A finite-element scheme for the vertical discretization of the semi-Lagrangian version of the ECMWF forecast model

A vertical finite-element (FE) discretization designed for the European Centre for Medium-Range Weather Forecasts (ECMWF) model with semi-Lagrangian advection is described. Only non-local operations are evaluated in FE representation, while products of variables are evaluated in physical space. With semi-Lagrangian advection the only non-local vertical operations to be evaluated are vertical integrals. An integral operator is derived based on the Galerkin method using B-splines as basis functions with compact support. Two versions have been implemented, one using piecewise linear basis functions (hat functions) and the other using cubic B-splines. No staggering of dependent variables is employed in physical space, making the method well suited for use with semi-Lagrangian advection. The two versions of the FE scheme are compared to finite-difference (FD) schemes in both the Lorenz and the Charney-Phillips staggering of the dependent variables for the linearized model. The FE schemes give more accurate results than the two FD schemes for the phase speeds of most of the linear gravity waves. Evidence is shown that the FE schemes suffer less from the computational mode than the FD scheme with Lorenz staggering, although temperature and geopotential are held at the same set of levels in the FE scheme too. As a result, the FE schemes reduce the level of vertical noise in forecasts with the full model. They also reduce by about 50% a persistent cold bias in the lower stratosphere present with the FD scheme in Lorenz staggering (i.e. the operational scheme at ECMWF before its replacement by the cubic version of the FE scheme described here) and improve the transport in the stratosphere.