Steady radiating gravity waves: an exponential asymptotics approach (Q6556839)

This research investigates the radiation of steady surface gravity waves induced by a uniform stream over localized bottom topography, a classical problem with applications in geophysical fluid dynamics, such as oceanic flows over seamounts and underwater ridges. The study specifically focuses on subcritical flow conditions in the low-Froude-number limit. In this regime, the downstream wave amplitude is exponentially small relative to the Froude number squared but is determined by a fully nonlinear mechanism, even for small topography amplitudes. This finding contrasts with the limited validity of linear theory in capturing the wave response.\N\NThe authors employ an exponential-asymptotics methodology in the wavenumber domain to address the nonlinear generation of radiating gravity waves. This approach, building on previous work by \textit{T. R. Akylas} and \textit{T. S. Yang} [Stud. Appl. Math. 94, No. 1, 1--20 (1995; Zbl 0823.35155)], is particularly effective for capturing the exponentially small wave amplitudes that elude traditional asymptotic analyses in the physical domain. The study shows that the downstream wave amplitude, although exponentially small, is governed by a nonlinear mechanism across a broad range of flow conditions, a significant departure from the linear predictions.\N\NThe paper's methodology involves analyzing the nonlinear response to localized topography using potential flow theory and asymptotic expansions. By focusing on the residues of poles in the response's Fourier transform, the authors derive expressions for the wave amplitude and phase shift of the radiated waves. The findings indicate that nonlinear effects play a crucial role in determining the wave amplitude, with the amplitude being exponentially small relative to the Froude number squared.\N\NThe main findings of the manuscript demonstrate that the nonlinear mechanism controls the wave response for various flow conditions, rendering the classical linear solutions largely invalid. The study also presents a comparison of the asymptotic results with numerical computations for flow over topography with a sech profile, confirming the robustness of the exponential-asymptotics approach.\N\NIn conclusion, this research significantly advances the understanding of steady radiating gravity waves in subcritical flows with low Froude numbers. By highlighting the limitations of linear theory and providing a comprehensive nonlinear analysis, the study offers valuable insights into the generation and behaviour of surface gravity waves over localized topography. This work has broad implications for geophysical fluid dynamics and contributes to the theoretical foundation necessary for accurately modelling and predicting wave phenomena in natural and engineered systems.