1. Global and Regional Atmospheric Modeling Using Spectral Elements
Ferdinand Baer
Aimé Fournier
Joe Tribbia (NCAR)
Mark Taylor (Los Alamos)

2. Some Outstanding Issues Associated with Climate Modeling
Climate model representation
Coupling of related models
Parameterization of forces
Methodology for computation
Computing hardware
Speed of computations
Time and space scales required
Regional climate modeling

3. SEAM: Spectral Element Atmospheric Model
A global model offering great flexibility and advantages in:
Geometric properties of finite element methods
Local mesh refinement and regional detail using various sized grids over the global domain
parallel processing
Accuracy of spectral models
Computational efficiency
No pole problem

4. Method
Tile spherical surface with arbitrary number and size of rectangular elements;
Inscribe polyhedron with rectangular faces inside sphere
Map surface of polyhedron to surface of sphere with gnomic projection
Subdivide each face (variable)
Define basis functions in each element
Define test functions over sphere
We use Adams-Bashforth 3rd order time stepping and 4th order Runge-Kutta to start.

5. Atmosphere Spectral Elements

6. Spectral Elements Ocean

7. Example + + ]

8. Spectral element discretization
Decompose sphere into rectangular regions
Within each rectangle, estimate integral equations
by Gauss-Lobatto Quadrature
8x8 grid is arbitrary

9. Spectral element discretization

10. Element boundary points:
Global Test Functions
Simple combinations of Legendre cardinal functions
One global test function for each grid point
Element interior points:
Element boundary points:

11. + + ] +

12. Summarizing: Gauss-Labatto quadrature
Legendre cardinal functions for the basis functions
Test functions based on Legendre cardinal functions
These choices result in an extremely simple finite element method with a diagonal mass matrix.
Thus the spectral element method appears to be a most efficient and natural way to achieve a high order finite element discretization and to simplify regional modeling.

13. Tests with various representations:
Shallow water model - the standard test set (Williamson et al., 1992)
Shallow water model - mesh-refine with topography
3-D dynamical core with Held-Suarez conditions
SWE high resolution turbulence studies
A high resolution 3-D model with a polar vortex*
3-D model with Held-Suarez and topography to test for blocking events*

15. SE - spectral element
TWIG - twisted icosahedral grid
A-L - Arakawa-Lamb

17. Shallow water test case

18. Justification for 8x8 grid
Shallow water test case

19. Mesh-refine on globe

20. Mesh
Refine
Global
Local

21. Meshrefine:
Zoom
Global
Local

22. Mesh-refine Andes topography
Local

23. 3-D Held-Suarez Case
red
blue
green
HP Exemplar SPP2000

24. 3-D test case/Held-Suarez * 384x8x8 ~ T85/L20
T*2 (K2)
U (m/s)
T63
G72
3-D test case/Held-Suarez * 384x8x8 ~ T85/L20

25. Held-Suarez Case

27. T85
T180
50L
100L
200L
Polar Vortex

28. Blocking Experiment 3-D SEAM model with Held-Suarez conditions
T-85, 10 levels, 8x8 degree in elements
Two ten year runs
One without topography
One with T42 topography
Search for statistics on persistent ridges
Compare to similar reported experiments

29. blocking event

30. SEAM D- NH w/o topog, Ñ- SH w/o topog 100 10 s- NH with topog.
t- SH with topog
Ñ- SH w/o topog
100 10
SEAM 1
Histogram of frequency distribution of blocking length interval.

31. Frequency distribution of blocking length on 5-20 day interval
Frequency distribution of blocking length on 5-20 day interval. After D’Andrea et al, Cli. Dynamics , 1998
Number of events in 14 winter seasons. After Dole & Gordon, 1983

32. SWE on Jupiter A study of decaying turbulence with high resolution
We use Jupiter dimensions;
g = 23 m s-2, radius = 7x104 km
One Jupiter day equals 9 Earth hours
Equivalent depth = m
Very weak dissipation
Dt = 25 Earth seconds
T170, T360, T533, T1033 runs on a CRAY T3E with 128 processors
T1033 has elements (8x8) ~ 3000 equatorial pts.
Expected result: Rhines scales with 15 jets pole to pole

33. Equatorial Jet Strength
(-)
200
(-)
(-)
(-)
1000
Jovian days

34. (-)
(-)
(-)
(-) S N
Latitude

35. Potential Vorticity on Jupiter
276 Jupiter days
T170
T1033
T1033

37. Demonstrated features of SEAM
Flexibility: from various applications
Regional detail and mesh refinement: from shallow water experiment
Advantages on parallel processors: from Jupiter experiment
Accuracy: from shallow water experiments
Computational efficiency: from all the experiments

38. Future plans Incorporate state-of-the-art physics
Embed dynamical core into CCM3 for physics and diagnostics
Participate in NCAR ‘bakeoff’
Companion studies with stretched grid model (Fox-Rabinovitz)
Compare dynamical core experiments with local grid refinement (stretching)
Use elements to calculate local sub-grid scale parameterization