a. Turbulence observations from the Surface Salinity Profiler

Microstructure data were collected during the SPURS-2 field campaign from the ship-towed Surface Salinity Profiler (SSP) and used to estimate ε. Data were collected during cruises on R/V Roger Revelle in August–September 2016 and October–November 2017 to the eastern tropical Pacific Ocean near 10°N, 125°W (Drushka 2019). Only data from the 2016 cruise were used in this study. The SSP was a modified version of the platform described by Asher et al. (2014a,b). The version of the SSP used during SPURS-2 is also described by Drushka et al. (2019) and consisted of a stand-up paddleboard (Lakeshore Paddleboard Company, Reno, Nevada) connected with two horizontal struts to a surfboard outrigger (Fig. 1). The paddleboard, providing most of the buoyancy of the system, was 3.8 m long, 0.74 m wide, and 0.13 m thick and contained all sensors. The surfboard, providing stability, was 2.2 m long. The paddleboard was attached to a rigid 1.2-m-deep and 0.6-m-wide keel to which conductivity–temperature–depth (CTD) instruments and microstructure sensors were affixed.

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(a) A side-view photo of the SSP’s keel (and sensors) after deployment. Note that the microstructure sensors are forward of the CTD intakes. (b) A close-up photo of the SSP’s sensors in the top 54 cm after deployment. CTDs can be seen on the starboard side of the keel (left side of photo). Microstructure temperature (μ_T) and conductivity (μ_C) sensors are located on either side of the keel at 37 cm depth between the 23 and 54 cm CTDs. (c) A photo of the SSP when deployed: The keel is mounted to the stand-up paddleboard (orange). The batteries and pump for the surface measurement are in sealed boxes atop the paddleboard. The direction of tow is to the left.

Citation: Journal of Atmospheric and Oceanic Technology 38, 1; 10.1175/JTECH-D-20-0002.1

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A Rockland Scientific MicroSquid package (Rockland Scientific, Victoria, British Columbia, Canada), consisting of probes and a datalogger, was integrated into the SSP. This package included Seabird SBE 7 microstructure conductivity (μ_C) and FP07 microstructure temperature (μ_T) sensors that were affixed to opposite sides of the keel of the SSP and sampled at 1024 Hz (Fig. 1b). The microstructure sensors were mounted on the keel at a nominal depth of 37 cm, though the actual depth varied by ±3 cm due to wave motion. Vertical vibration from the tow line had a negligible influence on sensor depths. Several measures were taken to avoid the effect of platform vibration on the microstructure (and CTD) data. First, the keel was composed of a 12.5-mm-thick layer of rubber sandwiched between two 3-mm-thick aluminum plates to reduce the vibration of the board being transmitted to the sensors. Second, as done by Asher et al. (2014b), the keel was attached to the SSP using angled struts. The SSP was also towed at speeds of approximately 2 m s⁻¹ to minimize the generation of bubbles and effects of vibration on microstructure measurements while maintaining platform stability. Underwater video taken from a GoPro camera showed that the platform was stable and vibrated minimally. The SSP was towed at a small angle of attack (approximately 4°) relative to the flow and the microstructure probes were positioned well forward of the CTD intakes (Fig. 1a). The flow angle and positioning of microstructure probes ensured that lift generated by the platform and the wakes of individual CTD sensors had minimal influence on turbulence measured by the microstructure sensors.

The SSP was deployed 18 times on the 2016 cruise for a total of 124 h. Individual SSP deployments lasted from 1 to 12 h and were relatively evenly distributed between daytime and nighttime, with 57.5% of data collected between 0600 and 1800 local time. During deployments, the SSP was towed from the starboard side of the ship using a 200 m long tow cable as part of a three-point bridle system. This system caused the tow line to be at an approximately 135° angle from the ship’s direction of travel. With this tow configuration, the SSP was generally slightly aft of the ship’s stern and outside of the ship’s wake (Drushka et al. 2019). A time offset was applied to the ship data to ensure ship and SSP observations were made over the same area of ocean. A relatively slow tow speed of 1.5 to 2.5 m s⁻¹ and the platform’s buoyant design allowed it to follow the swell in conditions without wave breaking (wind speeds below 5–6 m s⁻¹; Thorpe and Humphries 1980; Holthuijsen and Herbers 1986). A GPS on the platform measured the SSP’s position at a frequency of 10 Hz. For the majority of the time during the deployments, the ship moved in a straight line, with occasional wide turns.

b. Environmental conditions

To obtain salinity and temperature profiles near the surface, four Seabird SBE 49 FastCat CTDs (Seabird Electronics, Bellevue, Washington) were mounted on the keel of the SSP at approximately 12, 23, 54, and 110 cm depth (Fig. 1). These depths varied by roughly ±3 cm due to the motion of the platform caused by waves. The CTDs sampled at 6 Hz. Water was pumped through a hose from the surface to a Seabird SBE 45 thermosalinograph to measure very-near surface salinity, and a Seabird SBE 56 temperature logger was used to measure temperature near the surface. The SBE 45’s intake was between 0 and 5 cm depth, and the SBE 56 measured in the same depth range. We refer to measurements in this depth range as surface measurements.

