The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 33

2.3 CONSERVATION OF MASS, AND THE BOUSSINESQ AND
ANELASTIC APPROXIMATIONS

The equation of continuity, which is a statement of the conservation of mass, is as
follows for a compressible atmosphere:
1

= D = Dt ¼ JE v ð 2 : 28 Þ

When air converges in three dimensions, some of it is compressed so that its
density increases; it is also deformed such that some air is drawn in from some
sides and is expelled out other sides. To see this, think of what happens when you
squeeze a tube of toothpaste; if the toothpaste is compressible, then its density
increases somewhat, but some of the toothpaste squirts out. The time derivative in
(2.28) creates some problems in understanding convective clouds because it
permits the existence of sound waves, which can propagate and therefore must be
accounted for. Sound waves are not thought to be important for the dynamics of
convective storms, unless perhaps you believe you can shout at a storm and
change its behavior (something that storm chasers often do, by ''badmouthing'' a
storm so as to conjure up a more intense storm) or that a convective storm can
contain some aural information and can thereby communicate with you. If one
integrates pressure with time, then rapid pressure fluctuations associated with
sound waves overwhelm the slower pressure fluctuations associated with storm-
scale processes. Just as if one integrates pressure with time for the hydrostatic
primitive equations, the rapid pressure fluctuations associated with gravity waves
overwhelm the slower pressure fluctuations associated with synoptic-scale, quasi-
geostrophic processes. A technique must be devised for filtering out sound waves,
unless it can be shown the dynamics of some parts of severe convective storms
do respond to or are dependent on sound waves. The original motivation for
simplifying the dynamical equations that describe convection was to eliminate
sound waves.
Under certain circumstances a compressible fluid such as the atmosphere may
be treated like an incompressible fluid (i.e., one in which 1

= =

Dt ¼ 0 ¼ JE

v)
and sound waves are not allowed. In this situation, air parcels are deformed when
there is divergence or convergence in one plane, but convergence in one plane
must be compensated for by divergence in the direction normal to the plane.
Think of squeezing a tube of toothpaste but this time, no matter how hard
you squeeze the toothpaste,

D

it does not get any denser if the toothpaste is

incompressible.
Let us now see if there are any conditions under which we can neglect the
time derivative term in the compressible form of the continuity equation. From the
adiabatic form of the thermodynamic equation (2.27), the equation of continuity
(2.28), and the ideal gas law, it is seen that
JE v ¼ C v = C p D ð ln p Þ= Dt ¼ C v = C p ð@=@ t þ v EJ z þ w @=@ z Þ ln p ð 2 : 29 Þ

The last term on the RHS of (2.29), with the aid of the hydrostatic approximation
and the ideal gas law, may be expressed as R @=@ z ð ln p Þg= T ¼ 1 = H, where H is