Gradient Wind Divergence: Hypotheticals and Realities

The gradient wind can be written as g ¶ z k V = - s ( V - c ) V gr f ¶ n f

(1a, 1b) k V = V - s ( V - c ) V gr g f

Consider a contour pattern (Fig. 1) in which equal height gradients are found

throughout. Figure 1: Contour/streamline pattern in upper troposphere in which the

height gradients are everywhere the same.

Discounting the variation of latitude, the geostrophic wind would be

everywhere the same on Fig. 1, since ∂z/∂n would be the same everywhere.

At the ridge axis, the curvature of the streamline is negative, and the local

centrifugal effects would produce a centrifugal force that adds to the

pressure gradient force, producing a real wind much stronger than the

geostrophic wind, as discussed in classs. Thus, V>>>Vg>>>>c, and the far

right hand term would produce strongly supergeostrophic flow.

In-class Question

Using similar reasoning, describe the gradient wind flow at the trough axis.

The equation for horizontal divergence in natural coordinates is

DIV = ¶V + V ¶q h ¶s ¶n (2)

where the two terms to the right of the equals sign are speed divergence and

diffluence, respectively (speed convergence, if negative, and confluence, if negative, respectively). Applying (2) to Fig. 1, one can see that there is no diffluence. Hence, for sinusoidal waves in the jet stream, in which the actual wind streams through the pattern (moves faster than the speed of the wave itself), speed divergence is found from trough axis to downstream ridge axis, and speed convergence occurs from ridge axis to downstream trough axis.

Therefore, to the extent that upper waves resemble Fig. 1, divergence is found east of troughs and convergence is found east of ridges, in the upper troposphere.

Note also that divergence shown in Fig. 1 would be centered in the midst of the greatest cyclonic vorticity advection, as implied from the figure, and convergence shown in Fig. 1 would be centered in the midst of the greatest anticyclonic vorticity advection. Thus, the divergence/convergence patterns associated with the instantaneous geometry of the flow (meaning, the shape of the flow at any one time, for example, as shown in Fig. 1), should be associated (or diagnosed) by the areas of cyclonic and anticyclonic vorticity advection.

In-class Question

To what extent do upper tropospheric patterns REALLY look like Fig. 1?