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{{broader|Atmospheric model}}
[[File:AtmosphericModelSchematic.png|thumb|300px|right|Weather models use systems of [[differential equations]] based on the laws of [[physics]], which are in detail [[Fluid dynamics|fluid motion]], [[thermodynamics]], [[radiative transfer]], and [[chemistry]], and use a coordinate system which divides the planet into a 3D grid.
'''Numerical weather prediction''' ('''NWP''') uses
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
Manipulating the vast datasets and performing the complex calculations necessary to modern numerical weather prediction requires some of the most powerful [[supercomputer]]s in the world.
A more fundamental problem lies in the [[Chaos theory|chaotic]] nature of the [[partial differential equation]]s that
== History ==
{{Main|History of numerical weather prediction}}
[[File:Two women operating ENIAC (full resolution).
The [[history of numerical weather prediction]] began in the 1920s through the efforts of [[Lewis Fry Richardson]], who used procedures originally developed by [[Vilhelm Bjerknes]]<ref name="Lynch JCP"/> 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.<ref>{{cite journal |last1=Simmons |first1=A. J. |last2=Hollingsworth |first2=A. |date=2002 |title=Some aspects of the improvement in skill of numerical weather prediction |url=https://doi.org/10.1256/003590002321042135 |journal=Quarterly Journal of the Royal Meteorological Society |volume=128 |issue=580 |pages=647–677 | doi=10.1256/003590002321042135|bibcode=2002QJRMS.128..647S |s2cid=121625425 |url-access=subscription }}</ref><ref name="Lynch JCP">{{cite journal|last=[[Peter Lynch (meteorologist)|Lynch]]|first=Peter|title=The origins of computer weather prediction and climate modeling|journal=[[Journal of Computational Physics]]|date=March 2008|volume=227|issue=7|pages=3431–44|doi=10.1016/j.jcp.2007.02.034|bibcode=2008JCoPh.227.3431L|url=http://www.rsmas.miami.edu/personal/miskandarani/Courses/MPO662/Lynch,Peter/OriginsCompWF.JCP227.pdf|access-date=2010-12-23|url-status=dead|archive-url=https://web.archive.org/web/20100708191309/http://www.rsmas.miami.edu/personal/miskandarani/Courses/MPO662/Lynch,Peter/OriginsCompWF.JCP227.pdf|archive-date=2010-07-08}}</ref><ref name="Lynch Ch1">{{cite book|last=Lynch|first=Peter|title=The Emergence of Numerical Weather Prediction|url=https://archive.org/details/emergencenumeric00lync|url-access=limited|year=2006|publisher=[[Cambridge University Press]]|isbn=978-0-521-85729-1|pages=[https://archive.org/details/emergencenumeric00lync/page/n11 1]–27|chapter=Weather Prediction by Numerical Process}}</ref>
As computers have become more powerful, the size of the initial data sets has increased and [[Atmospheric model#Types|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 [[Navier–Stokes equations|equations of motion]] in numerical simulations of the atmosphere.<ref name="Harper BAMS"/>
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]] (MOS).<ref name="MOS"/><ref>{{cite book|title=Air Weather Service Model Output Statistics Systems|author1=Best, D. L. |author2=Pryor, S. P. |year=1983|pages=1–90|publisher=Air Force Global Weather Central}}</ref>
==Data collection and initialization==
[[File:Lockheed WP-3D Orion.jpg|280px|thumb|right|Weather reconnaissance aircraft, such as this [[WP-3D Orion]], provide data that is then used in numerical weather forecasts.|alt=A [[WP-3D Orion]] weather reconnaissance aircraft in flight.]]
The [[atmosphere]] is a [[fluid]]. 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]] and [[thermodynamics]] to estimate the state of the fluid at some time in the future.
==Computation==
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[[File:GFS 850 MB.PNG|right|280px|thumb|A [[prognostic chart]] of the 96-hour forecast of 850 [[millibar|mbar]] [[geopotential height]] and [[temperature]] from the [[Global Forecast System]]|alt=A prognostic chart of the North American continent provides geopotential heights, temperatures, and wind velocities at regular intervals. The values are taken at the altitude corresponding to the 850-millibar pressure surface.]]
An atmospheric model is a computer program that produces [[meteorological]] information for future times at given locations and altitudes.
These equations are initialized from the analysis data and rates of change are determined.
== Parameterization ==
[[File:GoldenMedows.jpg|thumb|right|Field of [[cumulus cloud]]s, which are parameterized since they are too small to be explicitly included within numerical weather prediction]]
{{Main|Parametrization (
Some meteorological processes are too small-scale or too complex to be explicitly included in numerical weather prediction models. ''[[Parametrization (atmospheric modeling)|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 {{convert|5|km|mi|0|sp=us}} and {{convert|300|km|mi|-2|sp=us}} in length.
