Abstract
Abstract
Highlights
Linear waves in an inviscid perfectly conducting fluid permeated by a uniform magnetic field B0 in a frame rotating with rate Ω satisfy the dispersion relation (Lehnert 1954)ω = ± Ω · k ± (Ω · k)2 + |k|2(B0 · k)2/(ρμ0), (1.1) |k|where ω is the frequency, k is the wavenumber vector, ρ is the density and μ0 is the magnetic permeability
We demonstrate that: (i) the amplitude at the first order is described by the Korteweg–de Vries (KdV) equation for the chosen basic states; (ii) the problem is dictated by an ordinary differential equation (ODE), which has no singularities as the wave speed approaches the basic flow speed; and (iii) the single-soliton solution to the KdV equation implies an isolated eddy that progresses in a stable permanent form on magnetostrophic time scales
The model we considered is an annulus model (Busse 1976; Canet et al 2014) of the form utilised by Hide (1966) for linear magnetic Rossby (MR) waves
Summary
Linear waves in an inviscid perfectly conducting fluid permeated by a uniform magnetic field B0 in a frame rotating with rate Ω satisfy the dispersion relation (Lehnert 1954). Β denotes the beta parameter, k is the azimuthal wavenumber, and the minus sign indicates that waves travel opposite to the hydrodynamic Rossby wave, ω = βk/|k|2 This class is sometimes referred to as slow hydromagnetic–planetary or magnetic–Rossby (MR). London (2017) found a couple of cases in which nonlinear equatorial waves in the shallow-water MHD should be governed by Korteweg–de Vries (KdV) equations and so behave like solitary waves They were mostly fast MR modes, recovering the equatorial Rossby wave soliton (Boyd 1980) in the non-magnetic limit, but he reported one case in which the wave would slowly travel in the opposite azimuthal direction. The slow, weakly nonlinear waves led to evolution obeying the KdV equation unless the basic state – all the magnetic field, topography and zonal flow – is uniform. (ii) the problem is dictated by an ordinary differential equation (ODE), which has no singularities as the wave speed approaches the basic flow speed; and (iii) the single-soliton (solitary wave) solution to the KdV equation implies an isolated eddy that progresses in a stable permanent form on magnetostrophic time scales
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