All CTDs were calibrated by the manufacturer before each cruise and measurements were processed using methods described by Drushka et al. (2019) to remove data spikes caused by features such as bubbles. However, there were small measurement offsets in temperature and salinity between the CTD sensors at different depths on the keel of the SSP. These offsets were determined by using the 23 cm CTD as a reference and assuming that temperature and salinity were the same at all depths in the top meter during unstratified conditions at night, when convective overturning mixes the upper ocean. Offsets between CTD salinity and temperature measurements in these conditions were below 0.004 psu and 0.004°C. These values are within the sensor calibration specifications and were used to correct the raw data before analysis. There was also a time lag of 6 s between the surface salinity measurements and the measurements from the other CTDs due to the time it took to pump the water from the surface to the SBE 45. This time lag was estimated by calculating the maximum lagged correlation between salinities from the 23 cm CTD and from the SBE 45 during unstratified conditions. The time stamp of surface salinity data was adjusted by 6 s to account for this lag. A 60-s running-mean filter was applied to the 6 Hz salinity and temperature data, and one data point per minute was retained. Spatially, this frequency corresponds to approximately one measurement every 90–150 m of transect distance, for typical ship speeds.

Data from several ship-based instruments were used as well. A three-axis sonic anemometer made measurements of wind velocity, which were corrected to 10 m height and are available with a 1-min time stamp (Clayson et al. 2019; Clayson 2019). An acoustic Doppler current profiler (ADCP) measured currents at approximately 20 m depth (Sprintall 2019). Surface Wave Instrument Float with Tracking (SWIFT) drifters (Thomson 2012), equipped with sensors to measure profiles of ε in the top 0.75 m, were deployed for several hours during five SSP deployments to provide validation for ε estimates from the μ_T sensor. Because of the noise floor of velocity observations from SWIFT drifters, ε estimates are only reliable when ε > 10⁻⁴ m² s⁻³.

c. Wave conditions

Estimates of significant wave height were made at the SPURS-2 central mooring at 10°N, 125°W (NOAA NDBC Station 43010 owned and maintained by WHOI). Central mooring significant wave height estimates were derived from accelerometer and inclinometer measurements on the buoy. Wave measurements were made every hour and were used to gain insight into the wave conditions in the area where data were collected from the ship. Because of the low temporal resolution of the wave data and the fact that the ship was tens to hundreds of kilometers away from the central mooring, direct comparisons between wave measurements and ε were not made.

Wind and surface waves are known to contribute to near-surface ε. The SPURS-2 central mooring measured significant wave height and wave direction starting 24 August 2016, encompassing most of the time when the SSP was deployed. Median significant wave heights during the 2016 SPURS-2 field campaign were 1.87 m, with a mean of 1.85 m. Wave height did not deviate by a large amount from the mean and median values observed (standard deviation of 0.31 m) and was only weakly positively correlated with wind speed at the central mooring. This weak correlation may be because winds were low (wind speeds between 0 and 10 m s⁻¹ were present in 99.2% of the observations) and so swell contributed greatly to total wave amplitudes. Mean wave heights were 1.49 m at the lowest observed wind speeds (<1 m s⁻¹) and were 2.23 m at wind speeds between 9 and 10 m s⁻¹, some of the highest observed sustained wind speeds. Median and mean peak wave periods were 12.90 and 11.99 s, respectively, with a standard deviation of 3.45 s.

a. Spectral analysis and stratification conditions

The value of ε can be estimated using μ_T or μ_C data (Oakey 1982; Kocsis et al. 1999; Goto et al. 2016); this paper describes the methods and results from the μ_T data collected during 2016 only. We estimated ε from 1024 Hz data using a spectral fitting procedure. Before performing spectral analysis, μ_T gradient (dT/dx) data were calculated as the spatial derivative of μ_T based on the platform speed and sampling frequency. The μ_T gradient data were subdivided into 1-min segments to match the time stamp of the atmospheric data. Each 1-min segment consisted of 239 smaller blocks (with 512 points each) that overlapped by 50% with neighboring blocks. Individual blocks were despiked by removing points having temperature gradients larger than 3 times the standard deviation of the block. Performing a fast Fourier transform yielded frequency spectra of dT/dx, which were averaged together for 1-min segments. The response of the μ_T sensor (thermistor) is limited at frequencies above 50–100 Hz as a result of the platform’s tow speed, so spectra show suppressed variance at high frequencies (Gregg 1999). To correct for the sensor response at high frequencies, a double-pole transfer function was applied to the dT/dx spectra (Gregg and Meagher 1980; Gregg 1999; Nash et al. 1999). The dT/dx frequency spectra were then converted to wavenumber spectra using the 1-min averaged speed through water of the SSP and assuming Taylor’s frozen field hypothesis (Taylor 1938). The speed of the SSP through the water was calculated by subtracting the current speed (from the ship’s ADCP) from the speed of the SSP with respect to Earth (from the GPS), assuming speed was constant within 1-min time segments. We assumed that the current measured by the ADCP at 20 m is approximately the same as the surface current. Considering the complex structure of the upper ocean, this assumption is a nonnegligible source of error in estimates of speed through water. Specifically, invoking Taylor’s hypothesis using these speed-through-water estimates introduces error because of wave orbital velocities, which cause fluctuations in the speed of the SSP on time scales of seconds and are not negligible compared to the speed of the platform. Sensitivity tests run to evaluate this source of error are discussed in greater detail in section 4d.

