Stabilization of Nonlinear Vertical Diffusion Schemes in the Context of NWP Models

The stability of the nonlinear vertical diffusion equation such as commonly used for parameterizing the turbulence: in NWP models is examined. As a starring point, this paper adopts the idea of Girard and Delage and shows how their results can be modified when the problem is examined in a less restrictive framework, typical of practical NWP applications. in Girard and Delage's work, an optimal compromise between stability and accuracy was proposed to eliminate the "fibrillations" resulting from the instability, by applying a time decentering in the diffusion operator for the points likely to be unstable according to a local linear analysis of the stability. This key idea is pursued here, bur two important changes are examined: (i) an exact method for the relaxation of the identity between thermal and dynamical exchange coefficients, and (ii) the introduction of a modification to the Richardson number for simulating the destabilization of the top of the PBL in shallow convection conditions. Compared to an approximate solution proposed by Girard and Delage for the first change, the one proposed here is more accurate and possesses a formal justification. For the second one, it is shown that the only consequence is that the stability now depends on the largest value of the modified/not modified Richardson number. A formulation of the temporal decentering; consistent with these changes is then proposed and evaluated.