a. Instrumentation

Our moored turbulence package (MTP) consists of an acoustic Doppler velocimeter (ADV; Vector, Nortek AS), a fast-response temperature sensor (FP07, GE Thermometics), and a motion pack (MP; Rockland Scientific). The MP consists of a motion sensor with 6 degrees of freedom (O-Navi, Gyrocube); a MicroMag magnetometer to derive the instrument’s heading once its pitch and roll are reconstructed from the Gyrocube data; and a fast-response pressure sensor. The Gyrocube motion sensors provide orthogonal linear accelerations and rotation rates of the MP in 3D (Fig. 1), enabling the determination of the moored instruments’ translation velocity and the tangential velocity at the instruments’ sampling volume, resulting from the angular rotation of the frame. The MP logs the FP07’s temperature data via the Rockland Scientific Ltd. proprietary MicroSquid eddy correlation measurement system.

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(a) Plan view and (b) elevation view of the moored turbulence package drawn to scale. The battery pack (BP) and fast-response thermistor (FP07) are also illustrated. The frame of reference is located at the Gyrocube within the motion pack.

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1

We mounted the MP near the top of an in-line galvanized steel frame alongside the ADV, while the MicroSquid was located at the lower end of the frame with the FP07 tip located about 5 cm away from the ADV’s sampling volume (Fig. 1). The ADV is equipped with its own compass and a liquid level tilt sensor that provides heading, and redundant pitch and roll at 1 Hz, while velocities can be collected up to 64 Hz. The MP samples the ADV’s analog output at its specified sampling rate of 64 Hz, irrespective of the programmed sampling rate of the ADV. The measured ADV velocities are thus synchronized in time with the other data streams measured by the MP. The ADV’s data quality parameters and low-frequency (1 Hz) motion information, however, remain stored on its memory card. Within the frame, the ADV and the MP share the same vertical axis but not necessarily the same x and y axes (Fig. 1); however, the ADV’s horizontal plane can be rotated onto the MP’s frame of reference using the difference in their respective headings.

b. Data analysis procedures

1) Deriving the environmental velocities

In vector form, the environmental velocities are given by

where the measured velocity is and the instrument’s motion in space is . The motion contains velocity contributions from both translation and rotation around the MP’s centroid ,

The rotationally induced tangential velocities are calculated as

where represents the position of the ADV’s transmitters relative to the MP’s Gyrocube (Fig. 1). The ADV has one emitting transducer and three receivers that are not coincident in space. The vertical position is set to the receivers’ position (30 cm away), since these are 3 cm farther from the Gyrocube than the emitter (Fig. 1b). To set and , we use the emitting transducer that is centered between the receivers in the horizontal plane (Fig. 1a), essentially averaging their distances from the MP’s centroid, as these are smaller than . The translation velocities are derived from time integrating the accelerometer signal after removing the gravitational tilt contributions (e.g., Perlin and Moum 2012),

since accelerometers respond to both gravitational tilts and translation . We obtain the gravitational tilt contributions from subsequent rotations around the x, y, and z axes,

where the Euler angles α and ϕ are the pitch and roll of the MP, respectively. The gravitational acceleration m s⁻¹ is negative in our frame of reference (Fig. 1a).

Complementary filtering is recommended to derive the low-frequency content of the pitch and roll from the accelerometers and the high-frequency content from time integrating the recorded rotation rates (Edson et al. 1998; Fer and Paskyabi 2014). The advantage of complementary filtering over time integrating the rotation rate signal over all frequencies is that inherent problems with sensor drift and error accumulation from integrating the (noisy) rotation rates are avoided. The technique amounts to low-pass filtering the detrended rotation rates around the horizontal axes ( and ) with a first-order Butterworth filter, followed by scaling the filtered data by a factor of . The frequencies beyond the chosen low-pass filter cutoff are thus time integrated, providing the high-frequency content of the pitch and roll. The same filter is applied to the recorded horizontal accelerometer signals and , which are then substituted into Eq. (6) to yield the low-frequency content of the pitch and roll—implying the low-frequency motion of the MTP is only a function of the tangential velocity, that is, . Overall, complementary filtering assumes the accelerometers’ variance at low frequencies is attributable (mostly) to tilting of the frame (Edson et al. 1998). Thus, an appropriate for the filters must be chosen.

