Numerical weather prediction
Overview

Numerical weather prediction uses mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

s of the atmosphere and oceans to predict the weather
Weather forecasting
Weather forecasting is the application of science and technology to predict the state of the atmosphere for a given location. Human beings have attempted to predict the weather informally for millennia, and formally since the nineteenth century...

based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation
Computer simulation
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

in the 1950s that numerical weather predictions produced realistic results. A number of global and regional forecast models are run in different countries worldwide, using current weather observations relayed from radiosonde
A radiosonde is a unit for use in weather balloons that measures various atmospheric parameters and transmits them to a fixed receiver. Radiosondes may operate at a radio frequency of 403 MHz or 1680 MHz and both types may be adjusted slightly higher or lower as required...

s or weather satellites as inputs to the models.

Mathematical models based on the same physical principles can be used to generate either short-term weather forecasts or longer-term climate predictions; the latter are widely applied for understanding and projecting climate change
Climate change
Climate change is a significant and lasting change in the statistical distribution of weather patterns over periods ranging from decades to millions of years. It may be a change in average weather conditions or the distribution of events around that average...

.
Discussions
Encyclopedia
Numerical weather prediction uses mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

s of the atmosphere and oceans to predict the weather
Weather forecasting
Weather forecasting is the application of science and technology to predict the state of the atmosphere for a given location. Human beings have attempted to predict the weather informally for millennia, and formally since the nineteenth century...

based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation
Computer simulation
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

in the 1950s that numerical weather predictions produced realistic results. A number of global and regional forecast models are run in different countries worldwide, using current weather observations relayed from radiosonde
A radiosonde is a unit for use in weather balloons that measures various atmospheric parameters and transmits them to a fixed receiver. Radiosondes may operate at a radio frequency of 403 MHz or 1680 MHz and both types may be adjusted slightly higher or lower as required...

s or weather satellites as inputs to the models.

Mathematical models based on the same physical principles can be used to generate either short-term weather forecasts or longer-term climate predictions; the latter are widely applied for understanding and projecting climate change
Climate change
Climate change is a significant and lasting change in the statistical distribution of weather patterns over periods ranging from decades to millions of years. It may be a change in average weather conditions or the distribution of events around that average...

. The improvements made to regional models have allowed for significant improvements in tropical cyclone track
Tropical cyclone track forecasting
Tropical cyclone track forecasting involves predicting where a tropical cyclone is going to track over the next five days, every 6 to 12 hours. The history of tropical cyclone track forecasting has evolved from a single station approach to a comprehensive approach which uses a variety of...

and air quality forecasts; however, atmospheric models perform poorly at handling processes that occur in a relatively constricted area, such as wildfire
Wildfire
A wildfire is any uncontrolled fire in combustible vegetation that occurs in the countryside or a wilderness area. Other names such as brush fire, bushfire, forest fire, desert fire, grass fire, hill fire, squirrel fire, vegetation fire, veldfire, and wilkjjofire may be used to describe the same...

s.

Manipulating the vast datasets and performing the complex calculations necessary to modern numerical weather prediction requires some of the most powerful supercomputer
Supercomputer
A supercomputer is a computer at the frontline of current processing capacity, particularly speed of calculation.Supercomputers are used for highly calculation-intensive tasks such as problems including quantum physics, weather forecasting, climate research, molecular modeling A supercomputer is a...

s in the world. Even with the increasing power of supercomputers, the forecast skill
Forecast skill
Skill in forecasting is a scaled representation of forecast error that relates the forecast accuracy of a particular forecast model to some reference model....

of numerical weather models extends to about only six days. Factors affecting the accuracy of numerical predictions include the density and quality of observations used as input to the forecasts, along with deficiencies in the numerical models themselves. Although post-processing techniques such as model output statistics
Model output statistics
Model Output Statistics is an omnipresent statistical technique that forms the backbone of modern weather forecasting. The technique pioneered in the 1960s and early 1970s is used to post-process output from numerical weather forecast models...

(MOS) have been developed to improve the handling of errors in numerical predictions, a more fundamental problem lies in the chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

nature of the partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s used to simulate the atmosphere. It is impossible to solve these equations exactly, and small errors grow with time (doubling about every five days). In addition, the partial differential equations used in the model need to be supplemented with parameterizations
Parametrization (climate)
Parameterization in a weather or climate model within numerical weather prediction refers to the method of replacing processes that are too small-scale or complex to be physically represented in the model by a simplified process. This can be contrasted with other processes—e.g., large-scale flow of...

for solar radiation, moist processes (clouds and precipitation
Precipitation (meteorology)
In meteorology, precipitation In meteorology, precipitation In meteorology, precipitation (also known as one of the classes of hydrometeors, which are atmospheric water phenomena is any product of the condensation of atmospheric water vapor that falls under gravity. The main forms of precipitation...

), heat exchange
Heat transfer
Heat transfer is a discipline of thermal engineering that concerns the exchange of thermal energy from one physical system to another. Heat transfer is classified into various mechanisms, such as heat conduction, convection, thermal radiation, and phase-change transfer...

