A semi‐implicit non‐hydrostatic dynamical kernel using finite elements in the vertical discretization

Abstract

AbstractThis work is a first step in the direction of implementing a high‐order finite‐element discretization in the vertical in the non‐hydrostatic version of the HARMONIE model. The present dynamical core of the HARMONIE model is shared with the ECMWF and the ALADIN models and uses a horizontal spectral discretization and a semi‐implicit semi‐Lagrangian time stepping scheme, all of which are maintained in this work. Trying to implement a finite‐element discretization in the non‐hydrostatic version of the HARMONIE model has been found to be very difficult due to the set of prognostic variables used and the mass‐based vertical coordinate. A different set of prognostic variables and a hybrid vertical coordinate based on height are tested here on a vertical slice non‐hydrostatic kernel. A stability analysis of the linear model has been done. To evaluate the model stability and accuracy, a set of test cases from the literature are presented in the linear and nonlinear regimes, with and without orography. An iterative centred‐implicit scheme can be applied to avoid instability related to steep orography, although this reduces the efficiency of the model. The novel aspects with respect to existing non‐hydrostatic model kernels are the use of cubic finite elements in the vertical discretization, the use of a height‐based vertical coordinate in conjunction with a spectral discretization in the horizontal, and the coordinate‐independent formulation of each element of the model including the semi‐Lagrangian advection. Copyright © 2011 Royal Meteorological Society