The Thermal Wind
The geostrophic wind in natural coordinates (in this case, s, n, p coordinates) can be obtained by substitution of the hydrostatic approximation into the equation of motion. The geostrophic wind speed then can be determined by finite difference approximation of the expression
where ∂z/∂n is the height gradient normal to the streamlines. It’s useful to attempt to relate the geostrophic wind at a lower elevation to that at an upper elevation. Equation (2) gives the difference between the 500 mb and 1000 mb geostrophic winds.
If the increment along the n-axis is the same, equation (2) can be rewritten as follows:
But z500 – z1000 is just the 1000-500 mb thickness Z1000-500.
Thus, equation (3) can be rewritten as:
Note that the right hand side of (4) has the appearance of a geostrophic wind. In fact, the right hand side is known as the “Thermal Wind”, Vt.
Equation (4) can be rewritten
The concept that emerges is that the thermal wind is a correction factor which, when added to the lower level (say, surface) geostrophic wind field can produce an estimate of the geostrophic wind field aloft. Since the wind field at 500 mb is most nearly geostrophic then this procedure can give you a qualitative estimate of the mid-tropospheric wind field.
Just as the geostrophic wind in the Northern Hemisphere blows parallel to the height contours and counter clockwise (clockwise) around areas of low (high) values, the Thermal Wind blows parallel to the thickness contours and counter clockwise (clockwise) around areas of low (high) values.
The Thermal Wind in finite difference form is
where Z is the thickness.
The hypsometric equation states that the thickness of a layer bounded by two isobars is directly proportional to its mean virtual temperature.
Z = R/g ln p1/p2 Tv (7)
For our purposes, the virtual temperature can be assumed to be the same as the temperature. Equation (7) should be substituted into (6) in the following manner:
Z2 = R/g ln 2T2 (8a)
Z1 = R/g ln 2T1 (8b)
where T2 and T1 are the mean temperatures of the layer bounded by the 1000 and 500 mb isobars at locations 2 and 1, respectively. Inserting (8a) and (8b) into (6) gives
which states, simply, that thermal wind at a given spot is directly proportional to the mean temperature gradient normal to the flow. If one makes the assumption that deep temperature gradients (for example, across fronts) are consistent through the troposphere, equation (10a,b) allows one to answer the question: What is the 500 mb wind vector if the surface winds are calm and temperatures decrease northward at 5o per 500 km?