One-minute averaged stratification conditions were estimated using the SSP’s CTD data. The approximate buoyancy frequency at the 37 cm depth of the microstructure sensors (N), and ⟨dT/dz⟩, the vertical temperature gradient at 37 cm depth, were obtained from temperature (T), salinity (S), and pressure measurements from the SSP’s CTDs at approximately 23 and 54 cm depth and interpolated vertically to the depth of the microstructure sensors. Exact depths were determined from the CTD pressure sensors. For the purpose of these calculations, it was assumed that the vertical T and S gradients between these depths were linear. The frequency of the CTD data was 6 Hz, and N and ⟨dT/dz⟩ data were averaged to match the 1-min time stamp of the spectra.

b. χ_T–ε relation from diffusivity arguments

The use of Eq. (3) to calculate ε requires at least a small amount of stratification in the water column, because as ⟨dT/dz⟩ approaches zero, the ratio between ε and χ approaches unrealistic values. We therefore excluded all points where N < 0.005 s⁻¹ in calculations of ε. The uncertainty in N was on the order of 10⁻⁴ s⁻¹, which is much smaller than the minimum value of N in data that were retained. Moulin et al. (2018) used a similar criterion for excluding unstratified conditions when calculating ε from μ_T data.

c. Fitting to theoretical spectra

The dT/dx wavenumber spectra were fit to theoretical Batchelor (1959) spectra as a means of estimating χ_T and ε, the two parameters that can be used to define the shape of a spectrum. An example is shown in Fig. 2a. Fitting theoretical Batchelor spectra to observed dT/dx spectra to estimate ε has been shown to produce results consistent with direct calculations of ε from shear microstructure (Oakey 1982; Kocsis et al. 1999; Peterson and Fer 2014; Goto et al. 2016).

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(a) A raw observed temperature gradient wavenumber spectrum (light blue) and temperature gradient wavenumber spectrum with a double-pole frequency response function applied (black) for a 1-min period during stratified conditions of SSP deployment 13 in 2016. The best-fit theoretical Batchelor spectrum is shown in dark blue. Dashed vertical red lines show the spectral fitting wavenumber range (1 to 35 m⁻¹). (b) The χ_T–ε relation (pink) and the goodness of maximum likelihood estimation fit (colored background) for varying χ_T and ε for the observed temperature gradient spectrum shown in (a). The colors show ratios of an maximum likelihood estimation cost function, as described by Ruddick et al. (2000), to the best fit obtained. Values closer to 1 indicate the best fits. The black dot indicates χ_T and ε, the gray (black) bars show uncertainty in χ_T and ε values including (excluding) uncertainty in Γ and q.

Citation: Journal of Atmospheric and Oceanic Technology 38, 1; 10.1175/JTECH-D-20-0002.1

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High-frequency portions of the observed μ_T spectra were usually underresolved and the Batchelor cutoff wavenumbers (and ε) could not be determined from the spectra alone because of the response time of thermistors at high frequencies. Spectral fitting was only done at lower frequencies equating to wavenumbers from 1 to 35 m⁻¹ (cycles per meter), to minimize the influence of error due to underresolved spectra at high frequency and differences between the true sensor response (dependent on probe geometry) and the double-pole function. High-frequency noise did not appear to affect the spectra in the 1–35 m⁻¹ range: the example spectrum in Fig. 2a illustrates that noise-generated peaks are only seen at wavenumbers larger than 100 m⁻¹. Because ε was generally high at 37 cm depth, the fitted wavenumber range was not at scales influenced by diffusion and was in the viscous-convective subrange, where the slope of the spectrum scales approximately with k⁺¹, where k is wavenumber (Batchelor 1959; Dillon and Caldwell 1980).

