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{{Main|History of numerical weather prediction}}
[[File:Two women operating ENIAC.gif|thumb|280px|The ENIAC main control panel at the [[Moore School of Electrical Engineering]] operated by [[Jean Bartik|Betty Jennings]] and [[Frances Spence|Frances Bilas]]]]
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=
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"/> In 1966, [[West Germany]] and the United States began producing operational forecasts based on [[primitive equations|primitive-equation models]], followed by the United Kingdom in 1972 and Australia in 1977.<ref name="Lynch JCP"/><ref name="Leslie BOM">{{cite journal|last=Leslie|first=L.M.|author2=Dietachmeyer, G.S. |title=Real-time limited area numerical weather prediction in Australia: a historical perspective|journal=Australian Meteorological Magazine|date=December 1992|volume=41|issue=SP|pages=61–77|url=http://www.bom.gov.au/amoj/docs/1992/leslie2.pdf|access-date=2011-01-03}}</ref> The development of limited area (regional) models facilitated advances in forecasting the tracks of [[tropical cyclone]]s as well as [[air quality]] in the 1970s and 1980s.<ref name="Shuman W&F">{{cite journal|last=Shuman|first=Frederick G.|author-link=Frederick Gale Shuman|title=History of Numerical Weather Prediction at the National Meteorological Center|journal=[[Weather and Forecasting]]|date=September 1989|volume=4|issue=3|pages=286–296|doi=10.1175/1520-0434(1989)004<0286:HONWPA>2.0.CO;2|bibcode=1989WtFor...4..286S|doi-access=free}}</ref><ref>{{cite book|title=Air pollution modeling and its application VIII, Volume 8|author=Steyn, D. G.|publisher=Birkhäuser|year=1991|pages=241–242|isbn=978-0-306-43828-8}}</ref> By the early 1980s models began to include the interactions of soil and vegetation with the atmosphere, which led to more realistic forecasts.<ref>{{cite journal|url=http://www.geog.ucla.edu/~yxue/pdf/1996jgr.pdf |title=Impact of vegetation properties on U. S. summer weather prediction |page=7419 |author1=Xue, Yongkang |author2=Fennessey, Michael J. |journal=[[Journal of Geophysical Research]] |volume=101 |issue=D3 |date=1996-03-20 |access-date=2011-01-06 |doi=10.1029/95JD02169 |bibcode=1996JGR...101.7419X |url-status=dead |archive-url=https://web.archive.org/web/20100710080304/http://www.geog.ucla.edu/~yxue/pdf/1996jgr.pdf |archive-date=2010-07-10 |citeseerx=10.1.1.453.551 }}</ref>
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==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. The process of entering observation data into the model to generate [[initial value problem|initial conditions]] is called ''initialization''. On land, terrain maps available at resolutions down to {{convert|1|km|mi|1|sp=us}} globally are used to help model atmospheric circulations within regions of rugged topography, in order to better depict features such as downslope winds, [[Lee wave|mountain wave]]s and related cloudiness that affects incoming solar radiation.<ref>{{cite book|url=https://books.google.com/books?id=lMXSpRwKNO8C&pg=PA56|title=Parameterization schemes: keys to understanding numerical weather prediction models|author=Stensrud, David J.|page=56|year=2007|publisher=Cambridge University Press|isbn=978-0-521-86540-1}}</ref> The main inputs from country-based weather services are observations from devices (called [[radiosonde]]s) in weather balloons that measure various atmospheric parameters and transmits them to a fixed receiver, as well as from [[weather satellite]]s. The [[World Meteorological Organization]] acts to standardize the instrumentation, observing practices and timing of these observations worldwide. Stations either report hourly in [[METAR]] reports,<ref>{{cite web|title=Key to METAR Surface Weather Observations|url=http://www.ncdc.noaa.gov/oa/climate/conversion/swometardecoder.html|publisher=[[National Oceanic and Atmospheric Administration]]|access-date=2011-02-11|author=[[National Climatic Data Center]]|date=2008-08-20|archive-date=2002-11-01|archive-url=https://web.archive.org/web/20021101221848/http://www0.ncdc.noaa.gov/oa/climate/conversion/swometardecoder.html|url-status=dead}}</ref> or every six hours in [[SYNOP]] reports.