1. Introduction
The cospectrum of the horizontal and vertical components of the turbulent velocity is an essential tool for understanding the turbulent Reynolds shear stress and interpreting stress measurements. The cospectrum provides an estimate of the dominant stress-carrying turbulent length scale (e.g., Kaimal et al. 1972), an assessment of whether the stress-carrying scales have been resolved, and an indication of contamination of stress estimates by spurious contributions from sensor noise.
In the ocean, surface waves complicate frequency cospectra computed from measurements obtained by point sensors at fixed positions, even in the absence of dynamical wave–current interactions, such as the steady streaming (Longuet-Higgins 1953), Langmuir circulations (Craik and Leibovich 1976), or the turbulent wave–current boundary layer (Grant and Madsen 1979), because of at least two kinematical effects. First, even if the wave velocities are filtered perfectly from measurements obtained by a point sensor at a fixed position, advection of the turbulence by the wave velocities distorts the frequency spectrum computed from the filtered measurements of the turbulent velocity (Lumley and Terray 1983). Second, even a small uncertainty in the sensor orientation relative to the principal axes of the velocities associated with waves can create large uncertainties in stress estimates obtained from measurements by a single sensor (Grant and Madsen 1986) because the velocity variances produced by the surface waves are often orders of magnitude larger than those associated with boundary layer turbulence.
Lumley and Terray (1983) analyzed the first effect (advection of turbulence by velocities associated with waves), assuming frozen turbulence, statistically independent waves and turbulence, and linear waves with Gaussian statistics. The Lumley and Terray analysis maps the spatial structure of the turbulence to the temporal fluctuations that are measured by a fixed sensor in the presence of waves. Terray et al. (1996), Trowbridge and Elgar (2001), Feddersen et al. (2007), Gerbi et al. (2009), Feddersen (2012), Scully et al. (2016), and others have combined this analysis with an isotropic model of the underlying turbulence to describe high-frequency turbulent autospectra for estimation of the turbulent energy dissipation rate. Gerbi et al. (2008) and Rosman and Gerbi (2017) used the Lumley and Terray (1983) analysis and a turbulence model similar to (1) to simulate the effects of wave advection on the stress-carrying turbulent cospectrum for the case in which the advection by the current and waves is solely in the x1 direction.
Trowbridge (1998) proposed mitigation of the second effect (sensitivity of stress measurements to uncertainty in the sensor orientation in the presence of strong waves and weak turbulence) by differencing measurements obtained by two sensors separated by a distance larger than the correlation scale of the turbulence but much smaller than the surface wavelength. The analysis indicates that 
To enable analysis and interpretation of stress measurements based on the differencing method, this study proposes and tests a model of the frequency cospectrum of time series of Δu1 and Δu3 that are measured by a pair of fixed sensors in the presence of surface waves. The model has two elements. The first is a representation of the spatial structure of the turbulence, guided by the rapid distortion solution for initially isotropic turbulence in mean shear (Townsend 1980) and consistent with the one-dimensional wavenumber cospectrum equation [(1)]. The rapid distortion solution provides a dynamically consistent framework and has proven successful, in spite of its simplicity, in other contexts (e.g., Savill 1987; Hunt and Carruthers 1990; Cambon and Scott 1999). The second model element is the Lumley and Terray (1983) mapping of the spatial structure of the turbulence to the temporal fluctuations measured by fixed sensors in the presence of waves, adapted to the difference between measurements obtained by two fixed sensors. The proposed model is tested against previously reported measurements of the spatial correlation function (the Fourier transform of the cospectrum) under wave-free conditions in the laboratory and against new measurements near the seafloor over the inner continental shelf off of Martha’s Vineyard, in which the velocities associated with surface waves are much larger than those associated with turbulence. The following presents methods (section 2), results (section 3), a discussion (section 4), and a summary and conclusions (section 5), followed by two appendices with model details.
