Comparing the formulations of CCAM and VCAM and aspects of their performance

1. Comparing the formulations of CCAM and VCAM and aspects of their performance John McGregor CSIRO Marine and Atmospheric Research Aspendale, Melbourne PDEs on the Sphere Cambridge 28 September 2012

2. Outline • CCAM formulation • VCAM formulation • Some comparisons

3. Alternative cubic grids OriginalSadourny (1972) C20 grid Equi-angular C20 grid Conformal-cubicC20 grid

4. CCAM is formulated on the conformal-cubic grid Orthogonal Isotropic The conformal-cubic atmospheric model Example of quasi-uniform C48 grid with resolution about 200 km

5. 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 dynamics CCAM physics • cumulus convection: • - mass-flux scheme, including downdrafts, entrainment, detrainment • - 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

6. 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.

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

8. Dispersion behaviour for linearized shallow-water equations Typical atmosphere case - large radius deformation Typical ocean case - small radius deformation N.B. the asymmetry of the R grid response disappears by alternating the reversing direction each time step, giving the same response as Z (vorticity/divergence) grid

9. Transformation of 2, 3, 4, 6-grid waves

10. The reversible staggering technique allows a very consistent, and thus more accurate, calculation of pressure gradient terms. For example, in the staggered u equation the RHS pressure gradient term is first evaluated at the staggered position, then transformed to the unstaggered position for calculation of the whole RHS advected value on the unstaggered grid. That whole term is then transformed to the staggered grid, fully consistent with the subsequent implicit evaluation of the LHS on the staggered grid. Treatment of pressure-gradient terms

11. Pressure advection equation Define an associated variable, similar to MSLP which varies smoothly, even over terrain. It is thus suitable for evaluation by bi-cubic interpolation, whilst the other term is found “exactly” by bi-linear interpolation (to avoid any overshooting effects). Formally, get Treatment of psadvection near terrain

12. Similarly to surface pressure advection, define an associated variable which varies relatively smoothly on sigma surfaces over terrain. Again the second term can be found “exactly” by bi-linear interpolation. A suitable function is Formally, get This technique effectively avoids the requirement for hybrid coordinates. Treatment of T advection near terrain

13. a posteriori conservation of mass and moisture “global” scheme simultaneously ensures non-negative values during each time step applies correction to changes occurring during dynamics (including advection) correction is proportional to the “dynamics” increment, but the sign of the correction depends on the sign of the increment at each grid point. The above are all described in the CCAM Tech. Report a posteriori conservation

14. MPI implementation Remapped region 0 Original Remapping of off-processor neighbour indices to buffer region Indirect addressing is used extensively in CCAM - simplifies coding

15. Typical MPI performance • Showing both Face-Centred (FC) and Uniform Decomposition (UD) for global C192 50 km runs, for 1, 6, 12, 24, 48, 72, 96, 144, 192, 288 CPUs • VCAM a little slower, but is still to be fully optimised

16. Tuning/selecting physics options: In CCAM, usually done with 200 km AMIP runs, especially paying attention to Australian monsoon, Asian monsoon, Amazon region No special tuning for stretched runs An AMIP run 1979-1995 DJF JJA Obs CCAM

17. Variable-resolution conformal-cubic grid • The C-C grid is rotated to locate panel 1 over the region of interest • The Schmidt (1975) transformation is applied • this is a pole-symmetric dilatation, calculated using spherical polar coordinates centred on panel 1 • it preserves the orthogonality and isotropy of the grid • same primitive equations, but with modified values of map factor • Plot shows a C48 grid(Schmidt factor = 0.3) with resolution about 60 km over Australia

18. Schmidt transformation can be used to obtain even finer resolution Grid configurations used to support Alinghi in America’s Cup, Olympic sailing C48 8 km grid over New Zealand C48 1 km grid over New Zealand

19. The 200 km run is then downscaled to 20 km (say) by running CCAM with a stretched grid, but applying a digital filter every 6 h to preserve large-scale patterns of the 200 km run Preferred CCAM downscaling methodology • Coupled GCMs have coarse resolution, but also possess Sea Surface Temperature (SST) biases • A common bias is the equatorial “cold tongue” • First run a quasi-uniform 200 km (or modestly stretched) CCAM run driven by the bias-corrected SSTs Stretched C48 grid with resolution about 20 km over eastern Australia Quasi-uniform C48 CCAM grid with resolution about 200 km

