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- Author or Editor: Stephen Belcher x

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## Abstract

The mean wind profile and the Charnock coefficient, or drag coefficient, over mature seas are investigated. A model of the wave boundary layer, which consists of the lowest part of the atmospheric boundary layer that is influenced by surface waves, is developed based on the conservation of momentum and energy. Energy conservation is cast as a bulk constraint, integrated across the depth of the wave boundary layer, and the turbulence closure is achieved by parameterizing the dissipation rate of turbulent kinetic energy. Momentum conservation is accounted for by using the analytical model of the equilibrium surface wave spectra developed by Hara and Belcher. This approach allows analytical expressions of the Charnock coefficient to be obtained and the results to be examined in terms of key nondimensional parameters. In particular, simple expressions are obtained in the asymptotic limit at which effects of viscosity and surface tension are small and the majority of the stress is supported by wave drag. This analytical model allows us to identify the conditions necessary for the Charnock coefficient to be a true constant, an assumption routinely made in existing bulk parameterizations.

## Abstract

The mean wind profile and the Charnock coefficient, or drag coefficient, over mature seas are investigated. A model of the wave boundary layer, which consists of the lowest part of the atmospheric boundary layer that is influenced by surface waves, is developed based on the conservation of momentum and energy. Energy conservation is cast as a bulk constraint, integrated across the depth of the wave boundary layer, and the turbulence closure is achieved by parameterizing the dissipation rate of turbulent kinetic energy. Momentum conservation is accounted for by using the analytical model of the equilibrium surface wave spectra developed by Hara and Belcher. This approach allows analytical expressions of the Charnock coefficient to be obtained and the results to be examined in terms of key nondimensional parameters. In particular, simple expressions are obtained in the asymptotic limit at which effects of viscosity and surface tension are small and the majority of the stress is supported by wave drag. This analytical model allows us to identify the conditions necessary for the Charnock coefficient to be a true constant, an assumption routinely made in existing bulk parameterizations.

## Abstract

A model is developed to explain the observation made in several laboratory experiments that short wind-generated waves are suppressed by a train of long, mechanically generated waves. A sheltering mechanism is responsible for generation of the short wind waves, by which wave growth is proportional to the local turbulent wind stress. Hence, if the turbulent wind stress near the surface is reduced by the long wave, then the short wind wave amplitude, and hence also the energy in the short waves at a given fetch, is lower than in the absence of long wave. A quantitative model of this process is formulated to examine the ratios of the growth rate and the total energy density of wind waves with and without a long wave, which is shown to agree reasonably well with the laboratory experiments. The model also explains why this suppression of wind waves by a very long swell is not observed in the ocean where the effects of swell on wind waves are extremely difficult to detect. In the model, the reduction in the turbulent wind stress by the long wave is largest for small values of *C*
_{L}/*u** (where *C*
_{L} is the phase speed of the long wave and *u** is the friction velocity of the wind). When this ratio is larger than about 25 (typical of ocean swell), both the reduction of the turbulent wind stress by the long wave and, consequently, the reduction in the total energy density of the wind waves are very small, which explains why this phenomenon has not yet been observed on the ocean.

## Abstract

A model is developed to explain the observation made in several laboratory experiments that short wind-generated waves are suppressed by a train of long, mechanically generated waves. A sheltering mechanism is responsible for generation of the short wind waves, by which wave growth is proportional to the local turbulent wind stress. Hence, if the turbulent wind stress near the surface is reduced by the long wave, then the short wind wave amplitude, and hence also the energy in the short waves at a given fetch, is lower than in the absence of long wave. A quantitative model of this process is formulated to examine the ratios of the growth rate and the total energy density of wind waves with and without a long wave, which is shown to agree reasonably well with the laboratory experiments. The model also explains why this suppression of wind waves by a very long swell is not observed in the ocean where the effects of swell on wind waves are extremely difficult to detect. In the model, the reduction in the turbulent wind stress by the long wave is largest for small values of *C*
_{L}/*u** (where *C*
_{L} is the phase speed of the long wave and *u** is the friction velocity of the wind). When this ratio is larger than about 25 (typical of ocean swell), both the reduction of the turbulent wind stress by the long wave and, consequently, the reduction in the total energy density of the wind waves are very small, which explains why this phenomenon has not yet been observed on the ocean.

