Craik–Leibovich CL2 Instability

• Craik–Leibovich CL2 instability is the interaction between surface-wave Stokes drift and Eulerian shear vorticity that forms streamwise-oriented roll/streak structures, key to Langmuir circulations.
• The mechanism is modeled via the Craik–Leibovich vortex-force framework, coupling prescribed Stokes drift with classical shear flows to elucidate linear and nonlinear instability dynamics.
• Recent advancements incorporate turbulence closures, wave scattering corrections, and geometric reformulations to extend CL2 analysis to ship wakes, laboratory vortices, and fully coupled wave–current systems.

Craik–Leibovich CL2 instability is the classical wave–current instability described in later work as arising from the interaction of Eulerian shear vorticity with the Stokes drift of a surface gravity wave field, producing streamwise-oriented roll/streak structures associated with Langmuir circulations (Kim et al., 29 Aug 2025). In current usage, the term refers both to a specific linear instability problem and, more broadly, to the Craik–Leibovich vortex-force mechanism by which a prescribed or effective Stokes drift reorganizes vortical shear into Langmuir-type rolls. Contemporary research therefore spans canonical instability analyses, turbulence-closure revisions, wave-scattering corrections, nonlinear realizations in ship wakes and laboratory vortices, and geometric or variational reformulations of the Craik–Leibovich equation itself (Basovich et al., 2023, Somero et al., 2018).

1. Definition, scope, and terminological usage

In the most standard description, CL2 is the instability of a wind-driven or otherwise sheared current in the presence of surface-wave Stokes drift, with the unstable structures taking the form of streamwise rolls and streaks. A 2025 statistical-state-dynamics study states this directly, identifying CL2 as the instability that “arises from interaction of the Eulerian shear vorticity with the Stokes drift of a surface gravity wave velocity field” and treating it as the familiar explanation for Langmuir-circulation roll/streak structures in the ocean mixed layer (Kim et al., 29 Aug 2025). A 2025 resolvent study similarly describes “CL-II or CL2 instability” as the widely recognized primary generation process for Langmuir circulations, in which spanwise disturbances in a sheared flow are amplified through coupling to Stokes drift (Xuan et al., 20 Aug 2025).

The label itself is not universal. Several mathematically central papers discuss the Craik–Leibovich equation, Langmuir circulations, and wave–vorticity coupling without explicitly using the CL1/CL2 taxonomy, even when their content is structurally relevant to CL2 (Yang, 2016, Yang, 2016). Other papers are explicit that they are not CL2-instability analyses in the narrow linear-stability sense, yet still isolate the same physical coupling between Stokes drift and pre-existing vorticity, as in ship-wake dynamics, vortex recoil, and generalized wave–current models (Somero et al., 2018, Humbert et al., 2017, Onuki et al., 2 Jun 2026). This terminological dispersion matters because “CL2 instability” denotes both a classical eigenvalue problem and a broader mechanistic family centered on the Craik–Leibovich vortex force.

2. Governing equations and the vortex-force mechanism

The common mathematical core is the Craik–Leibovich equation with prescribed drift velocity. In one standard form,

$$\frac{\partial v}{\partial t} + (v,\nabla)v + \operatorname{curl} v \times V_0 = - \nabla p,$$

where $V_0$ is the prescribed Stokes drift velocity (Yang, 2016). The same paper also gives the geometric form

$\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$

which makes explicit that the drift acts as a shift from $v$ to $v+V_0$ in the advection of momentum classes (Yang, 2016). In the closely related geometric treatment of the CL equation on a Riemannian manifold with boundary, the vector-field form is

$$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$

and the two-dimensional vorticity equation becomes

$$\frac{\partial \omega}{\partial t} + \{\psi,\ \omega - \operatorname{curl}V_s\} = 0,$$

so the dynamically relevant quantity is the shifted vorticity $\omega-\operatorname{curl}V_s$ rather than raw Euler vorticity alone (Yang, 2016).

Within CL2-oriented formulations, the physical mechanism is a roll–streak feedback. In the S3T formulation of Langmuir turbulence, the streamwise-mean roll equation contains the explicit CL2 forcing term

$S \frac{dU_s}{dy}\frac{\partial U}{\partial z},$

which forces rolls from streak gradients in the presence of Stokes-drift shear (Kim et al., 29 Aug 2025). The same paper interprets the mechanism as an imbalance produced by advection of cross-stream Eulerian shear vorticity by Stokes-drift shear without a compensating advection of spanwise Stokes vorticity by Eulerian spanwise shear, generating roll forcing proportional to the Eulerian streak perturbation (Kim et al., 29 Aug 2025). In this sense, CL2 is not merely “waves plus shear,” but a specific vortex-force route from spanwise shear modulation to streamwise-oriented circulation.

