AMS insights
The difference in the degree of anisotropy observed on Fig. 5 suggests a fundamental difference in the mechanism of formation of the bombs studied here. One condition required to preserve large degrees of anisotropy is the continuous shearing of the lava very close to an effective solidification point (Cañón-Tapia and Pinkerton 2000). Even if deformation during flow is large enough to momentarily produce a relatively large degree of anisotropy, the degree of anisotropy of the solid sample might be rather small if the motion producing the deformation stops when the lava is still fluid. Consequently, the presence of large degrees of anisotropy on the spindle bombs indicates that deformation took place very close to the time of effective solidification of the lava on the bombs. This effective solidification took place not only near the surface, but also includes the internal (central) parts of the bombs. On the other hand, the small degree of anisotropy of the bread-crust bombs indicates that the lava on their internal parts was already stationary when it achieved its effective solidification point, independently of any previous deformation that it might have experienced. The slightly larger values of the degree of anisotropy displayed by the small-size specimens of the San Quintin bread-crust bomb are compatible with this interpretation, because they actually suggest that the external parts of the bombs (closer to the rind), or very close to its center, experienced some amount of deformation very close to their solidification, whereas the intermediate parts of the bomb did not.
The parallelism of the k
max axes on the spindle bombs and the predominance of prolate shapes of the susceptibility tensor are also compatible with the above interpretation. Furthermore, these other two sources of information provide additional clues concerning the type of deformation that the spindle bombs experienced. Although in some special circumstances k
max axes can be perpendicular to the local extension direction, the tendency is to have k
max axes parallel to local extension (Cañón-Tapia and Castro 2004; Cañón-Tapia and Pinkerton 2000). Also, susceptibility tensors displaying a preferred prolate shape tend to be associated with elongation regimes. In contrast, oblate susceptibility tensors and a marked preferred orientation of k
min axes commonly indicate the presence of compaction-regimes. Thus, the orientation of the principal susceptibilities and the preferred shape of the susceptibility tensor of spindle bombs suggest that the shape of the bomb was acquired as the result of elongation that took place very close to the solidification of the entire bomb; whereas, the principal susceptibility axes and susceptibility tensor of the bread-crust bombs is more compatible with either a compaction regime or with cooling in the absence of external stresses.
Based on the information provided by the AMS results, it is possible to go a step further and re-examine the various mechanisms of formation of volcanic bombs summarized in Fig. 2a, b. The action of air drag and surface tension forces is expected to be stronger on the fluid closest to the surface of the bomb, and to decrease in intensity as the lava inside the bomb is closer to its center. In this case, a systematic variation of magnetic fabric as a function of position relative to the surface of the bomb would be expected. If the interior of the bomb behaved as a Newtonian fluid, drag-related shearing should extend from the surface to its very center. Otherwise, its influence would be limited to a region closer to the surface, where shearing is more intense. The AMS results of the two morphologies studied here suggest that a variation in the amount of shear is only observed very close to the surface (~1 or 2 cm), and is detectable only if small size specimens are used on the measurement. If large, standard size specimens are used, the AMS signal becomes dominated by the fabric of the interior of the bomb and does not display a systematic variation as a function of depth. For this reason, the influence of drag-related forces as agents controlling the shape acquisition of large volcanic bombs seems to be restricted to their surface and cannot be considered to be responsible for any deformation on their internal parts.
Similarly, a rotational movement having the largest dimension of spindle bombs as axis (rotation 2 in Fig. 2a) can be eliminated, because the orientation of the shear responsible for the elongation of the bomb is not parallel with the largest direction of the bomb, and therefore it would not explain the observed orientation of k
max axes. In contrast, a rotational movement around an axis perpendicular to the longest direction of the bombs (rotation 1 in Fig. 2a) is more compatible with the AMS observations.
It must be noted that the measured orientation of the principal susceptibilities of samples SB1 and SB2 is also compatible with the orientation of shear associated to the mechanism shown in Fig. 2b. The only constrain imposed would be that such deformation also took place close to the solidification point of lava, so that the orientation of principal susceptibility axes could be preserved.