## Abstract

The interaction between ocean surface waves and the overlying wind leads to a transfer of momentum across the air–sea interface. Atmospheric and oceanic models typically allow for momentum transfer to be directed only downward, from the atmosphere to the ocean. Recent observations have suggested that momentum can also be transferred upward when long wavelength waves, characteristic of remotely generated swell, propagate faster than the wind speed. The effect of upward momentum transfer on the marine atmospheric boundary layer is investigated here using idealized models that solve the momentum budget above the ocean surface. A variant of the classical Ekman model that accounts for the wave-induced stress demonstrates that, although the momentum flux due to the waves penetrates only a small fraction of the depth of the boundary layer, the wind profile is profoundly changed through its whole depth. When the upward momentum transfer from surface waves sufficiently exceeds the downward turbulent momentum flux, then the near-surface wind accelerates, resulting in a low-level wave-driven wind jet. This increases the Coriolis force in the boundary layer, and so the wind turns in the opposite direction to the classical Ekman layer. Calculations of the wave-induced stress due to a wave spectrum representative of fast-moving swell demonstrate upward momentum transfer that is dominated by contributions from waves in the vicinity of the peak in the swell spectrum. This is in contrast to wind-driven waves whose wave-induced stress is dominated by very short wavelength waves. Hence the role of swell can be characterized by the inverse wave age based on the wave phase speed corresponding to the peak in the spectrum. For a spectrum of waves, the total momentum flux is found to reverse sign and become upward, from waves to wind, when the inverse wave age drops below the range 0.15–0.2, which agrees reasonably well with previously published oceanic observations.

## Abstract

The interaction between ocean surface waves and the overlying wind leads to a transfer of momentum across the air–sea interface. Atmospheric and oceanic models typically allow for momentum transfer to be directed only downward, from the atmosphere to the ocean. Recent observations have suggested that momentum can also be transferred upward when long wavelength waves, characteristic of remotely generated swell, propagate faster than the wind speed. The effect of upward momentum transfer on the marine atmospheric boundary layer is investigated here using idealized models that solve the momentum budget above the ocean surface. A variant of the classical Ekman model that accounts for the wave-induced stress demonstrates that, although the momentum flux due to the waves penetrates only a small fraction of the depth of the boundary layer, the wind profile is profoundly changed through its whole depth. When the upward momentum transfer from surface waves sufficiently exceeds the downward turbulent momentum flux, then the near-surface wind accelerates, resulting in a low-level wave-driven wind jet. This increases the Coriolis force in the boundary layer, and so the wind turns in the opposite direction to the classical Ekman layer. Calculations of the wave-induced stress due to a wave spectrum representative of fast-moving swell demonstrate upward momentum transfer that is dominated by contributions from waves in the vicinity of the peak in the swell spectrum. This is in contrast to wind-driven waves whose wave-induced stress is dominated by very short wavelength waves. Hence the role of swell can be characterized by the inverse wave age based on the wave phase speed corresponding to the peak in the spectrum. For a spectrum of waves, the total momentum flux is found to reverse sign and become upward, from waves to wind, when the inverse wave age drops below the range 0.15–0.2, which agrees reasonably well with previously published oceanic observations.

## Abstract

This study describes the turbulent processes in the upper ocean boundary layer forced by a constant surface stress in the absence of the Coriolis force using large-eddy simulation. The boundary layer that develops has a two-layer structure, a well-mixed layer above a stratified shear layer. The depth of the mixed layer is approximately constant, whereas the depth of the shear layer increases with time. The turbulent momentum flux varies approximately linearly from the surface to the base of the shear layer.

There is a maximum in the production of turbulence through shear at the base of the mixed layer. The magnitude of the shear production increases with time. The increase is mainly a result of the increase in the turbulent momentum flux at the base of the mixed layer due to the increase in the depth of the boundary layer. The length scale for the shear turbulence is the boundary layer depth. A simple scaling is proposed for the magnitude of the shear production that depends on the surface forcing and the average mixed layer current. The scaling can be interpreted in terms of the divergence of a mean kinetic energy flux.

