Mathematical Diversion - 3

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:

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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.