# Geostrophic wind

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The geostrophic wind ( or ) 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 balance. 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 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 wave.

## Origin

Air naturally moves from areas of high pressure to areas of low pressure, due to the pressure gradient force. As soon as the air starts to move, however, the Coriolis "force" deflects it. The deflection is to the right in the northern hemisphere, and to the left in the southern hemisphere. As the air moves from the high-pressure area, its speed increases, and so does its Coriolis deflection. The deflection increases until the Coriolis and pressure gradient forces are in geostrophic balance: at this point, the air flow is no longer moving from high to low pressure, but instead moves along an isobar. (Note that this explanation assumes that the atmosphere starts in a geostrophically unbalanced state and describes how such a state would evolve into a balanced flow. In practice, the flow is nearly always balanced.) The geostrophic balance helps to explain why, in the northern hemisphere, low-pressure systems (or cyclones) spin counterclockwise and high-pressure systems (or anticyclones) spin clockwise, and the opposite in the southern hemisphere.

## 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.

## 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 frictional effects, which is usually a good approximation for the synoptic scale instantaneous flow in the midlatitude mid-troposphere.[1] 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 Semigeostrophic theory are used to model flows in the atmosphere more widely. These theories allow for divergence to take place and for weather systems to then develop.

## Governing formula

Newton's Second Law can be written as follows if only the pressure gradient, gravity, and friction act on an air parcel, where the bold symbolizes a vector:

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

Here $\boldsymbol{U}$ is the velocity field of the air, $\boldsymbol{\Omega}$ is the angular velocity vector of the planet, $\rho$ is the density of the air, $p$ is the air pressure, $\boldsymbol{F}_r$ is the friction, $\boldsymbol{g}$ is the acceleration vector due to gravity and ${D \; \over Dt}$ 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:

${Du \over Dt} = -{1 \over \rho}{\partial P \over \partial x} + f \cdot v$

${Dv \over Dt} = -{1 \over \rho}{\partial P \over \partial y} - f \cdot u$

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

With $f = 2 \Omega \sin{\phi}$ the Coriolis parameter (approximately 10−4 s−1, varying with latitude).

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

$f \cdot v = {1 \over \rho}{\partial P \over \partial x}$

$f \cdot u = -{1 \over \rho}{\partial P \over \partial y}$

By substituting using the third equation above, we have:

$f \cdot v = g\frac{\partial P / \partial x}{\partial P / \partial z} = g{\partial Z \over \partial x}$

$f \cdot u = -g\frac{\partial P / \partial y}{\partial P / \partial z} = -g{\partial Z \over \partial y}$

with Z the height of the constant pressure surface (satisfying ${\partial P \over \partial x}dx + {\partial P \over \partial y}dy + {\partial P \over \partial z} dZ = 0$).

This leads us to the following result for the geostrophic wind components $(u_g,v_g)$:

$u_g = - {g \over f} {\partial Z \over \partial y}$

$v_g = {g \over f} {\partial Z \over \partial x}$

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 Φ on a surface of constant pressure:

$\overrightarrow{V_g} = {\hat{k} \over f} \times \nabla_p \Phi$

## References

1. ^ Holton, J.R., 'An Introduction to Dynamic Meteorology', International Geophysical Series, Vol 48 Academic Press.