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## Abstract

A characteristic of internal waves reflecting from sloping boundaries is that they form fronts that travel with the component of the phase speed of the waves up the boundary. The strength of the fronts is assessed by estimating the magnitude of nonlinear terms leading to the asymmetry of density gradients at the slope when waves travelling in a fluid of uniform buoyancy frequency are at nonnormal, or oblique, incidence to the slope. Strong nonlinearities, indicating fronts, are found for both supercritical (*β* > *α*) and subcritical (*β* < *α*) waves near critical slopes where the inclination of the boundary to the horizontal, *α,* matches that of the wave group velocity *β.* They are also found for subcritical waves when *β* is near sin^{−1}[(sin*α*)/2]. Fronts become weaker as the angle at which the wave approaches the slope, the azimuth or incident angle, increases from zero (i.e., when waves are nonnormal), but not significantly so until this angle exceeds 30°.

## Abstract

A characteristic of internal waves reflecting from sloping boundaries is that they form fronts that travel with the component of the phase speed of the waves up the boundary. The strength of the fronts is assessed by estimating the magnitude of nonlinear terms leading to the asymmetry of density gradients at the slope when waves travelling in a fluid of uniform buoyancy frequency are at nonnormal, or oblique, incidence to the slope. Strong nonlinearities, indicating fronts, are found for both supercritical (*β* > *α*) and subcritical (*β* < *α*) waves near critical slopes where the inclination of the boundary to the horizontal, *α,* matches that of the wave group velocity *β.* They are also found for subcritical waves when *β* is near sin^{−1}[(sin*α*)/2]. Fronts become weaker as the angle at which the wave approaches the slope, the azimuth or incident angle, increases from zero (i.e., when waves are nonnormal), but not significantly so until this angle exceeds 30°.

## Abstract

Strong interactions, often leading to wave breaking and mixing, are known to occur between a train of internal waves, incident on a plane sloping ocean boundary, and its reflection propagating away from the slope. Intermittent processes of internal-wave generation at the sea surface or ocean floor, however produce packets or *groups* of internal waves. The size of the volume within which an internal wave group of narrow bandwidth overlaps with its reflection from a plane boundary depends on the finite dimensions of the group. The height above a slope within which interactions can occur is calculated as a function of the direction of approach of the incident group to the plane. Wave groups traveling with positive downslope group velocity components, such as internal tidal rays generated at the shelf break, produce relatively thicker interaction regions than do groups of equal dimensions and frequency approaching a slope from deep water. In some regions, the thickness of layers of enhanced mixing over continental slopes or on the flanks of deep ocean ridges may be determined by the scale of reflecting internal wave groups rather than by, for example, the straining of the ambient wave field by radiating internal tidal waves.

## Abstract

Strong interactions, often leading to wave breaking and mixing, are known to occur between a train of internal waves, incident on a plane sloping ocean boundary, and its reflection propagating away from the slope. Intermittent processes of internal-wave generation at the sea surface or ocean floor, however produce packets or *groups* of internal waves. The size of the volume within which an internal wave group of narrow bandwidth overlaps with its reflection from a plane boundary depends on the finite dimensions of the group. The height above a slope within which interactions can occur is calculated as a function of the direction of approach of the incident group to the plane. Wave groups traveling with positive downslope group velocity components, such as internal tidal rays generated at the shelf break, produce relatively thicker interaction regions than do groups of equal dimensions and frequency approaching a slope from deep water. In some regions, the thickness of layers of enhanced mixing over continental slopes or on the flanks of deep ocean ridges may be determined by the scale of reflecting internal wave groups rather than by, for example, the straining of the ambient wave field by radiating internal tidal waves.

## Abstract

Simple analytical models are devised to describe the dispersion of a plume of buoyant material from a fixed source on the sea surface under the action of both mean current and the spatially variable flows induced by Langmuir circulation. A “free” plume meanders at source and later becomes broken into bands lying in the convergence regions produced in the circulation pattern. A model is used to compare free plume dispersion and that in which the floating material is constrained to lie in a band along the water surface, as described in a recent study by Therpe and Curé.

