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Further Developments of the Runge-Kutta Time Integration Scheme

Further Developments of the Runge-Kutta Time Integration Scheme Investigation of Convergence (task 5). Gabriella Ceci, Pier Luigi Vitagliano g.ceci@cira.it , p.vitagliano@cira.it. OUTLINE. OBJECTIVES AND MOTIVATIONS WORK PLAN TEST CASES DESCRIPTION

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Further Developments of the Runge-Kutta Time Integration Scheme

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  1. Further Developments of the Runge-Kutta Time Integration Scheme Investigation of Convergence (task 5) Gabriella Ceci, Pier Luigi Vitagliano g.ceci@cira.it, p.vitagliano@cira.it COSMO WG 2 - Runge Kutta

  2. OUTLINE • OBJECTIVES AND MOTIVATIONS • WORK PLAN • TEST CASES DESCRIPTION • NEW RESULTS 2D: CONSTANT TIME STEP, NON-TVD RK3 • 3D TEST CASE: effect of different spatial scheme (3th vs 5th order) • 3D HYDROSTATIC AND NON HYDROSTATIC MOUNTAIN FLOW • EFFECT OF MOISTURE ON MOUNTAIN FLOW • CONCLUSIONS COSMO WG 2 - Runge Kutta

  3. OBJECTIVES AND MOTIVATIONS OBJECTIVES • TEST OF 3 STAGES RUNGE KUTTA TVD SCHEME WITH 5th ORDER UPWIND ADVECTION • TEST OF NEW DYNAMICS with P' and T' • MOTIVATIONS • ALLOWS LARGER TIME STEPS • MORE ACCURATE • FASTER • CONVERGENCE PROPERTIES IN PRACTICAL APPLICATIONS UNKNOWN COSMO WG 2 - Runge Kutta

  4. WORK PLAN TEST CASES: 2D MOUNTAIN FLOWS WITHOUT PHYSICS 3D MOUNTAIN FLOWS WITHOUT PHYSICS 3D MOUNTAIN FLOW WITH MOISTURE COSMO WG 2 - Runge Kutta

  5. TEST CASES DESCRIPTION • Gaussian ridge h(r)=H 2-(r/a)2 • HYDROSTATIC FLOW (aN/U) >> 1 • NON HYDROSTATIC FLOW (aN/U) ~ 1 • NON LINEAR FLOW (HN/U) ~ 1 r = (x2 + y2)½ • Basic flow velocity U = 10 m/s • Brunt Väisälä frequency N = 0.01 s-1 • Rayleigh damping layer above 11 km • Vertical resolution 100 m (195 levels) COSMO WG 2 - Runge Kutta

  6. TEST CASES DESCRIPTION HYDROSTATIC LINEAR / NON LINEAR a = 10 km H = 10 m / 500 m Time = 60 h / 100 h dt = 2.5” Domain size 500x19.5 km2 Horizontal resolution = 4km, 2km, 1km, 500m, 250m, 125m NON HYDROSTATIC a = 500 m H = 10 m Time = 10 h dt = 2.5” Domain size 250x19.5 km2 Horizontal resolution = 1km, 500m, 250m, 125m, 62.5m COSMO WG 2 - Runge Kutta

  7. Comparison with analytical solutionlinear hydrostatic Left: solution with a damping layer of 85 levels and nRΔt=200. Right: analytical solution following Klemp-Lilly (J.Atmos.Sc. 35, 78-107, 1978) COSMO WG 2 - Runge Kutta

  8. ISSUES WITH LATERAL BOUNDARIES Disturbances at the side boundaries due to p’ T’ (left), removed by initialization of reference atmosphere p0 T0 with constant Brunt-Väisälä frequency N (right) COSMO WG 2 - Runge Kutta

  9. ISSUES WITH UPPER DAMPING LAYER Fine tuning of damping layer (both thichness and amount of damping) required to minimize wave reflection and distorsion. COSMO WG 2 - Runge Kutta

  10. NON HYDROSTATIC FLOW: w AND u COSMO WG 2 - Runge Kutta

  11. NON LINEAR HYDROSTATIC FLOW VERY DEEP RAYLEIGH DAMPING LAYER IS REQUIRED TO OBTAIN REASONABLE SOLUTIONS FOR HIGHER RIDGES (LEFT: 1.35 WAVE LENGTHS, RIGHT: 2 W.L.) COSMO WG 2 - Runge Kutta

  12. TIME CONVERGENCE STEADY FLOW IS NOT OBTAINED WHEN THE RIDGE IS HIGHER THAN 500m COSMO WG 2 - Runge Kutta

  13. POST PROCESSING • DRAG COEFFICIENT CD=∑ p'(x,0) dh/dx ∆x / PR • MOMENTUM FLUX Mx(z)=- ρ(z) ∑ u(x,z) w(x,z) ∆x / PR • KINETIC ENERGY = (u'(x,z)2 + w'(x,z)2) • ABSOLUTE ERROR |F - Fexact| • RELATIVE ERROR |F - Ffinest mesh| • ERROR NORM L0 max |F - Ffinest mesh| • ERROR NORM L1 1/N ∑ |F - Ffinest mesh| • ERROR NORM L2 [1/N ∑ (F - Ffinest mesh)2]½ COSMO WG 2 - Runge Kutta

