Page 119 - Fundamentals of Physical Volcanology

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96 CHAPTER 7

in the mixture that is expelled is larger. Taken sient explosions assumed that the air ahead of the
together, these factors mean that it is unlikely that expanding gas and magma was stationary and thus
the magma can buffer the gas temperature to any that a large amount of energy from the explosion
significant extent. Instead the expansion is likely to would be used in overcoming the air drag acting
occur nearly adiabatically. In this case the energy on individual pieces of solid material. However,
equation can be written as observations of transient explosions show that they
commonly send a shock wave outwards from the
nQT γ G 1 P 5 (γ−1)/γ J (1 − n) explosion point slightly ahead of the expanding
i H 1 − 4 f 4 K + (P − P ) cloud of gas and solids. This shock wave pushes the
m γ − 1 I 3 P i 7 L ρ m i f surrounding air out of the way and so, in the early
stages of the explosion, instead of thinking in terms
= 1 (U − U ) + gh + friction (7.1) of drag forces acting on individual solid fragments
2
2
2 f i we have to evaluate the energy used to displace the
atmosphere as a whole.
where n is the gas mass fraction, Q is the universal When all of these factors are taken into account,
gas constant, m is molecular weight of the gas, T is the results can be summarized as in Fig. 7.1, which
i
the initial temperature of the gas and magma, γ is shows the maximum speed of the ejected solid
the ratio of the specific heats of the gas, P is the material as a function of the pressure which built
i
initial gas pressure, P is the final pressure (equal to up in the trapped gas before the explosion and
f
the atmospheric pressure, P ), ρ is density of the the weight fraction which the gas represents of all
a m
magmatic material, U is the initial velocity of the of the materials ejected. Recall that the typical dis-
i
gas (which is essentially zero since the magma rise solved volatile contents of basaltic magmas likely to
speed is very small), U is the final velocity of the gas be involved in Strombolian explosions are less than
f
at the end of the expansion phase, g is the accele- 1 wt% and the volatile contents of the more evolved
ration due to gravity, and h is the vertical distance magmas commonly involved in Vulcanian explo-
risen by the magmatic material during the expan- sions are up to a few weight percent. The curves in
sion. Just as in eqn 6.4, the left-hand side of the Fig. 7.1 are given for much larger volatile contents
equation represents the energy released by expan- because of the expected accumulation of gas before
sion of the gas (in this case adiabatically) while the the explosion. The range of pressures used in the
right-hand side represents the three ways in which calculations is cut off at 10 MPa because no rocks

this released energy is used in the eruption. The are strong enough to allow pressures this large to
first term represents the change in the kinetic accumulate before they break.
energy of the system, in other words the energy Figure 7.1 shows the expected basic relationship
used to accelerate the gas and magma in the explo- between skin/plug strength and eruption velocity:
sion. The second term represents the energy used i.e., that the highest velocities will correspond to
to raise the magmatic material within the gravity the greatest strengths. Observations of Strombolian
field while it is being accelerated. The final term is eruptions at Heimaey and Stromboli give clast
concerned with the friction between the gas and velocities of ∼150ms −1 as typical for Heimaey
−1
magmatic material and the air through which it is (with a maximum of 230 m s ) and 50–100 m s −1
passing, i.e., the air drag. as typical for Stromboli. Estimates have been made
The treatment of the air drag is a critical issue in both cases of the weight percentage of gas in
in modeling these eruptions. When an explosion these eruptions: the minimum gas content in both
occurs the gas and magmatic materials move up- cases is ∼11 wt% and ranges up to values of 36–38
wards and outwards from the point of the explo- wt%. These combinations of ejecta velocity and gas
sion. In doing so they must push the surrounding content suggest that the strength of the skin prior
air out of the way, and the resistance of the air to to rupture is < 0.3 MPa. Such low “skin” strengths
this pushing is the air drag. Initial models of tran- are consistent with the idea that the skin is still