Robust Energy Transfer Mechanism via Precession Resonance in Nonlinear Turbulent Wave Systems

A robust energy transfer mechanism is found in nonlinear wave systems, which favors transfers toward modes interacting via triads with nonzero frequency mismatch, applicable in meteorology, nonlinear optics and plasma wave turbulence. We emphasize the concepts of truly dynamical degrees of freedom and triad precession. Transfer efficiency is maximal when the triads’ precession frequencies resonate with the system’s nonlinear frequencies, leading to a collective state of synchronized triads with strong turbulent cascades at intermediate nonlinearity. Numerical simulations confirm analytical predictions.

Figure 1

Schematic showing the source triad (S) ${\mathbf{k}}_1+{\mathbf{k}}_2={\mathbf{k}}_3$ with all first layer modes (${\mathbf{k}}_4$, ${\mathbf{k}}_5$, ${\mathbf{k}}_6$) and second layer modes ${\mathbf{k}}_7$ and ${\mathbf{k}}_8$. Circles denote algebraically dependent triads, eight triads are shown in total. Arrows indicate hypothetical transfers of enstrophy from S.

Figure 2

Numerical results from case (B) with ${\epsilon }_1=10^{-5}$ showing dimensionless precession (top) and enstrophy transfer efficiency to mode $n_{{\mathbf{k}}_4}(t)$ (bottom). Vertical lines indicate predicted resonances (as ${\epsilon }_1\to 0$) and show strong transfer efficiency at these values when condition (4) is satisfied (horizontal lines, top figure).

Figure 3

Numerical results from case (C) with ${\epsilon }_1={\epsilon }_2=0.1$ at $128^2$ resolution. Left: Enstrophy transfer efficiency against $\alpha$ in ${\text{bin}}_2$ and ${\text{bin}}_3$. Vertical lines denote peaks at $\alpha =26.01$ and $\alpha =180$. Right: Dimensionless precession ${\mathrm{\Omega }}_{\mathbf{a}\mathbf{b}}^{\mathbf{c}}/\sqrt{\mathcal{E}}$ for triad $(1,2)+(5,-2)=(6,0)$ and enstrophy transfer efficiency in ${\text{bin}}_3$, near efficiency peak $\alpha =180$.

Figure 4

Numerical results from full PDE model, case (D) at $128^2$ resolution. Left: Enstrophy transfer efficiency against $\alpha$ in ${\text{Bin}}_3$ and ${\text{Bin}}_4$. Vertical lines denote $\alpha =900$. Right: Dimensionless precession standard deviation (over all interacting triads) and enstrophy transfer efficiency in ${\text{Bin}}_4$, both near efficiency peak $\alpha =900$.

Figure 5

Time-averaged dimensionless enstrophy spectra, compensated for enstrophy equipartition, for various values of $\alpha$ from case (D) at $128^2$ resolution. The vertical line $|\mathbf{k}|=8$ denotes the high wave number boundary of ${\text{Bin}}_1$. This bin contains all the initial enstrophy.