thumb|[[Jet streams (shown in pink) are well-known examples of thermal wind. They arise from the horizontal temperature gradients between the warm tropics and the colder polar regions.]]
In atmospheric science, 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.
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.
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.
The term thermal wind was originally proposed by British meteorologist Ernest Gold. It 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. In addition, when forces acting in the vertical dimension are dominated by the vertical pressure-gradient force and the gravitational force, hydrostatic balance occurs.left|thumb|429x429px|The geostrophic wind on different isobaric levels in a barotropic atmosphere (a) and in a baroclinic atmosphere (b). The blue portion of the surface denotes a cold region while the orange portion denotes a warm region. This temperature structure is restricted to the surface in (a) but extends through the depth of the fluid in (b). The dotted lines enclose isobaric surfaces which remain at constant slope with increasing height in (a) and increase in slope with height in (b). Pink arrows illustrate the direction and amplitude of the horizontal wind. Only in the baroclinic atmosphere (b) do these vary with height. Such variation illustrates the thermal wind.
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.
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.