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An overview of numerical weatherprediction models
WORKSHOP BCAM June 7-8 2010 - Derio
Santiago Gaztelumendi.Dtor Meteorology Division, EUVE Foundation.
Coord EUSKALMET (Basque Meteorology Agency).
OUTLINE
An overview of numerical weather prediction models
1. The beginnings.2. NWP process.3. How meteorological forecast
process works?
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The beginnings.
Fundamental physics laws essential forweather aplications are there from “long time ago” :
- 1687 Isaac Newton , Principia Mathematica: laws of mechanics-1812 Pierre S. de Laplace : ‘Complete knowledge of the masses, positions and velocities of all particles at any single instant would enableprecise calculation of all past and future events.- 1840 R. Mayer, J. Joule & H. von Helmholtz : concept of conservation ofenergy, 1st law of thermodynamics- 1850 Rudolf Clausius : 2nd law of thermodynamics. Heat: warmer tocooler in absence of external constraints.-C. L. Navier (1785-1836) & G. G. Stokes (1819-1903) equations formotion of fluids
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The beginnings.
In 1908 a procedure for rational weather forecasting was defined by VilhelmBjerknes (considered first in recognize that weather forecast was possible).
“subsequent atmospheric states develop from the preceeding ones according to physical law, then it is apparent that the necessary and sufficient conditions for the rational solution of forecasting problems are the following:1. A sufficiently accurate knowledge of the state of the atmosphere at the initial time2. A sufficiently accurate knowledge of the laws according to which one state of the atmosphere develops from another.”(Step (1) is Diagnostic, Step (2) is Prognostic)
Laws of motion and conservation of energy (Newton's Second Law of Motion and the FirstLaw of Thermodynamics) that govern the evolution of the atmosphere can be convertedinto a series of mathematical equations that make up the core of a weather forecastprocess.Weather prediction could be seen as essentially an initial value problem in mathematics: since equations govern how meteorological variables change with time, if we know theinitial condition of the atmosphere, we can solve the equations to obtain new values ofthose variables at a later time (i.e., make a forecast).
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The beginnings.
Bjerknes on his paper on “Weather forecasting as a problem in mechanics and physics ”:
- Emphasizes the fact that the atmosphere may be described by seven scalar variables. The three components of wind, temperature, density, pressure and a measure of the content of atmospheric moisture.
- Point out that local changes of these variables are interconnected by seven equations: Three equations of motion, the thermodynamic equation, the continuity equation, the gas equation and a equation for the change of moisture in the atmosphere as influenced by evaporation and condensation, clouds and precipitation.
- Discuss the fact that no analytical solutions to these general non-linear set of equations are known.
Vilhelm Bjerknes
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The beginnings.1922. L.F. Richardson first try a numerical solution for Bjerknesequations.
Richardson formulates the equations in finite differences form. He used an initial state for witch he had surface values of some of the meteorological variables, but he had to make sever assumptions to generate the physical state of the upper atmosphere (since no meteorological measurements were available above the surface of the Earth). Based on the constructed initial state for his grid-point domain (200 km spatial resolution) he calculated (by hand) all terms in the equations, and therefore he was able to multiply by 6 hours to obtain the estimate of the state of the atmosphere 6 hours later.
- His calculations in general gave much larger changes than were observed (even unrealistic pressure changes for 145 mb in 6 hours where -1mb where observed).- Theoreticaly he estimate that for a future operational application for the globe 65000 computers (in the original sense of people computing bay hand) were needed (….“Richardson’s dream”).
The results of his experiment tendency calculations indicated that his approach to NWP was far for practical, and a couple of decades had to pass before the objective prediction problem could be considered again.
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The beginnings.
The essence of the problem with the full primitive equations, is that pressure changes are determined by the divergence field, which is calculated as a small residue resulting from near-cancellation of large terms, an inherently error-prone process. On the other hand Richardson’s data were contaminated by imbalances which gave rise to spurious large amplitude oscillations, and a computed initial pressure tendency of 145mb in 6 hours. He also used a large time step which would have resulted in instability of the integration, but this did not affect the initial tendencies.
(During the 1920s, Courant, Friedrichs and Lewy had studied the numerical solutions of partial differential equations, and had shown that there is a limitation on the time-step for a given space step; this is now known as the CFL criterion.)
