Dynamics and energetics of bubble growth in magmas: Analytical formulation and numerical modeling

We have developed a model of diffusive and decompressive growth of a bubble in a finite region of melt which accounts for the energetics of volatile degassing and melt deformation as well as the interactions between magmatic system parameters such as viscosity, volatile concentration, and diffusivity. On the basis of our formulation we constructed a numerical model of bubble growth in volcanic systems. We conducted a parametric study in which a saturated magma is instantaneously decompressed to one bar and the sensitivity of the system to variations in various parameters is examined. Variations of each of seven parameters over practical ranges of magmatic conditions can change bubble growth rates by 2–4 orders of magnitude. Our numerical formulation allows determination of the relative importance of each parameter controlling bubble growth for a given or evolving set of magmatic conditions. An analysis of the modeling results reveals that the commonly invoked parabolic law for bubble growth dynamics R∼t1/2 is not applicable to magma degassing at low pressures or high water oversaturation but that a logarithmic relationship R∼log(t) is more appropriate during active bubble growth under certain conditions. A second aspect of our study involved a constant decompression bubble growth model in which an initially saturated magma was subjected to a constant rate of decompression. Model results for degassing of initially water‐saturated rhyolitic magma with a constant decompression rate show that oversaturation at the vent depends on the initial depth of magma ascent. On the basis of decompression history, explosive eruptions of silicic magmas are expected for magmas rising from chambers deeper than 2 km for ascent rates \u3e1–5 m s−1

Reply [to “Comment on “Dynamics and energetics of bubble growth in magmas: Analytical formulation and numerical modeling” by A. A. Proussevitch and D. L. Sahagian”]

We have developed a model of diffusive and decompressive growth of a bubble in a finite region of melt which accounts for the energetics of volatile degassing and melt deformation as well as the interactions between magmatic system parameters such as viscosity, volatile concentration, and diffusivity. On the basis of our formulation we constructed a numerical model of bubble growth in volcanic systems. We conducted a parametric study in which a saturated magma is instantaneously decompressed to one bar and the sensitivity of the system to variations in various parameters is examined. Variations of each of seven parameters over practical ranges of magmatic conditions can change bubble growth rates by 2–4 orders of magnitude. Our numerical formulation allows determination of the relative importance of each parameter controlling bubble growth for a given or evolving set of magmatic conditions. An analysis of the modeling results reveals that the commonly invoked parabolic law for bubble growth dynamics R∼t1/2 is not applicable to magma degassing at low pressures or high water oversaturation but that a logarithmic relationship R∼log(t) is more appropriate during active bubble growth under certain conditions. A second aspect of our study involved a constant decompression bubble growth model in which an initially saturated magma was subjected to a constant rate of decompression. Model results for degassing of initially water‐saturated rhyolitic magma with a constant decompression rate show that oversaturation at the vent depends on the initial depth of magma ascent. On the basis of decompression history, explosive eruptions of silicic magmas are expected for magmas rising from chambers deeper than 2 km for ascent rates \u3e1–5 m s−1

Dynamics of coupled diffusive and decompressive bubble growth in magmatic systems

Bubble growth in an ascending parcel of magma is controlled both by diffusion of oversaturated volatiles and decompression as the magma rises. We have developed a numerical model which explores the processes involved in water exsolution from basaltic and rhyolitic melts rising at a constant rate from magma chamber depths of 4 and 1 km. While the model does not attempt to simulate natural eruptions, it sheds light on the processes which control eruptive behavior under various conditions. Ascent rates are defined such that a constant rate of decompression dP/dt is maintained. A variety of initial ascent rates are considered in the model, from 1 m/s to 100 m/s for basalts, and from a few centimeters per second to 10 m/s for rhyolite, at the base of the conduit. The model results indicate that for any reasonable ascent rate, basaltic melt degasses at a rate sufficient to keep the dissolved volatile concentration at equilibrium with the decreasing ambient pressure. Rhyolitic melt reaches the surface at equilibrium if its ascent rate is less than 1 m/s, but it can erupt with high oversaturation at greater ascent rates. The latter may lead to explosive eruptions. If the ascent rate of rhyolite is 10 m/s or more, then melt barely degasses at all in the conduit and erupts with the highest oversaturation possible. For the case of slow magma rise, bubble growth is limited by decompression. For the case of rapid magma rise, bubble growth is limited by diffusion. The results of our simple model do not accurately simulate natural volcanic eruptions, but suggest that subsequent, more complex models may be able to simulate eruptions using the insights regarding diffusive and decompressive bubble growth processes explored in this study. Numerical modeling of volcanic degassing may eventually lead to better prediction of eruption timing, energetics and hazards of active volcanoes