## Abstract

Simple analytical models are devised to describe the dispersion of a plume of buoyant material from a fixed source on the sea surface under the action of both mean current and the spatially variable flows induced by Langmuir circulation. A “free” plume meanders at source and later becomes broken into bands lying in the convergence regions produced in the circulation pattern. A model is used to compare free plume dispersion and that in which the floating material is constrained to lie in a band along the water surface, as described in a recent study by Therpe and Curé.

## Abstract

The alongslope currents flowing over topography of sufficiently length scale, typically less than 10 km, on the continental slopes generate internal lee waves. These carry their momentum predominently toward shallower water, that is up the slope toward and across the shelf break, and onto the continental shelf, at least when, in summer, stratification permits their propagation. Analytical results show that even when the lee waves are generated with a component of their group velocity directed toward deeper water, reflection at the sloping seabed may lead to a turning toward shallower water. A numerical model is used to examine internal wave propagation and to quantify the flux of their momentum across the shelf break. In the conditions consiidered here with *f/N*≪1 and slope angle, α a, near 5 deg, the flux is parameterized by a stress (momentum flux per unit vertical area along the shelf break) per unit length downslope,.τ_{*}, given by

_{*}

*kρ*

_{0}

*VNh*

^{2}

^{4}

*β*

*β*

_{0}

where po is the mean water density, *V* is the mean alongslope flow over the slope,*N* is the buoyancy frequency in the vicinity of the shelf break, *f* is the Coriolis parameter, and *h*
^{2}’ and, β are the mean square amplitude of the topography of wavenumber *l*, such that *VIIN * ≪ 1, and its mean orientation relative to the upslope direction, respectively. The constant β _{O} is 7 ±2 deg, and the formula is only valid if β <60 deg. A value of *k* of about 9 (±4) × 10^{−6} m^{−2} suggested, with values new 1.3 × 10 ^{−5} m ^{−2}’ when the topography is dominated by wavelengths less than 47π*VIN*, or 5 × 10 ^{−6} m ^{−2}’ when they exceed 2O*VIN*. This flux represents a transfer of momentum to the shelf currents in a direction contrary to the current over the slope that generates the internal waves. The magnitude of the flux is usually dominated by conditions near the top of the continental slope. Timescales of about 5 days are associated with this transfer on 5 deg slopes with 10−m high topography when *N*≈10 ^{−2} s ^{−1}

## Abstract

The alongslope currents flowing over topography of sufficiently length scale, typically less than 10 km, on the continental slopes generate internal lee waves. These carry their momentum predominently toward shallower water, that is up the slope toward and across the shelf break, and onto the continental shelf, at least when, in summer, stratification permits their propagation. Analytical results show that even when the lee waves are generated with a component of their group velocity directed toward deeper water, reflection at the sloping seabed may lead to a turning toward shallower water. A numerical model is used to examine internal wave propagation and to quantify the flux of their momentum across the shelf break. In the conditions consiidered here with *f/N*≪1 and slope angle, α a, near 5 deg, the flux is parameterized by a stress (momentum flux per unit vertical area along the shelf break) per unit length downslope,.τ_{*}, given by

_{*}

*kρ*

_{0}

*VNh*

^{2}

^{4}

*β*

*β*

_{0}

where po is the mean water density, *V* is the mean alongslope flow over the slope,*N* is the buoyancy frequency in the vicinity of the shelf break, *f* is the Coriolis parameter, and *h*
^{2}’ and, β are the mean square amplitude of the topography of wavenumber *l*, such that *VIIN * ≪ 1, and its mean orientation relative to the upslope direction, respectively. The constant β _{O} is 7 ±2 deg, and the formula is only valid if β <60 deg. A value of *k* of about 9 (±4) × 10^{−6} m^{−2} suggested, with values new 1.3 × 10 ^{−5} m ^{−2}’ when the topography is dominated by wavelengths less than 47π*VIN*, or 5 × 10 ^{−6} m ^{−2}’ when they exceed 2O*VIN*. This flux represents a transfer of momentum to the shelf currents in a direction contrary to the current over the slope that generates the internal waves. The magnitude of the flux is usually dominated by conditions near the top of the continental slope. Timescales of about 5 days are associated with this transfer on 5 deg slopes with 10−m high topography when *N*≈10 ^{−2} s ^{−1}

## Abstract

The rapid changes in density observed near the continental slope in association with internal waves are explained as nonlinear features of wave reflection.