Consequently, based on the information provided by AMS measurements, the two favored mechanisms of elongated bomb formation are those of rotation 1 in Fig. 2a and that illustrated on Fig. 2b.
Analytical assessment of current models of bomb formation
Since AMS results favor the occurrence of two mechanisms of formation (rotation1 and ejection in Fig. 2), it is reasonable to focus only on those two mechanisms to examine them from an analytical point of view. The analyses made on this section are independent of the AMS measurements and highly simplified. Nevertheless, the general comments presented here provide guidelines of several aspects that can be investigated in more detail in future studies.
In-flight rotation
The stresses in any body that rotates about a single axis are proportional to the square of its radius, its density, and the square of the angular velocity (Timoshenko 1934). Due to the influence of density, even small angular velocities might result in stresses of a few thousands of Pascals, making relatively easy to deform lavas of moderate viscosities (~1500 Pa s) in a few seconds. Because time of travel has been documented to be on the range 1 to 10 s (Vanderkluysen et al. 2012; Wright et al. 2007), it is reasonable to accept that some bombs might have formed through this mechanism. It must be noted, however, that the larger times required for cooling of the larger bombs should provide ample opportunity for relaxation of any fabric acquired during fly, unless the lava behaves as a non-Newtonian fluid. The preservation of the AMS fabrics described above is a strong indication of a non-Newtonian behavior that therefore needs to be considered in future studies.
Apart from the rheological behavior of the lava forming the bomb, there is another aspect concerning the time of cooling necessary for the preservation of the deformation that needs to be examined. Vanderkluysen et al. (2012) have shown that most pyroclastic particles experience a well-defined cooling trend during their flight, landing with a temperature 70% lower than the starting temperature. Fifteen percent of the pyroclasts landed with a temperature 80% of the starting, and 9% of the pyroclasts experienced an increase on their surface temperatures. If the glass transition is set to 700 °C, this means that the starting temperatures of bombs should be between 900 °C and 1000 °C. Otherwise, the deformation acquired during the initial expansion of gas would be destroyed upon impact on landing. This would explain why volcanic bombs tend to be more abundant on top of the deposits being associated with the increasing viscosity during the waning stages of an eruption, rather than being interspersed with scoria or other pyroclasts formed during the main phases (Wentworth and Macdonald 1953). This range in temperatures is likely to be different for the various bomb morphologies, as suggested by the differences on their respective degrees of anisotropy. Such assertion, however, needs to be evaluated by using a more extensive population of volcanic bombs, and is part of an ongoing study.
Deformation upon leaving the magma pool
I now refer to the mechanism illustrated in Fig. 2b. This mechanism has been less studied than the deformation upon rotation, and therefore deserves a more extended treatment. As mentioned above, this mechanism has been invoked mainly to explain the formation of a specific type of achnelith known as Pele’s hair.
A key element on the mechanism of formation of Pele’s hair pointed out by several authors (Duffield et al. 1977; Heiken 1972, 1974; Wentworth 1938) is that such structure forms from droplets that emanate directly from the magma pool and that are stretched into threads before completely breaking out from that pool. A similar mechanism was described to take place from lava falls associated to changes of slope affecting a moving lava flow. Despite the detailed descriptions that some of those authors offer, however, none of them provides a theoretical foundation that could serve to identify the most relevant parameters controlling the formations of those threads.
Following a similar line of reasoning, Shimozuru (1994) suggested that the conditions for the formation of Pele’s hair were similar to those encountered during the formation of ink jets. According to him, the conditions for the formation of Pele’s hair and tears depended on the ratio of the Weber and Reynolds numbers, which he called the Pele’s number (=v × η/σ; where v is the velocity of the fluid in the jet, η is its viscosity, and σ is its surface tension); formation of Pele’s hair is favored when Pele’s number is large, otherwise Pele’s tears form. In the remainder of this section I examine this mechanism in the context of a particle that approaches, deforms, pierces, and moves away from a deformable interface between two fluids, following a trajectory that is perpendicular to the originally undeformed interface(Fig. 6a).
