Title: Global Modeling on the Expanded Spherical Cube
1Global Modeling on theExpanded Spherical Cube
Alistair Adcroft MIT Jean-Michel Campin Chris
Hill John Marshall
2The 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
3The 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
4What 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?
5Regular 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?)
6Isotropic 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?)
7What 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
8Why 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
9Gnomonic transformationfrom cube to sphere
- Grid face of cube
- Project image of grid onto sphere
Sadourny, MWR 1972 Ronchi et al., JCP 1996
10Gnomonic 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
11Covariant / Contravariant
V
a2
a2
V
a1
a1
12Conformal 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
13Conformal 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
14Gnomonic vs Conformal
Amax/ Amin
?xmax/ ?xmin
?xmin/2?R
G-N
GC-M
2
CC-M
( MN/2 )
15More cubic grids
SCC-16
Rancic Purser 95
xtan 2/3 x
Uniform ?x _at_ x 7/8
Uniform ?x on edge
16Scaling
17Global bathymetry128x64 6x32x32
18Geographic projection
19Global bathymetry on tiles
S
N
Grid 6x 32x32 (equiv. to 2.8x2.8)
20Eqns 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
21Cubic topology
Exchange transform
Degenerate region of halo
Exchange
22Evaluating terms at corners I
23Evaluating terms at corners II
24Evaluating terms at corners II
25Finite volume expressions
26Vector 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
27Finite 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?
28Vector 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
29Summary 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
30Testing 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!
31Advection on a sphere
Error after 1 rotation
32Gravitational adjustment on a sphere
Should be a movie here!
33Testing dynamics Held Saurez 97
CC-32
- Zonal averages hide problems with grid
- Check that longer time-averages approach zonal
symmetry
34HS 97 Non-zonal structure I
- Forcing is zonally symmetric
- Solutions should approach zonal symmetry
35HS 97 Non-zonal structure II
- Real atmosphere has peak at modes 5-6
- Peak at mode 4 indicates influence of grid
36Full physics solution two orientations
0o
45o shift
37Surface 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?
38Sensitivity 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
39General curvilinear coordinates
- Covariant and contravariant flow components
40Ocean
and coupling
- Ocean and Atmosphere on same grid
- greatly simplifies coupling
- For higher resolution ocean
- aligned grids still simple to make conservative
41Conclusions
- 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