Geology Reference
In-Depth Information
v
r
μ
h
Δρ
μ
Figure 7.11. Sketch of a buoyant plume head of radius
r
rising with velocity
v
through a surrounding fluid of viscosity
μ
. The plume head has a density deficit
of
ρ
and a viscosity
μ
h
.
7.3.4 Rate of ascent of a plume head
A blob of buoyant viscous fluid rising slowly through another viscous fluid will
assume a spherical shape. This may seem plausible, but it is not so obvious when you
realise that a small bubble in water is spherical mainly because of surface tension.
Nevertheless, it is observed in experiments. A plume head is also observed to be
approximately spherical, though in the thermal plumes of numerical experiments
(Figure 7.7) the temperature gradients within the plume head distort it from the
spherical. Thus we may approximate a plume head as a buoyant sphere. We can
estimate its rate of rise by following the force balance approach that we have
already been using. Figure 7.11 is a sketch of a rising plume head.
The buoyancy force of the plume head is
4π
r
3
gρ/
3
.
B
=
The rising head drags some of the surrounding fluid up with it as it passes. Fluid
adjacent to the head will therefore rise with velocity
v
, whereas fluid one or two
radii away will have a smaller velocity. We may therefore use
v/r
as a useful
measure of the velocity gradients set up in the surrounding fluid by the passage
of the head. Therefore viscous shear stresses of the order of
μv/r
will act on
the surface of the sphere, generating a resistance force
R
. The area over which
the viscous stresses act is the surface area of the sphere, 4π
r
2
. Combining these
quantities gives
4π
r
2
μv/r.
R
=
Now requiring these forces to balance yields
v
=
gρr
2
/
3
μ.
(7.23)
Rigorous analyses yield a very similar result, depending on the viscosity within
the plume head. A solid or very viscous sphere rises with this velocity, whereas
a low-viscosity sphere rises 50% faster. Such differences are not important to our
rough estimates. Let us use some values appropriate to mid-mantle depths, where
viscosity is about 10
22
Pa s, density is about 4000 kg/m
3
and thermal expansion is
