1. A cell-integrated semi-Lagrangian dynamical scheme based on a step-function representation Eigil Kaas, Bennert Machenhauer and Peter Hjort Lauritzen Danish Meteorological Institute Lyngbyvej 100, DK-2100 Copenhagen, Denmark SRNWP-NT mini workshop in Toulouse 12-13 December 2002

2. The goals To construct a dynamical scheme for atmospheric dynamics and tracer transport with all the following properties: Indefinite order of accuracy for advection by a flow that is constant in time an space (except for initial truncation). Full local mass conservation. Positive definite. Monotonic. Numerically effective. Our solution: SF-CISL Step-Function Cell Integrated Semi-Lagragian scheme combined with a semi-implicit scheme for inertia- gravity wave terms.

3. OUTLINE The basic idea behind step function advection. The basic idea behind CISL. 2-D passive test simulations. A new semi-implicit formulation of CISL for the shallow water equations. Tests simulations. Discussion: efficiency, generalisation to 3-D and to spherical geometry …

4. What is CISL ? Nair and Machenhauer ( 2002 ) Integrate the continuity equation over a time dependent Lagrangian volume:

5. What is CISL ? Nair and Machenhauer ( 2002 )

6. The basic idea behind step-function advection Time x-direction i i+1 i-1

7. Spatial truncation (horizontal diffusion)

8. 2-D step function representation

9. Order of calculations 7. Calculate for each departure cell (=(result of 4.)/ ) 6. Calculate for each ”north-south” intersect 4. Perform the cell integration (result = ) 5. Calculate for each departure grid cell point corner point 8. Do the ”horizontal diffusion” (i.e. modify if needed) 2. Calculate the departure grid cell corner points. 1. Calculate dx i,j and dy i,j for each Eulerian grid cell. 3. Calculate the ‘s and define the re-mappings.

10. 10. Calculate the relative areal change for each step function Order of calculations (cont.) 9. Calculate new values of dx i,j and dy i,j based on upstream values of, and. 12. Calculate the final values of and from the values of dx i,j, dy i,j and. 11. Use this information to calculate the final value of

11. 2-D passive test simulations Solid body rotation, 6 rotations, 96 time steps per rotation 100 x 100 grid points/cells

12. A new semi-implicit formulation of CISL for the shallow water equations. depth of fluid height of topography velocity components

13. A new semi-implicit 2-D CISL formulation (1) ~ indicates time extrapolation from n and n-1 The traditional two-time level SL-scheme: CISL explicit forecast: Ideal semi-implicit CISL forecast: Elliptic equation too complicated !

14. A new semi-implicit 2-D CISL formulation (2) The basic explicit forecast Inconsistent implicit correction term Correction of the inconsistency in the previous time step

15. Tests of the semi-implicit SF-CISL in a shallow water channel model 20000 km

16. Spectral Eulerian model: 3 time level spectral transform scheme (double Fourier series) Semi-implicit formulation (Coriolis explicit) Reasonable implicit horizontal diffusion Eulerian grid-point model: 3 time level centered difference scheme Semi-implicit formulation (Coriolis explicit) Reasonable explicit horizontal diffusion Interpolating semi-Lagrangian (IPSL) model: 2 time level scheme based on bi-cubic interpolation Semi-implicit formulation (Coriolis implicit) No additional horizontal diffusion SF-CISL model: 2 time level scheme based on step-function representation Semi-implicit formulation (Coriolis implicit) No additional horizontal diffusion Four different model formulatoins

17. Spectral Eulerian model Eulerian grid-point model Interpol. semi Lagrangian model SF-CISL model 48 hour ”forecasts” at low resolution. Parameter: height field

18. Spectral Eulerian model Eulerian grid-point model Interpol. semi Lagrangian model SF-CISL model 48 hour ”forecasts” at high resolution. Parameter: height field

19. Spectral Eulerian model Eulerian grid-point model Interpol. semi Lagrangian model SF-CISL model 48 hour ”forecasts” at low resolution. Parameter: passive tracer

20. Spectral Eulerian model Eulerian grid-point model Interpol. semi Lagrangian model SF-CISL model 48 hour ”forecasts” at high resolution. Parameter: passive tracer

21. Interpol. semi Lagrangian model SF-CISL model 10 day ”forecasts” at high resolution. Parameter: passive tracer

22. Discussion and conclusion Cost Passive advection 1.7 times IPSL. Truncation/horizontal diffusion This is a critical point Memory consumption When step function geometries are defined from the total mass field they could in principle be used for all prognostic variables (i.e. only one prognostic variable per tracer variable) The passive advection tests in realistic flow demonstrate the monotonicity, mass conservation and positive definiteness The shallow-model works with the new scheme ! No noise due to step functions ”Bad” ”Good”

23. Discussion and conclusion Other possible formulations “horizontal diffusion/truncation” Choice of step-functions. Generalisation to 3-D Cascade interpolation (Nair et al. 1999) for the vertical problem. Prognostic variables: 3-D cell averages, horizontal averages at model levels, vertical averages at grid points, grid point values. Spherical geometry No real problem (reduced lat-lon (or Gaussian) grid).