A model equation for monochromatic standing Alfven waves in the axially symmetric magnetosphere

Abstract

A model equation is proposed to describe the radial (in the direction across magnetic shells) structure of standing Alfven waves with fixed frequency and azimuthal wave number in the axially ‐ symmetric magnetosphere. A semiphenomenological derivation of this equation is given. In those limiting cases when the solution of starting equations can be obtained rigorously, it coincides qualitatively with the solution of the model equation. Its chief merit is a relative simplicity and the possibility of investigating azimuthal harmonics with an arbitrary azimuthal wave number m (except for a few fundamental harmonics with m ∼ 1). An integral representation is obtained for the solution of the model equation, important limiting cases are studied analytically, and it is investigated numerically. The dependence of the radial structure on the azimuthal wave number and the magnitude of dissipation of standing waves in the ionosphere are analyzed. It is shown that, with increasing m, the radial structure changes from field line resonance to a traveling wave‐like structure, while with increasing dissipation, the mode changes from toroidal to poloidal one.