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2/11/20101Research Review
Geodesic Grids Finite volume transport Reconstruction functions Mass Flux Calculations Monotonic Limiting
2/11/2010 2Research Review
Source: NCAR
Source: Wikipedia
Geodesic grids: Tessellations of the sphere generated from platonic solids
Motivation : To avoid the POLE PROBLEM
Problem: Traditional finite difference methods are not applicable – grid lines not aligned to latitude/longitude
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Transport Equation
Integrating over the area
Integrating in time gives
0Vt
)(.
0Vqt
q
)(.
dnVtA
ˆ
dnVq
t
q
A
ˆ
dnVqA
qq ttt ˆ1
)()(
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dnVqA
qq ttt ˆ1
)()(
Note : In three cases, Depending on V the q estimate will be different. (Trajectory Calculation)
Reconstruction Function
cybxayxq ),(
fxyeydxcybxayxq 22),(
ayxq ),(Constant
Linear
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fluxoutgingfluxgincoqq ttt min)()(
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dnVqA
qq ttt ˆ1
)()(
Various Cases
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dnVqA
qq ttt ˆ1
)()(
Note : In three cases, Depending on V the q estimate will be different. (Trajectory Calculation)
Reconstruction Function
cybxayxq ),(
fxyeydxcybxayxq 22),(
ayxq ),(Constant
Linear
Quadratic2/11/2010 8Research Review
fluxoutgingfluxgincoqq ttt min)()(
yyxyxx
yx
fkyfkyhxfhx
fkyfhxkhf
fxyeydxcybxakyhxf
22
22
)())((2)(2
1
)()(),(
),(
2,1,,1 )/()2( xffff jijijixx
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223322 jxyyixhygxfxyeydxcybaf
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Area- Avg -Quad
Area Cubic
e^(x)
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SOLUTION : MONOTONICITY CONSTRAINT
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ZalesekKeep as much as weighing of higher order estimate as possible and at the same time preserving the monotonicity . (Details in Zalesek ,1979)
Vanleer :Gradients are limited for each hexagon as described in Dukowicz and Kodis(1987)
Thuburn:A multidimensional flux limiter which modifies the estimated mixing ratio values at the edge itself.( details in Thuburn 1996)
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M D
T Z
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Geodesic Grids are attractive because are computationally more efficient for AGCM
Steps involved in Tracer transport are a) Compute reconstruction; b) Compute Mass flux; c) Monotonic limiting ( if needed)
Accuracy of reconstruction function determines the convergence order of scheme
Mass flux calculation are important and are responsible for overall computational cost of scheme
Monotonic limiting is important and has been successfully tested
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