A simple bulk model of the boundary layer is developed to obtain equations describing the variation of the mixed layer and boundary layer depths with time. The model shows that the rate at which the boundary layer deepens does not depend on the stratification of the thermocline. The bulk model shows that the variation in the mixed layer depth is small as long as the surface buoyancy flux is small.

## Abstract

This study describes the turbulent processes in the upper ocean boundary layer forced by a constant surface stress in the absence of the Coriolis force using large-eddy simulation. The boundary layer that develops has a two-layer structure, a well-mixed layer above a stratified shear layer. The depth of the mixed layer is approximately constant, whereas the depth of the shear layer increases with time. The turbulent momentum flux varies approximately linearly from the surface to the base of the shear layer.

There is a maximum in the production of turbulence through shear at the base of the mixed layer. The magnitude of the shear production increases with time. The increase is mainly a result of the increase in the turbulent momentum flux at the base of the mixed layer due to the increase in the depth of the boundary layer. The length scale for the shear turbulence is the boundary layer depth. A simple scaling is proposed for the magnitude of the shear production that depends on the surface forcing and the average mixed layer current. The scaling can be interpreted in terms of the divergence of a mean kinetic energy flux.

A simple bulk model of the boundary layer is developed to obtain equations describing the variation of the mixed layer and boundary layer depths with time. The model shows that the rate at which the boundary layer deepens does not depend on the stratification of the thermocline. The bulk model shows that the variation in the mixed layer depth is small as long as the surface buoyancy flux is small.

## Abstract

Under high-wind conditions, breaking surface waves likely play an important role in the air–sea momentum flux. A coupled wind–wave model is developed based on the assumption that in the equilibrium range of surface wave spectra the wind stress is dominated by the form drag of breaking waves. By conserving both momentum and energy in the air and also imposing the wave energy balance, coupled equations are derived governing the turbulent stress, wind speed, and the breaking-wave distribution (total breaking crest length per unit surface area as a function of wavenumber). It is assumed that smaller-scale breaking waves are sheltered from wind forcing if they are in airflow separation regions of longer breaking waves (spatial sheltering effect). Without this spatial sheltering, exact analytic solutions are obtained; with spatial sheltering asymptotic solutions for small- and large-scale breakers are derived. In both cases, the breaking-wave distribution approaches a constant value for large wavenumbers (small-scale breakers). For low wavenumbers, the breaking-wave distribution strongly depends on wind forcing. If the equilibrium range model is extended to the spectral peak, the model yields the normalized roughness length (Charnock coefficient) of growing seas, which increases with wave age and is roughly consistent with earlier laboratory observations. However, the model does not yield physical solutions beyond a critical wave age, implying that the wind input to the wave field cannot be dominated by breaking waves at all wavenumbers for developed seas (including field conditions).

## Abstract

Under high-wind conditions, breaking surface waves likely play an important role in the air–sea momentum flux. A coupled wind–wave model is developed based on the assumption that in the equilibrium range of surface wave spectra the wind stress is dominated by the form drag of breaking waves. By conserving both momentum and energy in the air and also imposing the wave energy balance, coupled equations are derived governing the turbulent stress, wind speed, and the breaking-wave distribution (total breaking crest length per unit surface area as a function of wavenumber). It is assumed that smaller-scale breaking waves are sheltered from wind forcing if they are in airflow separation regions of longer breaking waves (spatial sheltering effect). Without this spatial sheltering, exact analytic solutions are obtained; with spatial sheltering asymptotic solutions for small- and large-scale breakers are derived. In both cases, the breaking-wave distribution approaches a constant value for large wavenumbers (small-scale breakers). For low wavenumbers, the breaking-wave distribution strongly depends on wind forcing. If the equilibrium range model is extended to the spectral peak, the model yields the normalized roughness length (Charnock coefficient) of growing seas, which increases with wave age and is roughly consistent with earlier laboratory observations. However, the model does not yield physical solutions beyond a critical wave age, implying that the wind input to the wave field cannot be dominated by breaking waves at all wavenumbers for developed seas (including field conditions).

