Geoscience Reference
In-Depth Information
∞
2
−
2
2
π
r
4
cos
3
Δ
I
33
=
ρ
φ
d
λ
d
φ
d
r
(45)
r
s
0
∞
2
−
2
2
π
r
3
u
cos
2
h
3
=
ρ
φ
d
λ
d
φ
d
r
.
(46)
r
s
0
Integration Over Pressure Increments
If a vertical atmospheric pressure gradient is assumed on the basis of the hydrostatic
equation d
p
=−
ρ
g
d
r
, the radial integral in the AAM formulae can be written
in terms of pressure increments, see Barnes et al. (
1983
).
g
=
(
)
is the gravity
acceleration and
p
s
, introduced in the integration limits, denotes the surface pressure.
g
r
2
−
2
p
s
2
π
r
4
g
Δ
I
cos
2
e
i
λ
d
=−
sin
φ
φ
λ
d
φ
d
p
(47)
0
0
2
−
2
p
s
2
π
r
3
g
(
h
e
i
λ
d
=−
u
sin
φ
+
i
v
)
cos
φ
λ
d
φ
d
p
(48)
0
0
2
p
s
2
π
r
4
g
cos
3
Δ
I
33
=
φ
d
λ
d
φ
d
p
(49)
−
2
0
0
2
p
s
2
π
r
3
g
u
cos
2
h
3
=
φ
d
λ
d
φ
d
p
.
(50)
−
2
0
0
Integration Over Pressure Increments, Constant Radius and Gravity
Given the relatively small extension of any surficial fluid layer compared to Earth's
radius,
i
t is justified to model the atmosp
he
re as thin spherical shell with constant
radius
r
and constant gravity acceleration
g
. This thin layer approximation enables
us to directly reduce the pressure integral of the matter terms in Eqs. (
47
) and (
49
)
to the plain surface pressure
p
s
(Barnes et al.
1983
)
2
r
4
g
2
π
Δ
I
cos
2
e
i
λ
d
=−
p
s
sin
φ
φ
λ
d
φ
(51)
−
2
0
2
r
3
g
p
s
2
π
h
e
i
λ
d
=−
(
u
sin
φ
+
i
v
)
cos
φ
λ
d
φ
d
p
(52)
−
2
0
0
2
r
4
g
2
π
p
s
cos
3
Δ
I
33
=
φ
d
λ
d
φ
(53)
−
2
0
2
r
3
g
p
s
2
π
u
cos
2
h
3
=
φ
d
λ
d
φ
d
p
.
(54)
−
2
0
0
