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## Abstract

An estimate of the dissipation rate for mechanical energy in a turbulent flow can be obtained by computing the variance of the temporal derivative of the fluid velocity measured at a single point. This technique is not suited to conventional laser-Doppler velocimetry because of inherent lower bounds on noise in the velocity measurements, which biases the dissipation estimates. A two-spot technique, described by George and Lumley, overcomes this difficulty. In this technique, laser-Doppler measurements of velocity are obtained at two points, which are separated in the spanwise direction by a distance larger than the size of the optical scatterers but smaller than the Kolmogorov scale. The covariance of the temporal derivative of the velocities measured at the two points provides an unbiased estimate of the dissipation, because the velocities at the two points are essentially identical, while the two noise records are uncorrelated. An oceangoing sensor based on this technique has been developed, and its success is demonstrated by experiments in the near-boundary region of a laboratory channel flow, where measurements of shear production provide a standard for evaluation of the dissipation estimates.

## Abstract

An estimate of the dissipation rate for mechanical energy in a turbulent flow can be obtained by computing the variance of the temporal derivative of the fluid velocity measured at a single point. This technique is not suited to conventional laser-Doppler velocimetry because of inherent lower bounds on noise in the velocity measurements, which biases the dissipation estimates. A two-spot technique, described by George and Lumley, overcomes this difficulty. In this technique, laser-Doppler measurements of velocity are obtained at two points, which are separated in the spanwise direction by a distance larger than the size of the optical scatterers but smaller than the Kolmogorov scale. The covariance of the temporal derivative of the velocities measured at the two points provides an unbiased estimate of the dissipation, because the velocities at the two points are essentially identical, while the two noise records are uncorrelated. An oceangoing sensor based on this technique has been developed, and its success is demonstrated by experiments in the near-boundary region of a laboratory channel flow, where measurements of shear production provide a standard for evaluation of the dissipation estimates.

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## Abstract

Fall and winter mean current profiles from a midshelf (water depth ∼90 m) northern California site exhibit a similar vertical structure for several different years. The alongshelf flow is poleward with a maximum velocity of 5–10 cm s^{−1} in the middle or upper water column. There is an offshore flow of about 2 cm s^{−1} in the upper 20–30 m, an onshore flow of about 2 cm s^{−1} in the interior (depths 35–65 m), and an offshore flow of about 1 cm s^{−1} within 20 m of the bottom. Profiles are similar for averages over timescales from weeks to months. Mean current profiles at other midshelf sites along northern California and two sites off Peru also have a similar vertical structure.

The vertical shear in the mean alongshelf flow is geostrophic throughout the water column, that is, in thermal wind balance with the cross-shelf density gradient. For timescales of a week or longer the thermal wind balance extends to within 1 m of the bottom and reduces the mean near-bottom alongshelf flow to 1 cm s^{−1} or less. These observations support recent theoretical work suggesting that, over a sloping bottom, adjustment of the flow and density fields within the bottom boundary layer may reduce the bottom stress. The alongshelf momentum balance is less clear. Weekly averages of offshore transports in the upper and lower water column, relative to the interior onshore flow, are correlated with the surface and bottom stresses, suggesting Ekman balances. However, both the surface and bottom stresses are generally too small by a factor of 2–3 to account for the offshore transports. Limited data suggest that alongshelf buoyancy gradients, estimated over scales of 15 km or less, can be a significant component of the alongshelf momentum balance within both the upper and lower water column.

## Abstract

Fall and winter mean current profiles from a midshelf (water depth ∼90 m) northern California site exhibit a similar vertical structure for several different years. The alongshelf flow is poleward with a maximum velocity of 5–10 cm s^{−1} in the middle or upper water column. There is an offshore flow of about 2 cm s^{−1} in the upper 20–30 m, an onshore flow of about 2 cm s^{−1} in the interior (depths 35–65 m), and an offshore flow of about 1 cm s^{−1} within 20 m of the bottom. Profiles are similar for averages over timescales from weeks to months. Mean current profiles at other midshelf sites along northern California and two sites off Peru also have a similar vertical structure.

The vertical shear in the mean alongshelf flow is geostrophic throughout the water column, that is, in thermal wind balance with the cross-shelf density gradient. For timescales of a week or longer the thermal wind balance extends to within 1 m of the bottom and reduces the mean near-bottom alongshelf flow to 1 cm s^{−1} or less. These observations support recent theoretical work suggesting that, over a sloping bottom, adjustment of the flow and density fields within the bottom boundary layer may reduce the bottom stress. The alongshelf momentum balance is less clear. Weekly averages of offshore transports in the upper and lower water column, relative to the interior onshore flow, are correlated with the surface and bottom stresses, suggesting Ekman balances. However, both the surface and bottom stresses are generally too small by a factor of 2–3 to account for the offshore transports. Limited data suggest that alongshelf buoyancy gradients, estimated over scales of 15 km or less, can be a significant component of the alongshelf momentum balance within both the upper and lower water column.

