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Michael Baldauf , Daniel Reinert, Günther Zängl (DWD, Germany)

An exact analytical solution for gravity wave expansion of the compressible, non-hydrostatic Euler equations on the sphere. PDEs on the sphere, Cambridge, 24-28 Sept. 2012. Michael Baldauf , Daniel Reinert, Günther Zängl (DWD, Germany). Motivation

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Michael Baldauf , Daniel Reinert, Günther Zängl (DWD, Germany)

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  1. An exact analytical solution for gravity wave expansion of the compressible, non-hydrostatic Euler equations on the sphere PDEs on the sphere, Cambridge, 24-28 Sept. 2012 Michael Baldauf, Daniel Reinert, Günther Zängl (DWD, Germany) Baldauf, Reinert, Zängl (DWD)

  2. Motivation For the development of dynamical cores (or numerical methods in general) idealized test cases are an important evaluation tool. • Idealized standard test cases with (at least approximated) analytic solutions: • stationary flow over mountains linear: Queney (1947, ...), Smith (1979, ...) Adv Geophys, Baldauf (2008) COSMO-Newsl.non-linear: Long (1955) Tellus for Boussinesq-approx. Atmosphere • Balanced solutions on the sphere: Staniforth, White (2011) ASL • non-stationary, linear expansion of gravity waves in a channelSkamarock, Klemp (1994) MWR for Boussinesq-approx. atmosphere • most of the other idealized tests only possess 'known solutions' gained from other numerical models. There exist even fewer analytic solutions which use exactly the same equations as the numerical model used, i.e. in the sense that the numerical model converges to this solution. One exception is given here: linear expansion of gravity/sound waves on the sphere Baldauf, Reinert, Zängl (DWD)

  3. Non-hydrostatic, compressible, shallow atmosphere, adiabatic,3D Euler equations on a sphere with a rigid lid most global models using the compressible equations should be able to exactly use these equations in the dynamical core for testing. For an analytic solution only one further approximation is needed: linearisation (= controlled approximation) around an isothermal, steady, hydrostatic atmosphere either at rest (0 possible) or with a constant background flow (=0) Boundary conditions: w(r=rs) = 0 w(r=rs+H) = 0 Baldauf, Reinert, Zängl (DWD)

  4. Solution strategy • Isothermal background state + shallow atmosphere approx. • Bretherton (1966) transformation •  all coefficients of the linearized PDE system are constant • Shallowatmosphereapproximation • replace all prefactors1/r 1/rs • in thedivergenceoperator: omitthemetriccorrectionterm~ w/r • apart fromthat all earthcurvaturemetrictermsareincluded • (optional) Coriolis force: ,globally on a local f-plane‘2 (,) = f  er (,), f=const. (andv0 = 0) Spectral representation of fields: Spherical harmonics: Time integration by Laplace transform Baldauf, Reinert, Zängl (DWD)

  5. Analyticsolution fortheverticalvelocityw (Fourier componentwithkz, sphericalharmonicwithl,m) analogous expressions for ûlm(kz, t), ... The frequencies ,  are the gravity wave and acoustic branch, respectively, of the dispersion relation for compressible waves in a spherical channel of height H; kz = ( / H)  n n=2    cs / g n=1 n=0  l Baldauf, Reinert, Zängl (DWD)

  6. Categoriesoftests • Onlygravitywaveandsoundwaveexpansion • Additional advectionby a solid bodyrotationvelocityfieldv0 = b r (andf=0) testthecouplingof fast (buoyancy, sound) andslow(advection) processes importantforsplit-explicit, semi-implicit, … schemes • Additional Coriolis force (,globally on a local f-plane‘) (andv0 = 0) test proper discretizationofinertia-gravitymodes, e.g. in a C-griddiscretization.Forproblemswith C-griddiscretizations on non-quadrilateralgridsseeNickowicz, Gavrilov, Tosic (2002) MWR, Thuburn, Ringler, Skamarock, Klemp (2009) JCP, Gassmann (2011) JCP Baldauf, Reinert, Zängl (DWD)

  7. Test case initialization expansion of gravity and sound waves by the initialisation of a weak warm bubble: scale height of the isothermal atmosphere p‘(,,r, t=0) = 0  finite expansion into Legendre-polynomials weak bubble T =0.01 K  linear regime … oh, please, leavemealonewithyetanothertestsetup, … • The initialization is quite similarto one of the DCMIP 2012 test cases(Jablonowski, Lauritzen, …) Baldauf, Reinert, Zängl (DWD)

