equations of incompressible inviscid flow, are investigated through construction of the second variations δ 2 M and δ 2 K of the magnetic energy and kinetic energy with respect to a virtual displacement field η( x ) about the equilibrium configuration. The expressions for δ 2 M and δ 2 K differ because in case (i) the magnetic lines of force are frozen in the fluid as it undergoes displacement, whereas in case (ii) the vortex lines are frozen, so that the analogy between magnetic field and velocity field on which the existence of steady flows is based does not extend to the perturbed states. It is shown that the stability condition δ 2 M > 0 for all η( x ) for the magnetostatic case can be converted to a form that does not involve the arbitrary displacement η( x ), whereas the condition δ 2 K > 0 for all η for the stability of the analogous Euler flow cannot in general be so transformed. Nevertheless it is shown that, if δ 2 M and δ 2 K are evaluated for the same basic equilibrium field, then quite generally \[ \delta^2 M + \delta^2 K > 0\quad \hbox{(all non-trivial}\;\eta). \] A number of special cases are treated in detail. In particular, it is shown that the space-periodic Beltrami field \[ {\boldmath B}^{\rm E} = (B_3\cos \alpha z+B_2\sin \alpha y, B_1\cos \alpha x + B_3\sin \alpha z, B_2\cos \alpha y + B_1 \sin \alpha x) \] is stable (i.e. δ2 M > 0 for all η) and that the medium responds in an elastic manner to perturbations on a scale large compared with α −1 . By contrast, it is shown that δ 2 K is indefinite in sign for the analogous Euler flow, and it is argued that the flow is unstable to certain large-scale helical perturbations having the same sign of helicity as the unperturbed flow. It is conjectured that all topologically non-trivial Euler flows are similarly unstable.