Unstable gaseous detonations as a deterministic paradigm for Self-Organized Criticality

We study the non-linear dynamics of galloping unstable detonations, characterized by a complex sequence of initiation and failure. We analyze the time-series of the lead shock obtained numerically over very long periods. Calculations are reported for the reactive Euler equations and the reduced Fickett reactive Burgers equation model. The detonation speed dynamics develop an approximate self-similar power law scaling, with a power spectrum scaling like $1/f$ over at least one decade of scales. This is characteristic of Self-Organized Criticality (SOC), i.e., the scale-free organization of the dynamics on large scales with no characteristic time, shared by many systems in nature governed by avalanche-like dynamics for energy release (e.g., earthquakes, evolution, sand-pile avalanches, droplet coalescence). We argue that detonations offer a unique deterministic paradigm to study SOC in avalanche dynamics and inverse energy cascades. We discuss the organization of the detonation front dynamics in terms of the physics controlling the wave front evolution. It is found that the largest "avalanches" trigger the longest lived shock relaxation by the gasdynamic relaxation, involving the expansion wave catching up to the front. Instability in the decaying front triggers a hierarchy of smaller avalanches superposed on the larger ones in a fractal-like structure. These dynamics are discussed in the context of our earlier high order non-linear oscillator model (Bellerive, ICDERS 2016).