2. Methods
a. Model of the spatial structure of the turbulence
b. Model of the temporal statistics of velocity differences
In the present application, the sensor separation is horizontal, and proximity to the seafloor dictates that the vertical component of the mean velocity and the vertical velocities associated with surface waves make a negligible contribution to (9). For these conditions, model equations (5), (8), and (9) determine 
c. Laboratory measurements
Tritton (1967) and Ganapathisubramani et al. (2005) reported measurements of the spatial covariance functions R13(r1, 0, 0) and R13(0, r2, 0) in unstratified turbulent boundary layers under wave-free conditions in laboratory wind tunnels. The measurements in the two studies were under similar conditions and agree well with each other. The present study uses the Ganapathisubramani et al. (2005) measurements because these, unlike the Tritton (1967) measurements, were obtained symmetrically for positive and negative r1 and r2, permitting extraction of the even part 
The elevation at the uppermost laboratory measurements did not satisfy x3 ≪ δ, required for wall layer scaling, where δ is the boundary layer thickness. Thus, λ at this elevation plausibly depends not only on x3 but also on δ. To provide a predictive relationship, λ is set to 1.5x3 for x3 < 0.15δ, consistent with (1), and to 0.23δ for x3 ≥ 0.15δ, in analogy with the turbulent mixing length, which is observed to be proportional to x3 for x3 ≪ δ and approximately constant for x3 > 0.15δ (e.g., Schlichting and Gersten 2000).
d. Field measurements
Field measurements were made on the inner continental shelf south of Martha’s Vineyard, Massachusetts, near the Martha’s Vineyard Coastal Observatory (MVCO). The study site is exposed to the Atlantic Ocean and has a predominantly sandy seafloor that fines with depth and has alternating ribbons of fine sand with small ripples and coarse sand with large ripples (Fig. 1). A quadpod (Fig. 2) was placed from early July to late August 2014 at site QS1, at a mean depth of 16.2 m, and from mid-November 2014 to mid-January 2015 at site QS3, at a mean depth of 17.9 m (Fig. 1). The seafloor at QS1 was gravelly coarse sand (median size from a grab sample = 790 μm) with large orbital ripples [wavelength from the sonar = 0.5 to 0.8 m; heights estimated from the Aquatec acoustic backscatter sensor (ABSS) = 0.08 to 0.15 m]. At QS3, no grab sample was taken, but the sonar shows small anorbital ripples (wavelength 0.10–0.15 m and height ̴0.02 m), indicating fine sand, consistent with the sidescan sonar imagery (Fig. 1).
Map of the coastal ocean south of Martha’s Vineyard, Massachusetts, with overlain sidescan sonar imagery (Denny et al. 2009; Ackerman et al. 2016), indicating selected MVCO infrastructure [the seafloor node and the Air–Sea Interaction Tower (ASIT)] and the quadpod deployment sites QS1 and QS3. Bright and dark sidescan images correspond to coarse and fine seafloor sediments, respectively.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Map of the coastal ocean south of Martha’s Vineyard, Massachusetts, with overlain sidescan sonar imagery (Denny et al. 2009; Ackerman et al. 2016), indicating selected MVCO infrastructure [the seafloor node and the Air–Sea Interaction Tower (ASIT)] and the quadpod deployment sites QS1 and QS3. Bright and dark sidescan images correspond to coarse and fine seafloor sediments, respectively.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Map of the coastal ocean south of Martha’s Vineyard, Massachusetts, with overlain sidescan sonar imagery (Denny et al. 2009; Ackerman et al. 2016), indicating selected MVCO infrastructure [the seafloor node and the Air–Sea Interaction Tower (ASIT)] and the quadpod deployment sites QS1 and QS3. Bright and dark sidescan images correspond to coarse and fine seafloor sediments, respectively.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
The quadpod being deployed from the University of Connecticut R/V Connecticut.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
The quadpod being deployed from the University of Connecticut R/V Connecticut.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
The quadpod being deployed from the University of Connecticut R/V Connecticut.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
The quadpod supported (i) two Nortek Vector acoustic Doppler velocimeter (ADVs) for measuring the current and making direct covariance estimates of the stress; (ii) a downward-looking Nortek Aquadopp acoustic Doppler profiler and an ABSS for measuring the vertical structure of the current, the suspended sediment concentration in the bottom meter of the water column, and ripple heights; (iii) an upward-looking, five-beam Nortek Signature acoustic Doppler current profiler (ADCP) for measuring waves and currents and detecting the presence and characteristics of Langmuir circulations (Gargett et al. 2004; Gargett and Wells 2007); (iv) temperature and conductivity sensors (RBR Solo-Ts and Sea-Bird MicroCATs) to measure the stratification; and (v) downward-looking, Imagenex, rotary, azimuth-drive, pencil-beam and rotary fan-beam sonars to quantify bed forms. The instrument cases were mounted at the top of the quadpod (Fig. 2), separated from the sample volumes of the velocity sensors.
