A Physical Explanation of Difficulties Appearing in the Theory of Toroidal Magnetohydrodynamic Oscillations

Toroidal magnetohydrodynamic oscillations frequently give rise to terms of the form t cos ( kt ), t sin ( kt ) (secular terms) in the component of the perturbed electric current parallel to the permanent magnetic field. This paper gives the physical explanation of the existence of such secular terms and shows that in practical applications the parallel component of electric current does not in fact grow indefinitely with time. Estimates of the time of decay of toroidal oscillations are found; it is shown, both in general, and by reference to the Plumpton–Ferraro oscillating shell problem, that for non-axial and non-equatorial shells the ratio of the time of decay of currents to the time of decay in a rigid conductor of the same size and conductivity is given by $$({\eta/a{V}_{A})^{2/3}}$$ : 1. Here a is a typical length, V _A the Alfvén velocity and η the magnetic viscosity. The quantity $$({\eta/a{V}_{A})^{2/3}}$$ is usually small and the decay of electric currents of toroidal oscillations is therefore much faster than the decay of currents in a rigid conductor of the same size and conductivity. This comment applies to many general systems in which waves propagate along field lines and are reflected many times from fixed boundaries, and must be borne in mind when considering the toroidal magnetohydrodynamic oscillation of stars.