ATMOSPHERIC MODELING NUMERICAL WEATHER PREDICTION

Instructor: Dr. SEVINC SIRDAS

ISTANBUL TECHNICAL UNIVERSITYFACULTY OF AERONAUTICS and

ASTRONAUTICS DEPARTMENT 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 Assimilation

Numerical Methods Finite Differences Taylor Series 1D Types of PDEs

MTO477

Brief 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’s Meteorologica

There are two reasons for there being more winds from the northerly than the southerly regions. First, our inhabited region lies toward the north; second, far more rain and snow is pushed up into this region because the other lies beneath the sun and its course. These melt and are absorbed by the earth and when subsequently heated by the sun and the earth’s own heat cause a greater and more extensive exhalation.

Let us now explain lightning and thunder, and then whirlwinds, firewinds and thunderbolts; for the cause of all of them must be assumed to be the same. As we have said, there are two kinds of exhalation, moist and dry; and their combination (air) contains both potentially. It condenses into cloud, as we have explained before, and the condensation of clouds is thicker toward their farther limit. Heat when radiated disperses into the upper region. But any of the dry exhalation that gets trapped when the air is in process of cooling is forcibly ejected as the clouds condense and in its course strikes the surrounding clouds, and the noise caused by the impact is what we call thunder.

The Continuity Equation

The Forces of a Parcel of Air

Scales of Motion Molecular scale (<< 2 mm)Molecular diffusionMolecular viscosity

Microscale (2 mm- 2 km)EddiesSmall plumesCar exhaustCumulus clouds

Mesoscale (2 - 2000 km)Gravity wavesThunderstormsTornadosLocal windsUrban air pollution

Synoptic scale (500-10,000 km)Pressure systemsWeather frontsTropical stormsHurricanesAntarctic ozone hole

Planetary 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 albedos

Meteorological processes Velocity Geopotential Pressure Water vapor Temperature Density Turbulence

Surface processes Temperatures and water content of

Soil WaterSnow

Sea ice VegetationRoads

Roofs Surface energy/moisture fluxes Ocean-atmosphere exchange Ocean dynamics, chemistry

Part 1: An Overview of Numerical Weather

Prediction

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 grid First model to have QPF (Quantitave Precipitation Forecasts)

1971 – Limited Fine-Mesh Model (LFM) (Howcroft) – First operational regional model. 190 km grid, 6 vertical layers Improved 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

Highlights 1991 (August) – Regional Analysis and Forecast System (RAFS) updated for last

time, NGM run with only two grids with inner grid doubled in size Implemented Regional Data Assimilation System (RDAS) – included 3-hourly updates from

an improved OI analysis using all off-time data including profiler and Evaluation of Aircraft (ACARS) wind reports & complex quality control procedures

1993 (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 levels

1994 (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 upgrades 1995 (August ) – Mesoscale version of the Eta model implemented at 03 and 15

UTC for an extended CONUS (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. Package

1995 (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) replaced by hourly RUC II system with extended CONUS domain 40 km gird and 40 level resolution Additional data sources and extensive physics upgrades

1998 (June) – Meso Eta runs 4 times a day for North America domain at 32 km grid and 45 vertical level resolution Used new snow analysis All 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 all four 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

Highlights

2001 (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 T62L28

2002 (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 AVN 2002 (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 to 384 hr 2003 (May) – Change from OI to 3D-VAR in RUC 2003 (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 RUC 2005 (January) – Eta model renamed as North American Model (NAM)

Barotropic Atmosphere

Surfaces of constant pressure coincide with surfaces of constant density

Temperature is the same at every point meaning there is no thermal wind.

No change in intensity with height and no slope of systems

No isotherms on a constant P chart

Equivalent-Barotropic Atmosphere

Constant-pressure surface now has isotherms on it everywhere parallel to the contours

Wind can change speed with height but not direction

Systems are vertically stacked, no temp advection exists

Equivalent-barotropic level presumed to be near at 500 mb

Atmosphere often close enough to the equivalent-baroclinic state (i.e. tropics) such that barotropic dynamics may be dominant

Baroclinic Atmosphere

No assumptions about the patterns of density or temperature on a pressure surface

Thermal wind can now change speed and direction

Systems slope with height

Real atmosphere always baroclinic

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, TW

The jet stream is a well-known example of the thermal wind. It arises from the horizontal temperature gradient from the warm tropics to the cold polar regions

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 do this

Written in general form:

Ai = dependent forecast variables such as u, v, T etc…

F(Ai) = advection and physical forcing terms that can be calculated from obs of Ai

To get a forecast, integrate from an initial time t0 to some time in the future t1

Using a forward difference expression for gives us:

Δt = time step on the order of minutes – repeated several hundred times to get a 24 and 48 hour forecast

iiAF

t

A

dtAFdtt

A t

t

i

t

t

i

1

0

1

0

We get dtAFAA

t

t

ii

t

i

t

1

0

01

t

Ai

it

i

t

i

ttAFtAA

Seek equations between variables you want to know and the forcing mechanisms that cause changes in these variables

