Nonlinear dynamics of water waves over nonuniformly periodic bottom

By numerical simulation of exact equations of motion (in terms of conformal variables) for planar non-stationary potential flows of an ideal fluid with a free surface over a strongly non-uniform bottom profile, the effect of nonlinear compression of a long wavepacket during its Bragg reflection from domain of gradually increasing, periodically placed barriers has been detected. In this case, a short and tall packet of standing waves with sharp crests is formed, and then it is transformed into the backward wave. It is essential that with variation of frequency of the incident wave, the effect is absent in the middle of the barrier-induced spectral gap, but it is quite prominent closely to the upper edge of the gap, when the forward wave penetrates deeply into the scattering domain and there, together with emerged backward wave, they form a semblance of Bragg soliton for some time interval.

💡 Research Summary

This paper presents a numerical investigation into the nonlinear dynamics of surface gravity waves interacting with a strongly non-uniform, periodic bottom topography. The primary objective is to explore whether soliton-like structures can emerge naturally from the scattering of a propagating wave packet by an array of bottom barriers, moving beyond idealized initial conditions used in prior studies.

The core methodology employs a high-precision numerical scheme based on conformal variables. This approach maps the time-dependent fluid domain with a free surface and an arbitrary bottom profile onto a simpler canonical domain (a lower half-plane or a horizontal strip). This transformation simplifies the treatment of boundary conditions and allows for the use of efficient spectral methods, leveraging Fast Fourier Transforms. The bottom profile is carefully constructed using analytic functions that enable control over barrier height, shape, and spacing, including the key feature of a gradually increasing barrier height. Wave generation is realistically simulated by applying a localized, time-limited pressure distribution to the initially calm free surface, creating a long wave packet that subsequently interacts with the barrier region.

The central finding is the observation of strong nonlinear wave packet compression during Bragg reflection. When the incident wave frequency lies inside the spectral gap induced by the periodic bottom but close to its upper edge, the forward wave penetrates deeply into the scattering region. There, it coexists with the generated backward wave, forming a transient packet of standing waves. Nonlinearity, coupled with the inherent dispersion, then causes this packet to compress spatially into a short, tall group of waves with sharp crests, reaching amplitudes comparable to the local water depth. This high-amplitude structure eventually transforms into a backward-propagating wave that leaves the barrier region. Crucially, the effect is resonant and frequency-dependent; for waves with frequencies at the middle of the gap, reflection is total and wall-like, with no significant penetration or nonlinear compression occurring.

The study systematically varies parameters, demonstrating the phenomenon for both scenarios with gradually varying water depth and with uniform depth but varying incident wavenumber. It also notes that a sharper transition into the barrier region slightly diminishes the effect. The results align qualitatively with predictions from simplified envelope equation models but provide a full nonlinear, non-hydrostatic simulation within the exact potential flow framework.

In conclusion, the research successfully demonstrates a mechanism for the transient, soliton-like concentration of wave energy through interaction with a non-uniform periodic bottom. However, it concludes that the observed structures are not long-lived Bragg solitons trapped within the array but rather a compelling nonlinear scattering outcome. The work highlights the complex interplay between topography-induced linear scattering (Bragg resonance) and inherent wave nonlinearity, paving the way for further studies on robust energy localization in water waves. The research was supported by a state assignment, and the author declares no conflicts of interest.