, soil, vegetation, surface water, and the effects of terrain. In an effort to quantify the large amount of inherent uncertainty remaining in numerical predictions, ensemble forecasts
Ensemble forecasting
Ensemble forecasting is a numerical prediction method that is used to attempt to generate a representative sample of the possible future states of a dynamical system...

have been used since the 1990s to help gauge the confidence in the forecast, and to obtain useful results farther into the future than otherwise possible. This approach analyzes multiple forecasts created with an individual forecast model or multiple models.

## History

The history of numerical weather prediction
History of numerical weather prediction
The history of numerical weather prediction considers how current weather conditions as input into mathematical models of the atmosphere and oceans to predict the weather and future sea state has changed over the years...

began in the 1920s through the efforts of Lewis Fry Richardson
Lewis Fry Richardson
Lewis Fry Richardson, FRS   was an English mathematician, physicist, meteorologist, psychologist and pacifist who pioneered modern mathematical techniques of weather forecasting, and the application of similar techniques to studying the causes of wars and how to prevent them...

, who used procedures originally developed by Vilhelm Bjerknes to produce by hand a six-hour forecast for the state of the atmosphere over two points in central Europe, taking at least six weeks to do so. It was not until the advent of the computer and computer simulation
Computer simulation
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

s that computation time was reduced to less than the forecast period itself. The ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....

was used to create the first weather forecasts via computer in 1950; in 1954, Carl-Gustav Rossby's group at the Swedish Meteorological and Hydrological Institute used the same model to produce the first operational forecast (i.e. routine predictions for practical use). Operational numerical weather prediction in the United States began in 1955 under the Joint Numerical Weather Prediction Unit (JNWPU), a joint project by the U.S. Air Force, Navy and Weather Bureau. In 1956, Norman Phillips developed a mathematical model which could realistically depict monthly and seasonal patterns in the troposphere; this became the first successful climate model
Climate model
Climate models use quantitative methods to simulate the interactions of the atmosphere, oceans, land surface, and ice. They are used for a variety of purposes from study of the dynamics of the climate system to projections of future climate...

. Following Phillips' work, several groups began working to create general circulation models. The first general circulation climate model that combined both oceanic and atmospheric processes was developed in the late 1960s at the NOAA Geophysical Fluid Dynamics Laboratory
Geophysical Fluid Dynamics Laboratory
The Geophysical Fluid Dynamics Laboratory is a laboratory in the National Oceanic and Atmospheric Administration /Office of Oceanic and Atmospheric Research . The current director is Dr. V...

.

As computers have become more powerful, the size of the initial data sets has increased and newer atmospheric models have been developed to take advantage of the added available computing power. These newer models include more physical processes in the simplifications of the equations of motion in numerical simulations of the atmosphere. In 1966, West Germany
West Germany
West Germany is the common English, but not official, name for the Federal Republic of Germany or FRG in the period between its creation in May 1949 to German reunification on 3 October 1990....

and the United States began producing operational forecasts based on primitive-equation models
Primitive equations
The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models...

, followed by the United Kingdom in 1972 and Australia in 1977. The development of limited area (regional) models facilitated advances in forecasting the tracks of tropical cyclone
Tropical cyclone
A tropical cyclone is a storm system characterized by a large low-pressure center and numerous thunderstorms that produce strong winds and heavy rain. Tropical cyclones strengthen when water evaporated from the ocean is released as the saturated air rises, resulting in condensation of water vapor...

s as well as air quality in the 1970s and 1980s. By the early 1980s models began to include the interactions of soil and vegetation with the atmosphere, which led to more realistic forecasts.

The output of forecast models based on atmospheric dynamics is unable to resolve some details of the weather near the Earth's surface. As such, a statistical relationship between the output of a numerical weather model and the ensuing conditions at the ground was developed in the 1970s and 1980s, known as model output statistics
Model output statistics
Model Output Statistics is an omnipresent statistical technique that forms the backbone of modern weather forecasting. The technique pioneered in the 1960s and early 1970s is used to post-process output from numerical weather forecast models...

(MOS). Starting in the 1990s, model ensemble forecasts have been used to help define the forecast uncertainty and to extend the window in which numerical weather forecasting is viable farther into the future than otherwise possible.

## Initialization

The atmosphere
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

is a fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

. As such, the idea of numerical weather prediction is to sample the state of the fluid at a given time and use the equations of fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

and thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

to estimate the state of the fluid at some time in the future. The process of entering observation data into the model to generate initial conditions
Initial value problem
In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

is called initialization. On land, terrain maps available at resolutions down to 1 kilometre (0.621372736649807 mi) globally are used to help model atmospheric circulations within regions of rugged topography, in order to better depict features such as downslope winds, mountain waves and related cloudiness that affects incoming solar radiation. The main inputs from country-based weather services are observations from devices (called radiosonde
A radiosonde is a unit for use in weather balloons that measures various atmospheric parameters and transmits them to a fixed receiver. Radiosondes may operate at a radio frequency of 403 MHz or 1680 MHz and both types may be adjusted slightly higher or lower as required...