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.
==Domains==
[[File:Sigma-z-coordinates.svg|thumb|280px|A cross-section of the atmosphere over terrain with a [[Sigma coordinate system|sigma
The horizontal [[Domain of a function|___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.
[[File:Spatial scales of cloud models.png|thumb|A comparison of different types of atmospheric models by spatial ___domain and model grid size.|alt=A plot of model ___domain size versus model grid size with several different types of numerical models arranged diagonally.|left]]
The vertical coordinate is handled in various ways. Lewis Fry Richardson's 1922 model used geometric height (<math>z</math>) as the vertical coordinate. Later models substituted the geometric <math>z</math> coordinate with a pressure coordinate system, in which the [[geopotential height]]s of constant-pressure surfaces become [[dependent variable]]s, greatly simplifying the primitive equations.<ref name="Lynch Ch2">{{cite book|last=Lynch|first=Peter|title=The Emergence of Numerical Weather Prediction|url=https://archive.org/details/emergencenumeric00lync|url-access=limited|year=2006|publisher=[[Cambridge University Press]]|isbn=978-0-521-85729-1|pages=[https://archive.org/details/emergencenumeric00lync/page/n55 45]–46|chapter=The Fundamental Equations}}</ref>
==Model output statistics==
{{Main|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.
Model output statistics differ from the ''perfect prog'' technique, which assumes that the output of numerical weather prediction guidance is perfect.<ref>{{cite book|url=https://books.google.com/books?id=QwzHZ-wV-BAC&pg=PA1144|page=1144|title=Fog and boundary layer clouds: fog visibility and forecasting|author=Gultepe, Ismail|publisher=Springer|year=2007|isbn=978-3-7643-8418-0|access-date=2011-02-11}}</ref>
==Ensembles==
{{main|Ensemble forecasting}}
[[File:WRF rita spread2.jpg|thumb|280px|''Top'': [[Weather Research and Forecasting model]] (WRF) simulation of [[Hurricane Rita]] (2005) tracks. ''Bottom'': The spread of NHC multi-model ensemble forecast.|alt=Two images are shown. The top image provides three potential tracks that could have been taken by Hurricane Rita. Contours over the coast of Texas correspond to the sea-level air pressure predicted as the storm passed. The bottom image shows an ensemble of track forecasts produced by different weather models for the same hurricane.]]
In 1963, [[Edward Lorenz]] discovered the [[chaos theory|chaotic nature]] of the [[fluid dynamics]] equations involved in weather forecasting.<ref name="Cox">{{cite book|title=Storm Watchers|pages=[https://archive.org/details/stormwatcherstur00cox_df1/page/222 222–224]|year=2002|author=Cox, John D.|publisher=John Wiley & Sons, Inc.|isbn=978-0-471-38108-2|url=https://archive.org/details/stormwatcherstur00cox_df1/page/222}}</ref>
[[Edward Epstein (meteorologist)|Edward Epstein]] 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 (fluid mechanics)|ensemble]] of [[stochastic process|stochastic]] [[Monte Carlo method|Monte Carlo simulations]] to produce [[arithmetic mean|means]] and [[variance]]s for the state of the atmosphere.<ref>{{cite journal|last=Epstein|first=E.S.|title=Stochastic dynamic prediction|journal=[[Tellus A]]|date=December 1969|volume=21|issue=6|pages=739–759|doi=10.1111/j.2153-3490.1969.tb00483.x|bibcode=1969Tell...21..739E}}</ref> 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]] was a representative sample of the probability distribution in the atmosphere.<ref>{{cite journal|last=Leith|first=C.E.|title=Theoretical Skill of Monte Carlo Forecasts|journal=[[Monthly Weather Review]]|date=June 1974|volume=102|issue=6|pages=409–418|doi=10.1175/1520-0493(1974)102<0409:TSOMCF>2.0.CO;2|bibcode=1974MWRv..102..409L|doi-access=free}}</ref>
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.
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.
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.<ref>{{cite journal|url=http://www.emc.ncep.noaa.gov/mmb/SREF/2222289_WAF_Feb-2010.official.PDF|title=Fog Prediction From a Multimodel Mesoscale Ensemble Prediction System|author1=Zhou, Binbin |author2=Du, Jun |volume=25|issue=1|date=February 2010|access-date=2011-01-02|journal=[[Weather and Forecasting]]|page=303|doi=10.1175/2009WAF2222289.1|bibcode=2010WtFor..25..303Z|s2cid=4947206 }}</ref>
== Applications ==
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===Air quality modeling===
{{see also|Atmospheric dispersion modeling}}
[[Air pollution forecasting|Air quality forecasting]] attempts to predict when the concentrations of pollutants will attain levels that are hazardous to public health.