The frequency is specific to the mooring design, prevailing site conditions, and the quality of the Gyrocube data. As a rule of thumb, for moored platforms, gravitational tilts dominate the accelerometer signal at Hz ( rad s⁻¹), but this must be verified for a specific deployment using information from both the measured rotation rate and accelerations. Surface wave “pumping” of the mooring’s buoyancy elements may induce both gravitational tilts and translation contributions to the accelerometer signal over the frequency band from 0.04 to 0.2 Hz. To identify , we estimate the pitch α and roll ϕ derived solely from time integrating all frequencies of , which we denote as and , respectively, to those obtained from the recorded accelerometer signal ,

The accelerometer-derived pitch and roll, denoted with a subscript A, assumes that the accelerometer signal is composed of only gravitational tilts. The spectra of (or ) will become larger than the spectra of (or ) with increasing frequency as more of the variance in the accelerometer signal results from translation. The frequency at which this occurs represents the highest frequency for that can be used to reconstruct the low-frequency pitch and roll from . Similarly, the frequency at which the pitch and roll derived from time integrating the rotation rate signal (or ) becomes larger than that obtained from the accelerometers using Eq. (7) sets the lowest frequency that can be used for .

We can also constrain by estimating the maximum peak displacement from the accelerometer spectra ,

over the frequency band of interest (e.g., swell) spanning from to . Equation (8) assumes that the accelerometer signal contains only translation contributions, and so the integration range is deemed to consist of mainly tilts when the returned is unrealistic given the mooring’s dimensions. For example, the MTP cannot be displaced by more than the distance to the anchor. The vertical displacement calculated from the measured pressure (dbars) observations ,

can also be used to gauge whether the estimated is mainly composed of gravitational tilts. This equation is especially useful when the tidal forcing and surface wave climate are known from nearby independent instruments, as the translation (motion) contribution to can be determined, leaving the vertical displacement of the MTP that can be compared to . We demonstrate in section 4 how the above-mentioned concepts and equations [Eq. (7)–(9)] for choosing are applied to our moored field observations.

2) Obtaining the environmental velocity spectra

We detail two separate techniques to recover the environmental velocity spectra. The first is the cospectral technique, which is the exact representation of Eq. (2) in the frequency domain, while the second is the squared-coherency technique that is commonly employed to remove motion contamination from turbulence measurement platforms (e.g., Levine and Lueck 1999; Zhang and Moum 2010). We first describe the two techniques, followed by a discussion of their advantages and drawbacks.

Our first technique, the cospectral technique, allows the environmental velocity spectra in a given direction j,

to be obtained by summation of the spectra of the measured velocities , the spectra of motion-induced velocities of the instrument’s frame , and the cospectra between the measured velocities and the motion-induced velocities . This generalized equation is obtained using the general definitions of a spectrum and cospectrum. Integrating the one-sided velocity spectra yields the variance of the original time series, for example, , which in the time domain can be computed from , where j is each of the n individual measurements. Equation (10) is then obtained by replacing υ in the equation for the variance with Eq. (2). The variance (and thus spectra) of the environmental velocities are now a function of the motion and measured velocities. During the expansion of the terms, the covariance between the motion-induced and measured velocities appears, which is then replaced by the cospectra, since . Thus, Eq. (10) is the mathematical representation of the environmental velocity spectra in the frequency domain, equal to the result obtained by computing the spectra directly with the environmental velocities reconstructed in the time domain via Eq. (2). By working directly in the frequency domain, however, Eq. (10) can be rewritten as

since our choice of for complementary filtering assumes over frequencies f < f_c. This version is preferred to Eq. (10), since we can reduce the accumulation of noise (error) variance with decreasing frequencies. These errors can arise when time integrating the (noisy) translation component of the accelerometer signal to obtain [Eq. (5)], when has been deemed negligible for .

