Blowup driven by critical balance in a differential kinetic model of gravity wave turbulence
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arXiv
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| Natura: | Preprint |
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2025
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| author | Schubring, Daniel Rosenhaus, Vladimir Thalabard, Simon |
| author_facet | Schubring, Daniel Rosenhaus, Vladimir Thalabard, Simon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23199 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blowup driven by critical balance in a differential kinetic model of gravity wave turbulence Schubring, Daniel Rosenhaus, Vladimir Thalabard, Simon Fluid Dynamics Mathematical Physics Chaotic Dynamics 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. |
| title | Blowup driven by critical balance in a differential kinetic model of gravity wave turbulence |
| topic | Fluid Dynamics Mathematical Physics Chaotic Dynamics |
| url | https://arxiv.org/abs/2509.23199 |