Abstract
The nonlinear Schrödinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile eta , and the fidelity of such profiles provided by the nonlinear Schrödinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential phi can be reconstructed in a similar way as the free surface profile. This observation opens up a range of potential applications since the nonlinear Schrödinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrödinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.
Highlights
Recent years have seen a flurry of activity aimed at understanding the motion of fluid particles in free surface flows
While there is an abundance of studies investigating the reconstruction of the surface profile η, and the fidelity of such profiles provided by the nonlinear Schrödinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid
The focus of this paper is to examine particle trajectories associated with wave motion which can be approximately described by the nonlinear Schrödinger (NLS) equation and a few of its generalizations
Summary
Recent years have seen a flurry of activity aimed at understanding the motion of fluid particles in free surface flows. Using both experiments and high-order asymptotics, a strong case is made in [9] that there is a net forward drift throughout the fluid column These results are in line with mathematical advances in the understanding of particle motion in free surface flow (see [16] for a review). The focus of this paper is to examine particle trajectories associated with wave motion which can be approximately described by the nonlinear Schrödinger (NLS) equation and a few of its generalizations. This equation arises in the case where a carrier wave of a certain frequency is modulated slowly. Somewhat different procedures have recently been used to understand properties of particle motion in the context of the narrow-banded spectrum approximation in the presence of shear flows [12] and in the presence of point vortices [11]
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