Our second technique, the squared coherency, has previously been employed to remove motion contamination from turbulence measurement platforms (e.g., Levine and Lueck 1999; Zhang and Moum 2010). For this technique, the measured velocity spectra is corrected using the squared-coherency spectrum between the measured and motion-induced velocities,

We denote this result for the environmental spectra as to distinguish it from that obtained with the cospectral technique described above. In its most general form, the squared-coherency spectra can be written as

where the numerator represents the cross spectrum between the measured velocities and the motion-induced velocities . The quadspectrum represents the covariance contributions that are out of phase, while the cospectrum are the covariance contributions that are in phase (see Emery and Thomson 2001). We use the superscript C and Q to denote the contributions of the cospectrum and the quadspectrum, respectively, to . With sufficient spectral averaging, provides information on the proportion of measured velocity variance that is attributable to the instrument’s motion (Emery and Thomson 2001). However, the squared-coherency technique relies on the assumption that for a given frequency band, either the motion-induced or environmental signal (but not both) contribute to the measured signal. Hence, unlike the cospectral technique presented above, the environmental spectra obtained with the squared-coherency technique are not an exact representation of the environmental velocity time series [Eq. (2)] in the frequency domain.

When the required assumptions of the squared-coherency technique are met, its use has several advantages over the cospectral technique. For instance, the squared-coherency spectra may be calculated directly using the measured acceleration or the rotation rate as a proxy for the motion-induced velocity, thus avoiding the use of complementary filtering to derive the instrument’s pitch and roll. The tangential velocities generally contribute very little to , particularly when the instrument’s sampling volume is close to the MTP’s centroid and so the instrument’s motion is then a function of the translation velocities , obtained through linear operations on the recorded accelerations [see Eq. (5)]. In these situations, the motion-induced velocities used in the calculation of in Eq. (13) can be replaced by the measured accelerations. We note that similarly, the cross spectrum is then calculated between the measured velocities and the measured accelerations to yield the squared coherency used in Eq. (12). Another advantage is that by correcting the measured velocity spectra with a factor between 0 and 1, the squared-coherency technique avoids introducing excessive noise variance from the motion sensors at low frequencies in the estimated environmental velocity spectra . The squared-coherency technique can generally be used at high frequencies where the measured velocity spectra are contaminated only by the mooring’s vibrations (Levine and Lueck 1999), since the underlying assumptions are met.

In the surface wave frequency band, however, the surface waves can induce motion by pumping of the mooring’s elements, while the surface wave–induced orbital motions can impart a (measurable) environmental contribution to the measured velocities. In this situation, the squared-coherency technique will underestimate the environmental spectral peak (variance), as the measured velocities contain variance from both the motion and environmental signal. The squared-coherency technique may even remove variance from the measurements that should in fact be added to the measured spectra to yield . This situation arises when the motion-induced and environmental velocities contain contributions at a given frequency that are in phase, such that the cospectra in Eq. (10) is positive, that is, . Thereby, the cospectral technique [Eq. (11)] should be used instead of the squared-coherency technique if the advection of turbulence by the orbital motions generated by surface waves is nonnegligible.

3) Determining turbulence dissipation from the environmental spectra

We estimated ϵ by fitting the inertial subrange spectral model ,

to our corrected environmental velocity spectral observations. Here, C denotes the empirical Kolmogorov universal constant of C = 1.5 and k denotes the wavenumber. The constant depends on the velocity direction; in the direction of mean advection = 18/55 and in the other transverse directions = 1.33 (Pope 2000). To convert our environmental spectral observations from f to k space, we invoked Taylor’s frozen turbulence hypothesis, which requires knowledge of the mean advection velocity past the sensor. We set this velocity to the time-averaged velocity magnitude measured by the ADV over the 8.53-min-long (4096 points) segments used to estimate the velocity spectra. We chose this segment length to be shorter than the average buoyancy period (i.e., in the turbulence range) and yet long enough to allow sufficient averaging to ensure the coherence and cospectral calculations are statistically meaningful (Emery and Thomson 2001). We did these calculations on subsets of 1024 points with a 50% overlap after applying a Hanning window to each subset. The spectra obtained were then band averaged over three points, yielding our final spectra with 64 degrees of freedom, while the coherence was significant for values greater than 0.12. We used the maximum likelihood estimator (MLE) as described by Bluteau et al. (2011b) to fit the model spectrum to our observations, accounting for the chi distribution and the degrees of freedom of the estimated spectra.

