michael baldauf , daniel reinert, günther zängl (dwd, germany)

24-28 Sept. 2012Baldauf, Reinert, Zängl (DWD) 1

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

PDEs on the sphere, Cambridge, 24-28 Sept. 2012

An exact analytical solution for gravity wave expansion of the compressible, non-hydrostatic Euler equations on the sphere


• 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 equationsas 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

Motivation

For the development of dynamical cores (or numerical methods in general)idealized test cases are an important evaluation tool.


Non-hydrostatic, compressible, shallow atmosphere, adiabatic,3D Euler equations on a sphere with a rigid lid

For an analytic solution only one further approximation is needed:linearisation (= controlled approximation) around an isothermal, steady, hydrostaticatmosphereeither at rest (0 possible)or with a constant background flow (=0)

most global models using the compressible equations should be able to exactly use these equations in the dynamical corefor testing.

Boundary conditions:w(r=rs) = 0w(r=rs+H) = 0


Spectral representation of fields:

Spherical harmonics:

Solution strategy

Isothermal background state + shallow atmosphere approx. Bretherton (1966) transformation all coefficients of the linearized PDE system are constant

Shallow atmosphere approximation•replace all prefactors 1/r 1/rs

•in the divergence operator: omit the metric correction term ~ w/r•apart from that all earth curvature metric terms are included•(optional) Coriolis force: ,globally on a local f-plane‘ 2 (,) = f er (,), f=const. (and v0 = 0)

Time integration by Laplace transform


Analytic solution for the vertical velocity w (Fourier component with kz, spherical harmonic with l,m )

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

l

cs /

g

n=0

n=1

n=2

analogous expressions for ûlm(kz, t), ...


Categories of tests

1.Only gravity wave and sound wave expansion

2.Additional advection by a solid body rotation velocity field v0 = b r (and f=0)

test the coupling of fast (buoyancy, sound) and slow (advection) processes important for split-explicit, semi-implicit, … schemes

3.Additional Coriolis force (,globally on a local f-plane‘) (and v0 = 0) test proper discretization of inertia-gravity modes, e.g. in a C-grid discretization.

For problems with C-grid discretizations on non-quadrilateral grids seeNickowicz, Gavrilov, Tosic (2002) MWR, Thuburn, Ringler, Skamarock, Klemp (2009) JCP,Gassmann (2011) JCP


Test case initializationexpansion of gravity and sound waves by the initialisation of a weak warm bubble:

weak bubble T =0.01 K linear regime

finite expansion into Legendre-polynomials

scale height of the isothermal atmosphere

p‘(,,r, t=0) = 0

The initialization is quite similarto one of the DCMIP 2012 test cases(Jablonowski, Lauritzen, …)

… oh, please, leave me alone

with yet another test setup,

…


‚Small earth‘-simulations

Wedi, Smolarkiewicz (2009) QJRMS

•rs= rearth / 50 ~ 6371 km / 50 ~ 127 kmsimulations with ~ 1°... 0.125° x ~ 2.2 km ... 0.28 km non-hydrostatic regime

•for runs with Coriolis force : f = fearth 10 ~ 10-4 1/s 10 ~ 10-3 1/s dimensionless numbers

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 icosahedronPredictor-corrector time integration

Talks by G. Zängl, F. Prill, Posters by P. Ripodas, D. Reinert

in z=5 km

ICON simulation, f=0

24-28 Sept. 2012 9Baldauf, Reinert, Zängl (DWD)


f=0

f0

Solid lines: analytic solutionColours: ICON simulation

Time evolution of T‘

NEquS


f=0

f0



NEquS


Convergence

NEquS

R2B4-L10

~ 1° ~ 2.2 kmz = 1000 m



Convergence

NEquS

R2B5-L20

~ 0.5° ~ 1.1 kmz = 500 m



Convergence

NEquS

R2B6-L40

~ 0.25° ~ 0.55 kmz = 250 m



Convergence

NEquS

R2B7-L80

~ 0.125° ~ 0.28 kmz = 125 m



Convergence rate of the ICON model

T‘ w‘

• 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

L2-error

L-error

L2-error

L-error

1st order

1st order


Some hints for a proper simulation of the w field

Time step t: must be chosen that sound waves are resolved!

•small earth x, y and z are of the same order ok

•real earth x, y >> z and for (vertically) implicit schemes,t is limited additionally by the acoustic cut-off frequency of ~ 1 min.

t is not only limited by stability but also by accuracy.

„ … We are not interested in sound waves and want to damp them …“

•Any off-centering for sound wave propagation should be reduced to a minimum!

•In split-explicit schemes: divergence damping to reduce compressible waves;not a problem for convergence: div ~ t 0 (Skamarock, Klemp (1992) MWR)


Summary

•An analytic solution of the compressible, non-hydrostatic Euler equations on the sphere was derived

• a reliable solution for a well known test exists and can be used not only for qualitative comparisons but even as a reference solution for convergence tests

•The test setup is quite similar to one of the DCMIP 2012 test cases

•'standard' approximations used: ‚globally local f-plane‘, shallow atmosphere,can be easily realised in every atmospheric model

•only one further approximation: linearisation (=controlled approx.)

•For fine enough resolutions ICON has a spatial-temporalconvergence rate of about 1, no drawbacks visible.

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.

For a similar solution on a plane 2D channel:

Partially suported by the METSTROEM priority program of DFG


Small scale testwith a basic flow U0=20 m/sf=0


Large scale testU0=0 m/sf=0.0001 1/s


Convergence properties of COSMO

michael baldauf , daniel reinert, günther zängl (dwd, germany)

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