Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 212

Figure 4.31. Illustration of how updrafts propagate (propagation vector shown) from where
the downward-directed (solid circle) dynamic pressure gradient is increasing the most (dashed
circle) to where the upward-directed dynamic pressure gradient is increasing the most.

vorticity increases vorticity, updraft propagation is also important in vorticity
amplification at low levels in a storm.
The divergence equation (2.62) may be expressed as:

2 p 0 ¼½ð@ 2 2 2

0 r u =@ x Þ þð@v=@ y Þ þð@ w =@ z Þ

2 ½@ =@ @v=@

x þ@

=@ @ =@

z þ@

=@ @v=@

z þ@

=@

z ð 4

:

30 Þ u y w x u w y B

where 0 1 = , which is treated as a constant for the base state atmosphere

2 2 2

(cf. (2.4)). The terms '' ½ð@ u =@ x Þ þð@v=@ y Þ þð@ w =@ z Þ '' are called ''fluid

extension terms'' and the terms '' ½@ u =@ y @v=@ x þ@ w =@ x @ u =@ z þ@ w =@ y @v=@ z ''

are called ''shear terms''. The shear terms may be expressed in terms of

2 2

'' 2 ½j D j

'', as may be verified by brute force and obstinacy using simple
algebra, where j D j is the magnitude of resultant three-dimensional deformation
and j!j is the magnitude of three-dimensional vorticity (

j!j

!

):

2

¼ D xy þ D zx þ D zy j D j ð 4 : 31 Þ

where

D xy ¼ @v=@ x þ@ u =@ y ; deformation in the x y-plane ð 4 : 32 Þ

D zx ¼ @ u =@ z þ@ w =@ x ; deformation in the z x-plane ð 4 : 33 Þ

D zy ¼ @ w =@ y þ@v=@ z ; deformation in the z y-plane ð 4 : 34 Þ

and

2 2 2 2

j!j ¼ þ þ ð 4 : 35 Þ

where

¼ @v=@ x @ u =@ y ; vorticity about the z-axis ð 4 : 36 Þ

¼ @ w =@ y @v=@ z ; vorticity about the x-axis ð 4 : 37 Þ

¼ @ u =@ z @ w =@ x ; vorticity about the y-axis ð 4 : 38 Þ