Using 3D finite volume for the pressure‐gradient force in atmospheric models

Abstract The horizontal pressure‐gradient force is traditionally discretized in partial‐differential equation (PDE) form in atmospheric models, but may alternatively be discretized as the net normal surface force divided by the mass, which is referred to as the Green–Gauss method in computational fluid dynamics (CFD). The latter approach is ubiquitous in engineering because of the utility of unstructured grids; in the meteorological context Lin derived in 1997 a two‐dimensional (2D), plane‐parallel, Green–Gauss algorithm and showed that it eliminated inaccuracies related to using terrain‐following (sigma) coordinates in steep terrain. Although a hierarchy of three‐dimensional (3D) algorithms have been developed by the CFD community, with varying cost‐to‐benefit trade‐offs, work remains to be done to take full advantage of the 3D approach in atmospheric models. Here, we move in this direction by extending Lin's 2D algorithm to 3D. As is true of all Green–Gauss implementations, our algorithm is the same regardless of choice of vertical coordinate or number of fluid constituents. We show that the modifications to the 2D algorithm may be written in the form of additive terms composed of differences of the squares of the top and bottom tilts of the finite‐volume faces; the cases for which the 2D and 3D algorithms are identical are then straightforward to discern. For plane‐parallel geometry, we include explicit finite‐volume formulae for the longitude–latitude and general convex quadrilateral planforms, which includes the cubed‐sphere grid, with arbitrary geopotentials on the eight corners. Compared with the 2D version, the 3D algorithm yields approximately a 10% reduction in truncation error and, in the context of a general circulation model, it requires only about a 1% increase in run time. Copyright © 2012 Royal Meteorological Society