  This paper investigates the mathematical properties of a stochastic version
of the balanced 2D thermal quasigeostrophic (TQG) model of potential vorticity
dynamics. This stochastic TQG model is intended as a basis for parametrisation
of the dynamical creation of unresolved degrees of freedom in computational
simulations of upper ocean dynamics when horizontal buoyancy gradients and
bathymetry affect the dynamics, particularly at the submesoscale (250m--10km).
Specifically, we have chosen the SALT (Stochastic Advection by Lie Transport)
algorithm introduced in [1] and applied in [2,3] as our modelling approach. The
SALT approach preserves the Kelvin circulation theorem and an infinite family
of integral conservation laws for TQG. The goal of the SALT algorithm is to
quantify the uncertainty in the process of up-scaling, or coarse-graining of
either observed or synthetic data at fine scales, for use in computational
simulations at coarser scales. The present work provides a rigorous
mathematical analysis of the solution properties of the thermal
quasigeostrophic (TQG) equations with stochastic advection by Lie transport
(SALT) [4,5].
