Three-Dimensional Mass-Conserving Semi-Lagrangian Scheme Employing Forward Trajectories

Abstract

Through the use of the dimensional splitting ?cascade? method of grid-to-grid interpolation, it is shown that consistently high-order-accurate semi-Lagrangian integration of a three-dimensional hydrostatic primitive equations model can be carried out using forward (downstream) trajectories instead of the backward (upstream) trajectory computations that are more commonly employed in semi-Lagrangian models. Apart from the efficiency resulting directly from the adoption of the cascade method, improved computational performance is achieved partly by the selective implicit treatment of only the deepest vertical gravity modes and partly by obviating the need to iterate the estimation of each trajectory's location. Perhaps the main distinction of our present semi-Lagrangian method is its inherent exact conservation of mass and passive tracers. This is achieved by adopting a simple variant of the cascade interpolation that incorporates mass (and tracer) conservation directly and at only a very modest additional cost. The conserving cascade, which is described in detail, is a generic algorithm that can be applied at arbitrary order of accuracy. Tests of the new mass-conserving scheme in a regional forecast model show small but consistent improvements in accuracy at 48 h. It is suggested that the benefits to extended global forecasting and simulation should be much greater.