Geology Reference
In-Depth Information
The variations in χ are unrestricted, except by the essential boundary conditions of
the problem. Stationarity of the functional
F
then implies that
∇·
(
f
u
)
=
0andthat
the
natural boundary condition
, (
u
0, holds as well.
The functional in the form (8.7) may be expanded, by substitution for
T
−
u
B
)
·
n
=
·∇
χ
=
u
from (7.16), to give
2
f
k
·∇
χ
k
·∇
χ
∗
d
∇
χ
·∇
χ
∗
d
fB
ψψ
∗
d
F =−
σ
f
V+
V+
V
V
V
V
2
σ
1
·
∇
χ
×∇
χ
∗
d
f
k
2
−
i
σ
V+
4π
b
σ
−
Σ
S
,
(8.19)
V
where ψ
=
C
·∇
χ/
B
and we have used the vector identity
∇
χ
∗
·
k
×∇
χ
·
∇
χ
×∇
χ
∗
.
k
=
(8.20)
The stability, or otherwise, of the density profile is represented by the Brunt-
Vaisala frequency defined by (6.167). It is convenient to work with the dimension-
less form of this quantity,
N
=
ω
v
/2
Ω
. Expression (7.15) for
B
then takes
the form
1
N
2
σ
2
2
σ
1
k
s
2
2
0
2
B
=−
g
−
+
·
−
σ
,
(8.21)
with the unit vector
s
in the orthometric direction (opposite to gravity, see Sec-
tion 5.3). The vector quantity
C
, defined by expression (7.4), becomes
C
=−
g
0
k
·
s
k
+
i
σ
2
s
.
k
×
s
−
σ
(8.22)
k
The scalar product
·
s
=
cosθ, whereθ is the geographic co-latitude. For
z
=
cosθ,
0
ζ
z
2
, where
2
2
1
we have
B
=
g
−
2
1
2
−
σ
−
1
1
=
σ
ζ
.
(8.23)
N
2
θ
1
z
2
1/2
,and
k
k
θ
sinθ
=
φ
sinθ
The unit vector
=
s
cosθ
−
s
z
−
−
×
s
=
z
2
1/2
, for unit vectors
θ
and
φ
in the direction of increasing geographic
co-latitude and in the azimuthal direction, respectively. Neglecting corrections of
the order of the flattening (see Section 5.2) in expressing the gradient in geograph-
ical co-ordinates, we have
φ
1
=
−
1
z
2
1/2
r
0
−
s
∂χ
∂χ
∂
z
+
im
θ
φ
∇
χ
=
∂
r
0
−
1/2
χ,
(8.24)
r
0
1
z
2
−
with
r
0
representing the equivolumetric radius (see Section 5.3). We then find that
z
1
z
2
⎣
⎦
.
σ
−
z
2
∂χ
−
C
·∇
B
=
1
∂χ
∂
z
+
m
σ
r
0
χ
2
ψ
=
g
0
ζ
1
−
z
2
∂
r
0
−
(8.25)
r
0
2
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