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Measuring the Dissipation Rate of Turbulent Kinetic Energy in Strongly Stratified, Low-Energy Environments… Scheifele, Benjamin; Waterman, Stephanie; Merckelbach, Lucas; Carpenter, Jeffrey R. Jul 2, 2018

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Journal of Geophysical Research: OceansMeasuring the Dissipation Rate of Turbulent Kinetic Energyin Strongly Stratified, Low-Energy Environments: A CaseStudy From the Arctic OceanBenjamin Scheifele1 , Stephanie Waterman1 , Lucas Merckelbach2,and Jeffrey R. Carpenter21Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Vancouver, British Columbia,Canada, 2Institute of Coastal Research, Helmholtz-Zentrum Geesthacht, Geesthacht, GermanyAbstract We compare estimates of the turbulent dissipation rate, ๐œ–, obtained independently fromcoincident measurements of shear and temperature microstructure in the southeastern Beaufort Sea,a strongly stratified, low-energy environment. The measurements were collected over 10 days in 2015 byan ocean glider equipped with microstructure instrumentation; they yield 28,575 shear-derived and 21,577temperature-derived ๐œ– estimates. We find agreement within a factor of 2 from the two types of estimateswhen ๐œ– exceeds 3 ร— 10โˆ’11 W/kg, a threshold we identify as the noise floor of the shear-derived estimates.However, the temperature-derived estimates suggest that the dissipation rate is lower than this thresholdin 58% of our observations. Further, the noise floor of the shear measurements artificially skews thestatistical distribution of ๐œ– below 10โˆ’10 W/kg, that is, in 70% of our observations. The shear measurementsoverestimate portions of the geometric mean vertical profile of ๐œ– by more than an order of magnitude andunderestimate the overall variability of ๐œ– by at least 2 orders of magnitude. We further discuss uncertaintiesthat arise in both temperature- and shear-derived ๐œ– estimates in strongly stratified, weakly turbulentconditions, and we demonstrate how turbulence spectra are systematically modified by stratification underthese conditions. Using evidence from the temperature-gradient spectral shapes and from the observed ๐œ–distributions, we suggest that the temperature-derived dissipation rates are reliable to values as small as2 ร— 10โˆ’12 W/kg, making them preferable for characterizing the turbulent dissipation rates in the weaklyturbulent environment of this study.1. MotivationThe purpose of this study is to examine the agreement between measures of the turbulent kinetic energydissipation rate, ๐œ–, derived from measurements of shear and temperature microstructure in a stratifiedlow-energy environment, that is, a stratified environment where the amount of turbulent kinetic energy inthe flow field is unusually small. It was motivated when, in an attempt to quantify turbulent mixing in theBeaufort Sea thermocline, we discovered that results from the twomeasurements diverged strongly at low ๐œ–.Where dissipation rate estimates from shear measurements frequently clustered near a clearly defined lowerlimit, estimates from temperature measurements distributed to much lower values that were often multipleorders of magnitude smaller. We noted that this may have serious implications for how shear microstructuremeasurements from the Arctic Ocean are interpreted. This study is therefore dedicated to describing thedivergenceweobserved anddiscussing its causeswith the goal of informing the collection and interpretationof microstructure measurements in the Arctic Ocean or similar stratified low-energy environments.ThewesternArcticOcean,wherewecollectedourmeasurements, is known tobeanexceptionally low-energy,and highly stratified, ocean environment with some of the lowest estimates of oceanic turbulence in theworld (e.g., Guthrie et al., 2013; Lincoln et al., 2016). Only a relatively small number of microstructure mea-surements from this region exist to date (e.g., Bourgault et al., 2011; Padman & Dillon, 1987; Rainville &Winsor, 2008; Rippeth et al., 2015; Shaw & Stanton, 2014; Shroyer, 2012), but this number is certain toincrease in the coming years owing to increased interest in constraining oceanic heat budgets in theArctic (Carmack et al., 2015). Constraining these budgets requires knowledge of turbulent mixing rates in theoceanwhich are obtainedmost directly frommicrostructuremeasurements; we demonstrate here why this isa challenging endeavor and why special considerations are needed when interpreting those measurementsRESEARCH ARTICLE10.1029/2017JC013731Key Points:โ€ข Dissipation rate estimates inlow-energy environments aresubject to increased uncertaintywhen stratification is strongโ€ข Shear microstructure measurementsmay overestimate dissipation rates inlow-energy environmentsโ€ข Resulting biases in geometricallyaveraged dissipation rates mayexceed an order of magnitudeCorrespondence to:B. Scheifele,bscheife@eoas.ubc.caCitation:Scheifele, B., Waterman, S.,Merckelbach, L., & Carpenter, J. R.