Breaking internal waves and turbulent dissipation

We explore what might be discovered about the breaking of progressive internal waves and the consequent mixing by following some of the methodologies and techniques used to study surface wave breaking. It is suggested that breaking is most likely to occur in wave groups, where the wave field is locally amplified. In a stratified fluid of uniform buoyancy frequency, N, the breaking regions of internal wave groups extend in approximately horizontal directions. Two classes of breaking, “convective overturn” and “shear instability,” are possible in progressive internal waves propagating in uniform stratification with no mean shear. Convective overturning and associated static instability occur at all wave frequencies, but only if the wave slope, s = am, exceeds unity, where a is the wave amplitude and m is the vertical wavenumber. Self-induced shear instability may take place in waves with slopes s \u3c 1, and therefore less than the slopes required for convective overturn, but only when a wave-related Richardson number is sufficiently small; to achieve this, the wave frequency must be close to the inertial frequency. Equations are derived to express the energy dissipated in breaking or the strength of breaking in terms of the characteristics of a breaking wave. A particular measure of breaking analogous to that used to quantify surface wave breaking is ΛI(cb)dcb, the mean area of the fronts of breaking regions, projected onto the vertical and per unit volume, that are produced by internal breakers traveling at speeds between cb and cb + dcb. Estimates are made of the values of ΛI required to sustain a vertical eddy diffusion coefficient of Kρ = 10–5 m2 s–1 through the breaking of internal waves of typical amplitude by convective overturn (with s \u3e 1) and by the self-induced shear instability of near-inertial waves when s \u3c 1. Values of ΛI are of order 1.0 × 10–2 m–1 (i.e., a vertical surface area of about 10 cm × 10 cm in each cubic meter). The predictions are tested by using them to find the fraction of the water column in which turbulence occurs and by comparing the predicted values with existing observations. Additional theoretical studies and laboratory experiments are required to test the proposed analytical relations. Existing sea-going measurement techniques are reviewed and further observations are suggested to advance the understanding of breaking internal waves

Foam triangles

Foam patches left by waves breaking as they approach a smooth and gently sloping beach from a near-normal direction sometimes have the distinctly triangular shape that has been studied by Turner and Turner (2011). Explanations of the size of the angle at the apex of the triangles observed by Turner and Turner are suggested in terms of physical processes that determine the speed at which the point of breaking travels along a wave crest. These explanations differ from the entrainment model proposed by Turner and Turner (2011). The range of sizes of the apex angles can most likely be explained in terms of the directional spreading of waves approaching the surf zone

The relation between the duration and shape of internal wave groups

This paper discusses the effect of the shape of internal wave groups on their “duration” or “lifetime”—how long they retain their form before their component waves disperse. The methodology devised by Smith and Brulefert (2010) to study the dispersion of surface wave groups is extended to examine the dispersion of internal wave groups. To provide tangible examples, it is supposed that wave groups of ellipsoidal shape, symmetrical about a vertical plane, are generated in a uniformly stratified thermocline by moving periodic disturbances perturbing the base of an overlying mixed layer. The dispersion relation for internal waves is used to determine the group duration, taken as the time required for the volume of the group to approximately double through the fastest separation of its component waves. As well as allowing the orientation (inclination of their larger, major axis to the horizontal) and aspect ratio (that of the minor to major axis in a vertical plane) of wave groups to vary, their lifetimes are compared in two particular cases: Case A in which the length of the initial minor axes in the vertical plane of the groups is the same, and Case B in which groups are initially composed of the same number of waves. Two-dimensional groups and “three-dimensional groups” (the latter predominantly two-dimensional but of limited extent in one horizontal direction) are considered. As has been found for surface waves, the duration of internal wave groups does indeed depend upon their shape. In both Cases, groups with relatively small aspect ratio and, in Case B, groups with many waves usually have greater lifetimes than relatively large aspect ratio groups with few waves. Two-dimensional groups have greater lifetimes than three-dimensional groups. In many cases, the groups with the longest lifetime have their longer (major) axis inclined at an angle to the horizontal that is close to the inclination of the group velocity vector; in these cases the lines of constant phase of waves composing the groups are not (as is found for some surface wave groups) slanted with respect to the major axis of the groups, but parallel. Some long-lifetime groups are found to have their major axes inclined to the horizontal at an angle that is very close to that of the front of a packet of waves generated by the moving periodic disturbance at the foot of the mixed layer. In Appendix B it is shown that the ratio, np/n or nm/n, of the number of waves that would be recorded as the group passes a fixed point or a vertical mooring, to the number of waves contained, instantaneously, within a wave group, depends on the shape of the group and on the ratio of the dominant wave phase speed to the group velocity of the group. A simple model described in Appendix C suggests how such slanted wave groups can be generated in the thermocline by moving, but transient, disturbances. The orientation of “scars,” regions left by waves breaking in the wave groups, is examined in Appendix D. Except for near inertial waves with small aspect ratio, scars are generally close to being horizontal

