The Equations of Magnetoquasigeostrophy

Abstract

The dynamics contained in magnetized layers of exoplanet atmospheres are important to understand in order to characterize what observational signatures they may provide for future observations. It is important to develop a framework to begin studying and learning the physical processes possible under those conditions and what, if any, features contained in them may be observed in future observation missions. The aims of this study is to formally derive, from scaling arguments, a manageable reduced set of equations for analysis, i.e. a magnetic formulation of the equations of quasigeostrophy appropriate for a multi-layer atmosphere. The main goal is to provide a simpler theoretical platform to explore the dynamics possible within confined magnetized layers of exoplanet atmospheres. We primarily use scaling arguments to derive the reduced equations of "magnetoquasigeostrophy" which assumes dynamics to take place in an atmospheric layer which is vertically thin compared to its horizontal scales. The derived equation set retains features existing in standard shallow-water magnetohydrodynamic equations but are absent in more classical derivations of the quasi-geostrophic limit, namely, the non-divergence of the in-plane components of the magnetic field. We liken this non-divergence of the in-plane magnetic fields as indicative of a quantity whose behaviour mimics a two-dimensional "pseudo"-magnetic monopole source. We also find, using the same scaling argument procedures, appropriate limits of the fundamental parameters of the system which yield reduced equations describing the flow dynamics primarily characterized by magnetostrophic balance. The standard scaling arguments employed here show how traditional magnetized quasigeostrophic equations connect to their magnetized shallow water forms. The equations derived are amenable to analysis using well-known techniques.