Convection - Mantle Convection for Geologists - page 47

T = 0

(b)

T m

(a)

T

t=0

d t

1

t

z

2

2d

T m

z

Figure 5.4. (a) A body of hot material placed against a cold boundary. (b) Sketch
of the temperature variation with depth, z , in crude approximation, as it develops
in time. The profile is shown at three times, as marked.

Figure 5.4(b). The logic of this is that heat will initially flow down the very
steep gradient at the surface, but heat will not flow at depth because the temper-
ature is still uniform down there: zero gradient gives zero heat flow, according to
Eq. (5.10). Thus heat will be removed from near the surface, but the surface will still
be at T 0, and deeper down the temperature will still be T m , so the profile must

connect these two temperatures. We are simplifying by assuming that the profile is
simply linear in this shallow range. A more realistic profile will be shown later. We
can repeat this logic: more heat will flow down the new, gentler gradient and the
cooling region will extend downwards, but at some greater depth the temperature
will still be uniform. Thus the temperature at a later time t 2 will look something
like the long-dashed profile shown in Figure 5.4(b).
The region where the temperature changes from the boundary value to the
interior value is called a thermal boundary layer. Thus it is a transitional region,
and typically the temperature gradient is relatively steep within it. We want to know
how the thermal boundary layer will develop with time.
Now, to get at the rate at which the temperature profile changes, we can ask
how long it has taken to go from the initial profile to that at time t 1 . We can get a
timescale by estimating a rate at which heat is being lost and combining that with
the amount of heat lost. The rate at which heat is being lost at time t 1 is

=

q 1 = KT m /d.

(5.11)

The rate of heat loss at earlier times will be greater than this, because the gradient
is steeper, so this is a lower bound . If we also know how much heat has been lost,
we can get an upper bound on the time it has taken to lose the heat.
The heat content of a body is determined by its temperature, its mass and a
material property called the specific heat , C P . This is the amount of heat it takes
to raise the temperature of a unit mass of the material by 1 ◦ C. For mantle rocks
this is around 1000 J/kg ◦ C. (The subscript P specifies that this is the specific