Analysis of Vesicular Basalts and Lava Emplacement Processes for Application as a Paleobarometer/Paleoaltimeter

We have developed a method for determining paleoelevations of highland areas on the basis of the vesicularity of lava flows. Vesicular lavas preserve a record of paleopressure at the time and place of emplacement because the difference in internal pressure in bubbles at the base and top of a lava flow depends on atmospheric pressure and lava flow thickness. At the top of the flow, the pressure is simply atmospheric pressure, while at the base, there is an additional contribution of hydrostatic lava overburden. Thus the modal size of the vesicle (bubble) population is larger at the top than at the bottom. This leads directly to paleoatmospheric pressure because the thickness of the flow can easily be measured in the field, and the vesicle sizes can now be accurately measured in the lab. Because our recently developed technique measures paleoatmospheric pressure, it is not subject to uncertainties stemming from the use of climate‐sensitive proxies, although like all measurements, it has its own sources of potential error. Because measurement of flow thickness presupposes no inflation or deflation of the flow after the size distribution at the top and bottom is “frozen in,” it is essential to identify preserved flows in the field that show clear signs of simple emplacement and solidification. This can be determined by the bulk vesicularity and size distribution as a function of stratigraphic position within the flow. By examining the stratigraphic variability of vesicularity, we can thus reconstruct emplacement processes. It is critical to be able to accurately measure the size distribution in collected samples from the tops and bottoms of flows because our method is based on the modal size of the vesicle population. Previous studies have used laborious and inefficient methods that did not allow for practical analysis of a large number of samples. Our recently developed analytical techniques involving high‐resolution x‐ray computed tomography (HRXCT) allow us to analyze the large number of samples required for reliable interpretations. Based on our ability to measure vesicle size to within 1.7% (by volume), a factor analysis of the sensitivity of the technique to atmospheric pressure provides an elevation to within about ±400 m. If we assume sea level pressure and lapse rate have not changed significantly in Cenozoic time, then the difference between the paleoelevation “preserved” in the lavas and their present elevation reflects the amount of uplift or subsidence. Lava can be well dated, and therefore a suite of samples of various ages will constrain the timing of epeirogenic activity independent of climate, erosion rates, or any other environmental factors. We have tested our technique on basalts emplaced at known elevations at the base, flanks, and summit of Mauna Loa. The results of the analysis accurately reconstruct actual elevations, demonstrating the applicability of the technique. The tool we have developed can subsequently be applied to problematic areas such as the Colorado and Tibetan Plateaus to determine the history of uplift

The size range of bubbles that produce ash during explosive volcanic eruptions

Volcanic eruptions can produce ash particles with a range of sizes and morphologies. Here we morphologically distinguish two textural types: Simple (generally smaller) ash particles, where the observable surface displays a single measureable bubble because there is at most one vesicle imprint preserved on each facet of the particle; and complex ash particles, which display multiple vesicle imprints on their surfaces for measurement and may contain complete, unfragmented vesicles in their interiors. Digital elevation models from stereo-scanning electron microscopic images of complex ash particles from the 14 October 1974 sub-Plinian eruption of Volcán Fuego, Guatemala and the 18 May 1980 Plinian eruption of Mount St. Helens, Washington, U.S.A. reveal size distributions of bubbles that burst during magma fragmentation. Results were compared between these two well-characterized eruptions of different explosivities and magma compositions and indicate that bubble size distributions (BSDs) are bimodal, suggesting a minimum of two nucleation events during both eruptions. The larger size mode has a much lower bubble number density (BND) than the smaller size mode, yet these few larger bubbles represent the bulk of the total bubble volume. We infer that the larger bubbles reflect an earlier nucleation event (at depth within the conduit) with subsequent diffusive and decompressive bubble growth and possible coalescence during magma ascent, while the smaller bubbles reflect a relatively later nucleation event occurring closer in time to the point of fragmentation. Bubbles in the Mount St. Helens complex ash particles are generally smaller, but have a total number density roughly one order of magnitude higher, compared to the Fuego samples. Results demonstrate that because ash from explosive eruptions preserves the size of bubbles that nucleated in the magma, grew, and then burst during fragmentation, the analysis of the ash-sized component of tephra can provide insights into the spatial distribution of bubbles in the magma prior to fragmentation, enabling better parameterization of numerical eruption models and improved understanding of ash transport phenomena that result in pyroclastic volcanic hazards. Additionally, the fact that the ash-sized component of tephra preserves BSDs and BNDs consistent with those preserved in larger pyroclasts indicates that these values can be obtained in cases where only distal ash samples from particular eruptions are obtainable

