The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 42

Figure 2.4. An idealized illustration of the generation of vorticity about the y-axis when
buoyancy decreases in the x-direction. Imagine a paddle wheel placed in the flow; it will begin
to rotate about the y-axis because the left side experiences an upward acceleration while the
right side experiences a downward acceleration.

Equations (2.50) and (2.51) are often used to analyze convective storm
dynamics—a similar equation for the x-component of vorticity ð@ z Þ

can also be used. The vertical vorticity equation (2.50) is most useful for analyzing
the formation, maintenance, and dissipation of vortices such as tornadoes, meso-
cyclones, and meso-anticyclones. It is most useful when expressed in a Lagrangian
framework, so that the processes contributing significantly to the vertical vorticity
associated with the vortex, which is being transported along with the mean flow,
can be determined. Trajectories must be calculated to follow air parcels along and
one must carefully select trajectories that are representative: one must select those
that begin/end in specific regions and pass through the vortex being analyzed. In
some instances small differences in the beginning (or, in the case of backward
trajectories, ending) positions may lead to trajectories that pass through very
different locations in space. 1
To overcome the problem of having to compute many parcel trajectories, it is
useful to analyze the macroscopic measure of vorticity, circulation C, where one
makes use of Stokes' theorem, so that

w =@ y @v=@

þ v E dl ¼ ðð

C ¼ ð JT v E n Þ dA ð 2 : 52 Þ

where the line integral of the component of wind velocity over the perimeter (l is
tangent to the edge of the fluid element) of a two-dimensional fluid element is
given by the component of vorticity normal to the fluid element (n is a unit vector
pointing normal to the plane surface of A, in the direction defined by the right-
hand rule) integrated over the area defined by the two-dimensional fluid element
(A) ( Figure 2.5 ). In other words, circulation is the vorticity normal to the fluid
element multiplied by its area or, equivalently, vorticity is circulation per unit
area.

1 When windspeeds are high, as in a tornado, the effects of small changes in position (and time
increment used to compute the trajectories) are amplified.