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### 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

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

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

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.

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

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)

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

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

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

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

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., Qv, Qr, Qs, Qi, Qg), and
- conservation of other gaseous and aerosol materials

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

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)

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

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)

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

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)

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

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

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

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

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)

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

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

Initial Condition Errors

Observational Data Coverage

Spatial Density

Temporal Frequency

Errors in the Data

Instrument Errors

Representativeness Errors

Errors in Quality Control

Errors in Objective Analysis

Errors in Data Assimilation

Missing Variables

Model Errors

Equations of Motion Incomplete

Errors in Numerical Approximations

Horizontal Resolution

Vertical Resolution

Time Integration Procedure

Boundary Conditions

Horizontal

Vertical

Terrain

Physical Processes

Sources of Errors - continuedSource: Fred Carr

Forecast Error Growth and Predictability

Source: Fred Carr

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)

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

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.

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.

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

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.

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

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.

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

Japan Meteorological Agency Models

JMA, in fact, uses spectral methods for all their models!

Global Spectral Model

Asia Spectral Model

Japan Spectral Model

Typhoon Spectral Model

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