In this paper, we investigate the two-dimensional extension of a recently introduced set of shallow water models based on a regularized moment expansion of the incompressible Navier-Stokes equations \cite{kowalski2017moment,koellermeier2020analysis}. We show the rotational invariance of the proposed moment models with two different approaches. The first proof involves the split of the coefficient matrix into the conservative and non-conservative parts and proves the rotational invariance for each part, while the second one relies on the special block structure of the coefficient matrices. With the aid of rotational invariance, the analysis of the hyperbolicity for the moment model in 2D is reduced to the real diagonalizability of the coefficient matrix in 1D. Then we analyze the real diagonalizability by deriving the analytical form of the characteristic polynomial. We find that the moment model in 2D is hyperbolic in most cases and weakly hyperbolic in a degenerate edge case. With a simple modification to the coefficient matrices, we fix this weakly hyperbolicity and propose a new global hyperbolic model. Furthermore, we extend the model to include a more general class of closure relations than the original model and establish that this set of general closure relations retains both rotational invariance and hyperbolicity.

翻译：本文研究了一类基于不可压缩Navier-Stokes方程正则化矩展开的浅水模型在二维情形下的扩展。我们通过两种不同方法证明了所提矩模型的旋转不变性。第一种证明将系数矩阵分解为守恒部分与非守恒部分，并分别证明各部分的旋转不变性；第二种证明则依赖于系数矩阵的特殊分块结构。借助旋转不变性，二维矩模型的双曲性分析可转化为一维系数矩阵的实可对角化性分析。随后，我们通过推导特征多项式的解析形式来分析实可对角化性。研究发现，该二维矩模型在多数情况下具有双曲性，仅在退化边缘情形下呈现弱双曲性。通过对系数矩阵进行简单修正，我们修复了该弱双曲性问题，并提出了一种新的全局双曲模型。此外，我们将模型扩展至比原始模型更一般的闭合关系类，并证明该广义闭合关系集同时保持旋转不变性与双曲性。