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## Abstract

Measurements from a horizontal array of velocity sensors indicate that the alongshore scales of turbulence contributing to the near-bottom Reynolds stress just seaward of the surf zone on an ocean beach range from 0 to approximately 4 times the height of the measurements above the seafloor, with shorter scales during stable stratification than during neutral or unstable stratification. The dependence of alongshore turbulence scales on the stratification, Reynolds stress, and height above the bottom is consistent with semiempirical results from the atmospheric surface layer, implying similar dynamics of near-boundary turbulence in the atmosphere and the shallow coastal ocean.

## Abstract

Measurements from a horizontal array of velocity sensors indicate that the alongshore scales of turbulence contributing to the near-bottom Reynolds stress just seaward of the surf zone on an ocean beach range from 0 to approximately 4 times the height of the measurements above the seafloor, with shorter scales during stable stratification than during neutral or unstable stratification. The dependence of alongshore turbulence scales on the stratification, Reynolds stress, and height above the bottom is consistent with semiempirical results from the atmospheric surface layer, implying similar dynamics of near-boundary turbulence in the atmosphere and the shallow coastal ocean.

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## Abstract

Velocity measurements within 1 m of the bottom in approximately 4.5-m water depth on a sand beach provide estimates of turbulent Reynolds shear stress, using a dual-sensor technique that removes contamination by surface waves, and inertial-range estimates of dissipation. When combined with wave measurements along a cross-shore transect and nearby wind measurements, the dataset provides direct estimates of the terms in simplified equations for alongshore momentum and turbulence energetics and permits examination of semiempirical relationships between bottom stress and near-bottom velocity. The records are dominated by three events when the measurement site was in the outer part of the surf zone. Near-bottom turbulent shear stress is well correlated with (squared correlation coefficient *r*
^{2} = 0.63), but smaller than (regression coefficient *b* = 0.51 ± 0.03 at 95% confidence), wind stress minus cross-shore gradient of wave-induced radiation stress, indicating that estimates of one or more of these terms are inaccurate or that an additional effect was important in the alongshore momentum balance. Shear production of turbulent kinetic energy is well correlated (*r*
^{2} = 0.81) and consistent in magnitude (*b* = 1.1 ± 0.1) with dissipation, and both are two orders of magnitude smaller than the depth-averaged rate at which the shoaling wave field lost energy to breaking, indicating that breaking-induced turbulence did not penetrate to the measurement depth. Log-profile estimates of stress are well correlated with (*r*
^{2} = 0.75), but larger than (*b* = 2.3 ± 0.1), covariance estimates of stress, indicating a departure from the Prandtl–von Kármán velocity profile. The bottom drag coefficient was (1.9 ± 0.2) × 10^{−3} during unbroken waves and approximately half as large during breaking waves.

## Abstract

Velocity measurements within 1 m of the bottom in approximately 4.5-m water depth on a sand beach provide estimates of turbulent Reynolds shear stress, using a dual-sensor technique that removes contamination by surface waves, and inertial-range estimates of dissipation. When combined with wave measurements along a cross-shore transect and nearby wind measurements, the dataset provides direct estimates of the terms in simplified equations for alongshore momentum and turbulence energetics and permits examination of semiempirical relationships between bottom stress and near-bottom velocity. The records are dominated by three events when the measurement site was in the outer part of the surf zone. Near-bottom turbulent shear stress is well correlated with (squared correlation coefficient *r*
^{2} = 0.63), but smaller than (regression coefficient *b* = 0.51 ± 0.03 at 95% confidence), wind stress minus cross-shore gradient of wave-induced radiation stress, indicating that estimates of one or more of these terms are inaccurate or that an additional effect was important in the alongshore momentum balance. Shear production of turbulent kinetic energy is well correlated (*r*
^{2} = 0.81) and consistent in magnitude (*b* = 1.1 ± 0.1) with dissipation, and both are two orders of magnitude smaller than the depth-averaged rate at which the shoaling wave field lost energy to breaking, indicating that breaking-induced turbulence did not penetrate to the measurement depth. Log-profile estimates of stress are well correlated with (*r*
^{2} = 0.75), but larger than (*b* = 2.3 ± 0.1), covariance estimates of stress, indicating a departure from the Prandtl–von Kármán velocity profile. The bottom drag coefficient was (1.9 ± 0.2) × 10^{−3} during unbroken waves and approximately half as large during breaking waves.

