Thermal Wind

            Thermal Wind is the vertical shear of the geostrophic wind cause by a horizontal temperature gradient—it “blows” parallel to the thickness contours, leaving low thickness to the left. The Thermal Wind Equation states that the vertically averaged shear of the geostrophic wind (within the layer between any two pressure surfaces) is related to the horizontal gradient of thickness of the layer, in the same manner in which geostrophic wind is related to geopotential height.

Expressed as a linear relationship between vertical wind shear of the geostrophic wind and the horizontal temperature gradient,

            In a barotropic atmosphere—where density is only a function of pressure—the slope of the isobaric surfaces are independent of temperature thus, the geostrophic wind doesn’t increase with height. In other words, there is a complete absence of the horizontal temperature (thickness) gradients such that on constant pressure surfaces. However, the slope of the isobaric surfaces and the speed of the geostrophic wind may vary from level to level due to those thickness variations.

            In an Equivalent Barotropic Atmosphere, isobars and isotherms, on a horizontal surface map, have the same shape.

            In a Baroclinic Atmosphere—where density is a function of both pressure and temperature—the height and thickness contours intersect such that the geostrophic wind exhibits a component normal to the isotherms (or thickness contours). In other words, the horizontal temperature gradients cause the thickness of the layers between isobaric surfaces to increase with higher temperatures. When multiple layers are stacked on each other the geostrophic wind and the slope of the isobaric surfaces increase with height.

Virtual Temperature

     A fictitious temperature that dry air would need to attain in order to have the same density as the moist air at the same pressure is considered virtual temperature. The fact that moist air is less dense than dry air was first clearly stated by Sir Isaac Newton in his “Opticks.” However, the basis for this relationship was not generally understood until the latter half of the 18^th century.

At any rate, due to that fact, and at the same temperature and pressure, the virtual temperature is always greater than the actual temperature. The use of virtual temp allows us to use the gas constant for dry air, R_d, saving us from constantly having to calculate gas constants for moist air—the value of which would vary with water vapor content.

            However, virtual temperature correction is usually neglected except in certain calculations relating to the boundary layer. Nonetheless, in moving from a given pressure surface to another pressure surface located above or below it, the geopotential height (used as the vertical coordinate in most atmospheric applications in which energy plays a role—i.e. large scale motions) is related geometrically to the thickness of the intervening layer which, in turn, is directly proportional to the mean virtual temperature of the layer. The mean virtual temperature is used for determining the thickness of a layer between two pressure surfaces (p₁ and p₂).

where e is vapor pressure, p is pressure, and ε is approximately equal to 0.622.