Craik-Leibovich Theory Research Articles

On the physical mechanisms of the two-way coupling between a surface wave field and a circulation consisting of a roll and streak

The governing equations of a surface wave field and a coexisting roll–streak circulation typical of Langmuir circulations or submesoscale frontal circulations are derived to better describe their two-way interactions. The gradients and vertical velocities of the roll–streak circulation induce wave refraction, amplitude modulation and higher-order waves. These changes then produce wave–wave nonlinear forces and divergence of the wave-induced mass transport, both of which in turn affect the circulation. To accurately represent these processes, both a wave theory and a wave-averaged theory are developed without relying on any extrapolation, any spatiotemporal mapping or an approximation that treats the wave-induced mass divergence as being concentrated at the surface. This wave theory finds seven types of current-induced higher-order wave motions. It also determines the wave dynamics such as the governing equation of the wave action density valid in the presence of the complex circulation. The evolution of the wave action density is clearly affected by the upwelling or downwelling. The new wave-averaged theory presents the governing equations of the wave-averaged circulation which satisfies the wave-averaged mass conservation. This circulation is different from the circulation considered to satisfy the mass conservation in the Craik–Leibovich theory, and the difference becomes critical when the wave field evolves due to refraction. In this case, compared to the Craik–Leibovich theory, long waves are more important and also the rolls are more weakly forced.

Reply to “Comments on ‘A Wave-Resolving Simulation of Langmuir Circulations with a Nonhydrostatic Free-Surface Model: Comparison with Craik–Leibovich Theory and an Alternative Eulerian View of the Driving Mechanism’”

AbstractFujiwara et al. explicitly simulated Langmuir circulations using a wave-resolving simulation (WRS) technique and found that the residual wave effect on vorticity was well represented by the vortex force of the Craik–Leibovich (CL) equation, at least in the simulated situation. In response to the simulation results, Mellor has proposed a view that ubiquitous applicability of the CL formulation is still questionable and that the three-dimensional radiation stress (3DRS) formulation that he has derived encompasses both of the vortex force effect and an effect that is lower order in terms of wave steepness. Here, these opinions are discussed in terms of the approximations used in the wave-averaged formulations. The asymptotic expansion of the Eulerian-averaged momentum equation allows the separate discussion of two different wave effects: pressure correction and torque. It is argued that the approximation adopted in Mellor’s 3DRS formulation is presumably not accurate enough to properly parameterize the wave torque effect, and possible approaches to examine its performance are proposed. We agree with the view that the applicability of the CL formulation needs further investigation. WRS will be a helpful tool for this purpose.

Comments on “A Wave-Resolving Simulation of Langmuir Circulations with a Nonhydrostatic Free-Surface Model: Comparison with Craik–Leibovich Theory and an Alternative Eulerian View of the Driving Mechanism”

AbstractThe results of the subject paper are reviewed wherein credible Langmuir cells are produced by a numerical solution of the primitive fluid dynamic equations with a free surface. Whereas it is a major achievement, the claim that the same results support the general application of the so-called vortex force equations is challenged.

A Wave-Resolving Simulation of Langmuir Circulations with a Nonhydrostatic Free-Surface Model: Comparison with Craik–Leibovich Theory and an Alternative Eulerian View of the Driving Mechanism

AbstractThe present study performs a wave-resolving simulation of wind-driven currents under monochromatic surface gravity waves using the latest nonhydrostatic free-surface numerical model. Here, phase speed of the waves is set much greater than the current speed. Roll structures very similar to observed Langmuir circulations (LCs) appear in the simulation only when both waves and down-wave surface currents are present, demonstrating that the rolls are driven by the wave–current interaction. A vorticity analysis of simulated mean flow reveals that the rolls are driven by the torque associated with wave motion, which arises from a correlation between wave-induced vorticity fluctuation and the wave motion itself. Furthermore, it is confirmed that the wave-induced torque is very well represented by the curl of the vortex force (VF), that is, the vector product of mean vorticity and Stokes drift velocity. Therefore, it is concluded that the simulated rolls are LCs and that the wave effects are well represented by the VF expression in the present simulation. The present study further revisits the scaling assumptions made by previous studies that derived VF formulation and shows that there is disagreement among the previous studies regarding the applicability of VF formulation when the wave orbital velocity (proportional to the amplitude times the frequency) is much smaller than the mean flow velocity. The result from the present simulation shows that the VF expression is still valid even with such small wave amplitudes, as long as phase speed of the waves is much greater than the current speed.

TRANSIENT DYNAMICS OF STREAK AND LONGITUDINAL VORTICES IN SHEAR FLOW WITH WAVE

Langmuir circulations are alternating right and left helical vortices in the ocean having horizontal axes parallel to the wind. They are manifested by floating material on the ocean surface as streaks. The laboratory wind driven shear flow demonstrate streak feature with/without wave condition. When the wave is present the explanation of results with Craik-Leibovich instability mechanism, originally proposed for weak current shear in ocean, is not appropriate for some problem as in the laboratory the current shear is strong. The existing normal mode analysis based on the generalized Lagrangian mean formulation for wave with strong shear is not able to capture the fully dynamics of streak and longitudinal vortices. We extend Craik-Leibovich theory to strong shear in Eulerian framework but focusing on the fully perturbation evolution dynamics. Transient dynamics of perturbation is studied with an initial value problem. In this approach the effect of shear and wave and its interaction are made clear. The transient growth in the time period without wave is significant and is crucial to the initial preferred wave number observed in the experiments. The optimal longitudinal vortex form perturbation is searched and the preferred spanwise wave number predicted is well consistent with experimental data of wind shear and wave.

Onset of Langmuir circulation when shear flow and Stokes drift are not parallel

The linear stability problem for Langmuir circulation in the mixed layer of the ocean is investigated according to the Craik-Leibovich theory, whereby motion is generated through an interaction between a shear flow and the Stokes drift due to the surface gravity waves. This paper, unlike most previous analyses, allows the shear flow and the Stokes drift to be in different directions. Such misalignment provides a simple physical mechanism for the observed lateral drifting of Langmuir circulations. A long-wave asymptotic expansion is used to investigate the orientation of the fastest growing modes of Langmuir circulation: in general they are predicted to take the form of long counter-rotating rolls, whose axes lie parallel to a direction between that of the shear flow and that of the Stokes drift. For a given shear flow, the mixed layer is most unstable when the shear flow and the Stokes drift are parallel to one another.

The Langmuir Circulation Instability as a Mixing Mechanism in the Upper Ocean

A numerical study of the fully nonlinear instability of the ocean to Langmuir circulations is reported. The extended Craik-Leibovich theory is used to compute the development of a mixed layer in an ocean of infinite depth as an initial value problem. A wind stress and surface wave field are imposed on a quiescent ocean with a linear temperature gradient. The initial response to the applied stress is a rectilinear current that is unstable to Langmuir circulations. The resulting convective motions appear to cascade energy from small-scale circulations to more vigorous ones of larger scale. Horizontal averages allow one to identify the Reynolds stress, heat flux and mixing efficiency of the Langmuir “eddies.” Mixing efficiencies several times (up to an order of magnitude) larger than those reported in laboratory experiments are possible. It is suggested that numerical experiments such as these may offer a means of parameterizing the effects of sea state and Langmuir circulations for use in one-dimensional and mixed-layer models.