1. Introduction
Turbulence in the ocean surface boundary layer results both from shear and convective instabilities similar to those found near rigid boundaries and from instabilities related to surface gravity waves, wave breaking, and Langmuir turbulence. While rigid-boundary turbulence has been extensively studied for nearly a century, turbulence driven by surface waves has been addressed in detail only in the past two decades. In particular, the relationships of turbulent fluxes and energies to wave breaking and Langmuir turbulence continue to be uncertain. Observations (Santala 1991; Plueddemann and Weller 1999; Terray et al. 1999b; Gerbi et al. 2008), laboratory experiments (Veron and Melville 2001), and large eddy simulations (LESs; Skyllingstad and Denbo 1995; Mc Williams et al. 1997; Noh et al. 2004; Li et al. 2005; Sullivan et al. 2007) have shown that vertical mixing is more efficient in wave-driven turbulence than in rigid-boundary turbulence alone. That is, given the same fluxes of momentum and buoyancy at the boundary, vertical gradients in the surface boundary layer are smaller, and turbulent viscosities and diffusivities are larger, than would be expected in a similarly forced flow beneath a rigid boundary. However, the relationship between the diffusivities and the forcing has not been established.
The energetics of turbulence provide important diagnostic and predictive tools and form the basis for most common turbulence closure models used in the ocean (Jones and Launder 1972; Mellor and Yamada 1982; Wilcox 1988; Burchard and Baumert 1995; Umlauf et al. 2003; Umlauf and Burchard 2003; Kantha and Clayson 2004). Because of the difficulty of measuring turbulent fluxes and kinetic energy in a wavy environment, observations of the energetics of ocean surface boundary layer turbulence have generally been confined to dissipation rates of turbulent kinetic energy (TKE) and the vertical velocity variance. Most studies have found dissipation rates of TKE that were enhanced over those expected beneath rigid boundaries (Kitaigorodskii et al. 1983; Agrawal et al. 1992; Anis and Moum 1995; Terray et al. 1996; Drennan et al. 1996; Greenan et al. 2001; Soloviev and Lukas 2003; Gemmrich and Farmer 2004; Stips et al. 2005; Feddersen et al. 2007; Jones and Monismith 2008b). These studies have successfully related the enhanced dissipation rates to fluxes of energy from the wave field, and have suggested that in the depth range where wave-breaking-induced turbulence is dominant, the vertical integral of the dissipation rate is equal to the amount of energy that the waves have lost to breaking. Turbulence closure models have dealt with the increased dissipation by assuming that breaking waves inject TKE at the sea surface and that that TKE is dissipated as it is transported downward by turbulence and pressure work (Craig and Banner 1994; Craig 1996; Terray et al. 1999b; Burchard 2001; Umlauf et al. 2003; Kantha and Clayson 2004). These models, in turn, predict that in the presence of breaking waves, the magnitude of TKE increases substantially within several wave heights of the surface relative to purely rigid-boundary turbulence. This prediction of enhanced TKE is consistent with the observations of D’Asaro (2001) and Tseng and D’Asaro (2004), who made measurements of the vertical component of TKE with Lagrangian floats and found it to be enhanced relative to expectations from rigid-boundary scaling. In a study similar to the one described here, Kitaigorodskii et al. (1983) measured two components of TKE beneath waves in a lake. They also found enhanced TKE near the surface, but their sums of only the vertical and down-wave horizontal components precluded a more complete analysis of the relationship between TKE, dissipation rate, and effective diffusivity.
A simple conceptual model has emerged from previous studies in the ocean surface boundary layer (Fig. 1). Nearest the surface, in what we refer to as the wave breaking layer, is the part of the boundary layer in which waves break and form turbulence. Below that, in what Stips et al. (2005) called the wave-affected surface layer (WASL), the boundary layer is affected by turbulence that is transported downward from the surface, but wave breaking does not inject turbulence directly. The WASL scaling relations developed by Terray et al. (1996) assume that the dissipation profile in the wave breaking layer is vertically uniform, an assumption that has been shown by Gemmrich and Farmer (2004) to break down very near the sea surface. In the WASL, near its upper boundary, the TKE balance is thought to be between dissipation and transport; at deeper depths, the relative importance of the transport of TKE from the surface diminishes and the TKE dynamics approach a production–dissipation balance, similar to that expected in rigid-boundary turbulence.
The present study of turbulence energetics was undertaken as a companion to a study of turbulent fluxes in the surface boundary layer (Gerbi et al. 2008) and was designed to address the following areas: closure of the TKE budget, our understanding of the relationship between TKE and dissipation, and the determination of the role of wave breaking in setting the turbulent diffusivity in the boundary layer. In the process, an analytical model of the vertical structure of TKE (Craig 1996; Burchard 2001) is tested. In the following, section 2 describes the observations, and section 3 shows the results of those observations. Section 4 analyzes the results in comparison to other observations and analytic studies, and section 5 offers conclusions. An appendix describes our method of estimating the dissipation rate of TKE using Eulerian measurements of turbulence in the presence of unsteady advection due to surface gravity waves.