We discarded segments with low turbulence levels compared to the mean advection velocity, as these violate Taylor’s frozen turbulence hypothesis (see Lumley 1965; Bluteau et al. 2011b). We also excluded segments where the advection of turbulence by surface waves was nonnegligible, that is, whenever the orbital velocities . When , the correction factor that must be applied to the inertial subrange model [Eq. (14)] is about 1.2 [see Eqs. (6) and (7) in Trowbridge and Elgar 2001], which would underestimate ϵ by at most 25%. To estimate the orbital velocities, we integrated the environmental spectra,

over the surface wave frequency band from to Hz (see section 4 on surface wave climate). We performed a nonlinear least squares fit to inhibit fitting portions of the spectra where the slope was near zero and thus dominated by instrument noise. We estimated ϵ from the vertical velocity direction, since it was the least affected by surface waves and instrumentation measurement noise. We note that the vertical velocity inertial subrange is more susceptible to a reduction in spectral bandwidth at low wavenumbers when the ratio of the largest to the smallest turbulent overturns is small due to anisotropy (Bluteau et al. 2011b).

a. Mooring deployment details

A 34-m-high mooring was anchored to the seafloor on the Australian North West Shelf in 105 m of water (Fig. 2) from 4 to 22 April 2012 and was equipped with two MTPs: one at 7.5 and the other at 20.5 m above the seabed (m ASB). The instruments ran out of power after about 10 days, one on 14 April (0600 UTC) and the other on 13 April (2100 UTC). The turbulent velocities were recorded on the ADV at 8 Hz, while the MPs sampled the analog output of the ADV at 64 Hz, resulting in eight redundant samples on the 16-GB compact flash memory card of the MP. All other available channels on the MP were recorded at 64 Hz, for example, Gyrocube motion sensor, magnetometer, and fast-response temperature and pressure sensors (Table 1). Other instruments on the mooring, detailed in Table 1, include an upward-pointing acoustic Doppler current profiler near the anchor (300-kHz Workhorse, Teledyne RD Instruments), providing current velocities from 7 to 40 m ASB in 1-m bins and 1-min averages; 22 temperature sensors (SBE56 and SBE39, Sea-Bird Electronics) sampling from 0.5 to 10 s; a temperature–pressure sensor sampling at 10 s (SBE39, Sea-Bird Electronics); and a conductivity–temperature sensor at 5.2 m ASB (SBE37, Sea-Bird Electronics) sampling at 15 s. Each of the two syntactic buoys (1 m in diameter) provided 180 kg of buoyancy to the mooring. The bottommost buoy was located directly above the highest MTP in the water column at about 22 m ASB, while the other was the topmost element of the mooring at 34 m ASB (Fig. 2b). A nearby Directional Waverider buoy (DWR; Datawell), located 9.5 km north in 125 m of water (Fig. 2a), provided directional surface wave displacement spectra, and hence significant wave heights and periods (Fig. 3). We used this information to identify periods when the moored instruments were expected to be pumped by the action of surface waves and for assessing the various means to recover the environmental velocity spectra.

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(a) Regional bathymetry with the location of MTPs (diamond), Datawell Waverider buoy (X), and topographical upslope direction indicated. (b) Schematic of the mooring with location of the MTPs and buoyancy.

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1

Table 1.

Sampling program details for the instruments on the 34-m-long mooring anchored on the seabed in 105 m of water (19°41.6′S, 116°06.6′E). The shallowest SBE39 at 32.4 m ASB, and the MTPs also measure pressure.

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Surface wave observations from the nearby Waverider buoy. (a) Significant wave height ; (b) wave period ; and (c) wave direction , where 0° (180°) indicates waves originating from the east (west).

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1

b. Velocity microstructure vertical profiles

A total of 121 microstructure velocity shear profiles were collected nearby, providing independent ϵ estimates to compare with those obtained from the MTPs. The majority of these shear profiles were within 2 km of the mooring, with a median distance of 900 m. These velocity shear profiles were collected with a vertical microstructure profiler (VMP500, Rockland Scientific Ltd.) during a 24-h period starting at 0700 UTC 10 April 2012. The VMP recorded the velocity shear from two airfoil probes, in addition to information from the following sensors: 3D accelerometers, a pressure sensor, high-accuracy temperature and conductivity sensors (SBE-3F and SBE-4C, Sea-Bird Electronics), and a fast-response temperature sensor (FP07). All data channels sampled at 512 Hz. Drop speeds were around 0.5–0.8 m s⁻¹ with lower and more variable drop speeds near the surface and seafloor. The slowdown of the VMP near the seabed resulted in fewer dissipation estimates around the deepest MTP at 7.5 m ASB than at 20.5 m ASB.

