Geoscience Reference
In-Depth Information
the type of problem that can be solved by using the two-dimensional version of
the heat-conduction equation in a moving medium (Eq. (7.19)). The boundary
conditions can be specified in a number of ways: these necessarily lead to different
solutions and thus to different estimates of heat flow and bathymetric depth. The
bathymetric depth is calculated from the temperature by assuming that the plate
is in isostatic equilibrium, and the heat flow is calculated from the temperature
gradient at the surface of the lithosphere. In this way, an understanding of the
thermal structure and formation of the plates has been built up. As in all scientific
work, the best model is the one which best fits the observations, in this case
variations of bathymetric depth and heat flow with age.
A simple model
The simplest thermal model of the lithosphere is to assume that the lithosphere
is cooled asthenospheric material, which, at the ridge axis, had a constant tem-
perature
T
a
and no heat generation. If we assume the ridge to be infinite in the
y
direction and the temperature field to be in equilibrium, then the differential
equation to be solved is
∂
∂
z
2
2
T
∂
x
2
2
T
k
ρ
c
P
+
∂
=
u
∂
T
∂
x
(7.59)
where
u
is the horizontal velocity of the plate and the term on the right-hand
side of the equation is due to advection of heat with the moving plate. A further
simplification can be introduced by the assumption that horizontal conduction
of heat is insignificant in comparison with horizontal advection and vertical
conduction of heat. In this case, we can disregard the
2
T
x
2
∂
/∂
term, leaving
the equation to be solved as
2
T
∂
z
2
ρ
c
P
∂
k
=
u
∂
T
∂
x
(7.60)
This equation, however, is identical to Eq. (7.43)ifwewrite
t
u
,which
means that we reintroduce time through the spreading of the ridge. Approximate
initial and boundary conditions are
T
=
x
/
=
T
a
at
x
=
0 and
T
=
0at
z
=
0. According
to Eq. (7.44), the solution to Eq. (7.60)is
T
(
z
,
t
)
=
T
a
erf
z
2
√
κ
t
(7.61)
The surface heat flow at any distance (age) from the ridge axis is then obtained
by differentiating Eq. (7.61):
z
=
0
Q
(
t
)
=−
k
∂
T
∂
z
kT
a
√
πκ
t
=−
(7.62)
The observed
t
1
/
2
relationship between heat flow and age is thus a feature of this
model which is called a half-space cooling model.
