Geology Reference
In-Depth Information
Substituting into (
13.105
)gives:
¡
D
¡
h
1
'
T
T
C
“.p
p/
In the static conductive state,
p
0
D
T
0
D
¡
0
D
0.
Then:
¡
0
D
¡
'T
0
C
“p
0
(13.113)
@T
2
T
T
2
@
2
¡
1
2¡
C
¡
0
D
¡
'T
0
C
“p
0
(13.114)
@T@p
T
T
.p
p/
1
2¡
@
2
¡
C
Let us consider now the continuity Eq. (
13.7
).
Substituting (
13.98
) and solving for the diver-
gence of
v
gives:
C
:::
@
2
¡
@p
2
.p
p/
2
1
2¡
C
(13.107)
1
C
1
d
¡
0
¡
dt
¡
0
C
¡
0
(13.115)
1
¡
¡
0
¡
C
Therefore,
r
v
D
¡
D
'
T
T
C
“.p
p/
¡
¡
The factor within the brackets can be ex-
panded into a geometric series, giving:
@T
2
T
T
2
@
2
¡
1
2¡
C
C
::: (13.108)
1
C
1
¡
0
¡
C
2
¡
0
¡
¡
0
¡
C
¡
0
¡
¡
0
¡
C
¡
0
¡
N
o
w,
because
j
¡
0
j
¡
0
,
it
follows
that
D
1
¡
0
=¡
O
(©).
Consequently,
by
(
13.103
) t
results:
D
1
C
O.©/
(13.116)
¡
0
C
¡
0
¡
O.©/
¡
¡
¡
D
(13.109)
Therefore, using (
13.102
) we conclude that:
This relation implies that also the right-hand
side of (
13.108
)isoforder
O
(
©
). Therefore, we
must have:
dt
¡
0
C
¡
0
C
O
©
2
(13.117)
©
¡
0
d
r
v
D
Consequently, to the first order in ©,
@T
2
T
T
2
<O
©
2
I
1
2¡
@
2
¡
1
2¡
r
v
Š
0
(13.118)
@T@p
T
T
.p
p/ <O
©
2
etc:
(13.110)
@
2
¡
This surprising result says that to the first order
in - the velocity field has the same solenoidal
property of velocity fields associated with in-
compressible fluids. Such conclusion should not
lead to think that we are modelling the mantle
as an incompressible fluid. Equation (
13.118
)
only specifies a property of the velocity field in
the Boussinesq approximation for
compressible
fluids. Now let us turn to the equations of motion.
In absence of motion, we have that ¡
0
D
p
0
D
0,
thereby substitution of (
13.98
) into the vertical
component of Navier-Stokes equations gives:
Hence, to order - the expression (
13.107
) can
be rewritten as follows:
¡
D
¡
h
1
'
T
T
C
“.p
p/
i
(13.111)
This is a linearized version of the equation of
state in the Boussinesq approximation. Substitut-
ing (
13.98
)into(
13.111
)gives:
¡
0
C
¡
0
D
'¡
T
0
C
T
0
C
“¡
p
0
C
p
0
(13.112)
@p
0
@z
C
.¡
C
¡
0
/g
0
D
(13.119)
