Designing turbulence through hierarchies

Abstract

Turbulence is defined by a set of emergent and singular phenomena: a constant-flux energy cascade, intermittency, a dissipation rate that survives the vanishing of viscosity, and the spontaneous loss of determinism of the inviscid limit. Which physical ingredients produce them remains unclear, because in fluids they always come together with incompressibility, spatial structure and vortex dynamics. Here we show that a hierarchy of inertia, coupled by the purely geometric nonlinearity of rigid constraints, is enough. A planar chain of rigid pendula whose lengths and masses decrease geometrically, coupled only through nearest-neighbor constraints, develops a far-from-equilibrium cascade that replaces thermal equipartition, with statistically local and time-irreversible energy transfer despite intrinsically nonlocal coupling. Its fluctuations are intermittent, yet self-similarity is restored by a hidden scale symmetry. As the viscosity vanishes, the dissipation rate saturates at a finite value, and thermal noise consistent with fluctuation-dissipation balance is amplified from the smallest to the largest scales within a fraction of a large-scale period. Because the hierarchy can be tuned and the same dynamics can be realized mechanically or electrically, turbulence becomes a property that can be designed and measured in the laboratory.