A Consistent Time-Split Numerical Scheme Applied to the Nonhydrostatic Compressible Equations*

Abstract

Abstract The primary interest of the paper is to apply a two-time-level split explicit time scheme developed by one of the authors to the Lokal-Modell (LM) of the German Weather Service (DWD). This model belongs to the operational NWP system at DWD, which makes it particularly interesting for this study. To better understand the implementation of this time scheme in a compressible nonhydrostatic model type, and so in the LM, a linear analysis is presented demonstrating how the equations are to be split up into fast- and slow-mode parts. For the fast-mode part, this analysis demonstrates how the connected short time-step scheme is necessary for a consistent treatment of gravity modes on the one side and a sufficient damping of acoustic modes on the other side. An extended linear stability analysis for the new splitting scheme follows then to establish its application in a full model. An advantage of the given time scheme is that any forward-in-time and stable advection scheme can be linked with the reformulated fast-mode equation part. A Runge–Kutta third-order-in-time and second-order-in-space scheme (RK3/2) has been applied to the horizontal advection, and the vertical advection terms are treated implicitly. A new consistent lower boundary condition and a radiative upper boundary condition are taken into account. Steady airflow simulations over an isolated mountain (Schär test) and the successful incorporation of the Klemp–Durran–Bougeault radiative upper boundary condition in the vertically implicit fast-mode scheme confirm the given approach as necessary and effective for the application of the time scheme.