2. Methods
a. Data collection
The observations reported here were made using instruments deployed in the ocean and atmosphere at the Martha’s Vineyard Coastal Observatory’s (MVCO’s) Air–Sea Interaction Tower, during the Coupled Boundary Layers and Air Sea Transfer low winds experiment (CBLAST-Low) in the fall of 2003. The tower is located about 3 km south of Martha’s Vineyard, Massachusetts, in approximately 16 m of water (Fig. 2). Currents are dominated by semidiurnal tides and are dominantly shore parallel (east–west). The mean wind direction is from the southwest. Velocity measurements were made by six Sontek 5-MHz Ocean Probe acoustic Doppler velocimeters (ADVs) deployed at 1.7, 2.2, and 3.2 m below the mean sea surface (Fig. 3). The 3.2-m sensor also contained a pressure sensor and was only used to compute wave statistics, not turbulence statistics. High-frequency temperature measurements were made with fast-response thermistors located within the ADV sample volumes, and the mean temperature and density were measured with Seabird MicroCATs at 1.4-, 2.2-, 3.2-, 4.9-, 6-, 7.9-, 9.9-, and 11.9-m depths. The measurements were described in detail by Gerbi et al. (2008).
Because of measurement sensitivity, estimates of the dissipation rate and TKE were limited to a subset of environmental conditions. We used several criteria to choose acceptable data for inclusion in our analysis. As stated by Gerbi et al. (2008), the instruments were mounted on the west side of the Air–Sea Interaction Tower; so to eliminate distortion from flow through the tower, we analyzed data only for flows from the west. Because this study focuses on boundary layer processes, we analyzed data only when the bottom of the surface boundary layer [defined as the depth at which the temperature difference from the shallowest MicroCAT exceeded 0.02°C (Lentz 1992)] was at least 3.2 m below mean sea level. The ADVs have finite sensitivity and as a criterion for eliminating large wave orbital velocities, we only took bursts for which the vertical velocity variance was less than 0.025 m2 s−2. The oscillating motions due to surface waves caused the wakes of the ADVs to be advected into the sample volumes of the instruments at times when the mean current was not strong enough to sweep the wakes from the ADVs before the waves carried them back to the sample volumes. Therefore, bursts were rejected when the wakes were likely to be advected back into the sample volumes for even a small fraction of the time. In practice, for estimates of the dissipation rate and TKE, we required Ud/σUd > 3, where Ud is the magnitude of the mean velocity and σUd is the velocity variance in the downstream direction. This restriction left times when the significant height of the wind waves was less than 1 m (Fig. 4). Finally, there are times when white noise dominates the measurements at frequencies above the wave band. These were identified during the estimation of the dissipation rates and were removed as described in section 2b.
The measurement period lasted about 34 days, with about 2500 bursts. The restriction on flow direction eliminated about 60% of those bursts. Of the remaining bursts, about 35% were eliminated by the vertical velocity variance threshold, and an additional 20% were eliminated by the boundary layer thickness criterion. Finally, the Ud/σUd threshold and the noise limit removed about 90% of the remaining observations. Thus, the energy and dissipation estimates shown in this study account for about 6% of the times when the mean flow was in a direction favorable to making turbulence measurements, and the bulk of that restriction was due to wave velocities being large enough that turbulent wakes from the ADVs were advected through the measurement volumes.
b. Terms in the TKE budget
This TKE equation assumes no mean vertical flow. Because Gerbi et al. (2008) were able to close the heat and momentum budgets without wave contributions to the fluxes, terms like 
The rate of change of the turbulent kinetic energy was estimated from one-sided finite differences between 20-min bursts when we had successive estimates of q2. A detailed discussion of how we computed TKE is found in section 2c.
In the flux of TKE F, the pressure work 
c. Turbulent kinetic energy estimates
Estimating TKE in the presence of surface waves is difficult, and a spectral approach was used to separate turbulent motions from wave motions. In brief, this approach ignored velocity fluctuations in the wave band and accounted for unsteady advection effects below and above the wave band to estimate the frequency spectra that would have been observed under steady advection. These spectra were transformed to wavenumber space by assuming that the turbulence field was frozen, and a model turbulence spectrum was used to interpolate between the high- and low-frequency portions of the observed spectra (Fig. 5). Fitting this model spectrum to the observed spectra allowed for the estimation of the variance explained by turbulent velocity fluctuations. The reader not interested in the details of this calculation can proceed to section 2d below.
In addition to elevating the high-frequency parts of the spectra, unsteady advection also affects the turbulence spectra at frequencies immediately below the wave band by drawing them down relative to what is expected in steady advection. At sufficiently low frequencies, however, the unsteady effect is minimal (Lumley and Terray 1983) and the spectrum observed in unsteady motion approaches the steady form. To avoid as much of the unsteady effect as possible below the wave band, the analysis included only motions with periods greater than about 3.5 min, using only the five lowest nonzero frequencies in the below-wave band part of the fits. Following Lumley and Terray [(1983), their Eqs. (4.9) and (4.10)] a two-dimensional model of unsteady advection was used to verify that at these long periods unsteady advection has minimal impact on the frequency spectra of the turbulence.