An iterative method was used to determine the best-fit χ_T and ε to each observed 1-min temperature spectrum. Each spectrum was first fit to the low-wavenumber portion (within a range of 1 to 35 m⁻¹) of 50 theoretical Batchelor spectra having evenly log-spaced values of ε between 10⁻¹⁰ and 10⁻¹ m² s⁻³ and χ_T consistent with the χ_T–ε relationship determined from Eq. (3) and outlined in section 3c (the pink line in Fig. 2b). This range encompassed realistic values of ε. The best fit was then determined using a maximum likelihood fitting procedure (Ruddick et al. 2000). This is represented by the colored background in Fig. 2b. Spectral fitting was again done to 50 theoretical spectra, this time evenly log spaced between the ε values of the two spectra that were nearest to the ε value of the best fit from the first set of fits. Because the spectra had a large number of degrees of freedom, the maximum likelihood method is approximately equivalent to a weighted least squares approach (Ruddick et al. 2000). This process of fitting to 50 evenly spaced spectra and finding the best fit was repeated a third time. The best fit values of ε and χ_T from the third iteration were taken as the final values of ε and χ_T. Selected values of ε and χ_T only differed by 1% from adjacent values, demonstrating that χ_T and ε generally converged after three iterations.

Following Ruddick et al. (2000), poor spectral fits were determined by calculating the mean absolute deviation of each observed spectrum from its best-fit theoretical spectrum. Ruddick et al. (2000) used a rejection criterion of 2(2/dof)^1/2, with dof being the number of degrees of freedom of the spectrum. Each spectrum had 242 degrees of freedom. We found that a mean absolute deviation rejection criterion that retained spectra with visually good fits and rejected the poorest fits was 6(2/242)^1/2 ≈ 0.5455. We rejected data with a mean absolute deviation that exceeded this value. We expected that 10⁻¹⁰ m² s⁻³ < ε < 10⁻¹ m² s⁻³, which is a range consistent with previous laboratory and field studies of the near surface ocean in a variety of conditions (Harrison et al. 2012; Thomson 2012; Callaghan et al. 2014; Walesby et al. 2015; Drushka et al. 2016; Harrison and Veron 2017; Moulin et al. 2018). Spectral fits where ε > 10⁻¹ or < 10⁻¹⁰ m² s⁻³ were unrealistic and discarded. At wind speeds under 2 m s⁻¹, instances where ε > 10⁻³ m² s⁻³ were also excluded. Excluding these points did not significantly affect the results.

Gargett (1985) asserted that the universality of Batchelor spectra shape is dependent on the isotropy of turbulence. Others (e.g., Goto et al. 2016) found that ε derived from Batchelor spectral fitting procedures was consistent with direct estimates of ε, regardless of the degree of isotropy. Isotropic conditions might be expected during energetic periods, and anisotropic conditions might be expected when there was strong near-surface vertical stratification. It is assumed that nonisotropic turbulence would be associated with poor spectral fits. No significant difference was observed between the proportion of poor fits during unstratified versus stratified conditions, so we conclude that assumption of isotropic turbulence in stratified conditions is valid. Therefore, calculating ε from Batchelor spectra fitting is reasonable. We hypothesize that poor spectral fits may be the result of an evolving turbulence structure or large contributions from the wave spectrum.

The use of Eq. (3) to calculate ε requires background stratification; we only calculated ε when N > 0.005 s⁻¹. At the site of the SPURS-2 experiment, this condition was usually satisfied near the surface. This is particularly the case because the timing of deployments was intentionally selected to coincide with conditions of rain-induced near-surface stratification: the SSP was often opportunistically deployed during and following rainfall. We rejected 18.1% of the μ_T data (14.2% of daytime data and 20.2% of nighttime data) because N < 0.005 s⁻¹, mostly during conditions of high wind or nighttime convection in the absence of rain. At wind speeds under 6 m s⁻¹, 15.2% of data were collected during unstratified conditions and thus rejected, increasing to 18.8% from 6 to 8 m s⁻¹ and 26.7% from 8 to 10 m s⁻¹. We acknowledge that because of the stratification requirement, estimating ε with the SSP is best suited for use in the stratified tropical ocean, particularly in rainy regions such as the intertropical convergence zone. The method is likely not usable at significantly higher wind speeds than 10 m s⁻¹ or high wave energy environments with a fully mixed layer near the surface, as might be expected at higher latitudes. An additional 12.8% of the data were rejected because of poor spectral fits or unreasonable values of ε. Acceptable ε estimates were obtained for 69.1% of the data.

