The QG limit of the rotating thermal shallow water equations

The rotating thermal shallow water (RTSW) equations is a model for studying thin layer fluids with horizontal variations in material properties. A mathematically rigorous framework is developed for deriving new reduced slow dynamics equations of the multiple-scale RTSW equations with both balanced and unbalanced initial data over the spatial periodic domain. It is rigorously proved that the reduced slow dynamics of the RTSW equations is the thermal quasi-geostrophic (thermal QG) equations by proper rescaling. Moreover, by introducing appropriate elliptic equations and using the time-averaging technique, we further obtained the convergence rates of the solutions with unbalanced initial data.