The present analysis is based on data from the ADVs, which provided measurements of the currents, wave velocities, and turbulence, and the temperature and conductivity sensors, which provided estimates of the stratification. The ADV sample volumes were approximately 0.50 m above bottom and separated horizontally by 1.20 m. The ADV separation (Fig. 3) was roughly along isobath during the first deployment (QS1) and at an angle of approximately 60° with respect to the isobaths during the second deployment (QS3). The ADVs sampled synchronously at 32 Hz, with 28-min bursts recorded each half hour. The temperature and conductivity sensors were at heights of approximately 0.24, 1.44, and 2.23 m above bottom. The temperature sensors sampled at 1 Hz and the conductivity sensors at 3 min.
Diagram showing the orientation of the quadpod and ADVs with the respect to the dominant current orientation and wave directions during deployments (a) QS1 and (b) QS3.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Diagram showing the orientation of the quadpod and ADVs with the respect to the dominant current orientation and wave directions during deployments (a) QS1 and (b) QS3.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Diagram showing the orientation of the quadpod and ADVs with the respect to the dominant current orientation and wave directions during deployments (a) QS1 and (b) QS3.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
The ADV data were quality controlled by rejecting measurements with correlations of successive acoustic pings less than 80%, similar to criteria recommended by Elgar et al. (2005) and Feddersen (2010) for applications in the surfzone. Rejected measurements were replaced with the burst mean of the retained measurements and bursts with more than 10% rejected were excluded from further analysis. This procedure resulted in exclusion of 41% of the bursts from the first deployment (QS1) and 4% from the second deployment (QS3). The large fraction of rejected bursts during the first deployment resulted from electronic noise from a source that was not identified despite extensive communications with the manufacturer.
For each 28-min burst, the quality-controlled ADV measurements were rotated into coordinates with x1 aligned with the mean velocity. The spectra 
Effects of stable stratification were quantified using the Ozmidov scale LO = ε1/2/N3/2 (e.g., Phillips 1980), an upper bound imposed by stratification on the scale of the turbulent eddies. Here, N is the buoyancy frequency, and ε is the dissipation rate for turbulent kinetic energy. The buoyancy frequency was estimated using the seawater equation of state (Fofonoff and Millard 1983) with the observed temperature differences and the array-averaged mean salinity, the conductivity cells having drifted sufficiently that differences between vertically separated conductivity measurements were not meaningful. The dissipation rate was estimated from inertial-range velocity autospectra using the method described by Scully et al. (2016).
3. Results
a. Model computations
Computations of the spatial structure of the turbulence, represented by 
Sample computations of 
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Sample computations of
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Sample computations of
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Model computations of the frequency spectrum of the turbulence for an idealized wave spectrum (uniform within a finite frequency bandwidth) indicate strong dependence of the spectral shape on the model parameters. In particular, the shape of the cospectrum, including the breadth of the cospectral peak, changes significantly depending on whether the two sensors for the spatial differencing are aligned parallel or perpendicular to the mean current and whether the wave direction is parallel or perpendicular to the mean current (Fig. 5). The dependence of the model on the parameters is sufficiently complex that general features, even qualitative features, are difficult to predict in the absence of detailed numerical computations.