## Abstract

The rapid changes in density observed near the continental slope in association with internal waves are explained as nonlinear features of wave reflection.

## Abstract

The presence and pattern of Langmuir circulation can be detected using side-scan sonar. The circulation creates bands of subsurface bubbles, scatterers of high-frequency sound, in the downwelling region beneath the surface convergence. The bands are clearly visible in sonographs. A common process of development is for them to join in pairs.

The stability of the circulation pattern is examined, making a number of simplifying assumptions. In particular, we represent the Langmuir cells as linear vortices. These are subjected to small disturbances. When these are restricted to two-dimensional motions normal to the axes of the vortices, stable modes are found in part of the parameter range in which the windrow separation is large in comparison to an appropriate depth scale, such as the depth of the vortex core in a very deep mixed layer or the depth of the thermocline or lake when this is finite. These modes are destabilized to collective instabilities when three-dimensional motions are permitted. The dominant mode of instability in the parameter range in which Langmuir circulation is mostly found is, however, a pairing mode (consistent with the sonar observations), having an axial wavelength similar to the observed downwind extent of windrows.

The growth rates of the instability agree favorably with those expected from observations. Further study is appropriate in view of the possible importance of this instability as a mechanism for dispersion of floating material or diffusion of soluble matter in the sea.

## Abstract

The presence and pattern of Langmuir circulation can be detected using side-scan sonar. The circulation creates bands of subsurface bubbles, scatterers of high-frequency sound, in the downwelling region beneath the surface convergence. The bands are clearly visible in sonographs. A common process of development is for them to join in pairs.

The stability of the circulation pattern is examined, making a number of simplifying assumptions. In particular, we represent the Langmuir cells as linear vortices. These are subjected to small disturbances. When these are restricted to two-dimensional motions normal to the axes of the vortices, stable modes are found in part of the parameter range in which the windrow separation is large in comparison to an appropriate depth scale, such as the depth of the vortex core in a very deep mixed layer or the depth of the thermocline or lake when this is finite. These modes are destabilized to collective instabilities when three-dimensional motions are permitted. The dominant mode of instability in the parameter range in which Langmuir circulation is mostly found is, however, a pairing mode (consistent with the sonar observations), having an axial wavelength similar to the observed downwind extent of windrows.

The growth rates of the instability agree favorably with those expected from observations. Further study is appropriate in view of the possible importance of this instability as a mechanism for dispersion of floating material or diffusion of soluble matter in the sea.

## Abstract

The onset of instability in monochromatic two-dimensional internal near-inertial gravity waves propagating in an ocean of constant buoyancy frequency and no mean shear is examined for increasing values of the wave steepness *s,* the product of the wave amplitude, and the vertical wavenumber of the waves. Stability of disturbances to the quasi-steady flow depends on the minimum Richardson number of the flow in the direction of the disturbance vector.

The minimum Richardson numbers both of the quasi-steady *x*-directed flow, *J*
_{
x
} (that is, in the horizontal direction of wave propagation), and of the transverse *y*-directed flow, *J*
_{
y
}, may be <1/4 for steepness, *s* < 1, provided that *σ*/*f* is sufficiently close to unity, where *σ* is the wave frequency and *f* the Coriolis frequency. For waves of increasing steepness but fixed frequency, it is found that the minimum Richardson number of flow in the *y* direction, *J*
_{
y
}, becomes less than 1/4 before those of flows in other directions, suggesting that disturbances in the *y* direction, transverse to the wave, may be first to become unstable. Analytical and numerical solutions of the Taylor–Goldstein equation however show that there is a singular neutral mode of the *y*-directed flow with a maximum Richardson number, *J*
_{
y
}, necessary and sufficient for instability, which *decreases* as *s* increases. This mode is stationary. Nonstationary singular *x*-directed neutral modes exist when the minimum Richardson numbers equals 1/4, independent of *s* (<1). These critical disturbances move in the *x* direction at speeds that increase roughly linearly with *s.* In consequence and depending on the relative growth rates and the manner in which wave steepness increases, moving *x*-directed disturbances will be the first to become unstable as *s* increases; they may characterize the onset of instability and consequent wave breaking at all values of *s* < 1. The wavelengths of the first disturbances to become unstable in near-inertial internal waves of a fixed frequency as their steepness increases are then slightly in excess of the vertical wavelength of the internal waves. These disturbances propagate in the positive *x* direction with speeds which increases with *s,* approximately as 0.325*cs,* where *c* is the horizontal phase speed of the internal wave. The study draws attention to the need to examine whether collective instability, leading to mixing on a scale *exceeding* that of a single wave, is possible.

