Degrees of universality in wave turbulence

TL;DR

This work investigates universality in wave turbulence across weak and strong regimes, focusing on inverse cascades from UV pumping to IR dissipation. At leading order, weak turbulence yields universal Kolmogorov–Zakharov spectra that are largely independent of interaction details, but next-to-leading corrections reveal UV/IR dependencies that break universality for spin waves due to vertex renormalization. By employing a large-$N$ framework, the authors demonstrate a transition to strong turbulence governed by critical balance, with spectra that depend on the pumping scale and renormalized vertices, sometimes yielding flux-independent, UV-controlled states. The results illuminate how nonlocality can enhance nonlinearity, and they map out how strong turbulence unfolds in spin-wave systems, NSE analogs, and MMT-like models, offering insights with potential relevance to plasma and optical turbulence. The findings underscore the nuanced roles of UV and IR cutoffs in shaping turbulence spectra beyond the traditional weak-turbulence paradigm.

Abstract

Turbulence of weakly interacting waves displays a great deal of universality: independence of the details of the interaction and of the pumping and dissipation scales. Here we study how inverse turbulent cascades (from small to large scales) transition from weak to strong. We find that while one-loop corrections can be dependent on excitation and dissipation scales, new types of universality appear in strong turbulence. We contrast turbulence of spin waves in ferromagnets with turbulent cascades in the Nonlinear Schrödinger Equation (NSE) and in an MMT-like model in higher dimensions having a multiplicative interaction vertex: vertex renormalization gives rise to dependence on the pumping (UV scale) in the former but not in the latter. As a result of this spectral nonlocality, spin-wave turbulence stops being weak if one is sufficiently far from the pumping scale, even when the interaction of waves with comparable wavenumbers is weak. We paraphrase this as: nonlocality enhances nonlinearity. We then describe strong turbulence in a multi-component version of these models with a large number of components. We argue that strong spin-wave turbulence is similar to turbulence of the focusing NSE, as it realizes a critical-balance state. However, UV nonlocality causes the level of spin-wave turbulence at large scales to decrease with increasing pumping level, culminating in a state that is independent of the level of pumping.

Figures (6)

• Figure 1: The leading order kinetic equation (\ref{['KE1']}) encodes the tree-level process of two modes directly scattering into two other modes.
• Figure 2: The contribution of the sextic interaction to the fourth cumulant.
• Figure 3: Feynman diagrams for (a) $\mathcal{L}_+$ and (b) $\mathcal{L}_-$ .
• Figure 4: The large $N$ kinetic equation is found by summing bubble diagrams.
• Figure 5: A sketch of $n_k$ for (a) spin waves (b) focusing nonlinear Schrödinger equation. In (a) $n_k$ is given by the weak turbulence scaling (\ref{['KZspin']}) at high $k$, while for low $k$ it is given by the critical balance scaling (\ref{['CB0']}), so increasing the flux increases the turbulence level at high $k$ but decreases it a low $k$. For the focusing nonlinear Schrödinger equation, studied in Rosenhaus:2025tjxRF2, the spectrum at low $k$ has the same scaling with $k$, but the prefactor is flux independent.