Canonical scale separation in two-dimensional incompressible hydrodynamics

Abstract

The rules that govern a two-dimensional inviscid incompressible fluid are simple. Yet, to characterise the long-time behaviour is a knotty problem. The fluid fulfils Euler's equations: a nonlinear Hamiltonian system with an infinite number of conservation laws. In both experiments and numerical simulations, coherent vortex structures emerge after an initial stage. These formations dominate the large-scale dynamics, but small scales also emerge and persist. The resulting scale separation resembles Kraichnan's theory of forward and backward cascades of enstrophy and energy. Previous attempts to model the double cascade use filtering techniques that enforce separation from the outset. Here, we show that Euler's equations possess an intrinsic, canonical splitting of the vorticity function. The splitting is remarkable in four ways: (i) it is defined solely by the Poisson bracket and the Hamiltonian; (ii) it characterises steady flows; (iii) it innately separates scales, enabling the dynamics behind Kraichnan's qualitative description; and (iv) it accounts for ‘broken line’ energy spectra 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 enables stochastic model reduction, where multiplicative noise models small scales.

Hydrodynamic Entropy and Emergence of Order in Two-dimensional Euler Turbulence

“…Bouchet and Venaille [12] argued that even though nonequilibrium steady states of 2D Euler turbulence often break detailed balance, under weak force and zero viscosity, they may be described by microcanonical measures and entropy functional. Bouchet and coworkers [11,12,24], and Modin and Viviani [23] explained the structures of 2D Euler turbulence in this framework. In this paper, we advance this theme by carefully examining the energy transfers and energy flux of 2D Euler turbulence.…”

“…The works of Fox and Orszag [4], Seyler et al [5], Pakter and Levin [13], Bouchet and Simonnet [11], Dritschel et al [21], and Modin and Viviani [23] indicate that 2D Euler turbulence is out of equilibrium, contrary to the assumptions of Onsager [6] and Kraichnan [16]. Bouchet and Venaille [12] argued that even though nonequilibrium steady states of 2D Euler turbulence often break detailed balance, under weak force and zero viscosity, they may be described by microcanonical measures and entropy functional.…”

On the Infinite Dimension Limit of Invariant Measures and Solutions of Zeitlin’s 2D Euler Equations

“…The answer may also change when we discuss suitable finite dimensional truncations instead of the true PDE. We do not have any precise conjecture but, based on numerical experiments reported in [4] and [22] we think that the true Euler equations (not necessarily certain Fourier trucations) may display the superposition of different scaling structures, some of them related to energy and enstrophy cascades, plus one related to very small vortex structures which has a k −1 scaling law. It is a sort of weak but visible background of "pointwise" vortices.…”

Statistical Equilibrium Principles in 2D Fluid Flow: From Geophysical Fluids to the Solar Tachocline

“…Finally we note that equilibration dynamics can be strongly affected by the topology of the 2D domain. Thus, equilibration on the surface of a sphere [ 49 ] (rather than in a flat bounded or doubly periodic plane) is found to fail much more catastrophically, with a macroscopically fluctuating chaotic vorticity field surviving for all achievable computation times [ 25 , 26 , 27 , 28 ]. Conservation of the full angular momentum vector in the spherical geometry (rather than just a vertical component) ensures that the vorticity cannot condense into a dipole pattern if the initial state has zero total angular momentum.…”

“…Conversely, lack of consistency, if indeed robustly borne out by the numerics, could point to existence interesting equilibration barriers and metastable behaviors. There is already significant evidence that such barriers are much more common in such highly constrained 2D flows than in, e.g., conventional particle systems, through a variety of mechanisms [ 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 ].…”