The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 43

Figure 2.5. An idealized illustration of circulation computation. Area A of a plane fluid
element, sense of direction of line integral of v E dl (arrow around closed curve), unit vector
n normal to the surface of A, using the right-hand rule to determine the direction it's pointing.

According to V. Bjerknes's first circulation theorem

þ Bk

=

Dt ¼

ð 2 :

53 Þ

DC E dl

which is obtained by using the definition of circulation (2.52), the three-

dimensional equation of motion (2.36), and noting that
þ

J ð p

= Þ dl ¼ 0 ð 2 :

54 Þ

because is treated as a constant. The same two-dimensional fluid element must

be followed along in space and its change in shape must be accounted for. When
the fluid element is well behaved, the fluid element can be followed. In some
instances, however, the fluid element with time becomes so distorted that it is
dicult to work with accurately.
From Stokes' theorem, (2.52) may be expressed as

ðð

DC = Dt ¼ ð JT Bk Þ E dA ð 2 : 55 Þ

In the absence of friction, in a Boussinesq atmosphere (which acts as if it were
incompressible) without gradients in buoyancy, circulation is conserved. Also, if
the material curve always lies in the x-y-plane, circulation is conserved.
Circulation is generated when there are horizontal buoyancy gradients and the
fluid element has a nonzero projection onto the vertical plane.
To analyze circulation dynamics, one is careful to choose a circuit for a
material curve that encompasses just the vortex under consideration. If one
chooses a circuit that contains two counterrotating vortices, then the total circula-
tion may be nearly zero; if one chooses a circuit that contains less than the
complete vortex, then the analysis may be in error because a significant part of
the vortex has been omitted. It is an art to determine circuits to be analyzed and
one must be sure that the orientation of the plane chosen is normal to the local
vorticity vector. Also, one cannot trace a material circuit too far back in time