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Two-dimensional fluids via matrix hydrodynamics

Klas Modin, Milo Viviani

TL;DR

The paper establishes a rigorous bridge between 2-D Euler hydrodynamics on the sphere and isospectral matrix flows via Zeitlin's matrix discretization, enabling a matrix–geometry dictionary that preserves Lie–Poisson structure and Casimirs. It constructs a representation-theoretic quantization $T_N$ of smooth vorticity, defines the Euler–Zeitlin dynamics on $\mathfrak{su}(N)$, and proves space–time convergence of the matrix model to the continuum Euler flow as $N\to\infty$, including convergence of empirical spectral measures to the level-set measures. By analyzing coadjoint orbits, closures, and Schur–Horn–Kostant convexity, it situates the finite-dimensional dynamics between smooth orbits and weak closures, and connects these ideas to mixing operators and canonical scale separation. The work also proposes a long-time behavior conjecture for vorticity, ties vortex mixing to weak-$*$ convergence, and highlights how Zeitlin’s model serves as a calculable framework to study vortex condensation, integrability, and the long-time structure of 2-D turbulence on the sphere.

Abstract

Two-dimensional (2-D) incompressible, inviscid fluids produce fascinating patterns of swirling motion. How and why the patterns emerge are long-standing questions, first addressed in the 19th century by Helmholtz, Kirchhoff, and Kelvin. Countless researchers have since contributed to innovative techniques and results. Yet, the overarching problem of swirling 2-D motion and its long-time behavior remains largely open. Here we shed light on this problem via a link to isospectral matrix flows. The link is established through V. Zeitlin's beautiful model for the numerical discretization of Euler's equations in 2-D. When considered on the sphere, Zeitlin's model offers deep connections between 2-D hydrodynamics and unitary representations of the rotation group. Consequently, it provides a dictionary that maps hydrodynamical concepts to matrix Lie theory, which in turn gives connections to matrix factorizations, random matrices, and integrability theory, for example. Results about finite-dimensional matrices can then be transferred to infinite-dimensional fluids via quantization theory, which is here used as an analysis tool (albeit traditionally describing the limit between quantum and classical physics). We demonstrate how the dictionary is constructed and how it unveils techniques for 2-D hydrodynamics. We also give accompanying convergence results for Zeitlin's model on the sphere.

Two-dimensional fluids via matrix hydrodynamics

TL;DR

The paper establishes a rigorous bridge between 2-D Euler hydrodynamics on the sphere and isospectral matrix flows via Zeitlin's matrix discretization, enabling a matrix–geometry dictionary that preserves Lie–Poisson structure and Casimirs. It constructs a representation-theoretic quantization of smooth vorticity, defines the Euler–Zeitlin dynamics on , and proves space–time convergence of the matrix model to the continuum Euler flow as , including convergence of empirical spectral measures to the level-set measures. By analyzing coadjoint orbits, closures, and Schur–Horn–Kostant convexity, it situates the finite-dimensional dynamics between smooth orbits and weak closures, and connects these ideas to mixing operators and canonical scale separation. The work also proposes a long-time behavior conjecture for vorticity, ties vortex mixing to weak- convergence, and highlights how Zeitlin’s model serves as a calculable framework to study vortex condensation, integrability, and the long-time structure of 2-D turbulence on the sphere.

Abstract

Two-dimensional (2-D) incompressible, inviscid fluids produce fascinating patterns of swirling motion. How and why the patterns emerge are long-standing questions, first addressed in the 19th century by Helmholtz, Kirchhoff, and Kelvin. Countless researchers have since contributed to innovative techniques and results. Yet, the overarching problem of swirling 2-D motion and its long-time behavior remains largely open. Here we shed light on this problem via a link to isospectral matrix flows. The link is established through V. Zeitlin's beautiful model for the numerical discretization of Euler's equations in 2-D. When considered on the sphere, Zeitlin's model offers deep connections between 2-D hydrodynamics and unitary representations of the rotation group. Consequently, it provides a dictionary that maps hydrodynamical concepts to matrix Lie theory, which in turn gives connections to matrix factorizations, random matrices, and integrability theory, for example. Results about finite-dimensional matrices can then be transferred to infinite-dimensional fluids via quantization theory, which is here used as an analysis tool (albeit traditionally describing the limit between quantum and classical physics). We demonstrate how the dictionary is constructed and how it unveils techniques for 2-D hydrodynamics. We also give accompanying convergence results for Zeitlin's model on the sphere.
Paper Structure (10 sections, 20 theorems, 93 equations, 6 figures, 1 table)

