The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 28

although it may vary in space, it does so on scales much larger than those of
convective clouds. Such an assumption is fine in most cases but not, for example,
when a strong, surface front or sharp outflow boundary/gust front is present.
Nevertheless, let

p ð z Þþ p 0 ð x p ¼ ; y ; z ; t Þ ð 2 : 3 Þ

and

0

¼ ð z Þþ ð x ; y ; z ; t Þ ð 2 : 4 Þ

and

@ 1 = p =@ z ¼ g ð 2 : 5 Þ

where p is the total pressure; are the pressure and

density of the hydrostatic ''base state'' (the horizontally averaged values of p and

is the total density; p and

0 are the ''perturbation'' pressure and density,
where in this case the perturbation quantities are at least an order of magnitude
less in magnitude than the base-state quantities. The reader is reminded that this
formulation requires modification when the basic state

over a broad area); and p 0 and

varies horizontally as well

(e.g., near a front or other baroclinic zone).
Because the perturbation quantities are so small, products of perturbation
terms with perturbation terms are much less than products of perturbation terms
with base-state terms, etc., it follows that

0 0

1 =ð þ Þ 1 ð = Þ ð 2 : 6 Þ

and the vertical equation of motion (2.2) may then be rewritten as

p 0

0

=

Dt ¼ð 1

= Þ@ =@

z ð

= Þg ð 2 :

7 Þ Dw

Archimedean buoyancy B is given by the second term on the RHS (right-hand
side) of (2.7)

0

= B ¼ð Þg ð 2 : 8 Þ

The vertical equation of motion is similar to the horizontal equation of motion,
except that there is an additional term that represents the acceleration induced by
the buoyancy force.

(If the atmosphere were in hydrostatic balance, then

p 0

=

Dw z.) The concept of Archimedean buoyancy is

nice because it is easy to visualize buoyancy in terms of a less dense object (i.e.,
less dense than water) released in water rising or a denser object sinking. In the
atmosphere, however, the distinction between ''the object'' and ''the water'' is not
always clear and the density of the surrounding air representing the water varies
horizontally as well as vertically.
The effect of Archimedean buoyancy may be understood quantitatively by
considering a box (of dimensions

= Dt ¼ 0, so that B ¼ 1 @ =@

D x D y D z) of fluid of density 1 embedded

within a fluid of density 2 ( Figure 2.1 ). If the fluid surrounding the buoyant box,

the ''environment'', is in hydrostatic balance, then
1 = 2 @ p 2 =@ z g ¼ 0 ð 2 : 9 Þ

where p 2 is the pressure in the environment. So, the vertical pressure gradient
force in the environment for a volume the size of the buoyant box is the