Stokes drift

Stokes drift is the net horizontal velocity experienced by fluid particles in a progressive wave field, resulting from the nonlinear superposition of oscillatory wave motion, which causes particles to follow closed orbital paths but with a systematic forward displacement in the direction of wave propagation.^[1] This phenomenon arises because particles spend more time traveling forward under wave crests than backward under troughs, leading to a mean Lagrangian velocity that exceeds the Eulerian mean flow.^[2] First derived theoretically by George Gabriel Stokes in 1847 for irrotational surface gravity waves, it quantifies the wave-induced transport absent in linear wave theory.^[1] In deep water, the surface Stokes drift velocity is approximately u_s = a^2 \omega k, where a is the wave amplitude, \omega is the angular frequency, and k is the wavenumber, decaying exponentially with depth as e^{-2kz} (with z increasing downward).^[1] For finite-amplitude waves, this drift drives significant mass transport, such as the movement of floating debris or plankton, and influences upper ocean mixing and circulation.^[3] The concept extends to various wave types, including internal waves and multidirectional seas, where it contributes to phenomena like Langmuir circulation through the Craik-Leibovich vortex force.^[2] Stokes drift plays a critical role in oceanographic modeling, affecting predictions of pollutant dispersion, oil slick trajectories, and marine ecosystem dynamics, with global estimates indicating it can enhance effective surface currents by up to several cm/s in windy conditions.^[3] Despite its foundational importance, debates persist regarding its precise definition in generalized Lagrangian mean frameworks, particularly for compressible flows or wave packets where vertical components may transiently appear.^[2]

Overview and Physical Principles

Definition and Historical Context

Stokes drift refers to the net drift velocity experienced by fluid particles in a wave field, arising from the asymmetry in the orbital motions of progressive waves, where particles move forward more than backward over each wave cycle. This phenomenon results in a steady, wave-induced transport that is distinct from the mean Eulerian flow, which represents the average velocity at fixed points in the fluid.^[4] The concept was first described by George Gabriel Stokes in his 1847 paper "On the Theory of Oscillatory Waves," presented to the Cambridge Philosophical Society. Stokes developed this idea while investigating irrotational wave theory, motivated in part by John Scott Russell's experimental observations that wave propagation velocity remains independent of wave height for small amplitudes. In the paper, Stokes noted that "the forward motion of the particles is not altogether compensated by their backward motion; so that, in addition to their motion of oscillation, the particles have a progressive motion in the direction of propagation of the waves." Originally formulated for small-amplitude waves in inviscid, incompressible fluids under the assumptions of irrotational flow, the theory has since been extended to broader contexts, including finite-amplitude waves and viscous effects, while retaining its foundational role in understanding wave-induced mean flows.^[4]

Physical Mechanism and Intuition

Stokes drift arises from the inherent asymmetry in the orbital motion of fluid particles within progressive surface gravity waves. In such waves, particles do not simply oscillate symmetrically around a fixed point; instead, they follow closed elliptical paths that result in a net forward displacement over each wave period. This occurs because the forward-moving phase under the wave crest involves greater horizontal excursion and higher velocities compared to the backward motion under the trough, primarily due to the steeper profile on the forward face of the wave. Consequently, particles spend more time and cover more distance in the forward direction, leading to an overall drift in the direction of wave propagation.^[5] To build intuition, consider the trajectories in the Lagrangian frame, which tracks individual particles. Here, the orbital paths appear as closed loops—nearly circular in deep water—but these loops slowly shift forward with each wave cycle, accumulating the net drift. This contrasts sharply with standing waves, where the symmetric superposition of opposing wave components results in purely oscillatory motion with no net displacement, as the forward and backward phases balance exactly. Visualizing this, a floating particle like a cork on the water surface bobs in an ellipse, rising and falling with the wave while gradually advancing with the wave train, a kinematic effect independent of viscosity.^[6]^[5] The key distinction lies between the Lagrangian mean velocity, which follows the particle and reveals the Stokes drift, and the Eulerian mean velocity, measured at fixed points in space and averaging to zero for irrotational waves. In the Eulerian view, the wave-induced velocities cancel out over a cycle at any stationary location, masking the true transport of material. For instance, observing a particle's path over multiple waves shows it returning close to but slightly ahead of its starting position, embodying the Lagrangian perspective that captures the drift as the difference between these two averaging methods.^[5]

