Slayt 1 - RTC, Regional Training Centre - Turkey

ATMOSPHERIC MODELINGNUMERICAL WEATHER PREDICTIONInstructor: Dr. SEVINC SIRDASISTANBUL TECHNICAL UNIVERSITYFACULTY OF AERONAUTICS and ASTRONAUTICSDEPARTMENT OF METEOROLOGICAL ENGINEERINGEmail address: sirdas@itu.edu.tr

Course Outlines: History of NWP Basic Equations Methods of Solution Boundary and Initial Conditions Model Evaluation Data AssimilationNumerical Methods Finite Differences Taylor Series 1D Types of PDEs

MTO477Brief History of Meteorology and NWPLecture 1

Brief History of Meteorology 340 B.C. Meteorologica - Aristotle 1400's Hygrometer - Cryfts (1450) Anemometer - Alberti (1450) 1500's Thermoscope - Galileo 1600's Barometer - Torricelli (1643) Les Meteores - Descarte (1637) 1700's Trade winds - Hadley (1730) 1800's Three-cell model - Ferrel (1855) Weather maps of surface pressure 1900's Weather prediction from maps -Bjerknes (1903) Polar front theory - Bjerknes(1921) Numerical weather prediction -Richardson (1922) First computer forecast - Charney /von Neumann (1948) Daily balloon observations (1940's) Weather satellites (Tiros I, 1960)

Excerpts from Aristotle’sMeteorologica There are two reasons for there being more winds from the northerly than thesoutherly regions. First, our inhabited region lies toward the north; second,far more rain and snow is pushed up into this region because the other liesbeneath the sun and its course. These melt and are absorbed by the earth andwhen subsequently heated by the sun and the earth’s own heat cause a greaterand more extensive exhalation. Let us now explain lightning and thunder, and then whirlwinds, firewinds andthunderbolts; for the cause of all of them must be assumed to be the same. Aswe have said, there are two kinds of exhalation, moist and dry; and theircombination (air) contains both potentially. It condenses into cloud, as wehave explained before, and the condensation of clouds is thicker toward theirfarther limit. Heat when radiated disperses into the upper region. But any ofthe dry exhalation that gets trapped when the air is in process of cooling isforcibly ejected as the clouds condense and in its course strikes thesurrounding clouds, and the noise caused by the impact is what we callthunder.

The Continuity Equation

The Forces of a Parcel of Air

Molecular scale (10,000 km)Global wind systemsRossby wavesStratospheric ozone lossGlobal warming

Atmospheric Model Aerosol processes (Microphysics) Nucleation/condensation Phase changes Cloud processes Conden./evap./deposition/sublim. Precipitation Stability (Vertical/Slantwise Ascent) Convection Entrainment Radiative transfer UV/visible/near-IR/thermal-IR Scattering/absorption Snow, ice, water albedosMeteorological processes Velocity Geopotential Pressure Water vapor Temperature Density TurbulenceSurface processes Temperatures and water content of Soil Water Snow Sea ice Vegetation Roads Roofs Surface energy/moisture fluxes Ocean-atmosphere exchange Ocean dynamics, chemistry

Part 1: An Overview ofNumerical WeatherPrediction

Highlights 1955 – First Operational Numerical Model – Barotropic Model (Charney) 1962 – First Operational Baroclinic Model (Cressman) 1966 – 6-layer Primitive Equation (PE) model (PEs by Shuman and Hovermale) –First Global PE Model.381 km gridFirst model to have QPF (Quantitave Precipitation Forecasts) 1971 – Limited Fine-Mesh Model (LFM) (Howcroft) –First operational regional model.190 km grid, 6 vertical layersImproved resolution had positive impact on QPF 1978 – 7-Layer PE mode: mesh size/time step halved from 6-layer model 1980 – Global Spectral Model (Sela) replaced 7-layer PE model 30 Spherical harmonic modes (resolves to 30 waves) and 12 levels 1985 – Nested Grid Model (NGM) (Phillips, Hoke) – Model part of Regional Analysis and Forecast System (RAFS) 3 grids: 381, 190, and 80 km and 16 levels Used optimum interpolation (OI) Later frozen for MOS (Model Output Statics) since 1990

Highlights1991 (August) – Regional Analysis and Forecast System (RAFS) updated for last time, NGM run withonly two grids with inner grid doubled in size Implemented Regional Data Assimilation System (RDAS) – included 3-hourly updates from an improved OIanalysis using all off-time data including profiler and Evaluation of Aircraft (ACARS) wind reports & complexquality control procedures1993 (June) – Early NCEP Eta (Mesinger, Janjic, Black) – Replaced the Lyon–Fedder–Mobarry (LFM)as the early run model (for 00 and 12 UTC) 80 km grid with 38 vertical levels1994 (September) – Rapid Update Cycle (RUC) (Forecasts System Lab (FSL), Benjamin) 60 km grid, 25 vertical levels, forecasts out to 12 hours 8 times a day CONUS domain with 3-hourly OI updates at 60 km resolution on 25 hybrid (sigma-theta) vertical levels.1994 (September) – Early Eta analysis upgrades1995 (August ) – Mesoscale version of the Eta model implemented at 03 and 15 UTC for an extendedCONUS (military) domain with 29 km and 50-layer resolution run out to 30 hours Included NMC’s first predictive cloud scheme and new coupled land-surface atm. Package1995 (October) – Major upgrade of Early Eta model 48 km grid with 38 vertical levels (replaced 80 km Eta as the early run)1996 (January) – New coupled land-surface-atmosphere scheme put into early Eta

Highlights 1996 – AVN/MRF changed to T126, 28 levels 1997 (February ) – Upgrade package implemented in the early and Meso Eta runs 1998 (February) – Early Eta upgraded to 32 km grid and 45 levels with 4 soil layers. OI analysis replaced by 3D-Variational Analysis (3D-VAR) method with new data sources 1998 (April) – (The Rapid Update Cycle (RUC) is a high-frequency weather forecast(numerical weather prediction) and data assimilation system) RUC (3-hourly) replacedby hourly RUC II system with extended CONUS domain40 km gird and 40 level resolutionAdditional data sources and extensive physics upgrades 1998 (June) – Meso Eta runs 4 times a day for North America domain at 32 km grid and45 vertical level resolutionUsed new snow analysisAll runs started from (Eta Data Assimilation System (EDAS) Eta Data Assimilation System(EDAS) has been run since 1995 and covers which is fully cycled for all variables 1998 (November) – Eta 03 UTC run moved to 06 UTC. 06 and 18 UTC productions run out to 48 hrs (instead of 33 and 30 hrs)

Highlights 2000 (January) – Resolution upgraded from T126L28 to T170L42 in AVN/MRF. MRF run at T170L42 through day 7, then at T62L28 through day 16. AVN run at T170L42 out to 84 hrs four times a day 2000 (March) – ETA 00 UTC and 12 UTC runs out to 60 hrs 2000 (May) – AVN available out to 126h at full T170 resolution at 00Z 2000 (June) Resolution of AVN/MRF (Medium Range Forecasts Model)ensemble members increased from T62 to T126 for first 60 hr of forecast. 2000 (September) – Eta model resolution changed to 22 km, 50 layers for allfour daily runs and domain expanded to match old 48-km domain. 2000 (December) – High-resolution satellite added to Eta assimilation 2001 (April) –Eta 00 UTC and 12 UTC runs extended to 84 hours 2001 (May) – List of changes implemented in the AVN/MRF