3. Canonical instability structure and its relation to Langmuir rolls

The canonical CL2 state is a roll/streak structure: streamwise rolls $(V,W)$ coupled to spanwise-modulated streamwise velocity $V_0$0. In the statistical-state-dynamics study, the base flow is a constant Eulerian shear $V_0$1 together with Stokes drift $V_0$2, at fixed $V_0$3, and the unstable structures are streamwise-oriented roll/streak structures throughout the parameter plane spanned by CL2 strength $V_0$4 and stochastic turbulence forcing amplitude $V_0$5 (Kim et al., 29 Aug 2025). That study emphasizes that Eulerian surface-stress-driven shear alone does not support an unstable roll/streak structure of the observed Langmuir form, whereas the addition of Stokes drift closes the missing feedback and makes the Langmuir RSS unstable (Kim et al., 29 Aug 2025).

The same framework also distinguishes pure CL2 from other mechanisms without denying its independent role. A pure CL2 eigenmode exists at $V_0$6, $V_0$7, while a pure Reynolds-stress-torque mode exists at $V_0$8, and the mixed mode retains essentially the same roll/streak morphology (Kim et al., 29 Aug 2025). This suggests that CL2 supplies a robust structural template for Langmuir rolls even when other turbulent feedbacks are present. At finite amplitude, the same paper reports that, with surface-confined Stokes drift, the pure CL2 eigenmode is shallow, nonlinear equilibration extends it downward, and additional Reynolds-stress torque strengthens full-depth penetration (Kim et al., 29 Aug 2025).

A later resolvent analysis retains the CL2 coupling term but reinterprets its role in fully developed Langmuir turbulence. That paper states that CL-II or CL2 instability explains onset, especially large-scale streamwise rolls, but finds that the linearized system about a turbulent mean state with vertically varying eddy viscosity is asymptotically stable; coherent structures in the equilibrated regime are therefore better interpreted as forced responses to sustained nonlinear forcing (Xuan et al., 20 Aug 2025). Large-scale counter-rotating rolls remain prominent, but smaller-scale three-dimensional near-surface vortices also appear, so the fully developed state is broader than the classical unstable-eigenmode picture.

4. Revisions and critiques of the classical CL2 model

One major revision concerns turbulent closure. A 2023 paper argues that the standard CL2 model is incomplete because it assumes constant eddy viscosity and therefore omits the feedback by which nascent circulations redistribute turbulence, alter near-surface $V_0$9, change the local wind-momentum transfer through

$\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$0

and thereby intensify the nonuniform streamwise current that the Craik–Leibovich vortex force acts upon (Basovich et al., 2023). In that model, the preferred spanwise wavelength $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$1 is extracted numerically and is reported as $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$2, $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$3, and $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$4 at $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$5, $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$6, and $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$7 minutes, respectively, with inverse growth rates approximately $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$8, $\frac{d}{dt}[u] = - \mathcal{L}_{v+V_0}[u],$9, and $v$0 s, and with $v$1 across the simulations (Basovich et al., 2023). The paper presents these results as evidence that scale selection requires an active turbulence-redistribution feedback absent from constant-$v$2 CL2.

A second revision concerns wave scattering by the slowly growing circulation itself. Another 2023 study reformulates the classical Langmuir-instability problem by allowing the spanwise-modulated slow current to scatter the primary surface wave into an oblique wave; interference between the primary and scattered waves then generates a modulated Stokes drift, which alters the vortex-force feedback (Vergeles et al., 2023). In that formulation, the perturbation vortex force is

$v$3

and the resulting growth rate is

$v$4

The paper states that this growth rate is approximately twice smaller than that obtained by A.D.D. Craik and that the vertical structure consists of two vortices corresponding to the next mode in Craik’s model (Vergeles et al., 2023). This is not a rejection of CL2; it is a wave-scattering-modified CL2 theory that changes the closure and, with it, the preferred mode.

These revisions share a common implication: the canonical CL2 problem with prescribed Stokes drift and fixed eddy viscosity is a useful reference model, but later work treats it as a reduced limit rather than a complete account of onset, growth, and mode selection.

5. Nonlinear, forced, and finite-amplitude realizations

A substantial literature uses the Craik–Leibovich vortex-force mechanism outside the canonical homogeneous mixed-layer setting. The ship-wake study “Structure and Persistence of Ship Wakes and the Role of Langmuir-Type Circulations” is explicit that it does not derive CL2 growth rates or solve a linear eigenvalue problem, but it uses the same wave–vorticity coupling in a highly nonuniform wake (Somero et al., 2018). Its governing equations are unsteady incompressible RANS with the CL vortex force

$v$5

and in a reduced two-dimensional wake picture the circulation-driving components are

$v$6

The paper finds that, in the presence of ambient waves, large-scale Langmuir-type circulations persist for $v$7, that the realistic wake produces a four-cell circulation pattern, and that head and following seas reverse roll orientation and therefore the remote-sensing signature of the wake (Somero et al., 2018). For CL2 interpretation, this is a nonlinear, forced manifestation of the same vortex-force physics in a localized shear field.