The VMP dissipation estimates were determined by two separate methods that were assessed by Bluteau et al. (2016). One method—the most traditional method of the two—involves integrating the viscous subrange of the velocity shear spectra ,

as a function of k. Here, ν is the kinematic viscosity for water, while and are the low and high wavenumber limits, respectively, for integrating the turbulent horizontal velocity gradients provided by the vertically dropped VMP, that is, and . To estimate the shear spectra, the raw profiles were despiked and then split into 50% overlapping segments of 2048 samples (i.e., 4 s long). Portions of the profiles were discarded when the angle of attack of the sensor’s tip with respect to the mean flow was large (Macoun and Lueck 2004). From the remaining segments, the spectra were estimated by applying the FFT on subsets of 512 points with a 50% overlap, after applying a cosine window to each subset. The VMP’s accelerometers were used to remove the motion contamination from the spectral observations using the multivariate technique of Goodman et al. (2006). These spectra were converted from frequency to wavenumber space using the drop speed calculated from the VMP’s pressure sensor. The lost variance at high k due to the finite spatial dimension of the shear probe was corrected with a single-pole transfer function (Oakey 1982; Macoun and Lueck 2004); here the half-power wavenumber for the VMP’s shear probe is cycles per meter (cpm). The low integration limit was set to the lowest k in the resulting spectra. The maximum integration limit in Eq. (16) was set to the lowest of the local minima, determined from the band-averaged spectra. This local minimum was usually in the viscous roll-off of the spectra except for high when the viscous roll-off was unresolvable given the finite size of the shear probe’s sensor. For these instances, was obtained by fitting the Nasmyth empirical spectrum to the shear spectral observations using the MLE (Bluteau et al. 2016).

a. Background ocean conditions

Our site experiences large semidiurnal tides, and shelfbreak-generated shoaling nonlinear internal waves whose arrival at the site is not phased lock with the local barotropic tide (e.g., Bluteau et al. 2011a). The moored ADCP measured total velocities as high as 0.8 m s⁻¹ that were directed on average at −27° from due east, almost aligning with the topographical slope at −60° from due east (Fig. 2a). The time-averaged velocities over the 8.5-min segments used for the turbulence analysis were on average 0.3 m s⁻¹ and were 90% of the time between 0.1 and 0.5 s⁻¹ at the measurement heights of the MTPs. The background stratification estimated from the moored thermistors was nearly linear with a buoyancy period of about 9 min ( 10⁻² rad s⁻¹).

During our deployment, the surface wave climate was dominated by the swell ( 10 s) generated by southwestern storms, with the exception of a few days, 8–12 April 2012, when a strong sea breeze induced shorter period seas ( s) with significant wave heights in excess of 1.5 m (Fig. 3). The large period swell waves were transitional in 105 m of water, as the vertical orbital velocities were smaller than the horizontal velocities (Fig. 4). With decreasing wave period, the surface waves behaved more like deep-water waves. For wave periods shorter than 8 s, linear surface wave theory predicts the orbital velocities over the vertical extent of our 30-m-long mooring to be less than 1% of surface values. This depth attenuation of the surface wave signature translates to predicted near-bed orbital velocities less than 1 cm s⁻¹, for the maximum measured of 2 m from the nearby Waverider buoy (Fig. 3). For the longer period swell, the resultant orbital velocities predicted at our instrument heights were larger. The maximum expected orbital velocity during our deployment was of the order 10 cm s⁻¹ in the horizontal (Fig. 4a) and 5 cm s⁻¹ in the vertical (Fig. 4b) over the mooring’s vertical extent given the observed 18 s and 1.25 m around 0300 UTC 5 April 2012 (Fig. 3). Given the surface wave climate, the environmental velocity spectra should contain only significant spectral peaks in the horizontal component for the storm-generated swell waves with frequencies smaller than 0.1 Hz, that is, wave periods s. At the local seas’ frequency band ( Hz), there should be negligible environmental contributions at the MTPs’ measurement depths.