<ref>{{cite web|title=SYNOP Data Format (FM-12): Surface Synoptic Observations|publisher=[[UNISYS]]|archive-url=https://web.archive.org/web/20071230100059/http://weather.unisys.com/wxp/Appendices/Formats/SYNOP.html|archive-date=2007-12-30|date=2008-05-25|url=http://weather.unisys.com/wxp/Appendices/Formats/SYNOP.html}}</ref> These observations are irregularly spaced, so they are processed by [[data assimilation]] and objective analysis methods, which perform quality control and obtain values at locations usable by the model's mathematical algorithms.<ref name="Krishnamurti Annu Rev FM">{{cite journal|last=Krishnamurti|first=T. N.|title=Numerical Weather Prediction|journal=[[Annual Reviews (publisher)|Annual Review of Fluid Mechanics]]|date=January 1995|volume=27|issue=1|pages=195–225|doi=10.1146/annurev.fl.27.010195.001211|bibcode=1995AnRFM..27..195K|s2cid=122230747 }}</ref> The data are then used in the model as the starting point for a forecast.<ref>{{cite web|title=The WRF Variational Data Assimilation System (WRF-Var)|publisher=[[University Corporation for Atmospheric Research]]|archive-url=https://web.archive.org/web/20070814044336/http://www.mmm.ucar.edu/wrf/WG4/wrfvar/wrfvar-tutorial.htm|archive-date=2007-08-14|date=2007-08-14|url=http://www.mmm.ucar.edu/wrf/WG4/wrfvar/wrfvar-tutorial.htm}}</ref>
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]] and well into the [[stratosphere]].<ref>{{cite web|last=Gaffen|first=Dian J.|title=Radiosonde Observations and Their Use in SPARC-Related Investigations|archive-url=https://web.archive.org/web/20070607142822/http://www.aero.jussieu.fr/~sparc/News12/Radiosondes.html|archive-date=2007-06-07|date=2007-06-07|url=http://www.aero.jussieu.fr/~sparc/News12/Radiosondes.html}}</ref> Information from weather satellites is used where traditional data sources are not available. Commerce provides [[pilot report]]s along aircraft routes<ref>{{cite journal|last=Ballish|first=Bradley A.|author2=V. Krishna Kumar |title=Systematic Differences in Aircraft and Radiosonde Temperatures|journal=[[Bulletin of the American Meteorological Society]]|date=November 2008|volume=89|issue=11|pages=1689–1708|doi=10.1175/2008BAMS2332.1|bibcode=2008BAMS...89.1689B|access-date=2011-02-16|url=http://amdar.noaa.gov/docs/bams_ballish_kumar.pdf}}</ref> and ship reports along shipping routes.<ref>{{cite web|author=National Data Buoy Center|url=http://www.vos.noaa.gov/vos_scheme.shtml|title=The WMO Voluntary Observing Ships (VOS) Scheme|access-date=2011-02-15|date=2009-01-28|publisher=[[National Oceanic and Atmospheric Administration]]}}</ref> Research projects use [[weather reconnaissance|reconnaissance aircraft]] to fly in and around weather systems of interest, such as [[tropical cyclone]]s.<ref name="Hurricane Hunters">{{cite web|year=2011|author=403rd Wing|url=http://www.hurricanehunters.com|title=The Hurricane Hunters|publisher=[[Hurricane Hunters|53rd Weather Reconnaissance Squadron]]|access-date=2006-03-30|archive-date=2012-05-30|archive-url=https://web.archive.org/web/20120530232904/http://www.hurricanehunters.com/|url-status=dead}}</ref><ref name="SunHerald">{{cite news|author=Lee, Christopher|title=Drone, Sensors May Open Path Into Eye of Storm|url=https://www.washingtonpost.com/wp-dyn/content/article/2007/10/07/AR2007100700971_pf.html|newspaper=The Washington Post|access-date=2008-02-22|date=2007-10-08}}</ref> 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.