A Mathematical Diversion - 2

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 zo is a constant and ro is the so-called core radius . From this definition, it can be seen that the vorticity is a constant value ( zo )within the core (which is in what is known as solid body rotation , where the rotational velocity increases directly with r ). This core is surrounded by a flow which has zero vorticity because the shear profile exactly cancels the curvature contribution. In many studies, the flow in real atmospheric vortices has been assumed to fit a Rankine Combined Vortex profile. From the proceeding, it can be seen that the vorticity has a discontinuity at r = ro .

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 ro = 0.6585 and the magnitude at that point is Vmax=0.3849. This can be normalized easily to give a peak velocity of unit value at unit radius, as shown in Fig. M2_1.

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 ro = 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 zo =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 = rc and larger than the curvature contribution at radii larger than about 2 units.

 

Reference

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