Stokes Drift in Wave Dynamics

• Stokes drift is the net Lagrangian displacement induced by wave motion, characterized by non-closing particle paths despite zero mean Eulerian velocity.
• Recent studies extend its framework to non-transverse waves, deriving second-order corrections and comparing crest and trough dynamics in various media.
• Oceanographic models rely on accurate vertical profiles of Stokes drift to simulate Langmuir turbulence and account for wave-induced momentum transport.

Searching arXiv for recent and foundational papers on Stokes drift. {"query":"Stokes drift arXiv Stokes drift and its discontents quantum fluid Phillips spectrum third-order deep-water theory", "max_results": 10} Stokes drift is the net Lagrangian drift induced by wave motion: although the Eulerian wave velocity oscillates and may have zero mean at a fixed point, the particle orbit does not close, so a residual displacement remains. In the classical formulation, the Stokes velocity is the wave-averaged difference between mean Lagrangian and Eulerian motion, but the same kinematic mechanism appears more broadly in non-transverse waves in fluids, including sound waves, surface gravity waves, compressible quantum-fluid waves, and wave fields modified by boundaries, turbulence, or electromagnetic forcing (Giuriato et al., 2022, Vanneste et al., 2022, Guha et al., 2023).

1. Definition and kinematic basis

In the standard deep-water gravity-wave setting, Stokes drift is the benchmark Lagrangian transport produced by a progressive wave. For tracers, the textbook deep-water result scales like

$u_S \sim \frac{U^2}{c} e^{2kz},$

where $U$ is the first-order wave-induced velocity amplitude at the surface, $c=\omega/k$ is the phase speed, and $z\le 0$ is depth (Boffetta et al., 2012). In the same setting, the exact spectral profile used in oceanography is

$$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$

with the associated Stokes transport

$V=\int_{-\infty}^0 v(z)\,dz$

(Breivik et al., 2016).

A central recent clarification is that the mechanism is not specific to water waves. Guha and Gupta show that Stokes drift is a generic consequence of particle kinematics in non-transverse waves and compare linear sound waves with intermediate-depth water waves. Using the Lagrangian phase $\theta=\omega t-kx(t)$, they reduce the 2D pathline equation of water waves to 1D and derive second-order-accurate expressions for crest-phase and trough-phase duration, displacement, and average horizontal velocity. In both sound and water waves, the crest phase has larger duration, larger horizontal displacement, and larger average horizontal velocity magnitude than the trough phase, and the imbalance produces a forward drift (Guha et al., 2023). For linear sound waves they obtain

$\overline{u^{\textnormal{SD}}=\frac{c}{2}\epsilon^2,$

while for intermediate-depth water waves they obtain

$\overline{u^{\textnormal{SD}}=\frac{c(1+\mathcal{C}_{\alpha}^2)}{2}\epsilon^2, \qquad \mathcal{C}_{\alpha}=\coth\alpha,\ \alpha=kH$

(Guha et al., 2023).

2. Mean-flow formulations and the ambiguity of “the” Stokes velocity

The classical identity

$$\bm{u}^{\mathrm{L}}=\bm{u}^{\mathrm{E}}+\bm{u}^{\mathrm{S}}$$

remains central, but recent work emphasizes that $U$0 depends on how the Lagrangian mean is defined (Vanneste et al., 2022). In standard GLM, the approximate Stokes velocity is

$U$1

yet this quantity is generally divergent even when the underlying fluid is incompressible: $U$2 For irrotational linear waves, the same paper shows the decomposition

$U$3

where

$U$4

is solenoidal, and the remainder is $U$5 for slowly modulated waves (Vanneste et al., 2022). In the volume-preserving glm theory of Soward & Roberts, specialized and implemented with a Lie-series expansion, $U$6 becomes the natural and sole Stokes velocity, and the resulting Lagrangian-mean momentum equation is formally identical to the Craik–Leibovich equation with $U$7 replacing the conventional $U$8 (Vanneste et al., 2022).

This issue has direct consequences in ocean modeling. One line of work argues that the common “Eulerian-mean hypothesis” for wave-agnostic OGCMs is inconsistent when Stokes drift is significant, because wave-agnostic models cannot accurately simulate the Eulerian-mean velocity if the missing Stokes-Coriolis term is $U$9. The alternative “Lagrangian-mean hypothesis” is then supported both theoretically and by comparison between a wave-agnostic global ocean simulation and an explicitly wave-averaged simulation: the wave-agnostic model accurately simulates the Lagrangian-mean velocity even though the Stokes drift is significant, and the conclusion is that Stokes drift should not be added to ocean general circulation model velocities (Wagner et al., 2022).

