|A meter sized bubble bursting|
above the vent of Stromboli (top)
and a jet (well-collimated) from a
typical explosion of such a bubble
Figure 1 in the Taddeucci et al paper
The Taddeucci et al. paper reports on direct measurements of pyroclasts ejected from Stromboli at velocities up to 405 m/s, during eruptions from the south-west vent area in 2009. The velocities were obtained from high-speed videos in which centimeter-sized fragments were tracked for 5-10 frames. The velocities cited are believed to be conservative for several reasons.
This velocity would correspond to a Mach number of 1.2 if the sound speed of air is taken as 330 m/s, and the authors point out that these velocities are "supersonic in ambient air." It isn't clear whether or not these pyroclasts were actually traveling through ambient air where measured because shock waves which would compress and heat the atmosphere were often observed, and it may also be possible that the blocks were in a steam cloud, in which case their Mach numbers might have been subsonic because the sound speed is higher in steam (e.g., 850 m/s at 1000 C). The authors do not discuss these possibilities.
The general picture that emerges for the eruption process is of the explosion of a large steam bubble that produces initially the high-velocity burst of small pyroclasts (cm -sized). Within tenths of a second, the velocities decrease and it is only later in the event that larger blocks and bombs are ejected. Only with the use of the high-speed cameras has there been resolution to detect the high-velocity phase of the eruption because lower-speed cameras only capture data later in the event when drag has already acted to slow down the small pyroclasts. The new measured values are a factor of four higher than previously measured. The authors were able to work out from the initial ejection pulses that the distance from the base of the pressurized gas pocket to the camera viewpoint at one of the vents (SW1) was 3-6 m, which corresponds to the size of the bursting bubbles driving this phase of the eruption. In contrast, application of the same analysis to the dominant pulses at the second vent, SW2, gave estimates of bursting length (similar to shock tube dimensions) of 100 m.
In many older models of reservoirs and the pressures that drive eruptions, including work of this blogger, reservoir pressures are taken as ambient (lithostatic or hydrostatic) pressures, and the pressure difference between the reservoir and atmosphere is assumed to drive the eruption. Alatorre-Ibarguengoitia et al. (2011 and 2010 below) have developed a conceptual model that says that the fragmentation of the magma itself may consume considerable energy, and that the kinetic energy available for driving pyroclasts is significantly less than from these older gas models. In their 2011 paper they put natural volcanic samples of differing porosities into a 1-D shock tube and measure both the fragmentation speed inside the shock tube and ejection velocities. They relate the results using 1-D conservation laws.
The lab experiments are instructive. In these experiments, a volcanic pumice is capped by a small volume of gas which, in turn, underlies a diaphragm. The duration of the shock-tube experiment is about 300-400 ms after a diaphragm separating the reservoir from the atmosphere is burst. The fragmentation threshold pressures were significant: about 4 Mpa (40 bars) for pumice; 8 Mpa (80 bars) for a denser sample. Below these pressures, the sample either does not fragment, or only partially fragments. From the experiments, the authors developed an empirical relation between the reservoir pressure and the fragmentation velocity (U is a logarithmic function of the ratio of the reservoir pressure to the fragmentation threshold pressure). Using this empirical relation in conjunction with the conservation laws for a pseudogas, they derive the theoretical velocity of eruption of the gas-particle mixture, and found it to be in excellent agreement with the lab measurements. This provides a way, then, to use measured or inferred field velocities to calculate reservoir pressures.
They then outline some conditions under which the experiments can be applied to volcanic eruptions, one of which is that it is restricted to magmatic eruptions, and produce the following graphs of ejection velocity versus initial reservoir pressure: