## 1. Introduction

Turbulent mixing driven by bottom drag is an essential component of shallow water flows. The dynamically significant effects of turbulence on Reynolds-averaged boundary layer flows are contained in the covariance terms that represent the vertical transport of heat and horizontal momentum (e.g., Monin and Yaglom 1971). As a result, the measurement of near-bottom, turbulence-induced shear stress and heat flux is a critical objective of coastal physical oceanography. High quality vertical flux estimates are necessary to formulate and test hypotheses regarding the dynamics of boundary layer turbulence. For example, models of currents and sediment transport on continental shelves rely heavily on how turbulent momentum flux is parameterized in the bottom boundary layer (e.g., Grant and Madsen 1986).

A number of methods are available for estimating turbulent fluxes in oceanic boundary layers. A direct vertical flux estimate is obtained by measuring the covariance between turbulent fluctuations of the transported quantity of interest (horizontal momentum or temperature) and fluctuations of vertical velocity (e.g., Bowden and Fairbairn 1956; Heathershaw 1979). Indirect methods include the profile technique (e.g., Charnock 1959; Sternberg 1968; Gross and Nowell 1983; Lueck and Lu 1997), in which fluxes are determined from the fit of semiempirical models to the vertical profiles of Reynolds-averaged quantities; the inertial-dissipation technique (e.g., Gross and Nowell 1985; Green and McCave 1995), in which fluxes are derived from Kolmogorov's theory of the inertial subrange and simplified variance budgets; and the dynamic technique (e.g., Bowden and Fairbairn 1952; Bowden et al. 1959), in which fluxes are estimated as the residual after other terms in simplified momentum or heat balances are measured. Both the profile technique and the inertial-dissipation technique rely on restrictive assumptions that often are hard to justify in the coastal ocean because of the presence of multiple length scales. In practice, the dynamic technique measures the influence of turbulence on a larger scale than the other techniques and it precludes the testing of dynamical balances.

We believe that direct measurements of turbulent fluxes will advance our understanding of the complex dynamics of the coastal bottom boundary layer. In the coastal ocean, however, direct turbulent flux measurements are often contaminated by the presence of energetic surface (Grant et al. 1984; Grant and Madsen 1986; Huntley and Hazen 1988; Trowbridge 1998) and internal waves that are often several orders of magnitude more energetic than turbulent eddies. Slight uncertainty in estimating the principle axes of the wave-induced velocity field can dominate the turbulence-induced component of the measured covariance (see section 2).

Recently, Trowbridge (1998) introduced a novel technique for removing surface wave contamination from shear stress estimates by taking the difference of records of velocity components measured by pairs of spatially separated current meters. The technique is based on the assumption that the spatial scale of surface waves is large relative to the correlation scale of the near-bottom turbulence. Essentially, Trowbridge (1998) assumed that the wave-induced velocities at the two sensors are equal and can be canceled by subtraction.

In this paper, alternative differencing strategies are considered and a new adaptive filtering technique is introduced to reduce further wave contamination of measured covariances. The adaptive filtering technique requires wave motions at spatially separated sensors to be coherent, but not necessarily equal, allowing useful flux estimates to be made in high-energy wave conditions and permitting a wider latitude in sensor placement. Important, the technique can be applied to a vertical array of sensors, which is a practical configuration for field experiments, and the technique results in flux estimates at the locations of each of the sensors rather than a spatial average between the two sensors.

Here, methods for removing surface and internal wave-induced contributions to estimates of vertical turbulent fluxes of heat and horizontal momentum in near-bottom flows are presented. A theoretical description of, and techniques to remove, wave bias are discussed in section 2. Measurements to test the techniques are described in section 3, followed by presentation and discussion of results (sections 4 and 5), and a summary and conclusions (section 6).

## 2. Theoretical framework

### a. Overview

A 2D model of the velocity field of the bottom boundary layer on the continental shelf, in the presence of wave motions, is developed to demonstrate wave bias and to provide a framework for the presentation of techniques for removing it. Although a simplification, a 2D analysis—in which the direction of the turbulent shear stress, the wave motions, and the components of the instrument coordinate system lie in a vertical plane—captures the important features of the wave bias problem. The 2D analysis presented here is a simplification of Trowbridge (1998), although it has been expanded to include the effects of waves on heat flux estimates and it relaxes assumptions about near-equality of wave-induced motions at the spatial scales of interest.

The analysis relies on several assumptions. The most fundamental is that the ratio of the spatial coherence scale of the wave-induced fluctuations to the spatial coherence scale of the turbulence-induced fluctuations is large, so a spatial separation exists at which the wave-induced fluctuations are coherent and the turbulence-induced fluctuations are incoherent. We also require that the wave- and turbulence-induced fluctuations are incoherent with one another (a justifiable assumption, because waves and turbulence are essentially independent processes outside of a thin wave boundary layer), and that the statistical properties of the waves and turbulence are stationary. For the purpose of illustration, we assume that the velocity sensors have perfect response, all components of the turbulent covariance tensor have equal magnitudes [which is a good assumption for boundary layer flows (e.g., Tennekes and Lumley 1972)], the waves are of small amplitude and narrow-banded in frequency, and stratification is due to a uniform vertical temperature gradient.

The horizontal component *x* of the model position vector **x** = (*x,* *z*) lies in the common direction of the wave propagation and the shear stress, and the vertical component *z* is defined to be positive upward with *z* = 0 at the bed (Fig. 1). The model velocity vector **u** = (*u,* *w*) where *u* and *w* are the horizontal and vertical components of the velocity, respectively, is composed of contributions from, respectively, mean current, waves, and turbulence: **u** = **u****ũ** + **u**′. The temperature similarly is denoted by *T* = *T**T̃* + *T*′. The wave-induced component of the fluctuations can include contributions from both surface and internal waves, although we do not consider interactions between them. The quantities of interest are the turbulent shear stress −*ρ**u*′*w*′*ρc*_{p}*T*′*w*′*ρ* is density and *c*_{p} is heat capacity.

In the remainder of this section, we first illustrate the problem of wave bias that results from a simple application of the direct eddy correlation technique to measurements obtained with an instrument that is tilted with respect to the principle axes of energetic surface wave motions (section 2b). We demonstrate that taking the difference of records from spatially separated sensors can be used to reduce the wave bias (section 2c). Then, in contrast to the previous estimates that assume the wave motions are essentially equal at the two positions, we introduce a linear filtering technique that further reduces wave bias by accounting for small differences in wave amplitude or phase between the two sensors (section 2d). Finally, we present an empirical constraint on the magnitude of the vertical separations that satisfies the critical assumption that the turbulent fluctuations at the two locations be uncorrelated (section 2e).

