1. CSIRO Marine and Atmospheric Research VCAM: the variable-cubic atmospheric model John McGregor Centre for Australian Weather and Climate Research CSIRO/BOM, Melbourne PDEs on the Sphere Potsdam 26 August 2010

2. CSIRO Marine and Atmospheric Research Outline Formulation of CCAM Formulation of VCAM AMIP results and some comparisons

3. CSIRO Marine and Atmospheric Research The conformal-cubic atmospheric model CCAM is formulated on the conformal-cubic grid Orthogonal Isotropic Example of quasi-uniform C48 grid with resolution about 200 km

4. CSIRO Marine and Atmospheric Research CCAM dynamics atmospheric GCM with variable resolution (using the Schmidt transformation) 2-time level semi-Lagrangian, semi-implicit total-variation-diminishing vertical advection reversible staggering - produces good dispersion properties a posteriori conservation of mass and moisture CCAM physics cumulus convection: - CSIRO mass-flux scheme, including downdrafts - up to 3 simultaneous plumes permitted includes advection of liquid and ice cloud-water - used to derive the interactive cloud distributions (Rotstayn 1997) stability-dependent boundary layer with non-local vertical mixing vegetation/canopy scheme (Kowalczyk et al. TR32 1994) - 6 layers for soil temperatures - 6 layers for soil moisture (Richard's equation) enhanced vertical mixing of cloudy air GFDL parameterization for long and short wave radiation Skin temperatures for SSTs enhanced for sunny, low wind speeds

5. CSIRO Marine and Atmospheric Research Location of variables in grid cells All variables are located at the centres of quadrilateral grid cells. However, during semi-implicit/gravity-wave calculations, u and v are transformed reversibly to the indicated C-grid locations. Produces same excellent dispersion properties as spectral method (see McGregor, MWR, 2006), but avoids any problems of Gibbs’ phenomena. 2-grid waves preserved. Gives relatively lively winds, and good wind spectra.

6. CSIRO Marine and Atmospheric Research Reversible staggering Where U is the unstaggered velocity component and u is the staggered value, define (Vandermonde formula) accurate at the pivot points for up to 4 th order polynomials gives periodic tridiagonal system - solved iteratively, or by cyclic tridiagonal solver excellent dispersion properties for gravity waves, as shown for the linearized shallow-water equations

7. CSIRO Marine and Atmospheric Research Dispersion behaviour for linearized shallow-water equations Typical atmosphere case Typical ocean case N.B. the asymmetry of the R grid response disappears by alternating the reversing direction each time step, giving same response as Z (vorticity/divergence) grid

8. CSIRO Marine and Atmospheric Research MPI performance Runs on many platforms, including Windows Mostly use 1, 6, 12, 24 processors but can use 1, 2, 3, 4, 6, 12, 16, 18, 24, … APAC SC N Time Speedup 1 127.1 1.0 2 65.0 2.0 3 44.6 2.9 4 34.7 3.7 6 23.0 5.5 12 12.3 10.3 16 10.6 12.0 24 6.6 19.3 54 3.7 34.0 Cherax – SGI Altix N Time Speedup 1 162.0 1.0 2 78.7 2.1 4 36.0 4.5 6 23.7 6.8 16 9.6 16.8 24 6.2 26.3

9. CSIRO Marine and Atmospheric Research Conformal-cubic C48 grid used for Australian simulations, Schmidt = 0.3 Resolution over Australia is about 60 km

10. CSIRO Marine and Atmospheric Research C48 8 km grid over New Zealand C48 1 km grid over New Zealand Grid configurations used to support Alinghi in America’s Cup: 60, 8, 1, km Digital filter used to provide broadscale fields from prior coarser-resolution run. Successfully use similar procedure for regional climate modelling. Schmidt transformation can be used to obtain very fine resolution

11. CSIRO Marine and Atmospheric Research Original Sadourny C20 grid Equi-angular C20 grid Alternative gnomonic grids

12. CSIRO Marine and Atmospheric Research Gnomonic grid showing orientation of the contravariant wind components Illustrates the excellent suitability of the gnomonic grid for reversible interpolation – thanks to smooth changes of orientation

