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Global Modeling on the Expanded Spherical Cube

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Finite volume good representation ... Now in 'Release 1' http://mitgcm.org/ Non-hydrostatic. Quasi-hydrostatic. Regional. Global ... (equiv. to 2.8 x2.8 ) N. S ... – PowerPoint PPT presentation

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Title: Global Modeling on the Expanded Spherical Cube


1
Global Modeling on theExpanded Spherical Cube
Alistair Adcroft MIT Jean-Michel Campin Chris
Hill John Marshall
2
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
3
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

4
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?

5
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?)
6
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?)
7
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

8
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

9
Gnomonic transformationfrom cube to sphere
  • Grid face of cube
  • Project image of grid onto sphere

Sadourny, MWR 1972 Ronchi et al., JCP 1996
10
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
11
Covariant / Contravariant
V
a2
a2
V
a1
a1
12
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
13
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

14
Gnomonic vs Conformal
Amax/ Amin
?xmax/ ?xmin
?xmin/2?R
G-N
GC-M
2
CC-M
( MN/2 )
15
More cubic grids
SCC-16
Rancic Purser 95
xtan 2/3 x
Uniform ?x _at_ x 7/8
Uniform ?x on edge
16
Scaling
17
Global bathymetry128x64 6x32x32
18
Geographic projection
19
Global bathymetry on tiles
S
N
Grid 6x 32x32 (equiv. to 2.8x2.8)
20
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

21
Cubic topology
Exchange transform
Degenerate region of halo
Exchange
22
Evaluating terms at corners I
23
Evaluating terms at corners II
24
Evaluating terms at corners II
25
Finite volume expressions
26
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

27
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?
28
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

29
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

30
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!

31
Advection on a sphere
Error after 1 rotation
32
Gravitational adjustment on a sphere
Should be a movie here!
33
Testing dynamics Held Saurez 97
CC-32
  • Zonal averages hide problems with grid
  • Check that longer time-averages approach zonal
    symmetry

34
HS 97 Non-zonal structure I
  • Forcing is zonally symmetric
  • Solutions should approach zonal symmetry

35
HS 97 Non-zonal structure II
  • Real atmosphere has peak at modes 5-6
  • Peak at mode 4 indicates influence of grid

36
Full physics solution two orientations
0o
45o shift
37
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?

38
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

39
General curvilinear coordinates
  • Covariant and contravariant flow components

40
Ocean
and coupling
  • Ocean and Atmosphere on same grid
  • greatly simplifies coupling
  • For higher resolution ocean
  • aligned grids still simple to make conservative

41
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
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