Physical Volcanology
Extra Credit
Due at Exam
1. You are tasked with considering different aspects of a rhyolitic, central vent eruption. For the prefragmentation portion of the conduit, assume you can treat the flow pattern using the fissure equation
derived in class:
v
1  P
  x2 w 


g
'
m
  2  2 x ,
  y


(1)
where the viscosity,  , is 106 Pa s, conduit width, w, is 100 m, magma density is 2400 kg/m 3, and the
excess pressure gradient is -26730 Pa/m. Assume that the solubility of water (dissolved water content) is
given by




H 2 0(wt%)  2.663  1016 P 2  6.28  108 P for pressures less than 1 x 108 Pa.
The total original water content of the magma is 3.5 wt. %. Assume the magma is approximately 1000 K.
A (5 pts) At what depth do bubbles start to form in this magma? Use magma density to go between
pressure and depth.
B. (15 pts) What is the velocity profile of magma at 1000 m? Graph your results. (Hint, you’ll need to
determine the weight percent exsolved water, then convert this to volume fraction using the ideal gas
relation to estimate gas density).
C. (10 pts) If the fragmentation criteria is .75 volume fraction of gas, what depth does the magma
fragment?
D. (10 pts) Use equation (1) to derive a relation for absolute value of the shear stress as a function of x.
E. (5 pts) Assume a critical shear stress is the criteria for fragmentation. Will the center or the near wall
regions be the first to fragment?
F. (5 pts) If the critical shear stress is
1  10 6 Pa where will the magma fragment?
2. (50 pts) Create a regime diagram for collapsing versus buoyant volcanic columns. Plot different
combinations of vent width and volume fraction of particles. If the column collapses with these parameters,
plot the point as a darkened circle. If the column is buoyant plot as an open circle. After calculating this for
several points (6 or so will work if you select well) draw a line separating the collapsing and buoyant fields.
Write down your equations and show your thought process, but feel free to use a spreadsheet for repeated
calculations.
In this problem, assume that the eruptive temperature is 1100 K, particle density is 3000 kg/m 3,the magma
has fragmented into very small ash (complete atomization assumption), the volcanic jet is pressure balance,
and the eruptive mixture exiting the conduit is choked (is at the mixture sound speed). You can also assume
that the only volatile in the eruption is water vapor. Assume that the background atmosphere is dry and has
a specific gas constant of 287 J/K kg. Assume all gas heat capacities are 1004 J/K kg. The atmospheric
pressure at zero height is 105 Pa and 300 K. You can assume an entrainment coefficient in the volcanic jet
of .1, and that the volcanic jet loses no heat and has a constant rise velocity in the jet thrust region.
(Hint. I suggest you first determine the pressure and density profile of the background atmosphere. Then
calculate the sound speed for a given volume fraction of particles, the jet thrust height, and then the
mixture density of the volcanic plume at this height. If the plume is denser than the atmosphere at the jet
thrust height it will collapse.)