On Forward-in-Time Differencing for Fluids: an Eulerian/Semi-Lagrangian Non-Hydrostatic Model for Stratified Flows

In this paper, we describe a non-hydrostatic anelastic model for simulating stratified flows in terrain-following coordinates. The model is based solely on non-oscillatory forward-in-rime integration schemes, and our primary goal is to demonstrate the utility of such methods for modelling small-scale atmospheric dynamics. We use the formal similarity of the Eulerian and semi-Lagrangian equations of two-time-level approximations to construct a unified model that readily allows selection of either formulation. We apply the model to two test problems of stratified flows past isolated obstacles. We use these tests to validate the forward-in-time approach against a traditional centred-in-time-and-space Eulerian model, and to discuss the relative accuracy and efficiency of the two formulations of our model. One problem illustrates the efficacy of flux-form Eulerian methods, while the other demonstrates strengths of the semi-Lagrangian approach.