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) 24-28 Sept. 2012 1

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 channel

Skamarock, 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) 24-28 Sept. 2012 2

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.

Boundary conditions:

w

(

r=r s

) = 0

w

(

r=r s +H

) = 0 For an analytic solution only one further approximation is needed:

linearisation

around an (=

controlled

approximation)

isothermal, steady, hydrostatic

atmosphere either at rest (   0 possible) or with a constant background flow (  =0 ) Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 3

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/

r s

• in the divergence operator: omit the metric correction term • apart from that all earth curvature metric terms are included • (optional) Coriolis force: ,globally on a local f-plane‘ 2  ( 

,

 ) =

f



e

r

( 

,

 ),

f

=const. (and

v

0 = 0 )

~ w

/

r

Spectral representation of fields: Spherical harmonics: Time integration by Laplace transform Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 4

Analytic solution

for the vertical velocity harmonic with

l,m

)

w

( Fourier component with

k z ,

spherical analogous expressions for

û lm (k z , 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

;

k z

= (  /

H

) 

n n

=0

n

=1

n

=2 

l

Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 5 

Categories of tests

1. Only gravity wave and sound wave expansion 2. Additional advection by a solid body rotation velocity field

v

0 =

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  test proper discretization of inertia-gravity modes, e.g. in a C-grid discretization.

v

0 = 0 ) For problems with C-grid discretizations on non-quadrilateral grids see

Nickowicz, Gavrilov, Tosic (2002) MWR, Thuburn, Ringler, Skamarock, Klemp (2009) JCP, Gassmann (2011) JCP

Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 6

Test case initialization

expansion of gravity and sound waves by the initialisation of a weak warm bubble:

p‘(



,



,r, t=

0

) =

0 scale height of the isothermal atmosphere  finite expansion into Legendre-polynomials weak bubble 

T

=0.01 K  linear regime  The initialization is quite similar to one of the DCMIP 2012 test cases (

Jablonowski, Lauritzen, …)

Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 7

‚Small earth‘-simulations

Wedi, Smolarkiewicz (2009) QJRMS

•

r s

= r earth / 50 ~ 6371 km / 50 ~ 127 km simulations with  ~ 1 °... 0.125°   x ~  non-hydrostatic regime 2.2 km ... 0.28 km • for runs 

with

Coriolis force : dimensionless numbers f = f earth  10 ~ 10 -4 1/s  10 ~ 10 -3 1/s

Ro f

/

N

= 0.2 

Ro earth

= 10 

f earth

/

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) 24-28 Sept. 2012 8

ICON simulation, f=0

in

z

=5 km Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 9

Time evolution of T‘

f

=0

f

 0 Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 10 Equ N

Time evolution of T‘

f

=0

f

 0 Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 11 Equ N

Time evolution of T‘

f

=0

f

 0 Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 12 Equ N

Time evolution of T‘

f

=0

f

 0 Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 13 Equ N

Time evolution of T‘

f

=0

f

 0 Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 14 Equ N

Time evolution of T‘

f

=0

f

 0 Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 15 Equ N

Time evolution of T‘

f

=0

f

 0 Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 16 Equ N

Convergence

R2B4-L10   ~ 1 ° ~ 2.2 km  z = 1000 m Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 17 Equ N

Convergence

R2B5-L20   ~ 0.5

° ~ 1.1 km  z = 500 m Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 18 Equ N

Convergence

R2B6-L40   ~ 0.25

° ~ 0.55 km  z = 250 m Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 19 Equ N

Convergence

R2B7-L80   ~ 0.125

° ~ 0.28 km  z = 125 m Solid lines: analytic solution C o l o u r s : ICON simulation Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 S 20 Equ N

Convergence rate of the ICON model

• The ICON simulation with/without Coriolis force produces almost similar L 2 , L  • L 2 , L  errors errors for w are generally higher than for T‘ • convergence order of ICON is ~ 1 T‘ w‘ L  -error L  -error L 2 -error L 2 -error Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 21

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 

t

  x , 

y

>> 

z

and for (vertically) implicit schemes, 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

) Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 22

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-temporal convergence rate of about 1, no drawbacks visible.

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) 24-28 Sept. 2012 23

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

Small scale test

with a basic flow U 0 =20 m/s f=0 25

Large scale test

U 0 =0 m/s f=0.0001 1/s 26

Convergence properties of COSMO

Baldauf, Reinert, Zängl (DWD) 24-28 Sept. 2012 27