Geology Reference
In-Depth Information
be used to further simplify (
13.114
). In fact,
the buoyancy term associated with pressure
fluctuations
p
0
in (
13.125
) is small relative to the
contribution coming from thermal fluctuations
T
0
.
Therefore, (
13.114
) can be rewritten as follows:
where
T
0
is the maximum change of the con-
ductive temperature variation
T
0
over the distance
H
. In agreement with (
13.103
) we also assume
that:
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
O.©/
T
0
T
(13.137)
¡
0
¡
Š
'T
0
(13.131)
The
re
fore, because
j
T
0
j
T
0
, it follows that
T
0
=T
O.©/.Now,using(
13.111
) we can
write, to order
O
(-):
This is the simplified equation of state in the
Boussinesq approximation. Now let us turn to
the equation of conservation of energy (
13.40
).
Neglecting the viscous dissipation and internal
heat production terms, this equations assumes the
form:
@¡
@T
p
D
'T
¡
T
¡
'T
D
¡
1
C
T
0
T
T
0
T
C
1
C
Š
'T (13.138)
¡c
p
T
'T p
D
k
r
D
'T
2
T
(13.132)
¡
0
¡
¡
0
¡
C
Substituting (
13.98
)for
T
, using the definition
(
13.2
) of material derivative, and dividing by ¡
c
v
gives:
Substituting (
13.138
)and(
13.134
) into the
energy Eq. (
13.133
) gives the final form of the
conservation of energy in the Boussinesq approx-
imation.
v
@p
0
@
z
C
p
0
T
0
C
v
@T
0
'T
¡c
p
2
T
@
z
D
r
(13.133)
T
0
C
v
@T
0
!
'Tg
c
p
2
T
0
@z
D
›
r
(13.139)
The variation of the hydrostatic pressure
p
0
with depth is simply ¡
g
. The other term in brack-
ets represents the time variation of pressure fluc-
tuation and can be neglected. The other variable
quantities in (
13.133
)aretheterm'
T
at the left-
hand side and the diffusivity ›
D
k
/(¡
c
p
)atthe
right-hand side (assuming
k
constant). To order
O
(©), by (
13.116
) we can set:
The term in brackets at the left-hand side of
this equation represents the static temperature
gradient in excess of the adiabatic gradient (see
be ensured that the thickness
H
of the fluid layer
be much less than the smallest among the scale
heights (
13.99
). Furthermore, the fluctuations of
temperature, pressure, and density induced by
fluid motions should not exceed the total varia-
tions of these quantities under static conditions.
We also note that temperature and velocity are
coupled through the equations of motion (
13.130
)
and the conservation of energy (
13.139
). This is
a consequence of the fact that in thermal convec-
tion the velocity field is governed by temperature
fluctuations, which in turn depend from velocity
through the advection of heat.
Numerical solutions to the governing
equations in the Boussinesq approximation have
been found by several authors, among which we
mention Turcotte and Oxburgh (
1967
), Richter
k
¡c
p
›
Š
›
D
(13.134)
Regarding the term '
T
, by hypothesis
H
T
H
¡
>>
H
. Accordingly,
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
1
T
dT
0
d
z
<<
1
H
(13.135)
Then, integrating from
z
D
0to
z
D
H
we
conclude that:
T
0
T
D
O.©/
(13.136)