d. Langmuir turbulence detection
The strength of the Langmuir turbulence, as reflected by the root-mean-square (RMS) amplitude of the surface velocity convergence, was estimated using a special-purpose acoustic Doppler current profiler (ADCP). This “fanbeam” ADCP (Plueddemann et al. 2001) was mounted on the seafloor about 50 m offshore of the Air–Sea Interaction Tower (Fig. 3). The instrument uses conventional ADCP electronics but has a modified transducer head that creates four narrow-azimuth beams (3°) spaced 30° apart in the horizontal plane. These beams are broad in elevation (24°), intersect the sea surface at a shallow angle, and have an intensity-weighted return that is dominated by scattering in the upper 1–3 m when bubbles injected by breaking waves are sufficiently strong (Crawford and Farmer 1987; Smith 1992). Standard range gating produces successive sampling cells along the sea surface with dimensions of about 2.5 m (along beam) × 5 m (cross beam). The along-beam aperture of the measurements varies with wind and wave conditions (Plueddemann et al. 2001). For this study, a conservative, fixed aperture of 90 m was used. The ADCP ping rate was 1 Hz, with 56-ping ensembles recorded every minute.
Each beam was processed separately to produce a velocity anomaly for 20-min time intervals and resolved spatial scales (5–90 m along beam). A temporal high-pass filter with a half-power point at 40 min was applied first. This removed the tidal variability that dominated the raw velocities. The high-passed velocities were then detrended in time and range within contiguous 20-min processing windows, after which wavenumber spectra were computed for each time step. The mean spectrum for the 20-min window was integrated over spatial scales from 40 to 5 m, giving a velocity variance. The square root of this quantity, denoted Vrms, was recorded for each beam.
When Vrms was above the estimated noise level of 1.2 cm s−1, the velocity anomaly often showed coherent structures (subparallel lines of convergence and divergence on a time–range plot and a broadly peaked wavenumber specrtrum) characteristic of Langmuir turbulence being advected past the sensor (Smith 1992; Plueddemann et al. 1996, 2001). A detailed investigation of Langmuir turbulence is beyond the scope of this paper. Instead, Vrms was used as an indicator of whether Langmuir turbulence was present during time periods when terms in the TKE budget could be estimated. A threshold of Vrms > 1.8 cm s−1 was found to be a robust indicator of the coherent structures in the fanbeam ADCP data and, in the results that follow, is used as the threshold for declaring that Langmuir turbulence was clearly detectable in the surface velocity field. Smaller-scale or weaker Langmuir turbulence could have been present at times when this Vrms threshold was not exceeded.
e. Directional wave spectra and windsea
To estimate Stokes drift and the characteristics of the windsea and swell, we used directional wave spectra derived from observations made with a 1200-kHz Teledyne RD Instruments (RDI) Workhorse ADCP located at the 12-m isobath, about 1 km shoreward of the Air–Sea Interaction Tower. The directional spectra were computed from contiguous 20-min segments of 2-Hz ADCP data using the RDI WavesMon software package. WavesMon uses a maximum likelihood estimator and linear wave theory to estimate the directional wave spectrum from individual beam velocities (Terray et al. 1999a; Strong et al. 2000; Krogstad et al. 1988). Comparisons of ADCP-derived frequency spectra to those of a laser altimeter mounted on the tower (Churchill et al. 2006) showed that the ADCP was influenced by noise at high frequencies. A cutoff of 2.5 rad s−1 was applied for the spectra used in this study. Because of the vertical decay (6) of the Stokes drift shear, the lack of directional wave spectra at frequencies above 2.5 rad s−1 is unlikely to lead to underestimates of Stokes shear production at the depths of the TKE estimates by more than 10%. Significant wave-height estimates from the ADCP and from the ADVs at the tower were well correlated, with a squared correlation coefficient of 0.87. This, combined with the qualitative agreement of one-dimensional spectra from the tower and the ADCP, suggests that the wave field at the ADCP location was similar to that at the tower.
The study region is influenced both by locally generated wind waves and by remotely generated swell (Fig. 8). The regional geography limits the swell to being predominantly from the south, and the presence of Martha’s Vineyard causes wind wave development at the site to depend on wind direction. During periods of weak wind forcing, the surface wave spectrum is often dominated by swell. To isolate the locally generated windsea from swell components, the method of Hanson and Phillips (2001) was applied by Churchill et al. (2006) using the APL Waves software package developed at the Applied Physics Laboratory of Johns Hopkins University. Spectral partitioning included the isolation of all peaks above a predefined threshold, identification of the windsea peak using the observed wind speed and direction, and the coalescence of adjacent swell peaks when certain criteria were met (Hanson and Phillips 2001). The output of the analysis includes the height, period, and direction of the windsea and one or more swell systems, as well as traditional measures of significant wave height and spectral peak period. During times of weak wind forcing, or when the expected windsea peak was at frequencies greater than 2.5 rad s−1, no windsea was identified. Unless otherwise noted, all subsequent analyses use windsea significant height Hs and windsea wave age cp/u*a, where cp is the phase speed of the peak of the wind wave spectrum, u*a =
f. Wind energy input
For turbulence generated by the wave breaking, the amount of energy transferred from the wave field to the turbulence has been suggested to play a role in setting dissipation rates, total TKE, and TKE flux (Terray et al. 1996; Drennan et al. 1996; Craig and Banner 1994; Craig 1996; Burchard 2001). Following Terray et al. (1996), we assume that most of the energy transferred from the wind to the waves is rapidly transferred from the waves to the water column, and that the wave field grows slowly compared to the rate of energy input from the wind. Thus, estimating the energy input from the wind to the waves is a proxy for estimating the energy input from the waves to the turbulence. To estimate the wind energy input, previous studies have used the directional wave spectrum and a growth rate formulation (Plant 1982; Donelan and Pierson 1987; Donelan 1999; Donelan et al. 2006). Unfortunately, this wave growth estimate is sensitive to frequencies above the 2.5 rad s−1 resolution of our directional spectra, so we were unable to make accurate estimates of the wind energy input by integrating the spectra. In addition, the growth rate formulas are untested in the complex wave fields present during this study, so it is not clear that even perfect directional wave spectra would have allowed precise estimates of wind energy input to the wave field.
