Tropically or locally fine triangular grids on a sphere based on conformal projections

1. Tropically or locally fine triangular grids on a sphere based on conformal projections Aug. 24, 2010 Shin-ichi Iga NICAM group, JAMSTEC, Japan

2. Introduction • Global simulation: • quasi-uniform (icosahedral, cubic, Yin-Yang, … etc) • Most convection occurs in the tropics. Is quasi-uniform most cost-effective? • We proposed tropically enhancing as an alternative method. • Regional simulation: • Grid nesting  noise around the boundary • Grid stretching • cubic, octagonal  corner problem • Stretched icosahedral • In this study, the other type of stretched triangular mesh is proposed. ( Iga 2010, submitted to MWR)

3. Newly proposed icosahedral grid different topology quasi-uniform Non-uniform resolution Stretched (Tomita2008)

4. Benefit of the tropically fine mesh Fine mesh in convective region High resolution is needed to represent cloud convection. Most deep convections occurs in the tropics. The new grid represents them cost-effectively.

5. Benefit of the locally fine mesh • Uniform resolution inside the target region. • No wave-reflection problem which is seen in nesting grid. Non-uniform uniform Resolution varies continuously New grid Stretched Icosahedral (Tomita2008)

6. PSP(polar stereo) １、two hexagon are projected on two Polar Stereographic map. They can be connected at an arbitrary latitudes. ( above example is 50S). 2, same alphabets are connected 3, smooth using spring grid of Tomita et al(2001） Equatorial resolution is twice Than that of pole

7. PSP (regional model) Connection latitude is 60N Quasi uniform 1, use PSP. 2, connected latitude is 60N ・half of grid points are inside the target region (red circle)

8. LCP (Lambert’s conformal conic) 1, use 2 Lambert conformal conic map. 2, angle of the map can be > 360 3, almost same to PSP Larger resolution contrast

9. MLCP (Mercator + LCP) LCP(low latitudes) + Mercator (high latitudes) Tropical belt of uniform resolution

10. Tropically fine cases resolution Normalized by maximum We can generate various kind of Resolution distribution Regionally fine cases S(1/3) is NICAMStretch Model

11. Topology • Hereafter, GPn: grid point which shares n lines • Euler’s formula: Vertex – Edge + Face = 2 • For the triangular grid: ∑n (6-n)Vn = 12 Where Vn is number of GPn (grid points which shares n lines). • Icosahedral mesh have 12 GP5

12. Relocate twelve GP5 Icosahedral grid New grid (locally fine) Located Around The Target region 12 GP5 are uniformly distributed GP5 Stretched icosahedral New grid (PSP) Located Around The equator Located Inside and Outside The Target region

13. 24 GP5 and 2 GP12 GP12 GP5 Quasi uniform Locate GP5 around high resolution area Locate GP(n>6) at or around low resolution area.

14. Shallow-water experiment • The new grid is implemented on NICAM Shallow-water model (Tomita et al 2001) • Williamson et al.(1992) test case 1 and 2are performed. • Testcase 1: cosine-bell type passive tracer on solid body rotation. 12days. • Test case 2: solid body rotation. 5days • We will show only grid cases (MLCP, PSP-region)

15. Time series MLCP12 pole-passing case Dx_min=53km t=0 t=3day t=4day t=12day Grid interval around poles is near the size of cosbel. But original bell shape is remain

16. Resolution dependency 1 Initial MLCP12 equator passing case After 12 days Dx_min=53km Dx_min=108km Dx_min=212km MLCP after 12days As resolution increase, original cosbel shape kept

17. Resolution dependency 2 PSP-region dx=102km PSP-region dx=51km PSP regional model. After 12 days. As resolution increases, error decreases PSP-region dx=25km Icosahedral-stretched dx=27km

18. error case1 PSP regional model Pole massing MLCP equator passing MLCP pole passing They are reasonable

19. error case2 They are reasonable

20. summary • Tropically or locally fine triangular grids are proposed using a new topology. • Shallow water tests were performed on NICAM Future works: • Now we are implementing the new grid to NICAM-3d model (maybe soon).