Four‐dimensional tensor equations for a classical fluid in an external gravitational field

A four‐dimensional tensor formalism suitable for the equations of motion of a classical fluid in the presence of a given external gravitational field is presented. The formalism allows for arbitrary time‐dependent transformations of spatial coordinates. Some well‐known conservation laws are derived in covariant form. The metric tensor and the associated Christoffel symbols are calculated for coordinate systems useful in meteorology. The vertical momentum equation employed in the Canadian operational weather forecasting model is obtained using the proposed tensor formalism.