Definitions

# Geostrophic wind

The geostrophic wind is the theoretical wind that would result from an exact balance between the Coriolis force 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 or the centrifugal force from curved fluid flow. 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.

## 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 due to the rotation of the earth. The 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 the deflection from the Coriolis force. The deflection increases until the Coriolis and pressure gradient forces are in geostrophic balance, at which point the air is no longer moving from high to low pressure, but instead moves along an isobar, a line of equal pressure (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 approximation has no predictive value since it does not contain any expression for change: it is purely diagnostic). The geostrophic balance helps to explain why low pressure systems spin counterclockwise and high pressure systems spin clockwise in the northern hemisphere (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. Geostrophic flow in air or water is a zero-frequency inertial wave.

## 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. 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.

## Governing formula

Assuming geostrophic balance, the geostrophic wind components $\left(u_g,v_g\right)$ on a constant-pressure surface can be derived as:

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

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

where g is the acceleration due to gravity (9.81 m.s-2), f is the Coriolis parameter (approximately 10−4 s−1, varying with latitude) and Z is the geopotential height of the constant pressure surface. 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:

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