Page 55 - Fundamentals of Physical Volcanology

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32 CHAPTER 3

body of the diapir into the surrounding rock, and a larger density contrast than in the previous
this process becomes ever more significant as the example if its composition had been influenced by
diapir rises because the surrounding rock is cooler incorporation of hydrated ocean floor sediment, in
−3
at shallower depths. The heat added to the host which case ∆ρ might be about 100 kg m . In this
rocks reduces their viscosity and makes the ascent case the diapir rise speed U would be about 0.25
of the diapir somewhat easier than it otherwise millimeters per year, more than 1000 times slower
would be. However, the diapir itself pays a price for than the mantle plume head.
this in terms of the heat it loses: the cooler it
becomes, the less buoyant it will be.
3.3 The change from diapir rise to
To put some of these relationships in perspect-
dike formation
ive we can consider two cases of diapiric rise: the
upward movement of the head of a large mantle
plume that would produce a major hot spot such as There is good evidence that in continental environ-
that forming the Hawaiian volcano chain, and the ments diapirically rising mantle plumes stop rising
rise of a much smaller diapiric body above a sub- when they get close to the base of the continental
duction zone at a continental margin or in an island crust. This is because they are no longer buoyant:
arc. We first need to specify the buoyancy force the higher silica content of the crustal rock means
acting on the diapir, which we approximate as that it is less dense than the mantle rocks in the
a sphere of radius R, and the drag force that it plume even though the plume is hotter than the
experiences as a result of deforming the plastic crust. In a case like this, where a body of rock
mantle rocks surrounding it. The buoyancy force reaches a level at which it is less dense than the
3
3
is F = (4/3)πR g∆ρ, where (4/3)πR is the volume of rocks below it but more dense than the rocks above
the sphere, g is the acceleration due to gravity, and it, we say that a neutral buoyancy level has been
∆ρ is the amount by which the sphere is less dense reached. When the rise of material is halted in this
than its surroundings. The drag force is D = 4πηR way, a significant amount of heat can be transfer-
U, where η is the viscosity of the host rocks and U red by conduction into the crustal rocks above the
is the rise speed. This second formula is similar plume and, because their solidus temperature is
to the more familiar version D = 6πηRU giving the significantly lower than that of the plume material,
drag force on a rigid sphere moving through a fluid, the crustal rocks may melt to form rhyolites, and
but the differing constant is due to the fact that the rhyolitic melt produced may itself ascend diap-

there is significant circulation of the fluid inside the irically some way into the shallower crust above.
sphere (see Fig. 3.1). If we equate D to F we find In other cases, however, it may not be the change
that the diapir rise speed is in rock density at the base of the crust that limits the
rise of a plume; instead it is the strain rate that the
2
U = (R g ∆ρ)/(3 η) (3.1) plume imposes on the surrounding rocks through
which it rises. The strain rate is a measure of the
For the diapiric head of a large mantle hot spot rate of deformation of the host rocks, and is found
plume, R might be 400 km. With a mantle density, by dividing the speed of the rising plume head by its
−3
ρ, equal to 3300 kg m , a typical volume expan- diameter. Equation 3.1 shows that large diapiric
−1
sion coefficient of rock, α, equal to 3 × 10 −5 K and bodies rise faster than small ones in proportion to
a temperature difference, ∆T, between the inside of the square of their diameter. Thus, other things
the plume and its surroundings of 200 K, the density being equal, the strain rate is just proportional to
−3
difference would be ∆ρ = (ρα∆T ) =∼20 kg m . the diameter. Using the values from the previous
The plastic viscosity of the mantle, η, is of order section, the strain rate imposed on the surrounding
10 21 Pa s, giving a diapir rise speed, U, of ∼0.3 rocks by a mantle plume head would be about
s , whereas that due to a small sub-
meters per year. In marked contrast to this, a diapir 1.25 × 10 −14 −1
s .
in a subduction zone setting might have a radius duction zone diapir would be about 1.7 × 10 −15 −1
of order 5 km. It could well be driven upward by These values are roughly ten times and two times