1. EGU General assembly 2014, AS 1.5 A three-dimensional Conservative Cascade semi-Lagrangian transport Scheme using the Reduced Grid on the sphere (CCS-RG) V. Shashkin 1,2 (vvshashkin@gmail.com), R. Fadeev 1, M. Tolstykh 1,2 April 29, 2014 1 - Institute of Numerical Mathematics, Russian Academy of Sciences 2 - Hydrometeorological centre of Russia

2. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Desirable features for transport schemes: (Rasch and Williamson, QJRMS, 1990 & Lauritzen et al., GMD, 2010) Accurate Transportive Local Invariant (mass etc.) conservation Monotonicity preserving Non-linear correlations preserving Computationally efficient No ideal scheme invented A lot of schemes! Let’s have a look at one more!

3. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Semi-Lagrangian method (in GCMs) To be … … or not to be? Stable for large CFL => large time-steps Large CFL => scalability problems (efforts to make scalable) Inherently multi-tracer efficient Non-conservative (mass, energy, enstrophy etc) Spurious orographic resonance … solved! The discussion is still open! Our believe: SL is ideal at least for relatively low-resolution simulations with relatively low (10 4 ) number of cores (Russian reality for future decade?)

4. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 CCS-RG basics: Mass-conservative SL (finite-volume SL) Integral formulation of transport equation: Air density Lagrangian air volume Tracer mass conservation provided no physical sources/sinks - Arrival volume = Grid cell - Departure volume Prognostic variable: Tracer density Time discretization : Tracer specific concentration

5. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 CCS-RG basics: 1D finite-volume SL PPM, Colella & Woodward, JCP, 1984 Subgrid reconstruction

6. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 CCS-RG: spatial approximation - Integral over departure volume Approximation of departure cell geometry O(Δx 2 ) Tracer density approximation O(Δx 3 ) 3D integral 3 x 1D integrals (remappings) (using cascade approach) (2D – Nair et al, MWR, 2002, 3D – Shashkin, HMC Proc, 2012)

7. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 CCS-RG Monotonicity Diagnostic filter (DF) Monotonicity violation Tracer mass Alternative option: Barth & Jespersen 1989 filter

8. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Reduced grid Regular lat-lon grid Meridian convergence Reduced grid Less points in latitude row near the poles

9. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Reduced grid: How to build it? Physical approach: keep longitudinal grid step constant (in length units) => works bad! Spectral approach: use asymptotic properties of associated Legendre polyn. => good for spectral models Interpolation accuracy approach (Fadeev, RCMMP, 2013) Given the fixed ration of central symmetric function interpolation errors on the regular and reduced grids minimize number of grid points: RMS Interpolation error Function center SL shallow water results with this rg design (Tolstykh, Shashkin, JCP, 2012)

10. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Reduced grid, structure reduced grid of 1 0 x1 0 resolution (at the equator) 15% less points 20% less points 25% less points 30 % less points … than in 1 0 x1 0 regular grid

11. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 DCMIP 1-1 testcase, Deformational flow (Kent et al, QJRMS, 2014) Tracer Q1. Cosine bells T=6 days (maximum deformation) T=0 days, T=12 days (initial distribution, exact solution) Tracer Q3. Slotted cylinder T=0 days, (vertical cross- section at 150 0 west) Initial distribution

12. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Deformational flow. Q1 No filter Diagnostic filter BJ filter

13. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Deformational flow. Q1 gridRegular 1 0 x1 0 x60 levs, time-step 1800sReduced 1 0 x1 0 (equator) x 60 levs, 30% less points filterl1l1 l2l2 l∞l∞ maxl1l1 l2l2 l∞l∞ No.159.132.275.078.167.134.275.078 DF.150.140.285 -0.015.155.143.285-0.014 BJ.223.177.305 -0.096.232.180.305-0.097 CAM-FV (from Kent et al.) MCore (from Kent et al.).121.0998.192.177.155.263 DF improves l 1 BJ is more diffusive than DF Reduced grid affect error norms slightly (in rotated test-variants too)

14. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Deformational flow. Q3 No filter Diagnostic filter B&J filter

15. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Deformational flow. Q3 No filter Diagnostic filter BJ filter Day 12 exact

16. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Deformational flow. Q3 gridRegular 1 0 x1 0 x60 levs, time-step 1800sReduced 1 0 x1 0 (equator) x 60 levs, 30% less points filterl1l1 l2l2 l∞l∞ maxl1l1 l2l2 l∞l∞ No.022.217.843.203.022.222.849.203 DF.026.263.817-0.145.026.265.835-0.145 BJ.029.281.827-0.245.029.282.850-0.249 CAM-FV (from Kent et al.) MCore (from Kent et al.) 0.0240.252.8590.0250.2350.844 DF improves l ∞ BJ is more diffusive than DF Reduced grid affect error norms slightly (in rotated test versions too)

17. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Deformational flow. Non-linear correlations Q1, Q2 No filter DF BJ real mixingrange pres unmixing overshooting UNLIM1.19e-32.78e-48.67e-4 DF1.24e-33.19e-40.00 BJ1.68e-31.68e-40.00 Т=6 days (maximum deformation) Correlation diagnostics (Lauritzen & Thuburn, QJRMS, 2012)

18. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 DCMIP test 1-2. Idealized Hadley cell No filter DF BJ

19. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 DCMIP test 1-2. Idealized Hadley cell CCS-RG UNLIMCCS-RG DF l1l1 l2l2 l∞l∞ maxl1l1 l2l2 l∞l∞ 2 0, 30 levs 0.16 0.358.68E-30.180.210.492.06E- 14 1 0, 60 levs 3.28E-24.05E-20.121.97E-34.14E-26.65E-20.234.75E- 14 0.5 0, 120 levs 4.72E-36.70E-32.54E-24.59E-57.15E-31.32E-26.29E-27.21E- 14 conv.2.542.281.892.322.011.48 CCS-RG BJMCore (from Kent et al.) 2 0, 30 levs 0.220.240.531.95E- 14 0.13680.16590.4214 1 0, 60 levs 6.44E-29.18E-20.303.73E- 14 0.02860.04620.1586 0.5 0, 120 levs 1.54E-22.56E-20.116.72E- 14 0.00630.01130.0435 conv.1.911.611.162.221.941.64

20. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Conclusions CCS-RG performs well and is competitive to CAM-FV and MCore (in terms DCMIP 1-x testcase diagnostics) Error norms grow only 5% when using reduced grid (maybe DCMIP case 1-1 even rotated is not a severe test for reduced grid desing) => reduced grid using isn’t limited by advection accuracy Two monotonic options are tested: Diagnostic filter is less diffusive and more accurate in terms of l 1, l 2, l ∞ error norms Diagnostic filter is better for species with rough distribution (hydrometeors etc) Barth & Jespersen filter is better when tracer correlation is important

21. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 Thank you for attention! More CCS-RG results (error norms, pictures) including DCMIP test 1-3 results can be found at: http://nwplab.inm.ras.ru/DCMIP-advResults-17.04.14.pdf

22. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 CCS-RG Monotonicity Barth & Jespersen filter 1989 (BJ filter) Scaling factor => No spurious max/min => => monotonic scheme

23. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 CCS-RG: tracer-mass coupling Barth & Jespersen filter: Unlimited or Diagnostic filter:

24. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 FV-SL scalability, overview SchemePublicationNum of coresCFL CCS-RGWork in progress?? CSLAM-HOMMEErath et al. Proc. Comp. Science., 2012 Lauritzen et al, JCP, 2010 4056 ( 16244 – high res ) < 1 * SPELT-HOMMEErath & Nair, JCP, 201316244< 1 * FARSIGHTWhite III & Dongarra, JCP, 201110000~ 10 * - CFL ~ 1 is still large from high-order Eulerian SE point of view

25. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 DCMIP test 1-3. Flow over orography Hybrid coordinates Sigma coordinates

26. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014 DCMIP test 1-3. Flow over orography Regular grid 1 0 x1 0 x60 levs. Dt = 3600 sec.