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

This 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 (c) 2011 Royal Meteorological Society