Geoscience Reference
In-Depth Information
equation (1.19)
, written for our motionless base state and with
x
3
positive upward
so that the gravity vector
g
i
=
(
0
,
0
,
−
g)
,is
dp
0
−
dx
3
−
ρ
0
g
=
0
.
(8.2)
We use the ordinary derivative because
x
3
is the only independent variable in the
base state.
Since we have three base-state variables, pressure, temperature, and density, we
need a third equation. We add the equation for specific entropy (entropy per unit
mass)
s
of air,
Dp
Dt
,
T
Ds
Dh
Dt
−
1
ρ
Dt
=
(8.3)
with
h
c
p
T
the specific enthalpy and
c
p
the specific heat per unit mass at constant
pressure. Here we use the substantial derivative
D/Dt
(
Chapter 1
) because we shall
use frictionless, adiabatic
virtual displacements
of fluid parcels to determine the
z
-
dependence of the base state. For such constant-entropy or
isentropic
displacements
we write
Eq. (8.3)
as
=
T
0
Ds
0
Dh
0
1
ρ
0
Dp
0
Dt
Dt
=
0
,
so that
Dt
=
.
(8.4)
This implies that the base-state profiles are related by
dh
0
dx
3
=
1
ρ
0
dp
0
dx
3
.
(8.5)
With
h
0
=
c
p
T
0
and the vertical equation of motion
(8.2)
,
(8.5)
yields
dT
0
g
c
p
,
dx
3
=−
(8.6)
the equation for the
adiabatic temperature profile
. The decrease in
T
0
is almost
exactly 1 K per 100 m height. From the set
(8.1)
,
(8.2)
,and
(8.6)
the profiles of
p
0
and
ρ
0
can also be calculated
(Problem 8.2)
.
8.2.2 Flow-induced deviations from the base state: mass conservation
As in
Part I
we denote properties of the moving, turbulent atmosphere with tildes.
We represent them as the sum of the base-state variables plus small deviations,
denoted with primes, about the base state:
p
(
x
,t)
T
+
T
(
x
,t)
ρ
(
x
,t).
(8.7)
p
˜
=
p
0
(z)
+˜
;
=
T
0
(z)
;
ρ
˜
=
ρ
0
(z)
+˜
The deviations also have tildes because they have both a mean and a fluctuating
part.
