Unsteady current-induced perturbation of a magnetically contained magnetohydrodynamic flow

Abstract

A time-dependent current distribution is abruptly switched on, initiating an unsteady disturbance of a two-phase magnetohydrodynamic configuration. This comprises a magnetically permeated, conducting fluid flow contained by a vacuum magnetic field, from within which the source current radiates. An exact solution to the proposed part-time problem is constructed for the magnetic line distortion of the vacuum field.

If the source current is oscillatory, two progressive, non-dissipative waves are normally encountered, these being superposed upon an infinite discrete set of terms obeying separate rules of decay. Propagation occurs longitudinally, parallel to the flow, but with amplitudes dependent on the transverse variable. The waves advance behind two fronts, the faster of which always travels down-stream. Depending on whether the flow speed exceeds or is exceeded by (or equals) √2 × the quadratic mean of both Alfvkn speeds involved, the slower front proceeds, respectively, downstream or upstream (or disappears, in which case, only one wave exists). Along a characteristic (a front path), appropriate contributions depend solely on the transverse co-ordinate, behaving otherwise like Riemann invariants. Contrary to expectation, the net perturbation is continuous across each characteristic. Various steady modes are ultimately attained after an infinite period. The radiation principle is satisfied.

Both travelling waves are vibrationally sustained, vanishing with the source frequency. In such an event, the infinite series result is summable to a closed form. From this, the general solution, corresponding to an arbitrary space–time source distribution, is deduced. Certain characteristic-associated equivalence laws are then established. An asymptotic approximation is made.