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'''Geopotential height''' is a vertical coordinate referenced to Earth's [[mean sea level]] — an adjustment to geometric height ([[elevation]] above mean sea level) using the variation of [[gravity]] with [[latitude]] and elevation. Thus it can be considered a "gravity-adjusted height". One usually speaks of the geopotential height of a certain '''pressure level''', which would correspond to the geopotential height necessary to reach the given [[atmospheric pressure|pressure]].

At an elevation of <math>h</math>, the '''[[geopotential]]''' is defined as
:<math>\Phi(h) = \int_0^h g(\phi,z)\,dz\, ,</math>
where <math>g(\phi,z)</math> is the acceleration due to gravity, <math>\phi</math> is latitude, and <math>z</math> is the geometric elevation.
Thus geopotential is the gravitational potential energy per unit mass at that elevation <math>h</math>.

The '''geopotential height''' is
:<math>{Z_g}(h) = \frac{\Phi(h)}{g_{0}}\, ,</math> 
which normalizes the geopotential to <math>g_0</math>, the [[standard gravity]] at mean sea level.

[[Geophysics|Geophysical scientists]] often use geopotential height as a function of pressure rather than pressure as a function of geometric height, because doing so in many cases makes analytical calculations more convenient. For example, the [[primitive equations]] which [[numerical weather prediction|weather forecast models]] solve are more easily expressed in terms of geopotential than geometric height. Using the former eliminates [[air density]] from the equations.

A plot of geopotential height for a single pressure level shows the troughs and ridges, [[high-pressure area|Highs]] and [[low-pressure area|Lows]], which are typically seen on upper air charts. The geopotential thickness between pressure levels — difference of the 850 [[Pascal (unit)|hPa]] and 1000 hPa geopotential heights for example — is proportional to mean [[virtual temperature]] in that layer. Geopotential height contours can be used to calculate the [[geostrophic wind]], which is faster where the contours are more closely spaced and tangential to the geopotential height contours.

The [[National Weather Service]] defines geopotential height as
{{Quotation|...roughly the height above sea level of a pressure level. For example, if a station reports that the 500 mb height at its location is 5600 m, it means that the level of the atmosphere over that station at which the atmospheric pressure is 500 mb is 5600 meters above sea level. This is an estimated height based on temperature and pressure data.<ref>{{cite web|title=Height|url=http://www.weather.gov/glossary/index.php?letter=h|work=NOAA's National Weather Service Glossary|publisher=NOAA National Weather Service|accessdate=15 March 2012}}</ref>}}

==See also==
* [[Above mean sea level]]

==References==

{{Reflist}}
* Hofmann-Wellenhof, B. and Moritz, H. "Physical Geodesy", 2005. ISBN 3-211-23584-1
* Eskinazi, S. "Fluid Mechanics and Thermodynamics of our Environment", 1975. ISBN 0-12-242540-5

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[[Category:Atmospheric dynamics]]