Instabilities in the Shallow-Water System with a Semi-Lagrangian, Time-Centered Discretization

Conventional wisdom suggests that the combination of semi-Lagrangian advection and an implicit treatment of gravity wave terms should result in a combined scheme for the shallow-water equations stable for high Courant numbers. This wisdom is well justified by linear analysis of the system about a uniform reference state with constant fluid depth and velocity, but it is only assumed to hold true in more complex scenarios. This work finds that this conventional wisdom no longer holds in more complicated flow regimes, in particular when the background state is given by steady-state flow past topography. Instead, this background state admits a wide range of instabilities that can lead to noise in atmospheric forecasts. Significance StatementThis work shows that solutions to the shallow-water equations with a semi-Lagrangian treatment of advection and an implicit, time-centered treatment of gravity wave terms can be unstable when there is a background state of flow over topography. This basic algorithm is used by many operational weather-forecasting models to simulate the meteorological equations, and showing an instability in the simplified, shallow-water system suggests that a similar mechanism may be responsible for "noise" in operational weather forecasts under some circumstances. If this problem can be addressed, it could allow numerical weather models to operate with less dissipation, improving forecast quality.