Bubble distributions and dynamics: The expansion‐coalescence equation
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".