The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 45

Since the flow in a Boussinesq fluid is non-divergent, this divergence equation is
time independent (diagnostic, as opposed to prognostic). It is used to compute
perturbation pressure from the three-dimensional distribution of wind. While the
circulations associated with convective storms are computed from the vorticity
equation, the three-dimensional pressure field that is consistent with the circula-
tions is computed from the divergence equation. It is therefore not appropriate to
infer that the pressure field causes the wind field, but rather that it is consistent
with it. We can, however, use the pressure field to compute pressure gradient
forces that will change the wind field in the future. Both (2.49) and (2.62) are used
in tandem to analyze storm dynamics.

2.5.1 Buoyancy-induced and dynamically induced pressure perturbations
Since the operator on p 0 is linear (a Laplacian operator), we can find solutions of
p 0 for each term on the right-hand side of (2.62) and add up all solutions to find
the total p 0 . To facilitate the physical interpretation of perturbation pressures, we
let

p 0 ¼ p 0 b þ p 0 d

ð 2 :

63 Þ

where p 0 b is the perturbation pressure due to the buoyancy term alone; and p 0 d is
the perturbation pressure due to the ''dynamic terms'' (i.e., those involving the
wind field and its variations).
Appropriate boundary conditions on p 0 must be selected so that (2.62) can be
solved. It may assumed that, far from a region of nonzero B, p 0 b ¼ 0. These lateral
and vertical boundaries may therefore be prescribed far from regions of signifi-
cantly nonzero B. Otherwise, one may use the equations of motion (2.13) and
(2.7) to compute the horizontal and vertical gradients of perturbation pressure at
the boundaries in terms of the three-dimensional wind field and its spatial and
temporal gradients (Neumann boundary conditions) and in terms of B. If the
outer boundaries are chosen so that the atmosphere is resting and there is no
buoyancy there, then the boundary conditions become p 0 d ¼ 0. The values of per-
turbation pressure are actually determined to within an arbitrary constant, but the
gradients of perturbation pressure are exact; the mean value of the perturbation
pressure is usually assumed to be zero.
If the fully compressible, time-dependent version of the equation of continuity
(2.28) were used, then the divergence equation would contain a term involving
time derivatives ( @=@ v) and it would therefore no longer be diagnostic.

Suppose now for the purposes of illustration that the atmosphere were fully
compressible. For simplicity, let's assume that the atmosphere has no variations in
the y-direction and that there is no mean (horizontal or vertical) flow. Then the
linearized equation of motion (2.13) in the x-direction is

t

JE

u 0 p 0 @ =@ t þ 1 = @ =@ x ¼ 0 ð 2 : 64 Þ

where u 0 is the perturbation part of the zonal wind. The compressible form of