The cost-effectiveness of semi-Lagrangian advection

The purpose of this paper is to ascertain the cost-effectiveness of semi-Lagrangian advection schemes for a wide variety of geophysical flows at all scales. The approach used is first to determine the minimum computational overhead associated with these schemes and then to examine temporal Variability in the Lagrangian and Eulerian frames by employing simple turbulent cascade phenomenologies. The goal is to evaluate whether the Lagrangian variability is sufficiently slower than that of the Eulerian frame to overcome the computational overhead. It is found that the most efficient semi-Lagrangian schemes require a factor of 5-10 times more floating point operations per grid point per time step than the classic second-order leapfrog scheme. In the enstrophy cascade of 2D or quasigeostrophic turbulence, evolution of flow quantities is considerably slower in the Lagrangian frame and semi-Lagrangian advection schemes can be very cost-effective. In an energy cascade such as the Kolmogorov range of 3D turbulence or the inverse cascade of QG or 2D turbulence, the Lagrangian evolution remains slower than the Eulerian evolution. However, the difference is very much less than in the enstrophy cascade. Since the computational overhead of semi-Lagrangian schemes is considerable, they are at best marginally cost-effective at current resolutions for these flows, which prevail in the atmosphere at scales below 300-400 km. In the presence of stationary forcing fields in the Eulerian frame, the time step must respect the advective timescale even in the Lagrangian frame, at length scales where the forcing is significant. Hen semi-Lagrangian schemes are not recommended.