SWIFT: A Monotonic, Flux-Form Semi-Lagrangian Tracer Transport Scheme for Flow with Large Courant Numbers

Abstract Local conservation of mass and entropy are becoming increasingly desirable properties for modern numerical weather and climate models. This work presents a flux-form semi-Lagrangian (FFSL) transport scheme, called Splitting With Improved FFSL for Tracers (SWIFT), that facilitates this conservation for tracer variables, while maintaining other vital properties such as preservation of a constant, monotonicity, and positivity. Importantly, these properties all hold for large Courant numbers and multidimensional flow, making the scheme appropriate for use within a dynamical core which takes large time steps. The SWIFT scheme presented here can be seen as an evolution of the FFSL methods of Leonard et al. and Lin and Rood. Two-dimensional and three-dimensional schemes consist of a splitting into a sequence of one-dimensional calculations. The new SWIFT splitting presented here allows monotonic and positivity properties from the one-dimensional calculations to be inherited by the multidimensional scheme. These one-dimensional calculations involve separating the mass flux into terms that correspond to integer and fractional parts of the Courant number. Key to achieving conservation is coupling the transport of tracers to the transport of the fluid density, through the reuse of the discrete mass flux that was calculated from the fluid density in the transport of the tracers. This work also describes how these properties can still be attained when the tracer is vertically staggered from the density in a Charney–Phillips grid. Significance Statement This work presents an algorithm to be used in weather and climate models to describe the transport of quantities by the wind. This algorithm is intended to be used for the transport of tracers such as cloud or chemicals in the atmosphere. The significance of the algorithm is that it conserves the mass of these tracers and can avoid the generation of unphysical negative concentrations. Importantly, these properties still hold when the wind is fast relative to the ratio of the grid spacing to the size of the time steps taken by the atmospheric model.