3. Results
a. Conditions of observation
The standard deviation of the tidal displacement of the sea surface was 0.35 m, so measurement depths were between about 1.35 and 2.55 m. Wind speeds during the study period were between 1 and 11 m s−1, with a mean of about 6.7 m s−1. The wind waves in our study were relatively mature, with ages cp/u*a between 18 and 44 (Fig. 4). Previous studies (McWilliams et al. 1997; Li et al. 2005) have shown that Langmuir turbulence usually occurs at turbulent Langmuir numbers, Lat =
b. Dissipation
In the complex seas in this study, the significant wave height could be that associated with the full spectrum (dominated by swell) or that computed from the energy in the wave field driven by the local wind (wind waves). The choice of significant wave height in (20) affects the agreement of the observations with the scaling (Fig. 9). For the data to collapse to the scaling, the significant wave height of the wind waves must be used, rather than that of the full spectrum. It has also been suggested that the wavelength of the dominant wind wave can be used as a depth scale (Drennan et al. 1996). Because the wavelength and significant height of the wind waves are correlated, our results are also consistent with this suggestion.
c. TKE balance
Although the enhanced dissipation rate has been observed many times in the surface boundary layer, the association of enhanced dissipation with the flux of TKE from a nonlocal source has remained an attractive, but, to our knowledge, untested, suggestion. By estimating (or bounding, in the case of shear production) terms in the TKE equation, we find that local production of TKE is not sufficient to balance the observed dissipation rates (Fig. 10). The buoyancy production and Stokes shear production terms both are consistently small compared to dissipation. We were unable to measure the storage term for all bursts because we did not always have sequential estimates of TKE, but when measurable, the storage term is also small compared to dissipation. Only the upper bound on the shear production occasionally approaches the magnitude of the dissipation rate at some times of low dissipation rates, suggesting that a local balance could hold. For most of the observations, the dissipation rate greatly exceeds even that upper bound on the shear production. Mean values of each term are given in Table 1.
d. Scaling of TKE and dissipation rate
The value of the parameter cμo has been determined to be 0.09 in neutral conditions near a rigid boundary, and has been assumed constant in other conditions (Umlauf and Burchard 2003), but this assumption has not been tested by observations. By making observations of ε, q, and z, this study makes estimates of the parameter Λ, but does not constrain the distinct values of cμo(3/4) or
4. Discussion
a. Vertical structure of TKE
Two comments are made here regarding the upper boundary in this model. As discussed by Terray et al. (1996) and Gemmrich and Farmer (2004), very near the sea surface, turbulent kinetic energy is injected directly by breaking waves, so the dissipation–transport–production balance is only likely to hold at depths that are below the wave troughs (the wave-affected surface layer). Therefore, the model solved by (27) is not valid above trough level. Accordingly, Burchard (2001) defined the origin of his model domain as being one roughness length below the mean sea surface. We continue to define the origin of our domain as the mean sea surface, a transformation that has been accounted for in (27). Assuming that dissipation is constant in the wave breaking layer, the scaling of Terray et al. (1996) suggests that the upper boundary of the WASL is at z = zb = −0.6Hs, and that one-half of the wind energy input is dissipated in the wave breaking layer, and the other half is exported to the WASL. The total turbulent kinetic energy injected via wave breaking is F0 = −Gtu*3. If only half of this TKE reaches the WASL, the upper-boundary condition leading to (27) must be F(z = zb) = F0/2 = −Gbu*3, where Gb = Gt/2. The ratio Gb/Gt is uncertain and is sensitive to the dissipation structure of the wave breaking layer.
Equation (27) does a reasonable job of reproducing the observations, particularly in reproducing the increase in energy at depths shallower than 5 times the significant wave height. However, the details of the agreement are sensitive to the choice of model constants that were discussed in section 3d (Fig. 12). The results presented here use cμ = cμo and σk = 1, as suggested by Burchard, and z0 = 0.6Hs, consistent with Terray et al. (1996) and Soloviev and Lukas (2003). Burchard used a similar value of z0 = 0.5Hs. The parameters cμ, cμo, and
For TKE and dissipation rate, the presence of stabilizing (|z|/L > 0.2) or destabilizing (|z|/L < 0.2) buoyancy forcings did not lead to substantial changes in the results. Observations that might have been expected to be affected by buoyancy forcing are distributed along with those that have minimal buoyancy forcings (Figs. 9, 11, and 12).
b. Effects of wave breaking on turbulent diffusivity
5. Conclusions
This study estimated selected terms in the turbulent kinetic energy budget of the ocean surface boundary layer: growth of TKE, shear production, Stokes shear production, buoyancy production, and dissipation (Fig. 10). Consistent with previous speculation, the local production terms do not balance dissipation. In the absence of a local balance, it is likely that the enhanced dissipation rates are balanced by the divergence of TKE flux. We were unable to separate turbulence from waves sufficiently to estimate the TKE flux, possibly because turbulent motions explained by wave-band frequencies are important in the TKE flux.
