Replicating Semi‐Implicit Semi‐Lagrangian Stability in Semi‐Implicit Time‐Stepping Schemes Using Forward‐in‐Time Eulerian Advection

ABSTRACT Semi‐implicit semi‐Lagrangian schemes have excellent stability properties, making them popular for operational numerical weather prediction. That stability is obtained by integrating the equations along a Lagrangian trajectory. It might be desirable, for example, as a route to achieving mass conservation, to avoid the integration along a Lagrangian trajectory and replace semi‐Lagrangian advection with an explicit, forward‐in‐time Eulerian scheme. However, a naïve replacement of certain Lagrangian‐average quantities by Eulerian‐average quantities, including the advecting velocity, results in an unstable time integration scheme. Here, this instability is understood through a linear stability analysis of different time integration schemes for the rotating shallow‐water equations. The analysis develops a general approach to restoring stability by a modified calculation of advection terms or advective fluxes in both the height and momentum equations, in which the advection operator acts on partially time‐stepped or “predicted” fields. The results of the analysis are supported by numerical experiments with the shallow‐water equations.