We introduce the magnetohydrodynamic thermal rotating shallow water (MTRSW) model as a framework to study magnetized rotating fluids influenced by thermal gradients and stratification. The model is derived from the full magnetohydrodynamic equations incorporating Coriolis forces, magnetic induction, and thermal effects. To extend its applicability to small-scale magnetic dynamics, Hall MTRSW corrections are included. The proposed MTRSW systems are particularly applicable for thin, magnetized, rotating stratified layers, such as the solar tachocline, neutron star oceans, and accretion disks with shallow vertical extent. Numerical simulations confirm that thermal and magnetic coupling substantially impacts the stability and evolution of vortices, with nontrivial temperature gradients fostering prolonged instabilities.

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• Hall effect
• Magnetohydrodynamic waves
• Magnetohydrodynamics