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Predicted orbital velocities from linear wave theory in the horizontal for (a) m and (c) m and in the vertical for (b) m and (d) m. The associated excursions in the horizontal for (e) m and (g) m and in the vertical for (f) m and (h) . The horizontal dashed lines represent the location of the top buoy (∘), the topmost MTP at 20.5 m ASB (), and the bottom MTP at 7.5 m ASB (□). The total depth of water is 105 m, and positive z is from the surface up.

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1

b. Mooring motion

The dominant frequencies over which the instruments tilt or translate in space were provided by the accelerometers (Fig. 5). The acceleration spectral observations from both MTPs showed significant peaks at the semidiurnal and diurnal tidal frequencies (not shown); two peaks in the sea-swell frequency band (0.04–0.2 Hz) from surface wave–induced pumping of the mooring’s buoyancy; and multiple peaks at higher frequencies due to shaking and strumming associated with the mooring’s design (Fig. 5). Over the surface wave frequency band, the horizontal acceleration spectra typically contained two peaks: one associated with the storm-generated swell ( s) and the other associated with the local wind-generated seas ( s). The vertical acceleration spectra contained similar peaks (Fig. 5), although the peaks associated with the local seas on the 8.5-min segments used for our turbulence calculations were often statistically insignificant.

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Example spectra of the time-integrated accelerometer signal at (a) 20.5 and (b) 7.5 m ASB during an 8-h period in the record. The spectra of the tangential velocities from the same 8-h period are shown for (c) 20.5 and (d) 7.5 m ASB. Note that the time-integrated accelerometer signal contains contributions from both tilts (gravity) and translation. The dashed line has a −2 slope, resulting from integrating white noise in the accelerometer signal.

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1

Unlike the accelerometers, the rotation rate sensors respond mainly to gravitational tilts, that is, the pitch and roll of the MTPs. We present in Fig. 5b the tangential velocity spectra derived from the rotation rate measurements using Eq. (4). Significant peaks in these spectra were confined to the surface wave frequency band (Fig. 5b) and lower tidal frequencies (not shown). Contrary to the accelerometers, no significant peaks were observed in the tangential velocity spectra at frequencies higher than the surface wave frequency band, thus implying that the high-frequency vibrations observed in the acceleration spectra resulted from the MTPs’ translation as opposed to gravitational tilting. The tangential velocities contained much less variance than the measured velocities, with the exception of certain times in the vertical direction where the tangential velocities contributed as much to the motion of the frame as the translation . These rare occurrences of large tangential velocities, however, were confined to the local seas’ frequency band. In general there is reduced movement in the vertical, particularly over the local seas’ frequency band, which is consistent with the predictions from linear wave theory (Fig. 4).

To summarize, the spectral peaks in the measured velocity at high frequencies, such as over the local seas’ frequency band and those contaminated by the mooring’s strumming and vibrations, resulted almost entirely from the frame’s motion. For these frequencies, the measured velocity spectra could be decontaminated using the squared-coherency technique [Eq. (12)]. In contrast, the long-period swell caused the frame to move (see Fig. 5), while the observed surface wave heights and periods produced measurable orbital velocities (i.e., an environmental signature) over the extent of our mooring, particularly in the horizontal direction (Fig. 4). The assumptions of the squared-coherency technique were not met, and so to recover the environmental spectra over these frequencies, we used the cospectral technique [Eq. (11)], after deriving the motion-induced velocities by reconstructing the pitch and roll via complementary filtering.

