Dimensional regimes in Kolmogorov Flow

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.