1. Conservative Cascade Remapping between Spherical grids (CaRS) Arunasalam Rahunanthan Department of Mathematics, University of Wyoming. NCAR, SIParCS Student Internship Presentations, Friday 10 th August, 2007.

2. 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.

3. 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

4. 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

5. 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

6. Gnomonic projection and cubed-sphere grid Gnomonic projection Great circle arcs are straight lines on gnomonic projection South pole North pole

7. First sweep Source grid

8. First sweep Source grid

9. First sweep Source grid

10. First sweep Source grid

11. First sweep Source grid Target grid (intermediate grid)

12. First sweep Source grid Target grid (intermediate grid)

13. First sweep Source grid Target grid (intermediate grid)

14. First sweep Source grid Target grid (intermediate grid) South pole North pole South pole

15. Second sweep Source grid (intermediate grid)

16. Second sweep Source grid (intermediate grid)

17. Second sweep Source grid (intermediate grid)

18. Second sweep Source grid (intermediate grid) Target grid (cubed- sphere grid)

19. Second sweep Source grid (intermediate grid) Target grid (cubed- sphere grid)

20. Second sweep Source grid (intermediate grid) Target grid (cubed- sphere grid)

21. Fields used for testing CaRS

22. Results – Mapping from a coarse RLL grid to a fine cubed-sphere grid (a) Field :: Y 32 16 Reconstruction :: PCMN c = 130,N λ = 128, N µ =64 Length based CaRS Area based CaRS

23. Results – Mapping from a fine cubed-sphere grid to a coarse RLL grid (b) Field :: Vortex, Reconstruction :: PPMN c = 130,N λ = 128, N µ =64 Length based CaRSArea based CaRS

24. Results – Mapping between a coarse RLL grid and a fine cubed-sphere grid CaRS Methodl2l2 l inf length based CaRS PCoM9.3136E-072.4562E-02 PCM4.9853E-071.5378E-02 Area based CaRS PCoM5.6647E-075.8725E-03 PCM4.3911E-075.1458E-03 (a) Field :: Y 32 16, Mapping :: RLL grid  Cubed-sphere grid N c = 130,N λ = 128, N µ =64 (b) Field :: Vortex, Mapping :: Cubed-sphere grid  RLL grid CaRS Methodl2l2 l inf length based CaRS PCoM5.8326E-048.3083E-02 PCM4.6124E-062.0156E-02 Area based CaRS PCoM5.5808E-048.3624E-02 PCM1.9817E-061.3403E-02

25. 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.

26. 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 - latitude Cubed- sphere grid :: top panel

27. 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)

28. 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) plane Second sweep – based on length measured on (λ,µ=sinθ) plane Field = Y 32 16, Reconstruction :: PCM N c = 33,N λ = 128, N µ =64