In Other Words:

Example of a prognostic equation:

In Meteorology we would solve for Du/Dt

OR

Written as:

Example of a diagnostic equation:

0

0

1DW pg

Dt z

0 and DW

Dtp pg

gz RT

Consider the vertical component of:

Right hand side balances perfectly for large-scale flow:

Hydrostatic eq. - used to deduce Z from T

Essential components of NWP models are:

Physical Process 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 PREDICTION

LECTURE

Instructor: Dr. SEVINC SIRDASEmail address: sirdas@itu.edu.tr

Model FundamentalsModel Components

Model Components Forecast: This represents the final product for which NWP was ultimately developed. The format,

meteorological variables, forecast period, and frequency are driven by customer needs. Verification: Forecasters use model verification data to identify specific limitations and statistical

biases of model guidance and to compensate for them. Modelers use verification data to help identify deficiencies so they can improve forecast model components. Model verification is an integral part of the NWP development process.

Forecast Process: In the forecast process, model output and current observations are combined with the

forecaster's understanding 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. The meteorological variables required in the forecast and the customer needs drive the types of guidance products and observations used in the forecast development process.

Understanding Meteorological Principles: A thorough understanding of basic meteorological principles and relationships is

necessary to intelligently use model guidance so one can, for example, identify when model output is not meteorologically sound or consistent. As models become more complex and predict more detailed and realistic-looking features, there is a greater need to understand meteorological principles in order to intelligently take advantage of NWP and avoid being misled. Knowledge of local climatology, terrain influences, and model performance in the local area is also important to developing the best possible forecast.

Observations: Observations of all types are needed to ascertain current atmospheric conditions and to

evaluate the accuracy of a model's analysis or forecast. Observations provide the ground-truth data and are used to help assess the reliability of model output and to make necessary adjustments.

Model Components Centralized Guidance: Using all of the forecasting tools at their disposal, NCEP meteorologists produce subjective, centralized

guidance products, such as hurricane track predictions, severe weather outlooks and discussions, and quantitative precipitation forecasts. 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 be readily 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 icing charts 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 be derived directly from the model forecast variables. Other parameters, such as surface maximum temperature, are sensitive to model weaknesses and vary locally. Statistical techniques, such as Model Output Statistics (MOS), have been developed to predict weather elements at particular point locations from direct and postprocessed model fields and other 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. These data are not interpolated for locations between model gridpoints and levels. The output data are used by forecasters to develop 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 data sources 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 by forecasters. Diagnostics and meteorological parameters are derived from the forecast variables. In addition, model variables are interpolated vertically to surfaces used by forecasters (isobaric, isentropic, and constant altitude) and interpolated 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 "numerical guidance."

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 scales Those involving exchanges of energy, water, and momentum between the atmosphere and external

sources (for example, radiation and land and sea surface processes) Cloud and precipitation microphysics

Model Components Dynamics: In NWP, dynamic processes refer to atmospheric processes that most often involve the forcing or

movement of air, such as advection, pressure gradient forces, and adiabatic heating and cooling. These processes are described by a set of horizontal 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 forecast variables and blended with short-range forecasts from an earlier model run to produce the initial conditions used to start a new forecast. The assimilation system tries to find the initial fields of the forecast variables that will optimize the accuracy of the forecast based on the available data.

Numerics: Model numerics refers to Model characteristics such as the mathematical formulation used to solve the model forecast equations How data is represented Model resolution Computational domain Coordinate system These all affect the handling of dynamics and how consistently the initial conditions and physical

processes are represented. Forecast Model: The forecast model contains all of the components needed to compute the current state and three-

dimensional evolution of basic weather variables. The components include the numerics, assimilation system, and treatment of atmospheric dynamics and physical processes.

Computer Resources: The capacity and speed of the computing resources available to run a forecast model govern the

amount and complexity of 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 the forecast model. This helps to ensure that inaccurate data are adjusted or removed before going into the analysis. The judgments 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 Nature NWP models cannot resolve weather features and/or processes that occur

within a single model grid box.

This example shows complex flow around a variety of surface features: Friction that is large over tall trees Turbulent eddies created around buildings or other obstacles Much less surface friction over open areas A model cannot resolve any of these local flows, swirls, or obstacles if they

exist within a grid box. However, the model must account for the aggregate effect of these surfaces on the low-level flow with a single number that goes into the friction (F) term in the forecast wind equation. The method of accounting for such effects without directly forecasting them is called parameterization.

Parameterized Processes and Parameters

The graphic depicts some of the physical processes and parameters that are typically parameterized, both because they cannot be explicitly predicted in full detail in model forecast equations regardless of the grid point or wave number resolution and because their effects on the forecast variables resolvable by a model are crucial to forecast realism. to identify 22 of these physical processes and parameters.

Parameterizing Physical Processes

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 several reasons: Computers are not yet 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

Parameterizing Sub Grid-Scale Processes The following examples of atmospheric processes illustrate the need to

parameterize sub grid-scale processes in order to account for their effects on the larger-scale forecast variables.