Charney in 1947 develop a scale theory for geostrophic motion with application to numerical weather prediction and the formulation of simplified barotropicvorticity equations. Barotropic model was first step in NWP under the new prevailing philosophy of using the simplest possible non-linear model simplified equations and not the general equations used without success prior to that time. Even prior to computers, the barotropic vorticity equation was used to calculate the vorticity tendencies by hand in many meteorological centers (as in the Rossby’s International Meteorological Institute in Stockholm).
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Jule Charney
The beginnings.In the mid 1930s Jon Von Neumann became interested in turbulent fluid flows. (The non-linear partial differential equations which describe such flows resist analytical assault and even qualitative insight comes hard). Von Neumann saw that progress in hydrodynamics would be greatly accelerated if a means of solving complex equations numerically were available. It was clear that very fast automatic computing machinery was required. He masterminded the design and construction of an electronic computer at the Institute for Advanced Studies (IAS) in Princeton. This machine was built between 1946 and 1952.- Von Neumann recognized weather forecasting, as a problem of both great practical significance and intrinsic scientific interest, as an ideal problem for an automatic computer.The Electronic Computer Project comprised four groups: (1) Engineering, (2) Logical design and programming, (3) Mathematical, and (4) Meteorological.The initial plan was to integrate the primitive equations; but the existence of high-speed gravity wave solutions required the use of such a short time step that the volume of computation might exceed the capabilities of the IAS machine. And there was a more fundamental difficulty: the impossibility of accurately calculating the divergence from the observations. The answers were not apparent; it remained for Charney to find a way forward.
The meteorological group was directed for the period 1948–1956 by JuleCharney.
Jon Von Neumann
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The beginnings.
Charney analysed the primitive equations using the technique of scale analysis, and was able to simplify them in such a way that the gravity wave solutions were completely eliminated. The resulting equations are known as the quasi-geostrophicsystem. (In the special case of horizontal flow with constant static stability, the vertical variation can be separated out and the quasi-geostrophic potential vorticityequation reduces to a form equivalent to the nondivergent barotropic vorticityequation.)
By early 1950 the Meteorology Group had completed the necessary mathematical analysis and had designed a numerical algorithm for solving the barotropic vorticityequation.
IAS computer was not ready so meteorology group(Fjortfot, Charney, Freeman, Platzman andSmagorinsky) participates in a 33 days NWP experimentfor barotropic computations with the Army´s Electronic Numerical Integrator and Computer (ENIAC).
The Electronic Numerical Integrator and Computer (ENIAC).
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There are essentially two ways round the problem in Richardson work: the initial data could be doctored to remove the imbalance, a process called initialization ; or the equations themselves could be doctored to eliminate the high frequency spurious gravity wave solutions, a process called filtering .
The beginnings.(During this experiment only four 24-h and two 12-h forecast was done (due to breakdown problems, theoretically ENIAC takes 24 hours for a prediction of 24-hours), 100.000 punched cards were used to record the evolving fields of geopotential and its derivatives.)- After this work the conclusion was that “with certain well marked exceptions, the features of the 500-mb flow can be forecasted barotropically”.
(ENIAC, operating in decimal system, used a vast number of radio tubes, with punch cards for input and output.)
The encouraging initial results of the Princeton team generated widespread interest and raised expectations that operationally useful computer forecasts would soon be a reality.
Several baroclinic models were developed in the few years after the ENIAC forecast. They were all based on the quasi-geostrophic system of equations.
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The beginnings.Within next years there were research groups at several centers throughout the world (including Charney´s group at Princenton) checking for barotropic single-level approach to weather forecast and other multilevel baroclinic models approach always based on simplified systems, with two appeals: lesser computational load and freedom from high frequency noise.
In the period 1952-1957 just 17 computers were available in the world, in much cases used for barotropic meteorological model experiments (as in the the case of Charneywith IAS Princenton Computer and Philips with the Swedish Computer BESK). As a results from different experiments, numerical teams concludes that it is difficult to prepare a 24 hour forecast for operational use (meaning before 24 hours of calculus), and that it will be essential to produce a procedure for an objective analysis by the machine (correct initial state determination is essential ).
-( 1955. In the USA case the Joint Meteorological Commitee, takes responsibility for the implementation of operational weather forecasting, and creates the Joint Numerical Weather Prediction Unit (JNWP). The quasi-geostrophic 3-level model was programmed and run on a operational schedule in a IBM 701 under the direction of G. Cressman. The first forecast was made on April 15 1955, results “not only was unable to predict reliably and accurately, but it also provides little or not useful information to the forecaster”. )
- 1958. First numerical weather prediction “useful” results with single-level barotropicmodel in Charney’s group, during this year an analysis program was implemented and automatic handling of data from the international WMO network was developed.
