Gridding the sphere

1
Gridding the sphere
Rancic et al., QJRMS 96
?x ?R/N cos(?) ?y ?R/N ?xpole ? ??y/N ?2R/N2

• Locally orthogonal
• (conformal mapping)
• More uniform resolution

G-N
C-M
Num. points 6?(M?M) 3/2 N2 Ratio ?xmax/?xmin
M1/3 Ratio Amax/Amin ¾M2/3 Aspect ratio ?y/?x
1
Num. Points 2?N?N 2N2 Ratio ?xeq/?xpole ?
N/(2?) Ratio Aeq/Apole ? N/(2?) Aspect ratio
?y/?xpole ? N/(2?)
2
Cubic topology
Exchange transform
Degenerate region of halo
Exchange
3
Evaluating terms at corners
4
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?
5
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

6
Free-surface height
Latitude-longitude grid G45 (4x4) 2NN 9045
4050 ?x??80 ? 77 km
Expanded spherical cube C24 (3¾x3¾) 6NN
6242 3456 ?xmin ? 150 km
7
Ocean circulation salinity and currents (670m)
G45
C24
8
Held and Saurez, 1997
Pot. temperature, ? (K)
Zonal wind speed (m/s)
G64
C32
9
Held and Saurez, 1997
Eddy variance, T2 (K2)
Zonal spectrum of T2
G64
C32
10
Questions

• How do we test ocean models
• Anisotropies of grid hidden by irregular domain
• Sector model?
• Effects of unwanted variable resolution
• versus
• need for regionally increased resolution?
• Would generalized (non-orthogonal) coordinates
  gain anything?