Global Modelling on the Expanded Spherical Cube

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Global Modelling on theExpanded Spherical Cube
Alistair Adcroft MIT
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The MIT GCM

• Unified dynamical kernel (z-p isomorphism)
• - both an ocean and atmospheric GCM
• Finite volume (topography) ? grid point model
• Parallel computing using tiles
• Growing user group
• - MIT, SCRIPPS, WHOI, JPL, U. Conn.,

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What are the issues?

• Grid-point model
• Time-step limited by grid-spacing
• Converging meridians ? very small ?x at poles
• Fill-in Artic Ocean or
• Filtering in Atmosphere (eg. zonal FFTs)
• wastes resolution
• difficult with topography
• Anisotropic grid
• Distorts dynamics

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Regular Latitude-Longitude Grid?? ?? 2?/N
Number of points N?N/2 ?x 2?R/N cos(?) ?y
2?R/N ?xpole ? 2??y/N 4?2R/N2
Uniformity of resolution ?xeq/?xpole ?
N/(2?) Ratio of max/min areas Aeq/Apole ?
N/(2?) Aspect ratio ?y/?xpole ? N/(2?)
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Isotropic Latitude-Longitude Grid?? 2?/N
?? min( ?? cos(?) , ??min )
Num. points N ? 5/3 N ln 1/ ??min ?y ?x
2?R/N cos(?) ? ?lt?o ?x?o ?y?o R??min
? ?gt?o ?xpole ? 2?R??min/N
Uniformity of resolution ?xeq/?x?o 2?/(N
??min) ?xeq/?xpole 1/??min Aspect ratio ?y/?x
? 1 ? ?lt?o ?ymin/?xpole N/(2?)
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What are the alternatives?

• Move North pole into Greenland (ocean)
• eg. POP (Los Alamos), OPA (LODYC)
• Spectral/semi-lagrangian methods (atmosphere)
• Unstructured grid (finite element)
• eg. SEOM (Rutgers), QUODDY (Dartmouth)
• Structured grids of hexagons, triangles, etc.
• Cubic or octagonal grids (square grid cells)
• Sadourny, 1972 Ronchi et al., 1995 Rancic et
  al., 1996
• McGregor, 1996

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Gnomonic transformationfrom cube to sphere

• Grid face of cube
• Project image of grid onto sphere

Sadourny, MWR 1972 Ronchi et al., JCP 1996
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Gnomonic transformationfrom cube to sphere
Num. points 6?(M?M) 3/8 N2 Ratio ?xmax/?xmin
2 Ratio Amax/Amin 33/2 (or ¼ 33/2)

• Uniform coverage of sphere
• Nearly isotropic resolution
• Need general curvilinear coordinates
• Angular discontinuity
• generates noise (Sadourny 72)

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General curvilinear coordinates

• Covariant and contravariant flow components

• g12g21 ? 0 gives different numerical algorithm

u,v
u,v
C-grid
B-grid
u,v
u,v
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Covariant / Contravariant
V
a2
a2
a1
a1
V
a2
V
a1
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Conformal mapping ofcube to the sphere

• Preserves angle between intersecting grid lines
• W(Z) and Z(W) expressed as Taylor series

Rancic et al., QJRMS 1996
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Conformal mapping ofcube to the sphere
Num. points 6?(M?M) 3/8 N2 Ratio ?xmax/?xmin
M1/3 Ratio Amax/Amin ¾M2/3

• Locally orthogonal
• Nearly isotropic
• Unbounded ?xmax/?xmin
• Much better scaling than ?-? grid

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Quasi-smooth conformal mapping
Ratio ?xmax/?xmin ? 2.30 Ratio Amax/Amin ? 3.86
Ratio ?xmax/?xmin ? 1.54 Ratio Amax/Amin ? 1.54
Purser Rancic, QJRMS 1999
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The comparison
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Global bathymetry128x64 6x32x32
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Mercator projection
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Global bathymetry on tiles
S
N
Grid 6x 32x32 (equiv. to 2.8x2.8)
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Implementation

• Regular pattern of
• exchange
• - odd-odd, even-even
• - odd-even

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Evaluating terms at corners I
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Evaluating terms at corners II
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Finite volume expressions
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Metric terms?
Original model equations on the sphere
Non-hydrostatic
Hydrostatic
Vector invariant equations (all coordinate
systems)
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Temperature (950 mb)
Day 2700.5
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Temperature (950mb)
Day 270.5
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Temperature (950 mb)
Day 2700.5
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Conclusions

• Larger time step
• 128 x 64 Lat-Lon (no filters) 30 seconds?
• 128 x 64 Lat-Lon (with filters) 450 seconds
• 32 x 32 x 6 Cubic-Sphere 12 minutes
• 28? Minutes with better time-stepping
• Less memory 1/3
• 2/3 due to grid
• 1/2 due to time-stepping vector invariant

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Next steps

• Write it up!
• Lateral boundary conditions (atmos/ocean)
• Parallelization, I/O
• Overhaul wrapper
• Improve vorticity/tracer advection
• Flux limited, semi-lagragian finite volume