On the dynamical consistency of geometrically approximated equations for geophysical fluids in arbitrary coordinates

Abstract

It is demonstrated that treating geometrically approximated equations for geophysical fluids as four-dimensional tensor equations in curved (or Riemannian) geometry automatically leads to dynamical consistency of the equations in arbitrary coordinates. Dynamical consistency is usually defined as the preservation of four-dimensional divergent-free conservation laws from the approximated equations for total energy, angular momentum and Ertel's potential vorticity. Two conditions must be met to preserve dynamical consistency under geometric approximations: (i) the same tensor equations developed in flat geometry are used in curved geometry and (ii) the covariant metric tensor and Christoffel symbols are not approximated, but rather recalculated from the geometrically approximated contravariant metric tensor, hence preserving the covariance of the equations. A more general definition of dynamical consistency under geometric approximations is proposed.