Highlights2001 (November) – Major changes to Eta resolution changed to 12 km, 60 layers for all four daily runs major upgrades to grid-scale precipitation.2002 (March) – AVN runs four times a day out to 384 hrs. Resolution T170L42 to180 hrs thereafter T62L282002 (April) – RUC-20 replaced RUC-2. 50 levels, 20 km grid spacing, improved microphysics and convection boundary conditions from 6 and 18 UTC Eta instead of using older 00 and12 UTC Eta.2002 (April) – MRF is replaced by the 00Z AVN2002 (Sept-Oct) – AVN now referred to as the Global Forecast System model (GFS)2002 (October) – Resolution in GFS changed to T254L64 to 84 hr, T170L42 to 180 hr, T126L28 to384 hr2003 (May) – Change from OI to 3D-VAR in RUC2003 (July) – Eta 06 UTC and 18 UTC runs extended to 84 hr. (all 4 daily runs to 84 hr)2004 (March) – GFS ensemble run 4 times daily. Resolution T126 0-180 hrs, then T62 to 384 hrs.2004 (April) – multiple changes and updates implemented in the RUC2005 (January) – Eta model renamed as North American Model (NAM)

Barotropic AtmosphereSurfaces ofconstant pressurecoincide withsurfaces of constantdensityTemperature is thesame at every pointmeaning there is nothermal wind.No change inintensity withheight and no slopeof systemsNo isotherms on aconstant P chart

Equivalent-Barotropic Atmosphere Constant-pressure surface now hasisotherms on it everywhere parallelto the contours Wind can change speed with heightbut not direction Systems are vertically stacked, notemp advection exists Equivalent-barotropic levelpresumed to be near at 500 mb Atmosphere often close enough tothe equivalent-baroclinic state (i.e.tropics) such that barotropicdynamics may be dominant

Baroclinic Atmosphere No assumptions about thepatterns of density ortemperature on a pressuresurface Thermal wind can now changespeed and direction Systems slope with height Real atmosphere alwaysbaroclinic

Thermal Wind (Remember?)The thermal wind is a measure of the Geostrophic wind shear:Written in vector form:Can also be related to:If the lines are Z, GWIf the lines are T, TWThe jet stream is a wellknownexample of thethermal wind. It arises fromthe horizontal temperaturegradient from the warmtropics to the cold polarregions

Basic Principles of NWP(Fred Carr) In 1905 Bjerknes recognized that NWP was possible in principle Eqs governing time rate of change of meteorological values are known Can integrate these eqs forward in time to get new values Must have “suitable” initial conditions (observations) in order to dothis

Written in general form:tAiA i = dependent forecast variables such as u, v, T etc…F(A i ) = advection and physical forcing terms that can becalculated from obs of A iTo get a forecast, integrate from an initial time t 0 to some time in thefuture t 1t11Ai dt Ftt0tt0Using a forward difference expression forAittAAiitdt tgives us:Δt = time step on the order of minutes – repeated severalhundred times to get a 24 and 48 hour forecasttFA1 0We get A A FAidti iFtAitAititt10

Seek equations between variables you want to know and the forcing mechanisms that causechanges in these variablesIn Other Words:Example of a prognosticequation:In Meteorology we would solve for Du/DtORWritten as:

Example of a diagnostic equation:Consider the vertical component of:DWDt g01 p zRight hand side balances perfectly for large-scale flow:0DW 0 andDtppg g zRTHydrostatic eq. - used to deduce Z from T

Physical ProcessEssential components of NWP models are: Equations i.e. PGF, friction, adiabatic & diabatic heating, advection terms … Numerical Procedures Approximations used to estimate each term Approximations used to integrate model forward in time Grid used over model domain (resolution) Boundary conditions Quality and quantity of obs are vital

NUMERICAL WEATHER PREDICTIONLECTUREInstructor: Dr. SEVINC SIRDASEmail address: sirdas@itu.edu.tr

Model FundamentalsModel Components

Model ComponentsForecast:This represents the final product for which NWP was ultimately developed. The format, meteorologicalvariables, forecast period, and frequency are driven by customer needs.Verification:Forecasters use model verification data to identify specific limitations and statistical biases of modelguidance and to compensate for them. Modelers use verification data to help identify deficiencies sothey can improve forecast model components. Model verification is an integral part of the NWPdevelopment process.Forecast Process:In the forecast process, model output and current observations are combined with the forecaster'sunderstanding of meteorological principles to develop a forecast for the area of responsibility.Centralized subjective guidance is used to help with specialized aspects of the forecast. Themeteorological variables required in the forecast and the customer needs drive the types of guidanceproducts and observations used in the forecast development process.Understanding Meteorological Principles:A thorough understanding of basic meteorological principles and relationships is necessary tointelligently use model guidance so one can, for example, identify when model output is notmeteorologically sound or consistent. As models become more complex and predict more detailed andrealistic-looking features, there is a greater need to understand meteorological principles in order tointelligently take advantage of NWP and avoid being misled. Knowledge of local climatology, terraininfluences, and model performance in the local area is also important to developing the best possibleforecast.Observations:Observations of all types are needed to ascertain current atmospheric conditions and to evaluate theaccuracy of a model's analysis or forecast. Observations provide the ground-truth data and are used tohelp assess the reliability of model output and to make necessary adjustments.

Model ComponentsCentralized Guidance:Using all of the forecasting tools at their disposal, NCEP meteorologists produce subjective, centralized guidanceproducts, such as hurricane track predictions, severe weather outlooks and discussions, and quantitative precipitationforecasts. These products are added to the mix of tools and resources used by forecasters.Numerical Guidance:Numerical guidance products are produced through postprocessing of the model output. They are in a form that can bereadily used by forecasters and are usually displayed on a grid with a different resolution than the original model.Examples include geopotential height charts, MSL pressure, and surface temperature. Aircraft turbulence and icingcharts are examples of fields calculated from numerical model output using physically based empirical relationships.Statistical Guidance:Some sensible weather elements, such as visibility and thunderstorms, are not predicted by the model and cannot bederived directly from the model forecast variables. Other parameters, such as surface maximum temperature, aresensitive to model weaknesses and vary locally. Statistical techniques, such as Model Output Statistics (MOS), have beendeveloped to predict weather elements at particular point locations from direct and postprocessed model fields andother pertinent data, including climatology.Direct Model Output:Direct model output typically refers to gridded forecast data provided at each model grid point and vertical level. Thesedata are not interpolated for locations between model gridpoints and levels. The output data are used by forecasters todevelop a wide variety of local forecast and diagnostic products and provide a look inside the model.Model Output:Model output products include all products that use model fields. The model forecast variables can be looked at directly,postprocessed into grids, plots, station predictions, etc., and used in combination with climatology and other datasources in statistical forecasts. Collectively, they are an important part of the forecast process.Postprocessing:In postprocessing, computations are made to the raw model output to transform it to a format readily usable byforecasters. Diagnostics and meteorological parameters are derived from the forecast variables. In addition, modelvariables are interpolated vertically to surfaces used by forecasters (isobaric, isentropic, and constant altitude) andinterpolated horizontally to forecast locations or output grids. Contour plots are also made. Additional postprocessing,such as using AWIPS algorithms, may be done later. The resulting products are collectively referred to as "numericalguidance."Physics:In NWP, physical processes refer to three types of processes:Those operating on scales smaller than the model resolution but which exert a cumulative effect felt at resolvable scalesThose involving exchanges of energy, water, and momentum between the atmosphere and external sources (forexample, radiation and land and sea surface processes)Cloud and precipitation microphysics