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## Abstract

Moored temperature and shipboard CTD observations from a northern California coastal upwelling region reveal variable bottom mixed-layer heights that are typically 5–15 m, but occasionally exceed 50 m. Observations from Oregon, northern California, and Peru, indicate that in coastal upwelling regions, maximum bottom mixed-layer heights tend to increase with water depth over the shelf, but rarely exceed half the water depth. Over the northern California shelf the bottom mixed-layer height is shown to depend on the stratification, the current magnitude, and the current direction. The dependence on current direction tends to dominate the response, with thicker bottom mixed layers during poleward flows and thinner bottom mixed layers during equatorward flows. This asymmetric response to poleward and equatorward currents is consistent with model results which indicate that the asymmetric response is due to the up- or downslope Ekman transport of buoyancy along the bottom.

## Abstract

Moored temperature and shipboard CTD observations from a northern California coastal upwelling region reveal variable bottom mixed-layer heights that are typically 5–15 m, but occasionally exceed 50 m. Observations from Oregon, northern California, and Peru, indicate that in coastal upwelling regions, maximum bottom mixed-layer heights tend to increase with water depth over the shelf, but rarely exceed half the water depth. Over the northern California shelf the bottom mixed-layer height is shown to depend on the stratification, the current magnitude, and the current direction. The dependence on current direction tends to dominate the response, with thicker bottom mixed layers during poleward flows and thinner bottom mixed layers during equatorward flows. This asymmetric response to poleward and equatorward currents is consistent with model results which indicate that the asymmetric response is due to the up- or downslope Ekman transport of buoyancy along the bottom.

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## Abstract

Stratification and turbulent mixing exhibit a flood–ebb tidal asymmetry in estuaries and continental shelf regions affected by horizontal density gradients. The authors use a large-eddy simulation (LES) model to investigate the penetration of a tidally driven bottom boundary layer into stratified water in the presence of a horizontal density gradient. Turbulence in the bottom boundary layer is driven by bottom stress during flood tides, with low-gradient (Ri) and flux (*R _{f}
*) Richardson numbers, but by localized shear during ebb tides, with Ri = ¼ and

*R*= 0.2 in the upper half of the boundary layer. If the water column is unstratified initially, the LES model reproduces periodic stratification associated with tidal straining. The model results show that the energetics criterion based on the competition between tidal straining and tidal stirring provides a good prediction for the onset of periodic stratification, but the tidally averaged horizontal Richardson number Ri

_{f}*has a threshold value of about 0.2, which is lower than the 3 suggested in a recent study. Although the tidal straining leads to negative buoyancy flux on flood tides, the authors find that for typical values of the horizontal density gradient and tidal currents in estuaries and shelf regions, buoyancy production is much smaller than shear production in generating turbulent kinetic energy.*

_{x}## Abstract

Stratification and turbulent mixing exhibit a flood–ebb tidal asymmetry in estuaries and continental shelf regions affected by horizontal density gradients. The authors use a large-eddy simulation (LES) model to investigate the penetration of a tidally driven bottom boundary layer into stratified water in the presence of a horizontal density gradient. Turbulence in the bottom boundary layer is driven by bottom stress during flood tides, with low-gradient (Ri) and flux (*R _{f}
*) Richardson numbers, but by localized shear during ebb tides, with Ri = ¼ and

*R*= 0.2 in the upper half of the boundary layer. If the water column is unstratified initially, the LES model reproduces periodic stratification associated with tidal straining. The model results show that the energetics criterion based on the competition between tidal straining and tidal stirring provides a good prediction for the onset of periodic stratification, but the tidally averaged horizontal Richardson number Ri

_{f}*has a threshold value of about 0.2, which is lower than the 3 suggested in a recent study. Although the tidal straining leads to negative buoyancy flux on flood tides, the authors find that for typical values of the horizontal density gradient and tidal currents in estuaries and shelf regions, buoyancy production is much smaller than shear production in generating turbulent kinetic energy.*

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## Abstract

The cospectrum of the horizontal and vertical turbulent velocity fluctuations, an essential tool for understanding measurements of the turbulent Reynolds shear stress, often departs in the ocean from the shape that has been established in the atmospheric surface layer. Here, we test the hypothesis that this departure is caused by advection of standard boundary layer turbulence by the random oscillatory velocities produced by surface gravity waves. The test is based on a model with two elements. The first is a representation of the spatial structure of the turbulence, guided by rapid distortion theory, and consistent with the one-dimensional cospectra that have been measured in the atmosphere. The second model element is a map of the spatial structure of the turbulence to the temporal fluctuations measured at fixed sensors, assuming advection of frozen turbulence by the velocities associated with surface waves. The model is adapted to removal of the wave velocities from the turbulent fluctuations using spatial filtering. The model is tested against previously published laboratory measurements under wave-free conditions and two new sets of measurements near the seafloor in the coastal ocean in the presence of waves. Although quantitative discrepancies exist, the model captures the dominant features of the laboratory and field measurements, suggesting that the underlying model physics are sound.

