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

Abstract

SUMMARYA 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.

Dynamical kernel of the Aladin–NH spectral limited‐area model: Revised formulation and sensitivity experiments

“…However, it is assumed that this potential source of spurious drift is acceptable, because its magnitude is smaller than the hydrostatic part of the source (the typical magnitude ofq is found to be about 10 −5 -10 −4 at the 10 km resolution used here, and about 10 −4 in real-case integrations made with 2.5 km and 0.5 km resolutions). Additionally, a lack of formal conservation for the hydrostatic part itself does not seem to create problems worth reporting in other NWP models: as an illustration, the relaxation of the formal conservation of angular momentum in the vertical finite-element scheme of the hydrostatic IFS model was found not to be detrimental to the quality of the forecasts (Untch and Hortal, 2004). Another element supporting this is that, to our knowledge, this physical constraint is not even taken into consideration in any other NWP NH model, for any part of the flow.…”

“…Moreover, both versions aim toward numerically efficient schemes such as 2-TL semi-Lagrangian schemes. In both Aladin and Aladin-NH models, two types of vertical discretization have been implemented as separate options: finite differences (Simmons and Burridge, 1981) and finite elements (Untch and Hortal, 2004). The finiteelements option is used in the operational HPE versions of Aladin, whereas for the EE versions of the model this discretization is still under research.…”

Using enthalpy as a prognostic variable in atmospheric modelling with variable composition

“…If, on the other hand, c p is allowed to vary according to Equation (9) (e.g. Untch and Hortal, 2004), then clearly Equation (11) is only an approximation and its exact form should contain additional terms depending on individual tracer tendencies dq i /dt via Equation (9). These tendencies depend on chemical or phase-transition production and loss processes and diffusive fluxes.…”

“…Dividing by c p and substituting definition (12) into the second term of Equation (11) results in a prognostic equation containing two different thermodynamic variables T and T v (cf. Untch and Hortal, 2004):…”

“…Most lower-atmospheric weather prediction and climate models still rely on an alternative formalism of virtual temperature T v to account for the presence of water vapour and other constituents in the air (e.g. Benjamin et al, 2004;Untch and Hortal, 2004;Davies et al, 2005). The purpose of this note is to revisit from first principles the derivation of the internal energy equation (IEE), also commonly referred to in meteorological literature as the thermodynamic energy equation (section 2), and to examine the applicability of the traditional approach to arbitrary gas mixtures.…”