### b. The wave bias problem

*u*′

*w*′

**U**= (

*U,*

*W*), measured in instrument coordinates that are rotated a small angle

*θ*from the model coordinate system (Fig. 1) to first order in

*θ,*is

*T*′

*w*′

*θ*is

*θ*is small. We can estimate the real wave bias in (2) as follows. For the case of a uniform vertical temperature gradient and small-amplitude motions, the wave-induced temperature fluctuations arise from advection of the vertical temperature gradient and, for progressive waves, may be estimated as

*c*is the wave phase speed.

*T*′

*w*′

*κzu*∗

*d*

*T*

*dz*(e.g., Businger et al. 1971), where

*u*∗ is the bottom friction velocity and

*κ*(=0.4) is von Kármán's constant. With these estimates, the ratio of wave-induced bias to turbulent heat flux is

*ũ*

^{2}

^{2}s

^{−2},

*c*= 20 m s

^{−1},

*u*∗ = 0.01 m s

^{−1}, and

*θ*= 0.01), the wave-induced bias is 3 orders of magnitude smaller than the turbulent heat flux.

If the measured velocity vector can be rotated into the principal axes of the wave-induced velocity field, the wave biases are removed. In practice, the orientation of the principal coordinates of the wave-induced velocity field usually is not known with sufficient accuracy to remove wave bias by rotation. The objective of the present analysis is thus to consider alternative techniques for removing the real and apparent wave biases from estimates of the turbulent covariances *u*′*w*′*T*′*w*′

### c. Differencing strategies

Trowbridge (1998) showed that the wave bias in estimates of momentum flux can be reduced to an acceptable level by taking the difference of (“differencing”) measurements obtained from two sensors separated by a distance large in comparison with the correlation scale of the turbulence but small in comparison with the inverse wavenumber of the waves, assuming that the correlation scale of the waves is much greater than the correlation scale of the turbulence and that the waves and turbulence are uncorrelated. Here, a slightly different approach in which only one of the measurements composing the covariance is differenced, is used. This approach is beneficial because the resulting covariance estimate corresponds to the position of a single sensor, rather than an average of the covariances at each sensor, and two estimates are available for each sensor. The resulting turbulence estimates are not independent, but the waves are removed with essentially independent information, so the redundant estimates provide a check on the technique.

In the following, a parenthesized subscript is used to identify the position at which a particular measurement is obtained. For example, *u*_{(1)} is the horizontal velocity measured at the location of the first sensor. The operations of differencing and averaging between two sensors are represented by an upper-case delta sign and angular brackets, respectively. For example, Δ*u* and 〈*u*〉 are the difference and average, respectively, of the horizontal velocities measured by a pair of sensors. We focus here on the differencing of velocity measurements.

We consider three estimates of *u*′*w*′*U,* Δ*W*), cov[Δ*U,* *W*_{(1)}], and cov[*U*_{(1)}, Δ*W*]. The assumption that wave- and turbulence-induced velocities are uncorrelated (justified in section 2a), immediately allows us to decompose contributions to the *u*′*w*′

*u*′

*w*′

*θ*

_{1}and

*θ*

_{2}; see Fig. 1) as

*u*′

*w*′

*θ*

_{1}and

*θ*

_{2}) as

*u*≃

*r*∂

*ũ*/∂

*z,*where

*r*is the separation between the sensors and ∂

*ũ*/∂

*z*is evaluated at a location between the two sensors for the estimate (1/2)cov(Δ

*Ũ,*Δ

*W̃*) and at the location of the first sensor for the estimates cov[Δ

*U,*

*W*

_{(1)}] and cov[

*U*

_{(1)}, Δ

*W*]. Also, we assume that

*kz*≪ 1, where

*k*is the wavenumber of the waves, allowing the hyperbolic functions describing the vertical variation of wave-induced velocity to be represented with first-order expansions. With these assumptions, (8)–(10) can be approximated as

*ũw̃*

*θ*

*ũ*

^{2}

*θ*

*ũ*

^{2}

*θ*

*w̃*

^{2}

*θ*

*w̃*

^{2}

*k*

^{2}

*rz,*while two of the large terms in (13) have only been reduced by the factor

*r*/

*z,*which is constrained by the assumption that the turbulence must be uncorrelated at the two positions to be of order 1 (see section 2e). The stress estimate (1/2)cov(Δ

*Ũ,*Δ

*W̃*) has a greater reduction in wave bias at the cost of averaging the turbulent covariances between the two locations. For near-bottom measurements, the reduction in apparent wave bias resulting from the rotation of horizontal velocity into the vertical is better for (1/2)cov(Δ

*Ũ,*Δ

*W̃*) by an additional factor of

*k*

^{2}

*rz.*

### d. Further reduction in wave bias with adaptive filtering

Failure of the differencing technique described in section 2c occurs under conditions of high wave energy if differences in amplitude or phase of the wave-induced motions between the two locations cause the difference terms Δ*ũ* or Δ*w̃* appearing in (8)–(10) to be significant. A solution is to minimize the difference terms with linear filtration techniques. Assuming that the wave-induced fluctuations are completely spatially coherent, least squares filtering can be used to estimate the coherent component of velocity at one position with velocity measurements at the second position. Thus, the assumption behind the simple differencing technique used earlier—that the wave-induced motions at the two sensors are essentially identical—is replaced by the assumption that the wave-induced motions at the two sensors are essentially perfectly coherent. This relaxation allows potential differences in wave phase and amplitude that contribute to the difference terms in (8)–(10) to be minimized. In other words, because the coherence scale of wave motions is expected to be greater than the correlation scale, the filtering technique extends the range of application of the differencing techniques. The assumption that the turbulence is spatially uncorrelated is replaced by the assumption that the turbulence is spatially incoherent.

*t*is time and

*h*(

*t*) is a filter that represents the relationship between the wave-induced fluctuations at the two locations. In effect, (14) states that if

*Ũ*

_{(1)}and

*Ũ*

_{(2)}are perfectly coherent, then

*Ũ*

_{(1)}is completely predictable from

*Ũ*

_{(2)}. The goal of this analysis is to use the total measured velocities to estimate

*h*(

*t*). A more nearly wave-free estimate of horizontal velocity at position (1) than Δ

*U*=

*U*

_{(1)}−

*U*

_{(2)}is then Δ

*Û*=

*U*

_{(1)}−

*Û*

_{(1)}, where

*ĥ*(

*t*) and the measured velocities at position (2). The estimated wave velocity

*Û*

_{(1)}contains a turbulence component, but it is of no consequence if the assumption that the turbulence is spatially incoherent is valid. If the estimates cov[Δ

*U,*

*W*

_{(1)}] and cov[

*U*

_{(1)}, Δ

*W*] are replaced by cov[Δ

*Û,*

*W*

_{(1)}] and cov[

*U*

_{(1)}, Δ

*Ŵ*], respectively, the problematic difference terms appearing in (9) and (10) are minimized to the extent that the model assumptions are valid.

