1. Numerical Weather Prediction on Linux-clusters – Operational and research aspects Nils Gustafsson SMHI

2. Weather Prediction as a problem in mechanics and physics (V. Bjerknes,1904)  7 differential equations: 3 momentum equations, the thermodynamic equation, the equations for conservation of mass and moisture and the equation of state for gases.  7 model state variables: 3 velocity components, pressure, temperature, moisture and density  Bjerknes suggested that these differential equations and these variables could be used for forecast the future development of the weather  Bjerknes also stated that the calculation problem was impossible to solve

3. Historical development of Numerical Weather Prediction (NWP) 1904: The idea of V. Bjerknes 1922: L.F. Richardson makes the first trial to integrate a 2D problem by manual calculations 1950: First NWP calculation with a quasi-geostrophic 2D model on a computer by Charney et al. 1950’s: First operational NWP on the BESK computer in Sweden 1960’s: Quasi-geostrophic multi-level models 1970’s: Primitive equation models (hydrostatic equations), development of physical parameterizations 1980’s: Global models 1990 - : Advanced data assimilation techniques, use of satellite data, ensemble prediction systems, non-hydrostatic model (back to Bjerknes non-approximated equations)

4. Popular numerical techniques for NWP Finite difference approximations, finite element approximations, spectral transform techniques Semi-implicit techniques: treat fast processes with implicit time integration Semi-Lagrangian techniques: Move air packages along backward trajectories that end in gridpoints every time step. Use accurate spatial interpolation at the starting point of the trajectories.

5. NWP in Europe today Medium range (2-10 days) forecasting is handled by ECMWF (European Centre for Medium Range Forecasting) Short range forecasting is handled by national weather services Collaboration in 4 groups for development of short range forecasting

7. The international HIRLAM project (HIgh Resolution Limited Area Model) Started in 1985 Collaboration between the Nordic countries, Spain, Netherlands, Ireland and France Development of a complete system for Numerical Weather prediction: Model and Data Assimilation Present emphasis on data assimilation and on high resolution model (1-3 km)

8. Opertional HIRLAM areas at SMHI, 22 and 44 km, 40 levels

9. HIRLAM data assimilation.....

10. Basic data assimilation problems Degrees of freedom of the model >> number of observation Balances between state variables in the atmosphere are important Forecast errors have preferred spatial scales Fast growth of small-scale perturbations (non-linear instabilities) Model have multiple time-scales Observations are made irregularly in space and time Many different quantities are observed Random and systematic observation errors

11. Basic idea of 4D-Var

12. 4D-Var cost function Basic non-incremental formulation: X = Model state to be determined by minimization of J X b = Background model state (e.g. 6 h forecast) B = Covariance matrix (background state error) y = observations H = observation operator (model state to observed variables) R = Covariance matrix (observation errors) In case observations are distributed over a time interval, H includes forward integration of the atmospheric model and backward integration of an adjoint of the atmospheric model.

14. Successful operational 4D-Var implementations ECMWF, global model, 12 h assimilation window, incremental formulation, (3D-Var and) 4D-Var has made it possible to introduce a wide range of satellite observations operationally, ECMWF is at present 12 h better than any other NWP center at all forecast ranges Japan Meteorological Agency, mesoscale incremental 4D- Var, emphasis on improving short range precipitation forecasting, use of radar and GPS observations

15. HIRLAM 3D-Var and 4D-Var developments 1995-1997: Tangent linear and adjoint of the Eulerian spectral adiabatic HIRLAM. Sensitivity experiments. ”Poor man´s 4D- Var”. 1997-1998: Tangent linear and adjoints of the full HIRLAM physics. 1996-2000: Development of HIRLAM 3D-Var: Background error constraint and observation processing. 2000: First experiments with ”non-incremental” 4D-Var. 2001-2002: Incremental 4D-Var. Simplified physics packages (Buizza vertical diffusion and Meteo France package). 2002: 4D-Var feasibility study. 2003: Semi-Lagrangian scheme (SETLS), outer loops (spectral or gridpoint HIRLAM) and multi-incremenal minimization.

16. HIRLAM 3D-Var and 4D-Var area extension  FFT’s are used in the spectral tangent-linear and adjoint models and in the back-ground constraint (utilizing the assumption of homogeneity with respect to horizontal correlations).  Double periodic variations through area extension

17. HIRLAM 3D-VAR – background error formulation Transform the model state increment vector in such a way that the corresponding transform of a model forecast error state vector could be assumed to have a covariance matrix equal to the identity matrix. The following series of transforms are applied in the reference HIRLAM 3D-VAR: 1.Normalize with forecast error standard deviations 2.Horizontal spectral transforms (using an extension zone technique) 3.Reducing dependencies between the mass field and the wind field increments by subtracting geostrophic wind increments from the full wind increments 4.Project on eigen-vectors of vertical correlation matrices 5.Normalize with respect to horizontal spectral densities and vertical eigen-values Structure function are non-separable; different horizontal scales at different levels and different vertical scales for different horizontal scales Forecast error statistics are derived by the NMC-method from differences between 48 h and 24 h forecasts valid at the same time. Forecast error standard deviations are re-scaled.

