Geometry of shallow-water dynamics with thermodynamics

We review the geometric structure of the IL$^0$PE model, a rotating shallow-water model with variable buoyancy, thus sometimes called ``thermal'' shallow-water model. We start by discussing the Euler--Poincar\'e equations for rigid body dynamics and the generalized Hamiltonian structure of the system. We then reveal similar geometric structure for the IL$^0$PE. We show, in particular, that the model equations and its (Lie--Poisson) Hamiltonian structure can be deduced from Morrison and Greene's (1980) system upon ignoring the magnetic field ($\vec{\mathrm B} = 0$) and setting $U(\rho,s) = \frac{1}{2}\rho s$, where $\rho$ is mass density and $s$ is entropy per unit mass. These variables play the role of layer thickness ($h$) and buoyancy ($\t$) in the IL$^0$PE, respectively. Included in an appendix is an explicit proof of the Jacobi identity satisfied by the Poisson bracket of the system.