Recognition and separation of discrete objects within complex 3D voxelized structures

3D voxelized images can be manipulated if their component parts can be identified, cataloged, and measured. To accomplish this, it is necessary to separate individual convex objects from the complex structures that result from digital observation techniques such as X-ray tomography. Toward this end, we have developed schemes that peel away sequential layers of voxels from complex structures until narrow waists that connect individual objects disappear, and each component object can be identified. These peeling schemes provide the most uniform possible cumulative thickness of removed layers regardless of the orientation of the voxel grid pattern. Consequently, they lead to the most accurate application regarding inter-object interfaces, medial axis analysis, and individual object statistics such as volumes, orientations and interconnectivity. Peeling schemes can be categorized by the number of steps involved in each peeling iteration. Each step removes voxels according to three possible criteria for defining the exterior of a voxel: exposed faces, edges, or corners. Each of these ultimately causes an initial sphere, for example, to evolve into a cube, dodecahedron, or octahedron, respectively. Combinations of steps can be used to create more complex polyhedra (tetrahexadra, trisoctahedra, trapezohedra, and hexoctahedra). The resulting polyhedron that most closely resembles a starting sphere depends on the appropriate definition of “sphericity”. Using a metric based on the standard deviation of the polyhedral surface from that of a concentric sphere of equal volume, the optimal scheme is peeling by faces 7 times, by edges 3 times, and by corners 4 times. This leads to a hexoctahedron with Miller indices (14 7 4) and a standard deviation of 0.025. Using a metric based on minimizing surface area, the optimal scheme is peeling by faces 9 times, by edges 6 times, and by corners 5 times, leading to a hexoctahedron with Miller indices (20 11 5). In the past, only 1-step peeling has been used (by faces or corners). If computational or conceptual constraints limit peeling to 1-step, the criterion of edges should be used, as the dodecahedron that results deviates from a sphere by only half the amount of either the cube or octahedron resulting from 1-step peeling of faces or corners, respectively. We also determined the best criteria for 2-step and 3-step peeling. The peeling schemes we identify can be used to separate objects from complex structures for application to a number of geological and other problems. Information that emerges from the analysis includes object volumes, which can be used for determining grain- or bubble-size distributions in volcanologic, petrologic, and sedimentary applications, among others

3D particle size distributions from 2D observations: stereology for natural applications

We have developed a general formulation for stereological analysis of particle distributions which is applicable to any particle size or size distribution (not limited to log-normal, unimodal, etc.). We have applied numerical techniques to define intersection probability distributions for any shapes (previously only known for spheres), and quantified the errors involved in using spherical coefficients for various non-spherical particles. This stereological technique is based on knowing the probability distribution of random cross-sections through various particles so that `small circle\u27 cross-sections can be subtracted from an observed population to provide the 3D size distribution of particles. The results indicate that the most important parameter controlling calculated size distribution is particle aspect ratio. For a distribution of particles with a specific aspect ratio or range of aspect ratios, variations of particle form (spherical vs. cubic; rectangular vs. elliptical) do not alter the results, so the technique can be applied to a range of particle shapes. Applications can be made in a petrology, volcanology, and other fields, only a few of which can also be treated using expensive X-ray techniques