Macroscopic control parameter for avalanche models for bursty transport

Similarity analysis is used to identify the control parameter $R_A$ for the subset of avalanching systems that can exhibit Self- Organized Criticality (SOC). This parameter expresses the ratio of driving to dissipation. The transition to SOC, when the number of excited degrees of freedom is maximal, is found to occur when $R_A \to 0$. This is in the opposite sense to (Kolmogorov) turbulence, thus identifying a deep distinction between turbulence and SOC and suggesting an observable property that could distinguish them. A corollary of this similarity analysis is that SOC phenomenology, that is, power law scaling of avalanches, can persist for finite $R_A$, with the same $R_A \to 0$ exponent, if the system supports a sufficiently large range of lengthscales; necessary for SOC to be a candidate for physical ($R_A$ finite) systems.