Thermal wind

The thermal wind is not actually a wind, but a wind difference between two pressure levels $p 1$ and $p 0$ , with $p 1 < p 0$ . It is only present in an atmosphere with horizontal gradients of temperature, i.e. baroclinicity. In a barotropic atmosphere the geostrophic wind is independent of height. The name stems from the fact that this wind flows around areas of low (and high) temperature in the same manner as the geostrophic wind flows around areas of low (and hight) pressure.

The thermal wind equation is

$\mathbf{v}_T = \frac{1}{f} \mathbf{k} \times \nabla ( \phi_1 - \phi_0 )$ ,

where the $φ x$ are geopotential height fields with $φ 1 > φ 0$ , $f$ is the Coriolis parameter , and $\mathbf{k}$ is the upward-pointing unit vector in the vertical direction. The equation also makes obvious that the thermal wind does not apply in the tropics: $f$ is zero at the equator, and generally small in the tropics.

Contents

1 Examples of thermal wind

1.1 Temperature contrast between equator and pole: Increasing westerlies with height
1.2 Advection of warm or cold air: Turning of the geostrophic wind with height

2 Further reading

Examples of thermal wind

Temperature contrast between equator and pole: Increasing westerlies with height

(This applies similarly to the Southern hemisphere, and because $f$ is negative there, yields the same result.)

On the Northern hemisphere, the polar region is cold (low temperature) while the equatorial region is warm (high termperature). Because the thermal wind circles the area of low temperature in the same manner as the geostrophic wind flows around a cyclone, namely counter-clockwise in the Northern hemisphere, the thermal wind in the NH midlatitudes is westerly (i.e. is directed eastward). This can be seen by looking at a globe from above the North Pole — a westerly current flows counter-clockwise around the globe.

What does this mean for the vertical wind profile in the mid-latitude NH troposphere? If the thermal wind is westerly, the atmospheric flow will become more westerly with height, as the thermal wind describes the wind change with height. Therefore, if at a certain level, say, at the top of the boundary layer, the wind speed is close to zero, the wind speed will have a strong eastward component at higher levels.

This simple argument basically describes the jet stream, a westerly current of air with maximum wind speeds close to the tropopause which is basically (even though other factors are also important) a result of the temperature contrast between equator and pole.

Advection of warm or cold air: Turning of the geostrophic wind with height

If the geostrophic wind at a level advects (i.e. transports) warm or cold air, the thermal wind causes a turning of wind direction with height. A similar argument as in the other example with regard to how the thermal wind is related to the temperature distribution can be made.

The outcome is that a geostrophic wind that advects warm air into a region of colder air causes the wind to turn right (clockwise, veering) with height, while cold air advection into a region of warmer air results in the wind turning left (counter-clockwise, backing).