Model ComponentsDynamics:In NWP, dynamic processes refer to atmospheric processes that most often involve the forcing or movement of air, suchas advection, pressure gradient forces, and adiabatic heating and cooling. These processes are described by a set ofhorizontal and vertical momentum, mass conservation, and thermodynamic equations within the forecast model.Assimilation:An assimilation system is a complex procedure in which observed meteorological parameters are converted to forecastvariables and blended with short-range forecasts from an earlier model run to produce the initial conditions used to starta new forecast. The assimilation system tries to find the initial fields of the forecast variables that will optimize theaccuracy of the forecast based on the available data.Numerics:Model numerics refers toModel characteristics such as the mathematical formulation used to solve the model forecast equationsHow data is representedModel resolutionComputational domainCoordinate systemThese all affect the handling of dynamics and how consistently the initial conditions and physical processes arerepresented.Forecast Model:The forecast model contains all of the components needed to compute the current state and three-dimensionalevolution of basic weather variables. The components include the numerics, assimilation system, and treatment ofatmospheric dynamics and physical processes.Computer Resources:The capacity and speed of the computing resources available to run a forecast model govern the amount and complexityof the data and forecast model components used. Thus, computer resources can be a significant limitation to NWP.Quality Control and Analysis:Through a series of checks and tests, data are quality controlled to ensure the viability of the information input into theforecast model. This helps to ensure that inaccurate data are adjusted or removed before going into the analysis. Thejudgments of trained meteorologists are a critical part of the process.Data:Data are collected to describe the initial state of the atmosphere. Data sources include observations from satellites,profilers, surface stations, aircraft, upper-air soundings, and radar.

Physical Processes in NatureNWP models cannot resolve weather features and/or processes that occur within a singlemodel grid box.This example shows complex flow around a variety of surface features:Friction that is large over tall treesTurbulent eddies created around buildings or other obstaclesMuch less surface friction over open areasA model cannot resolve any of these local flows, swirls, or obstacles if they exist within agrid box. However, the model must account for the aggregate effect of these surfaces onthe low-level flow with a single number that goes into the friction (F) term in the forecastwind equation. The method of accounting for such effects without directly forecastingthem is called parameterization.

Parameterized Processes and Parameters The graphic depicts some of the physical processes and parameters that aretypically parameterized, both because they cannot be explicitly predicted in fulldetail in model forecast equations regardless of the grid point or wave numberresolution and because their effects on the forecast variables resolvable by amodel are crucial to forecast realism. to identify 22 of these physical processesand parameters.

Parameterizing Physical Processes Parameterization is how we include the effects of physicalprocesses implicitly when we cannot include the processesthemselves explicitly. Parameterization can be thought of asemulation (modeling the effects of a process) rather thansimulation (modeling the process itself). Parameterization is necessary for several reasons: Computers are not yet powerful enough to treat many physicalprocesses explicitly because they are either too small (asdiscussed earlier) or complex to be resolved Many other physical processes cannot be explicitly modeledbecause they are not sufficiently understood to be representedin equation format or there are no appropriate data

Parameterizing Sub Grid-Scale ProcessesThe following examples of atmospheric processes illustrate the need to parameterize subgrid-scale processes in order to account for their effects on the larger-scale forecastvariables.Convective processes: Important vertical redistribution of heat and moisture byconvection can easily occur between mesoscale grid boxes. The animation shows thedevelopment of the rain shaft (white and gray) and the accompanying cold pool (blueshading). Notice that sub grid-scale variations in the convection will have an effect on themoisture and heating in some of the model grid boxes.

Parameterizing Sub Grid-Scale ProcessesMicrophysical processes: Even in very high-resolution models, microphysicalprocesses occur on a scale too small to be modeled explicitly. There areimportant variations in both the horizontal and vertical. In this example, thecloud microphysical processes of condensation and droplet growth areoccurring inside a 1-km model grid box.

Accounting for the Effects of PhysicalProcessesEach important physical process that cannot be directly predicted requires aparameterization scheme based on reasonable physical (for example, radiation) orstatistical (for example, inferring cloudiness from relative humidity) representations. Thescheme must derive information about the processes from the variables in the forecastequations using a set of assumptions. Closure refers to the link between the assumptionsin the parameterization and the forecast variables. (It closes the loop between theparameterization and forecast equations.)Several types of assumptions are used to "create" information.Empirical/statistical: This assumes that a given relationship holds in every case (forexample, surface layer wind speed variations with height for PBL processes and surfacewind forecasts). Note that for a normal statistical distribution, one of every 20 cases isexpected to be an outlier.Dynamical/thermodynamical constraining assumption: A complex process is summarizedthrough a simplified relationship, for example, equilibrium of instability for Arakawa-Schubert convective parameterization.Model within a model: Although the use of nested models (for example, one-dimensionalcloud models and soil models) pushes the assumption back to a finer detail, assumptionsmust still be made. Running a model within a model requires far more development bymodelers and takes longer to run.The key problem of numerical parameterization is trying to predict with incompleteinformation, for example, the effects of sub grid-scale processes with information at thegrid scale. Imagine using the wind forecast in a grid box to predict boundary-layerturbulence without knowing topography details, vegetation characteristics, or the detailsof structures at the surface.

Impacts of Parameterizing Problems associated with using parameterizations can resultfrom Interactions between parameterization schemes, where eachscheme contains its own set of errors and assumptions (forexample, a soil model and radiation scheme passing back andforth information about heating the boundary layer) The increasing complexity and interconnectedness ofparameterizations, which result in forecast errors that are moredifficult to trace back to specific processes The largest impact of using parameterization schemes is usuallyon predictions of sensible weather at the surface. These problems and impacts make generic forecast rules-ofthumbless useful and require that forecasters apply physicalreasoning on a case-by-case basis when the processes beingparameterized are important to the forecast.

