Global Modeling on the Expanded Spherical Cube

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Global Modeling on theExpanded Spherical Cube
Alistair Adcroft MIT Jean-Michel Campin Chris
Hill John Marshall
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The MIT (O)GCM

• Finite volume ? good representation of topography
• Parallel computing using tiles ? widely
  portable
• Non-hydrostatic capability ? wide range of scales
• Automatic adjoint (TAF) ? ocean state estimation
• Well established and versatile ocean model
• Now in Release 1 http//mitgcm.org/

Non-hydrostatic
Regional
Quasi-hydrostatic
Global
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The MIT GCM

• Unified dynamical kernel (z-p isomorphism)
• - both an ocean and atmospheric GCM
• Atmospheric dynamics is global
• - can not hide pole in Greenland
• Model time step limited critically by converging
  meridians

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What are the issues?

• MIT GCM is a grid-point model
• Explicit 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?) Maximum 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 Maximum 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.
• eg. ZM grid of Ringler and Randall, MWR 01
• Cubic or octagonal grids (square grid cells)
• Sadourny, 1972 Ronchi et al., 1995 Rancic et
  al., 1996
• McGregor, 1996

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Why the cubed sphere?

• Faced with the constraints of converging
  meridians
• Arctic ignored (masked out) OR use FFT filters
• need longer time-step in atmosphere
• Need a new grid with more uniform resolution
• Growing group of users, experiments and code
• A new grid must have minimal impact on code
• i.e. no hexagons, spectral elements, etc
• Finite volume method ? orthogonal curvilinear
  coords.
• Expanded cubes (Sadourny, 72, Ronchi, 96,
  Rancic Purser, 90) represent an opportunity
  for quadrilateral based model to uniformly grid
  the sphere

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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)

• Near uniform coverage of sphere
• ie. near uniform resolution
• Need general curvilinear coordinates
• Angular discontinuity at edges
• generates noise (Sadourny 72)

Sadourny, MWR 1972 Ronchi et al., JCP 1996
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Covariant / Contravariant
V
a2
a2
V
a1
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 uniform resolution
• Unbounded ?xmax/?xmin
• Much better scaling than ?-? grid

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Gnomonic vs Conformal
Amax/ Amin
?xmax/ ?xmin
?xmin/2?R
G-N
GC-M
2
CC-M
( MN/2 )
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More cubic grids
SCC-16
Rancic Purser 95
xtan 2/3 x
Uniform ?x _at_ x 7/8
Uniform ?x on edge
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Scaling
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Global bathymetry128x64 6x32x32
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Geographic projection
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Global bathymetry on tiles
S
N
Grid 6x 32x32 (equiv. to 2.8x2.8)
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Eqns in Orthogonal Curvilinear Coordinates

• Orthogonal curvilinear coordinates (x1, x2)
  described by scale factors e1, e2
• Tensorial form of SWEs
• Interpret on finite volume grid (?x ?s1, ?y
  ?s2)
• In MITgcm, only the metric terms need be
  further generalized

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Cubic topology
Exchange transform
Degenerate region of halo
Exchange
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Evaluating terms at corners I
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Evaluating terms at corners II
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Evaluating terms at corners II
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Finite volume expressions
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Vector Invariant Eqns

• Tensorial form of Vector Invariant SWEs
• expressed in terms of well defined quantities
• no metric terms
• Finite volume discretization
• described entirely in terms of lengths and areas

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Finite Volume vs tensorial formalism

• Gradients across corners occur in finite
  difference mindset
• Integral formulation avoids any ambiguity about
  discretization

e.g. e1r cos?, e2r, x1?, x2?
e.g. ?x ?? r cos?, ?yr ??, A? ?x r d?
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Vector Invariant Eqns

• Tensorial form of conservative SWEs
• Finite volume method applied to Vector Invariant
  SWEs
• described entirely in terms of lengths and areas
• no metric terms

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Summary of Implementation

• Expanded cube can be realized by using
• modified exchange between tiles
• finite volumes to describe curvilinear grid
• vector invariant equations
• All gridding issues addressed by SWEs
• 3D HPEs work in exactly same way

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Testing models on novel grids

• Williamson, JCP 1996
• tests of SWE models on sphere
• particularly good results
• (considering were using a implicit-linear
  free-surface)
• Held and Saurez, BAMS 1997
• tests 3D dynamics with idealized forcing
• excellent agreement in zonal average diagnostics
• deviations from zonalality prove interesting
• there shouldnt be any!

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Advection on a sphere
Error after 1 rotation
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Gravitational adjustment on a sphere
Should be a movie here!
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Testing dynamics Held Saurez 97
CC-32

• Zonal averages hide problems with grid
• Check that longer time-averages approach zonal
  symmetry

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HS 97 Non-zonal structure I

• Forcing is zonally symmetric
• Solutions should approach zonal symmetry

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HS 97 Non-zonal structure II

• Real atmosphere has peak at modes 5-6
• Peak at mode 4 indicates influence of grid

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Full physics solution two orientations
0o
45o shift
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Surface pressure deviation from zonal mean

• Clearly the cube is apparent in the solution
• due to variable resolution?
• locking of natural modes to the structure of the
  grid?
• convergence problem?
• presence of singularities?

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Sensitivity to the corners

• In the vicinity of the corner singularities the
  grid can not be orthogonal
• Need to allow for general curvilinear
  grid/coordinates
• add necessary terms only for corner cells

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General curvilinear coordinates

• Covariant and contravariant flow components

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Ocean
and coupling

• Ocean and Atmosphere on same grid
• greatly simplifies coupling
• For higher resolution ocean
• aligned grids still simple to make conservative

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Conclusions

• Expanded spherical cube is viable
• avoids needs for zonal filters
• allows significantly larger time step (7?20mins)
• uniform coverage of sphere (no wasted points)
• Vector invariant/finite volume combined allows
  unambiguous discretization
• avoids problems at singularities
• The singularities at corners still lead to errors
• needs general orthogonal coordinates terms