Bubble distributions and dynamics: The expansion-coalescence equation
Résumé
As magma rises from depth, it forms bubbles by nucleation, followed by diffusion-decompressive expansion. Expansion induces shearing, and shearing in turn causes coalescence. As the bubbles grow larger, coalescence gradually becomes more efficient and can be dominant. Coalescence first as a binary ( bubble-bubble) and later as a ( possibly singular percolating) multibody process may thus be central to eruption dynamics. Here we consider a binary coalescence model governed by the Smoluchowski or coalescence/ coagulation equation. The introduction of decompressive expansion is theoretically straightforward and yields the nonlinear partial integrodifferential expansion-coalescence equation; we argue that this is a good model for bubble-bubble dynamics in a decompressing magma. We show that when the collision/interaction kernel has the same form over a wide range of interaction volumes (i.e., it is scaling), exact truncated power law solutions are possible irrespective of the expansion and the collision rate histories. This enables us to reduce the problem to a readily solvable linear ordinary differential equation whose solutions primarily depend on the total interaction integral. In this framework, we investigate the behavior of several eruption models. The validity of the expansion coalescence model is empirically supported by analysis of samples of pumice and lava. Theoretically, the suggested power laws are indeed stable and attractive under a wide range of conditions. We finally point out the effect of small perturbations and new ways to test the theory.