s) in weather balloons that measure various atmospheric parameters and transmits them to a fixed receiver, as well as from weather satellite
Weather satellite
The weather satellite is a type of satellite that is primarily used to monitor the weather and climate of the Earth. Satellites can be either polar orbiting, seeing the same swath of the Earth every 12 hours, or geostationary, hovering over the same spot on Earth by orbiting over the equator while...

s. The World Meteorological Organization
World Meteorological Organization
The World Meteorological Organization is an intergovernmental organization with a membership of 189 Member States and Territories. It originated from the International Meteorological Organization , which was founded in 1873...

acts to standardize the instrumentation, observing practices and timing of these observations worldwide. Stations either report hourly in METAR
METAR
METAR is a format for reporting weather information. A METAR weather report is predominantly used by pilots in fulfillment of a part of a pre-flight weather briefing, and by meteorologists, who use aggregated METAR information to assist in weather forecasting....

reports, or every six hours in SYNOP
SYNOP
SYNOP is a numerical code used for reporting weather observations made by manned and automated weather stations. SYNOP reports are typically sent every six hours on shortwave using RTTY...

reports. These observations are irregularly spaced, so they are processed by data assimilation
Data assimilation
Applications of data assimilation arise in many fields of geosciences, perhaps most importantly in weather forecasting and hydrology. Data assimilation proceeds by analysis cycles...

and objective analysis methods, which perform quality control and obtain values at locations usable by the model's mathematical algorithms. Some global models use finite differences, in which the world is represented as discrete points on a regularly spaced grid of latitude and longitude; other models use spectral methods that solve for a range of wavelengths. The data are then used in the model as the starting point for a forecast.

A variety of methods are used to gather observational data for use in numerical models. Sites launch radiosondes in weather balloons which rise through the troposphere
Troposphere
The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 80% of the atmosphere's mass and 99% of its water vapor and aerosols....

and well into the stratosphere
Stratosphere
The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

. Information from weather satellites is used where traditional data sources are not available. Commerce provides pilot reports along aircraft routes and ship reports along shipping routes. Research projects use reconnaissance aircraft
Weather reconnaissance
Weather reconnaissance is the acquisition of weather data used for research and planning. Typically the term reconnaissance refers to observing weather from the air, as opposed to the ground .- Aircraft :Balloon...

to fly in and around weather systems of interest, such as tropical cyclone
Tropical cyclone
A tropical cyclone is a storm system characterized by a large low-pressure center and numerous thunderstorms that produce strong winds and heavy rain. Tropical cyclones strengthen when water evaporated from the ocean is released as the saturated air rises, resulting in condensation of water vapor...

s. Reconnaissance aircraft are also flown over the open oceans during the cold season into systems which cause significant uncertainty in forecast guidance, or are expected to be of high impact from three to seven days into the future over the downstream continent. Sea ice began to be initialized in forecast models in 1971. Efforts to involve sea surface temperature
Sea surface temperature
Sea surface temperature is the water temperature close to the oceans surface. The exact meaning of surface varies according to the measurement method used, but it is between and below the sea surface. Air masses in the Earth's atmosphere are highly modified by sea surface temperatures within a...

in model initialization began in 1972 due to its role in modulating weather in higher latitudes of the Pacific.

## Computation

An atmospheric model is a computer program that produces meteorological information for future times at given locations and altitudes. Within any modern model is a set of equations, known as the primitive equations
Primitive equations
The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models...

, used to predict the future state of the atmosphere. These equations—along with the ideal gas law
Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas. It is a good approximation to the behavior of many gases under many conditions, although it has several limitations. It was first stated by Émile Clapeyron in 1834 as a combination of Boyle's law and Charles's law...

—are used to evolve the density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

, pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

, and potential temperature scalar field
Scalar field
In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the...

s and the air velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

(wind) vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

of the atmosphere through time. Additional transport equations for pollutants and other aerosol
Aerosol
Technically, an aerosol is a suspension of fine solid particles or liquid droplets in a gas. Examples are clouds, and air pollution such as smog and smoke. In general conversation, aerosol usually refers to an aerosol spray can or the output of such a can...

s are included in some primitive-equation high-resolution models as well. The equations used are nonlinear partial differential equations which are impossible to solve exactly through analytical methods, with the exception of a few idealized cases. Therefore, numerical methods obtain approximate solutions. Different models use different solution methods: some global models and almost all regional models use finite difference method
Finite difference method
In mathematics, finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.- Derivation from Taylor's polynomial :...

s for all three spatial dimensions, while other global models and a few regional models use spectral method
Spectral method
Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Dynamical Systems, often involving the use of the Fast Fourier Transform. Where applicable, spectral methods have excellent error properties, with the so called "exponential...

s for the horizontal dimensions and finite-difference methods in the vertical.

