Page 102 - Fundamentals of Physical Volcanology

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STEADY EXPLOSIVE ERUPTIONS 79

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difference and in doing so will provide energy to u du =− dP − gdh − fu dh
push the plunger out. Also, as the compressed gas ρ B t (6.5)
expands, it will get a little cooler. This is not very
Momentum Pressure Gravity Friction
noticeable in the case of the bicycle pump if only a
little of the pressure is released, but if all of the air
is let out of the tire it will be more obvious. Going Here account is taken explicitly of the friction
back to our rising gas–magma mixture, as the mix- between the magma and the wall of the dike. Both
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ture rises and the pressure decreases, the gas in the g dh and ( fu dh)/t are positive quantities, and so
bubbles expands, cools a little, and releases energy. the minus signs mean that they represent a loss
It is this energy that goes into raising and accelera- of kinetic energy of the magma; however, the pres-
ting the gas–magma mixture through the dike, and sure is decreasing as the magma rises, so dP is
the distribution of energy in the system is given by negative, so (−dP) is a positive number, and this
the energy equation: makes it clear that the decrease in pressure is the
key factor causing the eruption to happen.
(1 − n )dP The dependence of the friction factor, f, in
0 = f + g dh + u du + c dT (6.4) eqn 6.5 on viscosity is important. Magmas are much
ρ p
m
more viscous than, say, water, which has a viscosity
Liquid internal Potential Kinetic Gas internal of about 10 −3 Pa s. Basaltic magmas have typical vis-
cosities of 10 to 100 Pa s, whereas magmas such
The zero on the left-hand side of this equation as dacites and rhyolites can have viscosities as large
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just means that the total energy of the system does as 10 to 10 10 Pa s. Below the fragmentation level,
not change. The terms on the right-hand side are the rising mixture consists of magma with some gas
labeled by the components of the energy that they bubbles suspended within it, so the fluid in contact
represent. The rise speed of the mixture of gas and with the walls is magma (Fig. 5.9a–c). The magma
magmatic liquid is u and du is the change in the has a significant viscosity, and so the friction factor,
speed, so that u du is the increase in the kinetic f, is relatively large and the energy needed to over-
energy of the mixture. dh is the distance that the come wall friction is also large. After fragmentation,
mixture is lifted against the gravitational field of the the rising mixture consists of a stream of gas
planet represented by the acceleration due to grav- with pyroclasts – clots of magma – suspended in it
ity, g (about 9.8 m s −2 on Earth), and so g dh is the (Fig. 5.9d), and so the fluid in contact with the dike

potential energy needed to do this. The two inter- walls is now mostly gas, which typically has a viscos-
nal energy terms represent energy locked into ity of ∼10 −5 Pa s, which is ten million times smaller
the physical state of the materials; c is the specific than the viscosity of a typical basaltic magma and ten
p
heat at constant pressure of the mixture (the amount billion times smaller than the viscosity of a typical
of heat each kilogram releases in cooling by one rhyolite. Of course, from time to time clots of liquid
kelvin), and so c dT represents the change in ther- magma will collide with the wall, and so the effect-
p
mal energy of the system. Note that both u du and ive bulk viscosity of the gas–clot stream is greater
g dh increase as the magma rises toward the sur- than that of the gas alone. However, this bulk vis-
face. However, the pressure P decreases, so dP is cosity is so small compared with the viscosity of the
negative; it is not, therefore, trivial to predict what liquid magma that after fragmentation the friction
happens to the temperature T. In fact, except factor, f, becomes negligibly small. This means that
under very special circumstances, T decreases – the partitioning of energy in the energy equation
the magma cools by some amount – as it ascends. (eqn 6.4) changes significantly. Prior to fragmenta-
In parallel with the conservation of energy it tion, a lot of energy is used in doing work to over-
is necessary to consider how the forces acting come friction between the magma and dike walls,
on a given batch of rising magma influence its especially in narrow dikes. After fragmentation, the
motion, and this leads to the so-called momentum friction term becomes very much smaller, and so
equation: the energy which was used to overcome wall