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The **thermal wind** is the vector difference between the geostrophic wind at upper altitudes minus that at lower altitudes in the atmosphere. It is the hypothetical vertical wind shear that would exist if the winds obey geostrophic balance in the horizontal, while pressure obeys hydrostatic balance in the vertical. The combination of these two force balances is called *thermal wind balance*, a term generalizable also to more complicated horizontal flow balances such as gradient wind balance*.*

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. 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 shear**, sometimes referred to as wind gradient, is a difference in wind speed or direction over a relatively short distance in the atmosphere. Atmospheric wind shear is normally described as either vertical or horizontal wind shear. Vertical wind shear is a change in wind speed or direction with change in altitude. Horizontal wind shear is a change in wind speed with change in lateral position for a given altitude.

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.

- Description
- Physical explanation
- Mathematical formalism
- Examples
- Advection turning
- Frontogenesis
- Jet stream
- References
- Further reading

Since the geostrophic wind at a given pressure level flows along geopotential height contours on a map, and the geopotential thickness of a pressure layer is proportional to virtual temperature, it follows that the thermal wind flows along thickness or temperature contours. For instance, the thermal wind associated with pole-to-equator temperature gradients is the primary physical explanation for the jet stream in the upper half of the troposphere, which is the atmospheric layer extending from the surface of the planet up to altitudes of about 12-15 km.

**Geopotential height** or **geopotential altitude** is a vertical coordinate referenced to Earth's mean sea level, an adjustment to geometric height that accounts for the variation of gravity with latitude and altitude. Thus, it can be considered a "gravity-adjusted height".

In atmospheric thermodynamics, the **virtual temperature** of a moist air parcel is the temperature at which a theoretical dry air parcel would have a total pressure and density equal to the moist parcel of air.

**Jet streams** are fast flowing, narrow, meandering air currents in the atmospheres of some planets, including Earth. On Earth, the main jet streams are located near the altitude of the tropopause and are westerly winds. Their paths typically have a meandering shape. Jet streams may start, stop, split into two or more parts, combine into one stream, or flow in various directions including opposite to the direction of the remainder of the jet.

Mathematically, the thermal wind relation defines a vertical wind shear – a variation in wind speed or direction with height. The wind shear in this case is a function of a horizontal temperature gradient, which is a variation in temperature over some horizontal distance. Also called **baroclinic flow**, the thermal wind varies with height in proportion to the horizontal temperature gradient. The thermal wind relation results from hydrostatic balance and geostrophic balance in the presence of a temperature gradient along constant pressure surfaces, or isobars.

A **temperature gradient** is a physical quantity that describes in which direction and at what rate the temperature changes the most rapidly around a particular location. The temperature gradient is a dimensional quantity expressed in units of degrees per unit time. The SI unit is kelvin per second (K/s). It can be found in the formula for dQ/dt, the rate of heat transfer per second.

The term *thermal wind* is often considered a misnomer, since it really describes the change in wind with height, rather than the wind itself. However, one can view the thermal wind as a geostrophic wind that varies with height, so that the term *wind* seems appropriate. In the early years of meteorology, when data was scarce, the wind field could be estimated using the thermal wind relation and knowledge of a surface wind speed and direction as well as thermodynamic soundings aloft.^{ [1] } In this way, the thermal wind relation acts to define the wind itself, rather than just its shear. Many authors retain the *thermal wind* moniker, even though it describes a wind gradient, sometimes offering a clarification to that effect.

The thermal wind is the change in the amplitude or sign of the geostrophic wind due to a horizontal temperature gradient. The geostrophic wind is an idealized wind that results from a balance of forces along a horizontal dimension. Whenever the Earth's rotation plays a dominant role in fluid dynamics, as in the mid-latitudes, a balance between the Coriolis force and the pressure-gradient force develops. Intuitively, a horizontal difference in pressure pushes air across that difference in a similar way that the horizontal difference in the height of a hill causes objects to roll downhill. However, the Coriolis force intervenes and nudges the air towards the right (in the northern hemisphere). This is illustrated in panel (a) of the figure below. The balance that develops between these two forces results in a flow that parallels the horizontal pressure difference, or pressure gradient.^{ [1] } In addition, when forces acting in the vertical dimension are dominated by the vertical pressure-gradient force and the gravitational force, hydrostatic balance occurs.