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The beginnings.
Notice that before to 1960 computing costs and the absence of effective system software made it inevitable to write all codes in absolute machine language resulting in very efficient but also extremely inflexible programingIn the Meteorology International Symposium in Boulder USA (1964). Specialist concludes that: “The future strategy for modelling the general circulation of the atmosphere will inevitably lead to the aproximation of primitive equations with high spatial resolution”.
The limitations of the filtered equations were reco gnized at an early stage……..Back to primitive equations set).
- Routine numerical forecasting was introduced in the Deutscher Wetterdienst in 1966; this was the first ever use of the primitive equations in an operational setting. - A six-level primitive equation model was introduced into operations at the National Meteorological Center in Washington in June, 1966, running on a CDC 6600 [26]. There was an immediate improvement in skill: the S1 score for the 500 hPa one-day forecast was reduced by about five points.- The view in the U.K. Met. Office was that single-level models were unequal to the task of forecasting. As a result, barotropic models were never used for forecasting in the UK and, partly for this reason, the first operational model was not in place until the end of 1965. In 1972 a 10-level primitive equation model was introduced. The new model incorporated a sophisticated parameterization of physical processes and with it the first useful forecasts of precipitation were produced.
Today nearly all operational weather and climate mo dels uses primitve equations approach.
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NWP process.
Prior to a forecast integration to obtain a unique solution for a future atmospheric state we must consider different aspects:
-Set of equations.-Solution techniques.-Domain implementation.-Initial conditions.-Boundary conditions.-Codification and integration.
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Set of equations
Many sets of different equations are used, depending on type of meteorological phenomena we are interested in. From basics conservation principles applied to the atmosphere many different approaches could be done to obtain particular set of equations for a particular case:
Examples:
- Barotropic equations, first used for a NWP integration on 1950. - Baroclinic equations, used at 60s. - Hydrostatic primitive equations set with partial consideration of physical processes aplied to global forecast on 80s.
(That is, they use the hydrostatic primitive equations, which assume a balance between the weight of the atmosphere and the vertical pressure gradient force. This means that no vertical accelerations are calculated explicitly. The hydrostatic assumption is valid for synoptic- and planetary-scale systems and for some mesoscale phenomena. A most notable exception is deep convection, where buoyancy becomes an important force. Hydrostatic models account for the effects of convection using statistical parameterizations approximating the larger-scale changes in temperature and moisture caused by non-hydrostatic processes.)
- Non-hidrostatic primitive equations set with a quasi-complete consideration of physical processes used in mesoscale local area models applications today.
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Set of equationsA particular set of equations for the atmosphere after some assumptions made to basical conservation principles, can be expresed as 11 nonlinear partial differential equation set with 11 dependent variables, that must be solved to provide a forecast. This set of equations has not analytical solution and must be simplified and numerically solved.
( )∂ρ∂
ρ ∂∂
ρ ∂∂
ρ ∂∂
ρt
Vx
uy
vz
w= − ∇ − + +
=
ρ ρ.
∂θ∂
θ ∂θ∂
∂θ∂
∂θ∂θ θt
S ux
vy
wz
S= − ∇ + = − − − +ρ ρV.
∂∂ ρ
ρ ρ ρ ρ ρ ρ ρ ρVt
V V p gk V= − ∇ − ∇ − − ×.1
2Ω
∂∂
∂∂
∂∂
∂∂
qt
V q S uqx
vqy
wqz
Snn q
n n nqn n
= − ∇ + = − − − +ρ ρ
.
p RdTv= ρ T q T v = +( . . )1 0 61 θ=
T
pp
RdCp0
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Set of equations
- Matemáticamente:
- Físicamente: teniendo en cuenta volúmenes suficientemente grandes para que tenga sentido evaluar aspectos termodinamicos del fluido, es decir debemos considerar conjuntos de moléculas suficientemente grandes para que tenga sentido evaluar nagnitudes termodinamicas. Esto nos coloca, entre otros problemas, en dominios con tal cantidad de puntos que no es posible plantear su solución operativa.-Operativamente: Es necesario integrar las ecuaciones de conservación en intervalos que permitan abordar computacionalmente la solución del problema, por lo que se aplican técnicas de simplificación promediando al volumen de rejilla de forma que todas las variables se transforman en parte promedio y perturbación.
