Percolating magmas in three dimensions

Abstract. The classical models of volcanic eruptions assume that they originate as a consequence of critical stresses or critical strain rates being exceeded in the magma followed by catastrophic fragmentation. In a recent paper (Gaonac'h et al., 2003) we proposed an additional mechanism based on the properties of complex networks of overlapping bubbles; that extreme multibubble coalescence could lead to catastrophic changes in the magma rheology at a critical vesicularity. This is possible because at a critical vesicularity P_c (the percolation threshold), even in the absence of external stresses the magma fragments. By considering 2-D percolation with the (observed) extreme power law bubble distributions, we showed numerically that P_2c had the apparently realistic value ≈0.7.

The properties of percolating systems are, however, significantly different in 2-D and 3-D. In this paper, we discuss various new features relevant to 3-D percolation and compare the model predictions with empirical data on explosive volcanism. The most important points are a) bubbles and magma have different 3-D critical percolation points; we show numerically that with power law bubble distributions that the important magma percolation threshold P_3c,m has the high value ≈0.97±0.01, b) a generic result of 3-D percolation is that the resulting primary fragments will have power law distributions with exponent B_3f≈1.186±0.002, near the empirical value (for pumice) ≈1.1±0.1; c) we review the relevant percolation literature and point out that the elastic properties may have lower – possibly more realistic – critical vesicularities relevant to magmas; d) we explore the implications of long range correlations (power law bubble distributions) and discuss this in combination with bubble anisotropy; e) we propose a new kind of intermediate "elliptical" dimensional percolation involving differentially elongated bubbles and show that it can lead to somewhat lower critical thresholds.

These percolation mechanisms for catastrophically weakening magma would presumably operate in conjunction with the classical critical stress and critical strain mechanisms. We conclude that percolation theory provides an attractive theoretical framework for understanding highly vesicular magma.