A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers

Oloo, Joseph O.; Shrira, Victor I.

Abstract

To get an insight into the dynamics of the oceanic surface boundary layer we develop an asymptotic model of the nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in weakly stratified boundary-layer flows. Although in nature the free-surface boundary layers in the ocean are often weakly stratified due to solar radiation and air entrainment caused by wave breaking, weak stratification has been invariably ignored. Here, we consider an idealized hydrodynamic model, where finite-amplitude three-dimensional perturbations propagate in a horizontally uniform unidirectional weakly stratified shear flow confined to a boundary layer adjacent to the water surface. Perturbations satisfy the no-stress boundary condition at the surface. They are assumed to be long compared with the boundary-layer thickness. Such perturbations have not been studied even in a linear setting. By exploiting the assumed smallness of nonlinearity, wavenumber, viscosity and the Richardson number, on applying triple-deck asymptotic scheme and multiple-scale expansion, we derive in the distinguished limit a novel essentially two-dimensional nonlinear evolution equation, which is the main result of the work. The equation represents a generalization of the two-dimensional Benjamin–Ono equation modified by the explicit account of viscous effects and new dispersion due to weak stratification. It describes perturbation dependence on horizontal coordinates and time, while its vertical structure, to leading order, is given by an explicit analytical solution of the linear boundary value problem. It shows the principal importance of weak stratification for three-dimensional perturbations.

Citation

Oloo, J. O., & Shrira, V. I. (2023). A novel (2+1)-dimensional nonlinear evolution equation for weakly stratified free-surface boundary layers. Journal of Fluid Mechanics, 973, Article A40. https://doi.org/10.1017/jfm.2023.773

• Journal Article Type: Article
• Acceptance Date: Oct 23, 2023
• Online Publication Date: Oct 23, 2023
• Publication Date: Oct 25, 2023
• Deposit Date: Dec 20, 2023
• Publicly Available Date: Dec 21, 2023
• Journal: Journal of Fluid Mechanics
• Print ISSN: 0022-1120
• Electronic ISSN: 1469-7645
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 973
• Article Number: A40
• DOI: https://doi.org/10.1017/jfm.2023.773
• Keywords: Mechanical Engineering; Mechanics of Materials; Condensed Matter Physics; Applied Mathematics
• Additional Information: Copyright: © The Author(s), 2023. Published by Cambridge University Press.; License: This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.; Free to read: This content has been made available to all.