PFTOOL Precipitation forecast toolbox SemiLagrangian MassIntegrating Transport Algorithm

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

Outline • Motivation • Algorithm • Experiments • Results • Future Work Seite 2 Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de

Motivation For the complete chain of precipitation forecast at DWD holds: what you have lost globally (GME-GME 2 LM) 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 therefore Fachhochschule Bonn-Rhein-Sieg improve the semi-Lagrangian scheme with regard to -- conservation properties -- preservation of shape -- reduction of error -- reduction of non-physical oscillations Seite 3 wolfgang. joppich@fh-bonn-rhein-sieg. de

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 1. conservation of mass 2. Invariance of variation of mass at two edges 3. Interpolation of transported quantity to the 3 corner points Seite 4 Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de

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: 1. approximation of departure points (previous time level) 2. intersection of icosahedral mesh and departure mesh 3. integrate mass, variation of mass (previous time level) on polygons of intersection or departure edges respectively Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de Seite 5

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 Seite 6 Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de

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

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 Seite 8 Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de

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

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 Seite 10 Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de

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

Currently doing üGME - 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) Seite 12 Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de

Future work what you have lost globally (GME-GME 2 LM) remains lost locally (LM) üGME - implementation of new scheme into GME - systematic comparison (benefit vs. numerical cost) GME 2 LM - develop conservative interpolation for transfer GME 2 LM LM - adapt scheme for the LM (implementation, evaluation, …) then the complete chain has improved conservation properties which also hold for low resolutions, especially for the climate verion of GME Seite 13 Fachhochschule Bonn-Rhein-Sieg wolfgang. joppich@fh-bonn-rhein-sieg. de