  8. ‚Small earth‘-simulations • Wedi, Smolarkiewicz (2009) QJRMS • rs= rearth / 50 ~ 6371 km / 50 ~ 127 kmsimulationswith ~ 1°... 0.125°  x ~ 2.2 km ... 0.28 km non-hydrostaticregime • forrunswith Coriolis force : f = fearth 10 ~ 10-4 1/s  10 ~10-3 1/s dimensionlessnumbers • Ro = 0.2 Roearth • f / N =10fearth/ N~ 0.05 ICON (icosahedral grid, non-hydrostatic) is a joint development of the Deutscher Wetterdienst (DWD) and the Max-Planck-Institute for Meteorology, Hamburg C-grid discretization on a triangular grid decomposition of the icosahedron Predictor-corrector time integration  Talks by G. Zängl, F. Prill, Posters by P. Ripodas, D. Reinert Baldauf, Reinert, Zängl (DWD)

  9. ICON simulation, f=0 in z=5 km Baldauf, Reinert, Zängl (DWD)

  10. Time evolution of T‘ f=0 f0 Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  11. Time evolution of T‘ f=0 f0 Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  12. Time evolution of T‘ f=0 f0 Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  13. Time evolution of T‘ f=0 f0 Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  14. Time evolution of T‘ f=0 f0 Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  15. Time evolution of T‘ f=0 f0 Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  16. Time evolution of T‘ f=0 f0 Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  17. Convergence R2B4-L10  ~ 1° ~ 2.2 km z = 1000 m Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  18. Convergence R2B5-L20  ~ 0.5° ~ 1.1 km z = 500 m Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  19. Convergence R2B6-L40  ~ 0.25° ~ 0.55 km z = 250 m Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  20. Convergence R2B7-L80  ~ 0.125° ~ 0.28 km z = 125 m Solid lines: analytic solution Colours: ICON simulation S N Equ Baldauf, Reinert, Zängl (DWD)

  21. Convergence rate of the ICON model • The ICON simulation with/without Coriolis force producesalmost similar L2, L errors • L2, L errors for w are generally higher than for T‘ • convergence order of ICON is ~ 1 T‘ w‘ L-error L-error L2-error 1st order L2-error 1st order Baldauf, Reinert, Zängl (DWD)

  22. Somehintsfor a proper simulationofthewfield • Time stept: must bechosenthatsoundwavesareresolved! • smallearth x, yandzareofthe same order  ok • real earth  x, y >> zandfor (vertically) implicitschemes,tis limited additionallybytheacousticcut-off frequencyof ~ 1 min. •  tis not only limited bystability but also byaccuracy. • „ … Weare not interested in soundwavesandwanttodampthem …“ • Any off-centeringforsoundwavepropagationshouldbereducedto a minimum! • In split-explicit schemes: divergencedampingtoreducecompressiblewaves;not a problemforconvergence: div ~ t 0 (Skamarock, Klemp (1992) MWR) Baldauf, Reinert, Zängl (DWD)

  23. Summary • An analyticsolutionofthecompressible, non-hydrostatic Euler equationson thesphere was derived •  a reliablesolutionfor a well knowntestexistsandcanbeused not onlyfor qualitative comparisons but evenas a referencesolutionforconvergencetests • The testsetupisquitesimilartooneofthe DCMIP 2012 testcases • 'standard' approximationsused: ‚globallylocal f-plane‘, shallowatmosphere,canbeeasilyrealised in everyatmospheric model • onlyonefurtherapproximation: linearisation (=controlledapprox.) • Forfineenoughresolutions ICON has a spatial-temporalconvergence rate ofabout 1, nodrawbacksvisible. For a similar solution on a plane 2D channel: M. Baldauf, S. Brdar: An Analytic solution for Linear Gravity Waves in a Channel as a Test for Numerical Models using the Non-hydrostatic, Compressible Euler Equations, submitted to Quart. J. Roy. Met. Soc. Partially suported by the METSTROEM priority program of DFG Baldauf, Reinert, Zängl (DWD)

  24. Baldauf, Reinert, Zängl (DWD)

  25. Small scale test with a basic flow U0=20 m/s f=0 Baldauf, Reinert, Zängl (DWD)

  26. Large scale test U0=0 m/s f=0.0001 1/s Baldauf, Reinert, Zängl (DWD)

  27. Convergence properties of COSMO Baldauf, Reinert, Zängl (DWD)

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