Editing Geostrophic wind

{{Short description|Concept in atmospheric science}}
{{Redirect|Geostrophic flow|oceanic wind|Geostrophic current}}
{{More citations needed|dare=August 2018|date=August 2018}}

In [[atmospheric science]], '''geostrophic flow''' ({{IPAc-en|ˌ|dʒ|iː|ə|ˈ|s|t|r|ɒ|f|ɪ|k|,_|ˌ|dʒ|iː|oʊ|-|,_|-|ˈ|s|t|r|oʊ|-}}{{refn|{{Dictionary.com|access-date=2016-01-22|geostrophic}}}}{{refn|{{Cite dictionary |url=http://www.lexico.com/definition/geostrophic |archive-url=https://web.archive.org/web/20211223131345/https://www.lexico.com/definition/geostrophic |url-status=dead |archive-date=2021-12-23 |title=geostrophic |dictionary=[[Lexico]] UK English Dictionary |publisher=[[Oxford University Press]]}} }}{{refn|{{MerriamWebsterDictionary|access-date=2016-01-22|geostrophic}}}}) is the theoretical [[wind]] that would result from an exact balance between the [[Coriolis effect|Coriolis force]] and the [[pressure gradient]] force. This condition is called ''[[geostrophic equilibrium]]'' or ''geostrophic balance'' (also known as  ''geostrophy''). The geostrophic wind is directed [[Parallel (geometry)|parallel]] to [[isobar (meteorology)|isobar]]s (lines of constant [[Atmospheric pressure|pressure]] at a given height). This balance seldom holds exactly in nature. The true wind almost always differs from the geostrophic wind due to other forces such as [[friction]] from the ground. Thus, the actual wind would equal the geostrophic wind only if there were no friction (e.g. above the [[atmospheric boundary layer]]) and the isobars were perfectly straight. Despite this, much of the atmosphere outside the [[tropics]] is close to geostrophic flow much of the time and it is a valuable first approximation. Geostrophic flow in air or water is a zero-frequency [[inertial waves|inertial wave]].

==Origin==

A useful heuristic is to imagine [[air]] starting from rest, experiencing a force directed from areas of high [[pressure]] toward areas of low pressure, called the [[pressure gradient]] force. If the air began to move in response to that force, however, the [[Coriolis force]] would deflect it, to the right of the motion in the [[northern hemisphere]] or to the left in the [[southern hemisphere]]. As the air accelerated, the deflection would increase until the Coriolis force's strength and direction balanced the pressure gradient force, a state called geostrophic balance. At this point, the flow is no longer moving from high to low pressure, but instead moves along [[isobar (meteorology)|isobar]]s. Geostrophic balance helps to explain why, in the northern hemisphere, [[low-pressure system]]s (or ''[[cyclone]]s'') spin counterclockwise and [[High-pressure area|high-pressure systems]] (or ''[[anticyclone]]s'') spin clockwise, and the opposite in the southern hemisphere.

==Geostrophic currents==
{{main|Geostrophic current}}

Flow of ocean water is also largely geostrophic. Just as multiple weather balloons that measure pressure as a function of height in the atmosphere are used to map the atmospheric pressure field and infer the geostrophic wind, measurements of density as a function of depth in the ocean are used to infer geostrophic currents. [[satellite altimetry|Satellite altimeters]] are also used to measure sea surface height anomaly, which permits a calculation of the geostrophic current at the surface.

==Limitations of the geostrophic approximation==

The effect of friction, between the air and the land, breaks the geostrophic balance. Friction slows the flow, lessening the effect of the Coriolis force. As a result, the pressure gradient force has a greater effect and the air still moves from high pressure to low pressure, though with great deflection. This explains why high-pressure system winds radiate out from the center of the system, while low-pressure systems have winds that spiral inwards.

The geostrophic wind neglects [[friction]]al effects, which is usually a good [[approximation]] for the [[synoptic scale meteorology|synoptic scale]] instantaneous flow in the midlatitude mid-[[troposphere]].<ref>{{cite book |first1=James R. |last1=Holton |first2=Gregory J. |last2=Hakim |chapter=2.4.1 Geostrophic Approximation and Geostrophic Wind |title=An Introduction to Dynamic Meteorology |chapter-url=https://books.google.com/books?id=hcxcqQp7XOsC&pg=PA42 |date=2012 |publisher=Academic Press |isbn=978-0-12-384867-3 |pages=42–43 |edition=5th |series=International Geophysics |volume=88}}</ref> Although [[ageostrophic]] terms are relatively small, they are essential for the time evolution of the flow and in particular are necessary for the growth and decay of storms. [[Quasi-geostrophic equations|Quasigeostrophic]] and semi geostrophic theory are used to model flows in the atmosphere more widely. These theories allow for a divergence to take place and for weather systems to then develop.

