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].

翻译：本文研究了势涡度动力学中平衡二维热准地转（TQG，Thermal Quasigeostrophic）模型随机版本的数学性质。该随机TQG模型旨在为上层海洋动力学数值模拟中，当水平浮力梯度和海底地形影响动力学（特别是在次中尺度250m-10km范围内）时，未解析自由度动态生成的参数化提供基础。具体而言，我们选择文献[1]提出并已在[2,3]中应用的SALT（随机李输运平流，Stochastic Advection by Lie Transport）算法作为建模方法。SALT方法保留了开尔文环流定理和TQG的无穷族积分守恒律。该算法的目标是在粗粒度化或尺度粗化过程中，量化对精细尺度观测或合成数据进行粗尺度数值模拟应用时的不确定性。本文通过严格的数学分析，论证了具有随机李输运平流（SALT）[4,5]的热准地转方程的解性质。