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

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 ) ∝ ω − 4 and either a strong-turbulence regime E ( ω ) ∝ ω − 2 or the Phillips critical-balance spectrum E ( ω ) ∝ 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 ∈ [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 * ≃ 2.7. To our knowledge, this is the first example of a discretely self-similar blowup in the kinetic theory of waves.