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![Page 1: Numerical Weather Prediction: An Overview Mohan Ramamurthy Department of Atmospheric Sciences University of Illinois at Urbana-Champaign E-mail: mohan@uiuc.edu.](https://reader036.fdocuments.in/reader036/viewer/2022062518/56649eec5503460f94bfd6f3/html5/thumbnails/1.jpg)
Numerical Weather Prediction:An Overview
Mohan Ramamurthy
Department of Atmospheric Sciences
University of Illinois at Urbana-Champaign
E-mail: [email protected]
COMET Faculty Course on NWP
June 7, 1999
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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
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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
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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.
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Data Assimilation
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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
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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)
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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
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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
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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
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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
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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
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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
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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)
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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
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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)
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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
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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)
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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
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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
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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
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Arakawa E-grid staggering (Eta model)
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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
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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)
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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
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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
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Sources of Errors - continued
Initial Condition Errors
1 Observational Data Coverage
a Spatial Density
b Temporal Frequency
2 Errors in the Data
a Instrument Errors
b Representativeness Errors
3 Errors in Quality Control
4 Errors in Objective Analysis
5 Errors in Data Assimilation
6 Missing Variables
Model Errors
1 Equations of Motion Incomplete2 Errors in Numerical
Approximationsa Horizontal Resolutionb Vertical Resolutionc Time Integration Procedure
3 Boundary Conditionsa Horizontalb Vertical
4 Terrain
5 Physical Processes
Source: Fred Carr
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Forecast Error Growth and Predictability
Source: Fred Carr
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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)
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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
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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.
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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.
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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
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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.
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Triangular Truncation
0 +m-m-N N=80
nN=80
L=0
L=n-m
A B
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Triangular Truncation
A B C
E
D
0 1 2 3 4m
n
4
3
2
10
m = Zonal wave numbern = 2-D wave number
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(0,3)
(0,2)
(0,1)
(0,0)
(1,3)
(1,2)
(1,1)
(2,3) (3,3)
(2,2)
HL
H
H H
H
H
H
H
H
H
L
L L
L
L
L
L H
H
HL L
L L
H H HL L LH
L
H
L
A CB
E
D
Distribution of nodal lines for spherical harmonics
EQ
EQ
EQ
EQ EQEQ
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Rhomboidal Truncation
0 +m-m-N N=40
n
L=0
L=N
N=40
A
B
80
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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
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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.
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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
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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