Particle paths in nonlinear Schrödinger models in the presence of linear shear currents

Abstract

We investigate the effect of constant-vorticity background shear on the properties of wavetrains in deep water. Using the methodology of Fokas (A Unified Approach to Boundary Value Problems, 2008, SIAM), we derive a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension. We show that the presence of shear induces a strong coupling between the carrier wave and the mean-surface displacement. The effects of the background shear on the modulational instability of plane waves is also studied, where it is shown that shear can suppress instability, although not for all carrier wavelengths in the presence of surface tension. These results expand upon the findings of Thomas et al. (Phys. Fluids, vol. 24 (12), 2012, 127102). Using a modification of the generalized Lagrangian mean theory in Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) and approximate formulas for the velocity field in the fluid column, explicit, asymptotic approximations for the Lagrangian and Stokes drift velocities are obtained for plane-wave and Jacobi elliptic function solutions of the nonlinear Schrödinger equation. Numerical approximations to particle trajectories for these solutions are found and the Lagrangian and Stokes drift velocities corresponding to these numerical solutions corroborate the theoretical results. We show that background currents have significant effects on the mean transport properties of waves. In particular, certain combinations of background shear and carrier wave frequency lead to the disappearance of mean-surface mass transport. These results provide a possible explanation for the measurements reported in Smith (J. Phys. Oceanogr., vol. 36, 2006, pp. 1381–1402). Our results also provide further evidence of the viability of the modification of the Stokes drift velocity beyond the standard monochromatic approximation, such as recently proposed in Breivik et al. (J. Phys. Oceanogr., vol. 44, 2014, pp. 2433–2445) in order to obtain a closer match to a range of complex ocean wave spectra.

Evolution of Spectral Distributions in Deep-Water Constant Vorticity Flows

“…However, the impact of vorticity was not taken into account in these works, thereby ignoring a central mechanism for the transfer of energy across length scales in oceanic flows. Moreover, as shown recently in [17], linear shear profiles can have strong impacts on the behavior of mean properties of surface wave flows with some shear profiles being able to suppress the Stokes Drift at the surface. Therefore, while a relatively idealized case of shear, the interplay between nonlinearity and constant vorticity can manifest in markedly different physics.…”

“…To describe the evolution of the slowly evolving envelope η 1 (ξ, τ ), we use two models. The first model is the nonlinear Schrödinger equation (NLSE) derived in [17], though see also [4], given by…”

“…Therefore, while a relatively idealized case of shear, the interplay between nonlinearity and constant vorticity can manifest in markedly different physics. Building then on the work in [4] and [17], and complementing the results in [18], we numerically explore the statistical properties of random-nonlinear-wave fields moving over deep water and linear shear profiles. Moving beyond just examining the properties of the NLSE with vorticity, we look at the statistical properties of solutions to a higher order model, the vor-Dysthe equation (VDE) derived in [17].…”

“…Building then on the work in [4] and [17], and complementing the results in [18], we numerically explore the statistical properties of random-nonlinear-wave fields moving over deep water and linear shear profiles. Moving beyond just examining the properties of the NLSE with vorticity, we look at the statistical properties of solutions to a higher order model, the vor-Dysthe equation (VDE) derived in [17]. As shown in [3,5] and [16], the higher order terms associated with the Dysthe equation can have significant impacts on the statistics of the waves, so it is a nontrivial question to examine how the vorticity interacts with these higher order nonlinearities.…”

On the Existence and Instability of Solitary Water Waves with a Finite Dipole

“…There have also been a number of numerical results about vortex pairs in a fluid. The closest to the current problem is the recent paper of Curtis, Carter, and Kalisch [7], who studied how constant vorticity shear profile affects the motion of the particles both at and beneath waves in infinitely deep water. Many authors have looked at the related scenario where a submerged dipole is sent moving towards the free surface rather than moving with the wave; see, for example, [26], [32], [28].…”