∂φ∂
∂φ∂t t x x
≅ ≅ ⇒ →∆φ∆
∆φ∆
∆ 0
∆ ∆x cm t seg ≈ ≈1 1
φ φ φ = + ′′
φφ
=∫∫∫∫ dzdydzdt
z y z tz
z+ z
y
y+ y
x
x+ x
t
t+ t ∆∆∆∆
∆ ∆ ∆ ∆
El conjunto de ecuaciones básicas está formulado empleando una serie de operadores diferenciales, compuestos fundamentalmente de derivadas parciales espaciales y temporales. Estos operadores diferenciales pueden ser considerados desde el punto de vista matemático, físico u operativo, veamos:
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Set of equationsAtmospheric scales – In the continuous atmospheric flow we can distinguish different organized system on different spatial-temporal scales.
∂∂
∂∂
α ∂∂
wt
uwx
pzj
j
+ = − − +g 2u cosΩ Φ
H
Lp
gx
α ∂∂
ρ≈ ⇒ = −< 1 z
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Set of equations
La técnica de simplificación del conjunto de ecuaciones básicas mas utilizada y una de las mas importantes es el a análisis de Escala . Esta técnica consiste en la determinación de una nueva forma de ecuaciones, mas operativa, a aplicar en nuestro sistema meso particular, a través del análisis de la importancia relativa de los distintos términos de las ecuaciones de conservación (básicas).
As an example……vertical motion equation…. hidrostatic equation :
∂∂
∂∂
α ∂∂
wt
uwx
pzj
j
+ = − − +g 2u cosΩ Φ
H
Lp
gx
α ∂∂
ρ≈ ⇒ = −< 1 z
(Assuming a balance between the weight of the atmosphere and the vertical pressure gradient force. This means that no vertical accelerations need to be calculated explicitly. )
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Set of equations
Aplicando las diferentes técnicas de promediado y de simplificación podemos obtener un nuevo conjunto de ecuaciones aplicado a un sistema mesoescalar.
∂∂
∂∂ ρ
∂∂
ρ α ∂∂
ut
uux x
u upx
v wjj j
j= − − ′′ ′′ − + −12 2
00 0 Ω Φ Ω Φsen cos
∂∂
∂∂ ρ
∂∂
ρ α ∂∂
vt
uvx x
u vpy
ujj j
j= − − ′′ ′′ − − −12
00 0 Ω Φsen
∂∂
∂∂ ρ
∂∂
ρ α ∂∂
αα
wt
uwx x
u wpzj
j jj= − − ′′ ′′ − ′ + ′ +1
00 0
0
g 2u cosΩ Φ
∂θ∂
∂θ∂ ρ
∂∂
ρ θ θtu
x xu Sj
j jj= − − ′′ ′′ +
1
00
∂∂
∂∂ ρ
∂∂
ρq
tu
q
x xu q Sn
jn
j jj n qn
= − − ′′ ′′ +1
00 n = 1,2,3
∂ρ∂
∂∂
ρ∂
∂ρ
t xu
xu
jj
jj= − − ′′ ′′
u u ui i i= + ′′ α α α= + ′0 u = u ; u i i i′′ = = =0 ; ; ;.........∂∂
∂∂
∂∂
∂∂
u
t
u
t
u
x
u
xi i i
i
i
i
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Set of equations
This terms should be parametrized.
′ ′u i φ SφTurbulent fluxSubgrid flux
Sink/source term
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Set of equationsIn some sense all forecast Models have 2 Essential Computational Components:
The Model Dynamics – Basically the equations of MotionThe Model Physics -- Physically forced Processes. Many occur at small scales and
must have their effects approximated (parametrized).
NWP models cannot resolve weather features and/or processes that occur within a single model grid box. The method of accounting for such effects withoutdirectly forecasting them is called parameterization .
Parameterization is how we include the effects of physical processes implicitly when we cannot include the processes themselves explicitly. Parameterization can be thought of as emulation (modeling the effects of a process) rather than simulation (modeling the process itself).
Parameterization is necessary for some reasons:- Computers are not powerful enough to treat many physical processes explicitly
because they are either too small (as discussed earlier) or complex to be resolved
- Many other physical processes cannot be explicitly modeled because they are not sufficiently understood to be represented in equation format or there are no appropriate data
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Set of equations
Each important physical process that cannot be directly predicted requires a parameterization scheme based on reasonable physical (for example, radiation) or statistical (for example, inferring cloudiness from relative humidity) representations. The scheme must derive information about the processes from the variables in the forecast equations using a set of assumptions. Closurerefers to the link between the assumptions in the parameterization and the forecast variables. (It closes the loop between the parameterization and forecast equations, and alows same number of equations than variables inviolved.)
