The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 57

words, as we demonstrated in Section 2.5.1, the aspect ratio of the size of the
buoyant fluid element determines how much buoyancy is counteracted by a ver-
tical perturbation pressure gradient force: Larger spherical bubbles of equal
buoyancy do not have greater accelerations inside them than smaller bubbles. It
should also be noted how localized buoyancy can affect the wind field away from
the spherical bubble, where there is no buoyancy. The effects of the buoyant
bubble have been communicated to the outside world instantaneously, as action at
a distance. Sound waves, which are not modeled by the Boussinesq equations,
carry this information (i.e., that there is a buoyant sphere) at infinite speed: news
travels quickly in a Boussinesq atmosphere.
To solve for the pressure field, we can take the equations of motion (2.92) and
(2.93) and form a divergence equation in spherical coordinates to yield the
following:

2 P ¼ 1

r 2 tr 2 þ r 2 B sin r = @=@ r ð@ u =@ 'Þ

=ð r 2 sin

þ 1 'Þ@=@'ð r @v=@ 'þ rB sin ' 'Þ ð 2 :

136 Þ t sin cos

We can then substitute in (2.136) for a

using (2.130) and (2.131), and find solutions to P. It is perhaps just as useful and
easier for a student to recognize that isobars are aligned perpendicular to the
pressure gradient, and just evaluate

@ u =@ t and @v=@ t for r < a and for r >

at various locations and
draw isobars accordingly by looking for regions where the gradients are zero
( Figure 2.11 ). It is noted that when doing numerical integrations in a cloud
model, one cannot arbitrarily specify the initial pressure field. A diagnostic

@ P =@ r and 1 = r @ P =@'

Figure 2.11. Qualitative depiction of the perturbation pressure field (isobars denoted by
dashed lines) induced by a buoyant, spherical bubble (solid circle).