Convective processes: Important vertical redistribution of heat and moisture by convection can easily occur between mesoscale grid boxes. The animation shows the development of the rain shaft (white and gray) and the accompanying cold pool (blue shading). Notice that sub grid-scale variations in the convection will have an effect on the moisture and heating in some of the model grid boxes.

Parameterizing Sub Grid-Scale Processes

Microphysical processes: Even in very high-resolution models, microphysical processes occur on a scale too small to be modeled explicitly. There are important variations in both the horizontal and vertical. In this example, the cloud microphysical processes of condensation and droplet growth are occurring inside a 1-km model grid box.

Accounting for the Effects of Physical Processes 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. Closure refers to the link between the assumptions in the parameterization and the forecast variables. (It closes the loop between the parameterization and forecast equations.)

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.

Model within a model: Although the use of nested models (for example, one-dimensional cloud models and soil models) pushes the assumption back to a finer detail, assumptions must still be made. Running a model within a model requires far more development by modelers and takes longer to run.

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.

Impacts of Parameterizing

Problems associated with using parameterizations can result from

Interactions between parameterization schemes, where each scheme contains its own set of errors and assumptions (for example, a soil model and radiation scheme passing back and forth information about heating the boundary layer)

The increasing complexity and interconnectedness of parameterizations, which result in forecast errors that are more difficult to trace back to specific processes

The largest impact of using parameterization schemes is usually on predictions of sensible weather at the surface.

These problems and impacts make generic forecast rules-of-thumb less useful and require that forecasters apply physical reasoning on a case-by-case basis when the processes being parameterized are important to the forecast.

Outlook As computer power increases and the number and complexity of

schemes grows, it is important to remember the following:

1. The improved simulation of natural detail important to atmospheric processes leads to greater forecast sensitivity to physical parameters whose values are poorly or not at all known. (This is especially true for detailed structure in short-range forecasts and long-term background means in long-range forecasts)

2. More sophisticated schemes and finer resolution will lead to more realistic-looking forecast detail but also more complicated model error characteristics

3. The increasing complexity of model error characteristics will result in greater reliance on model diagnostics to make adjustments to the model forecast fields

4. Model changes will take longer to develop and test because changes in one parameterization affect the behavior of other parameterizations through a complex web of interactions

5. Operational model changes will continue to be released in bundles, rather than individually, because of the need to test complex interactions together

6. Although model skill will improve and model phenomena will appear more realistic, 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 or spectral) does not have an obvious application to the interpretation of model forecast output as, for example, knowledge of a model's horizontal resolution does. Yet there are many important reasons for knowing the type of model you are using. See if you can identify some of them…

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

Discussion

The 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 basic mathematical formulations contribute to different characteristic errors in model guidance.

The differences in the basic mathematical formulations lead to different methods for representing data. Grid point models represent data at discrete, fixed grid points, whereas spectral models use continuous wave functions. Different types and amounts of errors are introduced into the analyses and forecasts due to these differences in data representation.

The characteristics of each model type along with the physical and dynamic approximations in the equations influence the type and scale of features that a model may be 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 in the 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 the limitations of available computing power.

The remainder of this section explores the characteristics and errors associated with grid point and spectral models in more detail.

1- NWP Equations

Certain physical laws of motion and conservation of energy (for example, Newton's Second Law of Motion and the First Law of Thermodynamics) govern the evolution of the atmosphere. These laws can be converted into a series of mathematical equations that make up the core of what we call numerical weather prediction.

Vilhelm Bjerknes first recognized that numerical weather prediction was possible in principle in 1904. He proposed that 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 the initial condition of the atmosphere, we can solve the equations 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 write

where A equals the change in a forecast variable at a particular point in space t 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 A The equation can be expressed in words as The change in forecast variable A during the time period t is equal to the

cumulative effects of all processes that force A to change.

1- NWP Equations In NWP, future values of meteorological variables are solved for by finding their initial

values and then adding the physical forcing that acts on the variables over the time period of the forecast. This is stated as

where 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. The specific forecast equations used in NWP models are called the primitive equations (not because they are crude or simplistic, but because they describe the fundamental processes that occur in the atmosphere). These equations govern the motion and thermodynamic changes that occur in the atmosphere and are derived from the complete conservation laws of momentum, mass, energy, and moisture.

The way in which primitive equations are derived from their complete theoretical form and converted to computer codes can contribute to forecast errors in NWP models.

1. The model forecast equations are simplified versions of the actual physical laws governing atmospheric processes, especially cloud processes, land-atmosphere exchanges, and radiation. The physical and dynamic approximations in these equations limit the phenomena that can be predicted.

2. Due to their complexity, the primitive equations must be solved numerically using algebraic approximations, rather than calculating complete analytic solutions. These numerical approximations introduce error even when the forecast equations completely describe the phenomenon of interest and even 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 model even 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 atmospheric variables in the model and solving the set of forecast equations. Each technique introduces different types of errors.