Observations of the dissipation rate are explained well by the scaling of Terray et al. (1996) that relates the dissipation rate to the energy input from the wind to the waves (Fig. 9). The significant wave height used in the scaling must be that of the wind waves, rather than that of the full spectrum. Energy input proportional to u*3 gives good agreement between these observations and previous observations. More precise estimates of the energy input from the wind would only have been obtainable with better estimates of the directional wave spectrum and wave growth rate formulas that are well constrained for the complex sea conditions studied here.
As assumed in simplified turbulence closure models, TKE and the dissipation rate in the ocean surface boundary layer are related through a length scale proportional to the distance to the sea surface. However, a proportionality constant smaller by a factor of about 2 than that in rigid-boundary turbulence relates the dissipation rate, depth, and the three-halves power of TKE in the ocean surface boundary layer (Fig. 11).
With an adjusted proportionality between q3 and εz, the vertical distribution of TKE is reasonably well explained by a one-dimensional model that incorporates the effects of surface gravity waves and shear instabilities (Fig. 12).
Similarly, the vertical turbulent heat flux is predicted well by a one-equation closure model that includes the effects of wave breaking, buoyancy forcing, and shear instability (Fig. 13).
Our estimates of boundary layer turbulence properties were restricted to times of weak to moderate surface forcing. As a result, there were few times when robust Langmuir turbulence was detected concurrently with turbulence energetics. Highlighting the times when Langmuir turbulence was detected did not indicate that it played a distinct role in the energetics or diffusivity. The times when Langmuir turbulence was present did not stand out from the overall distributions when examining the TKE balance or comparing observed and modeled heat fluxes (Figs. 10 and 13). Questions of whether, at what depths, and under what forcing conditions, Langmuir turbulence plays a significant role in surface boundary layer energetics are topics for future research.
The data used to make Figs. 9 –13 are available from the authors or online (http://www.whoi.edu/mvco/data/user_data.html). A subset of these data were also printed in Gerbi (2008).
Acknowledgments
Janet Fredericks, Albert J. Williams III, Ed Hobart, and Neil McPhee assisted in the development and deployment of the instruments and the collection of the data. The Office of Naval Research funded this work as a part of CBLAST-Low. Jim Edson led the project and provided data on meteorological forcing. Jim Churchill analyzed the wave measurements. Comments from two anonymous reviewers greatly improved this manuscript.
REFERENCES
Agrawal, Y. C., E. A. Terray, M. A. Donelan, P. A. Hwang, A. J. Williams III, W. M. Drennan, K. K. Kahma, and S. A. Kitaigorodskii, 1992: Enhanced dissipation of kinetic energy beneath surface waves. Nature, 359 , 219–220.
Anis, A., and J. N. Moum, 1995: Surface wave–turbulence interactions: Scaling ε(z) near the sea surface. J. Phys. Oceanogr., 25 , 2025–2044.
Batchelor, G. K., 1982: The Theory of Homogeneous Turbulence. Cambridge University Press, 197 pp.
Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor., 30 , 327–341.
Bryan, K. R., K. P. Black, and R. M. Gorman, 2003: Spectral estimates of dissipation rate within and near the surf zone. J. Phys. Oceanogr., 33 , 979–993.
Burchard, H., 2001: Simulating the wave-enhanced layer under breaking surface waves with two-equation turbulence models. J. Phys. Oceanogr., 31 , 3133–3145.
Burchard, H., and H. Baumert, 1995: On the performance of a mixed-layer model based on the k ε turbulence closure. J. Geophys. Res., 100 , 8523–8540.
Churchill, J. H., A. J. Plueddemann, and S. M. Faluotico, 2006: Extracting wind sea and swell from directional wave spectra derived from a bottom-mounted ADCP. Woods Hole Oceanographic Institution Tech. Rep. 200613, 34 pp.
Corrsin, S., and A. L. Kistler, 1955: Free-stream boundaries of turbulent flows. National Advisory Committee on Aeronautics Rep. 1244, Washington, DC, 32 pp.
Craig, P. D., 1996: Velocity profiles and surface roughness under breaking waves. J. Geophys. Res., 101 , (C1). 1265–1277.
Craig, P. D., and M. L. Banner, 1994: Modeling wave-enhanced turbulence in the ocean surface layer. J. Phys. Oceanogr., 24 , 2546–2559.
Crawford, C. B., and D. M. Farmer, 1987: On the spatial distribution of ocean bubbles. J. Geophys. Res., 92 , (C8). 8231–8243.