a. Comparisons with SWIFT drifter observations

Figure 3 shows wind speed and ε during an SSP deployment when a SWIFT drifter was deployed and remained within a few kilometers of the ship. Wind speed and ε from the μ_T sensor are highly positively correlated, as ε increases from roughly 10⁻⁵ to 10⁻³ m² s⁻³ as wind speeds increase from 4 to 8 m s⁻¹. Estimates of ε at 38 cm depth were obtained from the SWIFT using the second-order structure function of inertial-scale vertical velocity differences, as described by Thomson (2012) and Wiles et al. (2006). SWIFT data were collected in 12-min bursts; ε estimates were obtained as averages over that time period (orange points on Fig. 3). For comparison, μ_T-derived ε at 37 cm depth from the SSP were averaged over 12-min time periods to match the SWIFT data (blue points on Fig. 3). ε estimates were compared only during the most energetic conditions observed (μ_T-based ε > 10⁻⁴ m² s⁻³), as the SWIFT drifter used here was not capable of estimating substantially lower ε than this. SWIFT and μ_T-derived ε generally agreed well, although ε calculated from the μ_T data had significantly more high-frequency variability (Fig. 3). Lower variability of ε from SWIFT drifters may be due to sampling bias in the platform: SWIFTs have been observed to get trapped in areas of convergence (Zippel et al. 2020), which may explain the lower variability of ε observed by the SWIFT compared to the ship-towed SSP. Significant variability in ε on short time scales is consistent with previous observations showing that turbulence statistics can vary greatly over time scales of minutes (e.g., Derakhti et al. 2020). During the 360 min when ε > 10⁻⁴ m² s⁻³, median μ_T-derived and SWIFT ε values were 6.76 × 10⁻⁴ m² s⁻³ and 6.37 × 10⁻⁴ m² s⁻³, respectively. Observations of ε from the two platforms agreed within their uncertainty limits 87% of the time during these energetic conditions. This statistical agreement between observations from two independent platforms provides evidence that the ε estimates from μ_T measurements collected with the SSP are reasonable.

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(a) Wind speed and (b) ε during one SSP deployment (deployment 13), which lasted for several hours on 11 Sep 2016. ε (37 cm depth) calculated from 1-min resolution μ_T data is shown in black (shading denotes error bars). ε (38 cm depth) observed from a nearby SWIFT drifter with 12-min resolution is shown in orange. ε estimates from SWIFT drifters are at varying depths due to platform motion, so error bars on SWIFT ε represent ε estimates at 38 ± 8 cm. μ_T-derived ε averaged to the SWIFT’s 12-min time period is shown in blue. The distance between the SWIFT drifter and SSP varied between 0 and 5.5 km. Rain rates were over 20 mm h⁻¹ at the start of the time segment, but decreased to under 10 mm h⁻¹ after around 1400 local time. Wind data from the ship’s anemometer are shown when the ship was turning and sonic anemometer data were unreliable between 1510 and 1525 local time.

Citation: Journal of Atmospheric and Oceanic Technology 38, 1; 10.1175/JTECH-D-20-0002.1

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b. Spatial variability of TKE dissipation rate

Using a towed platform such as the SSP allows for observations of the spatial structure of ε on scales of tens of kilometers over a time scale of only a few hours. Figure 4 shows the spatial variability of ε at 37 cm depth during the final deployment of 2016. As the SSP was towed during a period of 11 h, the ship covered over 50 km of distance. ε varied by over five orders of magnitude during the entire deployment, which can also be seen in time series shown in Fig. 5. Over the course of the deployment, ε was significantly higher (>10⁻⁴ m² s⁻³) west of 140.9°W due to high winds (>7 m s⁻¹) in this region. The spike in ε just after 1730 UTC was due to a brief increase in wind speed (Fig. 5). Winds shifted from northerly to southerly west of 140.7°W. The changes in ε are likely due to a combination of temporal and spatial phenomena. Spatial variations of ε that appear independent of increases in wind speed are also observable; for example, between 140.85° and 140.75°W (2100 to 2300 UTC), a patch of increased turbulence was observed despite a slight decrease in wind speed. Figure 4 demonstrates that observations from the SSP enable the joint study of spatial and temporal patterns of ε variability.

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Spatial variations in log₁₀ε as measured during one example SSP deployment (deployment 18), which lasted for a period of over 11 h on 17–18 Sep 2016. Arrows denote wind speed and direction. Note that the plotted axes are not to scale and cover roughly 60 km of zonal distance and 8 km of meridional distance.

Citation: Journal of Atmospheric and Oceanic Technology 38, 1; 10.1175/JTECH-D-20-0002.1

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(a) Wind speed and (b) ε during SSP deployment 18, which lasted for over 11 h on 17–18 Sep 2016. ε (37 cm depth) calculated from μ_T data is shown in black (shading denotes error bars).

