Geology Reference
In-Depth Information
The highest point of the interface is depicted as rising at a velocity
v
(so
h
is
increasing with time). This motion will induce deformation in the fluid, and there-
fore viscous resistance. In Chapter 4, rate of deformation was measured by a
velocity gradient, which is equivalent to a strain rate. A representative velocity
gradient here is
v/w
, assuming the velocity falls away towards zero to the left or
right of the bulge. The resulting viscous stress is then
τ
=
μv/w.
This stress acts over the width of the bulge, so the total resisting force (per unit
length in the third dimension of this cross-section) is
R
=
wμv/w
=
μv.
(7.12)
In the mantle, flow is so slow that acceleration can be totally neglected, as we have
discussed previously. This means that there should be no net force acting on the
fluid, or in other words the two forces
B
and
R
should balance each other. Thus,
equating
B
and
R
,weget
μv
=
gρwh
or
v
=
(
gρw/μ
)
h.
(7.13)
This says that the velocity is proportional to
h
, or in other words that the higher
the bulge is, the faster it increases its height. This is a familiar equation in basic
calculus, and it signifies exponential growth (Appendix A, Section A.3). If the
bulge had an initial height
h
0
, then subsequently its height would be
h
=
h
0
exp(
t/τ
)
,
(7.14)
where
t
is time, exp denotes the exponential function and
τ
=
μ/gρw
(7.15)
has the dimensions of time. As explained in Section A.3,
τ
is related to a doubling
time:
τ
2
ln(2)
τ .
This means that the height
h
doubles every 0.693
τ
.
Let's pause to look at what this analysis means so far. It says, via Eq. (7.13),
that the higher the bulge, the faster it grows. It says the same thing via Eq. (7.15):
h
doubles with the passage of every time interval
t
=
=
τ
2
. In other words, once the
bulge starts to grow, it undergoes runaway growth: it is unstable. These equations
describe an
instability
. Although the buoyant layer moves very slowly at first, if its
upper interface is close to horizontal, any undulation with a horizontal scale similar
to
D
will undergo runaway growth, and the layer will eventually break up into a
series of rising blobs, much as we have seen in Figure 6.2.
