numex – numerical experiments for the gme fachhochschule bonn-rhein-sieg wolfgang joppich pftool -...

NUMEX – Numerical experiments for the GME

Fachhochschule Bonn-Rhein-Sieg

Wolfgang Joppich

PFTOOL - Precipitation forecast toolboxSemi-Lagrangian Mass-Integrating Transport Algorithm using the GME Gridwithin

Sabine Pott

Seite 2

Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

OutlineOutline

• Motivation• Algorithm• Experiments • Results• Future Work

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

improve the semi-Lagrangian scheme withregard to

-- conservation properties-- preservation of shape

-- reduction of error -- reduction of non-physical oscillations

MotivationMotivation

For the complete chain of precipitation forecast at DWD holds: what you have lost globally (GME-GME2LM)

remains lost locally (LM)

therefore

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

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

AlgorithmAlgorithm

A Semi-Lagrangian integrated-mass transport algorithm based on the integro-differential form of the continuity equation has been developed for the icosahedral grid

)(

)(

,

,0

),(),(

0),(

ttt

st

dMxthdMxtth

dMxthdt

dU

t

Hermitian interpolation of mass on each triangular cell, satisfying1. conservation of mass2. Invariance of variation of mass at two edges3. Interpolation of transported quantity to the 3 corner points

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

AlgorithmAlgorithm

on each triangle the transported quantity is represented as a second order polynomial in the local isothermal coordinates:

254

23210),( aaaaaap

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

mesh3. integrate mass, variation of mass (previous time

level) on polygons of intersection or departure edges respectively

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Experiments/ResultsExperiments/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

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Experiments/ResultsExperiments/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

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Experiments/ResultsExperiments/Results

advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days

Relative error in maximumnorm, old(yellow) vs. new(black) scheme

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Experiments/ResultsExperiments/Results

advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days

Relative massdefect, old(yellow) vs. new(black) scheme

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Experiments/ResultsExperiments/Results

advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days

Relative error in extremum, old(yellow) vs. new(black) scheme

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Experiments/ResultsExperiments/Results

advection of reversed cosine-bell, div = 0, pole-loop, ni = 48, 30 days

Nonphysical oscillations, old(yellow) vs. new(black) scheme

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Currently doing Currently doing

- 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)

GME

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Fachhochschule Bonn-Rhein-Sieg

wolfgang.joppich@fh-bonn-rhein-sieg.de

Future work Future work

- 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, …)

what you have lost globally (GME-GME2LM) remains lost locally (LM)

then the complete chain has improved conservation properties whichalso hold for low resolutions, especially for the climate verion of GME

GME

GME2LM

LM