A cell-integrated semi-Lagrangian semi-implicit integration scheme: Basic hydrostatic formulations and preliminary tes

1. A cell-integrated semi-Lagrangian semi-implicit integration scheme: Basic hydrostatic formulations and preliminary tests in the HIRLAM system. Eigil Kaas, University of Copenhagen (from 1st April). Peter Hjort Lauritzen, NCAR, Boulder CO. Karina Lindberg and Bennert Machenhauer, DMI

2. LAURITZEN, P.H., KAAS, E., MACHENHAUER, B. (2006). A mass-conservative semi-implicit, semi-Lagrangian limited-area shallow water model on the sphere. Mon. Wea. Rev., 134 (4), 1196–1212. LAURITZEN, P.H., LINDBERG, K., KAAS, K., MACHENHAUER, B. (2006b). A locally massconservative Semi-Lagrangian, Semi-implicit, Limited-Area Gridpoint Model of the Primitive Equations. Part I: Forecast Model. Mon. Wea. Rev., in prep.

3. To show up in ”Handbook on Numerical Analysis”. Elsevier

4. Motivations for development of the new scheme: • Inherent local mass conservation • Avoid CFL criterion • High order of accuracy and “shape conservation” • More consistent estimate of the vertical velocity and the energy conversion term • Applications: • Improved synoptic scale weather forecasting • Consistent/improved transport of multiple tracers incl. cloud constituents – i.e. avoid the wind-mass inconsistency in off-line tracer models. • General applications in fully integrated meteorological / chemistry models

5. Spatial discretisation Spatial discretisation Advective formulation of the continuity equation giving ( is an infinitesimal Lagrangian volume) with and a non-infinitesimal Lagrangian volume Flux form formulation of the continuity equation with

6. The exact update of the cell mean density : Time and space discretisation

7. DCISL (Departure Cell-Integrated Semi-Lagrangian scheme)

8. Different variants of CISL or finite volume schemes: • Traditional operator splitted flux based schemes (e.g. Bott (1989, 1992)) • Conservative operator splitted schemes also conserving a constant in a non-divergent flow (e.g. Lin and Rood (1996), Lin (2004), Leonard (1996), Adcroft et al. (2005)) • Non-split flux based schemes (e.g. Holm (1995)) • Lagrangian type (DCISL) schemes (Rančić, (1990), Machenhauer and Olk (1997), Nair and Machenhauer (2002), Lauritzen et al. (2006), Zerroukat et al. (2002). • The CFL problem – or not !

9. DCISL (Nair and Machenhauer (2002)

11. where Equations of motion. Primitive equations with standard η(p,ps) vertical coordinate

12. Lagrangian form divergence DCISL form of continuity equation in the η coordinate system: (Machenhauer, 1992) DCISL form of moisture (or tracer) continuity equation

13. Continuity equation, vertical winds, trajectories and vertical remappings Horizontal accelerations taken into account in estimation of horizontal trajectories

14. Staggered hybryd sigma – p vertical coordinate (SIMMONS and BURRIDGE [1981])

15. The traditional two-time level SL-scheme: ~ indicates time extrapolation from n and n-1 Semi-implicit formulation

16. The semi-implicit formulation of DCISL (1) CISL explicit forecast: ”Ideal” semi-implicit CISL forecast: Elliptic equation too complicated !

17. The basic explicit forecast Inconsistent implicit correction term Correction of the inconsistency in the previous time step. is the area-divergence based on the arrival winds in the previous time step. The semi-implicit formulation of DCISL (2) The most direct approach: Inconsistent !

18. Various options in the new dynamical core Consistent omega in thermodynamic equation (COP) Accuracy of horizontal trajectories and extrapolation filters Application of filters on upstream representation (monotonicity, positive definiteness). Cascade interpolation as e.g. in Nair et al. (2002) (CASC) Modification of vertical diffusion (VDIFF). General tuning of physics.

19. Case study: 18 hour forecast valid on 3th December 1999 (extratropical storm over Denmark)

20. Observed

21. HIRLAM

22. CISL

23. CISL - CO

24. CISL - COP - CASC

25. CISL - COP - CASC, 0.1*vdiff on all variables

26. CISL - COP - CASC, 0.1*vdiff on T and winds

27. CISL - COP - CASC, 0.1*vdiff on q only

28. HIRLAM

29. CISL

30. CISL - COP - CASC

31. CISL - COP - CASC, 0.1*vdiff on all variables

32. CISL - COP - CASC, 0.1*vdiff on T and winds

33. CISL - COP - CASC, 0.1*vdiff on q only

34. A new hydrostatic dynamical core of HIRLAM has been developed: • High order of accuracy. • It is possible to apply possitive definite and monotonic filters. • The departure area is defined with high accuracy, i.e. high degree of local mass conservation • It is generally possible to use long time steps, i.e. no CFL criterion • Non-hydrostatic version should be possible to formulate • Tuning of vertical diffusion needed (the scheme seems to have very little inherent numerical dispersion) • Disadvantages of DCISL schemes • The degree of conservation of a constant field in a non-divergent flow depends critically on the accuracy of the calculation of trajectories. • To do: • Tuning of vertical diffusion + optimise the code + test the different options. • Formulate conservation of enstrophy and total energy