Outlook As computer power increases and the number and complexity of schemesgrows, it is important to remember the following:1. The improved simulation of natural detail important to atmospheric processesleads to greater forecast sensitivity to physical parameters whose values arepoorly or not at all known. (This is especially true for detailed structure in shortrangeforecasts and long-term background means in long-range forecasts)2. More sophisticated schemes and finer resolution will lead to more realisticlookingforecast detail but also more complicated model error characteristics3. The increasing complexity of model error characteristics will result in greaterreliance on model diagnostics to make adjustments to the model forecast fields4. Model changes will take longer to develop and test because changes in oneparameterization affect the behavior of other parameterizations through acomplex web of interactions5. Operational model changes will continue to be released in bundles, rather thanindividually, because of the need to test complex interactions together6. Although model skill will improve and model phenomena will appear morerealistic, model output will still require human interpretation and adjustment

IMPACT OF MODEL STRUCTURES&DYNAMICS1-MODEL TYPE Introductory Question? Knowledge of model type (i.e., whether a model is grid point orspectral) does not have an obvious application to the interpretation ofmodel forecast output as, for example, knowledge of a model'shorizontal resolution does. Yet there are many important reasons forknowing the type of model you are using. See if you can identify some ofthem… Which of the following are affected by model type? a) How the model equations are solved b) How the data are represented c) The size of the model's domain d) The model's horizontal and vertical resolution e) The type and scale of weather features that can be resolved

DiscussionThe correct choices are (a), (b), and (e).Grid point and spectral models are based on the same set of primitive equations. However,each type formulates and solves the equations differently. The differences in the basicmathematical formulations contribute to different characteristic errors in model guidance.The differences in the basic mathematical formulations lead to different methods forrepresenting data. Grid point models represent data at discrete, fixed grid points, whereasspectral models use continuous wave functions. Different types and amounts of errors areintroduced into the analyses and forecasts due to these differences in data representation.The characteristics of each model type along with the physical and dynamicapproximations in the equations influence the type and scale of features that a model maybe able to resolve.Model type does not necessarily impact the size of a model's domain. Global models have,however, historically been spectral because the wave functions and spherical harmonics inthe spectral formulation operate over a spherical domain, a good match for global models.Global models are increasingly becoming grid point as computer resources increase.Model type has no direct impact on the choice of horizontal or vertical resolution.Theoretically, grid point and spectral models can be of any resolution, within thelimitations of available computing power.The remainder of this section explores the characteristics and errors associated with gridpoint and spectral models in more detail.

1- NWP EquationsCertain physical laws of motion and conservation of energy (for example, Newton'sSecond Law of Motion and the First Law of Thermodynamics) govern the evolution of theatmosphere. These laws can be converted into a series of mathematical equations thatmake up the core of what we call numerical weather prediction.Vilhelm Bjerknes first recognized that numerical weather prediction was possible inprinciple in 1904. He proposed that weather prediction could be seen as essentially aninitial value problem in mathematics: since equations govern how meteorological variableschange with time, if we know the initial condition of the atmosphere, we can solve theequations to obtain new values of those variables at a later time (i.e., make a forecast).To mathematically represent an NWP model in its simplest form, we writewhereA equals the change in a forecast variable at a particular point in spacet equals the change in time (how far into the future we are forecasting)F(A) represents terms that can cause changes in the value of AThe equation can be expressed in words asThe change in forecast variable A during the time period t is equal to the cumulativeeffects of all processes that force A to change.

1- NWP EquationsIn NWP, future values of meteorological variables are solved for by finding their initial values and thenadding the physical forcing that acts on the variables over the time period of the forecast. This is statedaswhere F(A) stands for the combination of all of the kinds of forcing that can occur.This stepwise process represents the configuration of the prediction equations used in NWP. Thespecific forecast equations used in NWP models are called the primitive equations (not because theyare crude or simplistic, but because they describe the fundamental processes that occur in theatmosphere). These equations govern the motion and thermodynamic changes that occur in theatmosphere and are derived from the complete conservation laws of momentum, mass, energy, andmoisture.The way in which primitive equations are derived from their complete theoretical form and convertedto computer codes can contribute to forecast errors in NWP models.1. The model forecast equations are simplified versions of the actual physical laws governing atmosphericprocesses, especially cloud processes, land-atmosphere exchanges, and radiation. The physical anddynamic approximations in these equations limit the phenomena that can be predicted.2. Due to their complexity, the primitive equations must be solved numerically using algebraicapproximations, rather than calculating complete analytic solutions. These numerical approximationsintroduce error even when the forecast equations completely describe the phenomenon of interest andeven if the initial state were perfectly represented.3. Computer translations of the model forecast equations cannot contain all details at all resolutions.Therefore, some information about atmospheric fields will be missing or misrepresented in the modeleven if perfect observations were available and the initial state of the atmosphere were known exactly.4. Grid point and spectral methods are techniques for representing information about atmosphericvariables in the model and solving the set of forecast equations. Each technique introduces differenttypes of errors.The ways in which NWP models produce forecast guidance and introduce forecast errors are exploredfurther throughout the rest of the Model Type section.

2-The Primitive Equations For forecasting purposes, this set of equations is consideredto be closed and complete (meaning that we can forecastvalues of all terms by solving each of the equations in theproper sequence) since All equations use the same basic forecast variables (u, v, , T,q, and z) The terms Fx, Fy, H, E, and P can also be described in termsof the six basic forecast variables We can specify initial conditions over the domain of themodel We can obtain suitable boundary conditions for all forecastvariables at the boundaries of the model

2-The Primitive Equations

Wind Forecast EquationsWest-to-East ComponentThis equation determines time changes in the west-to-east component of the wind caused by

Horizontal advection of west-to-east wind The graphic depicts an idealized situation to explain how advection of a quantityby the wind works.1. Vertical advection of west-to-east wind2. Deviations from the geostrophic balance ofthe south-to-north wind component.Imbalances between the west-to-eastpressure gradient force and the Coriolisforce acting on the south-to-north wind willchange the west-to-east wind3. Other physical processes, such as surfacefriction and turbulent mixing acting on thewest-to-east wind. The models also includeempirical approximations to try to accountfor atmospheric processes that cannot beforecast directly, although some of theeffects are indirect. For example, radiationand convection are applied only to thetemperature and moisture equations andare not included explicitly in the windforecast equations. However, the changesin temperature at one time will causechanges in the pressure gradient, which inturn will affect the wind at a later time

South-to-North ComponentThis equation determines time changes in the south-to-north component of the windcaused byHorizontal advection of south-to-north windVertical advection of south-to-north windDeviations from geostrophic balance of the west-to-east wind component. Imbalancesbetween the west-to-east pressure gradient force and the Coriolis force acting on thesouth-to-north wind will change the west-to-east windOther physical processes, such as surface friction and turbulent mixing acting on thesouth-to-north wind. The models also include empirical approximations to try to accountfor atmospheric processes that cannot be forecast directly, although some of the effectsare indirect. For example, radiation and convection are applied only to the temperatureand moisture equations and are not included explicitly in the wind forecast equations.However, the changes in temperature at one time will cause changes in the pressuregradient, which in turn will affect the wind at a later timeNote that the two wind components are interrelated -- each is affected by geostrophicimbalances in the other.

Additional Information: Coriolis ForceTo illustrate a conceptual example of the effects of the Coriolis force, the wind(momentum) equations are simplified by assuming that advection and frictionaleffects are equal to zero for an atmosphere initially at rest. Using thisassumption, the equations reduce to the following form. u/t and v/t are the rates of changes in the u and v components of the windduring the time step t fv and fu represent the Coriolis effect (g z/x and g z/y) represent the pressure gradient accelerations in the x and ydirections (the change in z over distance)

Additional Information: Coriolis Force As incoming solar radiation comes into play, the equator heats up morethan over the poles, creating a south-to-north temperature gradientwith temperature decreasing to the north, as illustrated in theanimation. Because warm air has greater thickness than colder, denserair, the upper-level pressure surfaces become higher over the equatorthan over the poles and a north-to-south pressure gradient develops. Inthe equation, the north-to-south gradient term becomes When this term is positive, a northward acceleration is created with airessentially moving down the pressure gradient from south to north (itmoves downhill). This means that the v component of the wind is alsopositive and a positive v component is physically manifest as a southerlywind.