**A**

*M*×

*N*windowed data matrix of velocity at position (2), where

*M*is the number of data points and

*N*is the number of filter weights (

*N*must be odd for the filter to be symmetric),

**h**is a vector of filter weights, and

**U**

_{(1)}is a vector of position (1) velocity. The

*m*th row of

**A**

*u*(

*m*− (

*N*− 1)/2), … ,

*u*(

*m*), … ,

*u*(

*m*+ (

*N*− 1)/2)], where

*u*(

*m*) is the

*m*th discrete sample of

*u.*The solution is

**ĥ**

**A**

^{T}

**A**

^{−1}

**A**

^{T}

**U**

_{(1)}

**Û**

_{(1)}of the wave-induced velocity at position (1) are found by convolving the measured velocity record with the estimated filter weights

**Û**

_{(1)}

**A**

**ĥ**

*M*and the number of filter weights

*N.*

### e. Turbulence correlation lengths

For these wave bias removal techniques to work in practice, the sensor separation must be large relative to the correlation length scale of the turbulence so that the turbulence cross-correlation terms *u*^{′}_{(1)}*u*^{′}_{(2)}*u*^{′}_{(1)}*w*^{′}_{(2)}*w*^{′}_{(1)}*u*^{′}_{(2)}*u*^{′}_{(1)}*w*^{′}_{(2)}*θ,* and therefore, assuming that the elements of the turbulent correlation function tensor have the same order of magnitude, we only need to ensure that *u*^{′}_{(1)}*w*^{′}_{(2)}*u*^{′}_{(1)}*w*^{′}_{(1)}

We have established an empirical guideline ensuring that *u*^{′}_{(1)}*w*^{′}_{(2)}*u*^{′}_{(1)}*w*^{′}_{(2)}*R*_{uw} = *u*(*z*)*w*(*z* + *r*)*r*/*z* must be greater than approximately 5 for the turbulence to be considered uncorrelated, that is, less than one-tenth of the value at zero separation. In this particular environment, the cross-correlation function decreases more rapidly as *z* increases, possibly because the scale of the eddies is limited by factors other than height above bottom (Trowbridge et al. 1999).

## 3. Measurements and analysis

The techniques described in section 2 are tested on a set of measurements collected as part of the Coastal Mixing and Optics (CMO) experiment during 1996–97. One component of the field experiment consisted of the deployment of a bottom tripod equipped with current meters to study the turbulence dynamics of the coastal bottom boundary layer. The techniques are applied to vertically separated pairs roughly satisfying the empirical constraint *r*/*z* > 5 discussed in section 2c. Three sets of turbulent shear stress and vertical heat flux estimates were made: raw covariance, covariance with differencing, and covariance with differencing and adaptive filtering.

For the purposes of this paper, we limit our attention to a 6-week deployment of the bottom tripod, which began 17 August 1996. The tripod was deployed at a depth of 70 m on the New England shelf (Fig. 3). The “SuperBASS” tripod deployed during the CMO experiment (Fredericks et al. 1999) was outfitted with a vertical array of seven BASS current meters (Williams et al. 1987), a horizontal array of acoustic Doppler velocimeters, and a pair of temperature and conductivity sensors (Fig. 4). The BASS sensor measures the three-dimensional velocity vector by determining the differential travel time of acoustic pulses traveling in opposite directions along four 15-cm acoustic axes. The BASS electronics were modified recently to measure the absolute travel time of acoustic pulses (Trivett 1991; Shaw et al. 1996), allowing the measurement of sound speed in addition to fluid velocity. The vertical BASS array consisted of seven sensors at heights of 0.4, 0.7, 1.1, 2.2, 3.3, 5.4, and 7.0 m above the bed. The sensors were sampled at 1.2 Hz in 27-min bursts for a total of 2060 samples per burst. The bursts were taken in 2-h cycles consisting of three 0.5-h periods, during which the instruments were sampled, followed by one 0.5-h period, during which the instruments were idle.

Velocity and sound-speed measurements from the BASS array were used to produce estimates of turbulent shear stress and vertical heat flux at the lower six measurement levels. Sound-speed fluctuations were converted to temperature fluctuations by assuming a linear relationship, *∂T*/*∂c*_{s} = 0.5°C m^{−1} s^{−1}, where *c*_{s} is the speed of sound, estimated from the equation of state (MacKenzie 1981) at a reference salinity of 32 psu, which is the average salinity during the deployment. All flux estimates were derived from individual 27-min bursts of 1.2-Hz data. Approximately satisfying the empirical constraint *r* > 5*z,* the sensors were paired for the purposes of differencing as follows: 0.38 and 2.20 m, 0.74 and 3.30 m, and 1.10 and 5.40 m.

To test the techniques described in section 2, we computed three sets of *u*′*w*′*U,* *W*); two “differenced” estimates, cov(Δ*U,* *W*) and cov(*U,* Δ*W*); and two “filtered” estimates, cov(Δ*Û,* *W*) and cov(*U,* Δ*Ŵ*). An analogous set of *T*′*w*′*u*(*t*) and *w*(*t*), denoted Co_{uw}, and the cospectrum of *T*(*t*) and *w*(*t*), denoted Co_{Tw}. All spectral and cospectral calculations were carried out in a coordinate system aligned with the burst-averaged flow and resulting shear stress estimates were then rotated into a coordinate system oriented in the along- and cross-shelf directions.

*M*× 3

*N*data matrix

**A**

*m*th row of the expanded data matrix

**A**

*u*(

*m*),

*υ*(

*m*), and

*w*(

*m*) are the

*m*th discrete sample components of the velocity vector

**u**.

Including all three velocity components in **A***T* replacing *U.* The filter weights were calculated with (17) and applied to the measured data according to (18). The length of the filter was *N* = 11, corresponding to a window length of 9.2 s, which is about one-half of the typical wave period at the CMO site, and resulted in a total of 33 filter weights for each estimated component of wave-induced velocity fluctuations, and 11 filter weights for estimated wave-induced temperature fluctuations. New filters were calculated for each burst, so the filtering technique was block adaptive to slowly changing wave conditions. As a result, there were 3*M* − 3*N* = 6447 degrees of freedom in the velocity regression and *M* − *N* = 2049 degrees of freedom in the temperature regression.

## 4. Results

### a. Conditions at the CMO site

Before proceeding to a test of the proposed techniques for removing wave bias, it is worthwhile to briefly describe the conditions at the CMO field site during the first deployment. Here, we describe the temporal variability of burst statistics and we present spectra representative of conditions of energetic surface waves and conditions of energetic internal motions.