The ways in which NWP models produce forecast guidance and introduce forecast errors

are explored further throughout the rest of the Model Type section.

2-The Primitive Equations For forecasting purposes, this set of equations is

considered to be closed and complete (meaning that we can forecast values of all terms by solving each of the equations in the proper 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 terms of the six basic forecast variables

We can specify initial conditions over the domain of the model

We can obtain suitable boundary conditions for all forecast variables at the boundaries of the model

2-The Primitive Equations

Wind Forecast Equations

West-to-East Component

This 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 quantity by the wind works.

1. Vertical advection of west-to-east wind 2. Deviations from the geostrophic balance of

the south-to-north wind component. Imbalances between the west-to-east

pressure gradient force and the Coriolis force acting on the south-to-north wind will change the west-to-east wind

3. Other physical processes, such as surface friction and turbulent mixing acting on the

west-to-east wind. The models also include empirical approximations to try to account for atmospheric processes that cannot be

forecast directly, although some of the effects are indirect. For example, radiation

and convection are applied only to the temperature and 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 pressure gradient, which in

turn will affect the wind at a later time

South-to-North Component

This equation determines time changes in the south-to-north component of the wind caused by

Horizontal advection of south-to-north wind Vertical advection of south-to-north wind Deviations from geostrophic balance of the west-to-east wind component.

Imbalances between the west-to-east pressure gradient force and the Coriolis force acting on the south-to-north wind will change the west-to-east wind

Other physical processes, such as surface friction and turbulent mixing acting on the south-to-north wind. The models also include empirical approximations to try to account for atmospheric processes that cannot be forecast directly, although some of the effects are indirect. For example, radiation and convection are applied only to the temperature and 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 pressure gradient, which in turn will affect the wind at a later time

Note that the two wind components are interrelated -- each is affected by geostrophic imbalances in the other.

Additional Information: Coriolis Force

To illustrate a conceptual example of the effects of the Coriolis force, the wind (momentum) equations are simplified by assuming that advection and frictional effects are equal to zero for an atmosphere initially at rest. Using this assumption, 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 wind during 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 y directions (the change in z over distance)

Additional Information: Coriolis Force

As incoming solar radiation comes into play, the equator heats up more than over the poles, creating a south-to-north temperature gradient with temperature decreasing to the north, as illustrated in the animation. Because warm air has greater thickness than colder, denser air, the upper-level pressure surfaces become higher over the equator than over the poles and a north-to-south pressure gradient develops. In the equation, the north-to-south gradient term becomes

When this term is positive, a northward acceleration is created with air essentially moving down the pressure gradient from south to north (it moves downhill). This means that the v component of the wind is also positive and a positive v component is physically manifest as a southerly wind.

Additional Information: Coriolis Force We have not yet considered the Coriolis effects. In the real atmosphere, the

earth is rotating and the Coriolis force is not zero except at the equator. Recall that the Coriolis effect 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 motion indicates a westerly component to the wind.

Now that the wind has an eastward component, the Coriolis term (fu) in equation 2 must also be considered. In this case, since u > 0, then -fu < 0, reducing the northward component of the wind.

This negative acceleration reduces the southerly wind component and eventually, over several hours, the wind becomes northerly (v < 0), deflecting the air parcel toward the south. This interaction of the pressure gradient forces and Coriolis effects results in an oscillation, as illustrated below.

It is important to note the following. In this example, the air parcel undergoes inertial oscillations in the absence

of other pressure gradient forces. In nature, however, these oscillations are usually quickly damped and a balanced flow regime is established.

The atmospheric response to a pressure gradient force has been presented as a series of sequential events. In the real atmosphere, these responses occur simultaneously as the wind and pressure fields continuously adjust to each other.

Continuity Equation

In this example, the continuity equation is calculated diagnostically from the horizontal wind fields without considering buoyancy effects. Horizontal divergence is determined from the spatial variations in both of the horizontal wind components. The divergence is then related to the change in vertical motion from the bottom to the top of a layer within the model. Areas of horizontal convergence must coincide either with areas where rising motion increases with height or where sinking motion weakens with height.

The continuity equation is used to calculate vertical motion in hydrostatic models. Non-hydrostatic models, on the other hand, do not use the continuity equation directly to calculate vertical motion. Rather, they use a combination of horizontal divergence and buoyancy to determine both vertical motions and vertical accelerations.

Temperature Forecast Equation

Time changes in the temperature are related to Horizontal advection of temperature by both wind components The difference between vertical advection of temperature and

cooling or warming caused by expansion or compression of rising or sinking air. This component of the temperature change is proportional to the intensity of vertical motion and the difference between the forecast lapse rate and the dry adiabatic lapse rate

The effects of all other processes, notably radiation, mixing, and condensation, including the effects of convection

Note the importance of the vertical velocity determined from the wind forecast and continuity equations. This equation is also dependent upon the moisture forecast equation because of the role of moisture in the amount of condensational heating and cooling and in the triggering of convection, which also contributes to condensational heating and cooling.

