Divergence Related to Gradient Wind

The equation of motion in natural coordinates is

(1)

where k is the trajectory curvature.

The relationship between the curvature of a trajectory and the curvature of a streamline is given by the so-called “Blaton’s Formula”

(2)

where k_s is the streamline curvature, V is the wind speed and c is the motion of the moving trough or ridge.

Put (2) into (1) and solve for V

(3a,b)

This says that the gradient wind differs from the geostrophic wind by a term proportional to the streamline curvature. What are the implications of this? Fool around with the equation by considering its implication in different portions of the troposphere.

Consider (see graphic below): (a) a low amplitude short wave trough ridge system in the upper troposphere in which the height gradient is everywhere constant;; and, (b) a low amplitude, but high wavelength trough ridge system in the upper troposphere in which the height gradient is everywhere constant. Sketch each pattern, and place a vector for the geostrophic wind and the gradient wind at trough and ridge axis.

Next, remember the definition of divergence in natural coordinates. Qualitatively assess the divergence at the inflection point for each case.