Time series of near-bottom mean currents at the CMO site (Fig. 5a) are usually dominated by a rotary semidiurnal tide with an amplitude of approximately 0.1 m s^{−1}. Subtidal events consist of predominantly westward flows with velocities up to 0.2 m s^{−1}. Time series of mean sound speed measured by the top and bottom BASS sensors (Fig. 5b) indicate that the bottom boundary layer at the CMO site usually is well mixed in sound speed. Occasionally, however, near-bottom stratification is strong, resulting in sound-speed differences between 0.38 and 5.50 m above the bed of up to 10 m s^{−1}, which roughly corresponds to 5°C. Time series of the standard deviation of horizontal velocity *σ*^{2}_{u} + *σ*^{2}_{υ}*σ*^{2}_{u} + *σ*^{2}_{υ}

Representative spectra of longitudinal and vertical velocity fluctuations, *S*_{uu} and *S*_{ww}, during conditions of strong surface waves (Fig. 6a) contain an energetic surface-wave peak centered at 0.07 Hz, corresponding to a wavelength greater than 300 m, and a phase speed of 20 m s^{−1}. The spectral levels of the *S*_{uu} wave peak are more than 2 orders of magnitude larger than the underlying stress-carrying eddies. The corresponding spectrum of temperature fluctuations, *S*_{TT} (Fig. 6b), contains a small peak at the dominant surface-wave frequency, consistent with the estimate (3), that is, the idea that *T̃*^{2}*T*′^{2}*ũ*^{2}*u*′^{2}

Representative spectra of longitudinal and vertical velocity fluctuations during conditions of strong internal waves (Fig. 7a) contain a low-frequency “hump” in *S*_{uu} below 0.002 Hz. Between 0.002 and 0.01Hz, the spectral levels of the horizontal velocity fluctuations fall off rapidly, giving the spectrum an overall concave upward shape. *S*_{TT} also contains energetic low-frequency variability (Fig. 7b), but, in contrast to *S*_{uu}, the temperature spectral levels do not fall off as rapidly between 0.002 and 0.01 Hz.

### b. Shear stress

It is instructive to consider the cospectra that are integrated to yield the three sets of shear stress estimates described in section 3, because wave-induced contributions can be recognized as deviations from the form expected in typical boundary layer flows. For reference, we present empirical forms for Co_{uw} and the running integral of Co_{uw}, ^{f}_{0}_{uw}(*f*′) *df*′ (known as an ogive curve) that were determined from measurements in the wall region of the atmospheric boundary layer (Kaimal et al. 1972) [and were shown to be consistent with cospectra obtained in the wall region of shallow-water, tidal-bottom boundary layers (Soulsby 1977)] in Fig. 8a. The energetic eddies responsible for transmitting stress lie approximately in the range of apparent nondimensional wavenumber 0.1 < 2*πfz*/*V* < 10 with a peak near 2*πfz*/*V* = 1, in which *V* is the burst-averaged, along-stream velocity and 2*πf*/*V* is the apparent wavenumber of the turbulent eddies advected past a point sensor by *V.* When energetic waves are present, this form of Taylor's hypothesis is inadequate because advection of eddies by the wave motions is important (Lumley and Terray 1983) and the actual wavenumber distribution is different than that given by 2*πf*/*V.* We present the results in the form of ogive curves, which are essentially low-pass-filtered cospectra, because cospectral estimates are inherently noisy.

To illustrate the filtering process, we present an example of the estimated filter weights between elevations of 5.4 and 1.1 m above the bed [positions (2) and (1), respectively] during energetic surface-wave conditions (Fig. 9) that demonstrates that the filters are physically meaningful. For clarity, we use a notation *h*_{ij}, where *i* denotes the component of velocity that is the output of the filter and *j* denotes the component of velocity that is the input to the filter. The estimation of *Ũ*_{(1)} is dominated by *h*_{uu} (Fig. 9a), which has a maximum at zero lag, as expected for vertically separated instruments, and decreases with increasing lag, with a zero crossing near the fourth lag, or about 3.3 s, which is roughly consistent with a dominant surface-wave period of 15 s. The weights of *h*_{uυ} and *h*_{uw} are near zero, indicating that *V*_{(2)} and *W*_{(2)} are not important inputs in the estimation of *Ũ*_{(1)}, although it is reassuring that they take near-zero values. In contrast, the estimation of *W̃*_{(1)} is not dominated completely by *h*_{ww} (Fig. 9b), and the weights of *h*_{uυ} and *h*_{uw} are significant, with skew-symmetric peaks near the third lag, indicating that *U*_{(2)} and *V*_{(2)} are coherent with *W̃*_{(1)} but are approximately in quadrature, as expected.

As an example of the effect of moderately energetic surface waves on Co_{uw}, we present a spectrum of horizontal velocity and cospectral ogive curves at 0.70 m above the bed, position (1), for a burst in which the surface-wave rms velocity equals 0.07 m s^{−1} (Fig. 10). The surface-wave spectral peak (Fig. 10a) is nearly collocated with the expected maximum in turbulence cospectral density (see Fig. 8a). The raw covariance estimate has a large contribution (visible as the region of steep slope in the ogive curve) at the frequency of the surface waves (Fig. 10b). This is anomalous for boundary layer turbulence (compare with Fig. 8a) and is attributable to wave contamination. Of the two differenced estimates, cov[Δ*U,* *W*_{(1)}] contains little or no surface-wave contamination, while cov[*U*_{(1)}, Δ*W*] contains a surface-wave contribution of magnitude roughly equal to the raw estimate (Fig. 10c), consistent with expectations of section 2c. Neither of the filtered estimates, cov[Δ*Û,* *W*_{(1)}] or cov[*U*_{(1)}, Δ*Ŵ*], contains a net contribution to the shear stress from surface waves (Fig. 10d). There is some contamination of Co_{UΔŴ}. However, the filtering has compensated in a manner that yields no net contribution to cov[Δ*Û,* *W*_{(1)}].

As a further test of the techniques in removing surface wave contamination, we present an example of the effect on Co_{uw} of highly energetic surface waves (rms velocity equal to 0.12 m s^{−1} at 0.70 m above the bottom; (Fig. 11). As in the example for moderate surface-wave energy, the estimates cov[*U*_{(1)}, *W*_{(1)}] and cov[*U*_{(1)}, Δ*W*] fail to remove wave contamination (Figs. 11b,c), whereas the estimates cov[Δ*Û,* *W*_{(1)}] and cov[*U*_{(1)}, Δ*Ŵ*] succeed at removing surface-wave contamination (Fig. 11d). In contrast, the differenced estimate cov[Δ*U,* *W*_{(1)}] also fails to remove surface-wave contamination (Fig. 11c), indicating that the filtering technique is necessary to remove wave contamination when highly energetic surface waves are present.

