A brief introduction to matrix hydrodynamics

TL;DR

This survey presents matrix hydrodynamics as a quantization-based discretization of the 2-D Euler equations on the sphere, centering on Zeitlin's Euler–Zeitlin model and its isospectral, Lie–Poisson structure. It details conservation laws (including angular momentum and Casimirs), a structure-preserving time integrator (the isospectral midpoint), and the long-time behavior that exhibits both coherent vortex dynamics and small-scale noise due to enstrophy conservation. The authors discuss a canonical dissipation that preserves energy while damping enstrophy, and provide extensive numerical demonstrations using the quflow package, highlighting the method's near-energy conservation and applicability to geophysical, plasma, and stochastic contexts. The work positions matrix discretizations as a bridge between geometric fluid dynamics and numerically tractable, long-time simulations, with extensions to reduced models, stochastic variants, and axisymmetric 3-D settings.

Abstract

This survey gives a basic demonstration of matrix hydrodynamics; the field pioneered by V. Zeitlin, where 2-D incompressible fluids are spatially discretized via quantization theory.

Figures (7)

• Figure 1: A skew-Hermitian matrix $W\in\mathfrak{su}(512)$, with eigenvalues $-\mathrm i\lambda_1,\ldots,-\mathrm i\lambda_{512}$ such that $-1\leq \lambda_m\leq 1$ and corresponding eigenvectors $\boldsymbol{e}_m$, is partially reconstructed by $\sum_{\mathrm i\lambda_m \geq \sigma} -\mathrm i\lambda_m \boldsymbol{e}_m\boldsymbol{e}_m^\dagger$ for $\sigma \in [-1,1]$. As seen in the figure, this decomposition corresponds to nullifying the level sets of the vorticity function with values below $\sigma$. The sphere is visualized via the area-preserving Hammer projection.
• Figure 2: Initial vorticity field, generated as a truncated spherical harmonic series $\sum_{\ell=0}^{\ell_{\it max}}\sum_{m=-\ell}^\ell \omega_{\ell,m}Y_{\ell,m}$, where $\ell_{\it max}=20$ and the coefficients $\omega_{\ell,m}$ are normally distributed.
• Figure 3: Evolution of the vorticity field for the quflow simulation in Section \ref{['sec:illustration']}, with initial data as in Figure \ref{['fig:initial_vorticity']}. Vorticity regions of equal sign undergo mixing until four "vortex blob condensates" remain: two positive and two negative. After that, the large-scale motion stabilizes in quasi-periodic interaction between the blobs.
• Figure 4: Visualization of the long-time spectral enstrophy components $(\omega_\ell^N)^2 = \sum_{m=-\ell}^\ell (\omega_{\ell,m}^N)^2$ for varying matrix sizes $N$. The initial data is the same as in Figure \ref{['fig:initial_vorticity']}. On small enough scales, empirically found to be $\ell > \ell^*\approx 20$, enstrophy gets uniformly distributed among the spherical harmonics coefficients $(\omega_{\ell,m}^N)^2$, such that $(\omega_\ell^N)^2 \approx (2\ell+1)\varepsilon^2$. For a smaller $N$, there is less available volume in phase space to distribute enstrophy, resulting in a larger background noise $\varepsilon^2$. The fact that the areas $a$ under the graphs are largely independent of $N$ confirms this discussion.
• Figure 5: Evolution of enstrophy for canonical dissipation with different values of $\kappa$. The initial data is the same as in Figure \ref{['fig:initial_vorticity']}, and the matrix size is $N=512$. The enstrophy is a convex Casimir and therefore decays, as predicted by Proposition \ref{['prop:dissipative_casimirs']}.

Theorems & Definitions (13)

• Proposition 2.1
• Corollary 2.2
• Proposition 3.1
• proof
• Proposition 5.1
• proof
• Proposition 5.2
• Lemma 5.3
• proof
• proof : Proof of Proposition \ref{['prop:dissipative_casimirs']}