Geography Reference
In-Depth Information
the logarithm of pressure. In the
log-pressure coordinates
, the vertical coordinate
is defined as
z
∗
≡−
H ln (p/p
s
)
(8.41)
where p
s
is a standard reference pressure (usually taken to be 1000 hPa) and H
is a standard scale height, H
RT
s
/g, with T
s
a global average temperature. For
the special case of an isothermal atmosphere at temperature T
s
, z
∗
is exactly equal
to geometric height, and the density profile is given by the reference density
ρ
0
z
∗
=
≡
ρ
s
exp
−
z
∗
/H
where ρ
s
is the density at z
∗
=
0.
For an atmosphere with a realistic temperature profile, z
∗
is only approximately
equivalent to the height, but in the troposphere the difference is usually quite small.
The vertical velocity in this coordinate system is
w
∗
≡
Dz
∗
/dt
The horizontal momentum equation in the log-pressure system is the same as that
in the isobaric system:
D
V
/Dt
+
f
k
×
V
=−
∇
(8.42)
However, the operator D/Dt is now defined as
w
∗
∂/∂z
∗
D/Dt
=
∂/∂t
+
V
·∇
+
The hydrostatic equation ∂/∂p
α can be transformed to the log-pressure
system by multiplying through by p and using the ideal gas law to get
=−
∂/∂ ln p
=−
RT
which upon dividing through by -H and using (8.41) gives
∂/∂z
∗
=
RT /H
(8.43)
The log-pressure form of the continuity equation can be obtained by transforming
from the isobaric coordinate form (3.5). We first note that
w
∗
≡−
(H/p) Dp/Dt
=−
Hω/p
so that
pw
∗
H
∂w
∗
∂z
∗
−
w
∗
H
=
∂ (ρ
0
w
∗
)
∂z
∗
∂ω
∂p
=−
∂
∂p
1
ρ
0
=