An example burst time series of longitudinal and vertical velocity 1.10 m above the bottom on day 253, during a period of measurable stratification (see Fig. 5b) and most likely associated with a horizontal intrusion, captures the passage of an energetic internal motion with near-bottom velocities greater than 0.1 m s^{−1} and a duration of approximately 5 min (Fig. 12a). The sound-speed record for this burst also records the passage of the “event” (see Fig. 15, described in next section) and indicates that the event had a borelike character. The internal motion made a large low-frequency contribution to the raw cospectrum below 2*πfz*/*V* = 0.02, and although the filtered estimate is better than the raw estimate, in contrast to the case of surface waves, the proposed techniques are not successful at removing the observed wave bias (Fig. 12b). The failure is partially owing to the long duration of the internal motion as compared with the period of surface waves and the length of the burst and to the intermittent nature of the internal motion. There is also a small surface-wave bias visible in the ogive curve of Co_{ΔÛW} (similar to those identified in Figs. 10 and 11), suggesting that the use of constant filter weights during a burst with nonstationary internal wave properties degrades the ability of the filtering technique to remove surface-wave bias. In practice, the range of internal motion contamination is nearly out of the expected nondimensional wavenumber range of the energy-containing eddies (Fig. 8a) and the unwanted internal motion bias can be removed by windowing Co_{uw}.

A comparison of *u*′*w*′*Û,* *W*) as a standard demonstrates that the raw shear stress estimates cov(*U,* *W*) are contaminated with surface-wave-induced covariance during the energetic surface-wave events (Fig. 13a). As anticipated in section 2c, the differenced shear stress estimate cov(Δ*U,* *W*) is mostly free of surface-wave contamination during the high-energy wave events, while cov(*U,* Δ*W*) is arguably worse than cov(*U,* *W*) (Figs. 13b,c). The two filtered estimates agree well (Fig. 13d). These results indicate that the three estimates cov(Δ*U,* *W*), cov(Δ*Û,* *W*), and cov(*U,* Δ*Ŵ*) are all successful at removing surface-wave bias when it exists in this dataset and that, as shown by comparison with the raw estimate, the successful techniques do not degrade the shear stress estimates when energetic surface waves are absent.

### c. Heat flux

As in the case of shear stress, we consider the cospectra that are integrated to yield the three sets of *T*′*w*′_{Tw} in typical boundary layer flows. Empirical forms for Co_{Tw} and the running integral of Co_{Tw}, ^{f}_{0}_{uw}(*f*′) *df*′ (Fig. 8b) from Kaimal et al. (1972) are similar to those for shear stress, except that the range of heat-flux-carrying eddies is shifted to higher 2*πfz*/*V* in comparison with the range of stress-carrying eddies (Fig. 8a).

For the most part, the bottom boundary layer is well mixed during periods of strong surface wave activity. However, near-bottom stratification existed during Hurricane Hortense (Fig. 5b), which resulted in temperature fluctuations produced by surface waves. The stratification was inferred to result from mixed layer entrainment and was accompanied by physically meaningful buoyancy fluxes [see Shaw et al. (2001) for further discussion]. As an example of the effect of these wave-induced temperature fluctuations on Co_{Tw}, we present a temperature spectrum and cospectral ogive curves at 3.3 m above the bed, position (1) in this case, for a burst in which the rms wave-induced temperature was 0.01°C (Fig. 14). In this particular example, the surface-wave peak in the temperature spectrum (Fig. 14a) is located at the upper end of the expected range of the heat-flux-carrying eddies (Fig. 8b). The raw estimate contains an anomalous contribution at the dominant surface wave frequency (Fig. 14b). The anomaly is visible in both of the differenced estimates (Fig. 14c) and the filtered estimate cov(Δ*T̂,* *W*) (Fig. 14d). Only in the filtered estimate cov(*T,* Δ*Ŵ*) is the observed bias removed (Fig. 14d). The heat-flux-carrying eddies in this example are located in a higher range of 2*πfz*/*V* than expected, which is probably caused by stable stratification limiting the size of eddies (compare with Fig. 8b). The pronounced effects of surface waves on estimates of heat flux contradict the theoretical scale analysis presented in section 2a.

Here, the burst containing an internal wave presented in Fig. 12 is considered in terms of the effect of the internal wave on estimates of heat flux. The time series of sound speed from 1.1 m above the bed is nearly constant until disturbed by the passage of the internal wave 16 min into the burst (Fig. 15a). The internal wave made large contributions to the raw estimate, and the contamination is not eliminated in the filtered estimate (Fig. 15b), as for the case of shear stress. In fact, both estimates result in a positive heat flux, which is countergradient for the stably stratified bottom boundary layer. Note that the sound-speed record (Fig. 15a) is inconsistent with the implicit assumption of stationarity of the statistical properties of waves within individual bursts. Unlike Co_{uw}, Co_{Tw} is contaminated well into the expected range of heat-flux-carrying eddies, above 2*πfz*/*V* = 1 (Fig. 8b), so that Co_{Tw} cannot simply be windowed to remove internal wave bias without a loss of turbulent flux.

## 5. Discussion

The results of section 4 confirm several of the theoretical predictions of section 2. The success of the shear stress estimate −*ρ* cov(Δ*U,* *W*) at removing surface wave bias, together with the failure of the estimate −*ρ* cov(*U,* Δ*W*), demonstrates the importance of taking advantage of the near-bottom properties of small-amplitude, irrotational surface waves, for which *ũ* is approximately independent of *z,* and *w̃* increases linearly with *z* for *kz* ≪ 1. The estimate −*ρ* cov(Δ*U,* *W*) is predicted to be more effective at reducing wave bias by a factor of *k*^{2}*z*^{2}, which is approximately 10^{−4} for near-bottom measurements at the CMO site with typical 15-s swell. The failure of −*ρ* cov(Δ*U,* *W*) during energetic surface wave conditions forced by hurricane Edouard (Fig. 11c) is likely owing to either surface waves with higher than usual nonlinearity or to local flow disturbances, perhaps associated with the tripod and instrumentation, that are enhanced by larger waves. As noted in section 4, the contamination of Co_{Tw} by surface waves (Fig. 14b) is larger than expected from the estimate (4), suggesting that the estimate for wave-induced temperature fluctuations used in section 2a is not appropriate. The failure of the estimate *ρc*_{p} cov(Δ*T,* *W*) probably is caused by a lack of temperature correlation owing to a nonuniform vertical temperature gradient. The bottom at the site is smooth and fairly flat, and the surface waves are weakly nonlinear, thus the surface-wave contamination likely resulted from errors in the leveling of the instruments. The case of internal waves is more complicated and will be discussed later.

