Canonical scale separation in two-dimensional incompressible hydrodynamics

TL;DR

The paper tackles the challenge of understanding the long-time behavior of two-dimensional Euler turbulence, where Kraichnan predicted a dual cascade of enstrophy to small scales and energy to large scales. It introduces a parameter-free, canonical split of vorticity into large-scale and small-scale components, defined via the Poisson structure and Hamiltonian, and shows that this split emerges dynamically in Zeitlin's Euler-Zeitlin discretization on the sphere and can be translated to the continuous Euler equations in a weak form. The main contributions are (i) a precise canonical splitting W = Ws + Wr tied to the stabilizer of the stream matrix, (ii) explicit reduced dynamics for Ws and Wr that reproduce scale separation and a broken-line energy spectrum, and (iii) numerical demonstrations of blob formation and scale separation, with theoretical groundwork for extending the splitting to the continuum and for stochastic model reduction. Together, these results provide a geometry-respecting framework to understand vortex condensation and to develop reduced models for 2D turbulence with potential practical impact in simulations and theory.

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.

Figures (8)

• Figure 1: Evolution of vorticity for Euler's equations on the sphere. Vorticity regions of equal sign undergo merging to form stable, interacting vortex condensates.
• Figure 2: Vanishing momentum simulation. Progression of the stream matrix $P$ and the components $W_s$ and $W_r$ for the same simulation as in \ref{['fig:evolution1']}. Initially $W_s$ and $W_r$ are similar in nature, but they evolve so $W_s$ traps the large-scale condensates whereas $W_r$ captures the small-scale fluctuations.
• Figure 3: Vanishing momentum simulation. Evolution of the decomposed enstrophies $E_s$ and $E_r$ (left), and decomposed energies $H_s$ and $H_r$ (right). The dashed, vertical lines indicate the sample times in Figures \ref{['fig:evolution1']}, \ref{['fig:Vorticity']}, \ref{['fig:Ene_spec']}. The energy $H_r$ decays almost to zero, so that most of the energy is contained in $H_s$ (reflecting the inverse energy cascade). On the other hand, the enstrophy $E_r$ increases over time (reflecting the forward enstrophy cascade).
• Figure 4: Vanishing momentum simulation. Spectrum in log-log scale for the energies $H$, $H_s$, and $H_r$ at the initial (left), intermediate (middle), and long time (right). The dashed lines indicate the slopes $l^{-3}$ and $l^{-1}$. The slope of $H_s$ is almost settled at the intermediate time. The slope of $H_r$ take much longer to settle. At long time, the broken line spectrum of $H$ is captured well by the components $H_s$ and $H_r$, which themselves have almost the same average slope at each scale.
• Figure 5: Non-vanishing momentum simulation. Progression of the stream matrix $P$ and the components $W_s$ and $W_r$ of the vorticity matrix $W$. Initially $W_s$ and $W_r$ are similar in nature, but they evolve so that $W_s$ contains the large-scale condensates whereas $W_r$ contains the small-scale fluctuations.

Theorems & Definitions (12)

• Remark 2.1
• Remark 3.1
• Theorem 3.1
• Remark 3.2
• Proposition 3.1
• Remark 3.3
• Proposition 5.1
• Proposition 5.2
• Proposition 5.3
• Remark 5.1