Page 58 - Fundamentals of Physical Volcanology

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MAGMA MIGRATION 35

multiplied by the square root of length, and in prac- region, and so ∆P must eventually decrease slowly.
tice values can range from a few million Pa m 1/2 for More important is the fact that ∆ρ will change a
m
rocks being fractured in the laboratory with no great deal if the dike tip rises into the relatively low
volatiles present inside the fracture to at least many density crust, and so we need to consider the issue
tens of millions of Pa m 1/2 for rocks at the tips of of the density difference between the magma and
dikes within the Earth. the surrounding rocks in more detail.
Tens of millions of Pa m 1/2 sounds like an impres- If a fracture starts some way below the top of a
sive number, but the size of the fracture toughness mantle plume head, then as it grows upward the
can be put in context by considering the stress surrounding rocks get slightly denser for a while
intensity at the tip of a dike due to a given internal due to their decreasing temperature (although this
excess pressure. Consider the head of a mantle is offset to some extent by the opposite effect of the
plume in which melting is occurring over a vertical decreasing pressure). The result is a slight increase
distance of H = 3 km. This would cause a few per- in the buoyancy of the magma. Eventually, when
cent of melt to be produced, enough to form a net- the base of the crust is reached (or at once if
work of interconnected veins which would allow the fracture starts at the base of the crust), the sur-
the melt to be extracted into a fracture if one rounding rocks become much less dense than the
formed. The effective excess pressure, ∆P, defined rocks of the mantle. However, this does not auto-
as the pressure in excess of the local lithostatic rock matically mean that they are less dense than the
load, in the melt in the middle of the 3 km high magma in the dike. Thus there are two possibilities:
zone of liquid is proportional to the buoyancy of if the crustal rocks are denser than the magma then
the melt and is given by the magma simply becomes less buoyant than it
was in the mantle, but if the crustal rocks are less
∆P =∆ρ g (H/2) (3.2)
m dense than the magma then it ceases to be buoyant
at all, in other words it becomes negatively buoy-
−2
where g is the acceleration due to gravity, ∼9.8ms ,
ant. There are two ways, which we now discuss, in
and ∆ρ is the amount by which the melt is less
m which this loss of magma buoyancy can lead to the
dense than the surrounding mantle rocks from
cessation of the upward growth of the dike, in
−3
which it is forming, typically ∼300 kg m . Thus ∆P
other words, to its trapping in the crust.
would be about 4.4 MPa. If a fracture starts to form
at the top of the melt zone, the stress intensity

at this point, available to overcome the apparent 3.5 Trapping of dikes
fracture toughness of the solid rock in front of the
fracture tip, would be K, which is given by
1 Stress traps. As soon as the magma in a growing
K =∆P (H/2) 1/2 + 0.5 g ∆ρ (H/2) 3/2 (3.3) dike passes through the level at which the magma is
m
neutrally buoyant, and thus becomes negatively
1/2
Using the above values, K =∼255 MPa m , more buoyant, the stress at the upper tip of the dike will
than enough to guarantee that fracture growth, decrease as it continues to grow upward, because
and hence the formation of a dike, does in fact the second term in eqn 3.3 changes sign and
get started, even for the largest likely value of the becomes negative. The details are quite complic-
effective fracture toughness. Furthermore, as the ated because, now that ∆ρ is not constant along
m
fracture extends and H gets larger, then as long as the dike, eqn 3.3 must be replaced by an integral
∆P and ∆ρ do not change significantly the stress equation which adds up the contributions to the
m
intensity at the fracture tip gets larger, making con- stress at the tip from each small vertical segment of
tinued growth of the dike ever easier. Of course in the dike using the local value of ∆ρ . Eventually the
m
practice, if the dike grows very far, both ∆P and stress intensity at the tip falls to a value that is
∆ρ will begin to change. Movement of magma into smaller than the fracture toughness and upward
m
the dike removes that magma from the source growth of the dike must cease.