On the higher-order wave-induced drift in deep water

The drift associated with the motion of inviscid, irrotational water waves was first derived by Stokes in the mid 19^th century, and is today called Stokes drift. In deep water this takes the form u_s=a²kωe^2kz₀, where a is the wave amplitude, k the wavenumber, ω the radian frequency and z₀ the initial particle depth. This formally second-order quantity is derived from linear theory, and is implemented in a wide variety of wave models to calculate the motion of marine contaminants and other passive tracers.

Adhering to linear wave theory, superposition allows for the immediate generalisation of the Stokes drift from a single wave to a wave spectrum. However, once more than one Fourier mode is included in the lowest order solution, nonlinear effects occurring at second and third order - chief among them the appearance of bound modes - should be considered when calculating Stokes drift.

We introduce a new, analytical correction to the Stokes drift

u_s= ∑_j a_j²ω_jk_je^2k_jz₀+∑_ki>kjω_ia_i²a_j²(k_i-k_j)²(ω_i-ω_j)^-1e^{2(k_i-k_j)z₀}

under assumptions of unidirectional waves and deep water for analytical simplicity - and test this using direct numerical integration of particle paths [1]. Velocity fields for numerical work up to third order are obtained from the reduced Hamiltonian formulation of the water-wave problem due to Zakharov [2], and allow for the inclusion or exclusion of bound harmonics, amplitude evolution and dispersion correction to distinguish among competing effects. In particular, on the typical scale of particle motion the amplitude evolution can be neglected, allowing us to use an algebraic expression for the velocity field in terms of the (initial) Fourier amplitude spectrum [1]. Such an approach has also been successfully employed for deterministic forecasts of the ocean surface [3].

To summarise: we show how higher order contributions to the Stokes drift have an effect throughout the water column. At the surface this is connected to the critical role of high frequencies in the Stokes drift, where dispersion corrections are most influential, as well as contributions from sum-harmonic terms. At greater depths difference harmonics can come to dominate the flow-field and therefore the Stokes drift, as previously demonstrated for wave groups. All of this points to a need to reconsider the common formulation stemming from linear wave theory.

References:

[1] R. Stuhlmeier, Wave-induced drift in third-order deep-water theory, arXiv:2507.15688 (2025).

[2] R. Stuhlmeier, An introduction to the Zakharov equation for modelling deep water waves, D. Henry (ed.) Nonlinear Dispersive Waves (Springer Lecture Notes in Mathematical Fluid Mechanics), Springer (2024), pp. 99-131.

[3] M. Galvagno, D. Eeltink, and R. Stuhlmeier, Spatial deterministic wave forecasting for nonlinear sea-states, Physics of Fluids, (2021) 33 102116