Model computations of the shape of the frequency cospectrum of Δu1 and Δu3. In all computations, the mean current velocity U1 is 0.3 m s−1, the turbulent length scale λ is 0.5 m, the waves are unidirectional, and the frequency spectrum of the wave velocities is nonzero only within bands centered on ωp = ±1 s−1 with width Δω = ωp/2 (i.e., a boxcar spectrum). The quantities θw and uw are, respectively, the direction of the waves relative to the direction of the current (in radians) and the standard deviation of the wave velocity. The curves labeled Kaimal are twice (1) with ω = k1U1, consistent with the standard frozen turbulence approximation.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Model computations of the shape of the frequency cospectrum of Δu1 and Δu3. In all computations, the mean current velocity U1 is 0.3 m s−1, the turbulent length scale λ is 0.5 m, the waves are unidirectional, and the frequency spectrum of the wave velocities is nonzero only within bands centered on ωp = ±1 s−1 with width Δω = ωp/2 (i.e., a boxcar spectrum). The quantities θw and uw are, respectively, the direction of the waves relative to the direction of the current (in radians) and the standard deviation of the wave velocity. The curves labeled Kaimal are twice (1) with ω = k1U1, consistent with the standard frozen turbulence approximation.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Model computations of the shape of the frequency cospectrum of Δu1 and Δu3. In all computations, the mean current velocity U1 is 0.3 m s−1, the turbulent length scale λ is 0.5 m, the waves are unidirectional, and the frequency spectrum of the wave velocities is nonzero only within bands centered on ωp = ±1 s−1 with width Δω = ωp/2 (i.e., a boxcar spectrum). The quantities θw and uw are, respectively, the direction of the waves relative to the direction of the current (in radians) and the standard deviation of the wave velocity. The curves labeled Kaimal are twice (1) with ω = k1U1, consistent with the standard frozen turbulence approximation.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
b. Model computations and laboratory measurements
At the uppermost height, the model computations and laboratory measurements of the along-stream correlation function 
c. Model computations and field measurements
The instruments experienced a range of conditions during the field deployments (Fig. 6). Wave heights ranged from approximately 0.5 m to more than 3 m, with dominant (spectral peak) wave periods typically between 5 and 10 s. Wave incidence was predominantly from the south during the first deployment and from the south and southwest during the second deployment. Near-bottom currents were predominantly east–west (parallel to the isobaths), with magnitudes of roughly 0.3 m s−1, dominated by semidiurnal tides, with occasional lower-frequency fluctuations of similar magnitude.
Conditions during the measurement periods QS1 (days 183 to 223) and QS3 (days 316 to 354), including (a) significant wave height, (b) dominant wave period, (c) dominant wave direction (from), and (d) near-bed current velocity, with the alongshore component in black and the cross-isobath component in red.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Conditions during the measurement periods QS1 (days 183 to 223) and QS3 (days 316 to 354), including (a) significant wave height, (b) dominant wave period, (c) dominant wave direction (from), and (d) near-bed current velocity, with the alongshore component in black and the cross-isobath component in red.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Conditions during the measurement periods QS1 (days 183 to 223) and QS3 (days 316 to 354), including (a) significant wave height, (b) dominant wave period, (c) dominant wave direction (from), and (d) near-bed current velocity, with the alongshore component in black and the cross-isobath component in red.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
The variability in the cospectral estimates based on the measurements is sufficiently large that averages over the deployments, segregated by the ratio of the standard deviation of the wave velocity uw to the current velocity U1, are most useful. As uw/U1 increases, the measured stress-carrying cospectrum shows pronounced departures from the shape that occurs under weak wave forcing (Figs. 7, 8), in particular declining at frequencies below the wave band and increasing at frequencies within and above the wave band. During the first deployment (QS1), the simple model (section 2a) and to a lesser extent the model based on rapid distortion theory (appendix A) capture the observed distortion of the cospectrum (Fig. 7). During the second deployment (QS3), the performance of the simple model is slightly better than that of the rapid distortion model, but neither model captures the observed cospectral shape, even under relatively weak forcing. However, the models capture some of the qualitative features of the QS3 measurements, including enhancement in the wave band (although less than observed), and the model is quantitatively accurate at frequencies above the wave band. Quantification in terms of root-mean-square model data differences indicates that the simple model performs better than the rapid distortion model in almost all cases (Table 1). The best-fit values of the stress-carrying turbulent scale λ indicate consistency with the Kaimal et al. (1972) result λ ≈ 1.5x3 in the limit of negligible stratification, corresponding to LO ≫ x3, and limitation to λ ≈ LO in strong stratification, corresponding to LO ≪ x3 (Fig. 9).