(Several types of assumptions are used to "create" information.Empirical/statistical: This assumes that a given relationship holds in every case (for example, surface layer wind speed variations with height for PBL processes and surface wind forecasts). Note that for a normal statistical distribution, one of every 20 cases is expected to be an outlier.Dynamical/thermodynamical constraining assumption: A complex process is summarized through a simplified relationship, for example, equilibrium of instability for Arakawa-Schubert convective parameterization)
The key problem of numerical parameterization is trying to predict with incomplete information, for example, the effects of sub grid-scale processes with information at the grid scale. (Imagine using the wind forecast in a grid box to predict boundary-layer turbulence without knowing topography details, vegetation characteristics, or the details of structures at the surface.)
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Set of equations
The graphic depicts some ofthe physical processes andparameters that are typicallyparameterized, both becausethey cannot be explicitlypredicted in full detail in model forecast equationsregardless of the grid point orwave number resolution andbecause their effects on theforecast variables resolvableby a model are crucial toforecast realism.
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Set of equations
(Precipitation is usually divided into two types in model:“Large Scale” or “Grid Scale” Precipitation - Which is used to simulate the occurance of Stratiform Precipitation“Convective” - which is used to account for the effects of deep convection on heating and moistening the atmosphereNot having a separation allows too much latent heating to concentrate at low levels in the atmosphere and produce overdevelopment.
Simplified solutions to the full radiative transfer process.Usually recalculated in models every hour or less.Highly interactive with other model conditions - namely: Cloudiness, Soil condition, Vegetationand Snow cover.
Turbulent mixing and ABL ………………….).
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Set of equations
HYDROSTATIC MODELS
Characteristics- Use the hydrostatic primitive equations, diagnosing vertical motion from predicted horizontal motions- Used for forecasting synoptic-scale phenomena, can forecast some mesoscale phenomena
Disadvantages
- Cannot predict vertical accelerations- Cannot predict details of small-scale processes associated with buoyancy
Advantages
- Can run fast over limited-area domains, providing forecasts in time for operational use- The hydrostatic assumption is valid for many synoptic- and sub-synoptic-scale phenomena
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Set of equationsNON-HYDROSTATIC MODELS
Characteristics- Use the non-hydrostatic primitive equations, directly forecasting vertical motion- Used for forecasting small-scale phenomena- Predict realistic-looking, detailed mesoscale structure and consistent impact on surrounding weather, resulting in either superior local forecasts or large errors
Disadvantages
- Take longer to run than hydrostatic models with the same resolution and domain size- Used for limited-area applications, so they require boundary conditions (BCs) from another model; if the BCs lack the structure and resolution characteristic of fields developing inside the model domain, they may exert great influence on the forecast- May predict realistic-looking phenomena, but the timing and placement may be unreliable
Advantages
- Calculate vertical motion explicitly- Explicitly predict release of buoyancy- Account for cloud and precipitation processes and their contribution to vertical motions- Capable of predicting convection and mountain waves
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Solution techniquesNumerical models solve the forecast equations using one of two basic model formulations: grid point or spectral .
Grid point models solve the forecast equations at regularly spaced grid points. The forecast variables are specified on a set of grid points. Derivatives are approximated at each grid point using a variety of arithmetic techniques called finite differences. The choice of finite difference method affects both the computational error and amount of computer time required to run the model. All the Local Area Models uses finite differences.
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Solution techniquesGrid point models actually represent the atmosphere in three-dimensional grid cubes. (The temperature, pressure, and moisture (T, p, and q), shown in the center of the cube, represent the average conditionsthroughout the cube. Likewise, the east-west winds (u) and the north-south winds (v), located at thesides of the cube, represent the average of the wind components between the center of this cube andthe adjacent cubes. Similarly, the vertical motion (w) is represented on the upper and lower faces of thecube. This arrangement of variables within and around the grid cube (called a staggered grid) has advantages when calculating derivatives. It is also physically intuitive; average thermodynamic propertiesinside the grid cube are represented at the center, while the winds on the faces are associated withfluxes into and out of the cube.)
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Solution techniquesWhile finite difference equations appear complex, they are relatively simple and fast for a computer to evaluate. The gridpoint model structure is then used so the equations can be solved in a straightforward way for every grid point to produce a weather forecast.
In practice, more complex expressions are used (that shown in figure) to increase the accuracy of the approximation. Typically, more grid points are also involved in the calculation of each term.
