Conservative Ca scade R emapping between S pherical grids ( CaRS )
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Conservative Cascade Remapping between Spherical grids (CaRS)
Arunasalam RahunanthanDepartment of Mathematics,University of Wyoming.
NCAR, SIParCS Student Internship Presentations, Friday 10th August, 2007.
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Introduction Need for CaRS
For a better higher order method.
For the coupling between different model components data.
Goal Higher order accurate
interpolation of field variables from one spherical grid to another without violating conservation and monotonicity.
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Implementation SCRIP – a conservative remapping procedure
on the sphere (Jones, Mon.Wea.Rev, 1999) Advantage - Great geometric flexibility and capable
of handling different types of spherical grids. Disadvantage – Low order method.
CaRS – a cascade remapping between cubed-sphere grids and the RLL grids CaRS – area based (Lauritzen & Nair,
Mon.Wea.Rev, July 2007) CaRS performance compared with SCRIP
CaRS – length based CaRS length based approach compared with CaRS
area based approach
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Cascade interpolation
Algorithm based on cascade interpolation method developed for semi-Lagrangian advection schemes
A two dimensional interpolation problem split into two one dimensional problems
First sweep - Interpolation from ‘o’ to ‘x’ along the source grid lines
Second sweep – The resulting field interpolated from ‘x’ to
Allows for high-order sub grid cell construction and advanced monotone filters
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Cubed-sphere latitudes and longitudes
The entire cubed-sphere grid reconstructed with a family of horizontal and vertical grid lines.
The cubed-sphere latitudes as vertically stacked closed curves (squared patterns)
The cubed-sphere latitudes belts intersected by a set of cubed-sphere longitudes
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Gnomonic projection and cubed-sphere gridGnomonic projection
Great circle arcs are straight lines on gnomonic projection
South pole
North pole
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First sweep
Source grid
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First sweep
Source grid
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First sweep
Source grid
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First sweep
Source grid
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First sweep
Source gridTarget grid
(intermediate grid)
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First sweep
Source gridTarget grid
(intermediate grid)
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First sweep
Source gridTarget grid
(intermediate grid)
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First sweep
Source gridTarget grid
(intermediate grid)
South pole
North pole
North pole
South pole
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Second sweep
Source grid (intermediate
grid)
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Second sweep
Source grid (intermediate
grid)
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Second sweep
Source grid (intermediate
grid)
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Second sweep
Source grid (intermediate
grid)
Target grid (cubed-sphere
grid)
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Second sweep
Source grid (intermediate
grid)
Target grid (cubed-sphere
grid)
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Second sweep
Source grid (intermediate
grid)
Target grid (cubed-sphere
grid)
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Fields used for testing CaRS
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Results – Mapping from a coarse RLL grid to a fine cubed-sphere grid (a) Field :: Y32
16 Reconstruction :: PCM
Nc = 130,Nλ= 128, Nµ=64
Length based CaRS
Area based CaRS
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Results – Mapping from a fine cubed-sphere grid to a coarse RLL grid
(b) Field :: Vortex, Reconstruction :: PPM
Nc = 130,Nλ= 128, Nµ=64Length based
CaRSArea based CaRS
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Results – Mapping between a coarse RLL grid and a fine cubed-sphere grid
CaRS Method l2 l inf
length based CaRS
PCoM 9.3136E-07 2.4562E-02
PCM 4.9853E-07 1.5378E-02
Area based CaRS
PCoM 5.6647E-07 5.8725E-03
PCM 4.3911E-07 5.1458E-03
(a) Field :: Y3216 , Mapping :: RLL grid Cubed-
sphere grid
Nc = 130,Nλ= 128, Nµ=64
(b) Field :: Vortex, Mapping :: Cubed-sphere grid RLL grid
CaRS Method l2 l inf
length based CaRS
PCoM 5.8326E-04 8.3083E-02
PCM 4.6124E-06 2.0156E-02
Area based CaRS
PCoM 5.5808E-04 8.3624E-02
PCM 1.9817E-06 1.3403E-02
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Conclusions
The length based CaRS is true representation of the one-dimensional remapping and it could be easily employed in more complicated problems.
We need to sacrifice little bit of accuracy while employing the length based CaRS compared to the area based CaRS.
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Future works
By projecting the spherical grids on a Tangent plane, the lengths will be measured for the cascade remapping. Area preserving projection. Both grid lines in unified co-ordinate system.
RLL grid
RLL grid - longitude
RLL grid - latitudeCubed- sphere grid :: top panel
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Appendix A - Remapping between 1D grids
Given source grid and cell average values on source grid remap to target grid.
Reconstruction of sub-grid scale distribution with mass-conservation and monotonicity as constraints.
Piecewise Constant Method (PCoM) Piecewise Linear Method (PLM) Piecewise Parabolic Method (PPM) Piecewise Cubic Method (PCM)
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Appendix B - Length measured on different projections
First sweep – based on arc length
Second sweep – based on arc length
First sweep – based on length measured on (λ,θ) plane
Second sweep – based on length measured on (λ,θ) plane
First sweep – based on length measured on (λ,µ=sinθ) planeSecond sweep – based on length measured on (λ,µ=sinθ) plane
Field = Y3216 , Reconstruction ::
PCM
Nc = 33,Nλ= 128, Nµ=64