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78 CHAPTER 6

dike system. The dike has length l and width t, and Table 6.1 Basaltic magma with a total gas content of 2 wt%
the gas–magma mixture flowing through it has a water rises through a dike of width 2 m and horizontal
bulk density ρ and rises at a speed u. The mass flux, length 2 km. The rise speed before any gas exsolves is
B 1ms and thus the mass flux through the dike system is
−1
M, through the dike is then given by
7
−1
1.12 × 10 kg s (eqn 6.1). As the magma rises, gas exsolves
from the magma according to the solubility law (eqn 5.1)
M = tl ρ u (6.1) and the bulk density of the gas–magma mixture decreases
B
f
(eqn 6.2). The rise speed, u, therefore increases (eqn 6.1).
The bulk density, ρ , of the gas–magma mixture ρ ρ (kg m ) −1
−3
B Depth (km) P (Mpa) B u (m s )
is given by
4 110.0 2800 1.0
2 55.0 2613 1.1
1 = n f + (1 − n ) 1 27.5 1758 1.6
f
ρ ρ ρ (6.2)
B g m 0.5 13.7 150 18.7
where n is the exsolved weight fraction of gas, ρ
f g
is the gas density and ρ is the density of the mag-
m speed of the magma is to increase, in other words
matic liquid.
if it is to accelerate, something must provide the
The gas density, ρ , varies with temperature and
g increase in kinetic energy (energy of movement)
pressure and usually can be approximated with
that this represents. Also, the magma is being lifted
reasonable accuracy by the perfect gas law:
in the Earth’s gravitational field, so some potential
energy must be supplied as well, and something
ρ = Pm (6.3) must compensate for the energy that the magma
g
QT is losing as a result of friction with the walls of
the dike.
where P is the pressure, m is the molecular
weight of the gas, Q is the universal gas constant
−1
(8310 J kmol ) and T is the absolute tempera- 6.3 Acceleration of the gas–magma mixture
ture of the gas, its temperature relative to the
absolute zero of temperature at about −273.15°C. The energy for the acceleration is provided by

Before any bubbles form within the magma, the the expansion of the gas which occurs as the
bulk density, ρ , is equal to the magma density, ρ . gas–magma mixture rises and the pressure on it
B m
As bubbles start to form, the bulk density of the decreases. The effect can be illustrated by thinking
gas–magma mixture decreases because the den- about what happens when a tire on a bicycle is
sity of the gas is much less than the density of the pumped up. The process forces air into the tire,
magma. Equation 6.1 shows that, if the mass flux, compressing the air and raising the pressure inside
M , stays constant throughout the dike system, then the tire. To do that takes energy, and with a hand-
f
as the bulk density of the gas–magma mixture pump the cyclist supplies that energy – it ultimately
decreases, the rise velocity, u, of the mixture must comes from the food the person eats! Anyone who
increase to compensate. Table 6.1 shows a simple has pumped up a tire in this way will recall that the
example to illustrate this effect. body of the pump becomes slightly warm – some of
As long as the decrease in the pressure acting the energy supplied goes into heating the gas. If,
on the magma is not too great, the main effect of after pumping-up the tire, the end of the pump is
the gas expansion is simply the one illustrated in released, the plunger is pushed out again by the
Table 6.1. However, as the magma approaches the gas in the tire. This happens because the pressure
surface, and the pressure decrease becomes very inside the tire is higher than outside. If there is noth-
large, there are other factors that must be taken ing stopping the air inside from expanding, then
into account. These are related to the fact that if the the air will expand to try to equalize the pressure