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## 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

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.

## 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

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

Bubbles produced by breaking wind waves are carried by, turbulence below the sea surface. In an earlier model of the distribution of bubble sizes with depth it was necessary to neglect certain terms in order to formulate a differential equation which was solved numerically. A model is devised in which this procedure is avoided. Turbulence is represented by a random walk or Monte Carlo simulation, and each bubble introduced at the surface is followed and a tally kept on its changing radius. Bubbles are continually introduced until a steady state is reached, when the distributions, gas fluxes, and acoustic scattering cross-sections are calculated. The results are compared with camera observations reported by Johnson and Cooke. The major contribution to both gas flux and to the acoustic scattering cross-section per unit volume at sonar frequencies of 248 KHz (corresponding to that which we have used to observe bubbles) comes from bubbles which, at the surface, have radii between ∼40 and 100 μm. The model successfully reproduces the variation of the total number of bubbles with depth, but fails to describe the observed shape of the size distribution. Factors contributing to this discrepancy are discussed. It is possible that bubble populations measured by floating cameras are biased because of the effects of Langmuir circulation both on the float and on the bubbles.

## Abstract

Bubbles produced by breaking wind waves are carried by, turbulence below the sea surface. In an earlier model of the distribution of bubble sizes with depth it was necessary to neglect certain terms in order to formulate a differential equation which was solved numerically. A model is devised in which this procedure is avoided. Turbulence is represented by a random walk or Monte Carlo simulation, and each bubble introduced at the surface is followed and a tally kept on its changing radius. Bubbles are continually introduced until a steady state is reached, when the distributions, gas fluxes, and acoustic scattering cross-sections are calculated. The results are compared with camera observations reported by Johnson and Cooke. The major contribution to both gas flux and to the acoustic scattering cross-section per unit volume at sonar frequencies of 248 KHz (corresponding to that which we have used to observe bubbles) comes from bubbles which, at the surface, have radii between ∼40 and 100 μm. The model successfully reproduces the variation of the total number of bubbles with depth, but fails to describe the observed shape of the size distribution. Factors contributing to this discrepancy are discussed. It is possible that bubble populations measured by floating cameras are biased because of the effects of Langmuir circulation both on the float and on the bubbles.

## 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 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 effect of rotation on the nonlinear reflection of internal waves from a sloping boundary is examined. The waves propagate at an angle *β* to the horizontal in an ocean of locally uniform buoyancy frequency *N,* and the boundary slopes at angle *α* to the horizontal. The following modifications are found when rotation is taken into account: 1) The modulus of the Lagrangian alongslope drift caused by the waves may be increased by an order of magnitude, and the level above the boundary at which the greatest drift is generated is no longer at *z* = 0, but depends on *f*/*N* where *f* is the Coriolis frequency, and the direction of the drift close to the boundary may be reversed. 2) Eulerian upslope currents associated with reflection are increased by a factor *O*(2). Particularly large currents are found to be generated for incident waves travelling almost directly downslope and when *β* > *α.* 3) The mean density and the vertical displacement of isopycnals caused by the waves are increased, possibly by factors *O*(2). 4) The generation of density fronts near the boundary is only slightly affected, except possibly when the incident wave direction *β* is close to values at which the second-order wave components are near critical when *f*/*N* = 0. Here rotation reduces the tendency for fronts to form.

## Abstract

The effect of rotation on the nonlinear reflection of internal waves from a sloping boundary is examined. The waves propagate at an angle *β* to the horizontal in an ocean of locally uniform buoyancy frequency *N,* and the boundary slopes at angle *α* to the horizontal. The following modifications are found when rotation is taken into account: 1) The modulus of the Lagrangian alongslope drift caused by the waves may be increased by an order of magnitude, and the level above the boundary at which the greatest drift is generated is no longer at *z* = 0, but depends on *f*/*N* where *f* is the Coriolis frequency, and the direction of the drift close to the boundary may be reversed. 2) Eulerian upslope currents associated with reflection are increased by a factor *O*(2). Particularly large currents are found to be generated for incident waves travelling almost directly downslope and when *β* > *α.* 3) The mean density and the vertical displacement of isopycnals caused by the waves are increased, possibly by factors *O*(2). 4) The generation of density fronts near the boundary is only slightly affected, except possibly when the incident wave direction *β* is close to values at which the second-order wave components are near critical when *f*/*N* = 0. Here rotation reduces the tendency for fronts to form.

## Abstract

Incident internal waves and those reflected from a uniform slope interact at second order. These interactions are considered for incident waves traveling obliquely to the slope in a uniformly stratified rotating fluid. It is found that (i) resonant interactions cannot occur unless the inclination of the slope to the horizontal, *α*, is less than the inclination of the wave group velocity vector to the horizontal, *β*; (ii) interactions generate a nonzero Eulerian upslope flow, which is exactly balanced by a Stokes drift, so leading to zero Lagrangian upslope flow; (iii) a Stokes drift parallel to the isobaths is generated by the reflecting waves, provided the incident waves are not in a plane normal to the slope. This drift is in addition to the possible Eulerian flows along the slope, which are in geostrophic balance with a correspondingly perturbed density field and are zero in a nonrotating flow. The order of magnitude of the drift is typically 0.01 m s^{−1}. These secondary flows do not depend on dissipation and are therefore distinct from alongslope currents associated with wave breaking in an “internal surf zone.”

## Abstract

Incident internal waves and those reflected from a uniform slope interact at second order. These interactions are considered for incident waves traveling obliquely to the slope in a uniformly stratified rotating fluid. It is found that (i) resonant interactions cannot occur unless the inclination of the slope to the horizontal, *α*, is less than the inclination of the wave group velocity vector to the horizontal, *β*; (ii) interactions generate a nonzero Eulerian upslope flow, which is exactly balanced by a Stokes drift, so leading to zero Lagrangian upslope flow; (iii) a Stokes drift parallel to the isobaths is generated by the reflecting waves, provided the incident waves are not in a plane normal to the slope. This drift is in addition to the possible Eulerian flows along the slope, which are in geostrophic balance with a correspondingly perturbed density field and are zero in a nonrotating flow. The order of magnitude of the drift is typically 0.01 m s^{−1}. These secondary flows do not depend on dissipation and are therefore distinct from alongslope currents associated with wave breaking in an “internal surf zone.”