1. Numerical Weather Prediction: An Overview Mohan Ramamurthy Department of Atmospheric Sciences University of Illinois at Urbana-Champaign E-mail: mohan@uiuc.edu COMET Faculty Course on NWP June 7, 1999

2. What is Numerical Weather Prediction? The technique used to obtain an objective forecast of the future weather (up to possibly two weeks) by solving a set of governing equations that describe the evolution of variables that define the present state of the atmosphere. Feasible only using computers

3. A Brief History Recognition by V. Bjerknes in 1904 that forecasting is fundamentally an initial-value problem and basic system of equations already known L. F. Richardson’s first attempt at practical NWP Radiosonde invention in 1930s made upper-air data available Late 1940s: First successful dynamical-numerical forecast made by Charney, Fjortoft, and von Neumann

4. NWP System NWP entails not just the design and development of atmospheric models, but includes all the different components of an NWP system It is an integrated, end-to-end forecast process system USWRP focus: “best practicable mix” of observations, data assimilation schemes, and forecast models.

5. Data Assimilation

6. Components of an NWP model 1. Governing equations 2. Physical Processes - RHS of equations (e.g., PGF, friction, adiabatic warming, and parameterizations) 3. Numerical Procedures : approximations used to estimate each term (especially important for advection terms) approximations used to integrate model forward in time boundary conditions 4. Initial Conditions: Observing systems, objective analysis, initialization, and data assimilation

7. Notable Trends Use of filtered models in early days of NWP Objective analysis methods Terrain-following coordinate system Improved finite-difference methods Availability of asynoptic data: OSSE and data assimilation issues Global spectral modeling Normal mode initialization Economic integration schemes (e.g., semi-implicit)

8. Trends - continued Parameterization of model physics Model output statistics Diabatic initialization Four-dimensional data assimilation Regional spectral modeling Introduction of adjoint approach Ensemble forecasting Targeted (or adaptive) observations

9. Computing trends NWP has evolved as computers have evolved The big irons: early 1950- late 1970 Vector supercomputers: Late 1970s Multi-processors: 1980s Massively parallel supercomputers High-performance workstations Personal computers as workstations

10. Hierarchy of models –Euler equations –Primitive equation –Hydrostatic vs. Non-hydrostatic –Filtered equations: Filter out sound and gravity waves Permits larger time-step for integration –Filtering sound waves: »Incompressible »Anelastic »Boussinesq –Filtering gravity waves: »Quasi-geostrophic »Semi-geostrophic »Equivalent barotropic

11. Governing Equations –It was recognized early in the history of NWP that primitive equations were best suited for NWP –Governing equations can be derived from the conservation principles and approximations. –It is important for students to understand the resulting wave solutions and their relationship to the chosen approximations. e. g., shallow-water models: one Rossby mode and two gravity modes

12. Key Conservation Principles –conservation of motion (momentum) –conservation of mass –conservation of heat (thermodynamic energy) –conservation of water (mixing ratio/specific humidity) in different forms (e.g., Q v, Q r, Q s, Q i, Q g ), and –conservation of other gaseous and aerosol materials

13. Prognostic variables Horizontal and vertical wind components Potential temperature Surface pressure Specific humidity/mixing ratio Mixing ratios of cloud water, cloud ice, rain, snow, graupel PBL depth or TKE Mixing ratio of chemical species

14. Vertical Representation Sigma (terrain following): e.g., NGM, MM5 Eta (step mountain): Eta model Theta (isentropic) Hybrid (sigma-theta): RUC Hybrid (sigma-z): GEM (Canadian model) Pressure (no longer popular in NWP) Height (mostly used in cloud models)

15. Map projections: Why? Equations are often cast on projections Output always displayed on a projection Data often available on native grids Projections used in NWP: Lambert-conformal Polar stereographic Mercator Spherical or Gaussian grid

16. Numerical Methods Finite difference (e.g., Eta, RUC-II, and MM5) Galerkin –Spectral (e.g., MRF, ECMWF, RSM, and all Japanese operational models) –Finite elements (Canadian operational models) Adaptive grids (COMMAS cloud model)

17. Time-integration schemes Two-level (e.g., Forward or backward) Three-level (e.g., Leapfrog) Multistage (e.g., Forward-backward) Higher-order schemes (e.g., Runge-Kutta) Time splitting (split explicit) Semi-implicit Semi-Lagrangian

18. Numerics: Important considerations Accuracy and consistency Stability and convergence Efficiency Monotonicity and conservation (e.g., positive definite advection) Aliasing and Nonlinear instability Controlling computational mode (e.g., Asselin filter) Other forms of smoothing (e.g., diffusion)

19. Eulerian or Semi-Lagrangian? Efficiency depends on applications Semi-Lagrangian methods require more calculations per time step S-L approach advantageous for tracer transport calculations (conservative quantities) S-L method is superior in models w/ spherical geometry Problems in which frequency of the forcing is similar in both Lagrangian and Eulerian reference

20. Eulerian vs. S-L methods - contd When the frequency of the forcing is similar in both Lagrangian and Eulerian reference frames, S-L approach loses its advantage S-L can be coupled with Semi-implicit schemes to gain significant computational advantage. ECMWF model S-L/SI example: –Eulerian approach: 3-min time step –S-L/SI approach: 20-min time step –S-L 400% more efficient including overhead

