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Amit Tandon and Sidney Leibovich

Abstract

Finite-amplitude Langmuir circulation in the form of rolls parallel to the wind direction is shown to be subject to three-dimensional instability under certain circumstances. Density stratification is not required for instability to manifest. The preferred form of this secondary instability appears to be traveling waves propagating in the direction of the wind. These cause the rolls, and their surface windrows, to deviate from the wind direction by a small angle for which estimates are given. The results of the paper show the value of secondary stability results for the design of numerical experiments to simulate Langmuir circulation.

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Sidney Leibovich and Samuel Paolucci

Abstract

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.

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Stephen M. Cox and Sidney Leibovich

Abstract

Langmuir circulations reside in, and are responsible in part for the existence and maintenance of, the mixed layer. It is, therefore, typical for the water containing Langmuir circulations to be bounded below by a thermocline. When this bounding thermocline is strong, it may be expected to act as an effective “slippery bottom” constraint. Such an assumption has been invoked previously, but failed to predict a preferred spacing for the windrows produced by the circulations. This model assumed that the momentum transfers across the horizontal boundaries of the mixed layer were independent of the water motion induced by Langmuir circulation. Here, mixed boundary conditions are explored. Estimates of the transfer coefficients in these boundary conditions suggest that the revised model differs only slightly from the earlier one, but allows for a more general and realistic stress model. Incorporating these effects into the theory gives windrows with a finite separation, in accord with the observations. The windrow spacing emerging from this modified theory depends in a simple way on the layer depth and the constant of proportionality in the stress boundary condition when the latter number takes physically plausible values. The analysis allows the water layer above the bounding thermocline to be homogeneous or either stably or unstably density stratified. The stably stratified case permits oscillatory convection under certain restricted circumstances.

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Gregory P. Chini and Sidney Leibovich

Abstract

Numerical simulations of the oceanic (atmospheric) boundary layer are complicated by the need to specify appropriate “outflow” or “radiation” boundary conditions at the artificial lower (upper) boundary of the computational domain. If the boundary layer is stratified, particular care is necessary to insure that internal-gravity-wave disturbances generated within the domain are not artificially reflected by the computational boundary. A major advance was made almost 20 years ago by Klemp and Durran; their radiation condition relates the Fourier transformed pressure fluctuation to the Fourier transformed vertical-velocity perturbation along the artificial boundary. Because it is local in time, the Klemp and Durran (KD) condition is easily incorporated into a wide variety of numerical models for only a minor computational expense. Indeed, it has been widely used in the atmospheric and oceanic sciences communities. For simulations of dissipative systems, however, perturbation-flux conditions must also be specified at the artificial boundary—these are in addition to the KD condition (or some other constraint) on the normal velocity component at that boundary. This article considers the performance of the KD condition in conjunction with zero perturbation stress and zero perturbation buoyancy-flux conditions (“KDZ” conditions, collectively), because the latter are generally assumed to be appropriate for simulations of boundary layer phenomena. Analysis of the response of a weakly dissipative, uniformly stratified fluid to forcing concentrated at a given depth reveals two potentially serious drawbacks of the KDZ conditions. First, nonhydrostatic dynamics are not adequately treated by the KD condition, itself. Moreover, the imposition of zero perturbation-flux conditions causes artificial boundary layers to form along the outflow boundary. Although these boundary layers are passive, they are unlikely to be resolved in numerical simulations; thus, discretization of the KDZ conditions may cause further errors in the simulated internal-wave dynamics. A consistent set of boundary conditions for simulations of dissipative, stratified fluids is proposed.

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