These equations are initialized from the analysis data and rates of change are determined. These rates of change predict the state of the atmosphere a short time into the future; the time increment for this prediction is called a time step. The equations are then applied to this new atmospheric state to find new rates of change, and these new rates of change predict the atmosphere at a yet further time step into the future. This time stepping is repeated until the solution reaches the desired forecast time. The length of the time step chosen within the model is related to the distance between the points on the computational grid, and is chosen to maintain numerical stability
Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....

. Time steps for global models are on the order of tens of minutes, while time steps for regional models are between one and four minutes. The global models are run at varying times into the future. The UKMET Unified Model
Unified Model
The Unified Model is a Numerical Weather Prediction and climate modeling software suite originally developed by the United Kingdom Met Office, and now both used and further developed by many weather-forecasting agencies around the world...

is run six days into the future, the European Centre for Medium-Range Weather Forecasts
European Centre for Medium-Range Weather Forecasts
The European Centre for Medium-Range Weather Forecasts is an independent intergovernmental organisation supported by 19 European Member States and 15 Co-operating States...

model is run out to ten days into the future, while the Global Forecast System
Global Forecast System
The Global Forecast System is a global numerical weather prediction computer model run by NOAA. This mathematical model is run four times a day and produces forecasts up to 16 days in advance, but with decreasing spatial and temporal resolution over time...

model run by the Environmental Modeling Center
Environmental Modeling Center
The Environmental Modeling Center , improves numerical weather, marine and climate predictions at the National Centers for Environmental Prediction , through a broad program of research in data assimilation and modeling...

is run sixteen days into the future. The visual output produced by a model solution is known as a prognostic chart, or prog.

## Parameterization

Some meteorological processes are too small-scale or too complex to be explicitly included in numerical weather prediction models. Parameterization is a procedure for representing these processes by relating them to variables on the scales that the model resolves. For example, the gridboxes in weather and climate models have sides that are between 5 kilometres (3 mi) and 300 kilometres (186.4 mi) in length. A typical cumulus cloud
Cumulus cloud
Cumulus clouds are a type of cloud with noticeable vertical development and clearly defined edges. Cumulus means "heap" or "pile" in Latin. They are often described as "puffy" or "cotton-like" in appearance. Cumulus clouds may appear alone, in lines, or in clusters...

has a scale of less than 1 kilometre (0.621372736649807 mi), and would require a grid even finer than this to be represented physically by the equations of fluid motion. Therefore the processes that such cloud
Cloud
A cloud is a visible mass of liquid droplets or frozen crystals made of water and/or various chemicals suspended in the atmosphere above the surface of a planetary body. They are also known as aerosols. Clouds in Earth's atmosphere are studied in the cloud physics branch of meteorology...

s represent are parameterized, by processes of various sophistication. In the earliest models, if a column of air in a model gridbox was conditionally unstable (essentially, the bottom was warmer and moister than the top) and the water vapor content at any point within the column became saturated then it would be overturned (the warm, moist air would begin rising), and the air in that vertical column mixed. More sophisticated schemes recognize that only some portions of the box might convect
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

and that entrainment
Entrainment (meteorology)
Entrainment is a phenomenon of the atmosphere which occurs when a turbulent flow captures a non-turbulent flow. It is typically used to refer to the capture of a wind flow of high moisture content, or in the case of tropical cyclones, the capture of drier air....

and other processes occur. Weather models that have gridboxes with sides between 5 and 25 km (3.1 and 15.5 mi) can explicitly represent convective clouds, although they need to parameterize cloud microphysics which occur at a smaller scale. The formation of large-scale (stratus
Stratus cloud
A stratus cloud is a cloud belonging to a class characterized by horizontal layering with a uniform base, as opposed to convective clouds that are as tall or taller than wide . More specifically, the term stratus is used to describe flat, hazy, featureless clouds of low altitude varying in color...

-type) clouds is more physically based; they form when the relative humidity
Relative humidity
Relative humidity is a term used to describe the amount of water vapor in a mixture of air and water vapor. It is defined as the partial pressure of water vapor in the air-water mixture, given as a percentage of the saturated vapor pressure under those conditions...

reaches some prescribed value. Sub-grid scale processes need to be taken into account. Rather than assuming that clouds form at 100% relative humidity, the cloud fraction
Cloud fraction
Cloud fraction is the percentage of each pixel in satellite imagery or each gridbox in a weather or climate model that is covered with clouds. A cloud fraction of one means the pixel is completely covered with clouds, while a cloud fraction of zero represents a totally cloud free pixel....

can be related a critical value of relative humidity less than 100%, reflecting the sub grid scale variation that occurs in the real world.

The amount of solar radiation reaching the ground, as well as the formation of cloud droplets occur on the molecular scale, and so they must be parameterized before they can be included in the model. Atmospheric drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...

produced by mountains must also be parameterized, as the limitations in the resolution of elevation
Elevation
The elevation of a geographic location is its height above a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface ....

contours produce significant underestimates of the drag. This method of parameterization is also done for the surface flux of energy between the ocean and the atmosphere, in order to determine realistic sea surface temperatures and type of sea ice found near the ocean's surface. Sun angle as well as the impact of multiple cloud layers is taken into account. Soil type, vegetation type, and soil moisture all determine how much radiation goes into warming and how much moisture is drawn up into the adjacent atmosphere, and thus it is important to parameterize their contribution to these processes. Within air quality models, parameterizations take into account atmospheric emissions from multiple relatively tiny sources (e.g. roads, fields, factories) within specific grid boxes.

