Editing Climate model (section)

=== Zero-dimensional models ===

Zero-dimensional models consider Earth as a point in space, analogous to the [[pale blue dot]] viewed by [[Voyager 1]] or an astronomer's view of very distant objects.    This [[zero-dimensional space|dimensionless]] view while highly limited is still useful in that the laws of physics are applicable in a bulk fashion to unknown objects,  or in an appropriate lumped manner if some major properties of the object are known.   For example, astronomers know that most planets in our own solar system feature some kind of solid/liquid surface surrounded by a gaseous atmosphere.

==== 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}}

==== Models with separated surface and atmospheric layers ====
[[file:greenhouse slab model.png|thumb|upright=1|right|One-layer EBM with blackbody surface]]

Dimensionless models have also been constructed with functionally separated atmospheric layers from the surface.   The simplest of these is the [[idealized greenhouse model|zero-dimensional, one-layer model]],<ref>{{cite web |url=https://www.acs.org/content/acs/en/climatescience/atmosphericwarming/singlelayermodel.html |title=ACS Climate Science Toolkit - Atmospheric Warming - A Single-Layer Atmosphere Model |publisher=[[American Chemical Society]] |accessdate=2 October 2022}}</ref>  which may be readily extended to an arbitrary number of atmospheric layers.   The surface and atmospheric layer(s) are each characterized by a corresponding temperature and emissivity value, but no thickness.  Applying radiative equilibrium (i.e conservation of energy) at the interfaces between layers produces a set of coupled equations which are solvable.<ref>{{cite web |url=https://www.acs.org/content/acs/en/climatescience/atmosphericwarming/multilayermodel.html |title=ACS Climate Science Toolkit - Atmospheric Warming - A Multi-Layer Atmosphere Model |publisher=[[American Chemical Society]] |accessdate=2 October 2022}}</ref>

Layered models produce temperatures that better estimate those observed for Earth's surface and atmospheric levels.<ref>{{cite web |url=https://www.e-education.psu.edu/meteo469/node/198 |title=METEO 469: From Meteorology to Mitigation - Understanding Global Warming - Lesson 5 - Modelling of the Climate System - One-Layer Energy Balance Model |publisher=[[Pennsylvania State University]] College of Mineral and Earth Sciences - Department of Meteorology and Atmospheric Sciences |accessdate=2 October 2022}}</ref>  They likewise further illustrate the radiative [[heat transfer]] processes which underlie the greenhouse effect.  Quantification of this phenomenon using a version of the one-layer model was first published by [[Svante Arrhenius]] in year 1896.<ref name="sa1896">{{Cite journal | author=Svante Arrhenius | year=1896 | title=On the influence of carbonic acid in the air upon the temperature of the ground | journal=Philosophical Magazine and Journal of Science | volume=41 | issue=251 | pages=237–276 | language=en| doi=10.1080/14786449608620846 | url=https://zenodo.org/record/1431217 }}</ref>