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EGU General assembly 2014, AS 1.5

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. Desirable features for transport schemes :

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EGU General assembly 2014, AS 1.5

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  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. Shashkin1,2(vvshashkin@gmail.com), R. Fadeev1, M. Tolstykh1,2 April 29, 2014

  2. 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! V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

  3. Semi-Lagrangian method (in GCMs) solved! The discussion is still open! Our believe: SL is ideal at least for relatively low-resolution simulations with relatively low (104) number of cores (Russian reality for future decade?) V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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

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

  6. CCS-RG: spatial approximation - Integral over departure volume • Approximation of departure cell geometry O(Δx2) • Tracer density approximationO(Δx3) 3D integral 3 x 1D integrals (remappings) (using cascade approach) (2D – Nair et al, MWR, 2002, 3D – Shashkin, HMC Proc, 2012) V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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

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

  9. 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 SL shallow water results with this rg design (Tolstykh, Shashkin, JCP, 2012) Function center V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

  10. Reduced grid, structure reduced grid of 10x10 resolution (at the equator) 15% less points 25% less points 30 % less points 20% less points … than in 10x10 regular grid V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

  11. DCMIP 1-1 testcase, Deformational flow (Kent et al, QJRMS, 2014) Tracer Q1. Cosine bells T=0 days, T=12 days (initial distribution, exact solution) T=6 days (maximum deformation) Tracer Q3. Slotted cylinder T=0 days, (vertical cross-section at 1500 west) Initial distribution V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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

  13. Deformational flow. Q1 • DF improves l1 • BJ is more diffusive than DF • Reduced grid affect error norms slightly (in rotated test-variants too) V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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

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

  16. Deformational flow. Q3 • DF improves l∞ • BJ is more diffusive than DF • Reduced grid affect error norms slightly (in rotated test versions too) V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

  17. Deformational flow. Non-linear correlations Q1, Q2 No filter DF Т=6 days (maximum deformation) Correlation diagnostics (Lauritzen & Thuburn, QJRMS, 2012) BJ V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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

  19. DCMIP test 1-2. Idealized Hadley cell V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

  20. 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 l1, l2, l∞error norms • Diagnostic filter is better for species with rough distribution (hydrometeors etc) • Barth & Jespersen filter is better when tracer correlation is important V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

  21. 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 V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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

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

  24. FV-SL scalability, overview * - CFL ~ 1 is still large from high-order Eulerian SE point of view V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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

  26. DCMIP test 1-3. Flow over orography Regular grid 10x10x60 levs. Dt = 3600 sec. V. Shashkin et al. CCS-RG, EGU General Assembly, AS 1.5 April 29, 2014

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