## Domains

The horizontal domain of a model is either global, covering the entire Earth, or regional, covering only part of the Earth. Regional models (also known as limited-area models, or LAMs) allow for the use of finer grid spacing than global models because the available computational resources are focused on a specific area instead of being spread over the globe. This allows regional models to resolve explicitly smaller-scale meteorological phenomena that cannot be represented on the coarser grid of a global model. Regional models use a global model to specify conditions at the edge of their domain in order to allow systems from outside the regional model domain to move into its area. Uncertainty and errors within regional models are introduced by the global model used for the boundary conditions of the edge of the regional model, as well as errors attributable to the regional model itself.

The vertical coordinate is handled in various ways. Lewis Fry Richardson's 1922 model used geometric height () as the vertical coordinate. Later models substituted the geometric coordinate with a pressure coordinate system, in which the geopotential height
Geopotential height
Geopotential height is a vertical coordinate referenced to Earth's mean sea level — an adjustment to geometric height using the variation of gravity with latitude and elevation. Thus it can be considered a "gravity-adjusted height"...

s of constant-pressure surfaces become dependent variables, greatly simplifying the primitive equations. This correlation between coordinate systems can be made since pressure decreases with height through the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

. The first model used for operational forecasts, the single-layer barotropic model, used a single pressure coordinate at the 500-millibar (about 5500 m (18,044.6 ft)) level, and thus was essentially two-dimensional. High-resolution models—also called mesoscale models—such as the Weather Research and Forecasting model
Weather Research and Forecasting model
The Weather Research and Forecasting model, , is a specific computer program with a dual use for forecasting and research. It was created through a partnership that includes the National Oceanic and Atmospheric Administration , the National Center for Atmospheric Research , and more than 150 other...

tend to use normalized pressure coordinates referred to as sigma coordinates. This coordinate system receives its name from the independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...

used to scale
Nondimensionalization
Nondimensionalization is the partial or full removal of units from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis...

atmospheric pressures with respect to the pressure at the surface, and in some cases also with the pressure at the top of the domain.

## Model output statistics

Because forecast models based upon the equations for atmospheric dynamics do not perfectly determine weather conditions, statistical methods have been developed to attempt to correct the forecasts. Statistical models were created based upon the three-dimensional fields produced by numerical weather models, surface observations and the climatological conditions for specific locations. These statistical models are collectively referred to as model output statistics
Model output statistics
Model Output Statistics is an omnipresent statistical technique that forms the backbone of modern weather forecasting. The technique pioneered in the 1960s and early 1970s is used to post-process output from numerical weather forecast models...

(MOS), and were developed by the National Weather Service
National Weather Service
The National Weather Service , once known as the Weather Bureau, is one of the six scientific agencies that make up the National Oceanic and Atmospheric Administration of the United States government...

for their suite of weather forecasting models in the late 1960s.

Model output statistics differ from the perfect prog technique, which assumes that the output of numerical weather prediction guidance is perfect. MOS can correct for local effects that cannot be resolved by the model due to insufficient grid resolution, as well as model biases. Because MOS is run after its respective global or regional model, its production is known as post-processing. Forecast parameters within MOS include maximum and minimum temperatures, percentage chance of rain within a several hour period, precipitation amount expected, chance that the precipitation will be frozen in nature, chance for thunderstorms, cloudiness and surface winds.

## Ensembles

In 1963, Edward Lorenz discovered the chaotic nature
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

of the fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

equations involved in weather forecasting. Extremely small errors in temperature, winds, or other initial inputs given to numerical models will amplify and double every five days, making it impossible for long-range forecasts—those made more than two weeks in advance—to predict the state of the atmosphere with any degree of forecast skill
Forecast skill
Skill in forecasting is a scaled representation of forecast error that relates the forecast accuracy of a particular forecast model to some reference model....

. Furthermore, existing observation networks have poor coverage in some regions (for example, over large bodies of water such as the Pacific Ocean), which introduces uncertainty into the true initial state of the atmosphere. While a set of equations, known as the Liouville equations
Liouville's theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics...

, exists to determine the initial uncertainty in the model initialization, the equations are too complex to run in real-time, even with the use of supercomputers. These uncertainties limit forecast model accuracy to about five or six days into the future.

Edward Epstein
Edward Epstein (meteorologist)
Edward Epstein was an American meteorologist who pioneered the use of statistical methods in weather forecasting and the development of ensemble forecasting techniques.During the 1960s he was professor of meteorology at the University of Michigan...

recognized in 1969 that the atmosphere could not be completely described with a single forecast run due to inherent uncertainty, and proposed using an ensemble
Ensemble (fluid mechanics)
In fluid mechanics, an ensemble is an imaginary collection of notionally identical experiments.Each member of the ensemble will have nominally identical boundary conditions and fluid properties...

of stochastic
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

Monte Carlo simulations
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

to produce means
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

s for the state of the atmosphere. Although this early example of an ensemble showed skill, in 1974 Cecil Leith showed that they produced adequate forecasts only when the ensemble probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

was a representative sample of the probability distribution in the atmosphere.