In general, the results indicate that wave bias can be reduced by accounting for spatial differences in wave-induced velocity fluctuations with empirically derived filters. In particular, the success of the estimate −*ρ* cov(*U,* Δ*Ŵ*) (Fig. 13d) demonstrates that the filtering technique is capable of accounting for the vertical variation of *w* that causes the estimate −*ρ* cov(*U,* Δ*W*) to fail (Fig. 13c). The capability of the filtering technique is further demonstrated by the success of −*ρ* cov(Δ*Û,* *W*) and −*ρ* cov(*U,* Δ*Ŵ*) when near-bottom surface wave motions were largest (Fig. 11d) and the estimate −*ρ* cov(Δ*Û,* *W*) failed (Fig. 11c). For practical purposes, the filtering step is not critical for this dataset, although the results do demonstrate the potential of the filtering step in harsher wave conditions.

In practice, the adaptive filtering technique is insensitive to the exact values chosen for the number of points *M* in the least squares problem or the number of filter weights *N,* so long as *M* ≫ *N.* A useful rule of thumb is to use a filter length such that *N dt* equals a significant fraction of the wave period and to use a number of data points such that *M dt* is much greater than the timescale of the largest eddies. The filtering technique requires that the turbulence is spatially incoherent at separations *r,* a stronger statement than requiring it to be uncorrelated. However, if turbulence is uncorrelated, it is almost certainly incoherent because eddies quickly lose their identity in turbulent flow. Including the full velocity vector in the prediction of each component of wave-induced velocity yields better stress estimates during demanding wave conditions than if only one velocity component is used for prediction. A fully recursive filtering technique (e.g., Haykin 1996) in which the filter weights are updated continuously throughout each burst during periods of internal wave activity resulted in a loss of turbulent covariance because, on short enough timescales, large eddies can be spatially coherent.

None of the proposed techniques were completely successful at removing internal wave bias from estimates of shear stress or heat flux, although the internal wave bias for stress estimates can usually be easily removed by high-pass filtering. The differenced stress estimates most likely failed because the near-bottom variation of internal wave-induced velocity was not small, as in the case for the horizontal motions induced by surface waves. The near-bottom structure of internal-wave-induced motion is complicated by an internal-wave boundary layer height that is not small relative to the measuring heights, the slow phase speed of internal waves (which increases potential for nonlinear effects), rotational effects, and internal wave periods comparable to the length of measurement. Nonlinear effects are especially likely for internal-wave-induced temperature fluctuations because the vertical displacements caused by the passage of large internal waves are large enough to advect the density interfaces that tend to develop at the top of the mixed layer (see Fig. 15a) a considerable distance, making *w̃dT̃*/*dz* large and invalidating the estimate (3). This nonlinear effect is demonstrated in the extreme example of the time series in Fig. 15a and in the spectrum and cospectrum presented in Figs. 7b and 15b for which the effect of internal waves on temperature is apparently transferred to higher frequencies. The filtered estimates fail because nonlinearity invalidates the assumptions of the linear filtration techniques and because intermittency within violates the assumption of stationary wave properties. In light of the complicated nature of the observed internal waves it is not clear how much of the internal-wave contribution to measured covariances is real or apparent. This is a question we hope to pursue in future work.

From the viewpoint of experimental design, the proposed techniques are valuable because they can be used to obtain surface-wave-free flux estimates with a vertical array of sensors, which allows the vertical structure of bottom boundary layer turbulence to be studied. In particular, the technique of differencing a single component of velocity results in a flux estimate at the height of a single sensor as opposed to an estimate that is spatially averaged. Without single differencing, it would be difficult to obtain flux estimates at a wide range of heights while satisfying the condition *r* > 5*z.* The results presented here suggest the differenced estimate −*ρ* cov(Δ*U,* *W*) is adequate for the removal of surface waves from stress estimates under all but the most demanding conditions on the outer shelf. For more energetic conditions, the filtered estimate −*ρ* cov(Δ*Û,* *W*) can be used. To obtain surface-wave-free heat flux estimates, the estimate *ρc*_{p} cov(*T,* Δ*Ŵ*) is necessary because wave-induced temperature fluctuations are not coherent if *d**dz* is strongly inhomogeneous. In practice, it is best to remove internal wave energy by high-pass filtering before the filter weights are calculated, because the presence of nonstationary internal waves can degrade the estimation of surface-wave-induced motions.

## 6. Summary and conclusions

This paper has presented a theoretical analysis and an oceanic test of techniques to estimate near-bottom, turbulence-induced fluxes in the presence of energetic wave motions with current meter arrays. A new adaptive filtering technique has been introduced to minimize the contributions of wave motions to measured covariances. The technique requires two sensors separated in space and requires that the coherency scale of wave motions be much larger than the coherency scale of turbulent motions. The techniques were applied to a 6-week set of coastal bottom boundary layer observations that include three energetic surface-wave events and a number of energetic internal-wave events.

Results from the oceanic test indicate the following conclusions. The proposed technique succeeds at removing surface-wave contamination from shear stress and heat flux estimates using pairs of sensors separated in the vertical dimension by a distance of approximately 5 times the height of the lower sensor, even during the close passage of hurricanes, but fails at removing longer-period, internal-motion contamination from shear stress and heat flux estimates. The presence of internal motions does not pose a significant problem for estimating turbulent shear stress because contamination caused by internal waves is limited to frequencies lower than the stress-carrying eddies. In contrast, the presence of internal motions does pose a problem for estimating turbulent heat flux because the contamination extends into the range of the heat flux–carrying eddies. The internal case is complicated by the facts that the motions are highly intermittent, the internal wave period is comparable to the period of measurement, the height of the internal wave boundary layer is on the order of the height of measurement, and, specifically for heat flux estimates, nonlinear effects are large.

## Acknowledgments

This study was supported by the Office of Naval Research under Grants N000149510373 and N000149610953. We thank James Priesig for suggesting the use of the least squares filtering method. We also thank Janet Fredericks for her preliminary processing of the data. We are grateful for the comments provided by Ole Madsen, Steve Elgar, and three anonymous reveiwers.

