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 parametrization 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 (250$$m–10$$km). Specifically, we have chosen the Stochastic Advection by Lie Transport (SALT) algorithm introduced in [D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. Roy. Soc. A: Math. Phys. Eng. Sci.471 (2015) 20140963, http://dx.doi.org/10.1098/rspa.2014.0963] and applied in [C. Cotter, D. Crisan, D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Found. Data Sci.2 (2020) 173, https://doi.org/10.3934/fods.2020010; Numerically modeling stochastic lie transport in fluid dynamics, SIAM Multiscale Model. Simul.17 (2019) 192–232, https://doi.org/10.1137/18M1167929] as our modeling 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 SALT [D. D. Holm and E. Luesink, Stochastic wave-current interaction in thermal shallow water dynamics, J. Nonlinear Sci.31 (2021), https://doi.org/10.1007/s00332-021-09682-9; D. D. Holm, E. Luesink and W. Pan, Stochastic mesoscale circulation dynamics in the thermal ocean, Phys. Fluids33 (2021) 046603, https://doi.org/10.1063/5.0040026].

本文研究了势涡动力学平衡二维热准地转 (TQG) 模型的随机版本的数学特性。当水平浮力梯度和测深影响动力学时，特别是在亚中尺度（250$$米–10$$公里）。具体来说，我们选择了[D. D. Holm，随机流体动力学的变分原理，Proc。罗伊. 苏克。答：数学。物理。工程师。科学。471 (2015) 20140963，http://dx.doi.org/10.1098/rspa.2014.0963]并应用于[C. Cotter、D. Crisan、D. Holm、W. Pan 和 I. Shevchenko，在 2 层准地转模型中使用随机传输噪声对不确定性进行建模，发现。数据科学。2（2020）173，https://doi.org/10.3934/fods.2020010；对流体动力学中的随机流体传输进行数值模拟，SIAM 多尺度模型。同时。17 号(2019) 192–232, https://doi.org/10.1137/18M1167929] 作为我们的建模方法。SALT 方法保留了开尔文循环定理和 TQG 的无限族积分守恒定律。SALT 算法的目标是量化精细尺度上观测数据或合成数据的放大或粗粒度过程中的不确定性，以用于较粗尺度的计算模拟。目前的工作使用 SALT 对热准地转 (TQG) 方程的解性质进行了严格的数学分析 [DD Holm 和 E. Luesink，热浅水动力学中的随机波-流相互作用，J. 非线性科学。31（2021），https://doi.org/10.1007/s00332-021-09682-9；DD Holm、E. Luesink 和 W. Pan，热海洋中的随机中尺度环流动力学，Phys。流体33 (2021) 046603，https://doi.org/10.1063/5.0040026]。