20. Uses a sequence of 1D passes over all panels to efficiently evaluate broad-scale digitally-filtered host-model fields (Thatcher and McGregor, MWR, 2009). Very similar results to 2D collocation method. These periodically (e.g. 6-hourly or 12-hourly) replace the corresponding broad-scale CCAM fields Gaussian filter typically uses a length-scale approximately the width of finest panel Suitable for both NWP and regional climate Digital-filter downscaling method

21. Being a semi-Lagrangian model, CCAM is able to absorb the extra phi terms into its Helmholtz equation solver, for “zero” cost The new dynamical core (VCAM) uses a split-explicit treatment, so the Miller-White treatment would need its own Helmholtz solver, so may use Laprise-style nonhydrostatic treatment for VCAM Nonhydrostatic treatment

22. CCAM simulations of cold bubble, 500 m L35 resolution, on highly stretched global grid

23. 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

24. Nonhydrostatic treatment Being a semi-Lagrangian model, CCAM is able to absorb the extra phi terms into its Helmholtz equation solver, for “zero” cost The new dynamical core of VCAM uses a split-explicit treatment, so the Miller-White treatment would need its own Helmholtz solver, Probably will use Laprise-style nonhydrostatic treatment for VCAM

25. uses equi-angular gnomonic-cubic grid - provides extremely uniform resolution - less issues for resolution-dependent parameterizations reversible staggering transforms the contravariant winds to the edge positions needed for calculating divergence and gravity-wave terms flux-conserving form of equations preferable for trace gas studies TVD advection can preserve sharp gradients forward-backward solver for gravity waves avoids need for Helmholtz solver linearizing assumptions avoided in gravity-wave terms New dynamical core for VCAM - Variable Cubic Atmospheric Model

27. Horizontal advection Low-order and high-order fluxes combined using Superbee limiter High order need covariant vels for LW term. Linear interp for edge values of q? Cartesian components (U,V,W) of horizontal wind are advected Flow=qyVj+1/2 vcov (qx,qy) q ucov Ui-1/2 Flow=qxUi+1/2 Vj-1/2 Transverse components (to be included in low/high order fluxes) calculated at the centre of the grid cells (loosely following LeVeque) qx: using dt/2 advection from vcov qy: using dt/2 advection from ucov

28. Solution procedure • Start t loop • Start Nx(Dt/N) forward-backward loop • Stagger (u, v) t+n(Dt/N) • Average ps to (psu, psv) t+n(Dt/N) • Calc (div, sdot, omega) t+n(Dt/N) • Calc (ps, T) t+(n+1)(Dt/N) • Calc phi and staggered pressure gradient terms, then unstagger these • Including Coriolis terms, calc unstaggered (u, v) t+(n+1)(Dt/N) • End Nx(Dt/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 t loop • Main MPI overhead is the reversible staggering at each substep, but this just needs nearest neighbours in its iterative tridiagonal solver. Also message passing is needed in the pressure gradient and divergence calcs

29. 500 hPa omega (Jan 1979) Hybrid coordinates introduced non-hybrid

30. CCAM VCAM 250 hPa windsin 1-year run

31. Same physics DJF JJA Obs climate VCAM 1-year CCAM 1-year

32. However, can can see some influence of panel edges on rainfall just south of Australia

33. Eastwards solid body rotation in 900 time steps Using superbee limiter Problem caused by spurious vertical velocities at vertices!

34. Spurious vertical velocities reduced by factor of 8 by more-careful calculation of pivot velocities near panel edges

35. With better staggered velocities at panel edges

36. VCAM advantages No Helmholtz equation needed Includes full gravity-wave terms (no T linearization needed) Mass and moisture conserving More modular and code is “simpler” No semi-Lagrangian resonance issues near steep mountains Simpler MPI (“computation on demand” not needed) VCAM disadvantages Restricted to Courant number of 1, but OK since grid is very uniform Some overhead from extra reversible staggering during sub time-steps (needed for Coriolis terms) Nonhydrostatic treatment will be more expensive Comparisons of VCAM and CCAM

37. Tentative conclusions • Reversible interpolation works well for both CCAM and VCAM • VCAM seems to perform better than CCAM in the tropics • better rain over SPCZ and Indonesia, possibly by avoiding linearizing ps term in pressure gradients, and better gravity wave adjustment by not using semi-implicit • rainfall presently not as good in midlatitudes

38. Thank you!