An analytical treatment of this type of movement for the specific case of spherical particles has been made by Geller et al. (2006). Here, I summarize the most important aspects of their work using conditions relevant for the formation of volcanic bombs. In this context, the spherical particle is the core of the bomb, and the wrapping of magma is the bomb body. According to Geller et al. (2006) the movement illustrated in Fig. 6a leads to two modes of interface deformation: a film drainage mode in which the envelope around the sphere thins out very rapidly (and consequently no bomb is produced), and a tailing mode in which the sphere deforms the interface remaining encapsulated by the fluid from which it is moving away, forming a thread-like tail. According to those authors, the difference between both situations can be assessed by using two main parameters. The first is λ, the ratio of the viscosity of the fluid towards which the particle moves (fluid 2 in Fig. 6a, hereafter assumed to be air) divided by the viscosity of the fluid in which the particle is originally (fluid 1 in Fig. 6a, hereafter considered to be magma). The second parameter of interest is Ca, the ratio of the characteristic viscous stress at the interface relative to surface tension (Ca = μ × v/σ; where μ is the viscosity of magma, v the velocity of the particle, and σ the surface tension at the interface). A third parameter, Cg, defined as the ratio of characteristic viscous stress relative to buoyancy forces (Cg = μ × v/(a
2 × g × Δρ)); where a is the radius of the particle that is envisaged as a sphere, g is the acceleration due to the action of gravity, and Δρ is the density difference between the magma and the air) is also of some interest, but since Cg is related to the form in which the shape of the interface deforms upon approach of the sphere from below, not exerting a large influence on the total amount of fluid that can be carried across the plane by entrainment, it can be neglected for the present purposes. Thus, although the exact outcome depends on the details of the particular situation, Geller et al. (2006) indicate that as long as λ is close to zero and Ca is of the order of 0.1 or larger, a tail will be formed. In contrast, λ equal to 1 and a very small Ca lead to film drainage behavior.
Due to the large differences that exist between the viscosity of the air and that of the magma from where the sphere is ejected, λ is expected to be very small on situations relevant for the formation of volcanic bombs. The viscosity of basalt can be as low as 30 Pa s (if pure liquid at temperatures larger than 1150 °C), but most typically will be on the range 200–1300 Pa s for temperatures lower than 1140 °C (Ishibashi and Sato 2007; Parfitt and Wilson 2008); in contrast the viscosity of air is equal to 1.71 × 10−5 Pa s. Using those values, λ is expected to be on the range 1.31 × 10−8 to 8.5 × 10−8. It remains to estimate the value of Ca. Taking typical values of surface tension between 0.05–0.1 N m−1 (Parfitt and Wilson 2008) and exit velocities of particles between 26 and 71 m s−1 (Vanderkluysen et al. 2012), Ca is found to take values of the order of 105. These values indicate that in most cases of volcanic interest, a particle approaching the surface of a magma pool from below will leave the pool of magma wrapped by a film carrying behind a tail of magma. The next step in the analysis is therefore to assess whether such exit velocities are reachable in volcanic conditions.
The main force responsible for the movement of a particle (gas bubble or solid with a smaller density than the surrounding lava) towards and beyond the deformable interface is in this case buoyancy. As a first approximation, disregarding gas expansion, the velocity of the particle will be controlled by the density difference between the particle and the magma, the size of the particle, and the magma viscosity. Entirely assuming a Stokes (laminar) flow, it is possible to calculate the terminal velocity of a spherical particle moving within the magma, as v = 2/9 × (a
2 × g × Δρ/μ) (Batchelor 1991). Figure 6b shows that the velocity of particles below 0.8 m will not reach 20 m s−1, which is a lower bound for the ejection velocity of particles, unless the density difference is extremely large (2500 kgm−3). The same figure shows that for very viscous magma (1300 Pa s) not even such a large density difference suffices to reach the minimum ejection velocity required to produce tails of magma that later become tear or spindle-shaped bombs. For this reason, even when it seems conceptually reasonable to invoke a mechanism of formation of volcanic bombs such as that shown in Fig. 2b, it is unlikely that this mechanism takes place in most circumstances if buoyancy alone is considered as the sole force controlling particle ascent within the magma pool. In addition, this mechanism cannot be invoked to explain the occurrence of bombs lacking a central core.