D’Asaro, E., 2001: Turbulent vertical kinetic energy in the ocean mixed layer. J. Phys. Oceanogr., 31 , 3530–3537.
Donelan, M. A., 1999: Wind-induced growth and attenuation of laboratory waves. Wind-over-wave Couplings. Perspectives and Prospects, S. G. Sajjadi, N. H. Thomas, and J. C. R. Hunt, Eds., Clarendon Press, 183–194.
Donelan, M. A., and W. J. Pierson, 1987: Radar scattering and equilibrium ranges in wind generated waves with applications to scatterometry. J. Geophys. Res., 92 , 4971–5029.
Donelan, M. A., A. V. Babanin, I. R. Young, and M. L. Banner, 2006: Wave-follower field measurements of the wind-input spectral function. Part II: Parameterization of the wind input. J. Phys. Oceanogr., 36 , 1672–1689.
Drennan, W. M., M. A. Donelan, E. A. Terray, and K. B. Katsaros, 1996: Oceanic turbulence dissipation measurements in SWADE. J. Phys. Oceanogr., 26 , 808–815.
Feddersen, F., J. H. Trowbridge, and A. J. Williams III, 2007: Vertical structure of dissipation in the nearshore. J. Phys. Oceanogr., 37 , 1764–1777.
Fung, J. C. H., J. C. R. Hunt, N. A. Malik, and R. J. Perkins, 1992: Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes. J. Fluid Mech., 236 , 281–318.
Gemmrich, J. R., and D. M. Farmer, 2004: Near-surface turbulence in the presence of breaking waves. J. Phys. Oceanogr., 34 , 1067–1086.
Gerbi, G. P., 2008: Observations of turbulent fluxes and turbulence dynamics in the ocean surface boundary layer. Ph.D. thesis, Woods Hole Oceanographic Institution–Massachusetts Institute of Technology, Woods Hole, MA, 119 pp.
Gerbi, G. P., J. H. Trowbridge, J. B. Edson, A. J. Plueddemann, E. A. Terray, and J. J. Fredericks, 2008: Measurements of momentum and heat transfer across the air–sea interface. J. Phys. Oceanogr., 38 , 1054–1072.
Greenan, B. J., N. S. Oakey, and F. W. Dobson, 2001: Estimates of dissipation in the ocean mixed layer using a quasi-horizontal microstructure profiler. J. Phys. Oceanogr., 31 , 992–1004.
Hannoun, I. A., H. J. S. Fernando, and E. J. List, 1988: Turbulence structure near a sharp density interface. J. Fluid Mech., 180 , 189–209.
Hanson, J. L., and O. M. Phillips, 2001: Automated analysis of ocean surface directional wave spectra. J. Atmos. Oceanic Technol., 18 , 277–293.
Jones, N. L., and S. G. Monismith, 2008a: Modeling the influence of wave-enhanced turbulence in a shallow tide- and wind-driven water column. J. Geophys. Res., 113 , C03009. doi:10.1029/2007JC004246.
Jones, N. L., and S. G. Monismith, 2008b: The influence of whitecapping waves on the vertical structure of turbulence in a shallow estuarine environment. J. Phys. Oceanogr., 38 , 1563–1580.
Jones, W. P., and B. E. Launder, 1972: The prediction of laminerization with a two-equation model of turbulence. Int. J. Heat Mass Transfer, 15 , 301–314.
Kaimal, J., J. C. Wyngaard, Y. Izumi, and O. R. Cote, 1972: Spectral characteristics of surface-layer turbulence. Quart. J. Roy. Meteor. Soc., 98 , 563–589.
Kantha, L. H., and C. A. Clayson, 2004: On the effect of surface gravity waves on mixing in the oceanic mixed layer. Ocean Modell., 6 , 101–124.
Kitaigorodskii, S. A., M. A. Donelan, J. L. Lumley, and E. A. Terray, 1983: Wave–turbulence interactions in the upper ocean. Part II: Statistical characteristics of wave and turbulent components of the random velocity field in the marine surface layer. J. Phys. Oceanogr., 13 , 1988–1999.
Klebanoff, P. S., 1955: Characteristics of turbulence in a boundary layer with zero pressure gradient. National Advisory Committee on Aeronautics Rep. 1247, Washington, DC, 19 pp.
Kolmogorov, A. N., 1941a: Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR, 31 , 538–540.
Kolmogorov, A. N., 1941b: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR, 30 , 301–305.
Krogstad, H., R. Gordon, and M. Miller, 1988: High-resolution directional wave spectra from horizontally mounted acoustic Doppler current meters. J. Atmos. Oceanic Technol., 5 , 340–352.
Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32 , 363–403.
Lentz, S. J., 1992: The surface boundary layer in coastal upwelling regions. J. Phys. Oceanogr., 22 , 1517–1539.
Li, M., C. Garrett, and E. Skyllingstad, 2005: A regime diagram for classifying turbulent large eddies in the upper ocean. Deep-Sea Res. I, 52 , 259–278.
Lumley, J., and E. Terray, 1983: Kinematics of turbulence convected by a random wave field. J. Phys. Oceanogr., 13 , 2000–2007.
McPhee, M. G., and J. D. Smith, 1976: Measurements of the turbulent boundary layer under pack ice. J. Phys. Oceanogr., 6 , 696–711.