c. Reconstructing the MTPs’ pitch and roll

Complementary filtering of the measured acceleration and rotation rate signals was used to estimate the pitch and roll of the frame. The cutoff frequency of the Butterworth filter was chosen such that the pitch and roll estimates relied as much as possible on the rotation rate signal while limiting the error accumulation from time integrating the rotation rates to low frequencies. Using the techniques detailed earlier, we compared the estimated pitch and roll derived solely from time integrating all frequencies of the rotation rate signals (i.e., and ) to that obtained from the original accelerometer signal using Eq. (7) (Fig. 6). As noted earlier, the highest peak within the sea-swell frequency band was at Hz and at higher frequencies the seas were dominated by translation given (Fig. 6). In contrast, at lower frequencies over the swell peak at Hz, there was more variance in than in . This excess variance was attributed to the error accumulation from time integrating the rotation rate signal. The −2 slope between significant peaks in further confirmed that the (white) noise floor of and was integrated. Overall, the accumulated errors by and precluded setting to frequencies smaller than the swell spectral peak, while the fairly large translation in over the “sea” frequency range prevented setting to frequencies greater than the swell. Hence, we set 0.1 Hz, which was located between the swell and sea wave peaks.

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Example spectra during a period of large swell for the pitch α at (a) 20.5 and (c) 7.5 m ASB, and example spectra for the roll ϕ at (b) 20.5 and (d) 7.5 m ASB. Each panel compares the spectra obtained from the ADV and from the Gyrocube data using different methods: by complementary filtering (subscript MTP) of the rotation rate and accelerometer signals with our chosen Hz; by relying solely on either the rotation rate sensors ω or the accelerometers’ A signal.

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1

To further justify that for the accelerometer signals were dominated by tilts, we estimated the peak displacements from the accelerometer spectral observations [Eq. (8)] and pressure spectral observations [Eq. (9)]. At tidal frequencies, these displacements were km, which is impossible given the 35-m-long mooring, indicating that these frequencies were dominated by gravitational tilts. At the swell-dominated frequencies ( 0.1 Hz), the peak displacements at both MPs were more realistic, as the horizontal displacements were on average 2–3 times larger than those predicted by linear theory using the region’s surface wave climate. During the period of large swell (0300 UTC 5 April; Fig. 3), the accelerometers yielded 0.40 m compared to a couple of centimeters in the vertical, while the pressure sensor yielded 0.15 m. The peak displacements estimated from the were consistent with the excursions predicted by linear wave theory (Fig. 4h) given the observed wave period 18 s and 1.75 m. This agreement implied that over the swell frequency band, the pressure sensor data were mostly composed of an environmental surface wave signature. Hence, very little vertical translation occurred over the swell, which was further supported by the small obtained from the accelerometer. The horizontal peak displacements estimated from the accelerometers, although realistic, also mostly contained contributions from gravitational tilting.

Figure 6 compares the MTP’s pitch and roll in frequency space as reconstructed by complementary filtering against the redundant 1-Hz pitch and roll from the liquid level tilt sensor aboard the ADV. The ADV’s pitch and roll spectra were within the confidence intervals of the spectra estimated for the reconstructed pitch and roll from the MTPs (Fig. 6), although the MTPs often predicted more (less) variance over the swell (seas) than the ADV. The MTPs’ slight overprediction for the swell frequency band may result from the accelerometer signal, including small contributions from the frame’s translation. However, the ADV’s tilt sensor suffers from the same problem as the accelerometer, as it also responds (albeit differently) to both gravitational tilts and lateral translation. This response of the ADV’s tilt sensor to translation was particularly apparent at frequencies beyond Hz, where the spectra of the ADV’s roll agreed best with that estimated purely from the MTP’s accelerometer (Fig. 6d).

d. Examples of the recovered environmental velocity spectra

We show examples of the spectral correction for the horizontal and vertical velocity components at both MTPs during a period of large swell with orbital velocities of ≈10 cm s⁻¹ at the measurement heights of the MTPs (Figs. 7a–d). As expected, the spectral peaks associated with the swell of both the environmental- and motion-induced velocities were largest in the horizontal for both MTPs, while the shallowest MTP was the most affected by the swell. The motion-induced velocities resulted largely from the translation of the frame—the spectral energy levels for the tangential velocities were orders of magnitude smaller than the spectral levels of , except for the horizontal example at 7.5 m ASB (Fig. 7b). For this example, however, the tangential velocity spectrum was still much smaller than the measured and environmental spectra. The motion-induced velocity spectrum in Fig. 7b has excessive energy at low frequencies, resulting from excessive error accumulation when integrating the translation component of the accelerometer signal [Eq. (5)]. To avoid this excess error, we recommend using Eq. (11) instead of Eq. (10) to obtain the environmental velocity spectra, thus relying solely on the tangential velocities to estimate the cospectra for , since complementary filtering with our selected cutoff Hz essentially assumes 0 for .

