The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 36

2.3.2 Anelastic approximation

For substantially subsonic convection that is ''deep'' (i.e., for which D H), the
equation of continuity (2.29) may be simplified as

v ¼ C v = C p ð w @=@ z Þð ln p Þ ð 2 : 42 Þ

JE

This approximation is referred to as the anelastic approximation; Yoshi Ogura
and Norman Phillips first coined this term in a seminal paper published in 1962.
Stigler's Law of Eponymy is evident here: according to this law, any scientific
discovery named for someone was actually discovered earlier by someone else. In
this case, the Australian applied mathematician and fluid dynamicist George
Batchelor had come up with a nearly identical formulation back in 1953, while
Jule Charney and Yoshi Ogura had used it in 1960 without naming it. (It also
appears that Boussinesq's work itself was pre-dated by A. Oberbeck in 1879,
though my knowledge of the German language is not adequate and I must rely on
translations by others.) For adiabatic flow, it is seen from (2.39) that (2.42) may
be expressed as

JE v ¼ 1

= @=@ z ð 2 : 43 Þ

For severe convective systems, which are typically deep (i.e., extend up to the
tropopause and slightly beyond, 10-15 km AGL), the anelastic continuity equa-
tion is often a better approximation. It is a simplification of (2.28) in which local
time derivatives and horizontal gradients are neglected, and only the vertical varia-
tions of density are retained. The anelastic continuity equation is used in some
numerical models and in some Doppler radar wind analysis schemes. To keep
analyses of the dynamics of convection as simple as possible, however, the
Boussinesq continuity equation is most frequently used. By ignoring the vertical
effects of compressibility, the overall physics are changed only slightly and funda-
mental results are not altered qualitatively. The results of numerical simulations
conducted with models that are fully compressible (using (2.28)) support the anal-
ysis of storm dynamics in a qualitative sense using the Boussinesq approximation.
Remember that we constrained the first and second terms in the continuity
equation (2.33) to be of the same scale. From the Eulerian form of the thermo-
dynamic equation (2.29) we estimate the time scale

W by assuming that
the time derivative term is the same order of magnitude as the vertical advection
term. For deep convection, D 10 km and W 10m s 1 , so that

D =

10 3 s, or
about 10 min. From the vertical equation of motion (2.7), we estimate

W =

B,

where B is the buoyancy. So, for B (10m s 2 ) (1K/300K), 300 s ¼ 5min. We

can now compare the magnitude of the time derivative term in (2.29) with the
horizontal advection term in (2.29) by assuming

5-10min, the reciprocal of
which is approximately the Brunt-Va ¨ isa ¨ la ¨ frequency ( 1/300-1/1,000Hz), which
is much lower than that of audible sound waves. So

@ ln p =@ t 1 = ð 2 : 44 Þ

and

v

EJ z ln p U

= L ð 2 : 45 Þ