Additional Information: Coriolis ForceWe have not yet considered the Coriolis effects. In the real atmosphere, the earth isrotating and the Coriolis force is not zero except at the equator. Recall that the Corioliseffect results in a deflection of the air to the right in the Northern Hemisphere (fv > 0).If fv > 0, then u/t > 0. This indicates an eastward acceleration since positive u motionindicates a westerly component to the wind.Now that the wind has an eastward component, the Coriolis term (fu) in equation 2 mustalso be considered. In this case, since u > 0, then -fu < 0, reducing the northwardcomponent of the wind.This negative acceleration reduces the southerly wind component and eventually, overseveral hours, the wind becomes northerly (v < 0), deflecting the air parcel toward thesouth. This interaction of the pressure gradient forces and Coriolis effects results in anoscillation, as illustrated below.It is important to note the following.In this example, the air parcel undergoes inertial oscillations in the absence of otherpressure gradient forces. In nature, however, these oscillations are usually quickly dampedand a balanced flow regime is established.The atmospheric response to a pressure gradient force has been presented as a series ofsequential events. In the real atmosphere, these responses occur simultaneously as thewind and pressure fields continuously adjust to each other.

Continuity Equation In this example, the continuity equation is calculated diagnostically fromthe horizontal wind fields without considering buoyancy effects.Horizontal divergence is determined from the spatial variations in bothof the horizontal wind components. The divergence is then related tothe change in vertical motion from the bottom to the top of a layerwithin the model. Areas of horizontal convergence must coincide eitherwith areas where rising motion increases with height or where sinkingmotion weakens with height. The continuity equation is used to calculate vertical motion inhydrostatic models. Non-hydrostatic models, on the other hand, do notuse the continuity equation directly to calculate vertical motion. Rather,they use a combination of horizontal divergence and buoyancy todetermine both vertical motions and vertical accelerations.

Temperature Forecast EquationTime changes in the temperature are related to Horizontal advection of temperature by both wind components The difference between vertical advection of temperature and cooling orwarming caused by expansion or compression of rising or sinking air. Thiscomponent of the temperature change is proportional to the intensity of verticalmotion and the difference between the forecast lapse rate and the dry adiabaticlapse rate The effects of all other processes, notably radiation, mixing, and condensation,including the effects of convectionNote the importance of the vertical velocity determined from the wind forecastand continuity equations. This equation is also dependent upon the moistureforecast equation because of the role of moisture in the amount ofcondensational heating and cooling and in the triggering of convection, whichalso contributes to condensational heating and cooling.

Moisture Forecast EquationTime changes in moisture are related to Horizontal advection of moisture Vertical advection of moisture Evaporation of liquid water or sublimation of ice crystals Condensation (precipitation). Models have many complicatedformulations for estimating condensation and subsequent precipitation.Note that conservation of moisture means that precipitation predictedby the model reduces the amount of moisture in the model atmosphere.Thus, when a model incorrectly forecasts precipitation, the amount ofmoisture downstream is affectedNote the importance of the vertical velocity determined from the windforecast and continuity equations. There is also interdependencybetween the forecast temperature equation and the amount ofevaporation that can be expected from the earth's surface.

Hydrostatic or Vertical MomentumEquation The hydrostatic equation preserves stability within the forecastmodel and is used to calculate the height field necessary fordetermining geostrophic balance in the wind forecast equations.This diagnostic equation links the mean temperature in a layer ofthe model to the difference in height between the upper andlower isobaric surfaces serving as the top and bottom of thelayer. Updated temperatures obtained from the temperatureforecast equation are used here to calculate heights, which arethen used in the wind forecast equations.

Prognostic/Diagnostic EquationsEquations (1a), (1b), (3), and (4) are called prognostic equations because time changes inforecast variables (u, v, T, and q) are determined explicitly using dynamic forcingequations. In equations (2) and (5), the remaining variables (and z) are determinedfrom the prognostic variables. Because they do not calculate time changesdirectly, they are known as diagnostic equations.

Physical ProcessesAll of the forecast equations must try to account for the effects of processes that cannot be forecast directly by the models, due to thecomplexity of the physical processes being simulated (for example, radiation) or because the actual processes occur at scales too smalto be included directly in the model (for example, convective clouds). Shorthand notations for the empirical approximations used in themodel appear as Fx, Fy, H, E, and P in the forecast equations.Fx and Fy (in equations 1a and 1b) are "friction" terms that modify the wind via surface drag but also incorporate other processes,including horizontal and vertical transport of momentum by turbulent eddies (generally called diffusion in large-scale models)."Friction" is affected by vegetation type (trees versus grass), surface type (snow and water), surface temperature, and otherconditions.

Physical ProcessesThe diabatic heating term H (in equation 3) incorporates several processes:H = HL + HC + HR + HSwhereHL is latent heating caused by condensation in large-scale ascent of saturated, stably-stratifiedair or cooling due to evaporation of falling precipitation and evaporation of water at the surfaceHC is latent heating due to condensation occurring in convection (which may itself beapproximated)HR is the radiative heating rate (primarily at the surface for solar radiation and within moistlayers of the atmosphere for infrared radiation; HR is negative for radiative cooling)HS represents sensible heat flux to and from the surface of the earthThe precipitation rate, P = PL + PC (stratiform and convective precipitation), is closely linked toHL and HC. Their calculation depends on details such as whether the model predicts clouds andwhich convective parameterization or microphysics parameterization is used.Evaporation (E) can be due either to evaporative moisture flux from the earth's surface or theevaporation of precipitation before it reaches the ground.To this point, the discussions have been based on flow over a flat surface. The effects ofmountains must also be included in a model. They are accounted for in the choice of a verticalcoordinate (discussed in the Vertical Coordinate and Vertical Resolution sections).The degree to which NWP models can simulate the real atmosphere using these approximationshas a direct impact on the amount of error in the model forecast in areas where these processesare occurring.

How Models Solve the ForecastEquations Numerical models solve the forecast equations using one of two basicmodel formulations: grid point or spectral. Grid point models solve the forecast equations at regularly spaced gridpoints. The forecast variables are specified on a set of grid points(illustrated below). Derivatives are approximated at each grid point using a variety ofarithmetic techniques called finite differences, as illustrated below. Thechoice of finite difference method affects both the computational errorand amount of computer time required to run the model.

How Models Solve the ForecastEquationsSpectral models are also based on the primitive equations, but their mathematicalformulation and numerical solutions are quite different from grid point models for some ofthe forecast variables. Spectral techniques were developed as a means of increasing thespeed and therefore enhancing the resolution of global models. Although thesetechniques can be adapted to limited-area (regional) forecast problems, they are mostsuited to global forecasting. However, as the resolution of global models increases overthe next decade, the advantages of using spectral techniques may lessen and more globalmodels may begin using grid point formulations.

