Thermal wind
Encyclopedia
Vertical direction
In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force at that point...
shear
Shearing (physics)
Shearing in continuum mechanics refers to the occurrence of a shear strain, which is a deformation of a material substance in which parallel internal surfaces slide past one another. It is induced by a shear stress in the material...
in the geostrophic wind
Geostrophic wind
The geostrophic wind 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 . This balance seldom holds exactly in nature...
caused by a horizontal
Horizontal plane
In geometry, physics, astronomy, geography, and related sciences, a plane is said to be horizontal at a given point if it is perpendicular to the gradient of the gravity field at that point— in other words, if apparent gravity makes a plumb bob hang perpendicular to the plane at that point.In...
temperature gradient
Temperature gradient
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 length...
. Its name is a misnomer
Misnomer
A misnomer is a term which suggests an interpretation that is known to be untrue. Such incorrect terms sometimes derive their names because of the form, action, or origin of the subject becoming named popularly or widely referenced—long before their true natures were known.- Sources of misnomers...
, because the thermal wind is not actually a wind
Wind
Wind is the flow of gases on a large scale. On 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...
, but rather a wind shear
Wind shear
Wind shear, sometimes referred to as windshear or wind gradient, is a difference in wind speed and direction over a relatively short distance in the atmosphere...
.
Physical Intuition
Geopotential
Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as minus the potential energy per unit mass, so that the gravity vector is obtained as the gradient of this potential, without the minus....
on a surface of constant pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
. In a barotropic
Barotropic
In meteorology, a barotropic atmosphere is one in which the pressure depends only on the density and vice versa, so that isobaric surfaces are also isopycnic surfaces . The isobaric surfaces will also be isothermal surfaces, hence the geostrophic wind is independent of height...
atmosphere
Atmosphere
An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
, one where density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
is a function only of pressure, the slope of isobaric
Isobaric
Isobaric may refer to:*in thermodynamics, an isobaric process, i.e. one that is carried out at constant pressure;...
surfaces is independent of temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
, so geostrophic wind does not increase with height.
This does not hold true in a baroclinic atmosphere, one where density is a function of both pressure and temperature. Horizontal temperature gradients cause the thickness of gas
Gas
Gas is one of the three classical states of matter . Near absolute zero, a substance exists as a solid. As heat is added to this substance it melts into a liquid at its melting point , boils into a gas at its boiling point, and if heated high enough would enter a plasma state in which the electrons...
layers between isobaric surfaces to increase with higher temperatures. When multiple atmospheric layers are stacked upon each other, the slope of isobaric surfaces increases with height. This also causes the magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of the geostrophic wind to increase with height.
Mathematical Formalism
The geopotential thickness of an atmospheric layer is described by the hypsometric equationHypsometric equation
The hypsometric equation relates an atmospheric pressure ratio to the equivalent thickness of an atmospheric layer under the assumptions of constant temperature and gravity...
:
,where
is the specific gas constantGas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
for air,
is the geopotentialGeopotential
Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as minus the potential energy per unit mass, so that the gravity vector is obtained as the gradient of this potential, without the minus....
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.If we differentiate the geostrophic wind,
(where 
is the CoriolisCoriolis
Coriolis may refer to:* Gaspard-Gustave Coriolis , French mathematician, mechanical engineer and scientist* Coriolis effect, the apparent deflection of moving objects from a straight path when viewed from a rotating frame of reference...
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 the thermal wind is at right angles to the horizontal temperature gradient, to the counter clockwise in the northern hemisphere. In the southern hemisphere, the change in sign of
flips the direction.Advection Turning
Advection
Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...
) the geostrophic wind will turn counterclockwise with height, a phenomenon
Phenomenon
A phenomenon , plural phenomena, is any observable occurrence. Phenomena are often, but not always, understood as 'appearances' or 'experiences'...
known as wind backing. Otherwise, if the geostrophic wind blows from warm air to cold air (warm advection) the wind will turn clockwise
Clockwise
Circular motion can occur in two possible directions. A clockwise motion is one that proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back to the top...
with height, also known as wind veering.