Measuring overturns with gliders

The accuracy of the estimation of the vertical size of eddies in turbulent stratified shear flows in the ocean from measurements obtained by gliders is examined. It is assumed that gliders move along paths inclined at moderate angles to the horizontal. Comparison is made with measurements by probes falling or lowered vertically through billows resulting from Kelvin-Helmholtz (K-H) or Holmboe instability and through the statically unstable regions formed at early stages of convective breaking of internal waves. The probable errors involved in estimating the overturn scale of a K-H billow along the track of a glider are greatest when the ratio of the billow\u27s vertical to horizontal scale, b/a, is greatest and when a glider\u27s inclination angle, α, is moderate, but the errors are generally relatively small. At small angles, α, the glider path may intersect more than one billow, reducing the errors. Larger errors are possible, however, in measuring eddies in turbulent stratified shear flows, and their magnitude depends on the orientation of eddies relative to the trajectory of the glider. False overturns may be apparent using gliders with small inclination angles, α, in internal waves, and consequently erroneous estimates of the displacement scales obtained, even when the slope of the waves, s, \u3c 1 and convective overturn is entirely absent. Quantification of overturns from glider measurements of the apparent vertical size of the regions in which the density increases upward can result in misleading estimates of the scale of overturns. Although, because of the wave-induced horizontal and vertical motions, the trajectory of free-fall probes will not be vertical when passing through an internal wave field, and nor will it be steady, the mean square displacements obtained from measurements are found to be the same as those that would be made by a probe passing vertically through a frozen wave field. Attention is drawn to the paucity of information about the structure of naturally occurring eddies in the stratified ocean

Dissipation in hydraulic transitions in flows through abyssal channels

There is growing evidence from observations that mixing occurs in hydraulic jumps, or flow transitions, downstream of sills within channels connecting deep ocean basins or within submarine canyons on the flanks of mid-ocean ridges. Models with continuous profiles of velocity and density upstream and downstream of a transition, but conforming to continuity conditions, are devised to represent the mixing that occurs in a hydraulic jump in a stratified shear flow of finite depth moving over a horizontal boundary in a deep fluid. These are used to assess the conditions in which transitions may occur and to provide an estimate of the loss in the flux of energy carried by the flow. An increase in the thickness of the stratified flow layer is necessary as water passes through a transition. The rate of loss of energy flux per unit channel width in a transition is typically of order 6ρh(gβh)3/2, where h is a measure of the thickness of the flow before transition, g the acceleration due to gravity and β = Δρ/ρ (≪1), where Δρ is half the difference in density between that in the flow approaching the transition and that in the overlying fluid, and ρ is the mean density. The mean rate of loss of energy in a transition in the flow of Antarctic Bottom Water over just one of the 6 – 8 sills in the Romanche Fracture Zone is estimated to be of order 60 MW (6 × 107 W)