In the real atmosphere, advection often occurs at very smallscales. This lack of resolution in the model representationintroduces errors into the solution of the finite difference equation. The greater the distance between grid points, the less likely themodel will be able to detect small-scale variations in thetemperature and moisture fields. Deficiencies in the ability of thefinite difference approximations to calculate gradients and higherorder derivatives exactly are called truncation errors , and may lead with inestability problems. The time-step interval also contributes to the truncation error: the larger the time-step interval, the greater the error introduced through the finite difference approximation.
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Solution techniquesTime step.The relationship governing the length of time between intermediate forecast steps is givenby the CFL criterion. (Time step between intermediate forecasts must be less than thetime it takes the fastest moving wave in the model (c) to cross a distance of model's grid point spacing . This constraint has a simple physical basis - one must look at a movingparcel often enough to keep track of its actual path. The time step equation computes thelargest time step interval for a given model resolution. Increasing the resolution requiresdecreasing the time step interval .)
(In the graphic, the time step interval (t1) is not short enough for parcel 1, since it has traveled a distance greater than the model's grid point spacing in one time step. This results in problems (noise) in the model solution. The time step interval ( t2) is sufficient for parcel 2, since it has traveled a distance less than model's grid point spacing.In reality, NWP models generally use much shorter time steps than would be computed by this equation. This is due to instabilities in the numerical methods and the use of physical parameterization schemes that require short time step intervals to adequately simulate smaller-scale processes.)
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Solution techniquesGRID POINT MODELS
Characteristics- Data are represented on a fixed set of grid points- Resolution is a function of the grid point spacing- All calculations are performed at grid points- Finite difference approximations are used for solving the derivatives of the model's equations- Truncation error is introduced through finite difference approximations of the primitive equations- The degree of truncation error is a function of grid spacing and time-step interval
Disadvantages
- Finite difference approximations of model equations introduce a significant amount of truncation error- Small-scale noise accumulates when equations are integrated for long periods- The magnitude of computational errors is generally more than in spectral models of comparable resolution- Boundary condition errors can propagate into regional models and affect forecast skill- Non-hydrostatic versions cover only very small domains and short forecast periods
Advantages
- Can provide high horizontal resolution for regional and mesoscale applications- Do not need to transform physics calculations to and from gridded space- As the physics in operational models becomes more complex, grid point models are becoming computationally competitive with spectral models- Non-hydrostatic versions can explicitly forecast details of convection, given sufficient resolution and detail in the initial conditions
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Solution techniques
Spectral models.Conceptually, spectral models instead of using grid points, they use a combination of continuous waves of differing wavelength andamplitude to specify the forecast variables and their derivatives at alllocations (not just at grid points).One of the nice features of the spectral formulation is that mosthorizontal derivatives are calculated directly from the waves and are therefore extremely accurate. This does not mean that spectral modelshave no truncation effects at all. The degree of truncation for a givenspectral model is associated with the scale of the smallest waverepresented by the model. A grid point model tries to include all scalesbut does a poor job of handling waves only a few grid points across. A spectral model represents all of the waves that it resolves perfectly butincludes no information on smaller-scale waves. If the number ofwaves in the model is small only larger features can be representedand smaller-scale features observed in the atmosphere will be entirelyeliminated from the forecast model. Therefore, spectral models withlimited numbers of waves can quickly depart from reality in situationsinvolving rapid growth of initially small-scale features.
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Solution techniques
Spectral models.
(Spectral models are also based on the primitiveequations, but their mathematical formulation andnumerical solutions are quite different from grid pointmodels for some of the forecast variables.)Spectral techniques were developed as a meansof increasing the speed and therefore enhancingthe resolution of global models.(These techniques are most suited to global forecasting. However, as the resolution of global models increasesover the next decade, the advantages of using spectraltechniques may lessen and more global models may begin using grid point formulations.)
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Solution techniquesSPECTRAL MODELSCharacteristics
- Data are represented by wave functions- Resolution is a function of the number of waves used in the model- Model resolution is limited by the maximum number of waves- The linear quantities of the equations of motion can be calculated without introducingcomputational error- Grids are used to perform non-linear and physical calculations- Transformations occur between spectral and grid point space- Equations can be integrated for large time steps and long periods of timeOriginally designed for global domains
Disadvantages- Transformations between spectral and grid point physics calculations introduce errors in the model solution- Generally not designed for higher resolution regional and mesoscale applications- Computational savings decrease as the physical realism of the model increases
Advantages- The magnitude of computational errors in dynamics calculations is generally lessthan in grid point models of comparable resolution- Can calculate the linear quantities of the equations of motion exactly- At horizontal resolutions typically required for global models (late 1990s), require lesscomputing resources than grid point models with equivalent horizontal resolution andphysical processes
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Domain implementation
HORIZONTAL : RESOLUTION AND DOMAIN SIZESynoptical models / Mesoscale models / Microscale models.
