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[[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> This correlation between coordinate systems can be made since pressure decreases with height through the [[Earth's atmosphere]].<ref>{{cite book|author=Ahrens, C. Donald|page=10|isbn=978-0-495-11558-8|year=2008|publisher=Cengage Learning|title=Essentials of meteorology: an invitation to the atmosphere|url=https://books.google.com/books?id=2Yn29IFukbgC&pg=PA244}}</ref> The first model used for operational forecasts, the single-layer barotropic model, used a single pressure coordinate at the 500-millibar (about {{convert|5500|m|ft|abbr=on}}) level,<ref name="Charney 1950">{{cite journal|last1=Charney|first1=Jule|last2=Fjørtoft|first2=Ragnar|last3=von Neumann|first3=John|title=Numerical Integration of the Barotropic Vorticity Equation|journal=Tellus|date=November 1950|volume=2|issue=4|bibcode=1950Tell....2..237C |doi=10.3402/tellusa.v2i4.8607|author-link1=Jule Charney|author-link2=Ragnar Fjørtoft|author-link3=John von Neumann|pages=237|doi-access=free}}</ref> and thus was essentially two-dimensional. High-resolution models—also called ''mesoscale models''—such as the [[Weather Research and Forecasting model]] tend to use normalized pressure coordinates referred to as [[sigma coordinates]].<ref>{{cite web|last=Janjic |first=Zavisa |title=Scientific Documentation for the NMM Solver |url=http://nldr.library.ucar.edu/collections/technotes/asset-000-000-000-845.pdf |publisher=[[National Center for Atmospheric Research]] |access-date=2011-01-03 |author2=Gall, Robert |author3=Pyle, Matthew E. |pages=12–13 |date=February 2010 |url-status=dead |archive-url=https://web.archive.org/web/20110823082059/http://nldr.library.ucar.edu/collections/technotes/asset-000-000-000-845.pdf |archive-date=2011-08-23 }}</ref> This coordinate system receives its name from the [[independent variable]] <math>\sigma</math> used to [[nondimensionalization|scale]] atmospheric pressures with respect to the pressure at the surface, and in some cases also with the pressure at the top of the ___domain.<ref>{{cite book|last=Pielke|first=Roger A.|title=Mesoscale Meteorological Modeling|url=https://archive.org/details/mesoscalemeteoro00srro|url-access=limited|year=2002|publisher=[[Academic Press]]|isbn=978-0-12-554766-6|pages=[https://archive.org/details/mesoscalemeteoro00srro/page/n147 131]–132}}</ref>
==Model output statistics==
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