13. CSIRO Marine and Atmospheric Research Solution procedure for VCAM In the following equations u, v denote wind components in the contravariant direction Components in the covariant direction are denoted by u T, v T u, v T are the stored variables as they are orthogonal (convenient for physics, displaying wind speed, etc.) uses flux form of primitive equations, with gravity-wave terms handled by forward-backward procedure, in conjunction with reversible staggering

14. CSIRO Marine and Atmospheric Research

15. Control volume notation for edge lengths and area v v u u  s/m Area = (  s*  s)/(m*m) Have kept map factor notation but with this interpretation

16. CSIRO Marine and Atmospheric Research Some time splitting details Strang splitting (Almut Gassmann) * Advection step, typically but not necessarily uses velocities at *

17. CSIRO Marine and Atmospheric Research Solution procedure Start  loop Start Nx(  t/N) forward-backward loop Stagger (u, v)  +n(  t/N) Average ps to (psu, psv)  +n(  t/N) Calc (div, sdot, omega)  +n(  t/N) Calc (ps, T)  +(n+1)(  t/N) Calc phi and staggered pressure gradient terms, then unstagger these Including Coriolis terms, calc unstaggered (u, v)  +(n+1)(  t/N) End Nx(  t/N) loop Perform TVD advection (of T, qg, Cartesian_wind_components) using average ps*u, ps*v, sdot from the N substeps Calculate physics contributions End  loop Advection of u and v As in the semi-Lagrangian advection of CCAM, the two u and v advection equations are replaced by three equations advecting the corresponding Cartesian wind components u c, v c, w c Benefit is that u c, v c, w c are simple variables, essentially behaving as scalars Then transform these back to u and v

18. CSIRO Marine and Atmospheric Research Comparisons with semi-Lagrangian method of CCAM Advantages No Helmholtz equation needed Includes full gravity-wave terms (no T linearization needed) Mass and moisture conserving More modular and “simpler” No semi-Lagrangian resonance issues near steep mountains Simpler MPI (“computation on demand” not needed) and runs faster - also MPI results always identical to single processor Disadvantages Restricted to Courant number of 1, but OK since grid is very uniform Some overhead from extra reversible staggering during sub time-steps (done for Coriolis terms) Non-hydrostatic version will take more effort

19. CSIRO Marine and Atmospheric Research Results for AMIP runs Unlike CCAM, VCAM dynamics seems to require hybrid coordinates to reduce spurious oscillations near high terrain (esp. Andes). There are less oscillations in PMSL near terrain (top) when using hybrid coordinates for 1-month January runs, with all other settings the same Also a clear signal in monthly-averaged omega at 500 hPa (next slide) hybrid non-hybrid

20. CSIRO Marine and Atmospheric Research 500 hPa omega (Jan 1979) hybrid non-hybrid

21. CSIRO Marine and Atmospheric Research hybrid VCAM CCAM

22. CSIRO Marine and Atmospheric Research AMIP 1979-95 CCAM VCAM Obs

23. CSIRO Marine and Atmospheric Research AMIP 1979-95 CCAM VCAM Obs

24. CSIRO Marine and Atmospheric Research AMIP 1979-95 CCAM VCAM Obs VCAM rainfall may be better over tropical land masses

25. CSIRO Marine and Atmospheric Research AMIP 1979-95 CCAM VCAM Obs

26. CSIRO Marine and Atmospheric Research wind speeds at 250 hPa CCAM VCAM CCAM VCAM DJFMAM JJASON

27. CSIRO Marine and Atmospheric Research wind speeds top level (4 hPa) CCAM VCAM CCAM VCAM DJFMAM JJASON

28. CSIRO Marine and Atmospheric Research Remaining tasks Check extra wind rotation terms in advection (run HS?) Check top boundary conditions May be able to do Coriolis in “long” time steps Finish coding pre-processing and post- processing

29. CSIRO Marine and Atmospheric Research Other plans Produce non-hydrostatic version Couple to PCOM (parallel cubic ocean model) of Motohiko Tsugawa from JAMSTEC

30. CSIRO Marine and Atmospheric Research Thank you!