Mathematical Derivation

General Formulation for Inviscid Fluids

The general formulation of Stokes drift is derived for progressive gravity waves propagating on the surface of an inviscid, incompressible fluid under the assumptions of irrotational flow and small-amplitude waves, where the wave steepness ε = ka ≪ 1, with k the wavenumber and a the wave amplitude.^[7] The irrotational condition allows the velocity field \mathbf{u} to be expressed as the gradient of a velocity potential \phi, satisfying Laplace's equation \nabla^2 \phi = 0 throughout the fluid domain.^[7] The governing equations are the Euler equations for inviscid flow, supplemented by kinematic and dynamic boundary conditions at the free surface, which introduce nonlinearity through the convective acceleration term (\mathbf{u} \cdot \nabla) \mathbf{u}.^[7] To solve these equations perturbatively, the velocity potential is expanded in powers of the small parameter ε representing the wave steepness: \phi = \varepsilon \phi_1 + \varepsilon^2 \phi_2 + \cdots, where the leading-order term \varepsilon \phi_1 corresponds to the linear wave solution, and higher-order terms account for nonlinear corrections.^[7] Similarly, the free-surface elevation \eta is expanded as \eta = \varepsilon \eta_1 + \varepsilon^2 \eta_2 + \cdots. The boundary conditions are applied at the mean surface level z = 0 (with z increasing upward), and the perturbation expansion is substituted into the dynamic and kinematic conditions, yielding a hierarchy of linear problems solved order by order.^[7] The second-order terms introduce steady contributions that represent the mean drift. The Stokes drift emerges as the difference between the mean Lagrangian velocity \langle \mathbf{u}_L \rangle—the average velocity following fluid particles—and the Eulerian mean velocity \langle \mathbf{u}_E \rangle at fixed points, with \langle \mathbf{u}_L \rangle = \langle \mathbf{u}_E \rangle + \langle \boldsymbol{\xi}^{(1)} \cdot \nabla \mathbf{u}^{(1)} \rangle, where \boldsymbol{\xi}^{(1)} is the first-order displacement and \langle \cdot \rangle denotes time-averaging over a wave period; this captures the cumulative effect of the oscillatory displacements on the velocity field.^[7] This drift term arises specifically from the second-order correction in the perturbation expansion, resulting from nonlinear interactions in the convective acceleration, which produce a nonzero time-averaged horizontal velocity in the direction of wave propagation despite the oscillatory nature of the linear solution.^[7] For surface gravity waves in deep water, the horizontal component of the Stokes drift velocity is given by

u_s = a^2 \omega k e^{2kz},

where \omega is the angular frequency, k the wavenumber, and z the vertical coordinate (z = 0 at the mean free surface, decreasing downward).^[7] This expression decays exponentially with depth, reflecting the localization of the drift near the surface, and scales with the square of the amplitude, consistent with its second-order origin.^[7]

