Nonlinear waves in rotating magnetohydrodynamics

Abstract

We consider an electrically conducting fluid in rotating cylindrical coordinates in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for the Earth's outer core. Fully nonlinear waves dominated by the nonlinear Lorentz forces are studied using the method of geometric optics (essentially WKB). These waves are assumed to be of the form of an asymptotic series expanded about ambient magnetic and velocity fields which vanish on the equatorial plane. They take the form of short wave, slowly varying wave trains. The first-order approximation is sinusoidal and basically the same as in the linear problem, with a dispersion relation modified by the appearance of mean terms. These mean terms, as well the undetermined amplitude functions, are found by suppressing secular terms in a “fast” variable in the second-order approximation. The interaction of the mean terms with the dispersion relation is the primary cause of behaviors which differ from the linear case. In particular, new singularities appear in the wave amplitude functions and an initial value problem results in a singularity in one of the mean terms which propagates through the fluid. The singularities corresponding to the linear ones are shown to develop when the corresponding waves propagate toward the equatorial plane.

Keywords:

• Asymptotics
• Earth's core
• Hydromagnetic waves
• Nonlinear waves