The Cost-E ectiveness of Semi-Lagrangian Advection

The purpose of this paper is to ascertain the cost-eeectiveness of semi-Lagrangian advection schemes for a wide variety of geophysical ows at all scales. Our approach is rst, 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 suuciently slower than that of the Eulerian frame to overcome the computational overhead. It is found that the most eecient semi-Lagrangian schemes require a factor of 5-10 times more oating point operations per grid point per time step than the classic second order leapfrog scheme. In the enstrophy cascade of 2D or quasi-geostrophic turbulence evolution of ow quantities is considerably slower in the Lagrangian frame and semi-Lagrangian advection schemes can be very cost-eeective. 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 diierence 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-eeective at current resolutions for these ows, which prevail in the atmosphere at scales below 300-400 km. In the presence of stationary forcing elds in the Eulerian frame, the time step must respect the advective time scale even in the Lagrangian frame, at length scales where the forcing is signiicant. Here semi-Lagrangian schemes are not recommended.-1-1. Introduction An evaluation of the cost-eeectiveness of semi-Lagrangian advection schemes is made on the basis of computational eeciency for a given level of accuracy. It has been argued that these schemes circumvent the numerical stability requirements of traditional Eulerian schemes and allow for a time step determined by accuracy considerations alone (Fjjrtoft 1952; Wiin-Nielson 1959; Sawyer 1963). In this work we use turbulent cascade phenomenology to estimate these accuracy limitations. Semi-Lagrangian methods are employed in other disciplines where they are variously referred to as Eulerian-Lagrangian methods or the modiied method of characteristics (Baptista 1987; Roache 1992). Finite-element methods where the elements are advected in this manner are known as Lagrange-Galerkin or characteristic-Although the early development of semi-Lagrangian schemes for atmospheric ows was performed on large-scale models (Robert 1981, Temperton and Staniforth 1987), where quasi-geostrophic enstrophy-cascade dynamics prevail near truncation scales, there seems to be increasing popularity of the method …