In physics, the **Coriolis force** is an inertial or fictitious force that acts on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the **Coriolis effect**. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term *Coriolis force* began to be used in connection with meteorology.

The **pressure-gradient force** is the force that results when there is a difference in pressure across a surface. In general, a pressure is a force per unit area, across a surface. A difference in pressure across a surface then implies a difference in force, which can result in an acceleration according to Newton's second law of motion, if there is no additional force to balance it. The resulting force is always directed from the region of higher-pressure to the region of lower-pressure. When a fluid is in an equilibrium state, the system is referred to as being in hydrostatic equilibrium. In the case of atmospheres, the pressure-gradient force is balanced by the gravitational force, maintaining hydrostatic equilibrium. In Earth's atmosphere, for example, air pressure decreases at altitudes above Earth's surface, thus providing a pressure-gradient force which counteracts the force of gravity on the atmosphere.

In a barotropic atmosphere, where density is a function only of pressure, a horizontal pressure gradient will drive a geostrophic wind that is constant with height. However, if a horizontal temperature gradient exists along isobars, the isobars will also vary with the temperature. In the mid-latitudes there often is a positive coupling between pressure and temperature. Such a coupling causes the slope of the isobars to increase with height, as illustrated in panel (b) of the figure to the left. Because isobars are steeper at higher elevations, the associated pressure gradient force is stronger there. However, the Coriolis force is the same, so the resulting geostrophic wind at higher elevations must be greater in the direction of the pressure force.^{ [2] }

In fluid dynamics, a **barotropic fluid** is a fluid whose density is a function of pressure only. The barotropic fluid is a useful model of fluid behavior in a wide variety of scientific fields, from meteorology to astrophysics.

In a baroclinic atmosphere, where density is a function of both pressure and temperature, such horizontal temperature gradients can exist. The difference in horizontal wind speed with height that results is a vertical wind shear, traditionally called the thermal wind.^{ [2] }

The geopotential thickness of an atmospheric layer defined by two different pressures is described by the hypsometric equation:

,

where is the specific gas constant for air, is the geopotential at pressure level , and is the vertically-averaged temperature of the layer. This formula shows that the layer thickness is proportional to the temperature. When there is a horizontal temperature gradient, the thickness of the layer would be greatest where the temperature is greatest.

Differentiating the geostrophic wind, (where is the Coriolis parameter, is the vertical unit vector, and the subscript "p" on the gradient operator denotes gradient on a constant pressure surface) with respect to pressure, and integrate from pressure level to , we obtain the thermal wind equation:

.

Substituting the hypsometric equation, one gets a form based on temperature,

.

Note that thermal wind is at right angles to the horizontal temperature gradient, counter clockwise in the northern hemisphere. In the southern hemisphere, the change in sign of flips the direction.

If a component of the geostrophic wind is parallel to the temperature gradient, the thermal wind will cause the geostrophic wind to rotate with height. If geostrophic wind blows from cold air to warm air (cold advection) the geostrophic wind will turn counterclockwise with height (for the northern hemisphere), a phenomenon known as wind backing. Otherwise, if geostrophic wind blows from warm air to cold air (warm advection) the wind will turn clockwise with height, also known as wind veering.

Wind backing and veering allow an estimation of the horizontal temperature gradient with data from an atmospheric sounding.

As in the case of advection turning, when there is a cross-isothermal component of the geostrophic wind, a sharpening of the temperature gradient results. Thermal wind causes a deformation field and frontogenesis may occur.

A horizontal temperature gradient exists while moving North-South along a meridian because curvature of the Earth allows for more solar heating at the equator than at the poles. This creates a westerly geostrophic wind pattern to form in the mid-latitudes. Because thermal wind causes an increase in wind velocity with height, the westerly pattern increases in intensity up until the tropopause, creating a strong wind current known as the jet stream. The Northern and Southern Hemispheres exhibit similar jet stream patterns in the mid-latitudes.

