We show how the moist-convective rotating shallow water model, where the moist convection and the related latent heat release are incorporated into the standard rotating shallow water model of the atmosphere, can be improved by introducing, in a self-consistent way, horizontal gradients of potential temperature and changes of the latter due to the condensation heating, radiative cooling, and ocean-atmosphere heat fluxes. We also construct the quasi-geostrophic limit of the model in mid-latitudes and its weak-gradient limits in the equatorial region. The capabilities of the new model are illustrated by the examples of convection-coupled gravity waves and equatorial waves produced by the relaxation of localized pressure and potential temperature anomalies in the presence of moist convection.

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In this work, we combine the idea of data-driven polynomial chaos expansions with the weighted least-square approach to solve uncertainty quantification (UQ) problems. The idea of data-driven polynomial chaos is to use statistical moments of the input random variables to develop an arbitrary polynomial chaos expansion, and then use such data-driven bases to perform UQ computations. Here we adopt the bases construction procedure by following [1], where the bases are computed by using matrix operations on the Hankel matrix of moments. Different from previous works, in the postprocessing part, we propose a weighted least-squares approach to solve UQ problems. This approach includes a sampling strategy and a least-squares solver. The main features of our approach are two folds: On one hand, our sampling strategy is independent of the random input. More precisely, we propose to sampling with the equilibrium measure, and this measure is also independent of the data-driven bases. Thus, this procedure can be done in prior (or in a offline manner). On the other hand, we propose to solve a Christoffel function weighted least-square problem, and this strategy is quasi-linearly stable -the required number of PDE solvers depends linearly (up to a logarithmic factor) on the number of (data-driven) bases. This new approach is thus promising in dealing with a class of problems with epistemic uncertainties. Several numerical tests are presented to show the effectiveness of our approach.