Solitary waves in shallow water magnetohydrodynamics

Abstract

We consider a perfect electrically conducting rotating fluid in the presence of an ambient toroidal magnetic field, governed by the shallow water magnetohydrodynamic (MHD) equations in a modified equatorial beta plane approximation. Using a multiple scale asymptotic technique, previously developed by Boyd (1980) for equatorial solitary hydrodynamic waves, we look for solitary MHD waves. In the case of a weak ambient magnetic field, the leading order equations governing the poleward velocities can be solved using associated Legendre functions. These functions are multiplied by an amplitude function of slow length and time variables. They are solved for at second order via a compatibility condition and have the form of equatorial Rossby solitary waves. When the ambient magnetic field is moderately strong, the equations governing the poleward velocities cannot be solved using special functions. Instead, we are able to apply a WKB type approximation to solve this problem using a large parameter, , which arises naturally in the governing equations. The solution has the form of an equatorial magneto-Rossby solitary wave. When the ambient magnetic field is very strong, the solutions are bounded away from the equator in the form of magnetostrophic solitary waves. The possible relationship between all of these waves and solar phenomena such as the solar cycle, sunspots and active longitudes is discussed.