Editing Climate model (section)

==== Model with combined surface and atmosphere ====

A very simple model of the [[radiative equilibrium]] of the Earth is

:<math>(1-a)S \pi r^2 = 4 \pi r^2 \epsilon \sigma T^4</math>

where
* the left hand side represents the total incoming shortwave power (in Watts) from the Sun
* the right hand side represents the total outgoing longwave power (in Watts) from Earth, calculated from the [[Stefan–Boltzmann law]]. 
The constant parameters include
* ''[[Solar constant|S]]'' is the [[solar constant]] – the incoming solar radiation per unit area—about 1367&nbsp;W·m<sup>−2</sup>
* ''[[Radius|r]]'' is Earth's radius—approximately 6.371×10<sup>6</sup>&nbsp;m
* ''[[pi|π]]'' is the mathematical constant (3.141...)
* ''<math> \sigma </math>'' is the [[Stefan–Boltzmann constant]]—approximately 5.67×10<sup>−8</sup> J·K<sup>−4</sup>·m<sup>−2</sup>·s<sup>−1</sup>

The constant <math> \pi\,r^2 </math> can be factored out, giving a nildimensional equation for the equilibrium

:<math>(1-a)S = 4 \epsilon \sigma T^4</math>

where
* the left hand side represents the incoming shortwave energy flux from the Sun in W·m<sup>−2</sup>
* the right hand side represents the outgoing longwave energy flux from Earth in W·m<sup>−2</sup>.
The remaining variable parameters which are specific to the planet include
* ''<math>a</math>'' is Earth's average [[albedo]], measured to be 0.3.<ref>{{cite journal |last=Goode |first=P. R. |year=2001 |title=Earthshine Observations of the Earth's Reflectance |journal=Geophys. Res. Lett. |volume=28 |issue=9 |pages=1671–4 |doi=10.1029/2000GL012580 |bibcode=2001GeoRL..28.1671G|s2cid=34790317 |display-authors=etal|url=https://authors.library.caltech.edu/50838/1/grl14388.pdf |archive-url=https://web.archive.org/web/20180722192421/https://authors.library.caltech.edu/50838/1/grl14388.pdf |archive-date=2018-07-22 |url-status=live }}</ref><ref>{{cite web |title=Scientists Watch Dark Side of the Moon to Monitor Earth's Climate |url=http://www.agu.org/sci_soc/prrl/prrl0113.html |work=American Geophysical Union |date=17 April 2001 |access-date=1 March 2010 |archive-date=27 February 2009 |archive-url=https://web.archive.org/web/20090227182139/http://www.agu.org/sci_soc/prrl/prrl0113.html |url-status=dead }}</ref>
* ''<math> T </math>'' is Earth's [[global surface temperature|average surface temperature]], measured as about 288&nbsp;[[Kelvin|K]] as of year 2020<ref>{{cite web |url=https://www.climate.gov/news-features/understanding-climate/climate-change-global-temperature |title=Climate Change: Global Temperature |publisher=NOAA |accessdate=6 July 2023}}</ref>
* ''<math> \epsilon </math>'' is the [[Emissivity#Effective emissivity due to atmosphere|effective emissivity]] of Earth's combined surface and atmosphere (including clouds).   It is a quantity between 0 and 1 that is calculated from the equilibrium to be about 0.61.   For the zero-dimensional treatment it is equivalent to an average value over all viewing angles.

This very simple model is quite instructive. For example, it shows the temperature sensitivity to changes in the solar constant, Earth albedo, or effective Earth emissivity.  The effective emissivity also gauges the strength of the atmospheric [[greenhouse effect]],  since it is the ratio of the thermal emissions escaping to space versus those emanating from the surface.<ref>{{cite web |url=http://eospso.gsfc.nasa.gov/ftp_docs/lithographs/CERES_litho.pdf |title=Clouds and the Earth's Radiant Energy System |publisher=NASA |archive-url=https://web.archive.org/web/20130218204711/http://eospso.gsfc.nasa.gov/ftp_docs/lithographs/CERES_litho.pdf |archive-date=18 February 2013 |year=2013 |url-status=dead}}</ref>

The calculated emissivity can be compared to available data. Terrestrial surface emissivities are all in the range of 0.96 to 0.99<ref>{{cite web|url=http://www.icess.ucsb.edu/modis/EMIS/html/seawater.html|title=Seawater Samples - Emissivities|work=ucsb.edu}}</ref><ref>{{cite journal |doi=10.1175/JCLI3720.1 |vauthors=Jin M, Liang S |title=An Improved Land Surface Emissivity Parameter for Land Surface Models Using Global Remote Sensing Observations |journal=J. Climate |volume=19 |issue=12 |pages=2867–81 |date=15 June 2006 |url=http://www.glue.umd.edu/~sliang/papers/Jin2006.emissivity.pdf |archive-url=https://web.archive.org/web/20070604185622/http://www.glue.umd.edu/~sliang/papers/Jin2006.emissivity.pdf |archive-date=2007-06-04 |url-status=live |bibcode = 2006JCli...19.2867J }}</ref> (except for some small desert areas which may be as low as 0.7). Clouds, however, which cover about half of the planet's surface, have an average emissivity of about 0.5<ref>{{cite conference |author1=T.R. Shippert |author2=S.A. Clough |author3=P.D. Brown |author4=W.L. Smith |author5=R.O. Knuteson |author6=S.A. Ackerman |title=Spectral Cloud Emissivities from LBLRTM/AERI QME  |book-title=Proceedings of the Eighth Atmospheric Radiation Measurement (ARM) Science Team Meeting March 1998 Tucson, Arizona |url=http://www.arm.gov/publications/proceedings/conf08/extended_abs/shippert_tr.pdf |archive-url=https://web.archive.org/web/20060925194147/http://www.arm.gov/publications/proceedings/conf08/extended_abs/shippert_tr.pdf |archive-date=2006-09-25 |url-status=live }}</ref> (which must be reduced by the fourth power of the ratio of cloud absolute temperature to average surface absolute temperature) and an average cloud temperature of about {{convert|258|K|abbr=on}}.<ref>{{cite conference |author1=A.G. Gorelik |author2=V. Sterljadkin |author3=E. Kadygrov |author4=A. Koldaev |title=Microwave and IR Radiometry for Estimation of Atmospheric Radiation Balance and Sea Ice Formation |book-title=Proceedings of the Eleventh Atmospheric Radiation Measurement (ARM) Science Team Meeting March 2001 Atlanta, Georgia |url=http://www.arm.gov/publications/proceedings/conf11/extended_abs/gorelik_ag.pdf |archive-url=https://web.archive.org/web/20060925174423/http://www.arm.gov/publications/proceedings/conf11/extended_abs/gorelik_ag.pdf |archive-date=2006-09-25 |url-status=live }}</ref>  Taking all this properly into account results in an effective earth emissivity of about 0.64 (earth average temperature {{convert|285|K|abbr=on}}).{{cn|date=July 2023}}