Global variable-resolution models for regional climate by pptfiles

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									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   1
    Outline

• CCAM formulation
• VCAM formulation
• Some comparisons




              CSIRO Marine and Atmospheric Research   2
                  Alternative cubic grids
Original
Sadourny (1972)
C20 grid




Equi-angular
C20 grid




Conformal-cubic
C20 grid


                               CSIRO Marine and Atmospheric Research   3
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   4
                             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. 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


                                           CSIRO Marine and Atmospheric Research   5
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   6
                              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 4th 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   7
Dispersion behaviour for linearized shallow-water equations




    Typical atmosphere case                                       Typical ocean case
       - large radius deformation                               - 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


                                                           CSIRO Marine and Atmospheric Research               8
Transformation of 2, 3, 4, 6-grid waves




                     CSIRO Marine and Atmospheric Research
  Treatment of pressure-gradient terms
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.


                                CSIRO Marine and Atmospheric Research
     Treatment of ps 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
                    MPI implementation




                                     Remapped region 0



         Original
Remapping of off-processor neighbour indices to buffer region

Indirect addressing is used extensively in CCAM
- simplifies coding
                                 CSIRO Marine and Atmospheric Research 14
              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




                                         CSIRO Marine and Atmospheric Research 15
                  An AMIP run 1979-1995
                     DJF                          JJA



    Obs




    CCAM




               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
                                    CSIRO Marine and Atmospheric Research 16
    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 17
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




                                           CSIRO Marine and Atmospheric Research 18
         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

•     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




    Quasi-uniform C48 CCAM grid with resolution about 200 km            Stretched C48 grid with resolution about
                                                                              20 km over eastern Australia

                                                               CSIRO Marine and Atmospheric Research 19
    Digital-filter downscaling method




•   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



                                        CSIRO Marine and Atmospheric Research 20
        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
        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 23
         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 24
            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


                                   CSIRO Marine and Atmospheric Research
CSIRO Marine and Atmospheric Research 26
                      Horizontal advection
Low-order and high-
order fluxes combined
using Superbee limiter                                      Flow=qyVj+1/2
High order need
covariant vels for LW
term. Linear interp for                             vcov
edge values of q?                      (qx, qy)

Cartesian components                      q
(U,V,W) of horizontal         Ui-1/2               ucov
wind are advected                                               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

                                                  CSIRO Marine and Atmospheric Research 27
                       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




                                            CSIRO Marine and Atmospheric Research 28
              500 hPa omega (Jan 1979)


Hybrid
coordinates
introduced




non-hybrid




                          CSIRO Marine and Atmospheric Research
            CCAM                     VCAM




250 hPa
winds
in 1-year
run




               CSIRO Marine and Atmospheric Research 30
          DJF   Same physics           JJA



  Obs
climate




VCAM
1-year




CCAM
1-year


                      CSIRO Marine and Atmospheric Research
                                                              31
However, can can see some influence of panel edges on
rainfall just south of Australia


                             CSIRO Marine and Atmospheric Research 32
                      Eastwards solid
                      body rotation in
                      900 time steps
                      Using superbee
                      limiter




                      Problem caused
                      by spurious
                      vertical
                      velocities at
                      vertices!



CSIRO Marine and Atmospheric Research 33
      Spurious vertical
      velocities reduced by
      factor of 8 by more-
      careful calculation of
      pivot velocities near
      panel edges




CSIRO Marine and Atmospheric Research 34
                      With better
                      staggered
                      velocities at
                      panel edges




CSIRO Marine and Atmospheric Research 35
      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 36
             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


                              CSIRO Marine and Atmospheric Research 37
Thank you!




       CSIRO Marine and Atmospheric Research 38

								
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