The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 29

Figure 2.1. Idealized illustration of Archimedean buoyancy for a box of fluid. Outside the box
the fluid is in hydrostatic balance. Since the pressure gradient force (PGF) is the same outside
the box as it is inside the box, but the density inside the box 1 in this case is less than the density
outside the box 2 , there is a net upward buoyancy force. Forces inside and outside the box are
indicated.

acceleration due to the vertical pressure gradient force multiplied by the mass
of the fluid in the same volume as that of the box; that is, the upward force
is 1 = 2 @ p 2 =@ z ðD x D y D z Þ 2 ¼@ p 2 =@ z ðD x D y D z Þ and the equal and opposite

downward force due to gravity is 2 gðD z Þ . If we assume that the pressure

in the buoyant box is the same as that in the environment (a good assumption at
the outset only), then the upward pressure gradient force experienced by the box
is still @

x D y D

z Þ . However, the buoyant box's downward force due
to gravity is only 1 gðD

p 2 =@ z ðD x D y D

x D y D z Þ if, for example, 1 is less than 2 . The box of

fluid is not in hydrostatic balance and there is a net upward force of

gðD z Þð 2 1 Þ . The vertical acceleration (force/mass) experienced by the

buoyant box is therefore

x D y D

Dw = Dt ¼ gð 2 1 Þ= 1 ¼ B ð 2 : 10 Þ

Buoyancy is therefore due to the net difference in weight between the denser
environment and that of the less dense box of fluid. The formulations for buoy-
ancy in (2.8) and (2.10) are essentially identical, but the latter is exact, while the
former is approximate: for the former, B ¼ð

0 0 and

= Þg , where 1 ¼ 2 þ is

slightly less than 2 , because the horizontal average of is weighted mostly by 2

0 is not
zero, as it usually is taken to be, because the only perturbation is the density of
the buoyant cube. In (2.10),

1 ; in this instance, the average value of

and much less by the less dense

1 appears in the denominator rather than 2 ,

reflecting the influence of the slightly lower density of the buoyant cube.