2007/10/08 by Didier Clamond · 53 citations
Earth and Planetary Sciences · Mathematics · #Ocean Waves and Remote Sensing #Coastal and Marine Dynamics #Oceanographic and Atmospheric Processes #Stokes drift #Conservative vector field #Lagrangian #Physics #Classical mechanics #Gravitational wave #Eulerian path #Lagrangian and Eulerian specification of the flow field #Perturbation (astronomy) #Mechanics #Mathematical analysis #Surface wave #Mathematics #Compressibility #Mathematical physics #Optics
paper · doi:10.1017/s0022112007007811
published in Journal of Fluid Mechanics 589, 433-454 (Cambridge University Press)
openalex publication_date 2007/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper concerns the mathematical formulation of two-dimensional steady surface gravity waves in a Lagrangian description of motion. It is demonstrated first that classical second-order Lagrangian Stokes-like approximations do not exactly represent a steady wave motion in the presence of net mass transport (Stokes drift). A general mathematically correct formulation is then derived. This derivation leads naturally to a Lagrangian Stokes-like perturbation scheme that is uniformly valid for all time – in other words, without secular terms. This scheme is illustrated, both for irrotational waves, with seventh-order and third-order approximations in deep water and finite depth, respectively, and for rotational waves with a third-order approximation of the Gerstner-like wave on finite depth. It is also shown that the Lagrangian approximations are more accurate than their Eulerian counterparts of the same order.