Page 122 - Fundamentals of Physical Volcanology

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TRANSIENT VOLCANIC ERUPTIONS 99

3
F = (4/3) π R ρ g (7.3) ranges of blocks of different sizes. For a given block
g b
size, initial speed, and launch angle from the hori-
where g is the acceleration due to gravity, about zontal, eqn 7.4b is used to find the total accel-
9.8ms −2 on Earth. These forces each produce an eration caused by drag. This total acceleration is
acceleration in the block given by Newton’s third broken down into its vertical and horizontal com-
law of motion, that the acceleration is the force ponents and the acceleration due to gravity is
divided by the mass of the block. The mass of a added to the vertical component. Next, the vertical
3
spherical block is [(4/3)πR ρ ], so eqn 7.3 is just a and horizontal accelerations are used to find how
b
statement of Newton’s law. Equation 7.2, however, much the vertical and horizontal speeds change in
shows that the air drag acceleration, A, is given by some small time interval, and the average speeds
multiplied by the time interval give the vertical and
2
3
2
A = (0.5 ρ C π R U )/[(4/3) π R ρ ] (7.4a) horizontal distances traveled. Thus we have a new
a d b
block position, speed and direction, and therefore
which simplifies to can repeat the calculation for another small time
interval. This procedure is repeated a large number
2
A = (3 ρ C U )/(8 R ρ ) (7.4b) of times until the block finally reaches the ground.
a d b
Here this process is illustrated by looking at the
The drag force F always acts in exactly the oppo- ranges of large blocks in four well-documented
d
site direction to that in which the block is moving, transient explosions and inferring what the erup-
but the gravity force F always acts straight down tion conditions must have been to allow these
g
toward the center of the Earth, and this makes cal- blocks to be ejected to their observed final posi-
culating the path of a block quite complicated. But tions (Table 7.1).
as an illustration consider two blocks of radii 1 m Consider the ejection of large blocks in the two
−3
and 0.1 m, both with density ρ = 2000 kg m , trav- Vulcanian explosions (Arenal and Ngauruhoe) first.
b
−1
eling straight upward at a speed of 100 m s relative In both cases the vent pressures and ejecta veloci-
to the air. The air density near the Earth’s surface is ties are inferred to be high and are consistent with
−3
about 1 kg m , and so eqn 7.4b shows that the accel- the range inferred for Vulcanian explosions from
erations (actually decelerations since the blocks are Fig. 7.1. In both cases the inferred gas content is
slowing down) are about 1.3 m s −2 for the larger moderate (4–6 wt%) but, as these are andesitic
−2
block and about 13 m s for the smaller one. eruptions, it probably represents a small amount

These numbers have to be compared with the of gas segregation prior to eruption. These results
−2
downward accelerations due to gravity, g = 9.8ms , contrast, as expected, with those for the two
that the blocks will each be feeling at the same Strombolian eruptions (Heimaey and Stromboli) in
time. Thus the large block notices only a 13% which inferred eruption speeds and vent pressures
difference from the situation if there were no are lower. In both Strombolian eruptions the
atmosphere around it, whereas the smaller block inferred gas content is ∼20 wt%, implying consider-
experiences a 130% difference! Of course, for a able gas concentration prior to eruption. These
block launched at an angle to the vertical, the drag values are consistent with the idea that gas segrega-
force will change in size and direction as the speed tion and concentration is easier and greater in the
of the block changes in both the upward and side- basaltic magmas of Strombolian eruptions than in
ways directions, but this simple calculation gives the more evolved magmas commonly involved in
a good idea of why meter-sized blocks travel on Vulcanian explosions.
nearly ballistic trajectories, similar to those they The final three columns in Table 7.1 allow a com-
would have if there were no atmosphere around parison to be made between the amount of energy
them, while ejected blocks less than a fraction of a used in accelerating the ejected material, in raising
meter in size travel only to much smaller distances. the material in the gravitational field and in over-
The above ideas can be used to simulate a given coming air drag, i.e., the partitioning of energy
set of eruption conditions and predict the maximum depicted in the energy equation (eqn 7.1). Note