Editing Geopotential height

{{Short description|Type of altitude above mean sea level}}

'''Geopotential height''', also known as '''geopotential altitude''' or '''geopotential elevation''',<ref>Forrester, W.D. 1983. ''Canadian Tidal Manual''. Chapter 5: Datums and Vertical Control. Department of Fisheries and Oceans, Ottawa, 138pp. [https://psmsl.org/train_and_info/training/reading/canadian_manual/chapter5.pdf]</ref> is a [[vertical coordinate]] (with [[dimension (physics)|dimension]] of length) representing the [[work (physics)|work]] involved in lifting one [[unit of mass]] over one [[unit of length]] through a hypothetical [[space (mathematics)|space]] in which the [[Gravitational acceleration|acceleration of gravity]] is assumed constant.<ref name=NASA>{{cite web|title=NASA Technical Report R-459: Defining Constants, Equations, and Abbreviated Tables of the 1976 Standard Atmosphere|date=May 1976 |url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf|archive-url=https://web.archive.org/web/20170307211228/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf |archive-date=2017-03-07 |last1=Minzner |first1=R. A. |last2=Reber |first2=C. A. |last3=Jacchia |first3=L. G. |last4=Huang |first4=F. T. |last5=Cole |first5=A. E. |last6=Kantor |first6=A. J. |last7=Keneshea |first7=T. J. |last8=Zimmerman |first8=S. P. |last9=Forbes |first9=J. M. }}</ref> Geopotential heights are referenced to [[Earth]]'s [[mean sea level]], taking its best-fitting [[equigeopotential]] as a reference surface or [[vertical datum]].
In [[SI units]], a geopotential height difference of one [[metre|meter]] implies the vertical transport of a parcel of one [[kilogram]]; adopting the [[standard gravity]] value (9.80665 [[Metres per second squared|m/s<sup>2</sup>]]), it corresponds to a constant work or [[potential energy]] difference of 9.80665 [[joule]]s. 

Geopotential height differs from geometric height (as given by a [[tape measure]]) because [[Earth's gravity]] is not constant, varying markedly with altitude and latitude; thus, a 1-m geopotential height difference implies a different [[vertical distance]] in [[physical space]]: "the unit-mass must be lifted higher at the equator than at the pole, if the same amount of work is to be performed".<ref name="Bjerknes 1910 p. 13">{{cite book | last=Bjerknes | first=V. | author-link=V. Bjerknes | title=Dynamic Meteorology and Hydrography: Part [1]-2, [and atlas of plates] | publisher=Carnegie Institution of Washington | series=Carnegie Institution of Washington publication | issue=v. 1 | year=1910 | url=https://books.google.com/books?id=ub5XAAAAYAAJ&pg=PA13 | access-date=2023-10-05 | page=13}}</ref>
It is a useful concept in [[meteorology]], [[climatology]], and [[oceanography]]; it also remains a historical convention in aeronautics as the altitude used for calibration of aircraft [[pressure altitude|barometric altimeters]].<ref>{{cite book|last=Anderson|first=John|date=2007|title=Introduction to Flight|publisher=McGraw-Hill Science/Engineering/Math|page=109}} </ref>

==Definition==
''[[Geopotential]]'' is the [[gravitational energy|gravitational potential energy]] per unit mass at elevation <math>Z</math>:
:<math>\Phi(Z) = \int_0^Z\ 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.<ref name=NASA/>

'''Geopotential height''' may be obtained from normalizing geopotential by the acceleration of gravity:
:<math>{H} = \frac{\Phi}{g_{0}}\ = \frac{1}{g_{0}}\int_0^Z\ g(\phi,Z)\,dZ</math>
where <math>g_0</math> = 9.80665 m/s<sup>2</sup>, the [[standard gravity]] at mean sea level.<ref name="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf">{{cite web|title=NASA Technical Report R-459: Defining Constants, Equations, and Abbreviated Tables of the 1976 Standard Atmosphere|date=May 1976 |url=https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf|archive-url=https://web.archive.org/web/20170307211228/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19760017709.pdf |archive-date=2017-03-07 |last1=Minzner |first1=R. A. |last2=Reber |first2=C. A. |last3=Jacchia |first3=L. G. |last4=Huang |first4=F. T. |last5=Cole |first5=A. E. |last6=Kantor |first6=A. J. |last7=Keneshea |first7=T. J. |last8=Zimmerman |first8=S. P. |last9=Forbes |first9=J. M. }} </ref> Expressed in differential form, 
:<math>{g_0}\ {dH} = {g}\ {dZ}</math>

==Role in planetary fluids==
Geopotential height plays an important role in atmospheric and oceanographic studies.
The differential form above may be substituted into the [[Hydrostatic equilibrium|hydrostatic equation]] and [[ideal gas law]] in order to relate pressure to ambient temperature and geopotential height for measurement by barometric altimeters regardless of latitude or geometric elevation:
:<math>{dP} = {-g}\ {\rho}\ {dZ} = {-g_0}\ {\rho}\ {dH} = \frac{-g_0\ P}{R\ T}\ {dH}</math>
:<math>\frac{dP}{P} = -\frac{g_0}{R\ T}\ {dH}</math>
where <math>P</math> and <math>T</math> are ambient pressure and temperature, respectively, as functions of geopotential height, and <math>R</math> is the specific [[gas constant]]. For the subsequent [[definite integral]], the simplification obtained by assuming a constant value of gravitational acceleration is the sole reason for defining the geopotential altitude.<ref>{{cite book|last=Anderson|first=John|date=2007|title=Introduction to Flight|publisher=McGraw-Hill Science/Engineering/Math|page=116}} </ref>

==Usage==
[[File:NAM 500 MB.PNG|thumb|upright=1.5|Geopotential height analysis on the [[North American Mesoscale Model]] (NAM) at 500 hPa.]]
[[Geophysics|Geophysical]] sciences such as meteorology often prefer to express the horizontal [[Pressure-gradient force|pressure gradient force]] as the gradient of [[geopotential]] along a constant-pressure surface, because then it has the properties of a [[conservative force]]. For example, the [[primitive equations]] that [[numerical weather prediction|weather forecast models]] solve use [[hydrostatic pressure]] as a vertical coordinate, and express the slopes of those pressure surfaces in terms of geopotential height. 

A plot of geopotential height for a single pressure level in the atmosphere 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.{{fact|date=February 2016}}

The United States [[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 [i.e. [[millibar]]] height at its location is 5600&nbsp;m, it means that the level of the atmosphere over that station at which the atmospheric pressure is 500&nbsp;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|access-date=2012-03-15}}</ref>}}

==See also==
* [[Atmospheric model]]
* [[Above mean sea level]]
* [[Dynamic height]], a similar quantity used in geodesy, based on a slightly different gravity value

==References==
{{Reflist}}

==Further reading==
* 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}}.

==External links==
*{{Commonscatinline}}

[[Category:Atmospheric dynamics]]
[[Category:Vertical position]]

[[fr:Hauteur du géopotentiel]]