Canonical scale separation in two-dimensional incompressible hydrodynamics

Abstract: A two-dimensional inviscid incompressible fluid is governed by simple rules. Yet, to characterise its long-time behaviour is a knotty problem. The fluid evolves according to Euler's equations: a non-linear Hamiltonian system with infinitely many conservation laws. In both experiments and numerical simulations, coherent vortex structures, or blobs, emerge after an initial stage. These formations dominate the large-scale dynamics, but small scales also persist. Kraichnan describes in his classical work a forward cascade of enstrophy into smaller scales, and a backward cascade of energy into larger scales. Previous attempts to model Kraichnan's double cascade use filtering techniques that enforce separation from the outset. Here we show that Euler's equations posses an intrinsic, canonical splitting of the vorticity function. The splitting is remarkable in four ways: (i) it is defined solely via the Poisson bracket and the Hamiltonian, (ii) it characterises steady flows, (iii) without imposition it yields a separation of scales, enabling the dynamics behind Kraichnan's qualitative description, and (iv) it accounts for the "broken line" in the power law for the energy spectrum, observed in both experiments and numerical simulations. The splitting originates from Zeitlin's truncated model of Euler's equations in combination with a standard quantum-tool: the spectral decomposition of Hermitian matrices. In addition to theoretical insight, the scale separation dynamics could be used for stochastic model reduction, where small scales are modelled by multiplicative noise.