Citation: Journal of Atmospheric and Oceanic Technology 38, 1; 10.1175/JTECH-D-20-0002.1

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c. The influence of wind and waves on TKE dissipation rate

During SSP deployments, wind speeds varied between approximately 0 and 12 m s⁻¹. We compared the SSP observations of ε, binned by wind speed, to expected values of ε for a theoretical “law of the wall” boundary layer to validate the SSP observations during these conditions. We also made comparisons with two turbulent scaling relationships as a means to validate the observations with two independently determined relationships: the widely cited Terray et al. (1996) scaling and the scaling of Esters et al. (2018) which was developed from multiple robust observational datasets. The Terray et al. (1996) and Esters et al. (2018) scalings account for the contributions of both wind and wave breaking to ε. A “law of the wall” layer is defined as

$\epsilon ={\displaystyle \frac{u_{*}^3}{\kappa z}},$(4)

where κ is the von Kármán constant (0.41), u_* is the water-side friction velocity, and z is the depth below the sea surface (depth taken as 37 cm, the average depth of the μ_T sensor). The value of u_* was calculated using observed wind speeds and assuming a constant drag coefficient of 1.2 × 10⁻³. In the Terray et al. (1996) scaling, ε in the wave breaking layer near the surface is parameterized as

$\epsilon =0.3\alpha \left({\displaystyle \frac{u_{*}^3}{H_s}}\right){\left({\displaystyle \frac{z_b}{H_s}}\right)}^{-2},$(5)

where H_s is the significant wave height, α is a function of the wave age, and z_b is the depth of the wave breaking layer, proportional to H_s. In conditions where wind and waves dominate turbulence, Esters et al. (2018) found a best-fit scaling in the wave breaking layer of

$\epsilon =\left(26-388A\right)\left({\displaystyle \frac{u_{*}^3}{H_s}}\right){\left({\displaystyle \frac{z}{H_s}}\right)}^{-1.29},$(6)

where A is the inverse wave age and z is depth. We compared measured ε values to these scalings for all of the data and not at individual points in time, so the contribution of buoyancy forcing to the Esters et al. (2018) scaling is not considered. Because wave parameters were not available at the ship’s location, we calculated scaling dissipation rates using parameters (α, A) suggested by Esters et al. (2018) and Terray et al. (1996) for each individual scaling. Specifically, we assumed an effective wave speed of 1.5 m s⁻¹ for the Terray et al. (1996) scaling and the variable, piecewise wave age of Wang and Huang (2004) for the Esters et al. (2018) scaling. These parameters are consistent with Esters et al. (2018) and Terray et al. (1996). Significant wave heights of between 0 and 3.0 m were used when calculating ε from scalings, consistent with the wave conditions observed at the SPURS-2 central mooring. The colored background in Fig. 6 represents this range of wave heights.

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Binned log₁₀ε vs wind speed calculated using μ_T data from 18 deployments in 2016. Wind speed bin width is 1 m s⁻¹, and data are plotted in the center of each bin. Each bin had a minimum of 10 points. Smaller error bars denote 95% confidence intervals of the mean value within each bin (calculated from standard error) and larger error bars show the theoretical uncertainty limits, as discussed in section 4d. The theoretical law of the wall log₁₀ε at 37 cm depth is shown by the gray line. Wave-dependent scalings, assuming significant wave heights between 0 and 3.0 m, as suggested by Terray et al. (1996) and Esters et al. (2018), are shown by the purple and light blue shadings, respectively. The step change in the Esters et al. (2018) scaling results from assuming a linear, piecewise wave age (Wang and Huang 2004).

Citation: Journal of Atmospheric and Oceanic Technology 38, 1; 10.1175/JTECH-D-20-0002.1

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Direct comparisons between SSP observations and the Terray et al. (1996) scaling will have some differences due to differences in the methods. First, the Terray et al. (1996) scaling was developed based on observations referenced to the mean sea level, while SSP observations were made in a wave-following reference frame. Second, Terray et al. (1996) suggests a layer of constant ε in the wave-breaking layer (which includes the 37 cm depth of the SSP ε observations). More recent studies have found that ε is depth dependent up to the surface and have attributed differences between Terray et al. (1996) and newer work to the reference frame of the observations (Gemmrich 2010; Sutherland and Melville 2015; Thomson et al. 2016; Esters et al. 2018), suggesting that differences between SSP observations of ε and the scaling of Terray et al. (1996) are partly due to discrepancies in the reference frame and parameterization of ε in the wave-breaking layer.

Figure 6 shows a statistically significant increase in ε with increasing wind speed. As expected, ε was elevated above values predicted by the law of the wall and generally consistent with previous observations (Agrawal et al. 1992; Terray et al. 1996; Soloviev and Lukas 2003; Thorpe 2005; Harrison et al. 2012; Callaghan et al. 2014; Esters et al. 2018). At wind speeds between 0 and 12 m s⁻¹, ε was consistent with the values suggested by the scalings of Terray et al. (1996) and Esters et al. (2018) for waves with significant wave heights of between 0 and 3.0 m. The agreement between observations of ε and observational scalings, along with the aforementioned agreement with ε observed from SWIFT drifters, confirms that microstructure observations from the SSP can be used to estimate near-surface turbulence. The overall spread in observed ε likely results from a combination of varying stratification (i.e., stratification above 37 cm depth suppressed ε at 37 cm), the natural variability of turbulence (Moum et al. 1995), and the range of wave conditions for a given wind speed. At wind speeds above 9 m s⁻¹, there is greater uncertainty in the average value of ε, as indicated by the larger confidence intervals shown in Fig. 6. This is because fewer data were collected at these high wind speeds. Low values of ε at wind speeds of 3–4 m s⁻¹ are possibly due to near-surface stratification created by rain or diurnal warming, as turbulence has been shown to be enhanced within and suppressed below these layers (Smyth et al. 1997; Wijesekera et al. 1999; Callaghan et al. 2014; Walesby et al. 2015; Sutherland et al. 2016; Moulin et al. 2018). The effects of stratification on ε will be discussed in greater detail in a forthcoming study. We note that ε variability in individual cases (e.g., Figs. 3–5) was sometimes much greater than the bin-averaged values shown in Fig. 6, especially at high wind speeds.

