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 prove 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 prove the real diagonalizability by deriving the analytical form of the characteristic polynomial. 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 retain both rotational invariance and hyperbolicity.

翻译：本文研究了一类基于不可压缩纳维-斯托克斯方程正则化矩展开的浅水模型在二维情况下的扩展（参见文献\cite{kowalski2017moment,koellermeier2020analysis}）。我们通过两种不同方法证明了所提出矩模型的旋转不变性：第一种证明涉及将系数矩阵分解为保守部分与非保守部分，并分别证明两部分的旋转不变性；第二种证明则依赖于系数矩阵的特殊块状结构。借助旋转不变性，二维矩模型的双曲性分析可简化为对一维系数矩阵实可对角化性的判定。随后，我们通过推导特征多项式的解析形式，证明了该实可对角化性质。此外，我们将模型扩展至包含比原始模型更广泛的闭合关系族，并证明这组广义闭合关系同时保持了旋转不变性与双曲性。