<ref>{{cite web|url=http://www.noaanews.noaa.gov/stories2010/20100112_plane.html|title=NOAA Dispatches High-Tech Research Plane to Improve Winter Storm Forecasts|date=2010-11-12|access-date=2010-12-22|author=[[National Oceanic and Atmospheric Administration]]}}</ref> Sea ice began to be initialized in forecast models in 1971.<ref>{{cite book|url=https://books.google.com/books?id=lMXSpRwKNO8C&pg=PA137|author=Stensrud, David J.|page=137|title=Parameterization schemes: keys to understanding numerical weather prediction models|publisher=[[Cambridge University Press]]|year=2007|isbn=978-0-521-86540-1}}</ref> Efforts to involve [[sea surface temperature]] in model initialization began in 1972 due to its role in modulating weather in higher latitudes of the Pacific.<ref>{{cite book|url=https://books.google.com/books?id=SV04AAAAIAAJ&pg=PA38|pages=49–50|title=The Global Climate|author=Houghton, John Theodore|publisher=Cambridge University Press archive|year=1985|isbn=978-0-521-31256-1}}</ref>
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[[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. This method involves analyzing multiple forecasts created with an individual forecast model by using different physical [[parametrization (climate)|parametrizations]] or varying initial conditions.<ref name="HPCens">{{cite web|url=http://www.wpc.ncep.noaa.gov/ensembletraining|author=Manousos, Peter|publisher=[[Hydrometeorological Prediction Center]]|date=2006-07-19|access-date=2010-12-31|title=Ensemble Prediction Systems}}</ref> Starting in 1992 with [[Ensemble forecasting|ensemble forecasts]] prepared by the [[European Centre for Medium-Range Weather Forecasts]] (ECMWF) and the [[National Centers for Environmental Prediction]], 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.<ref name="Toth"/><ref name="ECens"/><ref name="RMS"/> The ECMWF model, the Ensemble Prediction System,<ref name="ECens">{{cite web | url=http://ecmwf.int/products/forecasts/guide/The_Ensemble_Prediction_System_EPS_1.html <!--Added by H3llBot--> | title=The Ensemble Prediction System (EPS) | publisher=[[ECMWF]] | access-date=2011-01-05 | archive-url=https://web.archive.org/web/20101030055238/http://ecmwf.int/products/forecasts/guide/The_Ensemble_Prediction_System_EPS_1.html <!--Added by H3llBot--> | archive-date=2010-10-30}}</ref> uses [[Singular value decomposition|singular vectors]] to simulate the initial [[probability density function|probability density]], while the NCEP ensemble, the Global Ensemble Forecasting System, uses a technique known as [[Bred vector|vector breeding]].<ref name="Toth">{{cite journal|last=Toth|first=Zoltan|author2=Kalnay, Eugenia |title=Ensemble Forecasting at NCEP and the Breeding Method |journal=[[Monthly Weather Review]]|date=December 1997|volume=125|issue=12|pages=3297–3319|doi=10.1175/1520-0493(1997)125<3297:EFANAT>2.0.CO;2|bibcode=1997MWRv..125.3297T|citeseerx=10.1.1.324.3941|s2cid=14668576 }}</ref><ref name="RMS">{{cite journal|title=The ECMWF Ensemble Prediction System: Methodology and validation|journal=Quarterly Journal of the Royal Meteorological Society|date=January 1996|volume=122|issue=529|pages=73–119|doi=10.1002/qj.49712252905|bibcode=1996QJRMS.122...73M|author1=Molteni, F. |author2=Buizza, R. |author3=Palmer, T.N. |author3-link=Tim Palmer (physicist) |author4=Petroliagis, T. }}</ref> The UK [[Met Office]] runs global and regional ensemble forecasts where perturbations to initial conditions are used by 24 ensemble members in the Met Office Global and Regional Ensemble Prediction System (MOGREPS) to produce 24 different forecasts.<ref name="The Met Office ensemble system- MOGREPS">{{cite web | url=http://www.metoffice.gov.uk/research/areas/data-assimilation-and-ensembles/ensemble-forecasting/MOGREPS | title=MOGREPS | publisher=[[Met Office]] | access-date=2012-11-01 | url-status=dead | archive-url=https://web.archive.org/web/20121022215636/http://www.metoffice.gov.uk/research/areas/data-assimilation-and-ensembles/ensemble-forecasting/MOGREPS | archive-date=2012-10-22 }}</ref>