21. Staggered Meshes Spatial staggering (velocity and pressure) –Arakawa grid staggering (horizontal) –Lorenz staggering (vertical) Wave motions and dispersion properties better represented with certain staggered meshes e.g., important in geostrophic adjustment Temporal staggering

22. Arakawa E-grid staggering (Eta model)

23. Boundary Conditions Lateral B. C. essential for limited-area models Top and lower B. C. needed for all models Some Examples: Relaxation (Davis, 1976) Blending (Perkey-Kreitzberg, 1976) Periodic Radiation (Orlanski, 1975) Fixed, symmetric

24. Model Physics Grid-scale precip. (large scale condensation) Deep and shallow convection Microphysics (increasingly becoming important) Evaporation PBL processes, including turbulence Radiation Cloud-radiation interaction Diffusion Gravity wave drag Chemistry (e.g., ozone, aeorosols)

25. Model Performance Validation Verification –Skill score, RMS error, AC, ETS, biases, etc. Verification of probabilistic forecasts Mesoscale verification problem QPF verification Verification over complex terrain

26. Sources of error in NWP Errors in the initial conditions Errors in the model Intrinsic predictability limitations Errors can be random and/or systematic errors

27. Sources of Errors - continued Initial Condition Errors 1Observational Data Coverage aSpatial Density bTemporal Frequency 2Errors in the Data aInstrument Errors bRepresentativeness Errors 3Errors in Quality Control 4Errors in Objective Analysis 5Errors in Data Assimilation 6Missing Variables Model Errors 1Equations of Motion Incomplete 2Errors in Numerical Approximations aHorizontal Resolution bVertical Resolution cTime Integration Procedure 3Boundary Conditions aHorizontal bVertical 4Terrain 5Physical Processes Source: Fred Carr

28. Forecast Error Growth and Predictability Source: Fred Carr

29. Galerkin Method - Series Expansion Method The dependent variables are represented by a finite sum of linearly independent basis functions. Includes: –the spectral method –the pseudospectral method, and –the finite element method Less widely used in meteorology (ex. Canadian models) Basis functions are local Can provide non-uniform grid (resolution)

30. Spectral Methods The basis functions are orthogonal The choice of basis function dictated by the geometry of the problem and boundary conditions. Introduced in 1954 to meteorology, but it did not become popular until the mid 70s. Principal advantage: The spectral representation does not introduce phase speed or amplitude errors - even in the shortest wavelengths! Avoids nonlinear instability since derivatives are known exactly. Runs faster when coupled with SI/SL method

31. Spectral Model - continued Early spectral models calculated nonlinear terms using the so-called interaction coefficient method, which required large amount of memory and it was inefficient. In 1970, the transform method was introduced. Coupled with FFT algorithms, the spectral approach became very efficient. The transform method also made it possible to include “physics.” Main Idea: Evaluate all main quantities at the nodes of an associated grid where all nonlinear terms can then be computed as in a classical grid-point model.

32. Spectral Basis Functions Global models (e.g., MRF) use spherical harmonics, a combination of Fourier (sine and cosine) functions that represent the zonal structure and associated Legendre functions, that represent the meridional structure. The double sine-cosine series are most popular for regional spectral modeling (e.g., RSM) because of their simplicity.

33. Spectral Truncation In all practical applications, the series expansion of spherical harmonic functions must be truncated at some finite point. Many choices of truncation are available. In global modeling, two types of truncation are commonly used: –triangular truncation –rhomboidal truncation

34. Triangular Truncation Universal choice for high-resolution global models. Provides uniform spatial resolution over the entire surface of the sphere. The amount of meridional structure possible decreases as zonal wavelengths decrease Not optimal in situations where the scale of phenomena varies with latitude.

35. Triangular Truncation 0 +m -m-N N=80 n L=0 L=n-m A B

36. Triangular Truncation

37. Distribution of nodal lines for spherical harmonics EQ

38. Rhomboidal Truncation 0 +m -m-N N=40 n L=0 L=N N=40 A B 80

39. Rhomboidal Truncation Spatial resolution concentrated in the mid- latitudes Equal amount of meridional structure is allowed for each zonal wavenumber Therefore, the time-step in a R-model is greater than that in a T-model for the same truncation. Often used in low-resolution atmospheric models

40. Gaussian Grid Spectral models use a spherical grid array called a Gaussian grid for transformations back to physical space. Gaussian grid is a nearly regular latitude- longitude grid. Its resolution is chosen to ensure alias-free transforms between the spectral and physical domains.

41. Characteristic Resolution and Degrees of Freedom In a Typical Spectral Model MRF/AVN: T126 (104 km) out to 7/3 days; T62 thereafter Note:MRF will soon be @ T170 (dynamics) out to 7/3 days. ECMWF: T319L31 (42 km) out to 10 days

42. Global Spectral Model Asia Spectral Model Japan Spectral Model Typhoon Spectral Model JMA, in fact, uses spectral methods for all their models! Japan Meteorological Agency Models