Magnetic properties of microtektites Semiannual status report, 1 Jan. - 31 Jun. 1969

Magnetic susceptibility, magnetization, and Curie constants for normal and bottle-green microtektites found in deep-sea sediment core

Sphaerodactylus fantasticus

Number of Pages: 8Integrative BiologyGeological Science

Internal waves and temperature fronts on slopes

International audienceTime series measurements from an array of temperature miniloggers in a line at constant depth along the sloping boundary of a lake are used to describe the `internal surf zone' where internal waves interact with the sloping boundary. More small positive temperature time derivatives are recorded than negative, but there are more large negative values than positive, giving the overall distribution of temperature time derivatives a small negative skewness. This is consistent with the internal wave dynamics; fronts form during the up-slope phase of the motion, bringing cold water up the slope, and the return flow may become unstable, leading to small advecting billows and weak warm fronts. The data are analysed to detect `events', periods in which the temperature derivatives exceed a set threshold. The speed and distance travelled by `events' are described. The motion along the slope may be a consequence of (a) instabilities advected by the flow (b) internal waves propagating along-slope or (c) internal waves approaching the slope from oblique directions. The propagation of several of the observed 'events' can only be explained by (c), evidence that the internal surf zone has some, but possibly not all, the characteristics of the conventional 'surface wave' surf zone, with waves steepening as they approach the slope at oblique angles

Are cascading flows stable?

The stability of flows cascading down slopes as dense inclined plumes is examined, with particular reference to flows observed in Lake Geneva during winter periods of severe cooling. A previous conjecture by Turner that the flow may be in a state of marginal stability is confirmed: the observed mean velocity and density profiles are unstable to Kelvin-Helmholtz instability, but only marginally so; the growth rates of the most unstable small disturbances to the cascading flow in Lake Geneva are small, with e-folding periods of about 2 h. A reduction in the maximum velocity by about 20% is required to stabilize the flow. The possibility that stationary hydraulic jumps may occur in the observed flow is also considered. Several plausible flow states downstream of transitions are examined, allowing for mixing and density changes to occur, ranging from one that preserves the shape of the density and velocity profiles to one in which, as a consequence of mixing, the velocity and density become uniform in depth within the cascading flow. Neither of these extreme states is found to conserve the fluxes of volume, mass and momentum through a transition in which the energy flux does not increase, and to be unique or ‘stable' in the sense that no further transition is possible to a similar flow state without more entrainment. Stable transitions to intermediate downstream flows that conserve flow properties and reduce energy flux are, however, found, although the smallest value of the flow parameter, Fr≡ Umax2/gΔ h (where Umax is the maximum flow speed, g is the acceleration due to gravity, Δ is a fractional density difference within the flow and h is the flow thickness) at which transitions may occur is only slightly less than that of the cascading flow in Lake Geneva. In this sense, the observed flow is marginally unstable to a finite-amplitude transition or hydraulic jump. Velocity and density profiles of possible flows downstream of a transition are found. The amplitudes of possible transitions and the flux of water entrained from the ambient overlying water mass are limited to narrow range

On the use of the method of images to investigate nearshore dynamical processes

This note describes how the method of images may be used to determine the motion and evolution of two related kinds of phenomena within a wedge of inviscid fluid. The image field of a curved vortex within a wedge with vortex lines lying along sectors of circles around the apex of the wedge is that segment of a complete vortex ring which remains outside the wedge and of which the curved vortex forms a part. The image system can be used to describe the motion, interaction and stability of single or multiple vortices within the wedge. Axisymmetric jets form the image system for flow parallel to the edge of the wedge, akin to alongshore currents. Knowledge of the instability of jets provides information about the evolution of waves in the wedge domain. Existing results on the motion and instability of single or multiple co-axial ring vortices and of waves and instabilities in jets may be applied to describe the evolution of low Froude number eddies and waves in alongshore flow over a steadily shelving sea bed