Moisture Forecast Equation

Time 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 complicated

formulations for estimating condensation and subsequent precipitation. Note that conservation of moisture means that precipitation predicted by the model reduces the amount of moisture in the model atmosphere. Thus, when a model incorrectly forecasts precipitation, the amount of moisture downstream is affected

Note the importance of the vertical velocity determined from the wind forecast and continuity equations. There is also interdependency between the forecast temperature equation and the amount of evaporation that can be expected from the earth's surface.

Hydrostatic or Vertical Momentum Equation

The hydrostatic equation preserves stability within the forecast model and is used to calculate the height field necessary for determining geostrophic balance in the wind forecast equations. This diagnostic equation links the mean temperature in a layer of the model to the difference in height between the upper and lower isobaric surfaces serving as the top and bottom of the layer. Updated temperatures obtained from the temperature forecast equation are used here to calculate heights, which are then used in the wind forecast equations.

Prognostic/Diagnostic Equations

Equations (1a), (1b), (3), and (4) are called prognostic equations because time changes in forecast variables (u, v, T, and q) are determined explicitly using dynamic forcing equations. In equations (2) and (5), the remaining variables (and z) are determined from the prognostic variables. Because they do not calculate time changes directly, they are known as diagnostic equations.

Physical Processes

All of the forecast equations must try to account for the effects of processes that cannot be forecast directly by the models, due to the complexity of the physical processes being simulated (for example, radiation) or because the actual processes occur at scales too small to be included directly in the model (for example, convective clouds). Shorthand notations for the empirical approximations used in the model 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 other conditions.

Physical Processes The diabatic heating term H (in equation 3) incorporates several processes:

H = HL + HC + HR + HS

where  HL is latent heating caused by condensation in large-scale ascent of saturated,

stably-stratified air or cooling due to evaporation of falling precipitation and evaporation of water at the surface

HC is latent heating due to condensation occurring in convection (which may itself be approximated)

HR is the radiative heating rate (primarily at the surface for solar radiation and within moist layers of the atmosphere for infrared radiation; HR is negative for radiative cooling)

HS represents sensible heat flux to and from the surface of the earth The precipitation rate, P = PL + PC (stratiform and convective precipitation), is

closely linked to HL and HC. Their calculation depends on details such as whether the model predicts clouds and which convective parameterization or microphysics parameterization is used.

Evaporation (E) can be due either to evaporative moisture flux from the earth's surface or the evaporation of precipitation before it reaches the ground.

To this point, the discussions have been based on flow over a flat surface. The effects of mountains must also be included in a model. They are accounted for in the choice of a vertical coordinate (discussed in the Vertical Coordinate and Vertical Resolution sections).

The degree to which NWP models can simulate the real atmosphere using these approximations has a direct impact on the amount of error in the model forecast in areas where these processes are occurring.

How Models Solve the Forecast Equations

Numerical 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 (illustrated below).

Derivatives are approximated at each grid point using a variety of arithmetic techniques called finite differences, as illustrated below. The choice of finite difference method affects both the computational error and amount of computer time required to run the model.

How Models Solve the Forecast Equations

Spectral models are also based on the primitive equations, but their mathematical formulation and numerical solutions are quite different from grid point models for some of the forecast variables. Spectral techniques were developed as a means of increasing the speed and therefore enhancing the resolution of global models. Although these techniques can be adapted to limited-area (regional) forecast problems, they are most suited to global forecasting. However, as the resolution of global models increases over the next decade, the advantages of using spectral techniques may lessen and more global models may begin using grid point formulations.

Representing the Forecast Variables

Conceptually, spectral models emulate the process of drawing contours through a data field to represent the forecast variables. Instead of using grid points, they use a combination of continuous waves of differing wavelength and amplitude to specify the forecast variables and their derivatives at all locations (not just at grid points).

Consider, for example, the process of drawing contours through the same data shown in the grid point graphic 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 a shorter wavelength and smaller amplitude (two sets of more subtle variations stretching around the globe, indicated by the positive and negative departures from the first wave), which complete the mathematical description of this data set for this two-wave model.

Since spectral models represent some of the forecast variables with continuous waves (a combination of sines and cosines) rather than at separate points along a wave, they can use more accurate numerical techniques to solve some of the equations and much longer forecast time steps than the finite difference techniques used by grid point models. The larger time step compensates for the added complexity of the computations required to solve the trigonometric functions. Since some grid point calculations are required in spectral models, some computational errors associated with grid point models will still be present.

3-Grid Point: Data Representation In the real atmosphere, temperature, pressure, wind, and moisture

vary from location to location in a smooth, continuous way. In the graphic below, the continuous temperature field is depicted with the red contours, labeled in degrees Celsius. This is similar to how a spectral model would depict the field.

Grid point models, however, perform their calculations on a fixed array of spatially disconnected grid points. The values at the grid points actually represent an area average over a grid box. The continuous temperature field, therefore, must be represented at each grid point as shown by the black numbers in the right panel. The temperature value at the grid point represents the grid box volume average.