18. Single temperature observation impact experiment

19. 4D-Var feasibility and impact study at DMI DMI operational G area  DMI G area, 202 x 190 x 31 gridpoints, 50 km grid  4D-Var minimization with 150 km increments  Buizza vertical diffusion  One single outer loop  Eulerian dynamics  6 h assimilation window, 1 h observation windows  Several test runs: 1-10 Dec 1999, 20-30 Dec 1999, Feb 2002  Conventional observations + AMSU A for Feb 2002  On the average neutral impact in comparison with 3D- Var (positive for 06 and 18 UTC, negative for 00 and 12 UTC)

20. HIRLAM forecasts for the French Christmas storm, valid 28 Dec 1999 00UTC. 3D-Var (left), 4D-Var (right). c. 24h, d. 12h, e. 6h, f. analysis

21. Parallelization of HIRLAM 4D-Var FFTs: (1) For the FFT in x-direction the area is divided according to the y-direction; (2) For FFT in the y-direction the area is divided according to x and z (3) Transpose of all data in between the 1D FFT transforms. Area decomposition also for vertical transforms of spectral coefficients Semi-Lagrangian interpolation: Area decomposition is the same as for x-direction FFT; A halo zone includes values communicated from neighbor processors. Observation processing: Each processor has equally many observation of each type to take care of. Field values are obtained by interpolation in the same area decomposition as for x-direction FFT and communicated. Use of MPI and OPEN MP as standard (SHMEM as an option).

22. Operational HIRLAM 4D-Var on a linux-cluster at SMHI? Model/assimilation runNumber of PEs Wall clock time (s) SGI 3000 Non-linear forecast, +3h, 1.5 min timestep31425 4D-Var minimization, 40 km increment, Buizza physics, 20 iterations 207474 Computer timings, 438 x 310 x 40 grid-points, 22 km horizontal resolution: Computer timing estimates, 306 x 306 x 40 gridpoints, 22 km horizontal resolution, SL scheme, 64 PEs SGI 3000: Non-linear forecast, +48h, 5 min timestep14 min 4D-Var minimization, 3 x 20 iterations, 40 km increment36 min

23. Scaling of the spectral HIRLAM on Monolith Process8 PE16 PE31 PE62 PE Total model348 s185 s91 s48 s FFT’s66 s37 s23 s15 s FFT transpose29 s20 s17 s11 s SL calculations106 s48 s22 s10 s SL swap16 s11 s10 s Physics103 s51 s26 s13 s Large problem size: 438 x 310 x 40 points, 22 km resolution, Semi- Lagrangian time integration, 7,5 min timestep, calculation time in seconds for 3 h forecast

24. Scaling of HIRLAM 4D-Var on Monolith Small problem size: 202 x 178 x 31 points, 44 km resolution, 88 km resolution in minimization, 20 iterations over a 6 hour window Process9 PE15 PE18 PE30 PE NL forecast73 s45 s39 s25 s Minimize785 s522 s492 s400 s TL forecast288 s196 s180 s150 s AD forecast420 s250 s240 s190 s FFT transpose240 s180 s152 s162 s SL swap117 s104 s94 s89 s

25. HIRLAM work on use of remote sensing data in 3D-Var and 4D-Var ATOVS data: AMSU-A radiances over sea (data thinning and bias-correction). Operational at DMI and DNMI. EUMETSAT re-transmission of ATOVS data from the Atlantic and Arctic areas. Work with AMSU-A over land and ice, HIRS and AMSU-B. Radar radial wind vectors and radar VAD profiles: De-aliasing (wind speed ambiguity). Formation of super-observations. Spatial and temporal filters. Development of observation operators taking radar beam bending and spread into account. Impact studies. Ground-based GPS data: Assimilation of Zenith Total Delay. Bias correction (individual for each station) and possibly a horizontally correlated error. Impact studies. Use of slant delays. Other data: Scatterometer winds, MODIS/MERIS IWV and WV above clouds, wind profilers.

26. Forecast verification scores from implementation of AMSU-A radiances at DMI. Impact study for January 2003. NOA without AMSU A; WIA with AMSU A.

27. Ten-day assimilation experiment with radar radial wind data: 1-10 December, 1999 Integration area and radar sites Observation fit statistics Verification of time-series of +24 h wind forecasts (against observations)

28. Assimilation of ground-based GPS data “One person’s noise is another person’s signal” Advantages:  High resolution  55 to 60 stations (in Sweden) every 15 minutes  All weather, all the time  Very cheap Disadvantages:  Essentially only Integrated Water Vapour  Possible biases and spatially correlated errors

29. Challenges Operational 4D-Var on a computer that a small weather service can afford Use of mesoscale moisture related observations, for example, radar reflectivity Improved treatment of non-linear processes (utilize ideas from ensemble Kalman filters) 4D-Var for nowcasting and very short range forecasting with a NWP model at the km-scale