## Abstract

This study uses large-eddy simulation (LES) to investigate the characteristics of Langmuir turbulence through the turbulent kinetic energy (TKE) budget. Based on an analysis of the TKE budget a velocity scale for Langmuir turbulence is proposed. The velocity scale depends on both the friction velocity and the surface Stokes drift associated with the wave field. The scaling leads to unique profiles of nondimensional dissipation rate and velocity component variances when the Stokes drift of the wave field is sufficiently large compared to the surface friction velocity. The existence of such a scaling shows that Langmuir turbulence can be considered as a turbulence regime in its own right, rather than a modification of shear-driven turbulence.

Comparisons are made between the LES results and observations, but the lack of information concerning the wave field means these are mainly restricted to comparing profile shapes. The shapes of the LES profiles are consistent with observed profiles. The dissipation length scale for Langmuir turbulence is found to be similar to the dissipation length scale in the shear-driven boundary layer. Beyond this it is not possible to test the proposed scaling directly using available data. Entrainment at the base of the mixed layer is shown to be significantly enhanced over that due to normal shear turbulence.

## Abstract

This study uses large-eddy simulation (LES) to investigate the characteristics of Langmuir turbulence through the turbulent kinetic energy (TKE) budget. Based on an analysis of the TKE budget a velocity scale for Langmuir turbulence is proposed. The velocity scale depends on both the friction velocity and the surface Stokes drift associated with the wave field. The scaling leads to unique profiles of nondimensional dissipation rate and velocity component variances when the Stokes drift of the wave field is sufficiently large compared to the surface friction velocity. The existence of such a scaling shows that Langmuir turbulence can be considered as a turbulence regime in its own right, rather than a modification of shear-driven turbulence.

Comparisons are made between the LES results and observations, but the lack of information concerning the wave field means these are mainly restricted to comparing profile shapes. The shapes of the LES profiles are consistent with observed profiles. The dissipation length scale for Langmuir turbulence is found to be similar to the dissipation length scale in the shear-driven boundary layer. Beyond this it is not possible to test the proposed scaling directly using available data. Entrainment at the base of the mixed layer is shown to be significantly enhanced over that due to normal shear turbulence.

## Abstract

Generally, ocean waves are thought to act as a drag on the surface wind so that momentum is transferred downward, from the atmosphere into the waves. Recent observations have suggested that when long wavelength waves—which are characteristic of remotely generated swell—propagate faster than the surface wind, momentum can also be transferred upward. This upward momentum transfer acts to accelerate the near-surface wind, resulting in a low-level wave-driven wind jet. Previous studies have suggested that the sign reversal of the momentum flux is well predicted by the inverse wave age, the ratio of the surface wind speed to the speed of the waves at the peak of the spectrum. Data from the 40-yr ECMWF Re-Analysis (ERA-40) have been used here to calculate the global distribution of the inverse wave age to determine whether there are regions of the ocean that are usually in the wind-driven wave regime and others that are generally in the wave-driven wind regime. The wind-driven wave regime is found to occur most often in the midlatitude storm tracks where wind speeds are generally high. The wave-driven wind regime is found to be prevalent in the tropics where wind speeds are generally light and swell can propagate from storms at higher latitudes. The inverse wave age is also a useful indicator of the degree of coupling between the local wind and wave fields. The climatologies presented emphasize the nonequilibrium that exists between the local wind and wave fields and highlight the importance of swell in the global oceans.

## Abstract

Generally, ocean waves are thought to act as a drag on the surface wind so that momentum is transferred downward, from the atmosphere into the waves. Recent observations have suggested that when long wavelength waves—which are characteristic of remotely generated swell—propagate faster than the surface wind, momentum can also be transferred upward. This upward momentum transfer acts to accelerate the near-surface wind, resulting in a low-level wave-driven wind jet. Previous studies have suggested that the sign reversal of the momentum flux is well predicted by the inverse wave age, the ratio of the surface wind speed to the speed of the waves at the peak of the spectrum. Data from the 40-yr ECMWF Re-Analysis (ERA-40) have been used here to calculate the global distribution of the inverse wave age to determine whether there are regions of the ocean that are usually in the wind-driven wave regime and others that are generally in the wave-driven wind regime. The wind-driven wave regime is found to occur most often in the midlatitude storm tracks where wind speeds are generally high. The wave-driven wind regime is found to be prevalent in the tropics where wind speeds are generally light and swell can propagate from storms at higher latitudes. The inverse wave age is also a useful indicator of the degree of coupling between the local wind and wave fields. The climatologies presented emphasize the nonequilibrium that exists between the local wind and wave fields and highlight the importance of swell in the global oceans.

