Conservative Cascade Remapping between Spherical grids (CaRS)

Arunasalam RahunanthanDepartment of Mathematics,University of Wyoming.

NCAR, SIParCS Student Internship Presentations, Friday 10th August, 2007.

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.

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

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

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

Gnomonic projection and cubed-sphere gridGnomonic projection

Great circle arcs are straight lines on gnomonic projection

South pole

North pole

First sweep

Source grid

First sweep

Source grid

First sweep

Source grid

First sweep

Source grid

First sweep

Source gridTarget grid

(intermediate grid)

First sweep

Source gridTarget grid

(intermediate grid)

First sweep

Source gridTarget grid

(intermediate grid)

First sweep

Source gridTarget grid

(intermediate grid)

South pole

North pole

North pole

South pole

Second sweep

Source grid (intermediate

grid)

Second sweep

Source grid (intermediate

grid)

Second sweep

Source grid (intermediate

grid)

Second sweep

Source grid (intermediate

grid)

Target grid (cubed-sphere

grid)

Second sweep

Source grid (intermediate

grid)

Target grid (cubed-sphere

grid)

Second sweep

Source grid (intermediate

grid)

Target grid (cubed-sphere

grid)

Fields used for testing CaRS

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

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

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

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.

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

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)

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