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 ks 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. Assume that the contours are equally spaced everywhere and the amplitude of the waves are everywhere the same.
| Upper Troposphere | Why, from the above equation (3b)? |
| Winds are (supergeostrophic/subgeostrophic) (circle one) at the ridge axes. | |
| The greater the wind speed, the (more/less) (circle one) the winds are (supergeostrophic/subgeostrophic) (circle one) at the ridge axis. | |
| The higher the latitude, for a given wind and trough speed, the (greater/lesser) (circle one) the difference between the gradient wind and the geostrophic wind at the ridge axis. | |
| The shorter the wavelength, the (greater/lesser) the difference between the gradient wind speed between points at the ridge and the trough axes. |
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. You can assume that contours are equally spaced so that no diffluence or confluence occurs in the pattern.
Next, remember the definition
of divergence in natural coordinates.
Qualitatively assess the divergence at the inflection point for each case in the following manner: using the reasoning established above, state which of the two examples would have the greatest divergence at the inflection point and why.