Perturbation Expansion Approach

The perturbation expansion approach to deriving Stokes drift involves expanding the velocity potential \phi and the free surface elevation \eta in a series of small parameter \epsilon, typically the wave steepness ka, where a is the wave amplitude and k is the wavenumber, assuming \epsilon \ll 1. This method, originally developed by Stokes, allows for a systematic solution to the nonlinear boundary value problem governing irrotational, incompressible surface gravity waves by solving the equations order by order in \epsilon. The expansions take the form \phi = \epsilon \phi_1 + \epsilon^2 \phi_2 + \cdots and \eta = \epsilon \eta_1 + \epsilon^2 \eta_2 + \cdots, with the phase \theta = kx - \omega t, where \omega is the angular frequency. At leading order, \eta_1 = a \cos \theta.^[7]^[5] The derivation proceeds by satisfying Laplace's equation \nabla^2 \phi = 0 at each successive order, subject to appropriate boundary conditions. At first order (\mathcal{O}(\epsilon)), the solution yields the linear wave approximation: \phi_1 satisfies \nabla^2 \phi_1 = 0, with the linearized kinematic boundary condition \partial \eta_1 / \partial t = \partial \phi_1 / \partial z and dynamic condition from the Bernoulli equation \partial \phi_1 / \partial t + g \eta_1 = 0 evaluated at z = 0, where g is gravitational acceleration. This gives the standard linear velocity potential \phi_1 \propto e^{kz} \sin \theta (for deep water) and surface elevation \eta_1 = a \cos \theta, satisfying the dispersion relation \omega^2 = gk \tanh(kh) for water depth h. The first-order velocity field \mathbf{u}_1 = \nabla \phi_1 oscillates without a net mean over a wave period.^[7]^[5] At second order (\mathcal{O}(\epsilon^2)), the nonlinear terms in the boundary conditions, applied at the exact free surface z = \eta, are Taylor-expanded to z = 0, introducing interactions between first-order terms. Solving \nabla^2 \phi_2 = 0 with the updated kinematic condition \partial \eta / \partial t + \partial \phi / \partial x \partial \eta / \partial x = \partial \phi / \partial z at z = \eta and dynamic condition \partial \phi / \partial t + (1/2) |\nabla \phi|^2 + [g](/page/G) \eta = 0 at z = \eta, both linearized at z = 0, produces oscillatory corrections to \phi_2 and \eta_2 at twice the frequency (e.g., \propto \cos 2\theta). The mean drift arises from averaging the second-order Euler momentum equation over one wave period T = 2\pi / \omega, where the nonlinear convective term \langle \mathbf{u}_1 \cdot \nabla \mathbf{u}_1 \rangle generates a steady Lagrangian mean flow. This yields the second-order Stokes drift velocity \mathbf{u}_s \approx \langle (\partial \mathbf{u}_1 / \partial x) \int \mathbf{u}_1 \, dt \rangle, with the integral representing the first-order particle displacement \boldsymbol{\xi}_1 = \int \mathbf{u}_1 \, dt, and angular brackets \langle \cdot \rangle denoting the period average; for deep water, u_s \approx a^2 \omega k e^{2kz}.^[7]^[5]

Specific Examples and Applications

One-Dimensional Compressible Flow

In one-dimensional compressible flow, Stokes drift arises during the propagation of longitudinal acoustic waves in a fluid where density varies as \rho = \rho_0 + \rho_1, with \rho_0 denoting the uniform background density and \rho_1 the small oscillatory perturbation induced by the wave.^[8] This setup allows for wave motion, unlike the incompressible case where strict density constancy prohibits non-trivial one-dimensional waves. The phenomenon manifests as a net displacement of fluid particles over multiple wave periods, driven by nonlinear interactions between the oscillatory velocity and density fields.^[8] The derivation adapts the standard perturbation expansion by incorporating compressibility effects into the governing equations. The continuity equation is modified to \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} = 0, where u is the velocity, accounting for both density advection and variation. The momentum equation follows the Euler form \rho \left( \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} \right) = -\frac{\partial p}{\partial x}, with pressure p related to density via an equation of state, such as p = c^2 \rho for isentropic flow, where c is the speed of sound. To second order, the Stokes drift velocity emerges from the Lagrangian particle displacement \xi, satisfying \frac{\partial \xi}{\partial t} = u(\xi, t), expanded as \xi = \xi_0 + \epsilon \xi_1 + \epsilon^2 \xi_2 + \cdots, yielding u_s \approx \frac{1}{2} \int \left( u \frac{\partial u}{\partial x} - \frac{1}{\rho} \frac{\partial p}{\partial x} \right) dt after averaging over the fast oscillatory phase. For a monochromatic wave u = \hat{u} \cos(kx - \omega t), this simplifies to a drift u_s = \frac{1}{2} k \hat{u}^2 / \omega, quadratic in amplitude and directed along the propagation.^[8]043<2636:TSDDTV>2.0.CO;2) In the compressible regime, the drift incorporates acoustic wave contributions, resulting in a net particle displacement that accumulates linearly with time for longitudinal waves. For instance, in vertically propagating internal gravity waves within a stratified compressible atmosphere, the vertical Stokes drift is proportional to the vertical energy flux of the wave, u_{s,z} \propto F_z / \rho_0, highlighting transport of momentum and tracers upward or downward depending on wave direction. This contrasts with the incompressible theory, where no density perturbations exist, and one-dimensional flows lack wave propagation; here, terms from \rho_1 in the continuity equation introduce corrections that enhance or modify the drift magnitude by factors involving the Mach number \epsilon = \hat{u}/c, typically reducing it for low-amplitude acoustics.043<2636:TSDDTV>2.0.CO;2)^[8]

