Geoscience Reference
In-Depth Information
So far we have assumed that there is no relative motion between the small
volume of material and its immediate surroundings. Now consider how the tem-
perature of the small volume changes with time if it is in relative motion through
aregion where the temperature varies with depth. This is an effect not previously
considered, so Eq. (7.13) and its three-dimensional analogue, Eq. (7.15), must be
modified. Assume that the volume element is moving with velocity
u
z
in the
z
direction. It is now no longer fixed at depth
z
; instead, at any time
t
, its depth is
z
+
u
z
t
. The
∂
T
/∂
tinEq. (7.13) must therefore be replaced by
∂
T
∂
t
+
d
z
d
t
∂
T
∂
z
The first term is the variation of temperature with time at a fixed depth
z
in the
region. The second term
d
z
d
t
∂
T
∂
z
is equal to
u
z
∂
z
and accounts for the effect of the motion of the small volume
of material through the region where the temperature varies with depth. Equations
(7.13) and (7.15) become, respectively,
∂
T
∂
t
=
T
/∂
ρ
c
P
∂
k
2
T
∂
z
2
A
ρ
c
P
−
u
z
∂
T
∂
z
+
(7.18)
and
∂
T
∂
t
=
k
ρ
c
P
∇
A
ρ
c
P
−
u
·∇
T
2
T
+
(7.19)
In Eq. (7.19),
u
is the three-dimensional velocity of the material. The term
u
T
is the
advective-transfer
term.
Relative motion between the small volume and its surroundings can occur
for various reasons. The difficulty involved in solving Eqs. (7.18) and (7.19)
depends on the cause of this relative motion. If material is being eroded from
above the small volume or deposited on top of it, then the volume is becoming
nearer to or further from the cool surface of the Earth. In these cases,
u
z
is the rate
at which erosion or deposition is taking place. This is the process of advection
referred to earlier. On the other hand, the volume element may form part of a
thermal-convection cell driven by temperature-induced differences in density. In
the latter case, the value of
u
z
depends on the temperature field itself rather than
on an external factor such as erosion rates. The fact that, for convection,
u
z
is
a function of temperature means that Eqs. (7.18) and (7.19) are nonlinear and
hence significantly more difficult to solve (Section 8.2.2).
·∇
7.2.2 Radioactive heat generation
Heat is produced by the decay of radioactive isotopes (Table 6.2). Those radioac-
tive elements which contribute most to the internal heat generation of the Earth
are uranium, thorium and potassium. These elements are present in the crust in
