Canonical scale separation in two-dimensional incompressible hydrodynamics

Klas Modin, Milo Viviani

Characterization of the long-time behavior of an inviscid incompressible fluid evolving on a two-dimensional domain is a long-standing problem in physics. The motion is described by Euler's equations: a non-linear system with infinitely many conservations laws, yet non-integrable dynamics. In both experiments and numerical simulations, coherent vortex structures, or blobs, typically form after some stage of initial mixing. These formations dominate the slow, large-scale dynamics. Nevertheless, fast, small-scale dynamics also persist. Kraichnan, in his classical work, qualitatively describes a direct cascade of enstrophy into smaller scales and a backward cascade of energy into larger scales. Previous attempts to quantitatively model this double cascade are based on filtering-like techniques that enforce separation from the outset. Here we show that Euler's equations possess a natural, intrinsic splitting of the vorticity function. This canonical splitting is remarkable in four ways: (i) it is defined only in terms of the Poisson bracket and the Hamiltonian, (ii) it characterizes steady flows (equilibria), (iii) it genuinely, without imposition, evolves into a separation of scales, thus enabling the quantitative 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 a quantized version of Euler's equations in combination with a standard quantum-tool: the spectral decomposition of Hermitian matrices. In addition to theoretical insight, the canonical scale separation dynamics might be used as a foundation for stochastic model reduction, where the small scales are modelled by suitable multiplicative noise.

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