3. Beyond ideal tracers: inertia, buoyancy, and quantum-fluid impurities

Once the transported entity is not a perfect tracer, Stokes drift is modified in systematic ways. For inertial particles in deep-water gravity waves, a reduced Maxey–Riley-type model yields a horizontal drift

$c=\omega/k$0

together with a vertical drift

$c=\omega/k$1

Even in the hypothetical case $c=\omega/k$2, the vertical drift survives: $c=\omega/k$3 Thus inertia modifies the horizontal Stokes drift and creates a genuinely dynamical vertical drift (Boffetta et al., 2012).

Floating objects at the free surface depart even more strongly from tracer behavior. For ideal spherical floating marine litter in deep-water, weakly nonlinear, non-breaking waves, the paper by van den Bremer and colleagues shows that larger buoyant objects can have increased drift compared with Lagrangian tracers. The mechanism is variable submergence and the corresponding dynamic buoyancy force components in a direction perpendicular to the local water surface; averaged over the wave cycle, this amplifies the drift relative to the classical Stokes drift (Calvert et al., 2021). The effect becomes substantial when object size is no longer negligible compared with wavelength: a $c=\omega/k$4 m diameter object with density $c=\omega/k$5 in a $c=\omega/k$6 s wave with $c=\omega/k$7 has $c=\omega/k$8, whereas a $c=\omega/k$9 m diameter object in the same wave field has $z\le 0$0 (Calvert et al., 2021).

A different extension arises in compressible quantum fluids. In a Gross–Pitaevskii fluid driven by a Bogoliubov density wave, a finite-size classical impurity obeys a reduced equation

$z\le 0$1

and the mean drift velocity is

$z\le 0$2

The leading-order drift depends on the initial phase $z\le 0$3 and can point either with or against the wave; after phase averaging, the surviving second-order drift is

$z\le 0$4

which recovers the classical Stokes-drift structure but with a coefficient controlled by impurity inertia, added mass, and healing-layer effects (Giuriato et al., 2022).

4. Vertical structure and oceanographic parameterization

In ocean circulation and mixed-layer modeling, the vertical profile of Stokes drift matters as much as its surface value, because the Stokes–Coriolis force depends on $z\le 0$5 and Langmuir turbulence parameterizations depend sensitively on $z\le 0$6 [(Breivik et al., 2016); (Breivik et al., 2014)]. The exact profile is spectrum-dependent, and much recent work concerns accurate approximations from bulk wave quantities.

Profile	Expression	Characteristic
Monochromatic	$z\le 0$7	Too weak near-surface shear
Exponential-integral	$z\le 0$8	Better curvature and depth decay
Phillips-based	$z\le 0$9	Stronger near-surface shear

The monochromatic profile is operationally simple but systematically too smooth near the surface and too penetrating at depth. The exponential-integral profile was introduced as a deep-water alternative based on the same two bulk quantities, the surface Stokes drift $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$0 and the transport $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$1, and it reduces MSE strongly relative to the monochromatic approximation in parametric spectra, ERA-Interim spectra, and buoy data (Breivik et al., 2014). The later Phillips-based approximation exploits the exact Stokes profile for the Phillips $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$2 tail, captures the square-root singularity in the derivative, and uses the same bulk inputs with $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$3 as a robust representative value (Breivik et al., 2016).

The practical performance difference is measurable. In more than $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$4 twenty-minute spectra from a 2 Hz Waverider buoy at Ekofisk in the central North Sea, the average NRMS differences were approximately $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$5 for the monochromatic profile, $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$6 for the exponential-integral profile, and $$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$7 for the Phillips approximation (Breivik et al., 2016). This indicates that high-frequency spectral tails control the near-surface Stokes-drift shear and that bulk-profile closures must represent this tail to remain accurate (Breivik et al., 2016).

5. Modified media, boundaries, and electromagnetic analogues

Stokes drift changes qualitatively when the wave medium or lower boundary departs from the standard single-fluid, rigid-bed configuration. In a two-layer system consisting of an inviscid free-fluid layer over a saturated porous bed representing coral, coupling to Darcy flow makes the wavenumber complex,

$$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$8

and the resulting Stokes drift acquires both horizontal and vertical components in the water and in the porous layer. The upper-layer drift is

$$\mathbf{v}_{\mathrm s}(z) = \frac{2}{g} \int_0^{2\pi}\!\int_0^\infty \omega^3 \hat{\mathbf k}\, e^{2kz} E(\omega,\theta)\, d\omega\, d\theta,$$9

so the vertical drift is controlled by $V=\int_{-\infty}^0 v(z)\,dz$0 and vanishes in the classical rigid-bed limit where $V=\int_{-\infty}^0 v(z)\,dz$1 is real (Webber et al., 2020).