The strongest part of jet streams should be in proximity where temperature gradients are the largest. Due to land masses in the northern hemisphere, largest temperature contrasts are observed on the east coast of North America (boundary between Canadian cold air mass and the Gulf Stream/warmer Atlantic) and Eurasia (boundary between the boreal winter monsoon/Siberian cold air mass and the warm Pacific). Therefore, the strongest boreal winter jet streams are observed over east coast of North America and Eurasia. Since stronger vertical shear promotes baroclinic instability, the most rapid development of extratropical cyclones (so called bombs) is also observed along the east coast of North America and Eurasia.

The lack of land masses in the Southern Hemisphere leads to a more constant jet with longitude (i.e. a more zonally symmetric jet).

The **barotropic vorticity equation** assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. It also implies that thickness contours are parallel to upper level height contours. In this type of atmosphere, high and low pressure areas are centers of warm and cold temperature anomalies. Warm-core highs and cold-core lows have strengthening winds with height, with the reverse true for cold-core highs and warm-core lows.

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

The **Rossby number** (**Ro**) named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial force to Coriolis force, terms and in the Navier–Stokes equations, respectively. It is commonly used in geophysical phenomena in the oceans and atmosphere, where it characterizes the importance of Coriolis accelerations arising from planetary rotation. It is also known as the **Kibel number**.

**Air currents** are concentrated areas of winds. They are mainly due to differences in pressure or temperature. They are divided into horizontal and vertical currents; both are present at mesoscale while horizontal ones dominate at synoptic scale. Air currents are not only found in the troposphere, but extend to the stratosphere and mesosphere.

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.

**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.

**Frontogenesis** is a meteorological process of tightening of horizontal temperature gradients to produce fronts. In the end, two types of fronts form: cold fronts and warm fronts. A cold front is a narrow line where temperature decreases rapidly. A warm front is a narrow line of warmer temperatures and essentially where much of the precipitation occurs. Frontogenesis occurs as a result of a developing baroclinic wave. According to Hoskins & Bretherton, there are eight mechanisms that influence temperature gradients: horizontal deformation, horizontal shearing, vertical deformation, differential vertical motion, latent heat release, surface friction, turbulence and mixing, and radiation. Semigeostrophic frontogenesis theory focuses on the role of horizontal deformation and shear.

The **omega equation** is of great importance in meteorology and atmospheric physics. It is a partial differential equation for the vertical velocity, , which is defined as the Lagrangian rate of change of pressure with time. Mathematically, , where represents a material derivative. It is valid for large scale flows under the conditions of quasi-geostrophy and hydrostatic balance. In fact, one may consider the vertical velocity that results from solving the omega equation as that which is needed to maintain quasi-geostrophy and hydrostasy.

The equation reads:

**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 oceanography, **Ekman velocity** – also referred as a kind of the residual ageostropic velocity as it derivates from geostrophy – is part of the total horizontal velocity (* u*) in the upper layer of water of the open ocean. This velocity, caused by winds blowing over the surface of the ocean, is such that the Coriolis force on this layer is balanced by the force of the wind.

**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.

While geostrophic motion refers to the wind that would result from an exact balance between the Coriolis force and horizontal pressure-gradient forces, **quasi-geostrophic (QG) motion** refers to flows where the Coriolis force and pressure gradient forces are *almost* in balance, but with inertia also having an effect.

**Conditional symmetric instability**, or **CSI**, is a form of convective instability in a fluid subject to temperature differences in a uniform rotation frame of reference while it is thermally stable in the vertical and dynamically in the horizontal. The instability in this case develop only in a inclined plane with respect to the two axes mentioned and that is why it can give rise to a so-called "slantwise convection" if the air parcel is almost saturated and moved laterally and vertically in a CSI area. This concept is mainly used in meteorology to explain the mesoscale formation of intense precipitation bands in an otherwise stable region, such as in front of a warm front. The same phenomenon is also applicable to oceanography.

- Holton, James R. (2004).
*An Introduction to Dynamic Meteorology*. New York: Academic Press. ISBN 0-12-354015-1.

- Vasquez, Tim (2002).
*Weather Forecasting Handbook*. ISBN 0-9706840-2-9.

- Vallis, Geoffrey K. (2006).
*Atmospheric and Oceanic Fluid Dynamics*. ISBN 0-521-84969-1.

- Wallace, John M.; Hobbs, Peter V. (2006).
*Atmospheric Science*. ISBN 0-12-732951-X.

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