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 SWIFT, that facilitates this conservation for tracer variables, whilst maintaining other vital properties such as preservation of a constant, monotonicity and positivity. Importantly, these properties all hold for large Courant numbers and multi-dimensional 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 multi-dimensional 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 re-use 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.

Figures (8)

• Figure 1: An illustration on a two-dimensional mesh of the process used to compute the coordinate of the departure point $x^{dep}_{i+1/2}$, corresponding to the flux point $x_{i+1/2}$. The process involves finding the volume that is swept along coordinate lines over time interval $\Delta t$ through the facet of area $S_{i+1/2}$ at $x_{i+1/2}$. This swept volume is shaded in grey. The departure point is such that the swept volume is equal to $u_{i+1/2}S_{i+1/2}\Delta t$, thus ensuring that \ref{['eqn:dep_point']} is satisfied.
• Figure 2: The final transported fields from the constant velocity test of Section \ref{['sec:test1']}, using a spatially-varying density. These fields are taken from the large Courant number case, with $\Delta t=2$ s. The top row shows fields transported with the COSMIC splitting, while the bottom row contains fields transported with the SWIFT splitting. For the density field $\rho$ (left column) the two splittings give the same result. The central column displays a conservative mixing ratio, with no limiter applied. The limited tracer (right column) is monotonic with the SWIFT splitting but not with the COSMIC splitting. The density fields are contoured at intervals of $0.05$ kg m$^{-3}$, with the $0.8$ kg m$^{-3}$ contour omitted. The mixing ratio fields have contours every $0.1$ kg kg$^{-1}$, omitting the zero contour. Arrows on the density plots show the direction and magnitude of the transporting velocity.
• Figure 3: The transported fields at the end of the final time step ($t=100$ s) of the non-divergent deformational test of Section \ref{['sec:test2']} with a constant density. These fields are taken from the large Courant number case, with $\Delta t=2$ s. Different rows show the fields transported with the COSMIC splitting and SWIFT splittings. The mixing ratio fields have contours every $0.1$ kg kg$^{-1}$, omitting the zero contour. Arrows on the density plots show the direction and magnitude of the transporting velocity.
• Figure 4: The transported fields at the halfway point ($t=50$ s, top two rows) and at the end of the final time step ($t=100$ s, bottom two rows) of the non-divergent deformational test of Section \ref{['sec:test2']}. These fields are taken from the large Courant number case, with $\Delta t=2$ s. Different rows show the fields transported with the COSMIC splitting and SWIFT splittings. Even under strong deformation, the limited tracer with the SWIFT splitting remains monotonic. The density fields are contoured at intervals of $0.05$ kg m$^{-3}$, with the $0.8$ kg m$^{-3}$ contour omitted. The mixing ratio fields have contours every $0.1$ kg kg$^{-1}$, omitting the zero contour. Arrows on the density plots show the direction and magnitude of the transporting velocity.
• Figure 5: The transported fields at the halfway point ($t=50$ s, top two rows) and end of the final time step ($t=100$ s, bottom two rows) of the divergent test of Section \ref{['sec:test2']}. These fields are taken from the large Courant number case, with $\Delta t=2$ s. Different rows show the COSMIC and SWIFT splittings. The density fields are contoured at intervals of $0.05$ kg m$^{-3}$, with the $0.8$ kg m$^{-3}$ contour omitted. The mixing ratio fields have contours every $0.1$ kg kg$^{-1}$, omitting the zero contour. Arrows on the density plots show the direction and magnitude of the transporting velocity.