3-Grid Point: Data Representation

Grid point models actually represent the atmosphere in three-dimensional grid cubes, such as the one shown above. The temperature, pressure, and moisture (T, p, and q), shown in the 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 the adjacent cubes. Similarly, the vertical motion (w) is represented on the upper and lower faces 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; average thermodynamic properties inside the grid cube are represented at the center, while the winds 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

difference techniques to solve the forecast equations. In the simplified moisture forecast equation shown below, time changes in moisture at the center of a grid cube are caused by moisture advection across the cube. This, in turn, depends upon the changes in the moisture between the adjacent cubes and the average wind over the grid cube. The cube drawing graphically illustrates the conceptual moisture equation shown at the bottom.

4-Grid Point Models

In the real atmosphere, advection often occurs at very small scales. For example, sea breezes have strong advection but are usually confined to distances of only a few tens of kilometers from shore. In our example, the grid points are spaced about 80 km apart. This lack of resolution introduces errors into the solution of the finite difference equation. The greater the distance between grid points, the less likely the model will be able to detect small-scale variations in the temperature and moisture fields. Deficiencies in the ability of the finite difference approximations to calculate gradients and higher order derivatives exactly are called truncation errors.

Additional Information: Simplified Finite Difference Form

The top finite difference equation can be converted into the form below it to explicitly show that we are solving for the future value of q. This value depends on its current value and the moisture difference between the grid points to the east and west. This is illustrated conceptually in the bottom equation.

While finite difference equations appear complex, they are relatively simple and fast for a computer to evaluate. The grid point model structure 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 expressions are 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. These calculations will not necessarily be the same as those performed by the forecast model itself, since some variables have been averaged during model postprocessing. For instance, a complicated quantity such as potential vorticity, which requires an average of the gradients of winds and temperatures over several grid points, will appear to be smoother in the forecaster's diagnostic than was in fact the case in the forecast model itself.

5- SPECTRAL MODELHow Data are Represented

Spectral models represent the spatial variations of meteorological variables (such as geopotential heights) as a finite series of waves of differing wavelengths.

In the introduction, we considered the structure of a conceptual two-wave model. Let's now look at a real data set.

Consider the example of a hemispheric 500-hPa height field in the top portion of the graphic. If the height data are tabulated at 40°N latitude every 10 degrees of longitude (represented at each yellow dot on the chart), there are 36 points around the globe. It takes a minimum of five to seven points to reasonably represent a wave and, in this case, five or six waves can be defined with the data. The locations of the wave troughs are shown in the top part as solid red lines.

When the data are plotted in the graph, the five wave troughs are definable by the blue dots but are unequally spaced. This indicates the presence of more than one wavelength of small-scale variations. In this case, the shorter waves represent the synoptic-scale features, while the longer waves represent planetary features.

Use of Grid Point Methods in Spectral Models

Spectral models use a combination of computational techniques, both spectral and grid point. Parts of the forecast equations use information about the forecast variables and their derivatives obtained entirely from the wave representation. Examples of these linear components include the important pressure gradient and Coriolis forces. Horizontal gradients are precisely calculated from the wave representation, avoiding errors associated with finite differencing.

Use of Grid Point Methods in Spectral Models

Other parts of the forecast equations must be calculated on grids, for example, precipitation and radiative processes, vertical advection, and parts of the wind advection terms. Grid point calculation of time tendencies for forecast variables resulting from physical processes introduces truncation errors. These errors are not removed when time tendencies are transformed back to wave representation 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, special mathematical properties avoid the introduction of error for these terms.

The more accurate computational techniques used in spectral models can be integrated over much longer periods than those used in grid point models without the generation of small-scale noise and provide smoother longer-range forecasts. This is one of the reasons why spectral models are most often used in global medium-range forecasting.

Impacts of Grid Point Physics Calculations in Spectral Models

For the grid point calculations, the values of the forecast variables must be transformed from spectral representation to grid points. The exact location and spacing of the grid points is determined by the model's "resolution" (maximum number of waves). The location and spacing of points is chosen to closely match the model's spectral resolution (maximum wave number) and most accurately calculate the non-linear dynamic terms. However, since model physics are also calculated on this grid, problems can result when the local effects of physics introduce errors during the transformation from grid point back to spectral representation.

The graphic illustrates the process for calculations done on the grid in spectral models.

Impacts of Grid Point Physics Calculations in Spectral Models

Now suppose that convective precipitation is triggered at a single physics grid point. The graphic illustrates how the effects are felt within the model. The red line represents the convective parameterization that causes a forcing of magnitude 1.0 at a single grid point on the physics grid in a spectral model. The yellow line is the spectral representation of this forcing plotted back onto the physics grid. Note that the associated warming retained in the spectral representation is reduced by around 33% at that location and its influence spread throughout a long distance in an unphysical oscillating pattern, as illustrated here. As the maximum number of waves in the spectral model is doubled, the oscillation fades faster so the distance scale would read about half of what is shown. This example is for a spectral model with a maximum wave number of 170 and a location along 40°N.

