  In analogy with similar effects in adiabatic compressible fluid dynamics, the
effects of buoyancy gradients on incompressible stratified flows are said to be
`thermal'. The thermal rotating shallow water (TRSW) model equations contain
three small nondimensional parameters. These are the Rossby number, the Froude
number and the buoyancy parameter. Asymptotic expansion of the TRSW model
equations in these three small parameters leads to the deterministic thermal
versions of the Salmon's L1 (TL1) model and the thermal quasi-geostrophic (TQG)
model, upon expanding in the neighbourhood of thermal quasi-geostrophic balance
among the flow velocity and the gradients of free surface elevation and
buoyancy. The linear instability of TQG at high wave number tends to create
circulation at small scales. Such a high wave number instability could be
unresolvable in many computational simulations, but its presence at small
scales may contribute significantly to fluid transport at resolvable scales.
Sometimes such effects are modelled via `stochastic backscatter of kinetic
energy'. Here we try another approach. Namely, we model `stochastic transport'
in the hierarchy of models TRSW/TL1/TQG. The models are derived via the
approach of stochastic advection by Lie transport (SALT) as obtained from a
recently introduced stochastic version of the Euler--Poincar\'e variational
principle. We also indicate the potential next steps for applying these models
in uncertainty quantification and data assimilation of the rapid, high
wavenumber effects of buoyancy fronts at these three levels of description by
using the data-driven stochastic parametrisation algorithms derived previously
using the SALT approach.
