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In the manuscript we have derived the flux-form atmospheric governingequations in the general curvilinear coordinate system which is used by ahigh-order nonhydrostatic multi-moment constrained finite volume (MCV)dynamical core, and given the explicit formulations in the shallow-atmosphereapproximation. In general curvilinear coordinate x^i(i = 1,2,3), unlike theCartesian coordinate, the base vectors are not constants either in magnitude ordirection. Following the representations such as base vectors, vector andtensor and so on in general curvilinear coordinate, we can obtain thedifferential relations of base vectors, the gradient and divergence operatoretc. which are the component parts of the atmospheric governing equation. Thenwe apply them in the two specific curvilinear coordinate system: the sphericalpolar and cubed-sphere coordinates that are adopted in high-ordernonhydrostatic MCV dynamical core. By switching the geometrics such as themetric tensors (covariant and contravariant), Jacobian of the transformation,the Christoffel symbol of the second kind between the spherical polar andcubed-sphere coordinates, the resulting flux-form governing equations in thespecific coordinate system can be easily achieved. Of course, the Cartesiancoordinate can be recovered. Noted that the projection metric tensors likespherical polar system and Cartesian coordinate become simple due to orthogonalproperties of coordinate.

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