The dynamics of planetary and stellar objects are dominated by the fluid motions of electrically conducting media.
Often, such fluid is confined to a shallow layer, perhaps in an atmosphere or bounded by stratification.
One such layer is the solar tachocline: a thin layer of high velocity shear in the Sun, which is permeated by strong magnetic fields.
The discovery of the solar tachocline has inspired the derivation of the equations of shallow water
magnetohydrodynamics (SWMHD), in which the small aspect ratio of vertical to horizontal length scales is used to simplify the governing equations. This thesis takes these equations as a base, and aims to build on knowledge
of the wave-like dynamics supported in the shallow water system.
It will be shown that the analogy between the shallow water and 2D compressible hydrodynamic systems is broken with
the introduction of magnetic field, and the differences between the two systems discussed. An energy conservation law will be derived and be used to infer stability properties of the SWMHD system. We will then construct a multi-layer system, and consider linear wave-like perturbations to a motionless basic state with a uniform magnetic field. Particular focus will be on the 2- and 3-layer models, and the effect of magnetic field strength on wave properties.
A weakly nonlinear analysis reveals that the single layer and 2-layer rigid lid weakly non-hydrostatic models support
solitary and cnoidal waves. The effect of magnetic field in the single layer case translates to a long-time rescaling,
but has much more of an effect on the supported modes in the 2-layer model. The phenomenon of three-wave resonance is also supported in the single layer system, and magnetic influence is discussed.
In rotating SWMHD, we find that resonant triad interactions are supported only in the presence of magnetic field. Weakly nonlinear predictions are found to be accurate at low disturbance amplitudes, when compared to the results from a fully nonlinear numerical scheme. Exact nonlinear solutions are derived and categorised, and their stability addressed using numerics.
