Geoscience Reference
In-Depth Information
2.4 Statics of the Atmosphere
We assume a spherical, concentric Earth. The mutual attraction of the particles is
neglected as well as the thermodynamics. Thus, each gas is subject to only two
influences of opposite tendency:
•
the movement of the molecules, which tends to distribute the gases uniformly in
space (diffusion);
•
gravity, which tends to condense the particles close to the Earth surface.
If there are no effects like, e.g., turbulence, a stationary state will be reached where
the density of the gases decreases with height. Then, the gases are in hydrostatic
equilibrium and the sum of external and internal forces is equal to zero. The gases
have a pressure
p
, and the partial derivatives of the scalar function
p
are force
components per unit volume:
=−
f
grad p
(17)
or
f
x
dz
. In addition to the pressure,
we only have gravity. Equilibrium is reached if the sum of all forces acting on the
particles vanishes. If
g
is the vector of gravity, we get
=−
dp
/
dx
,
f
y
=−
dp
/
dy
, and
f
z
=−
dp
/
f
+
ρ
g
=
0
(18)
or when
U
denotes the gravity potential
grad p
=
ρ
·
grad U
.
(19)
Multiplying Eq.
19
with the vector
dr
=
(
dx
,
dy
,
dz
)
, we get the equation for
hydrostatic equilibrium:
dp
=
ρ
·
dU
.
(20)
0, these surfaces are also
surfaces of constant pressure and density. From potential theory we know that the
relation between potential, gravity and height is:
Since equipotential surfaces are characterized by
dU
=
dU
=−
g
·
dh
(21)
so that
=−
ρ
·
·
.
dp
g
dh
(22)
This equation holds if the air moves along straight horizontal lines without accel-
eration. In the following case for wet air, we eliminate density with the gas law using
the virtual temperature (Eq.
9
) and get with Eq.
22
:
dp
p
=−
g
R
d
·
dh
.
(23)
T
v
