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[[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 (climate)}}
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 {{convert|5|km|mi|0|sp=us}} and {{convert|300|km|mi|-2|sp=us}} in length. A typical [[cumulus cloud]] has a scale of less than {{convert|1|km|mi|1|sp=us}}, 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]]s represent are parameterized, by processes of various sophistication. In the earliest models, if a column of air within 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 [[convection|convect]] and that [[Entrainment (meteorology)|entrainment]] and other processes occur. Weather models that have gridboxes with sizes between {{convert|5|and|25|km|mi|0|sp=us}} can explicitly represent convective clouds, although they need to parameterize [[cloud microphysics]] which occur at a smaller scale.<ref>{{cite journal|url=http://ams.confex.com/ams/pdfpapers/126017.pdf|title=3.7 Improving Precipitation Forecasts by the Operational Nonhydrostatic Mesoscale Model with the Kain-Fritsch Convective Parameterization and Cloud Microphysics|author1=Narita, Masami |author2=Shiro Ohmori |name-list-style=amp |date=2007-08-06|access-date=2011-02-15|journal=12th Conference on Mesoscale Processes}}</ref> The formation of large-scale ([[stratus cloud|stratus]]-type) clouds is more physically based; they form when the [[relative humidity]] 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]] can be related to a critical value of relative humidity less than 100%,<ref>{{cite web|url=http://www.atmos.washington.edu/~dargan/591/diag_cloud.tech.pdf |pages=4–5 |title=The Diagnostic Cloud Parameterization Scheme |author=Frierson, Dargan |publisher=[[University of Washington]] |date=2000-09-14 |access-date=2011-02-15 |url-status=dead |archive-url=https://web.archive.org/web/20110401013742/http://www.atmos.washington.edu/~dargan/591/diag_cloud.tech.pdf |archive-date=2011-04-01 }}</ref> {{cnspan|reflecting the sub grid scale variation that occurs in the real world.|date=December 2021}}
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. [[Drag (physics)|Atmospheric drag]] produced by mountains must also be parameterized, as the limitations in the resolution of [[elevation]] contours produce significant underestimates of the drag.<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=6|year=2007|publisher=Cambridge University Press|isbn=978-0-521-86540-1}}</ref> 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.<ref>{{cite book|page=188|title=A climate modelling primer|author1=McGuffie, K. |author2=A. Henderson-Sellers |name-list-style=amp |publisher=John Wiley and Sons|year=2005|isbn=978-0-470-85751-9}}</ref> Sun angle as well as the impact of multiple cloud layers is taken into account.<ref>{{cite book|url=https://books.google.com/books?id=vdg5BgBmMkQC&pg=PA226|author1=Melʹnikova, Irina N. |author2=Alexander V. Vasilyev |name-list-style=amp |pages=226–228|title=Short-wave solar radiation in the earth's atmosphere: calculation, oberservation, interpretation|year=2005|publisher=Springer|isbn=978-3-540-21452-6}}</ref> 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.<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.|pages=12–14|year=2007|publisher=Cambridge University Press|isbn=978-0-521-86540-1}}</ref> Within air quality models, parameterizations take into account atmospheric emissions from multiple relatively tiny sources (e.g. roads, fields, factories) within specific grid boxes.<ref>{{cite book|url=https://books.google.com/books?id=wh-Xf0WZQlMC&pg=PA11|pages=11–12|title=Meteorological and Air Quality Models for Urban Areas|author=Baklanov, Alexander, Sue Grimmond, Alexander Mahura|access-date=2011-02-24|year=2009|publisher=Springer|isbn=978-3-642-00297-7}}</ref>
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===Horizontal coordinates===
{{cnspan|Horizontal position may be expressed directly in [[geographic coordinates]] ([[latitude]] and [[longitude]]) for global models or in a [[map projection]] [[planar coordinates]] for regional models. The German weather service is using for its global [https://www.dwd.de/EN/research/weatherforecasting/num_modelling/01_num_weather_prediction_modells/icon_description.html ICON model] (icosahedral non-hydrostatic global circulation model) a grid based on a [[Icosahedron|regular icosahedron]]. Basic cells in this grid are triangles instead of the four corner cells in a traditional latitude-longitude grid.
The advantage is that, different from a latitude-longitude cells are everywhere on the globe the same size. Disadvantage is that equations in this non rectangular grid are more complicated.|date=December 2021}}
===Vertical coordinates===
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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}}</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 produced using a [[Kalman filter]].<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>{{cnspan|There are 24 ensemble members in the Met Office Global and Regional Ensemble Prediction System (MOGREPS).|date=December 2021}}
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}}</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>
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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 }}</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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