Consider the following. The 3-dimensional, Cartesian coordinate
wind components (*u, v, w *) can be expanded in a Taylor's
Series according to:

If only the *linear* (first order) components of the flow are
considered, this reduces considerably to:

Finally, consider only the *horizontal* part, so that the
foregoing simplifies still further:

By defining the following quantities,

then it can be seen that the *linear* part of the horizontal
wind field (the Taylor Series out to first order) can be expressed
as:

Some obvious notational shortcuts have been made in the preceding. The kinematic (linear) properties of a flow field are

- d
_{1}=*stretching deformation*, - d
_{2}=*shearing deformation*, *D*=*horizontal divergence*, and- z =
*vertical component of vorticity*.

I will return to this simplified, horizontal view of things from time to time. In the purely horizontal view of things, the important part of the vorticity is the vertical component, as I will show now.

The three-dimensional vorticity (a vector), **w**, is given by

where **w**_{h} is the
horizontal component of **w** and **k** is the vertically-pointing unit vector. If we
make the hydrostatic assumption (typically, a pretty good assumption,
except in strong thunderstorms and tornadoes), then scale analysis
shows that the x- and y- (horizontal) variation of the vertical
component of the wind is *much* smaller than the horizontal
variations of *u* and *v*, and also the vertical variation
of the horizontal components (by about 2 orders of magnitude). If,
therefore, we neglect the x- and y-derivatives of *w*, then to a
very good approximation,

and using this assumption, some vector identities can be used to show that