d. Uncertainty in TKE dissipation rate estimates

There were several sources of uncertainty involved in estimating ε from μ_T data. These sources of uncertainty are represented by the error bars shown in Figs. 2b, 3b, and 5b. Because we used Eq. (3) to determine ε, ε was dependent on N and ⟨dT/dz⟩, which were obtained using the SBE 49 CTDs on the SSP and assuming linear vertical T and S gradients between the 23 and 54 cm sensors. This method introduces uncertainty into our ε estimates in two ways. First, the SBE 49 CTDs have stated accuracies of ±0.002°C and ±0.0003 S m⁻¹ and stabilities of ±0.0002°C month⁻¹ and ±0.0003 S m⁻¹ month⁻¹. This causes maximum errors of approximately ±0.0003 s⁻¹ and ±0.0003°C m⁻¹ in N and ⟨dT/dz⟩, respectively. These maximum errors assume that the error due to sensor inaccuracy is random and take into account the 1-min averaging of the 6 Hz SSP data. Because we rejected data where N < 0.005 s⁻¹, the contribution of sensor error to ε estimates through Eq. (3) was negligible. In addition, errors in estimates of N and ⟨dT/dz⟩ should at least partially cancel out because of the opposing effects of N and ⟨dT/dz⟩ on ε in Eq. (3).

A second source of uncertainty was the assumption that vertical T and S gradients between the 23 and 54 cm sensors were uniform and that N and ⟨dT/dz⟩ could be linearly interpolated to 37 cm. In the case of rain-induced salinity stratification in the upper tens of centimeters as observed during SPURS-2, T and S will vary significantly, and possibly nonlinearly, between 23 and 54 cm (Asher et al. 2014a; Walesby et al. 2015; Boutin et al. 2016; Drushka et al. 2016). Thus, assuming linear vertical T and S gradients may not represent the true gradients at 37 cm depth. To test the sensitivity of assuming that N and ⟨dT/dz⟩ vary linearly between 23 and 54 cm, a cubic interpolation method was used, in addition to linear interpolation, to calculate N and ⟨dT/dz⟩ at 37 cm. The cubic interpolation produced similar quantitative χ_T–ε relations over a wide range of conditions; the median absolute difference between ε calculated assuming a cubic and linear relationship was approximately 5%. This suggests that linear interpolation is a reasonable method of calculating N and ⟨dT/dz⟩ at 37 cm depth. We included an additional maximum error of 5% in N and ⟨dT/dz⟩ to account for nonlinearity in the T and S profiles between 0.2 and 0.5 m.

We assumed constant values for Γ, the mixing efficiency, and q, the Batchelor subrange constant, to estimate ε from Eq. (3) (Gibson and Schwarz 1963; Grant et al. 1968; Dillon and Caldwell 1980; Oakey 1982; Sanchez et al. 2011). Note that while q does not appear in Eq. (3), it affects ε and χ_T through the spectral fitting procedure. However, Γ and q are not truly constants. Any error in Γ leads to error in ε estimates through Eq. (3), and a percentage error in q leads to twice that percentage error in ε (Dillon and Caldwell 1980; Oakey 1982; Ruddick et al. 2000). We assumed that Γ = 0.2 ± 0.1, consistent with many observational estimates of Γ (e.g., Oakey 1982; Sanchez et al. 2011). Uncertainty in ε due to variability in Γ was roughly one order of magnitude (Fig. 2b). Grant et al. (1968) suggest that q = 3.9 ± 1.5, and we used these same uncertainty limits. Gargett (1985) found that when stratification was low (N ≈ 0.005 s⁻¹), spectra do not follow the universal Batchelor form. They used direct measurements of ε to show that q = 12 during unstratified conditions. Assuming q = 12 instead of the range suggested by Grant et al. (1968) modifies ε estimates by at most a factor of roughly 2.5, significantly less than the uncertainty resulting from variability in Γ. As a result, we consider this source of uncertainty small relative to other sources. Furthermore, observations from Oakey (1982), Kocsis et al. (1999), Goto et al. (2016), and Moulin et al. (2018) support the assertion that Batchelor fitting techniques of determining ε are within an order of magnitude of direct estimates of ε in a variety of stratification conditions (Kocsis et al. 1999; Moulin et al. 2018) and at varying N and depth in the ocean (Oakey 1982; Goto et al. 2016).

