Geoscience Reference
In-Depth Information
Now we form the derivatives of the extended Stokes function (6-22) with
respect to
r
and
ψ
. By differentiating (6-23), we get
∂l
∂r
=
r − R
cos
ψ
∂ψ
=
Rr
∂l
,
sin
ψ.
(6-31)
l
l
By means of these auxiliary relations, we find
∂S
∂r
R
(
r
2
R
2
)
−
4
R
rl
−
r
2
+
6
Rl
R
=
−
−
rl
3
r
3
cos
ψ
13 + 6 ln
r
,
+
R
2
r
3
−
R
cos
ψ
+
l
2
r
(6-32)
∂ψ
=sin
ψ
2
R
2
r
l
3
6
R
2
rl
+
8
R
2
r
2
∂S
−
−
r
2
r
.
+
3
R
2
−
R
cos
ψ
−
l
+ln
r
−
R
cos
ψ
+
l
2
r
l
sin
2
ψ
Somewhat more convenient expressions are obtained by substituting
t
=
R
r
,
(6-33)
=
1
l
r
D
=
−
2
t
cos
ψ
+
t
2
.
(6-34)
Then the extended Stokes function (6-22) and its derivatives (6-32) become
S
(
r, ψ
)=
t
2
t
cos
ψ
5+3ln
1
,
−
t
cos
ψ
+
D
2
D
+1
−
3
D
−
(6-35)
1
t
2
R
t
2
D
3
∂S
(
r, ψ
)
∂r
−
4
D
+1
=
−
+
−
6
D
− t
cos
ψ
13 + 6 ln
1
,
−
t
cos
ψ
+
D
2
(6-36)
=
−t
2
sin
ψ
2
∂S
(
r, ψ
)
∂ψ
6
D
−
8
D
3
+
.
3
1
−
t
cos
ψ
−
D
3ln
1
−
t
cos
ψ
+
D
2
−
−
D
sin
2
ψ
These expressions are used in (6-21) and (6-30) to compute
T
and
δ
g
.
The separation
N
P
of the geopotential surface through
P, W
=
W
P
,and
the corresponding spheropotential surface
U
=
W
P
is according to Bruns'
theorem given by
=
T
P
γ
Q
N
P
;
(6-37)
