Geology Reference
In-Depth Information
@p
0
@
z
¡
0
g
Now, assuming ǜ and œ constants in
the Navier-Stokes equations, substituting the
expressions (
13.98
) for the pressure and the
density at the right-hand side of (
13.16
), using the
approximation (
13.118
), and applying (
13.119
)
we obtain:
@p
0
@
z
C
¡
0
©
¡
0
1
¡
1
¡
g
D
@p
0
@
z
¡“p
0
g
1
¡
'T
0
g
D
@p
0
@
z
D
p
0
1
¡
1
'T
0
g
D
¡
P
v
Dr
p
0
C
ǜ
r
2
v
C
¡
0
gk
(13.125)
(13.120)
where:
This equation still contains the full density ¡.
Substituting (
13.98
) and applying the approxima-
tion (
13.116
) leads to the following simplified
equation of motion:
1
¡“g
D
(13.126)
has the dimensions of a length and can be re-
garded as the t
h
ickness of a fluid layer with
constant density ¡ and
h
ydrostatic pressure
p
that
varies from zero to 1=“. Substituting (
13.119
)in
(
13.99
)for
H
p
gives:
¡
0
©
¡
0
gk
1
¡
r
p
0
C
r
2
v
C
P
v
D
(13.121)
where
ǜ=¡ is the average (constant) kine-
matic viscosity. In this equation, the buoyancy
term is the only place where the infinitesimal
factor - is retained. This is a consequence of the
fact that convective motions arise from buoyancy
forces that are associated with fluctuations in the
density field. Therefore, the acceleration
j
@
v
/@
t
j
must have the same order of magnitude of the
acceleration associated with buoyancy:
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
p
.¡
C
¡
0
/g
LJ
LJ
LJ
LJ
1
1
1
p
dp
0
d
z
1
H
p
D
D
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
D
p
.¡
C
¡
0
/g
p
¡g
1
1
C
¡
0
=¡
D
p
¡g
C
O.©/
(13.127)
D
Therefore substituting into (
13.126
),
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
¡
0
g
LJ
LJ
LJ
LJ
¡
0
©
@
v
@t
H
p
p“
(13.122)
D
D
(13.128)
or,
I
n t
he Earth's upper mantl
e, “
10
12
Pa
1
and p
10
10
Pa, thereby p“
10
2
.Fur-
thermore, we have that in any case
p
0
/
H
@
p
0
/@
z
.
Thus, the quantity:
LJ
LJ
LJ
LJ
LJ
LJ
LJ
LJ
g
@
v
=@t
.¡
0
©=¡
0
/
(13.123)
p“
H
p
!
p
0
<<
p“
H
!
p
0
<
1
Therefore, in a convective system the accel-
eration of gravity is always much greater than
j
@
v
/@
t
j
. Equation (
13.121
) can be simplified fur-
ther taking the vertical component. If
v
is the
vertical component of the velocity, then:
1
D
p
0
D
H
p
0
(13.129)
is negligible compared to @
p
0
/@
z
. Consequently,
the Eq. (
13.121
) are simplified to:
@p
0
@
z
C
r
¡
0
©
¡
0
@
v
@t
C
v
r
v
D
1
¡
2
v
C
g
1
¡
r
p
0
C
r
2
v
'T
0
gk
P
v
D
(13.130)
(13.124)
Now let us consider the first and the last terms
at the right-hand side of (
13.124
). Using (
13.114
)
and (
13.102
) we obtain:
These
are
the
equations
of
motion
in
the
Boussinesq
approximation.
The
same
considerations
used
to
derive
(
13.130
)
can
