S. Leibovich

A rational model for Langmuir circulations

A realistic theoretical model of steady Langmuir circulations is constructed. Vorticity in the wind direction is generated by the Stokes drift of the gravity-wave field acting upon spanwise vorticity deriving from the wind-driven current. We believe that the steady Langmuir circulations represent a balance between this generating mechanism and turbulent dissipation.Nonlinear equations governing the motion are derived under fairly general conditions. Analytical and numerical solutions are sought for the case of a directional wave spectrum consisting of a single pair of gravity waves propagating at equal and opposite angles to the wind direction. Also, a statistical analysis, based on linearized equations, is developed for more general directional wave spectra. This yields an estimate of the average spacing of windrows associated with Langmuir circulations. The latter analysis is applied to a particular example with simple properties, and produces an expected windrow spacing of rather more than twice the length of the dominant gravity waves.The relevance of our model is assessed with reference to known observational features, and the evidence supporting its applicability is promising.

Disrupted states of vortex flow and vortex breakdown

Flow visualization studies and laser Doppler anemometer measurements on swirling water flows reveal six distinct types of very large amplitude disturbance modes of the vortex core. Three, ’’axisymmetric’’ and spiral vortex breakdowns, and the ’’double helix,’’ have been described by others. A definite order of evolution in parameter space (Reynolds number and circulations) occurs, and is described. Puzzling responses of the axisymmetric and spiral vortex breakdowns to imposed flow transients reported previously are confirmed here, and are traced to the shedding of starting and stopping vortices from swirl vanes. Conclusions bearing upon the validity of some theories of vortex breakdown are possible from the data.

A sufficient condition for the instability of columnar vortices

The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r) , azimuthal velocity V(r) , where r is distance from the axis, let Ω be the angular velocity V / r , and let Γ be the circulation rV . Then the flow is unstable if

An experimental map of the internal structure of a vortex breakdown

V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right]

Spectral characteristics of vortex breakdown flowfields

The flow field of an ‘axisymmetric’ vortex breakdown has been mapped using a laser-Doppler anemometer. The interior of the recirculation zone is dominated by energetic, non-axisymmetric, low frequency periodic fluctuations. Spectra for a number of points inside this zone, as well as time-averaged swirl and axial velocity profiles both inside and outside the recirculation zone, have been obtained. The time-averaged streamlines in the interior show an unexpected two-celled structure attributed to the action of the fluctuations. Although the present experiment deals with one particular breakdown, flow-visualization studies indicate that the case examined is typical of the ‘axisymmetric’ form of breakdown over a range of flow conditions.

On the evolution of the system of wind drift currents and Langmuir circulations in the ocean. Part 1. Theory and averaged current

Laser‐Doppler anemometer measurements upstream and in the wakes of vortex breakdowns of bubble and spiral types are described. Spectral analysis of the data indicates prominent oscillations in the wakes at less than 10 Hz. These oscillations correspond closely to theoretical predictions of the linearly most unstable normal modes of the time‐averaged mean flow profiles. The oscillations are more energetic, and vortex core expansions are greater for flows with a bubble form of vortex breakdown, which is therefore regarded as the stronger form of breakdown.

Convective instability of stably stratified water in the ocean

A theory for the evolution of the wind drift current and of the Langmuir circulations in infinitely deep water of constant density is presented. The model improves and extends a recent quasi-steady theory of Craik & Leibovich which asserts that the Langmuir circulations arise from a nonlinear interaction between surface waves and the frictional wind drift current. In turn, the development of the wind drift should be strongly influenced by Langmuir circulations, when they are present, and the two current systems are therefore treated here as a single inseparable system driven by a prescribed wind stress and surface wave field. Mixing by the vertical motions in the Langmuir circulations is shown to yield solutions for the wind drift, obtained both analytically and numerically, which are consistent with experiments and with field observations. The model yields a streaky flow pattern with a mean motion much like a turbulent wall layer, although the model is deterministic. In particular, it is found that a ‘viscous sublayer’ joins surface water to a logarithmic ‘inertial subIayer ’ below. The scaling rules that emerge from the theory allow the surface speed of the wind drift to reach nearly full development in a matter of minutes.

Weakly non-linear waves in rotating fluids

A recent theoretical description of interactions between surface waves and currents in the ocean is extended to allow density stratification. The interaction leads to a convective instability even when the density stratification is statically stable. An unspecified random surface wave field is permitted provided that it is statistically stationary. The instability can be traced to torques produced by variations of a ‘vortex force’. Non-diffusive instabilities produced by this mechanism in water of infinite depth are explored in detail for arbitrary distributions of the destabilizing force. Stability is determined by an eigenvalue problem formally identical to that determining normal modes of infinitesimal internal waves in fluid with a density profile that is not monotone and thereby has a statically unstable region. Some tentative remarks are offered about the problem when dissipation is allowed. Application of the present theory to Langmuir circulations is discussed. Also, according to the present theory, internal wave propagation should be modified by the vortex force arising from the interaction between the surface waves and the current.

On wave-current interaction theories of Langmuir circulations

Rotating inviscid incompressible fluids in tubes, investigating axisymmetric nonlinear waves motion in relation to vortex breakdown

Invariant helical subspaces for the Navier-Stokes equations

The Craik-Leibovich (CL) equations for Langmuir circulations are shown to be an Eulerian approximation to an exact theory of the generalized Lagrangian mean (GLM) due to Andrews and McIntyre. Derivation of the CL equations using the GLM formalism is decisively simpler than the original method. The CL theory is then compared to other wave-current interaction theories of Langmuir circulations, notably those of Garrett and of Moen.