McWilliams, J. C., P. P. Sullivan, and C-H. Moeng, 1997: Langmuir turbulence in the ocean. J. Fluid Mech., 334 , 1–30.
Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys., 20 , 851–875.
Melville, W. K., F. Veron, and C. J. White, 2002: The velocity field under breaking waves: Coherent structures and turbulence. J. Fluid Mech., 454 , 203–233.
Nezu, I., and W. Rodi, 1986: Open-channel flow measurements with a laser Doppler anemometer. J. Hydraul. Eng., 112 , 335–355.
Noh, Y., H. S. Min, and S. Raasch, 2004: Large eddy simulation of the ocean mixed layer: The effects of wave breaking and Langmuir circulation. J. Phys. Oceanogr., 34 , 720–735.
Plant, W. J., 1982: A relationship between wind stress and wave slope. J. Geophys. Res., 87 , 1961–1967.
Plueddemann, A. J., and R. A. Weller, 1999: Structure and evolution of the oceanic surface boundary layer during the Surface Waves Processes Program. J. Mar. Syst., 21 , 85–102.
Plueddemann, A. J., J. A. Smith, D. M. Farmer, R. A. Weller, W. R. Crawford, R. Pinkel, S. Vagle, and A. Gnanadesikan, 1996: Structure and variability of Langmuir circulation during the Surface Waves Processes Program. J. Geophys. Res., 101 , (C2). 3525–3543.
Plueddemann, A. J., E. A. Terray, and R. Merrewether, 2001: Design and performance of a self-contained fan-beam ADCP. IEEE J. Oceanic Eng., 26 , 54–59.
Santala, M. J., 1991: Surface-referenced current meter measurements. Ph.D. thesis, Woods Hole Oceanographic Institution–Massachusetts Institute of Technology, Woods Hole, MA, 280 pp.
Skyllingstad, E. D., and D. W. Denbo, 1995: An ocean large-eddy simulation of Langmuir circulation and convection in the surface mixed layer. J. Geophys. Res., 100 , (C5). 8501–8522.
Smith, J. A., 1992: Observed growth of Langmuir circulation. J. Geophys. Res., 97 , 5651–5664.
Soloviev, A., and R. Lukas, 2003: Observation of wave-enhanced turbulence in the near-surface layer of the ocean during TOGA COARE. Deep-Sea Res. I, 50 , 371–395.
Stips, A., H. Burchard, K. Bolding, H. Prandke, A. Simon, and A. Wuest, 2005: Measurement and simulation of viscous dissipation in the wave affected surface layer. Deep-Sea Res. II, 52 , 1133–1155.
Strong, B., B. Brumley, E. A. Terray, and G. W. Stone, 2000: The performance of ADCP-derived wave directional spectra and comparison with other independent measurements. Proc. Oceans 2000, Providence, RI, MTS/IEEE, 1195–1203.
Sullivan, P. P., J. C. McWilliams, and W. K. Melville, 2007: Surface gravity wave effects in the oceanic boundary layer: Large-eddy simulation with vortex force and stochastic breakers. J. Fluid Mech., 593 , 405–452.
Taylor, G. I., 1938: The spectrum of turbulence. Proc. Roy. Soc. London, 164A , 476–490.
Tennekes, H., and J. L. Lumley, 1972: A First Course in Turbulence. Massachusetts Institute of Technology Press, 300 pp.
Terray, E. A., M. A. Donelan, Y. C. Agrawal, W. M. Drennan, K. K. Kahma, A. J. Williams III, P. A. Hwang, and S. A. Kitaigorodskii, 1996: Estimates of kinetic energy dissipation under breaking waves. J. Phys. Oceanogr., 26 , 792–807.
Terray, E. A., B. H. Brumley, and B. Strong, 1999a: Measuring waves and currents with and upward-looking ADCP. Proc. Sixth Working Conf. on Current Measurements, San Diego, CA, IEEE, 66–71.
Terray, E. A., W. M. Drennan, and M. A. Donelan, 1999b: The vertical structure of shear and dissipation in the ocean surface layer. Proc. Symp. on the Wind-driven Air–Sea Interface—Electromagnetic and Acoustic Sensing, Wave Dynamics, and Turbulent Fluxes, Sydney, NSW, Australia, University of New South Wales, 239–245.
Thompson, S. M., and J. S. Turner, 1975: Mixing across an interface due to turbulence generated by an oscillating grid. J. Fluid Mech., 67 , 349–368.
Thomson, R. E., and I. V. Fine, 2003: Estimating mixed layer depth from oceanic profile data. J. Atmos. Oceanic Technol., 20 , 319–329.
Trowbridge, J., and S. Elgar, 2001: Turbulence measurements in the surf zone. J. Phys. Oceanogr., 31 , 2403–2417.
Tseng, R-S., and E. D’Asaro, 2004: Measurements of turbulent vertical kinetic energy in the ocean mixed layer from Lagrangian floats. J. Phys. Oceanogr., 34 , 1984–1990.
Umlauf, L., and H. Burchard, 2003: A generic length-scale equation for geophysical turbulence models. J. Mar. Res., 61 , 235–265.