## Abstract

The onset of instability in monochromatic two-dimensional internal near-inertial gravity waves propagating in an ocean of constant buoyancy frequency and no mean shear is examined for increasing values of the wave steepness *s,* the product of the wave amplitude, and the vertical wavenumber of the waves. Stability of disturbances to the quasi-steady flow depends on the minimum Richardson number of the flow in the direction of the disturbance vector.

The minimum Richardson numbers both of the quasi-steady *x*-directed flow, *J*
_{
x
} (that is, in the horizontal direction of wave propagation), and of the transverse *y*-directed flow, *J*
_{
y
}, may be <1/4 for steepness, *s* < 1, provided that *σ*/*f* is sufficiently close to unity, where *σ* is the wave frequency and *f* the Coriolis frequency. For waves of increasing steepness but fixed frequency, it is found that the minimum Richardson number of flow in the *y* direction, *J*
_{
y
}, becomes less than 1/4 before those of flows in other directions, suggesting that disturbances in the *y* direction, transverse to the wave, may be first to become unstable. Analytical and numerical solutions of the Taylor–Goldstein equation however show that there is a singular neutral mode of the *y*-directed flow with a maximum Richardson number, *J*
_{
y
}, necessary and sufficient for instability, which *decreases* as *s* increases. This mode is stationary. Nonstationary singular *x*-directed neutral modes exist when the minimum Richardson numbers equals 1/4, independent of *s* (<1). These critical disturbances move in the *x* direction at speeds that increase roughly linearly with *s.* In consequence and depending on the relative growth rates and the manner in which wave steepness increases, moving *x*-directed disturbances will be the first to become unstable as *s* increases; they may characterize the onset of instability and consequent wave breaking at all values of *s* < 1. The wavelengths of the first disturbances to become unstable in near-inertial internal waves of a fixed frequency as their steepness increases are then slightly in excess of the vertical wavelength of the internal waves. These disturbances propagate in the positive *x* direction with speeds which increases with *s,* approximately as 0.325*cs,* where *c* is the horizontal phase speed of the internal wave. The study draws attention to the need to examine whether collective instability, leading to mixing on a scale *exceeding* that of a single wave, is possible.

## Abstract

When wind waves break in deep water, clouds of small bubbles are produced which are diffused downwards by turbulence. We describe here how the vertical diffusion coefficient *K _{v}
* of the turbulence near the sea surface may be determined from measurements of the bubbles made using a subsurface, upward-directed, high-frequency sonar. The method consists of comparing the observed distributions of scattering cross section with those which can be predicted. Existing observations are used to demonstrate the use of the technique, but are not yet sufficient to allow precise determination of

*K*or of its variation with depth or wind speed.

_{v}## Abstract

When wind waves break in deep water, clouds of small bubbles are produced which are diffused downwards by turbulence. We describe here how the vertical diffusion coefficient *K _{v}
* of the turbulence near the sea surface may be determined from measurements of the bubbles made using a subsurface, upward-directed, high-frequency sonar. The method consists of comparing the observed distributions of scattering cross section with those which can be predicted. Existing observations are used to demonstrate the use of the technique, but are not yet sufficient to allow precise determination of

*K*or of its variation with depth or wind speed.

_{v}## Abstract

Analysis of data from a mooring with five vector-averaging current meters between 10 and 70 m above the bed of the Madeira Abyssal Plain reveals the existence of narrow regions with relatively large gradients of potential temperature, or “fronts.” The orientation and structure of the fronts is examined by combining the temperature and current data and plotting contours of equal potential temperature on the progressive vector diagrams, a procedure justified because of the known horizontal coherence of the currents and the relatively long time-scales of evolution of the benthic boundary layer. Sections through the fronts show that they are typically ∼300 m in width. They extend horizontally for at least 8 km. The temperature differences across the observed fronts are only 2–4 mdeg C. The frontal surfaces are tilted at ∼10 deg to the horizontal, the observed cold fronts being steeper and with isotherms more closely compacted in the lower levels, than warm fronts. These features possibly result from the straining of the temperature field by mesoscale motions as proposed by Armi and D'Asaro.