Since the 1990s, ensemble forecasts have been used operationally (as routine forecasts) to account for the stochastic nature of weather processes – that is, to resolve their inherent uncertainty. This method involves analyzing multiple forecasts created with an individual forecast model by using different physical parametrizations
Parametrization (climate)
Parameterization in a weather or climate model within numerical weather prediction refers to the method of replacing processes that are too small-scale or complex to be physically represented in the model by a simplified process. This can be contrasted with other processes—e.g., large-scale flow of...

or varying initial conditions. Starting in 1992 with ensemble forecasts
Ensemble forecasting
Ensemble forecasting is a numerical prediction method that is used to attempt to generate a representative sample of the possible future states of a dynamical system...

prepared by the European Centre for Medium-Range Weather Forecasts
European Centre for Medium-Range Weather Forecasts
The European Centre for Medium-Range Weather Forecasts is an independent intergovernmental organisation supported by 19 European Member States and 15 Co-operating States...

(ECMWF) and the National Centers for Environmental Prediction
National Centers for Environmental Prediction
The United States National Centers for Environmental Prediction delivers national and global weather, water, climate and space weather guidance, forecasts, warnings and analyses to its Partners and External User Communities...

, model ensemble forecasts have been used to help define the forecast uncertainty and to extend the window in which numerical weather forecasting is viable farther into the future than otherwise possible. The ECMWF model, the Ensemble Prediction System, uses singular vectors
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

to simulate the initial probability density
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

, while the NCEP ensemble, the Global Ensemble Forecasting System, uses a technique known as vector breeding.

In a single model-based approach, the ensemble forecast is usually evaluated in terms of an average of the individual forecasts concerning one forecast variable, as well as the degree of agreement between various forecasts within the ensemble system, as represented by their overall spread. Ensemble spread is diagnosed through tools such as spaghetti diagrams, which show the dispersion of one quantity on prognostic charts for specific time steps in the future. Another tool where ensemble spread is used is a meteogram
Meteogram
A meteogram is a time cross-section that produces and uses data for a specific weather station on the ground. It can display past weather conditions up to the current time or forecast conditions from the current time out into the future...

, which shows the dispersion in the forecast of one quantity for one specific location. It is common for the ensemble spread to be too small to include the weather that actually occurs, which can lead to forecasters misdiagnosing model uncertainty; this problem becomes particularly severe for forecasts of the weather about ten days in advance. When ensemble spread is small and the forecast solutions are consistent within multiple model runs, forecasters perceive more confidence in the ensemble mean, and the forecast in general. Despite this perception, a spread-skill relationship is often weak or not found, as spread-error correlations are normally less than 0.6, and only under special circumstances range between 0.6–0.7. The relationship between ensemble spread and forecast skill
Forecast skill
Skill in forecasting is a scaled representation of forecast error that relates the forecast accuracy of a particular forecast model to some reference model....

varies substantially depending on such factors as the forecast model and the region for which the forecast is made.

In the same way that many forecasts from a single model can be used to form an ensemble, multiple models may also be combined to produce an ensemble forecast. This approach is called multi-model ensemble forecasting, and it has been shown to improve forecasts when compared to a single model-based approach. Models within a multi-model ensemble can be adjusted for their various biases, which is a process known as superensemble forecasting. This type of forecast significantly reduces errors in model output.

### Air quality modeling

Air quality forecasting attempts to predict when the concentrations of pollutants will attain levels that are hazardous to public health. The concentration of pollutants in the atmosphere is determined by their transport, or mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

velocity of movement through the atmosphere, their diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

, chemical transformation
Chemical transformation
In chemistry a chemical transformation shows the conversion of a substrate to a product omitting the reagents and catalysts or underlying reaction mechanism as opposed to a chemical reaction...

, and ground deposition
Deposition (Aerosol physics)
In aerosol physics, Deposition is the process by which aerosol particles collect or deposit themselves on solid surfaces, decreasing the concentration of the particles in the air. It can be divided into two sub-processes: dry and wet deposition. The rate of deposition, or the deposition velocity,...

. In addition to pollutant source and terrain information, these models require data about the state of the fluid flow in the atmosphere to determine its transport and diffusion. Meteorological conditions such as thermal inversions can prevent surface air from rising, trapping pollutants near the surface, which makes accurate forecasts of such events crucial for air quality modeling. Urban air quality models require a very fine computational mesh, requiring the use of high-resolution mesoscale weather models; in spite of this, the quality of numerical weather guidance is the main uncertainty in air quality forecasts.

