The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 44

because the shape of the circuit may have been extremely distorted 2 and friction may
have changed the shape and/or fractured the material circuit. Friction, of course, can
also reduce circulation. Also, circulation tendencies may reverse sign if the material
curve is traced too far back in time.
For the special case of vertical vortices, the flux form of the Eulerian vertical
vorticity equation in a frictionless, Boussinesq atmosphere may be used;

it is

obtained by rewriting (2.50) less the solenoidal term as
@=@

t ¼ J h E ½ v h þ w ð JT v Þ h

ð 2 :

56 Þ

where

J h ¼ @=@

x i þ@=@ y j ð 2 : 57 Þ

v h ¼ u i þ v j ð 2 : 58 Þ

ð JT v Þ h ¼ð@ w =@ y @v=@ z Þ i þð@ u =@ z @ w =@ x Þ j ð 2 : 59 Þ

and the latter is the horizontal vorticity vector.
Equation (2.56) is integrated over an area (fluid element) in the horizontal
plane so that

ðð

t ðð

@=@

tdA ¼ @=@ dA ¼ @

=@ ð 2 :

60 Þ

C t

Using the divergence theorem, we find that (2.60) may be expressed as a line
integral:

þ

@ =@

t ¼

ð½

v h þ w ð JT v Þ h E

ð 2 :

61 Þ

C ndl

where n is a unit vector normal to the line composing the perimeter of the fluid
element and pointing outward from it. The terms on the RHS of (2.61) represent
the net horizontal flux of vertical vorticity into the area and the net vertical flux of
horizontal vorticity into the area. If we include Earth's vorticity, then f v h is
included inside the brackets on the RHS of (2.61). Subgrid-scale mixing may also
be included.
The horizontal vorticity equation (one component of it) (2.51) is used to
diagnose circulations in the vertical plane. It is very useful, for example, for under-
standing the dynamics of vertical circulations in two-dimensional squall lines, in
circularly symmetric clouds, and across gust fronts.

2.5 THE DIVERGENCE EQUATION AND THE BUOYANCY FORCE

By applying the divergence operator to the full three-dimensional equation of
motion—combination of (2.13) and (2.7)—we obtain the following equation:

2 p 0 ¼½ð@ 2 2 2

1 = J u =@ x Þ þð@v=@ y Þ þð@ w =@ z Þ

þ 2 ð@ u =@ y @v=@ x þ@ w =@ x @ u =@ z þ@ w =@ y @v=@ z Þþ@ B =@ z ð 2 : 62 Þ

2 As for trajectories, 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.