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Estimates of the normalized cross-correlation function *R*_{uw}(*r*)/*R*_{uw}(0), where *R*_{uw}(*r*) = *u*(*z*)*w*(*z* + *r*)*r* estimated from measurements obtained in the Hudson River estuary at four heights above the bottom. Each curve represents an average derived from 720 3.28-min bursts of data

Citation: Journal of Atmospheric and Oceanic Technology 18, 9; 10.1175/1520-0426(2001)018<1540:TDEONB>2.0.CO;2

Estimates of the normalized cross-correlation function *R*_{uw}(*r*)/*R*_{uw}(0), where *R*_{uw}(*r*) = *u*(*z*)*w*(*z* + *r*)*r* estimated from measurements obtained in the Hudson River estuary at four heights above the bottom. Each curve represents an average derived from 720 3.28-min bursts of data

Citation: Journal of Atmospheric and Oceanic Technology 18, 9; 10.1175/1520-0426(2001)018<1540:TDEONB>2.0.CO;2

Estimates of the normalized cross-correlation function *R*_{uw}(*r*)/*R*_{uw}(0), where *R*_{uw}(*r*) = *u*(*z*)*w*(*z* + *r*)*r* estimated from measurements obtained in the Hudson River estuary at four heights above the bottom. Each curve represents an average derived from 720 3.28-min bursts of data

Citation: Journal of Atmospheric and Oceanic Technology 18, 9; 10.1175/1520-0426(2001)018<1540:TDEONB>2.0.CO;2

Map of the CMO experiment site. Water depth is contoured in meters. The site is approximately 100 km south of Martha's Vineyard, MA, on a portion of the New England shelf known as the “Mud Patch.” Along with the bottom tripod, an array of four moorings were deployed. The tripod was within 200 m of the central mooring (C), in 70 m of water. Here, I, O, and A represent the locations of onshore, offshore, and alongshore moorings, respectively. [Reproduced from Fredericks et al. (1999).]

Map of the CMO experiment site. Water depth is contoured in meters. The site is approximately 100 km south of Martha's Vineyard, MA, on a portion of the New England shelf known as the “Mud Patch.” Along with the bottom tripod, an array of four moorings were deployed. The tripod was within 200 m of the central mooring (C), in 70 m of water. Here, I, O, and A represent the locations of onshore, offshore, and alongshore moorings, respectively. [Reproduced from Fredericks et al. (1999).]

Map of the CMO experiment site. Water depth is contoured in meters. The site is approximately 100 km south of Martha's Vineyard, MA, on a portion of the New England shelf known as the “Mud Patch.” Along with the bottom tripod, an array of four moorings were deployed. The tripod was within 200 m of the central mooring (C), in 70 m of water. Here, I, O, and A represent the locations of onshore, offshore, and alongshore moorings, respectively. [Reproduced from Fredericks et al. (1999).]

Scale drawing of the “SuperBASS” tripod deployed as a part of the CMO experiment. The tripod was 8 m tall, containing a vertical array of BASS current meters, a horizontal array of acoustic Doppler velocimeters, and a pair of temperature and conductivity sensors. The heights of the BASS sensors above the bottom are marked. [Reproduced from Fredericks et al. (1999).]

Scale drawing of the “SuperBASS” tripod deployed as a part of the CMO experiment. The tripod was 8 m tall, containing a vertical array of BASS current meters, a horizontal array of acoustic Doppler velocimeters, and a pair of temperature and conductivity sensors. The heights of the BASS sensors above the bottom are marked. [Reproduced from Fredericks et al. (1999).]

Scale drawing of the “SuperBASS” tripod deployed as a part of the CMO experiment. The tripod was 8 m tall, containing a vertical array of BASS current meters, a horizontal array of acoustic Doppler velocimeters, and a pair of temperature and conductivity sensors. The heights of the BASS sensors above the bottom are marked. [Reproduced from Fredericks et al. (1999).]

Time series of (a) burst-averaged, along-shelf current, (b) burst-averaged sound speed, and (c) burst standard deviations of horizontal velocity *σ*^{2}_{u} + *σ*^{2}_{υ}

Time series of (a) burst-averaged, along-shelf current, (b) burst-averaged sound speed, and (c) burst standard deviations of horizontal velocity *σ*^{2}_{u} + *σ*^{2}_{υ}

Time series of (a) burst-averaged, along-shelf current, (b) burst-averaged sound speed, and (c) burst standard deviations of horizontal velocity *σ*^{2}_{u} + *σ*^{2}_{υ}

Energy density spectra of (a) longitudinal and vertical velocity fluctuations and (b) temperature fluctuations from BASS measurements at 0.70 m above the bed when energetic surface waves were present. The spectral densities are average values from 49 bursts when the rms horizontal surface-wave velocity in the frequency band 0.04 Hz ≤ *f* ≤ 0.2 Hz was greater than 0.04 m s^{−1}

Energy density spectra of (a) longitudinal and vertical velocity fluctuations and (b) temperature fluctuations from BASS measurements at 0.70 m above the bed when energetic surface waves were present. The spectral densities are average values from 49 bursts when the rms horizontal surface-wave velocity in the frequency band 0.04 Hz ≤ *f* ≤ 0.2 Hz was greater than 0.04 m s^{−1}

Energy density spectra of (a) longitudinal and vertical velocity fluctuations and (b) temperature fluctuations from BASS measurements at 0.70 m above the bed when energetic surface waves were present. The spectral densities are average values from 49 bursts when the rms horizontal surface-wave velocity in the frequency band 0.04 Hz ≤ *f* ≤ 0.2 Hz was greater than 0.04 m s^{−1}

Energy density spectra of (a) longitudinal and vertical velocity fluctuation and (b) temperature fluctuations from BASS measurements at 0.70 m above the bed when energetic internal waves were present. The spectral densities are average values from 49 bursts when the rms horizontal velocity minus the rms surface wave velocity was greater than 0.025 m s^{−1}

Energy density spectra of (a) longitudinal and vertical velocity fluctuation and (b) temperature fluctuations from BASS measurements at 0.70 m above the bed when energetic internal waves were present. The spectral densities are average values from 49 bursts when the rms horizontal velocity minus the rms surface wave velocity was greater than 0.025 m s^{−1}

Energy density spectra of (a) longitudinal and vertical velocity fluctuation and (b) temperature fluctuations from BASS measurements at 0.70 m above the bed when energetic internal waves were present. The spectral densities are average values from 49 bursts when the rms horizontal velocity minus the rms surface wave velocity was greater than 0.025 m s^{−1}

Empirical energy-preserving cospectrum and running integral of cospectrum of (a) *u* and *w* and of (b) *T* and *w* as determined from measurements during neutral conditions in the atmospheric boundary layer (Kaimal et al. 1972). The cospectra are normalized by *u*∗ = |*u*′*w*′^{1/2} and *T*∗ = −*T*′*w*′*u*∗

Empirical energy-preserving cospectrum and running integral of cospectrum of (a) *u* and *w* and of (b) *T* and *w* as determined from measurements during neutral conditions in the atmospheric boundary layer (Kaimal et al. 1972). The cospectra are normalized by *u*∗ = |*u*′*w*′^{1/2} and *T*∗ = −*T*′*w*′*u*∗