### Climate modeling

A General Circulation Model (GCM) is a mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

that can be used in computer simulations of the general circulation of a planetary atmosphere
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

or ocean. An atmospheric general circulation model (AGCM) is essentially the same as a global numerical weather prediction model, and some (such as the one used in the UK Unified Model) can be configured for both short-term weather forecasts and longer-term climate predictions. Along with sea ice
Sea ice
Sea ice is largely formed from seawater that freezes. Because the oceans consist of saltwater, this occurs below the freezing point of pure water, at about -1.8 °C ....

and land-surface components, AGCMs and oceanic GCMs (OGCM) are key components of global climate models, and are widely applied for understanding the climate
Climate
Climate encompasses the statistics of temperature, humidity, atmospheric pressure, wind, rainfall, atmospheric particle count and other meteorological elemental measurements in a given region over long periods...

and projecting climate change
Climate change
Climate change is a significant and lasting change in the statistical distribution of weather patterns over periods ranging from decades to millions of years. It may be a change in average weather conditions or the distribution of events around that average...

. For example, they can be used to simulate the El Niño-Southern Oscillation
El Niño-Southern Oscillation
El Niño/La Niña-Southern Oscillation, or ENSO, is a quasiperiodic climate pattern that occurs across the tropical Pacific Ocean roughly every five years...

and study its forcings
Forcing function (differential equations)
In a system of differential equations used to describe a time-dependent process, a forcing function is a function that appears in the equations and is only a function of time, not of any of the other variables. In effect, it is a constant for each value of t....

on global climate and the Asian monsoon circulation. For aspects of climate change, a range of man-made chemical emission scenarios can be fed into the climate models to see how an enhanced greenhouse effect
Greenhouse effect
The greenhouse effect is a process by which thermal radiation from a planetary surface is absorbed by atmospheric greenhouse gases, and is re-radiated in all directions. Since part of this re-radiation is back towards the surface, energy is transferred to the surface and the lower atmosphere...

would modify the Earth's climate. Versions designed for climate applications with time scales of decades to centuries were originally created in 1969 by Syukuro Manabe
Syukuro Manabe
is a Japanese meteorologist and climatologist who pioneered the use of computers to simulate global climate change and natural climate variations.-Scientific accomplishments:...

and Kirk Bryan
Kirk Bryan (oceanographer)
Kirk Bryan is an American oceanographer who is considered to be the founder of numerical ocean modeling. Starting in the 1960s at the Geophysical Fluid Dynamics Laboratory, then located in Washington, D.C., Bryan worked with a series of colleagues to develop numerical schemes for solving the...

at the Geophysical Fluid Dynamics Laboratory
Geophysical Fluid Dynamics Laboratory
The Geophysical Fluid Dynamics Laboratory is a laboratory in the National Oceanic and Atmospheric Administration /Office of Oceanic and Atmospheric Research . The current director is Dr. V...

in Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...

. When run for multiple decades, the models use a low resolution, which leaves smaller-scale interactions unresolved.

### Ocean surface modeling

The transfer of energy between the wind blowing over the surface of an ocean and the ocean's upper layer is an important element in wave dynamics. The spectral wave transport equation is used to describe the change in wave spectrum over changing topography. It simulates wave generation, wave movement (propagation within a fluid), wave shoaling
Wave shoaling
In fluid dynamics, wave shoaling is the effect by which surface waves entering shallower water increase in wave height . It is caused by the fact that the group velocity, which is also the wave-energy transport velocity, decreases with the reduction of water depth...

, refraction
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

, energy transfer between waves, and wave dissipation. Since surface winds are the primary forcing mechanism in the spectral wave transport equation, ocean wave models use information produced by numerical weather prediction models as inputs to determine how much energy is transferred from the atmosphere into the layer at the surface of the ocean. Along with dissipation of energy through whitecaps and resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

between waves, surface winds from numerical weather models allow for more accurate predictions of the state of the sea surface.

### Tropical cyclone forecasting

Tropical cyclone forecasting also relies on data provided by numerical weather models. Three main classes of tropical cyclone guidance models exist: Statistical models are based on an analysis of storm behavior using climatology, and correlate a storm's position and date to produce a forecast that is not based on the physics of the atmosphere at the time. Dynamical models are numerical models that solve the governing equations of fluid flow in the atmosphere; they are based on the same principles as other limited-area numerical weather prediction models but may include special computational techniques such as refined spatial domains that move along with the cyclone. Models that use elements of both approaches are called statistical-dynamical models.

In 1978, the first hurricane-tracking model based on atmospheric dynamics—the movable fine-mesh (MFM) model—began operating. Within the field of tropical cyclone track forecasting
Tropical cyclone track forecasting
Tropical cyclone track forecasting involves predicting where a tropical cyclone is going to track over the next five days, every 6 to 12 hours. The history of tropical cyclone track forecasting has evolved from a single station approach to a comprehensive approach which uses a variety of...

, despite the ever-improving dynamical model guidance which occurred with increased computational power, it was not until the 1980s when numerical weather prediction showed skill
Forecast skill
Skill in forecasting is a scaled representation of forecast error that relates the forecast accuracy of a particular forecast model to some reference model....

