Integration of the Fully Elastic Equations Cast in the Hydrostatic Pressure Terrain-Following Coordinate in the Framework of the ARPEGE/Aladin NWP System

Abstract

ARPEGE/Aladin is a limited-area 3D primitive equation model, which belongs to the integrated NWP ARPEGE/IFS system. Like its global counterpart, the limited-area version has a spectral representation of variables in the horizontal but uses double-Fourier series instead of the classical spherical harmonies, in the manner introduced by Machenhauer and Haugen. Following the suggestion of Laprise, a nonhydrostatic version of ARPEGE/Aladin has been developed using hydrostatic pressure as an independent variable. The dynamics employ the fully elastic Euler equations of motion, orography being introduced via a terrain-following hybrid coordinate. A semi-implicit scheme has been formulated to control both acoustic and gravity waves. The discrete linear operators appear to have the same form as in the hydrostatic dynamics, except an additional one representing the vertical part of the Laplacian operator. To keep an elegant elimination, it was necessary to modify the approximation of logarithmic thicknesses of the model layers. It is noteworthy that the Helmholtz matrix has a tridiagonal form, confirming a local character to the nonhydrostatic dynamics. The representation of the horizontal pressure gradient term fulfills the rules of conservation of energy and angular momentum. Some instability problems were encountered and it was thus necessary to introduce an additional semi-implicit type of correction of the nonlinear part of the total 3D divergence, a solution that calls for iterations of the semi-implicit scheme. The results of a few idealized numerical simulations and of a real situation are presented.