==Formulation==
{{See also|Geostrophic current#Formulation}}

[[Newton's second law]] can be written as follows if only the pressure gradient, gravity, and friction act on an air parcel, where bold symbols are vectors:

:<math>{D\boldsymbol{U} \over Dt} = - {1 \over \rho} \nabla p - 2\boldsymbol{\Omega} \times \boldsymbol{U} + \boldsymbol{g} + \boldsymbol{F}_r</math>

Here '''''U''''' is the velocity field of the air, '''Ω''' is the angular velocity vector of the planet, ''ρ'' is the density of the air, P is the air pressure, '''F'''<sub>r</sub> is the friction, '''g''' is the [[standard gravity|acceleration vector due to gravity]] and {{sfrac|D|D''t''}} is the [[material derivative]].

Locally this can be expanded in [[Cartesian coordinates]], with a positive ''u'' representing an eastward direction and a positive ''v'' representing a northward direction. Neglecting friction and vertical motion, as justified by the [[Taylor–Proudman theorem]], we have:

:<math>{Du \over Dt} = -{1 \over \rho}{\partial P \over \partial x} + fv + 0 + 0</math>
:
:<math>{Dv \over Dt} = -{1 \over \rho}{\partial P \over \partial y} - fu + 0 + 0</math>
:
:<math>{D w \over Dt}=-{1 \over \rho}{\partial P \over \partial z}+0-g+0</math>

With {{nowrap|1=''f'' = 2Ω sin ''φ''}} the [[Coriolis frequency|Coriolis parameter]] (approximately {{val|e=-4|u=s<sup>−1</sup>}}, varying with latitude).

Assuming geostrophic balance, the system is stationary and the first two equations become:

:<math>fv = {1 \over \rho}{\partial P \over \partial x}</math>
:
:<math>fu =  -{1 \over \rho}{\partial P \over \partial y}</math>
:
:<math> -g -{1 \over \rho}{\partial P \over \partial z}=0</math>

By substituting using the third equation above, we have:

:<math>\begin{align}
fv &= \frac{\; -g \;}{\; \frac{\partial P}{\partial z} \;} \frac{\partial P}{\partial x} := \frac{\; -g \;}{\; c \;} a \\[5px]
fu &= - \frac{\; -g \;}{\; \frac{\partial P}{\partial z} \;} \frac{\partial P}{\partial y} :=  + \frac{\; g \;}{\; c \;}b
\end{align}</math>

with ''z'' the [[geopotential height]] of the <u>constant pressure</u> surface, satisfying

:<math>{\rm d}P={\partial P \over \partial x}{\rm d}x + {\partial P \over \partial y}{\rm d} y + {\partial P \over \partial z}{\rm d}z  :=  a{\rm d} x+b{\rm d} y+c{\rm d} z = 0</math>

Further simplify those formulae above:

<math>\begin{align}
fv & = \frac{\; -g \;}{\; c \;} a = +g\biggl( \frac{{\rm d } z}{ {\rm d} x}\biggr)_{{\rm d}y=0} \\[5px]
fu &= + \frac{\; g \;}{\; c \;}b = -g\biggl(\frac{{\rm d} z}{{\rm d} y}\biggr)_{{\rm d} x=0}
\end{align}</math>

This leads us to the following result for the geostrophic wind components:

<math> v_g = {g \over f}  {{\rm d}z \over {\rm d} x}</math>

<math> u_g = - {g \over f}  {{\rm d} z \over {\rm d} y}</math>

The validity of this approximation depends on the local [[Rossby number]]. It is invalid at the equator, because ''f'' is equal to zero there, and therefore generally not used in the [[tropics]].

Other variants of the equation are possible; for example, the geostrophic wind vector can be expressed in terms of the gradient of the [[geopotential height|geopotential]] Φ on a surface of constant pressure:

:<math>\mathbf{V}_\mathrm{g} = \frac\hat\mathbf{k}{f} \times \nabla_p \Phi </math>

== See also ==
* [[Geostrophic current]]
* [[Thermal wind]]
* [[Gradient wind]]
* [[Prevailing winds]]

==References==
{{Reflist}}

== External links ==
* [http://atmos.nmsu.edu/education_and_outreach/encyclopedia/geostrophic.htm Geostrophic approximation]
* [http://nsidc.org/cryosphere/glossary/term/geostrophic-wind Definition of geostrophic wind]
* [https://web.archive.org/web/20110726123313/http://atmo.tamu.edu/class/atmo203/tut/windpres/wind8.html Geostrophic wind description]

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[[Category:Fluid dynamics]]
[[Category:Atmospheric dynamics]]