Umlauf, L., H. Burchard, and K. Hutter, 2003: Extending the k ω turbulence model towards oceanic applications. Ocean Modell., 5 , 195–218.
Veron, F., and W. K. Melville, 2001: Experiments on the stability and transition of wind-driven water surfaces. J. Fluid Mech., 446 , 25–65.
Wilcox, D., 1988: Reassessment of the scale-determining equation for advanced turbulence models. AIAA J., 26 , 1299–1310.
APPENDIX
Computing Dissipation Rate in Unsteady Advection in Three Directions
Schematic description of the boundary layer structure, including the wave breaking layer, above-trough level, and wave-affected surface layer, which is thought to approach rigid-boundary scaling at sufficient depths. The cartoon of the normalized dissipation profiles shows a constant region in the wave breaking layer, dissipation dominated by the transport of TKE at the top of the wave-affected surface layer, and a transition to rigid-boundary scaling at deeper depths. Here, z is the vertical coordinate, Hs is the significant wave height, ε is the dissipation rate, τw is the wind stress, and F0 is the wind energy input to the waves.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Maps showing the location of MVCO. Contours show isobaths between 10 and 50 m. The inset map shows the area in the immediate vicinity of the study site. [This figure is reprinted from Gerbi et al. (2008)]
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Photograph, looking north, and schematic plan-view drawing of the Air–Sea Interaction Tower at MVCO. In the photograph, the platform is 12 m above the sea surface. In the schematic diagram of the instrument tower, ellipses represent the tilted tower legs (which join at the platform). Small filled circles with three arms each represent ADVs and thermistors. The large filled circle represents the middepth ADCP. Mean wind and wave directions are shown by boldfaced arrows, and the range of flow directions (0°–120°) used in this study is shown to the left. [This figure is reprinted from Gerbi et al. (2008)]
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Environmental conditions during times of dissipation and TKE observations: (a) the significant height of the wind waves, (b) the wind speed at 10-m height, (c) the age of the wind waves, and (d) the Monin–Obukhov parameter at the lower (2.2 m) ADV computed from surface fluxes. Four points with values between −6 and 10 have been omitted from the histogram of |z|/L.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Example autospectra of velocity fluctuations for a 20-min burst: light gray, observed; dark gray (thick lines), observations used in model fitting; black, best fit to full spectrum model; and dashed, best fit to the inertial range model. The top spectrum is Sww and the bottom spectrum is Suu, reduced for clarity by a factor of 1000. As explained in the text, because of noise, the inertial ranges of Suu and Sυυ used in the fitting were determined from the inertial range w spectra and not from observations of u and υ. To minimize the effects of unsteady advection due to surface gravity waves at frequencies below the wave band, only the lowest wavenumbers were used in the model fit in the below-wave-band part of the spectrum.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Comparison of estimates of the dissipation rate as a test of internal consistency of two-parameter fits of the model spectrum to observed spectra: vertical axis, estimates derived from fitting the model to the full spectrum (17); horizontal axis, estimates derived using only the inertial range of the vertical velocity spectrum (8). The line is 1:1. Symbols are dissipation estimates from full-model fits to each component of velocity.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Comparison of the vertical distribution of vertical velocity variances measured in this study and those measured by autonomous floats from D’Asaro (2001) and Tseng and D’Asaro (2004). Error bars show 2 standard errors from the median in each bin.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Directional wave spectrum from a 20-min burst on 8 Oct 2003, showing distinct peaks due to swell and wind waves. The line at 54° from north shows the wind direction. In this burst, as is common during the study, the swell propagates toward the north-northwest and the wind waves propagate toward the northeast.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Observations of the dissipation rate, normalized as suggested by Terray et al. (1996). Depth is normalized by (a) the significant wave height associated with the wind waves and (b) the significant wave height computed from the full spectrum (usually dominated by swell). The thick lines are the expected dissipation rates using neutral rigid-boundary scaling, the thin lines show the scaling of Terray et al. (1996), and the dashed lines show the model predictions of Burchard (2001) and Craig (1996), with cμo = 0.2 and
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Estimates or upper bounds (on shear production only) on production, growth, and dissipation terms in the TKE budget. The dissipation term is usually larger than the sum of the other terms, suggesting that the terms not included here—the transport terms—are important in the TKE balance. Boxes show times when Langmuir turbulence was detected.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Test of the standard relationship between TKE, the dissipation rate, and a turbulent length scale. The quantity Λ is equal to
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Comparison of the observed energy profile (symbols) with that expected from analytic solutions to the TKE equation by Craig (1996) and Burchard (2001), and Eq. (27) (lines). These solutions were evaluated with cμ = cμo, σk = 1, z0 = 0.6Hs, and Gb = 84. Each solution uses different values for
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Vertical heat flux from cospectral observations (Gerbi et al. 2008) and models. The turbulent diffusivities used in modeling the temperature flux are explained in the text. Given the observed temperature gradient, the Monin–Obukhov model underpredicts the temperature fluxes. The composite model accounting for shear instability, buoyancy flux, and wave breaking gives much better agreement with the observations. Boxes show times when Langmuir turbulence was detected at the surface.
Citation: Journal of Physical Oceanography 39, 5; 10.1175/2008JPO4044.1
Mean values of each term in the TKE equation.