Deep Water Waves

In the deep water regime, where the product of the wavenumber k and water depth h satisfies kh \gg 1, surface gravity waves propagate such that the wavelength is much shorter than the depth, leading to negligible influence from the bottom boundary. The dispersion relation simplifies to \omega^2 = gk, where \omega is the angular frequency and g is the acceleration due to gravity.^[7]^[9] The horizontal orbital velocity in the linear approximation is given by

u = a \omega e^{kz} \cos\theta,

where a is the wave amplitude, \theta = kx - \omega t is the phase, x is the horizontal coordinate in the direction of propagation, and z is the vertical coordinate (positive upward, with z = 0 at the mean free surface and decreasing downward). This velocity describes circular particle orbits that decay exponentially with depth at rate k.^[7]^[9] Using a perturbation expansion to second order in wave amplitude, the Stokes drift velocity—the net Lagrangian mean flow induced by the waves—is

u_s = a^2 \omega k e^{2kz}.

This expression reveals an exponential decay with depth at twice the rate of the orbital motion ($2k), resulting in the maximum drift at the surface, where u_s(0) = a^2 \omega k. The drift arises from the correlation between particle displacements and the spatially varying orbital velocities, yielding a steady transport in the wave propagation direction after time-averaging over one period.^[7]^[10]^[9] The surface Stokes drift can equivalently be written as u_s(0) = c (ka)^2, where c = \omega / k is the phase speed, illustrating that the drift scales quadratically with the wave steepness ka. This quadratic dependence underscores its significance for finite-amplitude waves, where steeper waves produce disproportionately larger net transport despite the orbits remaining nearly closed in the linear limit.^[10]^[9]

