  Driven by growing momentum in two-dimensional geophysical flow modeling, this
paper introduces a general family of "thermal" rotating shallow-water models.
The models are capable of accommodating thermodynamic processes, such as those
acting in the ocean mixed layer, by allowing buoyancy to vary in horizontal
position and time as well as with depth, in a polynomial fashion up to an
arbitrary degree. Moreover, the models admit Euler--Poincare variational
formulations and possess Lie--Poisson Hamiltonian structure. Such a geometric
property provides solid fundamental support to the theories described with
consequences for numerical implementation and the construction of unresolved
motion parametrizations. In particular, it is found that stratification halts
the development of small-scale filament rollups recently observed in a popular
model, which, having vertically homogeneous density, represents a special case
of the models presented here.
