The general problem of Alfvén wave propagation in a dipole field has perplexed numerous authors and will continue to do so, since it has never been adequately solved or physically interpreted. Most work has concentrated on the special axisymmetric (longitude independent) case. Although this problem can only approximate the phenomena in an asymmetric magnetosphere, it has the appealing quality of being completely soluble. Dungey [1963] briefly studied the other extreme case of highly asymmetric modes.

The asymptotic temporal behavior of hydromagnetic waves in a model of the inner magnetosphere is shown to be characterized by guided modes. The basic hydromagnetic wave equation is derived and applied to a cylindrical model of the inner magnetosphere. This model offers a good representation of the spatially dependent field line frequencies present in a dipole field. The initial value problem for the symmetric toroidal mode is solved, and its singularities are treated by Fourier superposition. Singularities also are present in the asymmetric poloidal mode wave equation and are shown to be logarithmic. Fourier superposition leads to the solutions for the electric and magnetic fields, which are asymptotic in time. The results indicate decay of the poloidal modes and domination by the toroidal modes, i.e., field line control of the propagation. At the latitude at which the maximum amplitude of a particular frequency occurs, the wave becomes linearly polarized. The asymptotic micropulsation periods depend on the characteristic field line periods and are usually longer at higher latitudes. Undamped guided waves lead to some terms, such as change density and parallel current density, that increase linearly with time. The inclusion of loss mechanisms in the wave equation, e.g., conductivity, limits the guided modes and prevents such nonphysical effects from occurring.

The heuristic concept of vibrating field lines to explain micropulsation observations continues to be popular [Wilson, 1966]. It, therefore, seems appropriate to examine the equations governing this phenomenon to ascertain the mathematical reliability of such an appealing physical model. It is the contention of the author that, while there is no valid theoretical demonstration that permits the oscillation of individual field lines, there are strong theoretical reasons against such an occurrence. The basic wave equation describing low‐frequency propagation in a magnetized plasma is ∂2E/∂2 = A × A × curl curl E where E is the perturbed electric field and A = B(4πρ)1/2 is the Alf vén velocity

The usual magnetohydromagnetic approach is used to investigate axisymmetric perturbations in a plasma permeated by a static dipole field. The model plasmasphere used has stationary, perfectly reflecting boundaries concentric with the dipole, and the plasma density is assumed to be spherically symmetric. The poloidal wave equation is reduced to a hierarchy of coupled differential equations in the radial variable. The toroidal wave equation is solved exactly using a density law which gives a good order of magnitude fit to recent whistler data. Because the extent of the plasma is finite, the eigenperiods are everywhere finite, including the poles. The stretched string model of vibrating lines of force is derived using the Wentzel-Kramers-Brillouin approximation, which fortuitously gives the exact periods for the density law utilized. Two unusual results are the prediction of shorter periods in the polar zones than at some high middle latitudes and discontinuous eigenperiods across the line of force which just grazes the outer plasma boundary. A general discussion of the results and the special significance of the toroidal and poloidal modes because of their symmetry is provided.

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