1. Numerical simulations of inertia-gravity waves and hydrostatic mountain waves using EULAG model Bogdan Rosa, Marcin Kurowski, Zbigniew Piotrowski and Michał Ziemiański COSMO General Meeting, 7-11 September 2009

2. Outline 1. 1. Two dimensional 2D time dependent simulation of inertia-gravity waves (Skamarock and Klemp Mon. Wea. Rev. 1994) using three different approaches Linear numerical Linear numerical Incompressible Boussinesq Incompressible Boussinesq Quasi-compressible Boussinesq Quasi-compressible Boussinesq 2. 2D simulation of hydrostatic waves generated in stable air passing over mountain. (Bonaventura JCP. 2000)

3. COSMO General Meeting, 7-11 September 2009 Two dimensional time dependent simulation of inertia-gravity waves Skamarock W. C. and Klemp J. B. Efficiency and accuracy of Klemp-Wilhelmson time-splitting technique. Mon. Wea. Rev. 122: 2623-2630, 1994 Initial potential temperature perturbation Setup overview: domain size 300x10 km resolution 1x1km, 0.5x0.5 km, 0.25x0.25 km rigid free-slip b.c. periodic lateral boundaries constant horizontal flow 20m/s at inlet no subgrid mixing hydrostatic balance stable stratification N=0.01 s -1 max. temperature perturbation 0.01K Coriolis force included Constant ambient flow within channel 300 km and 6000 km long km outlet inlet

4. COSMO General Meeting, 7-11 September 2009 The Methods Quassi-compressible Boussinesq Incompressible Boussinesq Linear Initial potential temperature perturbation Initail velocity The terms responsible for the acoustic modes

5. COSMO General Meeting, 7-11 September 2009 Time evolution of flow field potential temperature and velocity (Incompressible Boussinesq) time Time evolution of  ’ (contour values between −0.0015K and 0.003K with a interval of 0.0005K) and vertical velocity (contour values between −0.0025m/s and 0.002m/s with a interval of 0.0005m/s). Grid resolution dx = dz = 1km. Channel size is 300km × 10km

6. COSMO General Meeting, 7-11 September 2009 Continuation... time Time evolution of  ’ (contour values between −0.0015K and 0.003K with a interval of 0.0005K) and vertical velocity (contour values between −0.0025m/s and 0.002m/s with a interval of 0.0005m/s). Grid resolution dx = dz = 1km. Channel size is 300km × 10km

7. COSMO General Meeting, 7-11 September 2009 Convergence study for resolution Analytical solution based on linear approximation ( Skamarock and Klemp 1994 ) dx = dz = 1km dx = dz = 0.5 km dx = dz = 250 m θ' (after 50min) Numerical solution from EULAG (incompressible Boussinesq approach) Contour values between −0.0015K and 0.003K with a contour interval of 0.0005K

8. COSMO General Meeting, 7-11 September 2009 Profiles of potential temperature along 5000m height Convergence to analytical solution

9. COSMO General Meeting, 7-11 September 2009 Time evolution of potential temperature in long channel (6000 km) time Time evolution of  ’ (contour values between −0.0015K and 0.003K with a interval of 0.0005K)

10. COSMO General Meeting, 7-11 September 2009 Solution convergence (long channel) Analytical solution based on linear approximation ( Skamarock and Klemp 1994 ) dx = 20 km dz = 1km dx = 10 km dz = 0.5 km dx = 5km dz = 250 m Numerical solution from EULAG (inocompressible Boussinesq approach)

11. COSMO General Meeting, 7-11 September 2009 Profiles of potential temperature along 5000m height Convergence to analytical solution Analytical Solution Δx = 5 km Δz = 0.25 km Δx = 10 km Δz = 0.5 km Δx = 20 km Δz = 1 km

12. COSMO General Meeting, 7-11 September 2009 Comparison of the results obtained from four different approaches (dx = dz = 0.25 km - short channel) Linear analytical Incompressible Boussinesq Compressible Boussinesq Linear numerical

13. COSMO General Meeting, 7-11 September 2009 Comparison of the results obtained from four different approaches (long channel) Linear analytical Incompressible Boussinesq Compressible Boussinesq Linear numerical

14. COSMO General Meeting, 7-11 September 2009 Quantitative comparison Differences between three numerical solutions: LIN - linear, IB - incompressible Boussinesq and ELAS quassi-compressible Boussinesq dx = dz = 1km dx = 1km dz = 20km

15. COSMO General Meeting, 7-11 September 2009 Quantitative comparison Differences of  ’ between solutions obtained using two different approaches incompressible Boussinesq and quassi-compressible Boussinesq. The contour interval is 0.00001K.

16. COSMO General Meeting, 7-11 September 2009 Comparison with compressible model EULAG (Incompressible Boussinesq) Klemp and Wilhelmson (JAS, 1978) (Compressible)

17. COSMO General Meeting, 7-11 September 2009 2D simulation of hydrostatic waves generated in stable air passing over mountain. Bonaventura L. A Semi-implicit Semi-Lagrangian Scheme Using the Height Coordinate for a Nonhydrostatic and Fully Elastic Model of Atmospheric Flows JCP. 158, 186–213, 2000 1000 km 25 km outlet inlet 1 m Initial horizontal velocity U = 32 m/s Grid resolution  x = 3km,  z = 250 m Terrain following coordinates have been used Problem belongs to linear hydrostatic regime Profiles of vertical and horizontal sponge zones from Pinty et al. (MWR 1995) Profile of the two-dimensional mountain defines the symmetrical Agnesi formula.

18. COSMO General Meeting, 7-11 September 2009 Horizontal and vertical component of velocity in a linear hydrostatic stationary lee wave test case. horizontal vertical EULAG (anelastic approximation)Bonaventura (JCP. 2000) (fully elastic) horizontal vertical

19. COSMO General Meeting, 7-11 September 2009 Horizontal component of velocity - comparison of numerical solution based on anelastic approximation (solid line) with linear analitical solution (dashed line) form Klemp and Lilly (JAS. 1978) In linear hydrostatic regime analytical solution has form where is surface level potential temperature

20. COSMO General Meeting, 7-11 September 2009 The vertical flux of horizontal momentum for steady, inviscid mountain waves. EULAG (2009) anelastic The flux normalized by linear analitic solution from (Klemp and Lilly JAS. 1978) Bonaventura (JCP. 2000) 0.97 Pinty et al. (MWR. 1995) fully compressible t =11.11 [h] H

21. COSMO General Meeting, 7-11 September 2009 Summary and conclusions Results computed using Eulag code converge to analitical solutions when grid resolutions increase. Results computed using Eulag code converge to analitical solutions when grid resolutions increase. In considered problems we showed that anelastic approximation gives both qualitative and quantitative agrement with with fully compressible models. In considered problems we showed that anelastic approximation gives both qualitative and quantitative agrement with with fully compressible models. EULAG gives correct results even if computational grids have significant anisotrophy. EULAG gives correct results even if computational grids have significant anisotrophy.