The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 47

Figure 2.6. An idealized illustration of how an air parcel (circle) that is accelerated upward by
positive buoyancy is opposed by a downward-directed perturbation pressure gradient force as
air is forced outward just above the parcel to make way for air in the buoyant parcel that is
accelerated upward and air is forced inward just below the parcel to replace the air that is
accelerated upward.

and negative in others. Since maxima (minima) in p 0 b tend to be correlated with
positive (negative) pressure perturbations and is always positive, p 0 b tends to be

2 p 0 b <

0 (i.e., when there is a local maximum in p 0 b ) and negative

positive when

J

0 (i.e., when there is a local minimum in p 0 b ). In other words, p 0 b is
negatively correlated with

2 p 0 b >

when

J

z. We arrive then at the following conclusion:
underneath a buoyant parcel of air the perturbation pressure is negative and
above it the pressure is positive. Thus, there is a downward-directed perturbation
pressure gradient force that acts to reduce the effects of Archimedean buoyancy
( Figure 2.6 ). It is only when the two do not cancel each other out that there is a
true buoyancy effect. When the two cancel each other out, the atmosphere is
hydrostatic. When p 0 b is neglected, as it is when employing the parcel method, B is
in fact the entire buoyancy force. Since there is a counteracting perturbation press-
ure gradient force, any estimates of acceleration based on Archimedean buoyancy
alone are an overestimate. The reader is reminded that our analysis has neglected
turbulent mixing with the environment of the buoyant air parcel (the entrainment
of environmental air); if mixing were taken into account, the value of B would be
reduced even more.
Let us now consider what vertical circulations are produced by buoyancy
gradients alone at the outer edges of the buoyant air parcel using the horizontal
component (for simplicity, only in the y-direction) of the Boussinesq vorticity
equation (2.61)

@ B =@

D = Dt ð@ u =@ z @ w =@ x Þ¼@ B =@ x ð 2 : 71 Þ