Geoscience Reference
In-Depth Information
From the definition of potential temperature (2.26) and the ideal gas law, the first
term on the RHS of (2.147) may be expressed as
=ð
Þ
d
ð
w
Þ=
dz
ð
2
:
150
Þ
We will now seek the condition under which the time derivative term can be
neglected in comparison with the other terms. Let us use the following scaling:
;
0
P
u
U
; v
U
;
w
W
;
1
;
T
;
t
;
x
y
L
;
and z
D
ð
2
:
151
Þ
where P
1. So the terms in the continuity equation (2.147) are approximately
j
JE
v
j
U
=
L
W
=
D
ð
2
:
152
Þ
j
C
v
=ð
R
Þð
wd
=
dz
Þj
WC
v
=ð
RD
Þð
C
v
=
R
Þð
U
=
L
Þ
ð
2
:
153
Þ
0
Dt
j
C
v
P
j
C
v
=ð
R
Þ
D
=
=ð
R
Þ
ð
2
:
154
Þ
The terms in the horizontal equation of motion (2.145) are scaled as follows:
j
Dv
h
=
Dt
j
U
=
ð
2
:
155
Þ
j
C
p
0
j
C
p
TP
J
h
=
L
ð
2
:
156
Þ
=
C
p
TP
So, U
=
L, and therefore
P
UL
=ð
C
p
T
Þ
ð
2
:
157
Þ
Now, c
2
C
v
Þ
RT. Substituting for P
in (2.154), we find that
the time derivative term may be neglected in comparison with the divergence term
when
¼ð
C
p
=
C
v
Þ
R
ð
C
p
=
L
=
c
ð
2
:
158
Þ
that is, when the time scale is much longer than the time it takes a sound wave
to travel the characteristic horizontal scale. So, when sound waves are filtered out
and only much longer time scales are considered the adiabatic form of
the
continuity equation may be expressed as
=ð
Þ
d
ð
v
¼
w
Þ=
dz
ð
2
:
159
Þ
JE
This equation is similar to the anelastic equation (2.42), except that
is now
included and is a function of height. We did not have to make any assumptions
about D
H, as we did earlier when using pressure and temperature. The anelastic
equation (2.42) in effect is valid when the basic state is isentropic (i.e., is one of
constant potential temperature and has a dry-adiabatic lapse rate), so that
=
dis-
appears explicitly from (2.159). This equation (2.159) is known as the ''pseudo-
incompressible equation''. Dale Durran first introduced it in 1989. It is valid when
sound waves are filtered out and when
j
0
jjj
. The main advantage of this
approximation is that the environment may be stratified in any way and is not
constrained to be dry adiabatic.
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