Blowup driven by critical balance in a differential kinetic model of gravity wave turbulence

TL;DR

The paper investigates a large-$N$ differential approximation model (N-DAM) for gravity-wave turbulence and shows that finite-time blowup is governed by a Phillips-like, critical-balance regime that drives energy from infrared to ultraviolet frequencies. Time-dependent self-similar analyses across weak, strong, and generic closures reveal that the blowup is robustly Phillips-driven, with a front that leaves behind a $ω^{-5}$ spectrum; a phase-dependent bifurcation to discrete self-similarity occurs around $φ_∗\approx2.7$. The work thus identifies a novel mechanism for finite-time energy transfer in wave turbulence and provides the first explicit example of discretely self-similar blowup in kinetic theory. The results generalize to broader N-DAM settings, offering a controlled framework to study critical balance and blowup in wave-kinetic systems beyond gravity waves.

Abstract

We describe the blowup scenarios in a phase-parametrized differential approximation kinetic model (N-DAM), inspired by the physics of deep water surface gravity waves and recently obtained using large-$N$ summation techniques under a local approximation in wavenumber space. Previous work showed that the model admits steady-state solutions interpolating between the Kolmogorov-Zakharov spectrum $E(ω)\propto ω^{-4}$ and either a strong-turbulence regime $E(ω)\propto ω^{-2}$ or the Phillips critical-balance spectrum $E(ω) \propto ω^{-5}$ at small scales. These solutions reproduce scaling regimes expected in gravity-wave kinetics, suggesting that the N-DAM may serve as an effective augmented version of an earlier differential approximation model introduced by Hasselmann. Here we investigate finite-time blowup in the N-DAM and show that it is generically governed by the critical-balance regime. This leads to a non-Kolmogorov finite-time transfer of the energy from the IR towards the UV for any value of the parameter $φ\in [0,π)$. We observe a bifurcation in the blowup dynamics from continuous to discrete self-similarity as $φ$ is increased above a critical value $φ_*\simeq 2.7$. To our knowledge, this is the first example of a discretely self-similar blowup in the kinetic theory of waves.

Figures (5)

• Figure 1: Left : Cartoon phenomenology of the direct cascade when $\phi \in [0,\pi)$. Right: Steady-state energy spectra observed in numerical simulations with forcing at $\omega \simeq 10^{-1}$, small-scale dissipation at $\omega> 2\times 10^3$, and large-scale damping at $\omega<8\times 10^{-2}$ for $\phi = 0, 0.99\pi, \pi$ in blue, purple, red respectively. The asymptotic scaling solutions $E_{W}, E_{S}, E_{P}$ are indicated by dashed lines. The inset shows the spectra compensated by the weak KZ solution $\propto \omega^4$. The dash vertical lines show the value $\omega_0 = 0.5 P_0^{-1/3}$, indicative of the weak to strong transition.
• Figure 2: Top left: Numerical continuation scheme in the weak regime: The similarity profile $\Omega$ is identified as a global bifurcation of \ref{['eq:ODEweak']} and tracked from the numerical continuation of a branch of limit cycles initiated from a Hopf bifurcation. Here, both axes are rescaled so that the Hopf point is at $(1,-1)$. Top right: Phase portrait of the similarity profile at $\phi=3\pi/4$ in the generic case, obtained from a shooting method --- see text. Inset shows the Phillips coefficient $\Omega_0$ for 5 different values of the phase parameter. Bottom: The anomalous exponent $x_\delta$ as a function of the nonlinearity exponent $\delta$ in the $\delta$-perturbed strong regime.
• Figure 3: Convergence to Phillips in vorticity representation (top). The bottom panels show the corresponding factors $|U|$ determining the transitions. The initial profile is centered at $\omega =10^{-8}$ and has initial peak vorticity $|W|_\infty \simeq 10^{-6}$ (left) and $|W|_\infty\simeq 10^{-4}$ (right).
• Figure 4: Pure or periodic solitonic profiles observed for the DG rescaled vorticity $\tilde{W}(\kappa,\tau) = W/|W|_\infty$. Left: $\phi= 2.675$; Right: $\phi=2.915$
• Figure 5: Left: Vorticity spectrum for $\phi=2.915$. The inset compensates the spectrum by the Phillips asymptotics to reveal the oscillatory part. Right: Extrema $\Omega_0^{\pm}$ of the oscillations as a function of the phase-parameter $\phi$. The black crosses indicate the self-similar values found by the shooting method of § \ref{['ssec:generic']}. Inset shows the oscillatory patterns for various values of $\phi$.