Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 209

frictionless, barotropic atmosphere, vorticity is changed only through tilting and
stretching. A vortex line is oriented along the three-dimensional vorticity vector. It
therefore moves along with the wind (advective terms on the LHS) and is tilted
(tilting terms on the RHS). Stretching does not alter the orientation of vortex
lines. According to Bob Davies-Jones, at NSSL, ''Barotropic vortex lines are
'frozen' into the fluid and behave like elastic strings that the flow moves, stretches,
and reorients.'' If baroclinic effects (i.e., gradients in buoyancy) are considered,
then vortex lines may change as a result and the analysis is more complicated;
thus, we typically ignore baroclinic effects for simplicity and consider qualitatively
the consequences of baroclinic effects: vortex lines cannot be broken in a
barotropic, frictionless atmosphere.
So, an initially horizontally oriented vortex line that points to the pole is
distorted by the updraft so that it is deformed into an upside-U shape; the vortex
line has a component that points upward on the equatorward side and downward
on the poleward side. Thus, the vertical component of vorticity on the equator-
ward (poleward) side has a component in the direction of (in the direction
opposite to that of) the rotation vector of the Earth ( Figure 4.29, top panel).
Another way to analyze the production of vertical vorticity in a vertically
sheared environment by an updraft is to make use of the conservation of Ertel's
potential vorticity (2.137 and 2.138). Inside a cloud, where the air is saturated, we
replace with e , so that

Z JT v

EJ e

ð 4 : 27 Þ

In other words, the component of three-dimensional vorticity in the direction of
the equivalent potential temperature gradient must remain constant.
Suppose that initially the three-dimensional vorticity vector is associated with
vertical shear alone due to a westerly thermal wind and that there is conditional
instability (i.e., that e decreases with height, but much more rapidly than it does

with y). It is seen in Figure 4.29 (middle panel) that Z ¼ 0 in this case, so that
vortex lines must always lie on surfaces of constant

e . (We will not be concerned
here if the potential vorticity vanishes so that the atmosphere is neutral with
respect to symmetric instability or if @ e =@ z > 0 so that the atmosphere is

convectively stable.)
In the absence of diabatic heating and friction, when a localized updraft
forms, surfaces of constant

e is conserved for adiabatic
processes) and so must the vortex lines, which in this case have a component
toward the north. We note that using Ertel's potential vorticity in our analysis, we
do not have to constrain ourselves to barotropic cases: Vortex lines always remain
on isentropic surfaces regardless of whether or not there is baroclinicity. However,
just as vortex lines cannot be broken in a barotropic atmosphere, surfaces of con-
stant potential vorticity cannot be fractured in a frictionless, adiabatic atmosphere.
When air approaches and enters the updraft at low levels from the
equatorward side, it begins with its vorticity vector pointing towards the pole. It is
therefore seen that the three-dimensional vorticity vector must change from being
directed from the equator to the pole only to having a component directed

e bulge upward (because