Transcript What is Numerical Weather Prediction?

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
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.
Data Assimilation
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
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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
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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
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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
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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
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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
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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
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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
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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
Arakawa E-grid staggering (Eta model)
Boundary Conditions
• Lateral B. C. essential for limited-area models
• Top and lower B. C. needed for all models
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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.
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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
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 Incomplete
2 Errors in Numerical
Approximations
a Horizontal Resolution
b Vertical Resolution
c Time Integration Procedure
3 Boundary Conditions
a Horizontal
b Vertical
4 Terrain
5 Physical Processes
Source: 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.
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.
Triangular Truncation
n
N=80
A
-N
-m
0
L=n-m
B
+m
N=80
Triangular Truncation
m = Zonal wave number
n = 2-D wave number
4
3
D
n 2 A B C
E
1
0
0 1 2 3 4
m
Distribution of nodal lines for spherical harmonics
(0,3)
D
H
EQ
L
(0,2)
A
L
H
H
L
L
H
L
L
L
(1,1)
L
H
(0,0)
EQ
H
H
L
(2,2)
H
E
L
L
C
L
(0,1)
L
EQ L H L
H
H
H
H
EQ
(1,2)
B
(3,3)
(2,3)
(1,3)
H
EQ
H
L
H
EQ
H L H
Rhomboidal Truncation
n
80
N=40
B
A
-N
-m
0
+m
N=40
Rhomboidal Truncation
• Spatial resolution concentrated in the midlatitudes
• 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 latitudelongitude 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