Interplay of blowups in a differential kinetic model of gravity wave turbulence

Abstract: We describe the blowup scenarios in a novel phase-parametrized differential approximation model of deep water surface gravity waves that was recently derived in the limit of a large number of multicomponent fields. Previous work showed that upon precisely tuning the phase parameter $\phi$ to $\pi$, the model formally supports steady-state solutions transitioning from Kolmogorov-Zakharov $\propto \omega^{-4}$ to Phillips' critical balance $\propto \omega^{-5}$ at small scales, a feature qualitatively consistent with oceanic measurements. Here we show that the critical balance is in fact a generic feature of the model, which prescribes a finite-time transfer of the energy from the IR towards the UV for any value of $\phi \in [0,\pi)$. We observe a bifurcation in the blowup scenario from continuous to discrete self-similarity as $\phi$ is increased above $\phi_*\simeq 2.7$. To our knowledge, this is the first example of a non-continuous blowup in the kinetic theory of waves.