Dynamics of diffusive bubble growth in magmas: Isothermal case

Abstract

The application of sophisticated numerical calculations to understanding complex natural phenomena is now widespread in earth science. Recently, Proussevitch et al. [1993] illustrated some of the advantages in the power of such methods to investigate complex processes and also some of the difficulties. Calculations are presented on the diffusive growth of bubbles in magmas following a sudden and large decrease in pressure. In terms of theory the approach is an advance over a previous numerical model [Sparks, 1978]. The new study calculates the full diffusion problem of spherical bubble growth, whereas the earlier study used empirical data to calculate a growth rate constant. The new model is only a beginning, because even more complex problems relevant to gas bubble growth in magmas can be tackled, such as concentration dependent diffusivity, correlation of viscosity with water content, and variable decompression rates. Before going to even more elaborate calculations, however, it is important that the present ones are well understood. A difficulty with numerical results of this kind is that the results themselves have to be interpreted, and this can be of difficulty comparable to interpreting the complex natural systems. Another difficulty is that there are no independent data with which to compare the calculated bubble growth rates, due to the experimental difficulties in studying violent bubble growth. In this comment I draw attention to the need for fuller explanation and verification of these models. Some of the results are not easily reconciled with the interpretations offered by Proussevitch et al. and indicate that a more detailed analysis of the physics of diffusive bubble growth is needed. I propose an alternative explanation of certain aspects of their results which would have important volcanological implications. Many of the numerical results shown by Proussevitch et al. display a characteristic sigmoidal growth curve. This result contrasts with the classical result [e.g., Scriven, 1959] where radius increases as the square root of time. At large bubble sizes toward the end of growth, the bubble growth rate should depart from the parabolic growth law, because the model involves a shell of finite volume. The departure from the parabolic law at small times and bubble sizes is substantial and is hard to reconcile with the interpretations offered. This period of growth shows radius increasing as some power n of time which is significantly larger than 1. For brevity, I use the term accelerating growth to describe this phase. This period is interpreted by Proussevitch et al. in terms of a time delay due to the effects of surface tension. The time delay in each calculation is listed in their Table 2,

Brittleness of fracture in flowing magma

“…[25] Prousevitch et al [1993] developed equations for modeling bubbly magma based on dynamics of an expanding viscous spherical shell bubble. Zhang [1999] applied this model to investigate the brittle fragmentation condition for ascending magma based on the critical stress criterion for brittle fracture.…”

When Does Magma Break?

“…The relaxation timescale is calculated assuming a composition independent value for G 1 of 10 10 Pa (Dingwell and Webb 1989) pressure in the bubble, relative to the hydrostatic pressure (Sparks 1978). This can be augmented for the diffusion-controlled gradient of viscosity in the immediate liquid shell around a growing bubble (Prousevitch et al 1993;Lensky et al 2001). The component of the rate of shear strain tangential to the bubble wall _ c h can then be computed throughout bubble growth (Ichihara et al 2002).…”

Equilibrium, stability and evolution of bubbles in a finite melt volume

“…Such a cylinder could be a vertical conduit through rock for volcanic magma, the dilatometer column used for assessing the properties of coking coal [3], or the head on a glass of carbonated drink, such as beer. For simplicity, we consider the bubbles to be locally positioned in a face centred cubic (fcc) arrangement, where the side of the standard cube is of length S. Then, taking the approach of, for example, Proussevich et al [5], we obtain a set of equations for the growth of the bubbles:…”

“…According to Proussevitch et al [5], η = 0.5 is appropriate for water dissolved in a basaltic melt, whereas η = 1 is appropriate for CO 2 dissolved in water [8]. We define the volume fraction of bubbles as…”

“…In each of these cases, it is necessary to consider the behaviour of a large number of bubbles, rather than a single, isolated bubble, in order to understand the behaviour of the system. Some studies of multiple bubble systems consider distributions of bubble sizes and their evolution through time via population equations [4], whereas others consider all of the bubbles to be the same, but arranged in a way that their interaction is taken into account [5,6]. We take the latter approach, in order to investigate the existence of equilibrium states for a representative multiple bubble system, and then examine the stability of the equilibria.…”