, and until the 1990s when it consistently outperformed statistical
Statistical model
A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...

or simple dynamical models. Predictions of the intensity of a tropical cyclone based on numerical weather prediction continue to be a challenge, since statistical methods continue to show higher skill over dynamical guidance.

### Wildfire modeling

On a molecular scale, there are two main competing reaction processes involved in the degradation of cellulose
Cellulose
Cellulose is an organic compound with the formula , a polysaccharide consisting of a linear chain of several hundred to over ten thousand β linked D-glucose units....

, or wood fuels, in wildfire
Wildfire
A wildfire is any uncontrolled fire in combustible vegetation that occurs in the countryside or a wilderness area. Other names such as brush fire, bushfire, forest fire, desert fire, grass fire, hill fire, squirrel fire, vegetation fire, veldfire, and wilkjjofire may be used to describe the same...

s. When there is a low amount of moisture in a cellulose fiber, volatilization of the fuel occurs; this process will generate intermediate gaseous products that will ultimately be the source of combustion
Combustion
Combustion or burning is the sequence of exothermic chemical reactions between a fuel and an oxidant accompanied by the production of heat and conversion of chemical species. The release of heat can result in the production of light in the form of either glowing or a flame...

. When moisture is present—or when enough heat is being carried away from the fiber, charring
Charring
Charring is a chemical process of incomplete combustion of certain solids when subjected to high heat. The resulting residue matter is called Char. By the action of heat, charring removes hydrogen and oxygen from the solid, so that the remaining char is composed primarily of carbon...

occurs. The chemical kinetics
Chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the study of rates of chemical processes. Chemical kinetics includes investigations of how different experimental conditions can influence the speed of a chemical reaction and yield information about the reaction's mechanism and transition...

of both reactions indicate that there is a point at which the level of moisture is low enough—and/or heating rates high enough—for combustion processes become self-sufficient. Consequently, changes in wind speed, direction, moisture, temperature, or lapse rate
Lapse rate
The lapse rate is defined as the rate of decrease with height for an atmospheric variable. The variable involved is temperature unless specified otherwise. The terminology arises from the word lapse in the sense of a decrease or decline; thus, the lapse rate is the rate of decrease with height and...

at different levels of the atmosphere can have a significant impact on the behavior and growth of a wildfire. Since the wildfire acts as a heat source to the atmospheric flow, the wildfire can modify local advection
Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...

patterns, introducing a feedback loop
Feedback
Feedback describes the situation when output from an event or phenomenon in the past will influence an occurrence or occurrences of the same Feedback describes the situation when output from (or information about the result of) an event or phenomenon in the past will influence an occurrence or...

between the fire and the atmosphere.

A simplified two-dimensional model for the spread of wildfires that used convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

to represent the effects of wind and terrain, as well as radiative heat transfer
Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

as the dominant method of heat transport led to reaction-diffusion systems of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s. More complex models join numerical weather models or computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

models with a wildfire component which allow the feedback effects between the fire and the atmosphere to be estimated. The additional complexity in the latter class of models translates to a corresponding increase in their computer power requirements. In fact, a full three-dimensional treatment of combustion
Combustion
Combustion or burning is the sequence of exothermic chemical reactions between a fuel and an oxidant accompanied by the production of heat and conversion of chemical species. The release of heat can result in the production of light in the form of either glowing or a flame...

via direct numerical simulation
Direct numerical simulation
A direct numerical simulation is a simulation in computational fluid dynamics in which the Navier-Stokes equations are numerically solved without any turbulence model...

at scales relevant for atmospheric modeling is not currently practical because of the excessive computational cost such a simulation would require. Numerical weather models have limited forecast skill at spatial resolutions under 1 kilometre (0.621372736649807 mi), forcing complex wildfire models to parameterize the fire in order to calculate how the winds will be modified locally by the wildfire, and to use those modified winds to determine the rate at which the fire will spread locally. Although models such as Los Alamos
Los Alamos National Laboratory
Los Alamos National Laboratory is a United States Department of Energy national laboratory, managed and operated by Los Alamos National Security , located in Los Alamos, New Mexico...

' FIRETEC solve for the concentrations of fuel and oxygen
Oxygen
Oxygen is the element with atomic number 8 and represented by the symbol O. Its name derives from the Greek roots ὀξύς and -γενής , because at the time of naming, it was mistakenly thought that all acids required oxygen in their composition...

, the computational grid cannot be fine enough to resolve the combustion reaction, so approximations must be made for the temperature distribution within each grid cell, as well as for the combustion reaction rates themselves.

• Atmospheric physics
Atmospheric physics
Atmospheric physics is the application of physics to the study of the atmosphere. Atmospheric physicists attempt to model Earth's atmosphere and the atmospheres of the other planets using fluid flow equations, chemical models, radiation balancing, and energy transfer processes in the atmosphere...

• Atmospheric thermodynamics
Atmospheric thermodynamics
Atmospheric thermodynamics is the study of heat to work transformations in the earth’s atmospheric system in relation to weather or climate...

• Tropical cyclone forecast model
• Types of atmospheric models