Empirical energy-preserving cospectrum and running integral of cospectrum of (a) *u* and *w* and of (b) *T* and *w* as determined from measurements during neutral conditions in the atmospheric boundary layer (Kaimal et al. 1972). The cospectra are normalized by *u*∗ = |*u*′*w*′^{1/2} and *T*∗ = −*T*′*w*′*u*∗

Example of the filters between 5.4 and 1.1 m above the bottom: (a) *h*_{uu}, *h*_{uυ}, and *h*_{uw} are the filter weights used to estimate *Ũ*_{(1)}, given measurements of *U*_{(2)}, *V*_{(2)}, and *W*_{(2)}; (b) *h*_{wu}, *h*_{wυ}, and *h*_{ww} are the filter weights used to estimate *W̃*_{(1)} with the same measurements as in (a). The filter weights plotted are average values from 49 bursts when the rms surface wave velocity was greater than 0.04 m s^{−1}

Example of the filters between 5.4 and 1.1 m above the bottom: (a) *h*_{uu}, *h*_{uυ}, and *h*_{uw} are the filter weights used to estimate *Ũ*_{(1)}, given measurements of *U*_{(2)}, *V*_{(2)}, and *W*_{(2)}; (b) *h*_{wu}, *h*_{wυ}, and *h*_{ww} are the filter weights used to estimate *W̃*_{(1)} with the same measurements as in (a). The filter weights plotted are average values from 49 bursts when the rms surface wave velocity was greater than 0.04 m s^{−1}

Example of the filters between 5.4 and 1.1 m above the bottom: (a) *h*_{uu}, *h*_{uυ}, and *h*_{uw} are the filter weights used to estimate *Ũ*_{(1)}, given measurements of *U*_{(2)}, *V*_{(2)}, and *W*_{(2)}; (b) *h*_{wu}, *h*_{wυ}, and *h*_{ww} are the filter weights used to estimate *W̃*_{(1)} with the same measurements as in (a). The filter weights plotted are average values from 49 bursts when the rms surface wave velocity was greater than 0.04 m s^{−1}

Estimates of *u*′*w*′^{−1}) from BASS measurements at 0.70 m above the bed taken on year day 246 as the energy of the surface waves forced by Hurricane Edouard diminished. (a) Energy density of longitudinal velocity fluctuations, smoothed with a 21-point boxcar filter, and ogive curves of (b) raw, (c) differenced, and (d) filtered stress estimates vs dimensionless wavenumber

Estimates of *u*′*w*′^{−1}) from BASS measurements at 0.70 m above the bed taken on year day 246 as the energy of the surface waves forced by Hurricane Edouard diminished. (a) Energy density of longitudinal velocity fluctuations, smoothed with a 21-point boxcar filter, and ogive curves of (b) raw, (c) differenced, and (d) filtered stress estimates vs dimensionless wavenumber

Estimates of *u*′*w*′^{−1}) from BASS measurements at 0.70 m above the bed taken on year day 246 as the energy of the surface waves forced by Hurricane Edouard diminished. (a) Energy density of longitudinal velocity fluctuations, smoothed with a 21-point boxcar filter, and ogive curves of (b) raw, (c) differenced, and (d) filtered stress estimates vs dimensionless wavenumber

Estimates of *u*′*w*′^{−1}) from BASS measurements at 0.70 m above the bed taken on yearday 246 as surface-wave energy forced by Hurricane Edouard peaked. (a) Energy density of longitudinal velocity fluctuations, smoothed with a 21-point boxcar filter, and ogive curves of (b) raw, (c) differenced, and (d) filtered stress estimates vs dimensionless wavenumber

Estimates of *u*′*w*′^{−1}) from BASS measurements at 0.70 m above the bed taken on yearday 246 as surface-wave energy forced by Hurricane Edouard peaked. (a) Energy density of longitudinal velocity fluctuations, smoothed with a 21-point boxcar filter, and ogive curves of (b) raw, (c) differenced, and (d) filtered stress estimates vs dimensionless wavenumber

Estimates of *u*′*w*′^{−1}) from BASS measurements at 0.70 m above the bed taken on yearday 246 as surface-wave energy forced by Hurricane Edouard peaked. (a) Energy density of longitudinal velocity fluctuations, smoothed with a 21-point boxcar filter, and ogive curves of (b) raw, (c) differenced, and (d) filtered stress estimates vs dimensionless wavenumber

Estimates of *u*′*w*′*U*_{(1)}, *W*_{(1)}] and cov[Δ*Û,* *W*_{(1)}]

Estimates of *u*′*w*′*U*_{(1)}, *W*_{(1)}] and cov[Δ*Û,* *W*_{(1)}]

Estimates of *u*′*w*′*U*_{(1)}, *W*_{(1)}] and cov[Δ*Û,* *W*_{(1)}]

Comparison of estimates of *u*′*w*′*Û,* *W*) is taken as a standard with which the other estimates are compared: (a) cov(*U,* *W*), (b) cov(Δ*U,* *W*), (c) cov(*U,* Δ*W*), and (d) cov(*U,* Δ*Ŵ*). Bursts with surface-wave rms > 0.04 m s^{−1} are plotted as open squares. The units on all axes are meters squared per second squared

Comparison of estimates of *u*′*w*′*Û,* *W*) is taken as a standard with which the other estimates are compared: (a) cov(*U,* *W*), (b) cov(Δ*U,* *W*), (c) cov(*U,* Δ*W*), and (d) cov(*U,* Δ*Ŵ*). Bursts with surface-wave rms > 0.04 m s^{−1} are plotted as open squares. The units on all axes are meters squared per second squared

Comparison of estimates of *u*′*w*′*Û,* *W*) is taken as a standard with which the other estimates are compared: (a) cov(*U,* *W*), (b) cov(Δ*U,* *W*), (c) cov(*U,* Δ*W*), and (d) cov(*U,* Δ*Ŵ*). Bursts with surface-wave rms > 0.04 m s^{−1} are plotted as open squares. The units on all axes are meters squared per second squared

Estimates of *T*′*w*′

Estimates of *T*′*w*′

Estimates of *T*′*w*′

Estimates under energetic internal wave conditions from a single burst of BASS measurements at 1.10 m above the bed taken on yearday 252 during a period of strong stratification. (a) Burst time series of sound speed, and (b) ogive curves for two heat flux estimates, for which the operations are performed on the vertical velocity component

Estimates under energetic internal wave conditions from a single burst of BASS measurements at 1.10 m above the bed taken on yearday 252 during a period of strong stratification. (a) Burst time series of sound speed, and (b) ogive curves for two heat flux estimates, for which the operations are performed on the vertical velocity component

Estimates under energetic internal wave conditions from a single burst of BASS measurements at 1.10 m above the bed taken on yearday 252 during a period of strong stratification. (a) Burst time series of sound speed, and (b) ogive curves for two heat flux estimates, for which the operations are performed on the vertical velocity component