Model data comparison for the summer deployment over coarse seafloor sediments (QS1), segregated by the ratio of the standard deviation of the wave velocity uw to the current velocity U1. The gray regions show the measurements with 95% confidence intervals for the cospectral estimates, the blue lines show computations based on the rapid distortion model (appendix A), the red lines show measurements based on the simple model (5), and the black lines show twice (1) with ω = k1U1. (a)–(d) Corresponds to quartiles of uw/U1. Table 1 shows the number of bursts in each quartile. Surface waves occur at radian frequencies ω of order unity, corresponding to dominant periods between 5 and 10 s (Fig. 6).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Model data comparison for the summer deployment over coarse seafloor sediments (QS1), segregated by the ratio of the standard deviation of the wave velocity uw to the current velocity U1. The gray regions show the measurements with 95% confidence intervals for the cospectral estimates, the blue lines show computations based on the rapid distortion model (appendix A), the red lines show measurements based on the simple model (5), and the black lines show twice (1) with ω = k1U1. (a)–(d) Corresponds to quartiles of uw/U1. Table 1 shows the number of bursts in each quartile. Surface waves occur at radian frequencies ω of order unity, corresponding to dominant periods between 5 and 10 s (Fig. 6).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Model data comparison for the summer deployment over coarse seafloor sediments (QS1), segregated by the ratio of the standard deviation of the wave velocity uw to the current velocity U1. The gray regions show the measurements with 95% confidence intervals for the cospectral estimates, the blue lines show computations based on the rapid distortion model (appendix A), the red lines show measurements based on the simple model (5), and the black lines show twice (1) with ω = k1U1. (a)–(d) Corresponds to quartiles of uw/U1. Table 1 shows the number of bursts in each quartile. Surface waves occur at radian frequencies ω of order unity, corresponding to dominant periods between 5 and 10 s (Fig. 6).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
As in Fig. 7, but for the fall and winter deployment over fine seafloor sediments (QS3).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
As in Fig. 7, but for the fall and winter deployment over fine seafloor sediments (QS3).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
As in Fig. 7, but for the fall and winter deployment over fine seafloor sediments (QS3).
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
Root-mean-square difference (10−6 m2 s−2) between observations and model computations of radian frequency ω times cospectral density 
(a) Near-bed stratification N2, where N is the buoyancy frequency and (b) binned-mean best-fit values of the turbulent length scale λ as a function of the height x3 above the bottom and the Ozmidov scale LO. In (b), the data are averaged in bins of LO/(1.5x3), and the error bars show plus and minus two standard errors, corresponding approximately to 95% confidence intervals; λ/(1.5x3) = 1 corresponds to the Kaimal et al. (1972) model [(1)] at neutral stratification; and the solid line shows equality for LO/(1.5x3) < 1 and a threshold for LO/(1.5x3) > 1.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
(a) Near-bed stratification N2, where N is the buoyancy frequency and (b) binned-mean best-fit values of the turbulent length scale λ as a function of the height x3 above the bottom and the Ozmidov scale LO. In (b), the data are averaged in bins of LO/(1.5x3), and the error bars show plus and minus two standard errors, corresponding approximately to 95% confidence intervals; λ/(1.5x3) = 1 corresponds to the Kaimal et al. (1972) model [(1)] at neutral stratification; and the solid line shows equality for LO/(1.5x3) < 1 and a threshold for LO/(1.5x3) > 1.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
(a) Near-bed stratification N2, where N is the buoyancy frequency and (b) binned-mean best-fit values of the turbulent length scale λ as a function of the height x3 above the bottom and the Ozmidov scale LO. In (b), the data are averaged in bins of LO/(1.5x3), and the error bars show plus and minus two standard errors, corresponding approximately to 95% confidence intervals; λ/(1.5x3) = 1 corresponds to the Kaimal et al. (1972) model [(1)] at neutral stratification; and the solid line shows equality for LO/(1.5x3) < 1 and a threshold for LO/(1.5x3) > 1.
