NUMEX – Numerical experiments for the GME

1. Fachhochschule Bonn-Rhein-Sieg NUMEX – Numerical experiments for the GME PFTOOL - Precipitation forecast toolbox Semi-Lagrangian Mass-Integrating Transport Algorithm using the GME Grid within Sabine Pott Wolfgang Joppich

2. Outline • Motivation • Algorithm • Experiments • Results • Future Work wolfgang.joppich@fh-bonn-rhein-sieg.de

3. Motivation For the complete chain of precipitation forecast at DWD holds: what you have lost globally (GME-GME2LM) remains lost locally (LM) DWD has reported that there is an obvious defect of mass in the GME , indication to insufficient accuracy specific humidity, specific cloud liquid water content, specific cloud ice content, ozone mixing ratio • improve the semi-Lagrangian scheme with • regard to • - conservation properties • - preservation of shape • -- reduction of error • -- reduction of non-physical oscillations therefore wolfgang.joppich@fh-bonn-rhein-sieg.de

4. Algorithm A Semi-Lagrangian integrated-mass transport algorithm based on the integro-differential form of the continuity equation has been developed for the icosahedral grid • Hermitian interpolation of mass on each triangular cell, satisfying • conservation of mass • Invariance of variation of mass at two edges • Interpolation of transported quantity to the 3 corner points wolfgang.joppich@fh-bonn-rhein-sieg.de

5. Algorithm on each triangle the transported quantity is represented as a second order polynomial in the local isothermal coordinates: equations for the coefficients are solved by LU decomposition • Key steps: • approximation of departure points (previous time level) • intersection of icosahedral mesh and departure mesh • integrate mass, variation of mass (previous time level) on polygons of intersection or departure edges respectively wolfgang.joppich@fh-bonn-rhein-sieg.de

6. Experiments/Results Several different constant wind fields Rotation of reversed cosine-bell, different rotation angles, loop across pole and equator Advection of cosine-bell in a meridional wind-field with div != 0 Both test cases are similar to Williamson‘s test case 1 Tests for ni = 12, 16, 24, 32, 48, 64, 96 wolfgang.joppich@fh-bonn-rhein-sieg.de

7. Experiments/Results advection of reversed cosine-bell, div = 0, pole-loop Initial state reached again after 12 days, ni = 48, dt = 540s, global error after 30 days wolfgang.joppich@fh-bonn-rhein-sieg.de

8. Experiments/Results advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days Relative error in maximumnorm, old(yellow) vs. new(black) scheme wolfgang.joppich@fh-bonn-rhein-sieg.de

9. Experiments/Results advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days Relative massdefect, old(yellow) vs. new(black) scheme wolfgang.joppich@fh-bonn-rhein-sieg.de

10. Experiments/Results advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days Relative error in extremum, old(yellow) vs. new(black) scheme wolfgang.joppich@fh-bonn-rhein-sieg.de

11. Experiments/Results advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days Nonphysical oscillations, old(yellow) vs. new(black) scheme wolfgang.joppich@fh-bonn-rhein-sieg.de

12. Currently doing • GME • implementation of the new scheme, developed within sequential SWE code using the GME mesh, • into parallel GME 2.10 (almost production code) • systematic comparison by means of realistic test • cases(benefit vs. numerical cost) wolfgang.joppich@fh-bonn-rhein-sieg.de

13. Future work what you have lost globally (GME-GME2LM) remains lost locally (LM) • GME • implementation of new scheme into GME • systematic comparison (benefit vs. numerical cost) • develop conservative interpolation for transfer GME2LM • adapt scheme for the LM (implementation, evaluation, …) GME2LM LM then the complete chain has improved conservation properties which also hold for low resolutions, especially for the climate verion of GME wolfgang.joppich@fh-bonn-rhein-sieg.de