# Geostrophic wind

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The geostrophic wind ( [1] [2] [3] ) 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. 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.

Wind is the flow of gases on a large scale. On the surface of the Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the Sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space. Winds are commonly classified by their spatial scale, their speed, the types of forces that cause them, the regions in which they occur, and their effect. The strongest observed winds on a planet in the Solar System occur on Neptune and Saturn. Winds have various aspects: velocity ; the density of the gas involved; energy content or wind energy. Wind is also an important means of transportation for seeds and small birds; with time things can travel thousands of miles in the wind.

In atmospheric science, the pressure gradient is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The pressure gradient is a dimensional quantity expressed in units of pascals per metre (Pa/m). Mathematically, it is obtained by applying the del operator to a pressure function of position. The negative gradient of pressure is known as the force density.

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.

## 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 cyclones ) spin counterclockwise and high-pressure systems (or anticyclones ) spin clockwise, and the opposite in the southern hemisphere.

In meteorology, a cyclone is a large scale air mass that rotates around a strong center of low atmospheric pressure. Cyclones are characterized by inward spiraling winds that rotate about a zone of low pressure. The largest low-pressure systems are polar vortices and extratropical cyclones of the largest scale. Warm-core cyclones such as tropical cyclones and subtropical cyclones also lie within the synoptic scale. Mesocyclones, tornadoes, and dust devils lie within smaller mesoscale. Upper level cyclones can exist without the presence of a surface low, and can pinch off from the base of the tropical upper tropospheric trough during the summer months in the Northern Hemisphere. Cyclones have also been seen on extraterrestrial planets, such as Mars, Jupiter, and Neptune. Cyclogenesis is the process of cyclone formation and intensification. Extratropical cyclones begin as waves in large regions of enhanced mid-latitude temperature contrasts called baroclinic zones. These zones contract and form weather fronts as the cyclonic circulation closes and intensifies. Later in their life cycle, extratropical cyclones occlude as cold air masses undercut the warmer air and become cold core systems. A cyclone's track is guided over the course of its 2 to 6 day life cycle by the steering flow of the subtropical jet stream.

A high-pressure area, high, or anticyclone, is a region where the atmospheric pressure at the surface of the planet is greater than its surrounding environment.

An anticyclone is a weather phenomenon defined by the United States National Weather Service's glossary as "a large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere". Effects of surface-based anticyclones include clearing skies as well as cooler, drier air. Fog can also form overnight within a region of higher pressure. Mid-tropospheric systems, such as the subtropical ridge, deflect tropical cyclones around their periphery and cause a temperature inversion inhibiting free convection near their center, building up surface-based haze under their base. Anticyclones aloft can form within warm core lows such as tropical cyclones, due to descending cool air from the backside of upper troughs such as polar highs, or from large scale sinking such as the subtropical ridge. The evolution of an anticyclone depends upon variables such as its size, intensity, and extent of moist convection, as well as the Coriolis force.

## 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. [4] 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..

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:

An approximation is anything that is intentionally similar but not exactly equal to something else.

The synoptic scale in meteorology is a horizontal length scale of the order of 1000 kilometers or more. This corresponds to a horizontal scale typical of mid-latitude depressions. Most high and low-pressure areas seen on weather maps such as surface weather analyses are synoptic-scale systems, driven by the location of Rossby waves in their respective hemisphere. Low-pressure areas and their related frontal zones occur on the leading edge of a trough within the Rossby wave pattern, while high-pressure areas form on the back edge of the trough. Most precipitation areas occur near frontal zones. The word synoptic is derived from the Greek word συνοπτικός, meaning seen together.

## 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:

${\displaystyle {\frac {\mathrm {D} {\boldsymbol {U}}}{\mathrm {D} t}}=-2{\boldsymbol {\Omega }}\times {\boldsymbol {U}}-{\frac {1}{\rho }}\nabla p+\mathbf {g} +\mathbf {F} _{\mathrm {r} }}$

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, Fr is the friction, g is the acceleration vector due to gravity and D/Dt is the material derivative.

The standard acceleration due to gravity, sometimes abbreviated as standard gravity, usually denoted by ɡ0 or ɡn, is the nominal gravitational acceleration of an object in a vacuum near the surface of the Earth. It is defined by standard as 9.80665 m/s2. This value was established by the 3rd CGPM and used to define the standard weight of an object as the product of its mass and this nominal acceleration. The acceleration of a body near the surface of the Earth is due to the combined effects of gravity and centrifugal acceleration from the rotation of the Earth ; the total is about 0.5% greater at the poles than at the Equator.

In continuum mechanics, the material derivative describes the time rate of change of some physical quantity of a material element that is subjected to a space-and-time-dependent macroscopic velocity field variations of that physical quantity. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.

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:

{\displaystyle {\begin{aligned}{\frac {\mathrm {D} u}{\mathrm {D} t}}&=-{\frac {1}{\rho }}{\frac {\partial P}{\partial x}}+f\cdot v\\[5px]{\frac {\mathrm {D} v}{\mathrm {D} t}}&=-{\frac {1}{\rho }}{\frac {\partial P}{\partial y}}-f\cdot u\\[5px]0&=-g-{\frac {1}{\rho }}{\frac {\partial P}{\partial z}}\end{aligned}}}

With f = 2Ω sin φ the Coriolis parameter (approximately 10−4 s−1, varying with latitude).

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

{\displaystyle {\begin{aligned}f\cdot v&=\;\;\,{\frac {1}{\rho }}{\frac {\partial P}{\partial x}}\\[5px]f\cdot u&=-{\frac {1}{\rho }}{\frac {\partial P}{\partial y}}\end{aligned}}}

By substituting using the third equation above, we have:

{\displaystyle {\begin{aligned}f\cdot v&=\;\;\,g{\frac {\;{\frac {\partial P}{\partial x}}\;}{\;{\frac {\partial P}{\partial z}}\;}}=\;\;\,g{\frac {\partial Z}{\partial x}}\\[5px]f\cdot u&=-g{\frac {\;{\frac {\partial P}{\partial y}}\;}{\;{\frac {\partial P}{\partial z}}\;}}=-g{\frac {\partial Z}{\partial y}}\end{aligned}}}

with Z the height of the constant pressure surface (geopotential height), satisfying

${\displaystyle {\frac {\partial P}{\partial x}}\mathrm {d} x+{\frac {\partial P}{\partial y}}\mathrm {d} y+{\frac {\partial P}{\partial z}}\mathrm {d} Z=0}$

This leads us to the following result for the geostrophic wind components (ug, vg):

{\displaystyle {\begin{aligned}u_{\mathrm {g} }&=-{\frac {g}{f}}{\frac {\partial Z}{\partial y}}\\[5px]v_{\mathrm {g} }&=\;\;\,{\frac {g}{f}}{\frac {\partial Z}{\partial x}}\end{aligned}}}

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 Φ on a surface of constant pressure:

${\displaystyle \mathbf {V} _{\mathrm {g} }={\frac {\hat {\mathbf {k} }}{f}}\times \nabla _{p}\Phi }$

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4. Holton, James R.; Hakim, Gregory J. (2012). "2.4.1 Geostrophic Approximation and Geostrophic Wind". An Introduction to Dynamic Meteorology. Internation Geophysics. 88 (5th ed.). Academic Press. pp. 42–43. ISBN   978-0-12-384867-3.