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Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta1 Further Developments of the Runge-Kutta Time Integration Scheme Investigation of Convergence (task.

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Presentation on theme: "Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta1 Further Developments of the Runge-Kutta Time Integration Scheme Investigation of Convergence (task."— Presentation transcript:

1 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta1 Further Developments of the Runge-Kutta Time Integration Scheme Investigation of Convergence (task 5) Gabriella Ceci, Pier Luigi Vitagliano g.ceci@cira.itg.ceci@cira.it, p.vitagliano@cira.itp.vitagliano@cira.it

2 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta2 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

3 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta3 OBJECTIVES AND MOTIVATIONS MOTIVATIONS ALLOWS LARGER TIME STEPS MORE ACCURATE FASTER CONVERGENCE PROPERTIES IN PRACTICAL APPLICATIONS UNKNOWN OBJECTIVES TEST OF 3 STAGES RUNGE KUTTA TVD SCHEME WITH 5 th ORDER UPWIND ADVECTION TEST OF NEW DYNAMICS with P' and T'

4 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta4 TEST CASES: 2D MOUNTAIN FLOWS WITHOUT PHYSICS 3D MOUNTAIN FLOWS WITHOUT PHYSICS 3D MOUNTAIN FLOW WITH MOISTURE WORK PLAN

5 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta5 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 Basic flow velocityU = 10 m/s Brunt Väisälä frequencyN = 0.01 s -1 Rayleigh damping layer above 11 km Vertical resolution100 m (195 levels) r = (x 2 + y 2 ) ½

6 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta6 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 km 2 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 km 2 Horizontal resolution = 1km, 500m, 250m, 125m, 62.5m

7 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta7 Comparison with analytical solution linear hydrostatic Left: solution with a damping layer of 85 levels and n R Δt=200. Right: analytical solution following Klemp-Lilly ( J.Atmos.Sc. 35, 78-107, 1978 )

8 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta8 ISSUES WITH LATERAL BOUNDARIES Disturbances at the side boundaries due to p’ T’ (left), removed by initialization of reference atmosphere p 0 T 0 with constant Brunt-Väisälä frequency N (right)

9 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta9 ISSUES WITH UPPER DAMPING LAYER Fine tuning of damping layer (both thichness and amount of damping) required to minimize wave reflection and distorsion.

10 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta10 NON HYDROSTATIC FLOW: w AND u

11 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta11 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.)

12 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta12 TIME CONVERGENCE STEADY FLOW IS NOT OBTAINED WHEN THE RIDGE IS HIGHER THAN 500m

13 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta13 POST PROCESSING DRAG COEFFICIENT CD=∑ p'(x,0) dh/dx ∆x / P R MOMENTUM FLUXMx(z)=- ρ(z) ∑ u(x,z) w(x,z) ∆x / P R KINETIC ENERGY  = (u'(x,z) 2 + w'(x,z) 2 ) ABSOLUTE ERROR|  -  exact | RELATIVE ERROR|  -  finest mesh | ERROR NORM L0max |  -  finest mesh | ERROR NORM L11/N ∑ |  -  finest mesh | ERROR NORM L2[1/N ∑ (  -  finest mesh ) 2 ] ½

14 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta14 OLD RESULTS: CD

15 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta15 OLD RESULTS: KINETIC ENERGY

16 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta16 OLD RESULTS: MOMENTUM FLUX Smaller DX

17 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta17 NEW RESULTS ALL TEST CASES RUNNED AGAIN WITH CONSTANT TIME STEP = 2.5” TEST CASES REPEATED WITH NON-TVD 3 STAGES RUNGE KUTTA

18 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta18 CONVERGENCE OF VERTICAL VELOCITY w

19 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta19 CONVERGENCE OF VERTICAL VELOCITY w

20 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta20 CONVERGENCE OF VERTICAL VELOCITY w

21 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta21 CONVERGENCE OF KINETIC ENERGY

22 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta22 CONVERGENCE OF KINETIC ENERGY

23 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta23 CONVERGENCE OF KINETIC ENERGY

24 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta24 CONVERGENCE OF WAVE DRAG

25 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta25 CONVERGENCE OF WAVE DRAG

26 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta26 CONVERGENCE OF WAVE DRAG

27 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta27 COMPARISON WITH ANALYTICAL SOLUTIONS

28 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta28 CONVERGENCE: CONCLUSIONS AFTER 2D TESTS 2 nd 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.

29 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta29 3D TEST CASES Gaussian mountain, hydrostatic flow, dry atmosphere effect of different spatial scheme (3 th vs 5 th order) and grid size Domain size: 256x128x19.5 km 3 195 vertical levels Rayleigh damping above 11 km Basic flow velocity U = 10 m/s Brunt Väisälä frequencyN = 0.01 s-1

30 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta30 3D TEST CASES

31 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta31 3D TEST CASES

32 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta32 3D TEST CASES

33 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta33 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 16 km 3 th order upwind 5 th order upwind

34 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta34 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 8 km 3 th order upwind 5 th order upwind

35 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta35 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 4 km 3 th order upwind5 th order upwind

36 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta36 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 16 km 3 th order upwind 5 th order upwind

37 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta37 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 8 km 3 th order upwind 5 th order upwind

38 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta38 3D TEST CASES: HYDROSTATIC FLOW Gaussian mountain height=750 m size=10 km Horizontal resolution 4 km 3 th order upwind 5 th order upwind

39 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta39 3D TEST CASES SOME CONCLUSIONS SMALLER INFLUENCE OF DAMPING LAYER ON 3D MOUNTAIN WAVES AND DRAG OPTIMAL DAMPING PARAMETER  t*nrdtau INCREASES TO 1000 s WITH POOR RESOLUTION DIFFERENT SCHEME CAN GIVE DIFFERENT SOLUTIONS WITH POOR RESOLUTION HIGHER ORDER UPWIND CAN IMPROVE RESULTS

40 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta40 3D TEST CASES: NON HYDROSTATIC FLOW Convergence analysis

41 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta41 3D TEST CASES: NON HYDROSTATIC FLOW Convergence analysis Smaller DX 3 D2 D

42 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta42 3D TEST CASES: NON HYDROSTATIC FLOW Convergence analysis 2 D 3 D

43 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta43 3D TEST CASES: EFFECT OF MOISTURE STEADY SOLUTION NOT ACHIEVED EVEN ON H=10m TEST ON H=300m RH=100% SHOWS INSTABLE LOWER LAYER SIMILAR TEST CASE IN 2D SHOWN BY Durran-Klemp ( J.Atmos.Sc. 39, 2490-2506, 1982 ) WITH 3D SIMULATION LESS INFLUENCE OF BOUNDARIES

44 Krakow - September, 15th 2008COSMO WG 2 - Runge Kutta44 3D TEST CASES: EFFECT OF MOISTURE FURTHER WORK (?) MOUNTAIN HEIGHT TIME STEP SPATIAL STEP B.C.

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