Model domain refers to the model's area of coverage.
The horizontal resolution of an NWP model is related to the spacing between grid points forgrid point models or the number of waves that can be resolved for spectral models.( In spectral models, the horizontal resolution is designated by a "T" number (for example, T80), which indicates the number of waves used to represent the data. The "T" stands fortriangular truncation, which indicates the particular set of waves used by a spectral model.)
Note that the smallest features that can be accurately representedby a model are many times larger than the grid 'resolution.' In fact, phenomena with dimensions on the same scale as the grid spacingare unlikely to be depicted or predicted within a model. Whether a model is considered high or low resolution depends uponthe size of the domain and the scale of weather phenomenon thatthe model is trying to predict. (A resolution on the order of 20 to 50 km is considered high for a global model, while for a storm-scale model, a resolution of 100 to 500 m is considered highand necessary to resolve the internal processes of convection.)
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Domain implementation
To address the problem of discontinuous forecast surfaces Phillips (1957) developed a terrain-following coordinate called the sigma coordinate, (illustrated below. The sigma coordinate or variants are used in many mesoscale and synopticsmodels. In its simplest form, the sigma coordinate is defined byp/ps, where p is the pressure on a forecast level within the model and ps is the pressure at the earth's surface, not mean sea level pressure. The lowest coordinate surface (usually labeled 1) follows a smoothed version of the actual terrain. Note that the terrain slopes used in sigma models are always smoothed to some degree. The other sigma surfaces gradually transition from being nearly parallel to the smoothed terrain at the bottom of the model to being nearly horizontal to the constant pressure surface at the top of the model (0). The top layer of the model is typically placed well above the tropopause, usually between 25 and 1 hPa. )
VERTICAL STRUCTURE.
A model's vertical structure is as important in defining the model's behavior as the horizontal configuration and model type. Proper depiction of the vertical structure of the atmosphere requires selection of an appropriate vertical coordinate and sufficient vertical resolution. All operational models use discrete vertical structures.
The equations of motion, which form the basis for all NWP models, have their simplest form in pressure coordinates. Unfortunately, pressure coordinate systems are not particularly suited to solving the forecast equations because, like height surfaces, they can intersect mountains and consequently 'disappear' over parts of the forecast domain.
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Domain implementation
Other vertical coordinates are: sigma-p ,sigma-z,, eta , hibrid-isentropic, hibrid-sigma presure, etc.
(All of the vertical coordinate systems used by NWP models have strengths and weaknesses, bothcomputational and meteorological in nature. coordinate systems have different abilities to represent weather features in the vertical based on their vertical resolution characteristics. Some inherently increase resolution in desired locations, others allow modelers to easily adjust the levels, and still others are not very adaptive or flexible. It is important to understand how the vertical resolution characteristics of each coordinate system affect a given model's ability to represent weather features.)Layer thickness must change gradually from one layer to the next. Otherwise, errors in the model's vertical finite differencing schemes, such as those used to advect momentum, heat, and moisture vertically, will be amplified. Therefore, since resolution must be a few tens of meters near the ground to properly depict a variety of surface processes, there must be a large number of levels in the PBL layer to act as a transition zone with the less-closely spaced levels in the mid-troposphere.
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Initial conditions
INITIAL CONDITIONS (good observations …good first guess).
Objetive analysis brings data from observations into the model in every grid point .Data asimilation techniques improve the Analysis First Guess, including as much Data in the Model Producing the Best First-Guess Possible In the variational analysis approach, analyses are made of the observed variables in their original form. (For example, instead of analysing temperature “soundings” determined from satellite radiances, the model “first guess” fields are converted to radiances and then the differences between the “first-guess” and observed radiances are analysed and converted back to model forecast variables. This eliminates much “variable conversion error”.)Continuous Assimilation (called 4-D Variational Analysis ) allows the analysis to create the best possible fit to the total set ofobservations and their changes throughtout the entire analysis period - The technique, however, needs too many computer resources for widespread operational use today
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Boundary conditions
Nesting is possible via two way or one way.One-way interaction: Information flows in one direction, from thecoarser-mesh, larger domain to the finer-mesh, smaller domain. Computations within the finer-mesh model do not affect the largerdomain model. Two-way interaction: Information flows in both directions in theinterior grid interfaces of a nested model. The coarser-grid forecastsupplies boundary conditions to the finer-grid forecast, while thefiner-grid forecast is used by the coarser grid in determining theforecast variables.
