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+
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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