Geoscience Reference
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function
, which is non-dimensional:
R
=
C
p
R
=
C
p
¼ð
p
=
p
0
Þ
¼
T
= ¼ð
R
=
p
0
Þ
ð
2
:
139
Þ
This pressure variable is proportional to potential temperature (2.26), so that
potential temperature is used rather than temperature as the temperature variable.
Use of the Exner function makes mathematical analysis easier, but we think in
terms of pressure which we measure with a barometer, and temperature which we
measure with a thermometer; I have never heard of anyone devising an instrument
to measure the Exner function, which could well be called an ''exnerometer''.
The equations of motion (2.1) and (2.2) and the adiabatic form of the
continuity equation (2.28), with the Exner function as the pressure variable and
potential temperature as the density/temperature variable, are given by
Dv
h
=
Dt
¼ @
v
h
=@
t
þ
v
v
h
¼
C
p
J
h
ð
2
:
140
Þ
EJ
and
Dw
=
Dt
¼
C
p
@=@
z
g
ð
2
:
141
Þ
v
¼ð
1
1
=Þ
D ln
=
Dt
ð
2
:
142
Þ
JE
where
C
p
. Unlike the equations of motion expressed in terms of density,
the equations of motion in terms of
¼
R
=
contain products of variables only;
there are no variables appearing in the denominator. To derive these equations, we
have made use of
and
and
expressed in terms of p, and T and
, respectively;
does not appear because it can be expressed in terms of
and
. The continuity
equation may be derived by expressing 1
=
D
=
Dt as D
ð
ln
Þ=
Dt before substitut-
ing for
. The equations of motion and the adiabatic
form of the continuity equation expressed subject to the decomposition in terms of
a basic state and perturbations (as in (2.3) and (2.4))
¼
expressed in terms of
and
0
ð
z
Þþ
ð
x
;
;
;
t
Þ
ð
2
:
143
Þ
y
z
and
¼
0
ð
z
Þþ
ð
x
;
;
;
t
Þ
ð
2
:
144
Þ
y
z
are as follows:
0
Dv
h
=
Dt
¼ @
v
h
=@
t
þ
v
v
h
¼
C
p
J
ð
2
:
145
Þ
EJ
0
z
þ
B
4
Dw
=
Dt
¼
C
p
@
=@
ð
2
:
146
Þ
0
v
¼
C
v
=ð
R
Þð
wd
=
dz
þ
D
=
Dt
Þ
ð
2
:
147
Þ
JE
4
where buoyancy
=
0
148
Þ
It is noted that, unlike in the derivation of B expressed in terms of T
0
and p
0
(2.11), the expression for B does not involve any assumptions about the Mach
number. The base state
B
¼ g
ð
2
:
is hydrostatic, so that the hydrostatic equation is
C
p
@
=@
z
¼g
ð
2
:
149
Þ
4
The vertical perturbation pressure gradient force and B do not correspond exactly with those
in (2.7).
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