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

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

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

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.

Atmosphere Spectral Elements

Spectral Elements Ocean

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

Spectral element discretization

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:

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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.

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*

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

Shallow water test case

Justification for 8x8 grid Shallow water test case

Mesh-refine on globe

Mesh Refine Global Local

Meshrefine: Zoom Global Local

Mesh-refine Andes topography Local

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

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

Held-Suarez Case

T85 T180 50L 100L 200L Polar Vortex

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

blocking event

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.

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

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 = 20000 m Very weak dissipation Dt = 25 Earth seconds T170, T360, T533, T1033 runs on a CRAY T3E with 128 processors T1033 has 60000 elements (8x8) ~ 3000 equatorial pts. Expected result: Rhines scales with 15 jets pole to pole

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

(-) (-) (-) (-) S N Latitude

Potential Vorticity on Jupiter 276 Jupiter days T170 T1033 T1033

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

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