6- Spectral Model : Truncation Effects

What are the effects of truncation in a spectral model? Recall that in a grid point model, truncation error is associated with the finite difference approximations used to evaluate the derivatives of the model forecast equations. One of the nice features of the spectral formulation 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 is associated with the scale of the smallest wave represented by the model. A grid point model tries to include all scales but 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 but includes no information on smaller-scale waves. If the number of waves in the model is small (for example, T80), only larger features can 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 of initially small-scale features.

Several types of wave orientation are possible in spectral models. Triangular (T, as in T170) configuration is the most common in operational 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 the current operational NWP suites are hydrostatic. 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-Hydrostatic Models

Non-hydrostatic models can explicitly forecast the release of buoyancy in the atmosphere and its detailed effects on the development of deep convection. To accomplish this, non-hydrostatic models must include an additional forecast equation that accounts for vertical accelerations and vertical motions directly, rather than determining the vertical motion diagnostically, solely from horizontal divergence. The basic form of the equation is similar to that of the horizontal wind forecast equation. Conceptually, it states

In addition to changes in the vertical motion due to changes in orographic uplift and descent, changes in vertical motion from one time step to the next in a grid box are caused by

Advection bringing in air with a different vertical velocity Pressure deviations from hydrostatic balance resulting from

Changes in horizontal convergence/divergence Phenomena with non-hydrostatic pressure perturbations, such as thunderstorms and

mountain waves Buoyancy (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 surroundings

Downward 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,

non-hydrostatic models must include a great deal of detail about cloud and precipitation processes in their temperature and moisture forecast equations. Since hydrostatic models do not have a vertical motion forecast equation, none of these processes can directly affect the vertical motion in their predictions.

One disadvantage of non-hydrostatic models is longer computation time. Since the models must finish running in time for forecasters to use model products, hydrostatic models are more advantageous unless non-hydrostatic phenomena need to be simulated or unless resolution finer than around 10 km is needed.

Non-hydrostatic models run at very high resolution characteristically predict detailed mesoscale structure and associated forecast impacts on surrounding areas. For instance, a prediction of a mesoscale convective system will include a well-defined gust front, downstream thick anvil affecting surface temperature, and trailing mesohigh affecting winds for some distance from the active convection. These details will look like the kinds of features observed in real convective systems, but the forecast of convective initiation is subject to considerable error, possibly throwing off the whole forecast. Generally, mesoscale detail is most reliably predicted when forced by topography or coastlines. Otherwise, the detailed structure gives an idea of what to expect if the weather event causing it develops, but the timing and placement of that event may have considerable error.

Summary GRID 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

SPECTRAL MODELS

Characteristics 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 introducing

computational 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 time Originally 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 less than 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 less computing

resources than grid point models with equivalent horizontal resolution and physical processes

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 Used in both spectral and grid point models (for instance, the AVN/MRF

and Eta) 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

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 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

References Carr, F.H., 1988: Introduction to Numerical Weather Prediction Models

at the National Meteorological Center. University of Oklahoma, 63 pp. Conklin, R.J., 1992: Computer Models Used by AFGWC and NMC for

Weather Analysis and Forecasting. AFGWC/TN 92/001, Air Weather Service, 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 NMC production 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 disc training module featuring experts Dr. Fred Carr and Dr. Ralph Petersen.

Primitive equations of motion: Set of governing equations that describe large-scale atmospheric

motions derived from conservation laws governing momentum, mass, energy, and moisture

Best suited for development of comprehensive dynamical-physical models of the atmosphere

Equations expressed in the Eulerian (fixed obs) framework in x-y-p coordinates written as:

(6)

(5)

(3)

(1)

(4)

(2)

Horizontal Momuentum Eqs of u and v

Vertical Momentum Eq

Continuity Eq

First Law of Thermodynamics

Conservation of Moisture Eq

Eqs. (1), (2), (5), and (6) are prognostic equations (involve a time derivative) and thus require initial conditions. Initial conditions are derived from observations or the

use of some balance relationship

Eqs. (3) and (4) are diagnostic equations and can be computed once the initial conditions are provided

The dependent variables in this set of equations are u, v, , , T, and q which are assumed to be continuous functions of the

independent variables x, y, p, and t.

Thus, Eqs (1) to (6) constitute a set of 6 equations and 6 unknowns

6 primitive equations are considered a closed system if:

1. Expressions can be found for Fx, Fy, H, E, and P in terms of the known dependent variables

2. There are suitable initial conditions over the domain

3. Suitable lateral boundary conditions for the dependent variables are formulated (for regional models); all models need boundary conditions at the top and bottom levels

Problems in finding suitable lateral boundary conditions and expressions for Fx, Fy, H, E and P

For lateral conditions – effect of (topography) mountains have to be included in the model via the lower boundary condition and choice of vertical coordinate.

Fx and Fy are “friction” terms which modify the momentum equations – raises need for the addition of physics to primitive eqs.