Towards a general model of formation of volcanic bombs and achneliths
On the analysis completed above, the forces associated to the sudden expansion of gas as it reaches the surface, and its subsequent bursting, have been neglected. Nevertheless, those forces provide an additional impetus that under certain circumstances might lead to the formation of volcanic bombs, as will be outlined next. In addition, the bursting of bubbles has been shown to have a direct relationship with the shapes of the particles at a microscopic level (Heiken 1972). Thus, it is of interest to explore the possibility to invoke such a mechanism in a macroscopic context.
To assess quantitatively the relevance of the forces involved, it is convenient to rely on the analysis made by Wilson (1980) of Strombolian eruptions. According to him, the sudden expansion of the upper part of a gas bubble that has reached the surface of a magma pond is related to the ratio of the internal and atmospheric pressures (P
i/P
s), and to the ratio of the specific heat of the gases involved (γ).
$$ d={r}_{\mathrm{i}}\ {\left(\frac{P\mathrm{i}}{P\mathrm{s}}\right)}^{\frac{1}{3\gamma}}\dots $$
(1)
where d is the distance reached by a bubble with original radius r
i (Fig. 7). From this relationship, it is possible to estimate the amount of deformation (or fractional change in area) experienced by the surface of the sphere as
$$ \frac{\mathrm{final}\ \mathrm{s}\mathrm{urface}-\mathrm{original}\ \mathrm{s}\mathrm{urface}}{\mathrm{original}\ \mathrm{s}\mathrm{urface}}=\frac{4\ \pi\ {d}^2-4\ \pi\ {r}_{\mathrm{i}}^2}{4\ \pi\ {r}_{\mathrm{i}}^2}={\left(\frac{P\mathrm{i}}{P\mathrm{s}}\right)}^{\frac{2}{3\ \gamma}}-1\dots . $$
(2)
In a sphere, surface deformation is related to linear deformation by the following expression:
$$ \mathrm{Linear}\ \mathrm{deformation}={\left(\mathrm{Surface}\ \mathrm{deformation}+1\right)}^{1/2}\kern0.5em -1\dots $$
(3)
In this case, the linear deformation can be envisaged as the stretching or shortening of a segment of a circle passing through the center of the sphere, and having the same initial and final radius. The advantage of considering a linear deformation by combining Eqs. 2 and 3 is that this type of deformation is easier to obtain from solidified bombs than the fractional change in area, for which the three dimensional form of the bomb becomes extremely important. Consequently, for the present purposes a rough indication of deformation was obtained by comparing the final shape of elongated bombs with an assumed bomb of circular cross section at the time of ejection from the magma pond, further assuming that all elongation took place along a single axis. In this form, it is possible to calculate an extreme deformation value corresponding to the linear elongation of the longest axis of the bomb (elongation = [final length − initial length]/initial length). The initial length of the axis can be assumed to be either the smallest or the intermediate axis of the bomb, or (as preferred here) an average of those two. Elongation of spindle bombs estimated from measurements made on 25 bombs range between 0.4 and 2.0, with an average slightly above 1. These values are used as reference when examining the results shown in Fig. 7b. The curves of that figure show the linear deformation obtained by combining Eqs. 2 and 3 and taking P
s = 100 kPa, and γ = 1.28 (Wilson 1980). As it can be seen in the figure, the amount of linear deformation experienced by a thin veneer of magma is large enough to explain the deformation measured in a typical spindle bomb, and consequently, the proposed mechanism is a viable alternative that shall be explored in more detail in future studies, preferably with more data than those presented here. In any case, the orientation of the principal susceptibilities and the associated degree of anisotropy measured in the two spindle bombs of this work, are consistent with this working hypothesis.