I developed the full 3-dimensional vorticity vector **w** in Mathematical
Diversion - 1. Recall that

It was asserted that under the hydrostatic assumption, the
horizontal vorticity **w**_{h}
satisfies (to a very good approximation) the relationship that

This relationship shows that the horizontal vorticity vector is,
indeed, 90 deg to the left of the horizontal shear vector (see
Doswell 1991). The vorticity components in a Cartesian system, [i.e.
(*x, h, z*)] are not necessarily the
most useful. A natural coordinate system can be used again, where
*s* now can be called the *streamwise* direction and
*n* can be called the *crosswise* direction. Unit vectors
in these directions can be defined in the following way:

By its definition, the *streamwise vorticity* is that
component of the horizontal vorticity that is parallel to the
horizontal wind:

where the division by the magnitude of the horizontal wind is
needed to put w_{s} into the
proper units for a vorticity. Obviously, from what has been given, it
is clear that w_{s} =
**e**_{s} **.** **w**_{h}. Analogously, w_{c} = **e**_{c} **.** **w**_{h}. Putting **w**_{h} into this natural coordinate
system with components given by (w_{s}, w_{c }) seems more physically insightful
than the Cartesian coordinate components (*x,
h*), where the partitioning among the vector components
depends on the artibrary orientation of the coordinate framework.