  14. OLD RESULTS: CD COSMO WG 2 - Runge Kutta

  15. OLD RESULTS: KINETIC ENERGY COSMO WG 2 - Runge Kutta

  16. OLD RESULTS: MOMENTUM FLUX Smaller DX COSMO WG 2 - Runge Kutta

  17. NEW RESULTS • ALL TEST CASES RUNNED AGAIN WITH CONSTANT TIME STEP = 2.5” • TEST CASES REPEATED WITH NON-TVD 3 STAGES RUNGE KUTTA COSMO WG 2 - Runge Kutta

  18. CONVERGENCE OF VERTICAL VELOCITY w COSMO WG 2 - Runge Kutta

  19. CONVERGENCE OF VERTICAL VELOCITY w COSMO WG 2 - Runge Kutta

  20. CONVERGENCE OF VERTICAL VELOCITY w COSMO WG 2 - Runge Kutta

  21. CONVERGENCE OF KINETIC ENERGY COSMO WG 2 - Runge Kutta

  22. CONVERGENCE OF KINETIC ENERGY COSMO WG 2 - Runge Kutta

  23. CONVERGENCE OF KINETIC ENERGY COSMO WG 2 - Runge Kutta

  24. CONVERGENCE OF WAVE DRAG COSMO WG 2 - Runge Kutta

  25. CONVERGENCE OF WAVE DRAG COSMO WG 2 - Runge Kutta

  26. CONVERGENCE OF WAVE DRAG COSMO WG 2 - Runge Kutta

  27. COMPARISON WITH ANALYTICAL SOLUTIONS COSMO WG 2 - Runge Kutta

  28. CONVERGENCE: CONCLUSIONS AFTER 2D TESTS • 2nd ORDER SPATIAL CONVENGENCE (FAST WAVE SCHEME DOMINATES) • TVD AND NON-TVD 3 STAGES RUNGE KUTTA SHOW SIMILAR BEHAVIOUR • TIME STEP HAS MINOR EFFECT (IF ANY) ON SPATIAL CONVERGENCE • IMPORTANT ISSUES WITH UPPER BOUNDARY CONDITION • ISSUE IN LATERAL BOUNDARY CONDITIONS FOR p’ T’ • DIFFICOULT TO COMPARE WITH ANALYTICAL SOLUTIONS, DUE TO B.C. COSMO WG 2 - Runge Kutta

  29. 3D TEST CASES Gaussian mountain, hydrostatic flow, dry atmosphere effect of different spatial scheme (3th vs 5th order) and grid size Domain size: 256x128x19.5 km3 195 vertical levels Rayleigh damping above 11 km Basic flow velocity U = 10 m/s Brunt Väisälä frequency N = 0.01 s-1 COSMO WG 2 - Runge Kutta

  30. 3D TEST CASES COSMO WG 2 - Runge Kutta

  31. 3D TEST CASES COSMO WG 2 - Runge Kutta

  32. 3D TEST CASES COSMO WG 2 - Runge Kutta

  33. 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 16 km 3th order upwind 5th order upwind COSMO WG 2 - Runge Kutta

  34. 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 8 km 3th order upwind 5th order upwind COSMO WG 2 - Runge Kutta

  35. 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 4 km 3th order upwind 5th order upwind COSMO WG 2 - Runge Kutta

  36. 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 16 km 3th order upwind 5th order upwind COSMO WG 2 - Runge Kutta

  37. 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 8 km 3th order upwind 5th order upwind COSMO WG 2 - Runge Kutta

  38. 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 4 km 3th order upwind 5th order upwind COSMO WG 2 - Runge Kutta

  39. 3D TEST CASES • SOME CONCLUSIONS • SMALLER INFLUENCE OF DAMPING LAYER ON 3D MOUNTAIN WAVES AND DRAG • OPTIMAL DAMPING PARAMETER Dt*nrdtau INCREASES TO 1000 s • WITH POOR RESOLUTION DIFFERENT SCHEME CAN GIVE DIFFERENT SOLUTIONS • WITH POOR RESOLUTION HIGHER ORDER UPWIND CAN IMPROVE RESULTS COSMO WG 2 - Runge Kutta

  40. 3D TEST CASES: NON HYDROSTATIC FLOW Convergence analysis COSMO WG 2 - Runge Kutta

  41. 3D TEST CASES: NON HYDROSTATIC FLOW Convergence analysis Smaller DX 2 D 3 D COSMO WG 2 - Runge Kutta

  42. 3D TEST CASES: NON HYDROSTATIC FLOW Convergence analysis 2 D 3 D COSMO WG 2 - Runge Kutta

  43. 3D TEST CASES: EFFECT OF MOISTURE SIMILAR TEST CASE IN 2D SHOWN BY Durran-Klemp (J.Atmos.Sc. 39, 2490-2506, 1982) WITH 3D SIMULATION LESS INFLUENCE OF BOUNDARIES • STEADY SOLUTION NOT ACHIEVED EVEN ON H=10m • TEST ON H=300m RH=100% SHOWS INSTABLE LOWER LAYER COSMO WG 2 - Runge Kutta

  44. 3D TEST CASES: EFFECT OF MOISTURE • FURTHER WORK (?) • MOUNTAIN HEIGHT • TIME STEP • SPATIAL STEP • B.C. COSMO WG 2 - Runge Kutta

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