Development of a unified high-order nonhydrostatic multi-moment constrained finite volume dynamical core: derivation of flux-form governing equations in the general curvilinear coordinate system

In the manuscript we have derived the flux-form atmospheric governing equations in the general curvilinear coordinate system which is used by a high-order nonhydrostatic multi-moment constrained finite volume (MCV) dynamical core, and given the explicit formulations in the shallow-atmosphere approximation. In general curvilinear coordinate x^i(i = 1,2,3), unlike the Cartesian coordinate, the base vectors are not constants either in magnitude or direction. Following the representations such as base vectors, vector and tensor and so on in general curvilinear coordinate, we can obtain the differential relations of base vectors, the gradient and divergence operator etc. which are the component parts of the atmospheric governing equation. Then we apply them in the two specific curvilinear coordinate system: the spherical polar and cubed-sphere coordinates that are adopted in high-order nonhydrostatic MCV dynamical core. By switching the geometrics such as the metric tensors (covariant and contravariant), Jacobian of the transformation, the Christoffel symbol of the second kind between the spherical polar and cubed-sphere coordinates, the resulting flux-form governing equations in the specific coordinate system can be easily achieved. Of course, the Cartesian coordinate can be recovered. Noted that the projection metric tensors like spherical polar system and Cartesian coordinate become simple due to orthogonal properties of coordinate.