A Vertical Finite-Difference Scheme for Hydrostatic and Nonhydrostatic Equations

Abstract

Vertical finite-differences schemes that preserve the important integral properties of the continuum equations are derived for the basic hydrodynamic and thermodynamic equations governing the atmosphere. The vertical coordinate is altitude. The first scheme applies in fully nonhydrostatic cases and therefore can be used to study smaller motion when the hydrostatic approximation cannot be justified. The finite-difference equations are then reduced to a hydrostatic set that continues to satisfy the integral constraints. In this simplified form, the vertical difference scheme guarantees local compliance with hydrostatic balance, in contrast to some sigma-coordinate models in which nonlocal treatments of this approximation may be required.