On the Accuracy of Semi-Lagrangian Numerical Simulation of Internal Gravity Wave Motion in the Atmosphere

Abstract

We have investigated the accuracy of the semi-implicit semi-Lagrangian (SISL) method in simulating internal gravity wave (IGW) motion. We have focused on the relative accuracy of the hydrostatic, and nonhydrostatic IGW solutions. The analysis is based on a linearized model and a Global Circulation Model-Dynamic Core (GCM-DC) with a stretched grid.The nonhydrostatic version of the GCM-DC model produces the familiar IGW train disturbance anchored to an isolated hypothetical mountain. The wave has a distinct tilt away from the vertical direction, which is consistent with classical theory. For the hydrostatic version of the model, the axis of the resulting IGW train rests nearly perpendicular to the mountain top, thus again consistent with classical theory. Increasing the time step from 10 s; Courant number ðCnÞ ¼ 0:5; to 60 s ðCn ¼ 3:0Þ, results in stable solutions for both the hydrostatic and nonhydrostatic versions of the model. The nonhydrostatic solution is in close agreement with the control run however, the hydrostatic solution exhibits large phase truncation errors.The solutions for the one-dimensional linearized SISL model confirm the GCM-DC results that the nonhydrostatic IGW train is less damped and shifted by the SISL scheme than the corresponding hydrostatic IGW motion. The linear solutions indicate very high accuracy of the physical mode of the solution, but it rapidly deteriorates when Cn exceeds unity. As Dt ! 0 the amplitude of the computational mode tends to zero and its frequency to infinity. However, as Dt ! y, the frequency of the computational SISL mode asymptotically approaches the value of the frequency of the corresponding SISL physical mode. Furthermore, the amplitude of the SISL computational mode is directly proportional to the size of the time step. Therefore, at large time steps, the amplification of the computational mode could offset some of the numerical damping of the physical mode by the SISL scheme.

The Utility of Upper-Boundary Nesting in NWP

“…First, wind speeds in the winter stratospheric jet can exceed 350 kt (180 m s 21 , Fig. 1), which leads to large Courant numbers that may result in instability in Eulerian advection schemes and reduced accuracy in semi-Lagrangian advection schemes (Héreil and Laprise 1996;Semazzi et al 2005). Technical problems such as nesting-zone exceedances may also arise for long back trajectories computed in implementations using semi-Lagrangian techniques.…”