CSIRO Marine and Atmospheric Research 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

CSIRO Marine and Atmospheric Research 2 Outline CCAM formulation VCAM formulation Some comparisons

CSIRO Marine and Atmospheric Research 3 Original Sadourny (1972) C20 grid Equi-angular C20 grid Alternative cubic grids Conformal-cubic C20 grid

CSIRO Marine and Atmospheric Research 4 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

CSIRO Marine and Atmospheric Research 5 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: - 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. TR ) - 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

CSIRO Marine and Atmospheric Research 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.

CSIRO Marine and Atmospheric Research 7 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 solved iteratively, or by cyclic tridiagonal solver excellent dispersion properties for gravity waves, as shown for the linearized shallow-water equations

CSIRO Marine and Atmospheric Research 8 Dispersion behaviour for linearized shallow-water equations Typical atmosphere case - large 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 Typical ocean case - small radius deformation

CSIRO Marine and Atmospheric Research Transformation of 2, 3, 4, 6-grid waves

CSIRO Marine and Atmospheric Research 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

CSIRO Marine and Atmospheric Research Treatment of p s advection near terrain 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

CSIRO Marine and Atmospheric Research Treatment of T advection near terrain 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.

CSIRO Marine and Atmospheric Research a posteriori conservation 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

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

CSIRO Marine and Atmospheric Research 15 Typical MPI performance  Showing both Face-Centred (FC) and Uniform Decomposition (UD) for global C 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

CSIRO Marine and Atmospheric Research 16 An AMIP run CCAM Obs 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 DJFJJA

CSIRO Marine and Atmospheric Research 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

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

CSIRO Marine and Atmospheric Research 19 Quasi-uniform C48 CCAM grid with resolution about 200 km Stretched C48 grid with resolution about 20 km over eastern Australia 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

CSIRO Marine and Atmospheric Research 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

CSIRO Marine and Atmospheric Research 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 (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

CSIRO Marine and Atmospheric Research CCAM simulations of cold bubble, 500 m L35 resolution, on highly stretched global grid

CSIRO Marine and Atmospheric Research 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

CSIRO Marine and Atmospheric Research 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

CSIRO Marine and Atmospheric Research New dynamical core for VCAM - Variable Cubic Atmospheric Model 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

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CSIRO Marine and Atmospheric Research 27 Horizontal advection F low =q y V j+1/2 V j-1/2 u cov U i-1/2 v cov (q x, q y ) q F low =q x U i+1/2 Transverse components (to be included in low/high order fluxes) calculated at the centre of the grid cells (loosely following LeVeque) q x : using dt/2 advection from v cov q y : using dt/2 advection from u cov 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

CSIRO Marine and Atmospheric Research 28 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 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

CSIRO Marine and Atmospheric Research 500 hPa omega (Jan 1979) Hybrid coordinates introduced non-hybrid

CSIRO Marine and Atmospheric Research 250 hPa winds in 1-year run 30 VCAMCCAM

CSIRO Marine and Atmospheric Research 31 DJFJJA VCAM 1-year CCAM 1-year Same physics Obs climate

CSIRO Marine and Atmospheric Research 32 However, can can see some influence of panel edges on rainfall just south of Australia

CSIRO Marine and Atmospheric Research 33 Problem caused by spurious vertical velocities at vertices! Eastwards solid body rotation in 900 time steps Using superbee limiter

CSIRO Marine and Atmospheric Research 34 Spurious vertical velocities reduced by factor of 8 by more- careful calculation of pivot velocities near panel edges

CSIRO Marine and Atmospheric Research 35 With better staggered velocities at panel edges

CSIRO Marine and Atmospheric Research 36 Comparisons of VCAM and CCAM 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

CSIRO Marine and Atmospheric Research 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 37

CSIRO Marine and Atmospheric Research 38 Thank you!