## Abstract

The cospectrum of the horizontal and vertical turbulent velocity fluctuations, an essential tool for understanding measurements of the turbulent Reynolds shear stress, often departs in the ocean from the shape that has been established in the atmospheric surface layer. Here, we test the hypothesis that this departure is caused by advection of standard boundary layer turbulence by the random oscillatory velocities produced by surface gravity waves. The test is based on a model with two elements. The first is a representation of the spatial structure of the turbulence, guided by rapid distortion theory, and consistent with the one-dimensional cospectra that have been measured in the atmosphere. The second model element is a map of the spatial structure of the turbulence to the temporal fluctuations measured at fixed sensors, assuming advection of frozen turbulence by the velocities associated with surface waves. The model is adapted to removal of the wave velocities from the turbulent fluctuations using spatial filtering. The model is tested against previously published laboratory measurements under wave-free conditions and two new sets of measurements near the seafloor in the coastal ocean in the presence of waves. Although quantitative discrepancies exist, the model captures the dominant features of the laboratory and field measurements, suggesting that the underlying model physics are sound.

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## Abstract

Measurements of velocity, density, and pressure gradient in the lower Hudson River estuary were used to quantify the dominant terms in the momentum equation and to characterize their variations at tidal and spring–neap timescales. The vertical momentum flux (assumed to be due mainly to turbulent shear stress) was estimated indirectly, based on the residual from the acceleration and pressure gradient terms. The indirect estimates of stress compared favorably to bottom stress estimates using a quadratic drag law, supporting the hypothesis that the tidal momentum equation involves a local balance between tidal acceleration, pressure gradient, and stress divergence.

Estimates of eddy viscosity indicated that there was significant tidal asymmetry, with flood tide values exceeding ebb values by a factor of 2. As a consequence of the asymmetry, the vertical structure of the tidally averaged stress bore no resemblance to the tidally averaged shear. In spite of the asymmetry of vertical mixing, the tidally averaged, estuarine circulation was found to depend simply on the intensity of bottom turbulence, which could be parameterized by a Rayleigh drag formulation based on the tidal velocity magnitude and the tidally averaged near-bottom flow. This seemingly paradoxical result indicates that the estuarine circulation can be modeled without detailed knowledge of the effective eddy viscosity, only requiring an estimate of the bottom drag coefficient, the tidal forcing conditions, and the baroclinic pressure gradient. A notable characteristic of this solution is an inverse dependence of the estuarine circulation on the amplitude of the tides.

## Abstract

Measurements of velocity, density, and pressure gradient in the lower Hudson River estuary were used to quantify the dominant terms in the momentum equation and to characterize their variations at tidal and spring–neap timescales. The vertical momentum flux (assumed to be due mainly to turbulent shear stress) was estimated indirectly, based on the residual from the acceleration and pressure gradient terms. The indirect estimates of stress compared favorably to bottom stress estimates using a quadratic drag law, supporting the hypothesis that the tidal momentum equation involves a local balance between tidal acceleration, pressure gradient, and stress divergence.

Estimates of eddy viscosity indicated that there was significant tidal asymmetry, with flood tide values exceeding ebb values by a factor of 2. As a consequence of the asymmetry, the vertical structure of the tidally averaged stress bore no resemblance to the tidally averaged shear. In spite of the asymmetry of vertical mixing, the tidally averaged, estuarine circulation was found to depend simply on the intensity of bottom turbulence, which could be parameterized by a Rayleigh drag formulation based on the tidal velocity magnitude and the tidally averaged near-bottom flow. This seemingly paradoxical result indicates that the estuarine circulation can be modeled without detailed knowledge of the effective eddy viscosity, only requiring an estimate of the bottom drag coefficient, the tidal forcing conditions, and the baroclinic pressure gradient. A notable characteristic of this solution is an inverse dependence of the estuarine circulation on the amplitude of the tides.

