Geography Reference
In-Depth Information
9.80665 m s
−
2
is
the global average of gravity at mean sea level. Thus in the troposphere and lower
stratosphere,
Z
is numerically almost identical to the geometric height z. In terms
of
Z
the hypsometric equation becomes
Here Z
≡
(z)/g
0
,isthe
geopotential height
, where g
0
≡
p
1
R
g
0
Z
T
≡
Z
2
−
Z
1
=
Tdln p
(1.22)
p
2
where Z
T
is the
thickness
of the atmospheric layer between the pressure surfaces
p
2
and p
1
. Defining a layer mean temperature
p
1
Tdln p
p
1
p
2
d ln p
−
1
T
=
p
2
≡
and a layer mean scale height H
R
T
/g
0
we have from (1.22)
Z
T
=
H ln(p
1
/p
2
)
(1.23)
Thus the thickness of a layer bounded by isobaric surfaces is proportional to the
mean temperature of the layer. Pressure decreases more rapidly with height in a
cold layer than in a warm layer. It also follows immediately from (1.23) that in an
isothermal atmosphere of temperature
T
, the geopotential height is proportional to
the natural logarithm of pressure normalized by the surface pressure,
Z
=−
H ln(p/p
0
)
(1.24)
where p
0
is the pressure at Z
0. Thus, in an isothermal atmosphere the pressure
decreases exponentially with geopotential height by a factor of e
−
1
per scale height,
=
p(0)e
−
Z/H
p(Z)
=
1.6.2
Pressure as a Vertical Coordinate
From the hydrostatic equation (1.18), it is clear that a single valued monotonic
relationship exists between pressure and height in each vertical column of the
atmosphere. Thus we may use pressure as the independent vertical coordinate and
height (or geopotential) as a dependent variable. The thermodynamic state of the
atmosphere is then specified by the fields of (x, y, p, t) and T (x, y, p, t).
Now the horizontal components of the pressure gradient force given by (1.1)
are evaluated by partial differentiation holding z constant. However, when pres-
sure is used as the vertical coordinate, horizontal partial derivatives must be
