Thermal wind equation

The Thermal wind is wind that varies with height due to a balance between the Coriolis and pressure-gradient forces in the atmosphere. It is the primary physical mechanism for the jet stream and plays an important role in other large-scale atmospheric phenomena. The thermal wind ensures the jet stream is typically strongest in the upper half of the troposphere, which is the atmospheric layer extending from the surface of the planet up to a height of 12 km to 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 is often considered a misnomer, since it really describes the change in wind with height, rather 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 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. In addition, when forces acting in the vertical dimension are dominated by the vertical pressure-gradient force and the gravitational force, hydrostatic balance occurs.

...
Wikipedia