===Climate modeling===
{{See also|Global climate model}}
A General Circulation Model (GCM) is a [[mathematical model]] that can be used in computer simulations of the global circulation of a planetary [[atmosphere]] or ocean.
===Ocean surface modeling===
[[File:NOAA Wavewatch III Sample Forecast.gif|right|thumb|280px|NOAA Wavewatch III 120-hour wind and wave forecast for the North Atlantic|alt=A wind and wave forecast for the North Atlantic Ocean. Two areas of high waves are identified: One west of the southern tip of Greenland, and the other in the North Sea. Calm seas are forecast for the Gulf of Mexico. Wind barbs show the expected wind strengths and directions at regularly spaced intervals over the North Atlantic.]]
{{main|Marine weather forecasting|Ocean dynamics|Wind wave model}}
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.<ref>{{cite journal|last=Chalikov|first=D. V.|title=The numerical simulation of wind-wave interaction|journal=[[Journal of Fluid Mechanics]]|date=August 1978|volume=87|issue=3|pages=561–82|doi=10.1017/S0022112078001767|bibcode=1978JFM....87..561C|s2cid=122742282 }}</ref>
===Tropical cyclone forecasting===
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Tropical cyclone forecasting also relies on data provided by numerical weather models. Three main classes of [[Tropical cyclone forecast model|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.<ref>{{cite web|title=Technical Summary of the National Hurricane Center Track and Intensity Models|url=http://www.nhc.noaa.gov/pdf/model_summary_20090724.pdf|publisher=National Oceanic and Atmospheric Administration|access-date=2011-02-19|author=[[National Hurricane Center]]|date=July 2009}}</ref>
In 1978, the first [[tropical cyclone forecast model|hurricane-tracking model]] based on [[Atmospheric dynamics#Dynamic meteorology|atmospheric dynamics]]—the movable fine-mesh (MFM) model—began operating.<ref name="Shuman W&F"/>
===Weather forecasts===
Because weather drifts across the world, producing forecasts a week or more in advance typically involves running a numerical prediction model for the entire planet. Agencies use various software to do this, including:
* [[North American Ensemble Forecast System]], which combines results from:
** [[Global Forecast System]] from the US [[National Weather Service]]
** [[Global Environmental Multiscale Model]] from the [[Canadian Meteorological Centre]]
* [
* [[Unified Model]], produced by a partnership of:
** UK [[Met Office]]
** Australia [[Bureau of Meteorology]]
** (South) [[Korea Meteorological Administration]]
** India [[National Centre for Medium Range Weather Forecasting]]<ref>[https://ncmrwf.gov.in/ncmrwf/NCUMG-Writeup-for-WEB-June2020.pdf Global NCMRWF Unified Model (NCUM-G) System]</ref>
** New Zealand [[National Institute of Water and Atmospheric Research]]
* [[Icosahedral Nonhydrostatic]] (ICON) from [[Deutscher Wetterdienst]], the German Meteorological Service
* [
* Global Spectral Model and Global Ensemble Prediction System from the [[Japan Meteorological Agency]]<ref name="JMA">[https://www.jma.go.jp/jma/en/Activities/nwp.html Numerical Weather Prediction Activities]</ref>
* [[China Meteorological Administration]] Global Assimilation Forecasting System<ref name="CMA">[https://www.cma.gov.cn/en/forecast/highlight/202311/t20231117_5892086.html Numerical Weather Prediction]</ref>
* Brazilian Global Atmospheric Model (BAM) from [[Centro de Previsão do Tempo e Estudos Climáticos]] (CPTEC)
The global models can be used to supply [[boundary conditions]] to higher-resolution models that provide more accurate forecasts for an area of interest, such as the country served by a government agency, or an area where military action or rescue efforts are planned.
* Users of the Unified Model re-run the same system (hence the name) for a specific country or crisis zone at a higher horizontal resolution, feeding it the output of the global Unified Model run. This is given a different name, such as the UKV model or the New Zealand Limited Area Model.<ref>[https://www.nesi.org.nz/case-studies/improving-new-zealands-weather-forecasting-ability A 36 hour forecast by NZCSM takes 130 minutes to complete using 810 processors spread across 13 nodes of FitzRoy]</ref>
* The US National Weather Service runs the [[Weather Research and Forecasting Model]] with different parameters to create:
** [[North American Mesoscale Model]] (NAM) every six hours (with an ensemble called Short Range Ensemble Forecast, SREF)
** [[Rapid Refresh (weather prediction)|Rapid Refresh]] (RAP) and High Resolution Rapid Refresh (HRRR), every hour<ref>[https://rapidrefresh.noaa.gov/ Rapid Refresh (RAP)]</ref><ref>[https://rapidrefresh.noaa.gov/hrrr/ The High-Resolution Rapid Refresh (HRRR)]</ref>
* The Japan Meteorological Agency runs:<ref name="JMA" />
** Meso-Scale Model (MSM) every 3 hours, looking 39 and 78 hours ahead
** Meso-scale Ensemble Prediction System every 6 hours, looking 39 hours ahead (providing uncertainty estimation)
** Local Forecast Model every hour, looking 10-18 hours ahead
* The China Meteorological Administration runs the Regional Numerical Forecasting Model (CMA-MESO)<ref name="CMA" />
* CPTEC runs the Brazilian Regional Atmospheric Modelling System (BRAMS) and ETA Regional Model (ETA) for South America
The output of higher-resolution models may be further modified by [[model output statistics]] to take into quirky local phenomena that general models do not handle well, such as [[mountain waves]].