The diabatic heating term H also consists of several effects which can be written: H = HL (ascent) + HC (convection) + Hr + HS

Parameterization problem – trying to express subgrid–scale processes in terms of the large-scale dependent variables

Evaporation (E) can be due to moisture flux from surface and evaporation of precipitation

Precipitation Rate—related to HL and HC, precipitation efficiency…

Once three conditions are “suitably” met, primitive equations are a closed system and can be solved by:

1. Obtain observations of the prognostic variables u, v, T, and q over the domain

2. Compute from (3) and from (4)3. Compute Fx, Fy, H, E, P and the other terms on the

right-hand sides of (1), (2), (5), and (6)4. Integrate the four prognostic equations forward in

time to obtain new values of u, v, T, and q5. Repeat steps 2 to 4 until the forecast is complete

One BIG Caution Conditions which make the primitive equations a closed system 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 the physics and many numerical procedures for the solution of the eqs

Each model may have its own systematic errors or biases

Important to be aware of these limitations in order to make intelligent use of model data

5 Major Steps in the Production of an NWP Model

Observations 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 to nearly all countriesAnalysis Objective analysis – obs checked for errors and interpolated to grid on

which model atmosphere 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 not directly forecast by the model Forecasts verified to document model errors and biases in order to

formulate improvements in the future.

Courant-Friederichs-Lewy criteria

Criteria states that the (Maximum) time step of the model must be small enough to capture the fastest moving wave on the model grid

This is determined by

t = time step, d = grid distance, c = speed of fastest wave

c

dt

2

Example: If you know d = 150 miles and the speed of the fastest wave is 700 mph, the length of the time step needed to capture the wave can be calculated.

min1.9min60152.7002

150

hr

hrmi

mit

Courant-Friederichs-Lewy criteria cont.

If the time step and grid spacing is known, this criteria can be used to determine the fastest wave the model will be able to resolve

Example d = 30 km and t = 90 sec (MM5 model)

hr

kmkmkm

t

d8503600

sec236.

2sec90

30

2

If this criteria is not obeyed, small scale waves amplify rapidly and overwhelm the solution leading to

computational instability

Note – smaller than the speed of sound which is 1152 km/hr

Setting up a Numerical Model Grid point models – models that solve the

forecast equations at equally spaced grid points. Forecast variables specified on a set of grid points

Spectral models – models that emulate the process of drawing contours through a data field to represent the forecast variables. Forecast variables at all locations using a combination

of continuous waves of differing wavelength and amplitude

Hydrostatic vs. Non-Hydrostatic Hydrostatic models assume hydrostatic

equilibrium Valid for most synoptic and global systems and some

mesoscale phenomena

Non-hydrostatic models include equations for vertical motions that hydrostatic models lack

Most grid-point models and all spectral models in operation are hydrostatic

Many mesoscale models are non-hydrostatic Non-hydrostatic processes and effects become

important when length of a feature is approximately equal to its height Typically features 10 km and less in size

Horizontal resolution Horizontal resolution is related to the spacing between

grid points for grid point models or the number of waves that can be resolved for 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 point model) Example – a model with 20 km grid spacing cannot resolve

anything less than 100 km in length Increasing horizontal resolution increases computation

Additional intermediate forecast steps are required to make same length 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 processes and near and below tropopause to accurately predict jet stream

Not as detailed between 600-300 mb Variety of vertical coordinate types used to represent

atmospheric layers Most common is sigma coordinate system

• Defined as ; p= pressure level, ps=station pressure

• 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)

Sigma Coordinate System (σ)

Near surface sigma levels closely mimic terrain

Aloft sigma levels flatten out horizontally

Sigma levels eliminate problem of constant height or pressure surfaces intersecting the ground

sp

p

Parameterization

The representation of the effects of sub-grid scale processes in terms of grid-scale variables predicted by

the model

NWP models cannot resolve features and/or processes within a grid box realistically

Parameterization has its greatest impact on predictions of sensible weather at the surface

Physical processes typically parameterized Soil moisture/temperature Longwave radiation Solar isolation/reflection Evaporation Convection Cloud and precipitation processes Friction/turbulence

Convective Parameterization Schemes

Most NWP models use these Designed to reduce atmospheric instability in

the model Prediction of precipitation is a by-product of

how the scheme reduces instability Expectations of schemes to accurately

predict location and timing of convective precipitation is usually low

3 reasons processes need to be parameterized

1. Phenomena are too small or too complex to be resolved numerically – computers aren’t powerful enough to directly treat them

2. Processes are often not understood well enough to be represented by an equation

3. Effects profoundly impact model fields and are crucial for making realistic forecasts

Problems associated with using parameterizations result from:

1. Increasing complexity of parameterization

2. Interactions between parameterization schemes – these are harder to trace than errors occurring in a single scheme

«Computations indicate that a perfect model should produce three-day forecasts ... Which are generally good; one week forecasts ... Which are occacionally good; and two-week forecasts ... Which, although not very good, may contain some useful information»

E. LORENZ, 1993: The Essence of Chaos. Univ. Of Washington Press, Seattle. 227 pp.