Convection - Mantle Convection for Geologists - page 41

are quite small, of the order of 1%. Thus we would not make much of an error if
we used the initial density instead of the final density in the denominator. Since it
simplifies the later mathematics quite a lot, we normally do that, and get
ρ
ρ 0 =− αT.

(5.3)

Now (5.3) is very similar in form to (5.2). The minus sign means that an increase
in temperature causes a decrease in density. Just to be clear, the approximation we
made, replacing the final density with the initial density in the denominator, causes
a 1% error in ρ , not a 1% error in ρ .Iftheerrorwere1%of ρ , that would imply
a 100% error in ρ , which would be serious, but that is not what we did.
In the upper mantle, α is about 3

10 − 5 ◦ C − 1 . Notice that the units are per
degree Celsius . When that is multiplied by a temperature difference in Eq. (5.3)
the result is dimensionless. This matches the left-hand side, which is a ratio of
densities and therefore also dimensionless.
The temperature in the upper mantle is about 1300 ◦ C. Because the temperature
at the Earth's surface is around 0 ◦ C, this means that a mantle rock taken from the
surface to the upper mantle would undergo a fractional change in density, using
Eq. (5.3), of

×

10 − 5 10 − 2 − αT =− 3 × × 1300 =− 3 . 9 ×

or 3.9%. This is the largest thermal density difference we will meet in mantle

convection. The absolute change in density is

−

130 kg/m 3 .
Notice that I have only been giving results to two significant figures. This is
sufficient, since the thermal expansion is not known to any greater accuracy. This is
another reason why the approximation made in getting to Eq. (5.3) is not a serious
problem.

− 0.039 × 3300 =−

5.2 Buoyancy
A rock is denser than water, and if you release the rock in water it will sink.
However, the downward force on the rock due to gravity is less than if the rock
were in air. You can compare the vertical pressure gradients inside and outside the
rock and reach this conclusion. However, the simple result is that gravity acts, in
effect, only on the difference between the density of the rock and the density of
water. It is an extension of Archimedes' principle: the net force is the weight of the
rock minus the weight of the water displaced by the rock.
Ice is less dense than water, and if you release a piece of ice under water it
will rise. In this case we say the ice is buoyant. In fluid dynamics, the technical
term buoyancy, B , refers to the force due to the action of gravity on the density