PHASE MIXED ROTATION MAGNETOCONVECTION AND TAYLORS CONDITION .3. WAVE-TRAINS

Nonlinear amplitude equations governing the radial modulation of quasi -geostrophic convective rolls, which occur in a rapidly rotating self- gravitating sphere permeated by a weak azimuthal magnetic held (small Elsasser number), were derived in Part I. Stationary and travelling pu lse solutions were obtained in Part II. That analysis is extended here ; wave train solutions are sought and their stability tested, Special features of the equations include: nonlinear diffusion and dispersion; also phase mixing, which leads to a lack of translational invariance of the system. In spite of the latter, the underlying structure of the wave trains sought is spatially periodic on a length L, but modulated by a time dependent Floquet exponent. Consequently, a Fourier represe ntation is employed and the rime evolution of the Fourier coefficients is determined numerically. It is shown that pulses confined to length s L(< L) can be superimposed non-interactively to form wave trains. Th e numerical demonstration relies on establishing that the pseudo-energ y (E) over bar based on the time averaged wave train amplitude coincid es with the corresponding pulse energy E calculated in Part II. When L and L are comparable some pulse interaction can be inferred, Availabl e numerical evidence suggests that wave trains, and by implication pul ses, are unstable. The geophysical implications are discussed. All fin ite amplitude solutions pertain to the Ekman regime in which the modif ied Taylor's condition is satisfied by small magnetic field perturbati ons. Only in the infinite amplitude limit do the solutions determine t rue Taylor states. It is anticipated that following instability in the Ekman regime convection equilibrates in some large amplitude Taylor s tate, which is determined when additional ageostrophic effects are tak en into account, Analysis of that slate lies outside the range of vali dity of our amplitude equations.