Representing the Forecast VariablesConceptually, spectral models emulate the process of drawing contours through a data field torepresent the forecast variables. Instead of using grid points, they use a combination of continuouswaves of differing wavelength and amplitude to specify the forecast variables and their derivatives at alllocations (not just at grid points).Consider, for example, the process of drawing contours through the same data shown in the grid pointgraphic for use in a two-wave model.Now, the model has added detail by including two sets of local increases and decreases (+ and - areas,respectively) to the "first-guess" contour. In effect, the model has defined a second set of waves with ashorter wavelength and smaller amplitude (two sets of more subtle variations stretching around theglobe, indicated by the positive and negative departures from the first wave), which complete themathematical description of this data set for this two-wave model.Since spectral models represent some of the forecast variables with continuous waves (a combinationof sines and cosines) rather than at separate points along a wave, they can use more accurate numericaltechniques to solve some of the equations and much longer forecast time steps than the finitedifference techniques used by grid point models. The larger time step compensates for the addedcomplexity of the computations required to solve the trigonometric functions. Since some grid pointcalculations are required in spectral models, some computational errors associated with grid pointmodels will still be present.

3-Grid Point: Data Representation In the real atmosphere, temperature, pressure, wind, and moisture vary fromlocation to location in a smooth, continuous way. In the graphic below, thecontinuous temperature field is depicted with the red contours, labeled indegrees Celsius. This is similar to how a spectral model would depict the field. Grid point models, however, perform their calculations on a fixed array ofspatially disconnected grid points. The values at the grid points actuallyrepresent an area average over a grid box. The continuous temperature field,therefore, must be represented at each grid point as shown by the blacknumbers in the right panel. The temperature value at the grid point representsthe grid box volume average.

3-Grid Point: Data RepresentationGrid point models actually represent the atmosphere in three-dimensional grid cubes, suchas the one shown above. The temperature, pressure, and moisture (T, p, and q), shown inthe center of the cube, represent the average conditions throughout the cube. Likewise,the east-west winds (u) and the north-south winds (v), located at the sides of the cube,represent the average of the wind components between the center of this cube and theadjacent cubes. Similarly, the vertical motion (w) is represented on the upper and lowerfaces of the cube.This arrangement of variables within and around the grid cube (called a staggered grid)has advantages when calculating derivatives. It is also physically intuitive; averagethermodynamic properties inside the grid cube are represented at the center, while thewinds on the faces are associated with fluxes into and out of the cube.

4-Grid Point Models As discussed earlier, grid point models must use finite differencetechniques to solve the forecast equations. In the simplified moistureforecast equation shown below, time changes in moisture at the centerof a grid cube are caused by moisture advection across the cube. This, inturn, depends upon the changes in the moisture between the adjacentcubes and the average wind over the grid cube. The cube drawinggraphically illustrates the conceptual moisture equation shown at thebottom.

4-Grid Point ModelsIn the real atmosphere, advection often occurs at very small scales. For example, seabreezes have strong advection but are usually confined to distances of only a fewtens of kilometers from shore. In our example, the grid points are spaced about 80km apart. This lack of resolution introduces errors into the solution of the finitedifference equation. The greater the distance between grid points, the less likely themodel will be able to detect small-scale variations in the temperature and moisturefields. Deficiencies in the ability of the finite difference approximations to calculategradients and higher order derivatives exactly are called truncation errors.

Additional Information: SimplifiedFinite Difference FormThe top finite difference equation can be converted into the form below it to explicitly show that we are solving for the future value ofq. This value depends on its current value and the moisture difference between the grid points to the east and west. This is illustratedconceptually in the bottom equation.While finite difference equations appear complex, they are relatively simple and fast for a computer to evaluate. The grid point modelstructure is then used so the equations can be solved in a straightforward way for every grid point to produce a weather forecast.Note that this is the simplest possible finite difference approximation for the original equation. In practice, more complex expressionsare used to increase the accuracy of the approximation. Typically, more grid points are also involved in the calculation of each term.Additionally, note that forecasters often calculate diagnostic quantities from model output as part of the forecasting process. Thesecalculations will not necessarily be the same as those performed by the forecast model itself, since some variables have been averagedduring model postprocessing. For instance, a complicated quantity such as potential vorticity, which requires an average of thegradients of winds and temperatures over several grid points, will appear to be smoother in the forecaster's diagnostic than was in factthe case in the forecast model itself.

5- SPECTRAL MODELHow Data are RepresentedSpectral models represent the spatial variationsof meteorological variables (such asgeopotential heights) as a finite series of wavesof differing wavelengths.In the introduction, we considered thestructure of a conceptual two-wave model.Let's now look at a real data set.Consider the example of a hemispheric 500-hPaheight field in the top portion of the graphic. Ifthe height data are tabulated at 40°N latitudeevery 10 degrees of longitude (represented ateach yellow dot on the chart), there are 36points around the globe. It takes a minimum offive to seven points to reasonably represent awave and, in this case, five or six waves can bedefined with the data. The locations of thewave troughs are shown in the top part as solidred lines.When the data are plotted in the graph, the fivewave troughs are definable by the blue dots butare unequally spaced. This indicates thepresence of more than one wavelength ofsmall-scale variations. In this case, the shorterwaves represent the synoptic-scale features,while the longer waves represent planetaryfeatures.

Use of Grid Point Methods in Spectral ModelsSpectral models use a combination of computational techniques, both spectraland grid point. Parts of the forecast equations use information about the forecastvariables and their derivatives obtained entirely from the wave representation.Examples of these linear components include the important pressure gradientand Coriolis forces. Horizontal gradients are precisely calculated from the waverepresentation, avoiding errors associated with finite differencing.

Use of Grid Point Methods in Spectral ModelsOther parts of the forecast equations must be calculated on grids, for example, precipitation andradiative processes, vertical advection, and parts of the wind advection terms. Grid point calculationof time tendencies for forecast variables resulting from physical processes introduces truncationerrors. These errors are not removed when time tendencies are transformed back to waverepresentation and noise is introduced in the transformation process.While vertical advections are calculated using finite differencing, which generates truncation errors,horizontal advections, including wind advection, are also calculated on grids. However, specialmathematical properties avoid the introduction of error for these terms.The more accurate computational techniques used in spectral models can be integrated over muchlonger periods than those used in grid point models without the generation of small-scale noise andprovide smoother longer-range forecasts. This is one of the reasons why spectral models are mostoften used in global medium-range forecasting.

Impacts of Grid Point Physics Calculations inSpectral ModelsFor the grid point calculations, thevalues of the forecast variables mustbe transformed from spectralrepresentation to grid points. Theexact location and spacing of the gridpoints is determined by the model's"resolution" (maximum number ofwaves). The location and spacing ofpoints is chosen to closely match themodel's spectral resolution (maximumwave number) and most accuratelycalculate the non-linear dynamicterms. However, since model physicsare also calculated on this grid,problems can result when the localeffects of physics introduce errorsduring the transformation from gridpoint back to spectral representation.The graphic illustrates the process forcalculations done on the grid inspectral models.

Impacts of Grid Point Physics Calculations inSpectral ModelsNow suppose that convectiveprecipitation is triggered at a singlephysics grid point. The graphicillustrates how the effects are feltwithin the model. The red linerepresents the convectiveparameterization that causes a forcingof magnitude 1.0 at a single grid pointon the physics grid in a spectral model.The yellow line is the spectralrepresentation of this forcing plottedback onto the physics grid. Note thatthe associated warming retained in thespectral representation is reduced byaround 33% at that location and itsinfluence spread throughout a longdistance in an unphysical oscillatingpattern, as illustrated here. As themaximum number of waves in thespectral model is doubled, theoscillation fades faster so the distancescale would read about half of what isshown. This example is for a spectralmodel with a maximum wave numberof 170 and a location along 40°N.