## Abstract

Analysis of data from a mooring with five vector-averaging current meters between 10 and 70 m above the bed of the Madeira Abyssal Plain reveals the existence of narrow regions with relatively large gradients of potential temperature, or “fronts.” The orientation and structure of the fronts is examined by combining the temperature and current data and plotting contours of equal potential temperature on the progressive vector diagrams, a procedure justified because of the known horizontal coherence of the currents and the relatively long time-scales of evolution of the benthic boundary layer. Sections through the fronts show that they are typically ∼300 m in width. They extend horizontally for at least 8 km. The temperature differences across the observed fronts are only 2–4 mdeg C. The frontal surfaces are tilted at ∼10 deg to the horizontal, the observed cold fronts being steeper and with isotherms more closely compacted in the lower levels, than warm fronts. These features possibly result from the straining of the temperature field by mesoscale motions as proposed by Armi and D'Asaro.

## Abstract

The pattern of disturbance left by internal wave groups traveling in a uniformly stratified ocean is examined. Particular attention is given to the temporal and spatial reoccurrence of extreme values of some parameter *Q,* such as the Richardson number or the wave slope, which may determine, for example, the onset of wave breaking in the group or the wave group’s refraction of smaller-scale waves. Extreme values reoccur with a period *T,* equal to the period of the internal waves, and are sustained along a direction that depends on the wave frequency, but that, over much of the frequency range from *f* (the Coriolis frequency) to *N* (the constant buoyancy frequency) of the internal waves, is nearly horizontal. The size of regions in which extreme values are achieved depends on the aspect ratio of the region of a wave group, termed the “group breaking region,” *V,* within which values of *Q* exceed some threshold *Q*
_{
c
}. Conditions in which regions of past exceedence of *Q*
_{
c
} (“scars” left by waves in passing wave groups) overlap, so as to be always observed by vertical or horizontal profile measurements, depends on the ratio *τ*/*T,* where *τ* is the time for which *Q* > *Q*
_{
c
} as a wave passes through *V.* Near-inertial and semidiurnal tidal internal waves are more likely to leave overlapping scars and may lead to more general mixing of the ocean than, for example, internal wave groups generated by tidal flow over small horizontal scale (1–3 km) topography. It is suggested that wave groups may be evident, and consequently their effects in promoting turbulence may be largest, near the site of internal wave generation, just where recent observations suggest is the region of enhanced turbulent dissipation in the abyssal ocean.

## Abstract

The pattern of disturbance left by internal wave groups traveling in a uniformly stratified ocean is examined. Particular attention is given to the temporal and spatial reoccurrence of extreme values of some parameter *Q,* such as the Richardson number or the wave slope, which may determine, for example, the onset of wave breaking in the group or the wave group’s refraction of smaller-scale waves. Extreme values reoccur with a period *T,* equal to the period of the internal waves, and are sustained along a direction that depends on the wave frequency, but that, over much of the frequency range from *f* (the Coriolis frequency) to *N* (the constant buoyancy frequency) of the internal waves, is nearly horizontal. The size of regions in which extreme values are achieved depends on the aspect ratio of the region of a wave group, termed the “group breaking region,” *V,* within which values of *Q* exceed some threshold *Q*
_{
c
}. Conditions in which regions of past exceedence of *Q*
_{
c
} (“scars” left by waves in passing wave groups) overlap, so as to be always observed by vertical or horizontal profile measurements, depends on the ratio *τ*/*T,* where *τ* is the time for which *Q* > *Q*
_{
c
} as a wave passes through *V.* Near-inertial and semidiurnal tidal internal waves are more likely to leave overlapping scars and may lead to more general mixing of the ocean than, for example, internal wave groups generated by tidal flow over small horizontal scale (1–3 km) topography. It is suggested that wave groups may be evident, and consequently their effects in promoting turbulence may be largest, near the site of internal wave generation, just where recent observations suggest is the region of enhanced turbulent dissipation in the abyssal ocean.