Laboratory wave–vortex interaction studies isolate a different finite-amplitude regime. In “Wave-induced vortex recoil and nonlinear refraction,” incoming surface gravity waves act on a pre-existing vortex, with the relevant control parameter

$v$8

where $v$9 is Stokes drift and $v+V_0$0 is a characteristic vortex velocity (Humbert et al., 2017). The paper interprets the vortex-center displacement through a balance between Stokes-drift advection and self-induced vortex motion, and the near-surface vorticity decrease through a reduced advection–diffusion model

$v+V_0$1

leading to the asymptotic scaling $v+V_0$2 for strong waves (Humbert et al., 2017). This is not an instability calculation, but it shows that CL-type forcing can laterally displace vortices and expel vorticity from the surface region, thereby modifying the base state on which any subsequent CL2-type process would act.

A further caveat arises from work on recoil and remote interactions. “Wave-vortex interactions, remote recoil, the Aharonov-Bohm effect and the Craik-Leibovich equation” argues that the local CL force $v+V_0$3 captures the total wave-induced recoil only in one exceptional limiting case; in generic cases, remote recoil mediated by pressure and Bretherton-flow return currents is also present (McIntyre, 2019). That result does not negate CL2, but it constrains how literally one should interpret a local vortex-force term as the complete wave forcing in more general wave–vortex geometries.

6. Geometric structure, consistency issues, and two-way generalizations

Several papers recast the Craik–Leibovich equation in geometric or variational language. One 2016 paper shows that the CL equation is the Euler equation on the dual of a central extension of the Lie algebra of divergence-free vector fields tangent to the boundary, with the averaged drift entering as a shift in the coadjoint dynamics and with the effective drift generated by

$v+V_0$4

The same paper proves that the averaged equation approximates the perturbed fast-slow system with $v+V_0$5 error over $v+V_0$6 times and derives adiabatic invariants, providing structural tools rather than direct CL2 growth rates (Yang, 2016). A companion geometric paper proves an Arnold-type nonlinear stability theorem for two-dimensional steady CL flows, with the key stability estimate

$v+V_0$7

under a monotone relation between streamfunction and shifted vorticity (Yang, 2016). These results are not CL2-instability theorems, but they delimit stable CL regimes and identify the geometric objects that a CL2 analysis perturbs.

Another technical issue concerns the Stokes drift itself. “Stokes drift and its discontents” shows that the standard GLM Stokes velocity $v+V_0$8 is generally divergent even in an incompressible fluid, and introduces an exactly divergence-free solenoidal component $v+V_0$9 such that, for irrotational surface waves,

$$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$0

The corresponding mean-momentum equation is formally identical to the CL equation with $$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$1 replacing $$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$2, while the difference between the two is $$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$3 for slowly modulated waves, so classical CL and CL2 analyses remain asymptotically valid in that regime (Vanneste et al., 2022). This clarifies a frequent misconception: the vortex-force-driving part of Stokes drift is the solenoidal component, and the non-solenoidal remainder is pressure-like at leading order for irrotational surface waves.

The most recent generalizations remove the assumption that Stokes drift is externally prescribed. One 2026 reduced model couples a CL-type Lagrangian-mean current equation,

$$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$4

to a companion wave-amplitude equation in physical space without spatial scale separation, so current-induced advection, refraction, and multidirectional scattering modify the wave field and hence the Stokes drift that re-enters the mean equation (Onuki et al., 2 Jun 2026). Another 2026 paper derives a phase-averaged wave–current model from a variational principle, in which the CL current equation is written for the Lagrangian-mean velocity $$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$5 and the wave effect enters as pseudomomentum

$$\frac{\partial v}{\partial t} + \nabla_v v + \operatorname{curl} v \times V_s = -\nabla p,$$6

That model couples wave action transport to the CL current through a weighted Doppler velocity and preserves momentum and energy of the coupled system (Vanneste et al., 25 Feb 2026). Both formulations point toward a generalized CL2 program in which the wave field is no longer a passive background but an evolving part of the instability mechanism itself.

In aggregate, the modern literature presents Craik–Leibovich CL2 instability less as a closed historical theory than as a hierarchy of models built around the same vortex-force coupling. At one end lies the canonical linear instability of a prescribed shear and prescribed Stokes drift; at the other lie turbulence-mediated, wave-scattering, stochastic, geometric, and fully coupled wave–current formulations that preserve the core CL mechanism while changing what counts as the base state, the feedback loop, and the relevant stability problem.