Geoscience Reference
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the continuity equation when v ¼ w ¼ 0 (i.e, when the motion of the fluid is in the
x-direction only) is
1
=
D
=
Dt ¼@
u
=@
x
ð 2
:
65 Þ
The adiabatic form of
the thermodynamic equation (2.27) may therefore be
expressed as
66 Þ
Linearized about a resting atmosphere, (2.66) may be combined to yield the
following:
@
u
=@
x þ 1
p Þ Dp
=
Dt ¼ 0
ð 2
:
u 0
p 0
67 Þ
Eliminating u 0 from the equation of motion (2.64) and the combined continuity
and adiabatic thermodynamic equation (2.66), the following wave equation for
perturbation pressure is obtained:
p
@
=@
x þ 1
= @
=@
t ¼ 0
ð 2
:
2 p 0
x 2
c 2
2 p 0
t 2
@
=@
1
=
@
=@
¼ 0
ð 2
:
68 Þ
where c is the speed of sound (2.17). When the Boussinesq version of the
divergence equation (2.62) is simplified so that the mean winds in the atmosphere
are calm (i.e., a ''resting'' atmosphere), there are no variations in y, and B ¼ 0, we
find that in this case the linearized version of (2.62) is
2 p 0
x 2
69 Þ
Thus, the Boussinesq divergence equation does not contain the time derivative
term. In nature, sound waves transmit information relating the pressure field to
the wind field. By eliminating them, we in effect assume that their speed is
infinite—since the speed of sound appears in the denominator of the factor multi-
plying the time derivative term in (2.68)—so that information linking the pressure
field to the wind field is instantaneous and they are linked by a Poisson equation.
(An analogous situation exists in synoptic meteorology, when the atmosphere is
analyzed using quasi-geostrophic theory. Information linking the pressure field to
the wind field is assumed to be instantaneous, through the geostrophic wind rela-
tion. However, in nature inertial gravity waves carry the information linking them
together and it is assumed there is a Poisson equation linking them, the time
derivative term being eliminated when the speed of
@
=@
¼ 0
ð 2
:
inertial gravity waves is
infinite.)
The pressure field retrieved from the divergence equation can be used to
separate the effects of the vertical perturbation pressure gradient force due to
dynamics (by means of terms involving the wind field) from those due to buoy-
ancy (the vertical derivative of buoyancy) as in (2.63). The vertical equation of
motion (2.7) may be expressed using (2.63) as
Dw
p 0 d =@
p 0 b =@
=
Dt ¼ 1
=
@
z þ½ð 1
=
Þ@
z þ B
ð 2
:
70 Þ
p 0 b =@
then ð 1
=
@
z þ B Þ the
The vertical acceleration due to buoyancy is
''effective'' buoyancy—not
just B alone. For example, underneath a buoyant
2 p 0 b >
@
=@
>
=
J
parcel of air,
0. Within a
volume, p 0 b averages out to zero and therefore must be positive in some locations
B
z
0; from (2.62) it is seen that 1
 
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