===Wildfire modeling===
{{
[[File:Propagation model wildfire (English).svg|thumb|280px|right|A simple wildfire propagation model]]
On a molecular scale, there are two main competing reaction processes involved in the degradation of [[cellulose]], or wood fuels, in [[wildfire]]s.
A simplified two-dimensional model for the spread of wildfires that used [[convection]] to represent the effects of wind and terrain, as well as [[Thermal radiation|radiative heat transfer]] as the dominant method of heat transport led to [[reaction–diffusion system]]s of [[partial differential equation]]s.<ref name="Asensio-2002-WFM">{{cite journal|author1=Asensio, M. I. |author2=L. Ferragut |name-list-style=amp |title=On a wildland fire model with radiation|journal=International Journal for Numerical Methods in Engineering|volume=54|issue=1 |pages=137–157|year=2002|doi=10.1002/nme.420|bibcode = 2002IJNME..54..137A |s2cid=122302719 }}</ref><ref name="Mandel-2008-WMD">{{cite journal|author=Mandel, Jan, [[Lynn Schreyer|Lynn S. Bennethum]], Jonathan D. Beezley, [[Janice Coen|Janice L. Coen]], Craig C. Douglas, Minjeong Kim, and Anthony Vodacek|title=A wildfire model with data assimilation|journal=Mathematics and Computers in Simulation|volume=79|pages=584–606|year=2008|doi=10.1016/j.matcom.2008.03.015|arxiv=0709.0086|bibcode=2007arXiv0709.0086M|issue=3|s2cid=839881}}</ref> More complex models join numerical weather models or [[computational fluid dynamics]] models with a wildfire component which allow the feedback effects between the fire and the atmosphere to be estimated.<ref name="Sullivan wildfire"/>
== See also ==
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{{Refbegin}}
*{{cite book |last=Beniston |first=Martin |title=From Turbulence to Climate: Numerical Investigations of the Atmosphere with a Hierarchy of Models |___location=Berlin |publisher=Springer |year=1998 |isbn=978-3-540-63495-9 }}
*{{cite book |last=
* {{cite book | last=Kalnay | first=Eugenia | title=Atmospheric Modeling, Data Assimilation and Predictability | publisher=Cambridge University Press | publication-place=New York | date=2003 | isbn=978-0-521-79629-3|author-link=Eugenia Kalnay }}
* {{cite book |author1=Roulstone, Ian |author2=Norbury, John |name-list-style=amp |title=Invisible in the Storm: the role of mathematics in understanding weather |url=https://books.google.com/books?id=qnMrFEHMrWwC|year=2013 |publisher=Princeton University Press|isbn=978-0691152721 }}
*{{cite book |last=Thompson |first=Philip |title=Numerical Weather Analysis and Prediction |url=https://archive.org/details/numericalweather00thom |url-access=registration |___location=New York |publisher=The Macmillan Company |year=1961 }}
*{{cite book |editor1=U.S. Department of Commerce |editor2=National Oceanic |editor3=Atmospheric Administration |editor4=National Weather Service |title=National Weather Service Handbook No. 1 – Facsimile Products |___location=Washington, DC |publisher=Department of Commerce |year=1979 }}From Turbulence to sCl
{{Refend}}
==External links==
* [https://web.archive.org/web/20191012133855/https://www.noaa.gov/media-release/noaa-kicks-off-2018-with-massive-supercomputer-upgrade NOAA Supercomputer upgrade]
* [
* [https://web.archive.org/web/20180111155953/http://www.metoc.navy.mil/fnmoc/fnmoc.html Fleet Numerical Meteorology and Oceanography Center]
* [https://www.ecmwf.int/ European Centre for Medium-Range Weather Forecasts]
▲* [http://www.usno.navy.mil/FNMOC/ Fleet Numerical Meteorology and Oceanography Center]
* [
▲* [http://www.ecmwf.int/ European Centre for Medium-Range Weather Forecasts]
▲* [http://www.metoffice.gov.uk/research/modelling-systems/unified-model/weather-forecasting UK Met Office]
{{Atmospheric, Oceanographic and Climate Models}}
{{Authority control}}
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