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## Abstract

Field observations of turbulent kinetic energy (TKE), dissipation rate ε, and turbulent length scale demonstrate the impact of both density stratification and nonlocal turbulent production on turbulent momentum flux. The data were collected in a highly stratified salt wedge estuary using the Mobile Array for Sensing Turbulence (MAST). Estimates of the dominant length scale of turbulent motions obtained from the vertical velocity spectra provide field confirmation of the theoretical limitation imposed by either the distance to the boundary or the Ozmidov scale, whichever is smaller. Under boundary-limited conditions, anisotropy generally increases with increasing shear and decreased distance to the boundary. Under Ozmidov-limited conditions, anisotropy increases rapidly when the gradient Richardson number exceeds 0.25. Both boundary-limited and Ozmidov-limited conditions demonstrate significant deviations from a local production–dissipation balance that are largely consistent with simple scaling relationships for the vertical divergence in TKE flux. Both the impact of stratification and deviation from equilibrium turbulence observed in the data are largely consistent with commonly used turbulence closure models that employ “nonequilibrium” stability functions. The data compare most favorably with the nonequilibrium version of the L. H. Kantha and C. A. Clayson stability functions. Not only is this approach more consistent with the observed critical gradient Richardson number of 0.25, but it also accounts for the large deviations from equilibrium turbulence in a manner consistent with the observations.

## Abstract

Field observations of turbulent kinetic energy (TKE), dissipation rate ε, and turbulent length scale demonstrate the impact of both density stratification and nonlocal turbulent production on turbulent momentum flux. The data were collected in a highly stratified salt wedge estuary using the Mobile Array for Sensing Turbulence (MAST). Estimates of the dominant length scale of turbulent motions obtained from the vertical velocity spectra provide field confirmation of the theoretical limitation imposed by either the distance to the boundary or the Ozmidov scale, whichever is smaller. Under boundary-limited conditions, anisotropy generally increases with increasing shear and decreased distance to the boundary. Under Ozmidov-limited conditions, anisotropy increases rapidly when the gradient Richardson number exceeds 0.25. Both boundary-limited and Ozmidov-limited conditions demonstrate significant deviations from a local production–dissipation balance that are largely consistent with simple scaling relationships for the vertical divergence in TKE flux. Both the impact of stratification and deviation from equilibrium turbulence observed in the data are largely consistent with commonly used turbulence closure models that employ “nonequilibrium” stability functions. The data compare most favorably with the nonequilibrium version of the L. H. Kantha and C. A. Clayson stability functions. Not only is this approach more consistent with the observed critical gradient Richardson number of 0.25, but it also accounts for the large deviations from equilibrium turbulence in a manner consistent with the observations.

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## Abstract

Measurements just beneath the ocean surface demonstrate that the primary mechanism by which energy from breaking waves is transmitted into the water column is through the work done by the covariance of turbulent pressure and velocity fluctuations. The convergence in the vertical transport of turbulent kinetic energy (TKE) balances the dissipation rate of TKE at first order and is nearly an order of magnitude greater than the sum of the integrated Eulerian and Stokes shear production. The measured TKE transport is consistent with a simple conceptual model that assumes roughly half of the surface flux of TKE by wave breaking is transmitted to depths greater than the significant wave height. During conditions when breaking waves are inferred, the direction of momentum flux is more aligned with the direction of wave propagation than with the wind direction. Both the energy and momentum fluxes occur at frequencies much lower than the wave band, consistent with the time scales associated with wave breaking. The largest instantaneous values of momentum flux are associated with strong downward vertical velocity perturbations, in contrast to the pressure work, which is associated with strong drops in pressure and upward vertical velocity perturbations.

## Abstract

Measurements just beneath the ocean surface demonstrate that the primary mechanism by which energy from breaking waves is transmitted into the water column is through the work done by the covariance of turbulent pressure and velocity fluctuations. The convergence in the vertical transport of turbulent kinetic energy (TKE) balances the dissipation rate of TKE at first order and is nearly an order of magnitude greater than the sum of the integrated Eulerian and Stokes shear production. The measured TKE transport is consistent with a simple conceptual model that assumes roughly half of the surface flux of TKE by wave breaking is transmitted to depths greater than the significant wave height. During conditions when breaking waves are inferred, the direction of momentum flux is more aligned with the direction of wave propagation than with the wind direction. Both the energy and momentum fluxes occur at frequencies much lower than the wave band, consistent with the time scales associated with wave breaking. The largest instantaneous values of momentum flux are associated with strong downward vertical velocity perturbations, in contrast to the pressure work, which is associated with strong drops in pressure and upward vertical velocity perturbations.