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The **geostrophic wind** ( /ˌdʒiːəˈstrɒfɪk, ^{ [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, an important one being its velocity ; another the density of the gas involved; another its energy content or wind energy. Wind is also a great source 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 Pa/m

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.

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.

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 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 on a few variables such as its size, intensity, moist-convection, Coriolis force etc.

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.

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 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*.

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:

Here * U* is the velocity field of the air,

The **standard acceleration due to gravity**, sometimes abbreviated as **standard gravity**, usually denoted by *ɡ*_{0} or *ɡ*_{n}, 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/s^{2}. 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:

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

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

By substituting using the third equation above, we have:

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

This leads us to the following result for the geostrophic wind components (*u*_{g}, *v*_{g}):

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:

In physics, the **Navier–Stokes equations**, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous fluid substances.

A **cylindrical coordinate system** is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

In the calculus of variations, a field of mathematical analysis, the **functional derivative** relates a change in a functional to a change in a function on which the functional depends.

The **primitive equations** are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:

- A
*continuity equation*: Representing the conservation of mass. *Conservation of momentum*: Consisting of a form of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere- A
*thermal energy equation*: Relating the overall temperature of the system to heat sources and sinks

In fluid dynamics, the **Euler equations** are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity. In fact, Euler equations can be obtained by linearization of some more precise continuity equations like Navier–Stokes equations in a local equilibrium state given by a Maxwellian. The Euler equations can be applied to incompressible and to compressible flow – assuming the flow velocity is a solenoidal field, or using another appropriate energy equation respectively. Historically, only the incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – of the more general compressible equations together as "the Euler equations".

**Hydraulic head** or **piezometric head** is a specific measurement of liquid pressure above a vertical datum.

In physics, **ram pressure** is a pressure exerted on a body moving through a fluid medium, caused by relative bulk motion of the fluid rather than random thermal motion. It causes a drag force to be exerted on the body. Ram pressure is given in tensor form as

The **Sverdrup balance**, or **Sverdrup relation**, is a theoretical relationship between the wind stress exerted on the surface of the open ocean and the vertically integrated meridional (north-south) transport of ocean water.

The **Ekman layer** is the layer in a fluid where there is a force balance between pressure gradient force, Coriolis force and turbulent drag. It was first described by Vagn Walfrid Ekman. Ekman layers occur both in the atmosphere and in the ocean.

**Ekman transport**, part of Ekman motion theory first investigated in 1902 by Vagn Walfrid Ekman, refers to the wind-driven net transport of the surface layer of a fluid that, due to the Coriolis effect, occurs at 90° to the direction of the surface wind. This phenomenon was first noted by Fridtjof Nansen, who recorded that ice transport appeared to occur at an angle to the wind direction during his Arctic expedition during the 1890s. The direction of transport is dependent on the hemisphere: in the northern hemisphere, transport occurs at 90° clockwise from wind direction, while in the southern hemisphere it occurs at a 90° counterclockwise.

**Potential vorticity** (PV) is seen as one of the important theoretical successes of modern meteorology. It is a simplified approach for understanding fluid motions in a rotating system such as the Earth's atmosphere and ocean. Its development traces back to the circulation theorem by Bjerknes in 1898, which is a specialized form of Kelvin's circulation theorem. Starting from Hoskins et al., 1985, PV has been more commonly used in operational weather diagnosis such as tracing dynamics of air parcels and inverting for the full flow field. Even after detailed numerical weather forecasts on finer scales were made possible by increases in computational power, the PV view is still used in academia and routine weather forecasts, shedding light on the synoptic scale features for forecasters and researchers.

A **geostrophic current** is an oceanic current in which the pressure gradient force is balanced by the Coriolis effect. The direction of geostrophic flow is parallel to the isobars, with the high pressure to the right of the flow in the Northern Hemisphere, and the high pressure to the left in the Southern Hemisphere. This concept is familiar from weather maps, whose isobars show the direction of geostrophic flow in the atmosphere. Geostrophic flow may be either barotropic or baroclinic. A geostrophic current may also be thought of as a rotating shallow water wave with a frequency of zero. The principle of geostrophy is useful to oceanographers because it allows them to infer ocean currents from measurements of the sea surface height or from vertical profiles of seawater density taken by ships or autonomous buoys. The major currents of the world's oceans, such as the Gulf Stream, the Kuroshio Current, the Agulhas Current, and the Antarctic Circumpolar Current, are all approximately in geostrophic balance and are examples of geostrophic currents.

The **shallow water equations** are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid. The shallow water equations in unidirectional form are also called **Saint-Venant equations**, after Adhémar Jean Claude Barré de Saint-Venant.

The intent of this article is to highlight the important points of the **derivation of the Navier–Stokes equations** as well as its application and formulation for different families of fluids.

In atmospheric science, **balanced flow** is an idealisation of atmospheric motion. The idealisation consists in considering the behaviour of one isolated parcel of air having constant density, its motion on a horizontal plane subject to selected forces acting on it and, finally, steady-state conditions.

The **Cauchy momentum equation** is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum. In convective form it is written:

In nonideal fluid dynamics, the **Hagen–Poiseuille equation**, also known as the **Hagen–Poiseuille law**, **Poiseuille law** or **Poiseuille equation**, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing through a long cylindrical pipe of constant cross section. It can be successfully applied to air flow in lung alveoli, or the flow through a drinking straw or through a hypodermic needle. It was experimentally derived independently by Jean Léonard Marie Poiseuille in 1838 and Gotthilf Heinrich Ludwig Hagen, and published by Poiseuille in 1840–41 and 1846.

**Ocean dynamics** define and describe the motion of water within the oceans. Ocean temperature and motion fields can be separated into three distinct layers: mixed (surface) layer, upper ocean, and deep ocean.

In fluid dynamics, the **radiation stress** is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor.

**Q-vectors** are used in atmospheric dynamics to understand physical processes such as vertical motion and frontogenesis. Q-vectors are not physical quantities that can be measured in the atmosphere but are derived from the quasi-geostrophic equations and can be used in the previous diagnostic situations. On meteorological charts, Q-vectors point toward upward motion and away from downward motion. Q-vectors are an alternative to the omega equation for diagnosing vertical motion in the quasi-geostrophic equations.

- ↑ "geostrophic".
*Dictionary.com Unabridged*. Random House . Retrieved 2016-01-22. - ↑ "geostrophic".
*Oxford Dictionaries*. Oxford University Press . Retrieved 2016-01-22. - ↑ "Geostrophic".
*Merriam-Webster Dictionary*. Retrieved 2016-01-22. - ↑ 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.

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