The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 41

Figure 2.3. An idealized illustration of positive vorticity about the y-axis when @ u =@ z > 0 and
@ w =@ x < 0. The dashed vector indicates vorticity. The curved arrow denotes the sense of
rotation, while the direction of the dashed vector may be determined by curling one's fingers
about the sense of the curl such that the thumb points in the direction of the vorticity vector
according to the ''right-hand rule''.

horizontal wind in the x and y-directions, respectively. Terms 2 and 3 on the left-
hand side of (2.50) (when multiplied by 1) represent horizontal and vertical
advection of vorticity, while terms 1, 2, and 3 on the right-hand side represent
stretching (the divergence term), tilting, and the solenoidal effect. The latter occurs
when the atmosphere is baroclinic (i.e., when isobars are not parallel to lines of
constant density, so that there is a nonzero horizontal gradient in the pressure
gradient force). The solenoidal term (3 on the RHS), however, is zero in a Boussi-
nesq atmosphere. The vorticity equation is in part more convenient than the
equations of motion for dynamical analyses because pressure does not appear ex-
plicitly. As before, Earth's vorticity is not included, but could be added to the
vertical vorticity to account for the behavior of convective phenomena that last
for a relatively long period of time.
The component of (2.49) along the y axis ( Figure 2.3 ) is

D = Dt ð@ u =@ z @ w =@ x Þ¼ð@ x Þ

| {z }
1

u =@ x þ@ w =@ z Þð@ u =@ z @ w =@

þð@v=@ y Þ

| {z }
2

z @ u =@ y @v=@ x @ w =@ @ x

| {z }
3

B =@ ð 2 : 51 Þ

where x is the component of vorticity about the y axis. Term 1

represents the stretching of vorticity about the y axis, 2 represents tilting of
vorticity about the z and/or x axes onto the y axis, and 3 represents baroclinic
generation. The latter is created when there is a horizontal gradient of vertical
acceleration due to a horizontal gradient of buoyancy ( Figure 2.4 ). Again, Earth's
vorticity is not included, but its effects could be accounted for by adding it to the
vertical vorticity.

@ u =@ z @ w =@