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Example velocity spectra from the topmost MTP in the (a) horizontal and (c) vertical directions during a period of large swell and from the lower MTP in the (b) horizontal and (d) vertical directions for the same period. (e)–(h) The squared coherency between the motion-induced velocities and the measured velocities used to obtain [Eq. (12)] shown in (a)–(d). The squared coherency associated with the cospectrum is also illustrated. Shown in (a)–(d) are the environmental velocity spectra recovered by both the cospectral and the squared-coherency techniques.

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1

The examples from the shallowest MTP illustrate that the squared-coherency technique removed too much variance over the swell frequency band, particularly considering how small the motion-induced velocity spectral energy levels were compared to the measured velocities in the vertical (Fig. 7c). Most of the variance over the swell band was removed because of the large squared coherency (Fig. 7g), despite the relatively small contribution from the cospectrum (Fig. 7g). In contrast, in the horizontal direction, both the squared-coherency technique and the cospectral technique yielded similar results for the environmental spectrum (Fig. 7a). The squared coherency obtained over the swell band was mostly composed of contributions from the cospectrum , and so it did not remove excessive variance (Fig. 7e). From this result it may appear attractive to use instead of , but this is inadvisable and inconsistent with the general definitions of the spectrum and cospectrum used to recover the environmental spectrum via Eq. (10). For instance, the squared coherency yields a value between 0 and 1, and so it does not account for the sign of the cospectra, nor can it recover (add) variance to the environmental signal when the surface waves generate motion of the instruments that is in phase with the wave orbital velocities.

e. Estimated orbital velocities from the Waverider and the MTPs

Figure 8 compares the horizontal orbital velocities estimated from the Waverider’s measured surface heights and periods during the storm-generated swell (Fig. 3) against those estimated from the environmental spectra recovered using both the squared-coherency technique and the cospectral technique. The assessment is presented only for the shallowest MTP at 20.5 m ASB, which is the most susceptible to motion induced by surface waves. During our 3-week deployment, the orbital velocities derived from the spectra corrected with the squared coherency were on average 30% smaller than those obtained from (Fig. 8a). The orbital velocities derived from the environmental spectra corrected using the cospectral technique generally agreed with the orbital velocities estimated from the Waverider’s statistics (Fig. 8a). The agreement was particularly good at the beginning of the deployment, when the horizontal orbital velocities for the swell were highest. During the period between 8 and 12 April, when the swell produced smaller wave heights, the orbital velocities obtained from both methods underpredicted the Waverider’s orbital velocities. This underprediction likely resulted from the lower signal-to-noise ratio of the MTP measurements at depth, given that the Waverider’s statistics yield orbital velocities of only 1–2 cm s⁻¹. For completeness, we also illustrate in Fig. 8b the ratio of the horizontal orbital velocities against the time-averaged velocities . The orbital velocities at 20.5 m ASB were generally much smaller than the time-averaged velocities, which were often in excess of 0.4 m s⁻¹ (Fig. 8b). We excluded the few instances when the orbital velocities were larger than the time-averaged velocities from our analysis of ϵ, since the inertial subrange model spectrum used to obtain ϵ [Eq. (14)] does not account for the advection of turbulence by surface waves (see, e.g., Trowbridge and Elgar 2001; Feddersen et al. 2007).

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(a) Time series of the horizontal orbital velocities at 20.5 m ASB estimated from the waverider’s measured wave heights, and from the environmental velocity spectra over the swell frequency band [Eq. (15)] for both the squared-coherency method and the cospectral method [Eq. (11)]. (b) Time series of the measured velocity magnitude over the same 30-min period used to estimate the Waverider’s statistics, along with the ratio between the orbital velocities and the time-averaged velocities. The Waverider’s orbital velocities were estimated via linear wave theory using the surface wave heights and periods presented for the swell in Fig. 3.

Citation: Journal of Atmospheric and Oceanic Technology 33, 11; 10.1175/JTECH-D-16-0041.1