Abstract
Within the context of exoplanetary atmospheres, we present a comprehensive
linear analysis of forced, damped, magnetized shallow water systems, exploring
the effects of dimensionality, geometry (Cartesian, pseudo-spherical and
spherical), rotation, magnetic tension and hydrodynamic and magnetic sources of
friction. Across a broad range of conditions, we find that the key governing
equation for atmospheres and quantum harmonic oscillators are identical, even
when forcing (stellar irradiation), sources of friction (molecular viscosity,
Rayleigh drag and magnetic drag) and magnetic tension are included. The global
atmospheric structure is largely controlled by a single, key parameter that
involves the Rossby and Prandtl numbers. This near-universality breaks down
when either molecular viscosity or magnetic drag varies significantly across
latitude or a poloidal magnetic field is present, suggesting that these effects
will introduce qualitative changes to the familiar chevron-shaped feature
witnessed in simulations of atmospheric circulation. We also find that
hydrodynamic and magnetic sources of friction have dissimilar phase signatures
and affect the flow in fundamentally different ways, implying that using
Rayleigh drag to mimic magnetic drag is inaccurate. We exhaustively lay down
the theoretical formalism (dispersion relations, governing equations and
time-dependent wave solutions) for a broad suite of models. In all situations,
we derive the steady state of an atmosphere, which is relevant to interpreting
infrared phase and eclipse maps of exoplanetary atmospheres. We elucidate a
pinching effect that confines the atmospheric structure to be near the equator.
Our suite of analytical models may be used to decisively develop physical
intuition and as a reference point for three-dimensional, magnetohydrodynamic
(MHD) simulations of atmospheric circulation.
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