Weakly nonlinear magnetic equatorial Kelvin waves in rotating spherical coordinates

Abstract

ABSTRACTIn a recent paper, Marquez-Artavia, Jones and Tobias (referred to as MJT) studied wave propagation on a rotating sphere using the shallow water ideal magnetohydrodynamic (MHD) equations. These equations were linearised about a toroidal magnetic field (the Malkus field) which depends only on the latitude and is proportional to where θ is the co-latitude. These waves may play an important role in the Earth's outer core and the solar tachocline. The purpose of this paper is to extend MJT's solution for magnetic equatorial Kelvin waves to the case of weakly nonlinear waves. To do this, we use a formal asymptotic approximation when two main parameters are separately assumed to be large. The approximation recovers MJT's results in the linear case and extends those results to the weakly nonlinear case. The main result of this paper is that the zonal wave function is obtained from a first-order linear wave equation so that, unlike in the hydrodynamic case, these waves do not break or form shocks. We demonstrate a mechanism by which the presence of a magnetic field regularises the governing equations in the sense that a sufficiently large magnetic field can suppress the nonlinear terms in the Burgers equation of the hydrodynamic problem. We also find weak north–south currents, not present in the linear problem. The nonlinearities do not appear to significantly modify the linear results.