Dimensional regimes in Kolmogorov Flow

TL;DR

This study quantifies the dimensionality of two-dimensional Kolmogorov flow over a broad range of $Re$ and forcing scales $k_f$ using both convolutional autoencoders and Kaplan–Yorke dimensions from Lyapunov analysis. It identifies two distinct transitions: first, loss of stability of a periodic orbit, and second, saturation of large-scale motions, with both transitions collapsing onto a universal value when expressed through the forcing Reynolds number $Re_f$ across $k_f\in\{2,4,8\}$. Autoencoder-based dimensionality saturates at the second transition, revealing that large-scale dynamics are fully developed, while subsequent increases are dominated by small-scale nonlinearities; the Kaplan–Yorke dimension saturates earlier, reflecting the baseline linear dynamics. Importantly, the saturation levels scale linearly with $k_f$, indicating that active degrees of freedom grow with forcing scale rather than with the total number of Fourier modes, and highlighting a strong link between data-driven and dynamical-systems perspectives in 2D turbulence. These findings have implications for reduced-order modeling by clarifying how dimensionality distributes across scales and forcing configurations in Kolmogorov-type flows.

Abstract

We study the dimensionality of two-dimensional Kolmogorov flows over a wide range of Reynolds numbers and forcing wavenumbers $k_f=\{2,4,8\}$ using two complementary approaches: convolutional autoencoders and a Kaplan-Yorke estimation based on Lyapunov analysis. As the Reynolds number increases, two distinct transitions are observed: the first corresponds to the destabilization of a periodic orbit, while the second marks the saturation of the large-scale motions. When expressed in terms of the forcing Reynolds number, these transitions occur at nearly the same value for all forcing wavenumbers, suggesting a universal scaling with respect to the forcing scale. By filtering the data to retain only the large-scale range ($k < k_f$), we show that the dimensionality estimated by the autoencoders also saturates at the second transition, implying that once the large scales are fully developed, the subsequent increase in dynamical activity occurs predominantly at smaller scales. At higher Reynolds numbers, the Kaplan-Yorke dimension ceases to grow, revealing its limited sensitivity to the nonlinear interactions that dominate in this regime. Both the Kaplan-Yorke saturation dimension and the filtered large-scale dimensionalities exhibit a linear dependence on $k_f$, indicating that the number of active degrees of freedom scales with the forcing scale rather than with the total number of available Fourier modes.

Figures (13)

• Figure 1: Minimum latent dimension $d^*$ and Kaplan-Yorke dimension $d^\dag$ as function of the Reynolds number $\mathrm{Re}$ for $k_f=4$.
• Figure 2: Lyapunov spectra $\lambda_i$ vs. index $i$ for two Reynolds numbers, ordered from largest to smallest. Only the leading exponents are shown. Dashed vertical lines mark their respective Kaplan–Yorke dimensions $d^\dag$.
• Figure 3: Vorticity fields for Kolmogorov flows at $k_f=4$ and $\mathrm{Re}=28$, $60$, and $200$. Each snapshot uses its own color scale (colorbars are not shared).
• Figure 4: Dissipation $D$ for Kolmogorov flow at $k_f = 4$ and $\mathrm{Re} = 28,\,60,\,200$. Time evolution of $D$ in units of turnover times; horizontal dashed lines mark the time-averaged values $\langle D \rangle_t$ for each case.
• Figure 5: Maximum Floquet multiplier $\max|\lambda|$ versus $\mathrm{Re}$ for the same orbit family.