In electrohydrodynamic surface waves on a dielectric fluid under a normal electric field, numerical conformal-mapping calculations show a different outcome: increasing the electric Bond number reduces the wave speed $V=\int_{-\infty}^0 v(z)\,dz$2, increases the drift time $V=\int_{-\infty}^0 v(z)\,dz$3, and changes particle velocities, but the trajectory shapes and the measured Stokes drift

$V=\int_{-\infty}^0 v(z)\,dz$4

remain essentially unchanged (Flamarion et al., 2024). The paper’s summary is that the electric field changes the temporal parametrization of the path but not the spatial orbit geometry or the drift magnitude (Flamarion et al., 2024).

The Stokes-drift concept also appears in plasma electrodynamics. In the inverse Faraday effect for a transversely finite circularly polarized beam, the peripheral “demagnetization current” is reinterpreted as a Stokes-drift current,

$V=\int_{-\infty}^0 v(z)\,dz$5

and inclusion of the longitudinal field doubles the drift current found previously. However, the total drift current is canceled by an equal and opposite magnetization current, so the inverse Faraday magnetization is not reduced (Bekshaev, 2022).

6. Unsteady, higher-order, and fully Lagrangian transport

A distinct development treats Stokes drift not merely as a perturbative correction to Eulerian velocities but as a directly Lagrangian dynamical quantity. For two-dimensional inviscid deep-water gravity waves, one paper identifies the mean drift with the phase-averaged horizontal momentum density,

$V=\int_{-\infty}^0 v(z)\,dz$6

and derives the exact identity

$V=\int_{-\infty}^0 v(z)\,dz$7

with $V=\int_{-\infty}^0 v(z)\,dz$8. At the free surface this gives

$V=\int_{-\infty}^0 v(z)\,dz$9

linking surface mean Lagrangian drift to the geometric mean water level of surface particles (Blaser et al., 2024).

Weakly nonlinear third-order theory further separates the classical Stokes approximation from actual Lagrangian drift. Using the reduced Hamiltonian formulation of Zakharov and Krasitskii and direct integration of particle trajectory mappings, recent work finds that near the surface the linear theory yields the largest forward drift, followed by second and third order, whereas at greater depth the trend reverses and linear theory yields the smallest forward drift. In bichromatic and multichromatic seas, adding difference-harmonic terms to the classical Stokes-drift formula improves agreement with the drift computed from nonlinear wave theory, particularly at depth (Stuhlmeier, 21 Jul 2025).

Freely decaying gravity waves introduce another modification. In high-resolution two-phase simulations and a perturbative model with

$\theta=\omega t-kx(t)$0

wave decay alters the classical Lagrangian drift by introducing first- and second-order corrections and generates a net vertical transport. The long-time horizontal displacement scales like $\theta=\omega t-kx(t)$1, whereas the long-time vertical displacement scales like $\theta=\omega t-kx(t)$2, showing that viscous decay changes both the order structure and the anisotropy of transport relative to the steady-wave Stokes-drift picture (Izawa et al., 27 Apr 2026).

7. Cancellations, anti-Stokes responses, and controversy

Not all treatments regard the observable transport associated with Stokes drift as a simple forward current. Chafin argues that the classical drift predicted by ideal Airy/Stokes theory is typically canceled in actual wave systems by nearly compensating Eulerian flows or by packet-edge processes, especially microbreaking at packet ends. The paper does not deny the mathematical Stokes drift of idealized periodic solutions, but it claims that in physically realizable wave packets a clean uncompensated transport is often absent from experiment because mass and momentum conservation require accompanying return flows or dissipative packet-edge adjustments (Chafin, 2015).

A more recent experimental and theoretical refinement concerns turbulence-induced anti-Stokes flow. When regular waves or wave groups propagate over tailored ambient subsurface turbulence, particle-image-velocimetry measurements show an Eulerian-mean current change concentrated near the surface and directed opposite to the Stokes drift. In the reported experiments, $\theta=\omega t-kx(t)$3 nearest the surface, so the cancellation is partial rather than exact, and the quasi-equilibrium relation

$\theta=\omega t-kx(t)$4

implies that the near-surface ratio between $\theta=\omega t-kx(t)$5 and $\theta=\omega t-kx(t)$6 approximately equals the ratio between the streamwise and vertical Reynolds normal stresses (Ellingsen et al., 11 May 2025). This suggests that the observable Lagrangian transport near the surface can be reduced substantially by the Eulerian response of turbulent water, even though the classical kinematic Stokes drift remains present.

Taken together, these developments show that Stokes drift is best understood as a family of wave-induced Lagrangian transport phenomena rather than a single immutable formula. The classical forward drift of ideal tracers under steady surface gravity waves remains the canonical case, but the measurable transport depends on how the mean is defined, whether the transported object is a tracer, how the wave field evolves, and how boundaries, porosity, turbulence, or additional fields modify the underlying wave kinematics.