A TWO-TIME-LEVEL VERSION OF THE IFS - SEMI-LAGRANGIAN SPECTRAL MODEL

Aug 25, 2026

Uploaded by Aurelia

An overview is presented of ongoing work to apply a technique that converts a three-time-level semi-Lagrangian spectral discretization into a two-time-level version for the ECMWF global spectral forecast model within the IFS. The introduction motivates improved efficiency enabling higher horizontal and vertical resolution while maintaining stability and accuracy. Recent numerical changes include modified vertical and horizontal interpolation, use of virtual temperature as the spectral variable, updated moisture treatment, and incorporation of Coriolis terms into semi-Lagrangian advection, followed by a compact two-time-level formulation using extrapolation and iterative departure-point evaluation.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages15

Reading Time38 min

AccessFree to download

1. INTRODUCTION

2. RECENT CHANGES TO THE THREE-TIME-LEVEL VERSION

2.1 Scheme numerics and interpolation changes

2.2 Spectral and moisture variable treatment

2.3 Incorporation of Coriolis terms and physical parameterization updates

3. FORMULATION OF THE TWO-TIME-LEVEL SCHEME

A TWO-TIME-LEVEL VERSION OF THE IFS SEMI-LAGRANGIAN SPECTRAL MODEL Clive Temperton European Centre for Medium-Range Weather ForecastsShinfield Park,Reading,UK # 1. INTRODUCTION In September 1991,a three-time-level semi-Lagrangian spectral model was implementedoperationally at ECMWF.The increased efficiency of the semi-Lagrangian scheme relativeto that of the previous Eulerian version was crucial in making feasible an increase inhorizontal resolution from T106 to T213 and in vertical resolution from 19 to 31 layers(Ritchie et al.,1995).At the new resolution,the semi-Lagrangian scheme is stable andaccurate with timesteps several times longer than for the corresponding Eulerian scheme,andthis is achieved with a computational overhead of only about 20%per timestep. In principle,it should be possible to obtain a further doubling of efficiency by converting thethree-time-level semi-Lagrangian scheme to a two-time-level scheme(Staniforth and Côté,1991).Such two-time-level schemes have been described for a limited-area finite-elementshallow-water model(Temperton and Staniforth,1987),a global spectral shallow-water model(Côté and Staniforth,1988),a global finite-element shallow-water model(Côté et al.,1993),global finite-difference shallow-water models(Bates et al.,1990;Bates et al.,1995),limited-area finite-difference multi-level models(McDonald and Haugen 1992,1993),and a globalfinite-difference multi-level model(Bates et al.,1993). In this paper,we report on work in progress to apply a similar technique to the ECMWFglobal spectral forecast model,which is now part of the IFS(Integrated Forecast System)developed in collaboration with Météo-France. # 2. RECENT CHANGES TO THE THREE-TIME-LEVEL VERSION Since it was documented by Ritchie et al.(1995),the three-time-level scheme,which formsthe basis for the new two-time-level version,has undergone some important changes as listedbelow. TEMPERTON,C.:ATWO-TIME LEVELVERSION OF THE IFS.…... (1)The semi-Lagrangian scheme reverted to the "fully interpolating in the vertical"form.Inthe original version,this form gave rise to a growth of edy kinetic energy during the forecastand a consequent deterioration of the verification scores.This problem has been overcome bytwo changes to the details of the numerical formulation.In each of the model equations,theinterpolations for the "advected variable"and the "right-hand side"terms are now carried outseparately.The right-hand-side term is taken as an average between the values at thedeparture point and the arrival point for all the equations,and the interpolation at thedeparture point is linear.Simplified cubic interpolation is still used to evaluate the advectedvariable at the departure point,but in the horizontal a quasi-monotone limiter is applied,basedon the scheme proposed by Bermejo and Staniforth(1992).The quasi-monotone limiter is alsoapplied in the vertical for the moisture field only.These modifications to the semi-Lagrangiannumerics were described by Hortal(1994). (2)The spectral variable is now the virtual temperature T,instead of the temperature T (thisoption was described in Ritchie et al.1995),and the moisture field q is treated entirely ingridpoint space. (3)The Coriolis terms in the momentum equations are incorporated in the semi-Lagrangianadvection,as first proposed by Rochas(1990);see also Temperton(1995). Meanwhile there have also been important changes to the physical parametrization schemes,notably the introduction of a prognostic cloud scheme(Tiedtke,1993)and a newrepresentation of sub-grid scale orographic effects coupled with a return to "mean"rather than"envelope"orography(Lott and Miller,1996). With the modifications to the numerics described above,and with the semi-implicit parameter βset to 1(see Section 2(c)of Ritchie et al.,1995),the three-time-level semi-Lagrangiandiscretization for each of the model equations can be written compactly in the form (2.1) where X+=X(x,t+△t)X-=X(x-2α,t-△t)

FAQs

Why was the semi-Lagrangian scheme adopted for the ECMWF spectral model?

It increases efficiency relative to the earlier Eulerian version, making higher horizontal and vertical resolutions feasible while remaining stable and accurate with longer timesteps and modest computational overhead.

What key changes were made to the three-time-level semi-Lagrangian version before forming the two-time-level scheme?

Changes include reverting to fully interpolating in the vertical, separating “advected variable” and “right-hand side” interpolations, using virtual temperature as the spectral variable, treating moisture entirely in gridpoint space, and incorporating Coriolis terms into the semi-Lagrangian advection.

How does the two-time-level scheme modify the timestep structure?

It replaces the double timestep used in the three-time-level scheme with a single timestep from t to t+Δt, and obtains quantities formerly evaluated at the central time level via extrapolation to t+1/2Δt, including time-extrapolated nonlinear terms and iterative solution of the displacement (departure point) equation.

Aug 25, 2026

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