Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 215

The dynamic perturbation pressure is decomposed as follows into linear and
nonlinear parts

p 0 d ¼ p 0 L þ p 0 NL

ð 4 : 47 Þ

We consider the nonlinear terms first because they are often the most
important. The nonlinear shear terms in (4.45) can be expressed,

like before

(4.40), as the following:

u 0

2

þð@v 0

2

w 0

2

2 ½j D 0 j

2 0 j 2

1

½ð@ =@ x Þ =@ y Þ þð@ =@ z Þ ½j! ð 4 : 48 Þ

where D represents the perturbation (storm-related) resultant three-dimensional
deformation; and

0 represents the perturbation (storm-related) three-dimensional
vorticity. In particular

!

D 0 2 w 0 y þ@v 0

2

u 0 w 0

2

þð@v 0 u 0

2

¼ð@ =@ =@ z Þ þð@ =@ z þ@ =@ x Þ =@ x þ@ =@ y Þ ð 4 : 49 Þ

and

0 2

w 0 y @v 0

2

u 0 w 0

2

þð@v 0 u 0

2

j! j ¼ð@ =@ =@ z Þ þð@ =@ z @ =@ x Þ =@ x @ =@ y Þ ð 4 : 50 Þ

The forcing function in (4.48) involving vorticity alone is called ''spin''. Bob
Davies-Jones in 2002 proposed that the nonlinear terms be decomposed as the
sum of the fluid extension and shear terms involving deformation, or ''splat''; the
remaining terms are spin. Davies-Jones argued that this decomposition is more
physical because the terms are invariant with respect to rotations of the coordinate
axes.
The fluid extension part of the nonlinear term contributes to positive
perturbation pressure and the deformation part contributes to positive perturba-
tion pressure, while the spin part contributes to negative perturbation pressure,
because the forcing functions associated with the fluid extension and deformation
are each negative definite, while that associated with spin is positive definite. The
main nonlinear effects are therefore as follows: regions of sharp horizontal
gradients in the horizontal wind field, sharp vertical gradients of the vertical com-
ponent of the wind field, or strong deformation are associated with positive
perturbation pressure. Regions of strong vorticity (either cyclonic or anticyclonic
in the vertical, or horizontal of any sign) are associated with negative perturbation
pressure; cyclones and anticyclones are therefore associated with centers of nega-
tive perturbation pressure. To understand this relationship qualitatively without
resorting to a divergence equation, just consider the special case of a vortex in
cyclostrophic balance ( Figure 4.33 ): An outward-directed centrifugal force must
always be balanced by a radially inward-directed pressure gradient force, so that
the pressure at the center of a vortex must be relatively low. 5
We now consider the linear terms. To analyze the dynamics of convective
storms when an updraft interacts with vertical shear, it is easiest first to consider
an environment characterized by a hodograph. When the hodograph is ''straight'',

5 A parcel must always encounter a force normal to its motion if the flow is curved; when the
flow, for example, is counterclockwise, there must be a pressure gradient force acting to the left.
This situation is consistent with flow about a region of relatively low pressure.