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Geostrophic wind

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Concept in atmospheric science

"Geostrophic flow" redirects here. For oceanic wind, seeGeostrophic current.

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Inatmospheric science,geostrophic flow (/ˌdʒiːəˈstrɒfɪk,ˌdʒiːoʊ-,-ˈstroʊ-/^[1]^[2]^[3]) is the theoreticalwind that would result from an exact balance between theCoriolis force and thepressure gradient force. This condition is calledgeostrophic equilibrium orgeostrophic balance (also known asgeostrophy). The geostrophic wind is directedparallel toisobars (lines of constantpressure 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 asfriction from the ground. Thus, the actual wind would equal the geostrophic wind only if there were no friction (e.g. above theatmospheric boundary layer) and the isobars were perfectly straight. Despite this, much of the atmosphere outside thetropics 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-frequencyinertial wave.

Origin

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A useful heuristic is to imagineair starting from rest, experiencing a force directed from areas of highpressure toward areas of low pressure, called thepressure gradient force. If the air began to move in response to that force, however, theCoriolis force would deflect it, to the right of the motion in theNorthern Hemisphere or to the left in theSouthern 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 alongisobars. Geostrophic balance helps to explain why, in the Northern Hemisphere,low-pressure systems (orcyclones) spin counterclockwise andhigh-pressure systems (oranticyclones) spin clockwise, and the opposite in the Southern Hemisphere.

Geostrophic currents

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Main article: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 altimeters are also used to measure sea surface heightanomaly, which permits a calculation of the geostrophic current at the surface.

Limitations of the geostrophic approximation

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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 neglectsfrictional effects, which is usually a goodapproximation for thesynoptic scale instantaneous flow in the midlatitude mid-troposphere.^[4] Althoughageostrophic 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.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

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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:

{D{\boldsymbol {U}} \over Dt}=-{1 \over \rho }\nabla p-2{\boldsymbol {\Omega }}\times {\boldsymbol {U}}+{\boldsymbol {g}}+{\boldsymbol {F}}_{r}

HereU 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_r is the friction,g is theacceleration vector due to gravity and⁠D/Dt⁠ is thematerial derivative.

Locally this can be expanded inCartesian coordinates, with a positiveu representing an eastward direction and a positivev representing a northward direction. Neglecting friction and vertical motion, as justified by theTaylor–Proudman theorem, we have:

{Du \over Dt}=-{1 \over \rho }{\partial P \over \partial x}+fv+0+0

{Dv \over Dt}=-{1 \over \rho }{\partial P \over \partial y}-fu+0+0

{Dw \over Dt}=-{1 \over \rho }{\partial P \over \partial z}+0-g+0

Withf = 2Ω sinφ theCoriolis parameter (approximately10⁻⁴ s⁻¹, varying with latitude).

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

fv={1 \over \rho }{\partial P \over \partial x}

fu=-{1 \over \rho }{\partial P \over \partial y}

-g-{1 \over \rho }{\partial P \over \partial z}=0

By substituting using the third equation above, we have:

{\begin{aligned}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{aligned}}

withz thegeopotential height of theconstant pressure surface, satisfying

{\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

Further simplify those formulae above:

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

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

$v_{g}={g \over f}{{\rm {d}}z \over {\rm {d}}x}$

$u_{g}=-{g \over f}{{\rm {d}}z \over {\rm {d}}y}$

The validity of this approximation depends on the localRossby number. It is invalid at the equator, becausef is equal to zero there, and therefore generally not used in thetropics.

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