Spatiotemporal Organization of Correlated Local Activity within Global Avalanches in Slowly Driven Interfaces

We study the jerky response of slowly driven fronts in disordered media, just above the depinning transition. We focus on how spatially disconnected clusters of internally correlated activity lead to large-scale velocity fluctuations in the form of global avalanches and identify three different ways in which local activity clusters may organize within a global avalanche, depending on the distance to criticality. Our analysis provides new scaling relations between the power-law exponents of the statistical distributions of sizes and durations of local bursts and global avalanches. Fluid fronts of imbibition in heterogeneous media are taken as a case study to validate these scaling relations.

Figure 1

(a) Activity map of local front velocities in an imbibition displacement with injection velocity $\overline{v}=0.057\mathrm{mm}/\mathrm{s}$ in an observation window of lateral size $\ell =20\mathrm{mm}$. (b) Global velocity signal obtained from integrating this local activity map with respect to $x$. The discontinuous line is the mean velocity. (c) Activity map of local velocities clipped with $v_{\mathrm{th}}=3\overline{v}$. (d) Corresponding global signal obtained from integrating this clipped activity map with respect to $x$.

Figure 2

Top: Scheme of horizontal (left) and vertical (right) distributions of local activity clusters. Bottom: Ratios $\max (D_i)/T$ (filled symbols) and $\sum _i{D}_i/T$ (open symbols) for a given clip $C=3$ and different injection velocities $\overline{v}$ as a function of $T\overline{v}/{\ell }_d$.

Figure 3

Total number of local events that occur in a global avalanche of duration $T$ as a function of $T\overline{v}/{\ell }_d$. Data from experiments at $\overline{v}=0.057\mathrm{mm}/\mathrm{s}$ with $C=3$. Open dots are individual global avalanches and solid dots their average in equally spaced bins. The dashed line is a guide to the eye.