The Flux-Form Semi-Lagrangian Spectral Element (FF-SLSE) method for tracer transport

The spectral element dynamical core has been demonstrated to be an accurate and scalable approach for solving the equations of motion in the atmosphere. However, it is also known that use of the spectral element method for tracer transport is costly and requires substantial parallel communication over a single time step. Consequently, recent efforts have turned to finding alternative transport schemes which maintain the scalability of the spectral element method without its significant cost. This article proposes a conservative semi-Lagrangian approach for tracer transport which uses upstream trajectories to solve the transport equation on the native spectral element grid. This formulation, entitled the Flux-Form Semi-Lagrangian Spectral Element (FF-SLSE) method, is highly accurate compared to many competing schemes, allows for large time steps, and requires fewer parallel communications over the same time interval than the spectral element method. In addition, the approach guarantees local conservation and is easily paired with a filter which can be used to ensure positivity. This article presents the dispersion relation for the 1D FF-SLSE approach and demonstrates stability up to a Courant number of 2.44 with cubic basis. Several standard numerical tests are presented for the method in 2D to verify correctness, accuracy and robustness of the method, including a new test of a divergent flow in Carteisan geometry.