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'll 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