If the natural coordinate system is denoted by (*s *,
*n* ), where *s* is the coordinate in the direction of the
flow at a point and *n* is the coordinate direction normal to
*s* such that (*s* , *n* ) produces a left-handed
coordinate system ... turning *s* into *n* produces an
upward-pointing vector when using a left-hand rule ... then the
vertical vorticity associated with horizontal wind shear in this
coordinate system is given by

In this system, then an *irrotational vortex* (i.e., one
which has zero vorticity) results when

For a flow with counterclockwise rotation, da/ds is negative,
which means that the flow speed in an irrotational vortex decreases
in the normal direction. It can be shown that if a circular flow has
a purely tangential flow speed that varies as 1/*r* (where
*r* is the radius from the center of the vortex), then it is an
irrotational vortex. Such a vortex has an *infinite* velocity at
*r* = 0, of course.

An idealized circular vortex of some significance is the so-called
*Rankine Combined Vortex *, which is purely tangential flow,
with that tangential velocity *V(r)* defined by

where z_{o} is a constant and
*r _{o}* is the so-called

A continuously-varying analog to the Rankine Combined Vortex, employing hyperbolic functions, can be used (see Doswell 1984) that avoids the minor problem of the discontinuity at the core radius but retains many the features of the Rankine vortex model. Its tangential velocity is described by

In this formula, the peak velocity is at *r _{o}* =
0.6585 and the magnitude at that point is

Fig. M2_1. Tangential velocity profiles (normalized to a peak velocity of 1.0 units at a radius of 1.0 units) as a function of radius for the Rankine Combined Vortex and the "hyperbolic" vortex introduced in Doswell (1984).

Observe that the tangential velocity profile of the Rankine
Combined Vortex falls off rather slowly outside the core (i.e.,
outside *r _{o}* = 1 for the normalized case shown) in
comparison to the model using hyperbolic functions. The vertical
vorticity for the two models is shown in
Fig. M2_2,

Fig. M2_1. Vorticity profiles (normalized to give a peak vorticity of 1.0 units at a zero radius) as a function of radius for the Rankine Combined Vortex and the "hyperbolic" vortex.

with z_{o }=1, and the peak
vorticity in the hyperbolic model also set equal to unity. Observe
that the vorticity in the hyperbolic model becomes negative outside r
= 2, but it is quite small and approaches zero asymptotically as the
radius increases. The negative vorticity is the result of the shear
contribution being negative outside *r* = *r _{c}*
and larger than the curvature contribution at radii larger than about
2 units.

Doswell, C.A. III, 1984: A kinematic analysis of frontogenesis
associated with a nondivergent vortex. *J. Atmos. Sci *.,
**41**, 1241-1248.