Mesoscale convective systems - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 286

Figure 5.19. Idealized model of two-dimensional vertical circulation in a squall line. Buoyant
air stream, ascending front-to-rear flow, (blue streamline) with speed in the x-direction noted in
the lower-right to upper-left direction; rear-inflow jet (green streamline) at mid-levels at left
without buoyancy at speed in the x-direction noted at the beginning and end of the streamline;
negatively buoyant return flow at lower left. Red line separates two separate branches of the
flow (modified from Emanuel, 1994).

to be steady state and two dimensional, 3 and therefore of limited application
to real MCSs which have some degree of two dimensionality and some three
dimensionality.
The object of this exercise is to find out what values of U 0 , U c , and h c are
dynamically and thermodynamically consistent with the model and if they are
reasonably close to what is observed. The acceleration from the pressure gradient
force at x ¼1 is found from the hydrostatic equation—expressed as the vertical
component of (4.11) with Dw = Dt ¼ 0

8
<

B u z 2h c

p 0

0 @ =@ z ¼ 0 h c z < 2h c ð 5 : 1 Þ

:

B d z < h c

We define

p 0 ð x ¼1; z ¼ H Þ¼ p t ð 5 : 2 Þ

Integrating (5.1) with respect to pressure at x ¼1 , we find that

8
< 0 p t B u ð H z Þ z 2h c

0 p 0 ¼ 0 p t B u ð H 2h c Þ h c z < 2h c ð 5 : 3 Þ

:

0 p t B u ð H 2h c Þ B d ð h c z Þ z < h c

Consider now the x-equation of motion (2.13) subject to the simplifications of
no friction (inviscid flow), two dimensionality, steady flow, and a Boussinesq
atmosphere, with 1 = ¼ 0

u

p 0 @ u =@ x þ w @ u =@ z ¼ 0 @ =@ x ð 5 : 4 Þ

3 Two-dimensional circulations in squall lines are sometimes referred to as examples of ''slab
overturning''.