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 plot|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]], 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;<ref name="ensbook"/> this problem becomes particularly severe for forecasts of the weather about ten days in advance.<ref>{{cite journal|last1=Palmer |first1=T.N. |author1-link=Tim Palmer (physicist) |first2=G.J. |last2=Shutts |first3=R. |last3=Hagedorn |first4=F.J. |last4=Doblas-Reyes |first5=T. |last5=Jung |first6=M. |last6=Leutbecher|title=Representing Model Uncertainty in Weather and Climate Prediction|journal=[[Annual Review of Earth and Planetary Sciences]]|date=May 2005|volume=33|pages=163–193|doi=10.1146/annurev.earth.33.092203.122552|bibcode=2005AREPS..33..163P}}</ref> 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.<ref name="ensbook">{{cite book|url=https://books.google.com/books?id=6RQ3dnjE8lgC&pg=PA261|title=Numerical Weather and Climate Prediction|author=Warner, Thomas Tomkins |publisher=[[Cambridge University Press]]|year=2010|isbn=978-0-521-51389-0|pages=266–275}}</ref> Despite this perception, a ''spread-skill relationship'' is often weak or not found, as spread-error [[Correlation and dependence#Correlation and linearity|correlations]] are normally less than 0.6, and only under special circumstances range between 0.6–0.7.<ref>{{cite web|url=http://www.atmos.washington.edu/~ens/pdf/WEM_WKSHP_2004.epgrimit.pdf|title=Redefining the Ensemble Spread-Skill Relationship from a Probabilistic Perspective|author1=Grimit, Eric P.|author2=Mass, Clifford F.|publisher=[[University of Washington]]|date=October 2004|access-date=2010-01-02|archive-url=https://web.archive.org/web/20081012094121/http://www.atmos.washington.edu/~ens/pdf/WEM_WKSHP_2004.epgrimit.pdf|archive-date=2008-10-12|url-status=dead}}</ref> {{cnspan|The relationship between ensemble spread and [[forecast skill]] varies substantially depending on such factors as the forecast model and the region for which the forecast is made.|date=December 2021}}
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> 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.<ref>{{cite journal|url=http://www.nat-hazards-earth-syst-sci.net/10/265/2010/nhess-10-265-2010.pdf|title=Multimodel SuperEnsemble technique for quantitative precipitation forecasts in Piemonte region|author1=Cane, D. |author2=Milelli, M. |date=2010-02-12|access-date=2011-01-02|journal=Natural Hazards and Earth System Sciences|doi=10.5194/nhess-10-265-2010|bibcode=2010NHESS..10..265C|volume=10|page=265|issue=2|doi-access=free}}</ref>
== Applications ==
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[[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> 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]], [[refraction]], energy transfer between waves, and wave dissipation.<ref>{{cite book|page=270|url=https://books.google.com/books?id=yBtOwfUG6cgC|title=Numerical modeling of water waves|author=Lin, Pengzhi|publisher=Psychology Press|year=2008|isbn=978-0-415-41578-1}}</ref> 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 [[Wind wave|whitecaps]] and [[resonance]] between waves, surface winds from numerical weather models allow for more accurate predictions of the state of the sea surface.<ref>{{cite journal|last=Bender|first=Leslie C.|title=Modification of the Physics and Numerics in a Third-Generation Ocean Wave Model|journal=[[Journal of Atmospheric and Oceanic Technology]]|date=January 1996|volume=13|issue=3|pages=726–750 |doi=10.1175/1520-0426(1996)013<0726:MOTPAN>2.0.CO;2|bibcode=1996JAtOT..13..726B|doi-access=free}}</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"/> Within the field of [[tropical cyclone track forecasting]], despite the ever-improving dynamical model guidance which occurred with increased computational power, it was not until the 1980s when numerical weather prediction showed [[Forecast skill|skill]], and until the 1990s when it consistently outperformed [[statistical model|statistical]] or simple dynamical models.