Wind backing and veering allow us to estimate the horizontal temperature gradient with data from an atmospheric sounding
Atmospheric sounding
An atmospheric sounding is a measurement of vertical distribution of physical properties of the atmospheric column such as pressure, temperature, wind speed and wind direction , liquid water content, ozone concentration, pollution, and other properties...
.
Frontogenesis
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. The thermal wind causes a deformation field and frontogenesisFrontogenesis
Frontogenesis, in meteorology, refers to the formation or strengthening of an atmospheric front. During frontogenesis, the temperature gradient tightens and as a result the thermal wind becomes imbalanced. To maintain balance, the geostrophic wind aloft and below adjust, such that regions of...
may occur.
Jet Stream
A horizontal temperature gradient exists while moving NorthNorth
North is a noun, adjective, or adverb indicating direction or geography.North is one of the four cardinal directions or compass points. It is the opposite of south and is perpendicular to east and west.By convention, the top side of a map is north....
-South
South
South is a noun, adjective, or adverb indicating direction or geography.South is one of the four cardinal directions or compass points. It is the opposite of north and is perpendicular to east and west.By convention, the bottom side of a map is south....
along a meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...
because the curvature of the Earth allows for more solar heating
Insolation
Insolation is a measure of solar radiation energy received on a given surface area in a given time. It is commonly expressed as average irradiance in watts per square meter or kilowatt-hours per square meter per day...
at the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
than at the poles. This creates a westerly
Westerlies
The Westerlies, anti-trades, or Prevailing Westerlies, are the prevailing winds in the middle latitudes between 30 and 60 degrees latitude, blowing from the high pressure area in the horse latitudes towards the poles. These prevailing winds blow from the west to the east, and steer extratropical...
geostrophic wind pattern to form in the mid-latitudes. Because thermal wind causes an increase in wind velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...
with height, the westerly pattern increases in intensity up until the tropopause
Tropopause
The tropopause is the atmospheric boundary between the troposphere and the stratosphere.-Definition:Going upward from the surface, it is the point where air ceases to cool with height, and becomes almost completely dry...
, creating a strong wind current known as the jet stream
Jet stream
Jet streams are fast flowing, narrow air currents found in the atmospheres of some planets, including Earth. The main jet streams are located near the tropopause, the transition between the troposphere and the stratosphere . The major jet streams on Earth are westerly winds...
. The Northern
Northern Hemisphere
The Northern Hemisphere is the half of a planet that is north of its equator—the word hemisphere literally means “half sphere”. It is also that half of the celestial sphere north of the celestial equator...
and Southern Hemisphere
Southern Hemisphere
The Southern Hemisphere is the part of Earth that lies south of the equator. The word hemisphere literally means 'half ball' or "half sphere"...
s exhibit similar jet stream patterns in the mid-latitudes.
Using the same Thermal Wind argument, the strongest part of the jet stream should be in proximity where temperature gradients are the largest. Due to the setup of the continents in the North America, 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). Indeed, the strongest part of the boreal winter Northern Hemisphere jet is observed over east coast of North America and Eurasia as well. Since stronger vertical shear promotes baroclinic instability
Baroclinity
In fluid dynamics, the baroclinity of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid...
, so the most rapid development of extratropical cyclones (so called bombs
Bomb (meteorology)
Bomb is a meteorological term for a baroclinic low pressure area characterized by a rapid pressure fall of at least 24 millibars in 24 hours. It is applied only to cold-air storms...
) is also observed along the east coast of North America and Eurasia.
A similar argument can be applied to the Southern Hemisphere. The lack of continents in the Southern Hemisphere should lead to a more constant jet with longitude (i.e. a more zonally symmetric jet), and that is indeed the case in observations.