The 2.0 m s⁻¹ tow speed of the platform contributed to error in the ε estimates in two primary ways: First, through the contribution of wave orbital motions to surface currents. We estimated surface currents by assuming that the current speed measured at 20 m depth represents the current speed at the depth of the SSP, and used this to calculate speed through water and convert frequency spectra to wavenumber spectra. This assumption introduces error due to the differences in current speed between these depths, which are largely due to wave orbital motions near the surface. Because direct wave measurements were unavailable, this assumption was investigated using a sensitivity test based on data from deployment 18 (Fig. 5). The sensitivity test first involved subdividing μ_T data into 2.5-s segments, instead of 1-min segments used in the main analysis, and calculating frequency spectra. An artificial speed through water was then generated by adding the velocity observations to an artificial fluctuating component. The artificial component was consistent with waves with a period of 12.9 s and maximum orbital velocity of 0.5 m s⁻¹, typical of observations at the SPURS-2 central mooring. The 2.5-s segment length (instead of 1 min), significantly shorter than the wave period, ensured that velocity fluctuations from waves were not averaged out. As before, a double-pole transfer function was used to correct frequency spectra for the sensor response: the double-pole function is speed dependent (Nash et al. 1999), so the artificial velocity is incorporated into this correction. Frequency spectra were then converted to wavenumber spectra using the artificial velocity and ε was calculated from spectral fitting as before. Finally, ε data were averaged to match the 1-min time period of the original data. During deployment 18, ε calculated with and without this fluctuating velocity component differed by an average of 27%, with a median difference of 29%. This source of uncertainty may be an overestimate because spectra were averaged within 1-min intervals, a period of time significantly longer than the wave period. Second, the tow speed of the platform introduces error through limitations in the thermistor response. The high frequency sensor response is limited at fast tow speeds, so spectra were corrected using a double-pole transfer function. This response function was chosen because it has been shown to be reasonable within the frequency range used for fitting (Gregg and Meagher 1980; Goto et al. 2016). However, other studies have suggested a single-pole thermistor response at frequencies at the lower end of the frequency range (Lueck et al. 1977). To quantify a maximum error associated with using a double-pole response function, ε was recalculated using a single-pole transfer function for deployment 18 (Fig. 5). Applying a single-pole response decreased log₁₀ε, on average, by a factor of 2.7. This is significant, although response function errors are associated with a relatively constant offset in ε, rather than a large random error. Overall, while the uncertainties in ε due to wave orbital motion and sensor response are nonnegligible, these values are relatively small compared to the primary aforementioned source of uncertainty, i.e., variability in Γ. The 2.0 m s⁻¹ SSP tow speed minimizes both sources of uncertainty: a faster tow speed would increase uncertainty due to the sensor response, and a slower tow speed would increase uncertainty due to wave orbital motions.

Waves impinging on the SSP from a particular direction did not bias ε observations. This was done by comparing observed ε, relative to the SSP’s direction of motion, during parts of an SSP deployment where the ship made large turns. Assuming that wave direction was constant over the comparison period of a few hours, we found that there was not a significant correlation between the heading of the SSP platform and ε (not shown). We also determined from the SSP’s pressure sensors that depth relative to the free surface only varied by ±3 cm (±5 cm at wind speeds over 8 m s⁻¹) due to wave motion, because the SSP was wave following. This may have contributed to the greater noise at wind speeds over 9 m s⁻¹.

The uncertainty in our ε estimates was on average ±1 order of magnitude, depending on the background stratification. Figures 3 and 5 provide an indication of how uncertainty limits varied during SSP deployments. In conditions of low stratification, the uncertainty was greater because small changes in N and ⟨dT/dz⟩ contributed proportionally more to the ratio between ε and χ_T through Eq. (3). In these conditions, uncertainty was typically ±1.5 orders of magnitude. In stratified conditions, the uncertainty in q and Γ was by far the most significant source of uncertainty. This can be seen from Fig. 2b, which shows uncertainties separated into contributions from q and Γ (gray) and from all other factors combined (black). The uncertainty bounds reported here are considerably larger than in several previous studies (e.g., Oakey 1982; Kocsis et al. 1999; Callaghan et al. 2014; Moulin et al. 2018), largely because we considered variability in Γ and q in our estimates, which led to greater uncertainty in ε. If variability in Γ and q was not considered, the uncertainty in ε estimates would be roughly 0.5 orders of magnitude in stratified conditions due primarily to CTD sensor inaccuracy and the assumption of linear T and S vertical gradients (Fig. 2b).