## Abstract

The interaction between the Coriolis force and the Stokes drift associated with ocean surface waves leads to a vertical transport of momentum, which can be expressed as a force on the mean momentum equation in the direction along wave crests. How this *Coriolis–Stokes forcing* affects the mean current profile in a wind-driven mixed layer is investigated using simple models, results from large-eddy simulations, and observational data. The effects of the Coriolis–Stokes forcing on the mean current profile are examined by reappraising analytical solutions to the Ekman model that include the Coriolis–Stokes forcing. Turbulent momentum transfer is modeled using an eddy-viscosity model, first with a constant viscosity and second with a linearly varying eddy viscosity. Although the Coriolis–Stokes forcing penetrates only a small fraction of the depth of the wind-driven layer for parameter values typical of the ocean, the analytical solutions show how the current profile is substantially changed through the whole depth of the wind-driven layer. It is shown how, for this oceanic regime, the Coriolis–Stokes forcing supports a fraction of the applied wind stress, changing the boundary condition on the wind-driven component of the flow and hence changing the current profile through all depths. The analytical solution with the linearly varying eddy viscosity is shown to reproduce reasonably well the effects of the Coriolis–Stokes forcing on the current profile computed from large-eddy simulations, which resolve the three-dimensional overturning motions associated with the turbulent Langmuir circulations in the wind-driven layer. Last, the analytical solution with the Coriolis–Stokes forcing is shown to agree reasonably well with current profiles from previously published observational data and certainly agrees better than the standard Ekman model. This finding provides evidence that the Coriolis–Stokes forcing is an important mechanism in controlling the dynamics of the upper ocean.

## Abstract

The interaction between the Coriolis force and the Stokes drift associated with ocean surface waves leads to a vertical transport of momentum, which can be expressed as a force on the mean momentum equation in the direction along wave crests. How this *Coriolis–Stokes forcing* affects the mean current profile in a wind-driven mixed layer is investigated using simple models, results from large-eddy simulations, and observational data. The effects of the Coriolis–Stokes forcing on the mean current profile are examined by reappraising analytical solutions to the Ekman model that include the Coriolis–Stokes forcing. Turbulent momentum transfer is modeled using an eddy-viscosity model, first with a constant viscosity and second with a linearly varying eddy viscosity. Although the Coriolis–Stokes forcing penetrates only a small fraction of the depth of the wind-driven layer for parameter values typical of the ocean, the analytical solutions show how the current profile is substantially changed through the whole depth of the wind-driven layer. It is shown how, for this oceanic regime, the Coriolis–Stokes forcing supports a fraction of the applied wind stress, changing the boundary condition on the wind-driven component of the flow and hence changing the current profile through all depths. The analytical solution with the linearly varying eddy viscosity is shown to reproduce reasonably well the effects of the Coriolis–Stokes forcing on the current profile computed from large-eddy simulations, which resolve the three-dimensional overturning motions associated with the turbulent Langmuir circulations in the wind-driven layer. Last, the analytical solution with the Coriolis–Stokes forcing is shown to agree reasonably well with current profiles from previously published observational data and certainly agrees better than the standard Ekman model. This finding provides evidence that the Coriolis–Stokes forcing is an important mechanism in controlling the dynamics of the upper ocean.

## Abstract

The differences between the conclusions of Noh and Choi and of Pearson et al., which are largely a result of defining different length scales based on different quantities, are discussed. This study shows that the layer over which Langmuir turbulence mixes (nominally *h*
_{TKE}) under a stabilizing surface buoyancy flux should be scaled by a combination of the Langmuir stability length *L*
_{L} and initial/nocturnal boundary layer depth *h*
_{0} rather than by the Zilitinkevich length.

## Abstract

The differences between the conclusions of Noh and Choi and of Pearson et al., which are largely a result of defining different length scales based on different quantities, are discussed. This study shows that the layer over which Langmuir turbulence mixes (nominally *h*
_{TKE}) under a stabilizing surface buoyancy flux should be scaled by a combination of the Langmuir stability length *L*
_{L} and initial/nocturnal boundary layer depth *h*
_{0} rather than by the Zilitinkevich length.