Citation: Journal of Physical Oceanography 48, 1; 10.1175/JPO-D-17-0016.1
4. Discussion
The model computations are successful in reproducing the laboratory measurements (Fig. 4) and the main features of the field measurements during the first field deployment (Fig. 7), indicating that the assumptions underlying the models were satisfied in these measurements. The wavenumber spectrum of the turbulence is represented reasonably well by both the simple expression in section 2a and the more complex expressions in appendix A, and the distortion of the observed cospectrum in the presence of surface waves is largely the effect of advection of approximately frozen turbulence by the wave velocities.
The qualitatively correct but poorer quantitative performance of the models during the second field deployment (Fig. 8) is not understood. The model data comparison does not improve if the measurements are segregated based on 
A possible reason for the observed model data discrepancies is a nonzero correlation between the turbulence and waves, which would be inconsistent with a fundamental assumption in the Lumley and Terray (1983) analysis. While possible, a significant correlation between the turbulence and waves is unlikely in the present application. The reasons are that wave nonlinearity (measured by the ratio of the near-bottom wave orbital velocity to the phase speed) is weak, the turbulence generated by wave breaking is unlikely to penetrate to the near-bottom measurement depth, and the turbulence and waves, while similar in temporal scales, have vastly different spatial scales (the distance above the boundary versus the inverse wavenumber of the surface waves), so that correlation of the two processes is unlikely.
The most likely explanation for the poorer quantitative performance of the model against the measured cospectra during the second deployment is inaccuracy of the underlying model of the spatial structure of the turbulence (section 2a). For the second deployment, the model does not capture the quantitative shapes of the measured cospectra even during relatively weak waves (Fig. 8a). The model representations of the spatial structure of the turbulence are likely less accurate in the cross-stream direction than in the along-stream direction, as indicated by the laboratory measurements (Fig. 4). The poorer model performance in Fig. 8, compared with Fig. 7, possibly results from the nearly cross-flow separation of the two ADVs in the second deployment (Fig. 3), so that the differencing operation likely compounded any model inaccuracies in the cross-stream direction.
However, the favorable agreement between model computations and measurements at high frequencies in both Figs. 7 and 8 suggests that the theoretically universal k−7/3 behavior captured by (1) and also the simple (section 2a) and rapid distortion (appendix A) models of the turbulent structure are quantitatively sound.
In spite of the possible model deficiencies regarding the spatial structure of the turbulence, the best-fit estimates of the turbulent length scale λ (Fig. 9) are consistent with expectations based on classical conceptions of stratified turbulence, as found in previous nearshore and estuarine studies (Trowbridge and Elgar 2003; Scully et al. 2011). It is noteworthy that stratification has a measurable effect on turbulence even at heights above bottom as small as 0.5 m.
5. Summary and conclusions
The present study has shown that relatively simple models of boundary layer turbulence, combined with an analysis of the advection of frozen turbulence by random surface waves, reproduce the main features of stress-carrying cospectra measured near the seafloor in the coastal ocean during one field deployment and, with less quantitative success, the main features of the cospectrum during a second deployment. A possible reason for the greater model data differences during the second deployment is potentially inaccurate model representation of the cross-stream structure of the stress-carrying turbulent wavenumber cospectrum, accentuated by spatial differencing in that direction, accentuated by the wave advection in the same direction. Model computations were quantitatively consistent with measurements at high frequencies (above the wave band) during both deployments, suggesting that the underlying turbulence model at high wavenumbers, founded in established theoretical concepts, is sound. The effect of stable stratification on the stress-carrying turbulent length scale is quantitatively consistent with expectations during both deployments.
Acknowledgments
The authors thank the captains and crews of the R/V Connecticut and the R/V Tioga, and the field teams from Woods Hole Oceanographic Institution and the U.S. Geological Survey, especially Jay Sisson, Andy Davies, Marinna Martini, Jon Borden, Ellyn Montgomery, Pat Dickhudt, and Sandra Brosnahan. This research was supported by National Science Foundation Ocean Sciences Division Award 1356060 and the U.S. Geological Survey Coastal and Marine Geology Program. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. government. The field measurements have been archived (Montgomery et al. 2016; http://dx.doi.org/10.5066/F7542KQR) and are publicly available online (at https://stellwagen.er.usgs.gov/mvco_14.html).
APPENDIX A
A Model of Φ13 Based on Rapid Distortion Theory
APPENDIX B
Derivation of (9)
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