[Paper Review] Canonical scale separation in two-dimensional incompressible hydrodynamics

Klas Modin, Milo Viviani|arXiv (Cornell University)|Feb 2, 2021

Fluid Dynamics and Turbulent Flows43 references19 citations

TL;DR

This paper introduces a canonical, parameter-free decomposition of vorticity in two-dimensional incompressible Euler flows, derived solely from the Poisson bracket and Hamiltonian. The method intrinsically splits vorticity into large-scale condensates ($\omega_s$) and small-scale fluctuations ($\omega_r$), dynamically reproducing Kraichnan's dual cascade: $\omega_s$ exhibits $k^{-3}$ energy spectrum and $\omega_r$ shows $k^{-5/3}$ to $k^{-1}$ scaling, with forward enstrophy and backward energy cascades. The splitting emerges naturally from Zeitlin's truncated model and spectral decomposition of Hermitian matrices.

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.

Motivation & Objective

• To resolve the long-standing challenge of explaining spontaneous scale separation in 2D incompressible Euler flows, particularly the emergence of coherent vortex structures and broken-line energy spectra.
• To develop a parameter-free, mathematically rigorous method for decomposing vorticity into large-scale and small-scale components based solely on the Hamiltonian and Poisson bracket structure.
• To provide a dynamical mechanism for Kraichnan's forward enstrophy and backward energy cascades, grounded in the intrinsic geometry of the Euler equations.
• To establish a foundation for stochastic model reduction by treating small-scale dynamics as multiplicative noise.
• To explain the origin of the $k^{-3}$ to $k^{-1}$ transition in energy spectra observed in simulations and geophysical flows, via a canonical decomposition that captures the 'broken line' behavior.

Proposed method

• Derives a canonical vorticity splitting $\omega = \omega_s + \omega_r$ from the Poisson bracket and Hamiltonian of the 2D Euler equations, using the Lie–Poisson structure on the sphere.
• Applies spectral decomposition of Hermitian matrices to Zeitlin's truncated model of Euler's equations, enabling a canonical splitting of the vorticity field.
• Defines $\omega_s$ as the orthogonal projection of $\omega$ onto the stabilizer of the stream function $\psi$, and $\omega_r$ as the complement, ensuring conservation of energy and enstrophy.
• Derives coupled evolution equations for $\omega_s$ and $\omega_r$ using the Lie derivative and commutator of projection operators, with $\omega_s$ evolving via $\dot{\omega}_s = [\Pi_\psi, \mathcal{L}_X]\omega$.
• Implements the splitting numerically on the sphere using a spectral method, solving the equations with a time-splitting scheme and tracking energy and enstrophy spectra.
• Uses the implicit relation $\{b, \psi\} = \Pi_\psi^\perp \Delta^{-1} \{\psi, \omega_r\}$ to define the vector field $b$ that generates the dynamics, ensuring consistency with the Poisson structure.

Experimental results

Research questions

• RQ1 Can a canonical, parameter-free decomposition of vorticity be derived directly from the Hamiltonian and Poisson bracket of the 2D Euler equations?
• RQ2 Does this intrinsic splitting naturally lead to large-scale vortex condensates and small-scale fluctuations, consistent with numerical and experimental observations?
• RQ3 Can the canonical splitting reproduce the dual cascade of energy and enstrophy—backward energy cascade and forward enstrophy cascade—as predicted by Kraichnan?
• RQ4 Does the resulting energy spectrum for $\omega_s$ and $\omega_r$ exhibit the experimentally observed 'broken line' behavior with $k^{-3}$ and $k^{-5/3}$ to $k^{-1}$ slopes?
• RQ5 Can this decomposition serve as a basis for stochastic model reduction, where small-scale dynamics are modeled as multiplicative noise?

Key findings

• The canonical vorticity splitting $\omega = \omega_s + \omega_r$ is uniquely determined by the Poisson bracket and Hamiltonian, requiring no external parameters or filtering.
• The decomposition is dynamically consistent: $\omega_s$ evolves into large-scale vortex condensates, while $\omega_r$ captures small-scale fluctuations, with no imposed scale separation.
• Numerical simulations show that $\omega_s$ develops an energy spectrum slope of approximately $k^{-3}$, matching the theoretical prediction for coherent structures.
• The small-scale component $\omega_r$ exhibits an energy spectrum slope between $k^{-5/3}$ and $k^{-1}$, consistent with the forward enstrophy cascade in 2D turbulence.
• The system displays a forward cascade of enstrophy (increasing average enstrophy in $\omega_r$) and a backward cascade of energy (decreasing average energy in $\omega_r$), confirming Kraichnan’s phenomenology.
• The energy and enstrophy are consistently split as $H(\omega) = H(\omega_s) - H(\omega_r)$ and $E(\omega) = E(\omega_s) + E(\omega_r)$, preserving the physical conservation laws.