Continuity equation for non-spherical geometries in mass-coordinates
This memo describes the transformation of physically-consistent atmospheric systems in non-spherical geometries from geopotential-based to mass-based coordinates. Only the continuity equation is examined here.
The physically consistent equations of the atmosphere for a general axially-symmetric and "geopotential" system of coordinates has been presented by White and Wood, 2012, Q. J. R. Meteor. Soc (WW12 hereafter). Here "geopotential" coordinates means that one coordinate follows the vertical direction everywhere, whilst the two other coordinates follow horizontal surfaces everywhere.
However, the various physically-consistent systems proposed in WW12 cannot be directly applied to mass-based coordinates, because these coordinates are not "geopotential" coordinates (iso-baric surfaces are not necessarily horizontal, and moreover, they are moving with respect to fixed iso-geopotential surfaces with time).
The classical technic of time- and space-dependent coordinate transformation used in Kasahara, MWR, 1974 is applied here to obtain the desired form of the continuity equation.