Spectral statistics of global avalanches along granular piles

– In a seminal experiment, Jaeger et al. (Phys. Rev. Lett., 62 (1989) 40) measured the power spectrum of the velocity of avalanches along a granular pile in a rotating drum. Although their findings started an actively discussed debate on whether self-organized criticality rules granular avalanche dynamics, their experimental results have not been explained in theoretical detail yet. By the use of a simple dynamical model that incorporates the basic deterministic mechanisms of global avalanche flow as well as their stochastic aspects, we find almost perfect agreement between the model results and Jaeger et al.’s experimental data. We also discuss in detail the significance of the underlying physical mechanisms. Granular matter such as beads, sand and powder can flow in a non-Newtonian way if driven out of its (metastable) static equilibrium by large enough external forces [1]. A striking example of the rheological complexity is that granular systems can exhibit stable static piling up to a maximum angle of repose φs of the inclined surface. For pilings with an inclination angle φ > φs the surface layer starts to slip in order to decrease the inclination angle until the minimum angle of repose φr < φs has been reached. Then, the avalanche stops again. So far, avalanching seems to be a fact of common sense. Highly non-trivial, however, is the problem of the spectral properties of sequences of avalanches that are generated by continuous addition of grains to the top of the pile or by slow rotation of the pile, e.g., in a drum [1]-[6]. A decade ago, Bak et al. [5] proposed the interesting scenario of self-organized criticality (SOC) that some spatially extended dissipative systems (such as sandpiles) might evolve into a critical state that does not possess characteristic time and length scales, and therefore, show a 1/f power spectrum [5]. In a subsequent seminal experiment, Jaeger et al. [2] probed the applicability of SOC to granular dynamics. They rotated a grain-filled semi-cylindrical drum (with a lower half being closed off) about its horizontally aligned axis with a constant rotation rate. By that, they created a well-defined sequence of discrete avalanches and measured the avalanche flow by detecting the flow over the rim at the lower end of the pile. The power spectrum of this velocity signal differs significantly from the 1/f power law decay suggested by SOC [5] (or the 1/f decay suggested in ref. [6]) and therefore this result started a still c © Les Editions de Physique 394 EUROPHYSICS LETTERS ongoing debate on the applicability of SOC to avalanches along granular piles [2]-[4], [6]. Despite its fundamental importance, there does not seem to exist a clear detailed simple theoretical explanation of Jaeger et al.’s result in the literature yet. In this letter, we reconsider Jaeger et al.’s result [2] from a theoretical point of view, although we start from a different concept than SOC. Our focus is to show that the power spectrum found by Jaeger et al. [2] can be explained theoretically in all its major details by using a dynamical model that incorporates the basic macromechanical mechanisms of the avalanche dynamics such as viscoplastic yield, macromechanical friction, and small macromechanical stochastics. Moreover, our investigation leads to a global picture of the influence of stochastics on the power spectra of the global avalanche velocity for granular flow-over-the-rim experiments. The model. – To model the statistics of sequences of discrete avalanches we start from the previously proposed deterministic minimal model (DMM) for the ensemble-averaged dynamics of avalanches [7] and combine it with the stochasticity of the individual avalanches. In the DMM (for details cf. ref. [7]), surface flow along granular piles is described by two global dynamical variables, the inclination angle φ(t) of the pile and the characteristic velocity v(t) of the avalanche flow. The latter is basically determined by the square root of the kinetic energy of the grain in motion. The DMM [7] generalizes Coulomb’s frictional motion of bodies on inclined planes as follows: i) a velocity-dependent friction coefficient kd(v) which interpolates between solid and Bagnold friction, kd(v) = b0 + b2v 2 (b0 > 0 and b2 > 0) [1], ii) a viscoplastic yield condition such that an avalanche can only start if φ > φs and stops again if v(t) reaches zero, and iii) a coupling of φ(t) to the velocity dynamics which counteracts the acceleration of the avalanche. Macromechanical stochasticity is rooted in the fact that moving granular matter consists of closely packed grains which interact by inelastic collisions and friction. This leads to local and (due to the finite extension of a pile) also to global fluctuations during avalanching. This effect, nicely demonstrated in the experiments [8], leads effectively to a macromechanical stochastics which can be modeled by a Langevin term, ζ̃(t), in the velocity equation. For the sake of simplicity, we suppose that ζ̃(t) represents Gaussian white noise fluctuations with zero mean and a correlation function 〈ζ̃(t)ζ̃(t′)〉 = ∆̃δ(t− t′). Here, ∆̃ denotes the fluctuation strength. The resulting model reads explicitly v̇ = g [ sinφ− (b0 + b2v ) cosφ+ ζ̃(t) ] χ(φ, v) , (1)