1. Priority project CDC Task 1.4: Choice of the anelastic equation system and Milestone 3.2: Suitability of fundamental approximations PP CDC-Meeting, 06.-10.09.2010, Moscow (SMC-meeting, 21./22.06.2010, Sopot) M. Baldauf (DWD)

2. The results from the idealised test cases (task 1.1) did not show any recognizable drawbacks of the anelastic approach in comparison with the compressible approach. In the following some basic properties of the equations sets are presented. Generally good news form the SMC-meeting in Sopot (June 2010): both branches of PP CDC ('EULAG' and 'compressible') will be continued!!

3. Boussinesq-approximation: replace cont.-eqn. by div v = 0 only usable for shallow flows anelastic approximation by Ogura, Phillips (1962) JAS: use div  0 v = 0 and isentropic base state  0 =const.  problem with deep convection anelastic approximation by Wilhelmson and Ogura (1972) JAS:  0  const. possible; but energy conservation lost anelastic approximation by Lipps, Hemler (1982) JAS  EULAG eqn. set allows weakly variing  0 =  0 (z); energy conserving pseudo-incompressible equations (Durran, 1989) JAS: replace cont.eqn. by div  0  0 v = H/c p /  0 (H=latent heating rate) is also usable in regions of strong static stability; energy conserving A collection of sound proof equation sets

4. Comparison between the compressible equations and the anelastic approximation; linear analysis (normal modes) divergence damping p=p0+p’T=T0+T’p=p0+p’T=T0+T’ Bretherton-Transformation: switches: compressible:all  i =1 compr. + div. damp.: all  i =1 anelastic:  2,3 =0,  1,4,5 =1 (inverse) scale height: ~ (10 km) -1

5. wave ansatz: u(x,z,t) = u ( k x, k z,  ) exp( i ( k x x + k z z -  t ) ), w ( x,z,t )= … Re  /  a aa N cos   =  ( k z, k x ) = 0° k * c s /  a c s sound velocity ( ~ 330 m/s) N Brunt-Vaisala-frequency ( ~ 0.01 1/s)  a acoustic cut off frequency (~ 0.03 1/s) Dispersion relation  =  ( k x, k z ) of internal waves sound waves gravity waves ~ 7 km ~ 3.5 km compr. + div.damp.

6. Im  /  a k * c s /  a ~ 7 km ~ 3.5 km Strong damping of short sound waves in the compressible equations due to artificial divergence damping. Dispersion relation  =  ( k x, k z ) of internal waves (timescale 1/  a ~ 30 sec.)

7. Dispersion relation  =  ( k x, k z ) of internal waves; only gravity waves /a/a N cos   =  ( k z, k x ) = 0° k * c s /  a ~ 7 km ~ 3.5 km quite similar dispersion relation for anelastic and compressible eqns. N ~ 0.01 1/s

8. Dispersion relation  =  ( k x, k z ) of internal waves only gravity waves compressible, with divergence damping compressible anelastic /a/a k * c s /  a  =  ( k z, k x ) = 0° ~ 70 km smaller differences for very long gravity waves N ~ 0.01 1/s

9. compressible, with divergence damping compressible anelastic Dispersion relation  =  ( k x, k z ) of internal waves only gravity waves  =  ( k z, k x ) = 30° ~ 70 km /a/a k * c s /  a N ~ 0.01 1/s

10. compressible, with divergence damping compressible anelastic Dispersion relation  =  ( k x, k z ) of internal waves focus on long gravity waves ~ 70 km /a/a k * c s /  a  =  ( k z, k x ) = 0° N ~ 0.018 1/s

11. summary of section '7. conclusions': Lamb modes (external acoustic): are filtered out by all sound proof approximations external Rossby modes: Lipps and Hemler (1982) set: distorts the height-scale pseudo-incompressible: handles height-scale correctly internal Rossby modes: anelastic sets misrepresent them at wavelengths typically encountered in atmospheric models. pseudo-incompressible handles them correctly. internal gravity modes: all sound proof sets mishandle deep vertical modes at large horizontal scale. Good representation of smaller horizontal scales Davies et al. (2003) QJRMS 'Validity of anelastic and other equation sets as inferred from normal-mode analysis'

12. Nance, Durran (1994) JAS 'A comparison of the accuracy of three anelastic systems and the pseudo-incompressible system' comparison between 4 different anelastic approximations and a compressible model (leapfrog, advection with centered differences 4th order horizontally and 2nd order vertically) the different tests resulted in bigger errors of the numerical scheme for the compressible equations compared to the errors of the anelastic approximation.  problem of 'Low-Mach number' solvers How would this comparison look like for currently used compressible solvers? Fig. 4 from Nance, Durran (1994) JAS W‘u‘ eqn- error numer. error

13. Smith, Bannon (2008) MWR 'A comparison of Compressible and Anelastic Models of Deep Dry Convection' Simulation of an initial warm bubble 2-dim., dry case, very high resolution (dx=dz=200 m) interesting aspect: different initialization of compressible and anelastic models needed for idealised setups Results: bigger differences in compressible and anelastic model for times t << 2  /N ~ 10 min. only small differences for times t  2  /N main difference: Lamb wave expansion better efficiency of the anelastic model

14. First conclusions about anelastic approximations for smaller scale models and dry Euler equations the anelastic approximation seems to work quite good keep in mind that short sound waves are also strongly damped in our compressible solver (divergence damping) all of the idealised tests studied in task 1.1 delivered satisfying results with the anelastic approx. objectives against the anelastic approach from Davies et al. (2003) mainly stem from larger scale applications what is the meteorological meaning of long sound waves and the Lamb mode? relevance of deviation for very long gravity waves?  tests with large scale mountain flows (upstream blocking,...) are there changes in the assessment when moist processes are studied?