Ordinary-cell convective storms - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 137

Figure 3.36. Illustration of how the cold side/warm side-directed, hydrostatic pressure gradi-
ent force (green vectors) decreases with height in a density current. Vertical cross section
showing edge of cold pool (solid red line), hydrostatic pressure (dashed thin blue line), and
sense of rotation induced at the leading edge (curved arrow). The cold pool lies to the left of the
solid red line, as in Figure 3.33.

Figure 3.37. Vertical cross section of idealized density current, in which a cold (dense) pool
having a density 1 and depth h (edge of which is shown as a solid red line) propagates into an
environment of less density 0 and depth H. The density current (cold pool) relative flow from
right to left is depicted by the streamlines; at the right edge, the density current relative flow is c
to the left; at the left edge above h the density current relative flow is u to the left.

After integrating the first term with respect to x and the second term with respect
to z, this equation becomes
ð H

u

ð 1

w

1 H

dz þ dx ¼ 0 ð 3 : 11 Þ

0 1

1 0

But we set w ð H Þ¼ 0 and w ð 0 Þ¼ 0 (there is no flow into the domain from the
surface or out from the top), so
ð H ð H

u 1 dz u 1 dz ¼ 0 ð 3 : 12 Þ

0 0