Wave-induced drift in third-order deep-water theory

TL;DR

This work investigates particle drift under unidirectional deep-water waves up to third order using the Zakharov-Krasitskii Hamiltonian to separate weakly nonlinear effects, including bound harmonics and mutual dispersion corrections. It introduces the constant-magnitude approximation to capture nonlinear frequency shifts and constructs the free-surface and bulk-potential expansions up to cubic order, enabling direct comparison of Stokes drift with full trajectory mappings across monochromatic, bichromatic, and multichromatic seas. A key finding is that incorporating difference harmonics substantially improves agreement with nonlinear wave theories, especially with depth, across regular and irregular spectra. The results have practical implications for ocean transport modeling, including energy spectra representations and surface-to-depth drift predictions in realistic sea states.

Abstract

The goal of this work is to investigate particle motions beneath unidirectional, deep-water waves up to the third-order in nonlinearity. A particular focus is on the approximation known as Stokes drift, and how it relates to the particle kinematics as computed directly from the particle trajectory mapping. The reduced Hamiltonian formulation of Zakharov and Krasitskii serves as a convenient tool to separate the effects of weak nonlinearity, in particular the appearance of bound harmonics and the mutual corrections to the wave frequencies. By numerical integration of the particle trajectory mappings we are able to compute motions and resulting drift for sea-states with one, two and several harmonics. We find that the classical formulation provides a slight underestimate of the drift at the surface, and a slight overestimate at depth. Incorporating difference harmonic terms into the formulation yields an improved agreement with the drift obtained from nonlinear wave theories, particularly at greater depth. The consequences of this are explored for regular and irregular waves, as well as parametric wave spectra.

Figures (18)

• Figure 1: Free surface of a bichromatic wave train with linear frequencies $\omega_1=1$ rad/s and $\omega_2=1.25$ rad/s and wave slopes $\epsilon_1=\epsilon_2=0.15,$ showing 1st order (linear), 2nd order, and 3rd order solutions and their bound wave (B.W.) constituents. Note the phase shift appearing at 3rd order due to the frequency correction \ref{['eq:Omega-nonlin-bichrom-short-wave-1']}--\ref{['eq:Omega-nonlin-bichrom-long-wave-2']}.
• Figure 2: A linear, monochromatic wave profile with $k=1$ 1/m and $H=0.4$ m propagating in the positive $x$-direction. Particle paths are denoted by coloured curves, with solid curves denoting the explicit integration of the particle trajectory ODEs \ref{['eq:lin-part-path-ODE-x']}--\ref{['eq:lin-part-path-ODE-z']} and dashed curves the circular trajectories with Stokes drift \ref{['eq:lin-monoch-Stokes-drift']} obtained from the approximate linear theory. The Stokes drift is shown in thin, dotted curves connecting the initial position (filled circle) and the final position (diamond) obtained from \ref{["eq:x'-2nd-order-Taylor"]}--\ref{["eq:z'-2nd-order-Taylor"]}.
• Figure 3:
• Figure 4: Comparison of Lagrangian drift with depth beneath monochromatic waves with $k=1$ 1/m and (a) $H=0.3$, (b) $H=0.45$ and (c) $H=0.6$. Particle trajectories are obtained at 1st, 3rd and 5th order from integration of the particle trajectory ODEs, and compared with the Stokes drift approximation \ref{['eq:lin-monoch-Stokes-drift']} (SD) and the fourth order Lagrangian solution blaser2025increased.
• Figure 5: Time series of a bichromatic wave with $k_1 = 1.2$ and $k_2=1$ 1/m, $\epsilon_1=\epsilon_2=0.1$ at $x=0$ (a), and accompanying particle trajectories at $z_0=-0.35$ m (b)--(c) and $z_0=-4$ m (d)--(e). Blue curves denote 1st order theory, red curves 2nd order theory, and yellow curves 3rd order theory in all panels. Note that panels (c) and (e) show particle paths from 1st order theory, 2nd order contributions only (red curves) and 3rd order contributions only (yellow curves), without the attendant lower order velocities. Markers '+' in panels (a) and (e) demarcate times between $t\approx -5.2..5.2$ s when the flow at depth is opposite the direction of wave propagation.