6- Spectral Model : Truncation EffectsWhat are the effects of truncation in a spectral model? Recall that in a grid point model, truncation error is associated with the finitedifference approximations used to evaluate the derivatives of the model forecast equations. One of the nice features of the spectralformulation is that most horizontal derivatives are calculated directly from the waves and are therefore extremely accurate.This does not mean that spectral models have no truncation effects at all. The degree of truncation for a given spectral model isassociated with the scale of the smallest wave represented by the model. A grid point model tries to include all scales but does apoor 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 of waves in the model is small (for example, T80), only larger featurescan be represented and smaller-scale features observed in the atmosphere will be entirely eliminated from the forecast model.Therefore, spectral models with limited numbers of waves can quickly depart from reality in situations involving rapid growth ofinitially small-scale features.Several types of wave orientation are possible in spectral models. Triangular (T, as in T170) configuration is the most common inoperational models since it has roughly the same resolution in the zonal and meridional directions around the globe.

7- Hydrostatic Models Most grid point models and all spectral models in thecurrent operational NWP suites are hydrostatic. That is, theyuse the hydrostatic primitive equations, which assume abalance between the weight of the atmosphere and thevertical pressure gradient force. This means that no verticalaccelerations are calculated explicitly. The hydrostatic assumption is valid for synoptic- andplanetary-scale systems and for some mesoscalephenomena. A most notable exception is deep convection,where buoyancy becomes an important force. Hydrostatic models account for the effects of convectionusing statistical parameterizations approximating the largerscalechanges in temperature and moisture caused by nonhydrostaticprocesses.

Non-Hydrostatic ModelsNon-hydrostatic models can explicitly forecast the release of buoyancy in the atmosphereand its detailed effects on the development of deep convection. To accomplish this, nonhydrostaticmodels must include an additional forecast equation that accounts for verticalaccelerations and vertical motions directly, rather than determining the vertical motiondiagnostically, solely from horizontal divergence. The basic form of the equation is similarto that of the horizontal wind forecast equation. Conceptually, it statesIn addition to changes in the vertical motion due to changes in orographic uplift anddescent, changes in vertical motion from one time step to the next in a grid box arecaused byAdvection bringing in air with a different vertical velocityPressure deviations from hydrostatic balance resulting from Changes in horizontal convergence/divergence Phenomena with non-hydrostatic pressure perturbations, such as thunderstorms and mountainwavesBuoyancy (B): Positive (negative) buoyancy generates a tendency toward upward(downward) motion. Positive buoyancy is caused by Warm temperature anomalies in a grid box compared to its surroundings Higher moisture content in a grid box compared to its surroundingsDownward drag caused by the weight of liquid or frozen cloud water and precipitation

Non-Hydrostatic Models In addition, to account for vertical motions and buoyancy properly, nonhydrostaticmodels must include a great deal of detail about cloud andprecipitation processes in their temperature and moisture forecast equations.Since hydrostatic models do not have a vertical motion forecast equation, noneof these processes can directly affect the vertical motion in their predictions. One disadvantage of non-hydrostatic models is longer computation time. Sincethe models must finish running in time for forecasters to use model products,hydrostatic models are more advantageous unless non-hydrostatic phenomenaneed to be simulated or unless resolution finer than around 10 km is needed. Non-hydrostatic models run at very high resolution characteristically predictdetailed mesoscale structure and associated forecast impacts on surroundingareas. For instance, a prediction of a mesoscale convective system will include awell-defined gust front, downstream thick anvil affecting surface temperature,and trailing mesohigh affecting winds for some distance from the activeconvection. These details will look like the kinds of features observed in realconvective systems, but the forecast of convective initiation is subject toconsiderable error, possibly throwing off the whole forecast. Generally,mesoscale detail is most reliably predicted when forced by topography orcoastlines. Otherwise, the detailed structure gives an idea of what to expect ifthe weather event causing it develops, but the timing and placement of thatevent may have considerable error.

SummaryGRID POINT MODELSCharacteristicsData are represented on a fixed set of grid pointsResolution is a function of the grid point spacingAll calculations are performed at grid pointsFinite difference approximations are used for solving the derivatives of the model'sequationsTruncation error is introduced through finite difference approximations of the primitiveequationsThe degree of truncation error is a function of grid spacing and time-step intervalDisadvantagesFinite difference approximations of model equations introduce a significant amount oftruncation errorSmall-scale noise accumulates when equations are integrated for long periodsThe magnitude of computational errors is generally more than in spectral models ofcomparable resolutionBoundary condition errors can propagate into regional models and affect forecast skillNon-hydrostatic versions cover only very small domains and short forecast periodsAdvantagesCan provide high horizontal resolution for regional and mesoscale applicationsDo not need to transform physics calculations to and from gridded spaceAs the physics in operational models becomes more complex, grid point models arebecoming computationally competitive with spectral modelsNon-hydrostatic versions can explicitly forecast details of convection, given sufficientresolution and detail in the initial conditions

HYDROSTATIC MODELS Characteristics Use the hydrostatic primitive equations, diagnosing vertical motionfrom predicted horizontal motions Used for forecasting synoptic-scale phenomena, can forecast somemesoscale phenomena Used in both spectral and grid point models (for instance, the AVN/MRFand Eta) Disadvantages Cannot predict vertical accelerations Cannot predict details of small-scale processes associated withbuoyancy Advantages Can run fast over limited-area domains, providing forecasts in time foroperational use The hydrostatic assumption is valid for many synoptic- and sub-synopticscalephenomena

NON-HYDROSTATIC MODELSCharacteristics 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 onsurrounding weather, resulting in either superior local forecasts or large errorsDisadvantages 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 anothermodel; if the BCs lack the structure and resolution characteristic of fields developing insidethe model domain, they may exert great influence on the forecast May predict realistic-looking phenomena, but the timing and placement may be unreliableAdvantages 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

References Carr, F.H., 1988: Introduction to Numerical Weather Prediction Models atthe National Meteorological Center. University of Oklahoma, 63 pp. Conklin, R.J., 1992: Computer Models Used by AFGWC and NMC forWeather Analysis and Forecasting. AFGWC/TN 92/001, Air WeatherService, 69 pp. Perkey, D.J., 1986: Formulation of mesoscale numerical models.Mesoscale Meteorology and Forecasting, P.S. Ray, Ed., Amer. Meteor.Soc., 573-596. Petersen, R.A., and J.D. Stackpole, 1989: Overview of the NMCproduction suite. Wea. Forecasting, 4, 314-322. Ross, B.B., 1986: An overview of numerical weather prediction.Mesoscale Meteorology and Forecasting, P.S. Ray, Ed., Amer. Meteor.Soc., 720-751. The COMET Program, 1993: Numerical Weather Prediction, a laser disctraining module featuring experts Dr. Fred Carr and Dr. Ralph Petersen.

