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 wh 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 ws into the proper units for a vorticity. Obviously, from what has been given, it is clear that ws = es . wh. Analogously, wc = ec . wh. Putting wh into this natural coordinate system with components given by (ws, wc ) 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.