This paper contains 10 sections, 20 theorems, 93 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background: geometry of the 2-D Euler equations
  3. Quantization and the Euler--Zeitlin equation
  4. Quantization via representation
  5. Coadjoint orbits and their closure
  6. Schur-Horn-Kostant convexity
  7. Canonical splitting and mixing operators
  8. A conjecture on the long-time behavior and integrability
  9. Proof of Theorem \ref{['thm:convergence']}
  10. Relation to Berezin-Toeplitz quantization

Key Result

Lemma 1

$\mathfrak{u}(N)\simeq V_0\oplus \cdots\oplus V_{N-1}$ where $V_\ell$ decomposes orthogonally as The components $V_{\ell m}$, corresponding to the weights $\pm 2 m$, are mapped to the $\pm m$th diagonals of the matrix in $\mathfrak{u}(N)$.

Figures (6)

  • Figure 1: A numerical simulation of the vorticity field for Euler's equations on $\mathbb{S}^2$. The results are displayed using the Mollweide area preserving projection. For random smooth initial data (a), the typical dynamical behavior is a mixing phase (b) where vorticity regions of equal sign undergo merging, followed by a long-time phase (c) of 3 or 4 remaining large, weakly interacting vortex condensates whose centers of mass move along nearly quasi periodic trajectories.
  • Figure 2: (a) Measured Reeb graph for a function $\omega$ on $\mathbb{S}^2$ with three local maxima and two local minima. Another function $\psi$ has the same measured Reeb graph if and only if there exists $\Phi\in\operatorname{Diff}_\mu(\mathbb{S}^2)$ such that $\psi = \omega\circ\Phi$ (cf. Theorem \ref{['thm:coadjoint_reeb']}). (b) Level-set measure $\lambda_\omega$. It is a flattening, via horizontal projection, of the Reeb graph. The measure $\lambda_\omega(I)$ of an interval $I$ is the area of the set $\{x\in \mathbb{S}^2\mid \omega(x)\in I \}$.
  • Figure 3: Given a vortex configuration $\omega_0$ with one blob, it is possible to find a sequence $\Phi_k$ of area-preserving diffeomorphisms which transports it $\omega_0\circ\Phi_k^{-1}$ such that in the limit $k\to\infty$ it is $L^\infty$-indistinguishable from a configuration with two blobs. Thus, the Reeb graph in Theorem \ref{['thm:coadjoint_reeb']} is not stable under the $L^\infty$ closure of $\mathcal{O}(\omega_0)$.
  • Figure 4: The simulation $W(t)\in \mathfrak{su}(512)$ in Fig. \ref{['fig:non-vanishing']} is projected to $\bar{W}(t) \in \mathfrak{su}(480)$. The empirical spectral measure $\lambda^{480}_{\bar{W}(t)}$ is then displayed at the initial (left) and final (right) time. We observe some tendency towards the Wigner semicircle distribution (dashed curve), except the rims survive due to vortex blobs (compare Fig. \ref{['fig:non-vanish-final']}).
  • Figure 5: Canonical splitting $W=W_s + W_r$ of the long-time state $W$ from Fig. \ref{['fig:non-vanish-final']}. The components capture the large and small scales. The projection $W\mapsto W_s$ is a mixing operator.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 2: HoYa1998
  • Remark 3
  • Theorem 3: ChPo2018
  • proof
  • Theorem 4
  • proof
  • ...and 27 more