Implications and Extensions

Oceanographic and Coastal Engineering Relevance

In oceanography, Stokes drift plays a crucial role in net mass transport within wave fields, as it represents the wave-averaged Lagrangian velocity that advects fluid parcels and tracers in the direction of wave propagation.^[1] This transport mechanism enhances alongshore currents by contributing to the overall momentum flux in nearshore environments, where it interacts with the mean flow to drive circulation patterns.^[11] Furthermore, Stokes drift influences pollutant dispersion by facilitating the offshore and alongshore spreading of surface contaminants, such as oil spills, with studies showing it can increase particle dispersal rates by up to 15-20% in wind-wave conditions.^[12] In sediment transport, it affects coastal morphology by promoting the onshore movement of suspended particles, particularly in the surf zone, where it balances with undertow to shape beach profiles over time.^[13] In coastal engineering, Stokes drift significantly impacts wave setup and runup on beaches, as the onshore mass flux it induces elevates the mean water level near the shore, altering flooding risks and shoreline stability.^[14] It is incorporated into models of nearshore circulation to simulate rip currents and undertow, where the drift's interaction with breaking waves generates return flows that prevent excessive shoreline erosion.^[11] For instance, in breaking wave scenarios, Stokes drift contributes to the shear in the surface layer, influencing sediment resuspension and the design of coastal defenses like breakwaters.^[15] Typical magnitudes of Stokes drift for ocean swells range from 3 to 13 cm/s at the surface, decreasing exponentially with depth and varying with wave height and period.^[16] It interacts with wind-driven currents by adding a wave-induced component that can amplify total surface velocities, particularly under fetch-limited conditions where local wind generates short waves.^[16] Historical measurements from ocean buoy observations starting in the 1970s confirmed these velocities, highlighting the drift's role in upper-ocean mixing.^[17] Extensions to higher-order approximations account for enhanced drift in steep waves, where nonlinear effects increase surface velocities by up to 20% compared to linear estimates.^[18] Numerical simulations in models like SWAN integrate Stokes drift profiles to predict wave-current interactions, improving forecasts of coastal inundation and tracer pathways with spectral wave data.^[19]

Relation to Eulerian and Lagrangian Descriptions

In fluid dynamics, the distinction between Eulerian and Lagrangian descriptions is fundamental to understanding Stokes drift. The Eulerian velocity \mathbf{u}_E represents the velocity field measured at fixed points in space, with its time average over a wave period typically zero for pure waves in the absence of mean flows. In contrast, the Lagrangian velocity \mathbf{u}_L tracks the motion of individual fluid particles, incorporating the net displacement due to wave oscillations. The Stokes drift \mathbf{u}_S arises as the difference between these descriptions, such that \mathbf{u}_L = \mathbf{u}_E + \mathbf{u}_S, capturing the mean drift of particles relative to fixed Eulerian points. This relation highlights how waves induce a systematic transport not evident in Eulerian observations alone. The general decomposition of the Lagrangian velocity in terms of Eulerian quantities involves the particle displacement \boldsymbol{\xi}, where the mean position of a particle labeled by \mathbf{x} is \langle \mathbf{x} \rangle = \mathbf{x} + \langle \boldsymbol{\xi} \rangle. Thus, \mathbf{u}_L(\mathbf{x}, t) \approx \mathbf{u}_E(\mathbf{x}, t) + \mathbf{u}_S(\mathbf{x} - \langle \mathbf{x} \rangle, t) + higher-order terms, with the Stokes drift serving as a pseudo-momentum correction that accounts for the wave-induced shift in particle paths. In the small-amplitude limit, \mathbf{u}_S = \langle (\boldsymbol{\xi}_1 \cdot \nabla) \mathbf{u}_1 \rangle, where \langle \cdot \rangle denotes the wave average, and \mathbf{u}_1, \boldsymbol{\xi}_1 are the first-order velocity and displacement. This formulation reveals the Stokes drift as an essential adjustment for reconciling fixed-point measurements with actual particle trajectories. In wave-averaged equations, the Stokes drift manifests in the momentum flux tensor, which includes wave-induced stresses that influence the evolution of the mean flow. Within the generalized Lagrangian mean (GLM) framework, this tensor, often denoted as \mathcal{R}_{ij}, incorporates the pseudo-momentum flux and asymmetric contributions from wave activity, leading to modified conservation laws for angular momentum and energy. The GLM approach, which bridges Eulerian and Lagrangian perspectives, ensures that the Stokes drift corrects the mean momentum equation to reflect the true mass transport, avoiding inconsistencies in Eulerian averaging alone. This theoretical framework connects directly to particle paths in nonlinear waves, where the Stokes drift quantifies the closed but drifting orbits of fluid parcels, with the net displacement per wave cycle given by the integral of \mathbf{u}_S over the period. In nonlinear regimes, higher-order terms in the expansion refine this drift, emphasizing its role in long-term transport without altering the basic asymmetry of wave motion.