Multiscale turbulence hierarchy in active matter: Polarization analysis of a zebrafish shoal

Fish shoals exemplify systems composed of interacting and self-propelled particles that exhibit the complex collective dynamics seen in active matter. The group average polarization measures the ability to synchronize collective motion. Here, we analyze the experimental time series of increments in the average polarization of zebrafish shoals confined in a shallow circular test tank. Based on a multiscale approach of turbulence hierarchy and assuming that the local variance of the polarization increments at a given timescale has a gamma conditional distribution, we reveal that a cascade of background events occurs at well-separated timescales in small groups. Up to N=5 cascade levels are identified. The number of distinct timescales decreases with the size of the system. The polarization fluctuations in large groups are shown to be well described by a single-scale background. The crossover from multiple- to single-scale turbulence hierarchy aligns with previous findings indicating a crossover from a strongly correlated non-Rayleigh regime with an anomalous finite-size scaling of polarization to a Rayleigh-like regime.