<ref>{{cite web|url=http://www.nhc.noaa.gov/verification/verify6.shtml|publisher=[[National Hurricane Center]]|date=2010-04-20|access-date=2011-01-02|author=Franklin, James|title=National Hurricane Center Forecast Verification|author-link=James Franklin (meteorologist)}}</ref> 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.<ref>{{cite journal|author=Rappaport, Edward N. |author2=Franklin, James L. |author3=Avila, Lixion A. |author4=Baig, Stephen R. |author5=Beven II, John L. |author6=Blake, Eric S. |author7=Burr, Christopher A. |author8=Jiing, Jiann-Gwo |author9=Juckins, Christopher A. |author10=Knabb, Richard D. |author11=Landsea, Christopher W. |author12=Mainelli, Michelle |author13=Mayfield, Max |author14=McAdie, Colin J. |author15=Pasch, Richard J. |author16=Sisko, Christopher |author17=Stewart, Stacy R. |author18=Tribble, Ahsha N.|title=Advances and Challenges at the National Hurricane Center|journal=[[Weather and Forecasting]]|date=April 2009|volume=24|issue=2|pages=395–419|doi=10.1175/2008WAF2222128.1|bibcode=2009WtFor..24..395R|citeseerx=10.1.1.207.4667 |s2cid=14845745 }}</ref>
===Wildfire modeling===
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On a molecular scale, there are two main competing reaction processes involved in the degradation of [[cellulose]], or wood fuels, in [[wildfire]]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]]. When moisture is present—or when enough heat is being carried away from the fiber, [[charring]] occurs. The [[chemical kinetics]] 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 to become self-sufficient. Consequently, changes in wind speed, direction, moisture, temperature, or [[lapse rate]] 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]] patterns, introducing a [[Feedback|feedback loop]] between the fire and the atmosphere.<ref name="Sullivan wildfire">{{cite journal|last=Sullivan|first=Andrew L.|title=Wildland surface fire spread modelling, 1990–2007. 1: Physical and quasi-physical models|journal=International Journal of Wildland Fire|date=June 2009|volume=18|issue=4|page=349|doi=10.1071/WF06143|arxiv=0706.3074|s2cid=16173400}}</ref>
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"/> 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]] via [[direct numerical simulation]] 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 {{convert|1|km|mi|1|sp=us}}, 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.<ref name="Clark-1996-CAFb">{{cite journal|author=Clark, T. L., M. A. Jenkins, J. Coen, and David Packham|title=A coupled atmospheric-fire model: Convective Froude number and dynamic fingering|journal=International Journal of Wildland Fire|volume=6|pages=177–190|year=1996|doi=10.1071/WF9960177|issue=4|url=https://zenodo.org/record/1236052}}</ref><ref name="Clark-1996-CAF">{{cite journal|author=Clark, Terry L., Marry Ann Jenkins, Janice Coen, and David Packham|title=A coupled atmospheric-fire model: Convective feedback on fire line dynamics|journal=Journal of Applied Meteorology|volume=35|pages=875–901|year=1996|doi=10.1175/1520-0450(1996)035<0875:ACAMCF>2.0.CO;2|bibcode=1996JApMe..35..875C|issue=6|doi-access=free}}</ref><ref name="Rothermel-1972-MMP">{{cite web|author=Rothermel, Richard C.|title=A mathematical model for predicting fire spread in wildland fires|publisher=[[United States Forest Service]]|date=January 1972|url=http://www.fs.fed.us/rm/pubs_int/int_rp115.pdf|access-date=2011-02-28}}</ref> {{cnspan| Although models such as [[Los Alamos National Laboratory|Los Alamos]]' FIRETEC solve for the concentrations of fuel and [[oxygen]], 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.|date=December 2021}}
== See also ==
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