LATERAL BOUNDARY CONDITIONS.Global models don`t need lateral boundary conditions.In LAM models lateral conditions comes from a bigger domain model data (a global model or superior domain in nesting case).LAM models historically tends to put lateral boundaries far enough for the forecast area of interest. By placing the upstream boundary well outside the forecast area, the influence of the boundary errors within the area of interest can be reduced.Physical parameterizations and dynamical formulations of the model providing lateral boundary conditions may be different from the smaller-domain model, which may lead to forecast errors.
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Boundary conditions
UPPER BOUNDARY CONDITIONS.All forecast models, including global models, require that boundaryconditions be specified at the top of the model domain.Model tops are placed well above the tropopause. Assumptionsmust be made as to how the forecast variables will change abovethe assigned top throughout.
Most models employ a rigid upper boundary condition, whichmeans that no vertical motion is allowed through the top of themodel. Problems occur when gravity waves (such as thosegenerated by convection in the model or flow over the modeltopography) reflect off the top of the model. If untreated, thesegravity waves can "bounce" around the entire model depth andseverely affect vertical motion and precipitation forecasts. Fortunately, special numerical treatments, such as the addition ofan "absorbing" or "damping" layer near the model top, have beendeveloped to avoid this problem. These treatments can only be applied when the model top is much higher than any weatherfeatures to be forecast, since the forecast for the highest modellayers will not be realistic.
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Boundary conditions
LOWER BOUNDARYAll forecast models, including global models, require thatboundary conditions be specified at the bottom of themodel domain.
Operational forecast models incorporate as many physicallymeaningful models of surface processes as possible (givencomputational constraints) to make accurate forecasts. Including coupled atmosphere-land interaction (soil moisture, vegetation, and snow forecasts). In the future, coupledatmosphere/land models will become even more sophisticatedto improve the depiction and forecast of surface conditions, andatmosphere/ocean models will likely be incorporated to allow a real linkage between the atmosphere and underlying watersurfaces.
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Boundary conditions
LOWER BOUNDARY( Since models use specified surface characteristics as part of their bottom boundary conditions, errors can be introduced because: - The surface may not be depicted with sufficient resolution to capture conditions necessary to produce an accurate forecast - Local-scale terrain features can be entirely contained within a grid box or two, resulting in poor forecasts of the phenomena they may force.- Processes that impact the model forecast are not properly represented in the model )
Surface processes exert a tremendous influence on sensible weather at the ground and on local variations in the forecast area. Unfortunately, representation of these processes is extremely difficult and contributes strongly to forecast error.
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Codification and integration .
- FORTRAN as programing language (not always).
- EFICIENT ALGORITHS AND CODIFICATION:All numerical integrations made on a computer sufers from: Roundoff errors –characteristics of hardware and float number representation. Increases when number of calculous increases. Truncation error – characteristics of discretization methods employed to solve a particular integration. Depends on algorithms used.
-COMPUTATIONAL RESOURCES.-NWP always on top on computational resources demands (calculus, storage and comunication). Since beginig, NWP has pushed the limits of available computing capacity. To complete one forecast, equations need to be solved for many variables and for thousands of grid points up to several thousands of times. (For example, in 1999, a 48-hr forecast from the Eta Model solves the forecast equations for 3.65 million grid points (223 x 365 for 45 layers) at 1,920 different times (48 hours at a time step of 90 seconds) for each variable)
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Codification and integration .
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1988 1993 20001997 2001
Cray-2S1st SC Systemuntil 1993
Cray C902nd SC Systemuntil 2001
Cray T3E
until 2002 NEC SX-5
IBM p690
4node
2002
PC Cluster
128node
2003
IBM p690+
17node
NEC SX-6TeraCluster
256node+
HP GS320
HPC160/320
Supercomputer performance1970s Mega flops1990s Giga flops2000s Tera flops2010s Peta flops2020s Exa flops?
Producing an accurateforecast is the result of a complex process thatinvolves carefulconsideration of manymeteorological data sources and principles, including numericalweather prediction (NWP) models.
How meteorological forecastprocess works?
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