A brief introduction to matrix hydrodynamics

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 $ {\texttt{quflow}}$ simulation in Section 4, with initial data as in Figure 2. 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 2. 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 2, and the matrix size is $ N = 512 $. The enstrophy is a convex Casimir and therefore decays, as predicted by Proposition 5.2

Figure 6.  Visualization of the long-time spectral enstrophy components $ (\omega_\ell^N)^2 = \sum_{m = -\ell}^\ell (\omega_{\ell, m}^N)^2 $ in the presence of canonical dissipation of varying magnitude $ \kappa $. The initial data is the same as in Figure 2, and the matrix size is $ N = 512 $. The forward cascade of enstrophy is damped out. In order to qualitatively preserve the first part of the spectrum, the dissipation parameter must be small; for this simulation, at least $ \kappa \leq 10^{-3} $. Notice that the canonical dissipation fails to conserve angular momenta; namely, the components with $ \ell = 1 $ are not conserved. We cannot explain the growth toward the end of the spectrum for $ 0 < \kappa \leq 10^{-4} $; it seems to be an artifact of the quantization

Figure 7.  Evolution of the relative energy error for varying matrix sizes $ N $. The initial data is the same as in Figure 2. Notice that the energy is "nearly conserved" in the sense compatible with the backward error analysis for symplectic integrators. There is no indication that the errors grow with $ N $. We currently lack a rigorous analysis of this behavior, since backward error analysis, as developed for finite-dimensional Hamiltonian systems, prevents control over $ N $