Geostrophic wind

In atmospheric science, geostrophic flow () is the theoretical wind that would result from an exact balance between the Coriolis effect and the pressure gradient force. This condition is called geostrophic equilibrium or geostrophic balance (also known as geostrophy). The geostrophic wind is directed parallel to isobars (lines of constant 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.

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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 isobars. Geostrophic balance helps to explain why, in the northern hemisphere, low-pressure systems (or ) spin counterclockwise and high-pressure systems (or ) spin clockwise, and the opposite in the southern hemisphere.

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Geostrophic currents 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 height anomaly, which permits a calculation of the geostrophic current at the surface.

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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 effects, which is usually a good approximation for the synoptic scale instantaneous flow in the midlatitude mid-troposphere.

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. 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.

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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:

$\{D\backslash boldsymbol\{U\}\; \backslash over\; Dt\}\; =\; -\; \{1\; \backslash over\; \backslash rho\}\; \backslash nabla\; p\; -\; 2\backslash boldsymbol\{\backslash Omega\}\; \backslash times\; \backslash boldsymbol\{U\}\; +\; \backslash boldsymbol\{g\}\; +\; \backslash boldsymbol\{F\}\_r$

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_r is the friction, g is the standard gravity and 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:

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

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

$\{D\; w\; \backslash over\; Dt\}=-\{1\; \backslash over\; \backslash rho\}\{\backslash partial\; P\; \backslash over\; \backslash partial\; z\}+0-g+0$

With the Coriolis parameter (approximately , varying with latitude).

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

$fv\; =\; \{1\; \backslash over\; \backslash rho\}\{\backslash partial\; P\; \backslash over\; \backslash partial\; x\}$

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

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

By substituting using the third equation above, we have:

$\backslash begin\{align\}$

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}

with z the geopotential height of the constant pressure surface, satisfying

$\{\backslash rm\; d\}P=\{\backslash partial\; P\; \backslash over\; \backslash partial\; x\}\{\backslash rm\; d\}x\; +\; \{\backslash partial\; P\; \backslash over\; \backslash partial\; y\}\{\backslash rm\; d\}\; y\; +\; \{\backslash partial\; P\; \backslash over\; \backslash partial\; z\}\{\backslash rm\; d\}z\; :=\; a\{\backslash rm\; d\}\; x+b\{\backslash rm\; d\}\; y+c\{\backslash rm\; d\}\; z\; =\; 0$

Further simplify those formulae above:

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