Abstract
Several alternative analyses of the steady state solutions of coupled modes have been proposed. In determining which, if any, steady state solution is of physical significance, the manner in which the hydromagnetic system evolves from an initial disturbance would be decisive. The basic initial value problems are those of field line excitation and relaxation. These two cases correspond to the energy being initially in the poloidal or toroidal modes. These difficult problems are investigated by establishing the relevance of a simple mechanical analogue which is studied in detail. This is accomplished in three stages: (a) a rectangular model of the plasmasphere is described and its physical characteristics are compared with those of a dipole field magnetized plasma; (b) a mechanical system is derived consisting of an elastic string with each point of which is associated an oscillator. For the symmetric modes with weak ioncyclotron coupling, the wave equations for the hydromagnetic and mechanical systems are essentially identical; (c) the mechanical system is reduced to an elementary wave-oscillator model representing the coupling between resonant poloidal and toroidal modes. The characteristic motion of the simplified system is independent of the strength of the coupling. The solutions suggest that resonant field lines tend to reflect an incident poloidal wave and transmit only an evanescent wave. The toroidal mode does not become singular but attains a finite amplitude in performing this reflection. In general, it is concluded that the behavior of coupled and uncoupled modes is essentially different. When any coupling is present, steady state solutions, as normally understood, are not possible.
