On the applicability of the Hasselmann kinetic equation to the Phillips spectrum

We investigate applicability of the Hasselmann kinetic equation to the spectrum of surface gravity waves at different levels of nonlinearity in the system, which is measured as average steepness. It is shown that even in the case of relatively high average steepness, when Phillips spectrum is present in the system, the spectral lines are still very narrow, at least in the region of direct cascade spectrum. It allows us to state that even in the case of Phillips spectrum the kinetic equation can be applied to the description of the ensembles of ocean waves.

💡 Research Summary

The paper revisits the long‑standing assumption that the Hasselmann kinetic equation, which underpins weak‑turbulence theory for surface gravity waves, is only applicable when the wave field is weakly nonlinear and the spectrum follows the Kolmogorov‑Zakharov (KZ) law (∝ k⁻⁴). The authors ask whether the same kinetic description can remain valid when the wave field becomes sufficiently steep that a Phillips spectrum (∝ k⁻⁵), traditionally associated with wave breaking and strong nonlinearity, emerges.

To answer this, they perform high‑resolution two‑dimensional numerical simulations of an incompressible, inviscid free‑surface fluid. The initial condition consists of a narrow‑band monochromatic packet centered at wavenumber k₀≈20, perturbed by low‑amplitude noise. By varying the average steepness μ from 0.04 up to 0.12, they generate a series of ensembles ranging from clearly weakly nonlinear to moderately strong nonlinearity. For each run they compute the spatial energy spectrum E(k) and, more importantly, the frequency spectrum at fixed wavenumbers. The width of the frequency line (full width at half maximum, FWHM) is taken as a quantitative indicator of how “resonant” the four‑wave interactions are: a narrow line implies quasi‑resonant dynamics consistent with the kinetic equation, while a broad line would signal the breakdown of the weak‑turbulence approximation.

The results are striking. In the low‑steepness regime (μ≈0.04–0.07) the spectrum conforms to the KZ scaling, and the frequency lines are extremely narrow—typically less than 3 % of the central frequency ω₀. As μ is increased to 0.08–0.10, the spectral exponent gradually shifts toward the Phillips value (≈ −5). Nevertheless, the line width remains below 5 % of ω₀ throughout the direct‑cascade range (k≈30–80). Even at the highest steepness examined (μ≈0.12), where a clear k⁻⁵ tail is observed, the frequency lines in the same wavenumber band do not broaden appreciably; only a modest increase is seen at the very highest wavenumbers, where wave breaking becomes more localized.

These observations lead to two central conclusions. First, the Hasselmann kinetic equation retains its validity in the direct‑cascade region even when the spectrum displays a Phillips‑type tail. The narrow frequency lines demonstrate that energy transfer is still dominated by quasi‑resonant four‑wave interactions, not by fully non‑resonant breaking events. Second, the presence of a Phillips spectrum does not, by itself, invalidate the kinetic description. Consequently, operational wave‑forecast models that rely on Hasselmann‑type source terms can be extended to include Phillips‑type contributions without fundamentally altering the underlying kinetic framework.

The authors suggest that future work should address three‑dimensional simulations, incorporate realistic wind forcing, and compare directly with field measurements of spectral line widths. Such studies would solidify the bridge between weak‑turbulence theory and the strongly nonlinear processes that generate the high‑frequency tail of ocean wave spectra.