Nonlinear phase synchronization and the role of spacing in shell models

TL;DR

The paper investigates how triadic phase synchronization governs energy transfer and intermittency in shell-model representations of turbulence. By deriving phase dynamics and introducing global and local Kuramoto order parameters, it shows that stronger phase locking among consecutive triads correlates with burst-like energy flux and enhanced intermittency, and that reducing inter-shell spacing amplifies this effect in GOY-like models. It also explores helical shell models, revealing topology-dependent variations where some classes suppress intermittency while others preserve forward cascades, and analyzes inverse cascades where phase organization is weaker but still detectable. The findings provide a mechanistic link between phase coherence and cascade statistics, offering diagnostic tools that could extend to more complex simulations and network-based perspectives on turbulence.

Abstract

A shell model can be considered as a chain of triads, where each triad can be interpreted as a nonlinear oscillator that can be mapped to a spinning top. Investigating the relation between phase dynamics and intermittency in a such a chain of nonlinear oscillators, it is found that synchronization is linked to increased energy transfer. In particular, the results provide evidence that the observed systematic increase of intermittency, as the shell spacing is decreased, is associated with strong phase alignment among consecutive triadic phases, facilitating the energy cascade. It is shown that while the overall level of synchronization can be quantified using a Kuramoto order parameter for the global phase coherence in the inertial range, a local, weighted Kuramoto parameter can be used for the detection of burst-like events propagating across shells in the inertial range. This novel analysis reveals how partially phase-locked states are associated with the passage of extreme events of energy flux. Applying this method to helical shell models, reveals that for a particular class of helical interactions, a reduction in phase coherence correlates with suppression of intermittency. When inverse cascade scenarios are considered using two different shell models including a non local helical shell model, and a local standard shell model with a modified conservation law, it was shown that a particular phase organization is needed in order to sustain the inverse energy cascade. It was also observed that the PDFs of the triadic phases were peaked in accordance with the basic considerations of the form of the flux, which suggests that a triadic phase of π/2 and -π/2 maximizes the forward and the inverse energy cascades respectively.

Figures (16)

• Figure 1: Plot of the average energy spectrum as a function of the wave number obtained for the 3D GOY model with inter-shell spacing $g=2$, driven by large-scale forcing. The bottom-left inset shows the average spectral energy flux as a function of the shell wave number $k_{n}$. In the top-right inset the normalized PDFs of the triad phases $\varphi_{n}$ are shown for different shells corresponding to the inertial range. The shell number is represented by different colors labeled by the continuous colorbar.
• Figure 2: Normalized probability distribution functions of the Kuramoto order parameter $R$ and $\Phi$, for various shell models exhibiting different type of cascades: a classical GOY model displaying a forward energy cascade; a helical shell model with elongated triads driving an inverse cascade (see Sec. \ref{['sec:Helical-shell-models']} and \ref{['subsec:Inverse-helical']} for a detailed discussion and the definition of the Kuramoto parameter in the context of helical shell models); and the model investigated in Ref. tom:2017 that drives an inverse cascade via local triads (see \ref{['subsec:inverse-ray-model']} below).
• Figure 3: Individual and joint statistics of energy flux, and the amplitude and phase of the Kuramoto parameter. In panel (a), located at the top-left, the PDF of the local spectral energy flux $\Pi_{n}^{E}$ is shown, exhibiting the typical behavior where the distribution develops increasingly heavy tails for larger shell indices.In panel (b) at the top right, the PDF of the local Kuramoto parameter $R_{n}$ is presented, displaying a scale-invariant universal statistics with higher probability of coherent local $R_{n}$. Curve colors correspond to different shell numbers, as indicated by the continuous colorbar. In panel (c) at the bottom left, the joint PDF between $\Pi_{n}^{E}$ and $R_{n}$ is shown, which illustrates how extreme flux events are predominantly correlated with higher local synchronization events. In panel (d) at the bottom right, the joint PDF between $\Pi_{n}^{E}$ and $\Phi_{n}$ is shown , complementary to panel (c), that present the clustering of triad phases around $\pi/2$ which is associated with larger flux values.
• Figure 4: Scaling of the structure functions $S_{p}(k_{n})$ (orders $p=2,3,\dots6,7$) as a function of the shell wave-number $k_{n}$, shown for two representative values of the inter-shell spacing $g=1.4$ and $g=2$. The right panel ($g=2$) presents the typical intermittency corrections, while the left panel ($g=1.4$) presents an enhancement of intermittency. The insets display the PDFs of the triadic phases $\varphi_{n}$ for different shells in the inertial range, with colors indicating shell index $n$ as shown in the colorbar.
• Figure 5: The scaling exponent $\xi(p)$ of the $p-$th order structure function as a function of $p$, for different shell spacings. The dashed line represents the Kolmogorov $p/3$ scaling.