Primitive equations of motion: Set of governing equations that describe large-scale atmospheric motionsderived from conservation laws governing momentum, mass, energy, andmoisture Best suited for development of comprehensive dynamical-physical models ofthe atmosphere Equations expressed in the Eulerian (fixed obs) framework in x-y-pcoordinates written as:(1)(2)(3)(4)(5)(6)Horizontal Momuentum Eqsof u and vVertical Momentum EqContinuity EqFirst Law of ThermodynamicsConservation of Moisture Eq

The dependent variables in this set of equations are u, v, , , T, andq which are assumed to be continuous functions of the independentvariables x, y, p, and t. Eqs. (1), (2), (5), and (6) are prognostic equations(involve a time derivative) and thus require initialconditions. Initial conditions are derived from observations or the use of somebalance relationship Eqs. (3) and (4) are diagnostic equations and can becomputed once the initial conditions are providedThus, Eqs (1) to (6) constitute a set of 6 equations and6 unknowns

6 primitive equations are considered a closed system if:1. Expressions can be found for F x , F y , H, E, and P in termsof the known dependent variables2. There are suitable initial conditions over the domain3. Suitable lateral boundary conditions for the dependentvariables are formulated (for regional models); allmodels need boundary conditions at the top andbottom levels

Problems in finding suitable lateral boundaryconditions and expressions for Fx, Fy, H, E and PFor lateral conditions – effect of (topography) mountains have to be included inthe model via the lower boundary condition and choice of vertical coordinate.Fx and Fy are ―friction‖ terms which modify the momentum equations – raisesneed for the addition of physics to primitive eqs.The diabatic heating term H also consists of several effects which can bewritten: H = H L (ascent) + H C (convection) + Hr + H SParameterization problem – trying to express subgrid–scaleprocesses in terms of the large-scale dependent variablesEvaporation (E) can be due to moisture flux from surface and evaporation ofprecipitationPrecipitation Rate—related to H L and H C , precipitation efficiency…

Once three conditions are “suitably” met, primitive equations area closed system and can be solved by:1. Obtain observations of the prognostic variables u, v, T, and qover the domain2. Compute from (3) and from (4)3. Compute F x , F y , H, E, P and the other terms on the right-handsides of (1), (2), (5), and (6)4. Integrate the four prognostic equations forward in time toobtain new values of u, v, T, and q5. Repeat steps 2 to 4 until the forecast is complete

Conditions whichOnemakeBIGthe primitiveCautionequations a closedsystem are never perfectly met Leads to a large part of the total forecast error seen in models No two numerical models are alike There are nearly an infinite number of ways to formulate thephysics and many numerical procedures for the solution of theeqs Each model may have its own systematic errors orbiasesImportant to be aware of these limitations in order tomake intelligent use of model data

5 Major Steps in the Production of an NWP ModelObservations All models require obs from an area larger than their forecast domain Forecasts longer than 2-3 days require global data sets Global Telecommunications System (GTS) gathers and disseminates conventional data tonearly all countriesAnalysis Objective analysis – obs checked for errors and interpolated to grid on which modelatmosphere is representedInitialization Adjusts the analyzed data so that the model and data are dynamically consistent Ensures no “noise” is generated when forecast beginsForecast System of forecast eqns marched forward in time until desired forecast length is reachedOutput Forecast maps produced and sent to users, including computations of many quantities notdirectly forecast by the model Forecasts verified to document model errors and biases in order to formulateimprovements in the future.

Courant-Friederichs-Lewy criteria Criteria states that the (Maximum) time step of the model mustbe small enough to capture the fastest moving wave on themodel grid This is determined bytd2ct = time step, d = grid distance, c = speed of fastest waveExample: If you know d = 150 miles and the speed of the fastest wave is 700mph, the length of the time step needed to capture the wave can be calculated.t150mi2700mihr.152hr60min9.1min

Courant-Friederichs-Lewy criteria cont.If the time step and grid spacing is known, this criteria can beused to determine the fastest wave the model will be able toresolveExample d = 30 km and t = 90 sec (MM5 model)dt230km90sec 2.236kmsec3600850kmhrNote – smaller than the speed of sound which is 1152 km/hrIf this criteria is not obeyed, small scale waves amplifyrapidly and overwhelm the solution leading tocomputational instability

Setting up a Numerical Model Grid point models – models that solve the forecastequations at equally spaced grid points. Forecast variables specified on a set of grid points Spectral models – models that emulate the process ofdrawing contours through a data field to represent theforecast variables. Forecast variables at all locations using a combination ofcontinuous waves of differing wavelength and amplitude

Hydrostatic vs. Non-Hydrostatic Hydrostatic models assume hydrostatic equilibrium Valid for most synoptic and global systems and some mesoscalephenomena Non-hydrostatic models include equations for verticalmotions that hydrostatic models lack Most grid-point models and all spectral models inoperation are hydrostatic Many mesoscale models are non-hydrostatic Non-hydrostatic processes and effects becomeimportant when length of a feature is approximatelyequal to its height Typically features 10 km and less in size

Horizontal resolution Horizontal resolution is related to the spacing between grid pointsfor grid point models or the number of waves that can be resolvedfor spectral models Directly related to size of weather feature it can simulate Higher the resolution – smaller the weather feature it can depict Typically takes at least 5 grid point to define a feature (grid pointmodel) Example – a model with 20 km grid spacing cannot resolve anythingless than 100 km in length Increasing horizontal resolution increases computation Additional intermediate forecast steps are required to make samelength of forecast

Vertical Resolution Can be set arbitrarily Highest vertical resolution used where it is needed most Highest is set near Earth’s surface to capture boundary layer processesand near and below tropopause to accurately predict jet stream Not as detailed between 600-300 mb Variety of vertical coordinate types used to represent atmosphericlayers Most common is sigma coordinate system

Sigma Coordinate System (σ)p• Defined as ; p= pressure level, p s =station pressureps• Bottom and top are levels where vertical motion are negligible• Bottom is near the earth’s surface (σ = 1.0)• Top is set to a very small pressure value (σ = 0.0) Near surface sigma levelsclosely mimic terrain Aloft sigma levels flattenout horizontally Sigma levels eliminateproblem of constant heightor pressure surfacesintersecting the ground

ParameterizationThe representation of the effects of sub-grid scale processes interms of grid-scale variables predicted by the model NWP models cannot resolve features and/or processes within agrid box realistically Parameterization has its greatest impact on predictions of sensibleweather at the surface Physical processes typically parameterized Soil moisture/temperature Longwave radiation Solar isolation/reflection Evaporation Convection Cloud and precipitation processes Friction/turbulence

Convective ParameterizationSchemes Most NWP models use these Designed to reduce atmospheric instability in themodel Prediction of precipitation is a by-product of how thescheme reduces instability Expectations of schemes to accurately predictlocation and timing of convective precipitation isusually low

3 reasons processes need to be parameterized1. Phenomena are too small or too complex to be resolvednumerically – computers aren’t powerful enough to directlytreat them2. Processes are often not understood well enough to berepresented by an equation3. Effects profoundly impact model fields and are crucial formaking realistic forecastsProblems associated with using parameterizationsresult from:1. Increasing complexity of parameterization2. Interactions between parameterization schemes – these areharder to trace than errors occurring in a single scheme

«Computations indicate that a perfect modelshould produce three-day forecasts ... Whichare generally good; one week forecasts ...Which are occacionally good; and two-weekforecasts ... Which, although not very good,may contain some useful information» E. LORENZ, 1993: The Essence of Chaos. Univ. OfWashington Press, Seattle. 227 pp.