(2018). Measuring the dissipationrate of turbulent kinetic energyin strongly stratified, low-energyenvironments: A case study fromthe Arctic Ocean. Journal ofGeophysical Research: Oceans, 123.https://doi.org/10.1029/2017JC013731Received 21 DEC 2017Accepted 21 JUN 2018Accepted article online 2 JUL 2018ยฉ2018. American Geophysical Union.All Rights Reserved.SCHEIFELE ET AL. 1Journal of Geophysical Research: Oceans 10.1029/2017JC013731in stratified low-energy environments. Our present study is therefore timely since turbulent mixing estimatesfrom microstructure measurements have become a key component in estimating heat fluxes through theBeaufort Sea thermocline.Both shear and temperature microstructure measurements are frequently used to estimate the dissipationrate ๐œ–, a quantity that characterizes the intensity of turbulent flows and can range overmore than 10 orders ofmagnitude in the ocean (Gregg, 1999; Lueck et al., 2002). Ours is not the first study to compare estimates of ๐œ–from coincident shear and temperature microstructure measurements: similar comparisons were performedbyOakey (1982), Kocsis et al. (1999), andPeterson andFer (2014). These three studies all foundexcellent agree-ment, generally within a factor of 2, between shear- and temperature-derived estimates. Our study, however,is distinct because we focus on comparing ๐œ– estimates at the very low end of reported values where shearprobes in particular operate at their lower sensitivity limit. The three previous comparative studies examinedprimarily dissipation rates in excess of 10โˆ’10 W/kg. We will demonstrate that it is necessary to resolve dissipa-tion rates lower than this in the Beaufort Sea, that ๐œ– estimates from shear and temperature measurements nolonger agree at these small values, and that this disagreement can lead to serious biases in the resulting mix-ing rate estimates in low-energy environments. In addition, wewill demonstrate the way in which turbulencespectra diverge systematically from commonly used reference shapes when turbulence becomes weak andstratification becomes strong.We attribute this divergence to a breakdownof the assumption that turbulencein the flow is stationary, homogeneous, and isotropic.2. Measurements2.1. Measurement Platform: Slocum GliderThe platform for our microstructure measurements was the 1,000-m-rated Slocum G2 ocean glider Comet,one of the gliders also used by Schultze et al. (2017). The glider samples autonomously in a vertical saw-tooth pattern, surfacing at predetermined intervals to establish a satellite link in order to update its positionestimate, send low-resolution flight and hydrography data, and receive updated sampling instructions froman onshore pilot. For a detailed review of the operation and utility of gliders, see Rudnick (2016).The gliderโ€™s onboard sensors include an SBE-41 (pumped) Seabird CTD measuring in situ conductivity,pressure, and temperature; a three-dimensional compass module measuring heading, pitch, and roll; andan altimeter measuring height-above-bottom. The turbulence measurements are taken with a specialized,externally mounted instrument, described in section 2.2. A moveable weight that controls the pitch of theglider was set to fixed positions and only moved during inflections at the top and bottom of profiles to avoidmechanical vibrations that affect the quality of turbulence measurements midprofile (Fer et al., 2014).The first published use of gliders as a platform for microstructure measurements is in the proof-of-conceptstudy by Wolk et al. (2009). Gliders have since successfully demonstrated their utility for microstructure mea-surements in studies by Fer et al. (2014), Peterson and Fer (2014), Palmer et al. (2015), and Schultze et al. (2017).These have shown that gliders are suitable low-noise platforms providing microstructure measurements ofcomparable quality to those obtained from free-falling profilers.Gliders are able to provide continuous measurements during a deployment, yielding a spatial and tempo-ral coverage in oceanic microstructure fields that is often unattainable from ship-based profiling, especiallyin inclement weather. The high density and large number of measurements obtained from Comet is animportant feature for our study because it allows us to calculate robust statistical measures of turbulencemetrics which are critical to interpreting microstructure measurements (Gibson, 1987); these have beenlargely unavailable from previous studies in the western Arctic Ocean where microstructure measurementsare sparse.2.2. Shear and Temperature MicrostructureComet is equipped with an externally mounted turbulence package (MicroRider) carrying two airfoil veloc-ity shear (SPM-38) and two fast-response temperature (FP07) probes. The shear probes are of the design byOsborn (1974) and sense transverse forces in a direction perpendicular to the direction of travel (Lueck et al.,2002; Osborn & Crawford, 1980). The probes are oriented such that each measures a distinct shear compo-nent, orthogonal to that measured by the other (as in Fer et al., 2014). The temperature probes are sensitivethermistors with response times of โˆผ10 ms and sensitivities better than 0.1 mK (Sommer, Carpenter, Schmid,Lueck, &Wรผest, 2013). Themicrostructure probes extend beyond the nose of the glider byโˆผ17 cm, outside ofSCHEIFELE ET AL. 2Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 1. (a) The measurement location at the entrance to the Amundsen Gulf in the southeastern Beaufort Sea.The glider path is shown by the line inside the dashed black rectangle. Bathymetry contours are drawn at 1,000-mintervals beginning at 200 m. (b) Enlarged view of the area inside the dashed rectangle indicated in panel a, showingthe glider path and the local bathymetry. Selected waypoints along the path are numbered consecutively and indicatedby squares for reference when reading Figures 2 and 3. Contours are drawn at 75-m intervals beginning at 50 m. In bothpanels, color indicates water depth (m); bathymetry data are from the International Bathymetric Chart of the ArcticOcean 3.0 (Jakobsson et al., 2012). Mean vertical profiles of (c) conservative temperature, (d) in situ density anomaly, and(e) buoyancy frequency are also shown; these are horizontally averaged over all casts where the gliderโ€™s maximum divedepth exceeded 325 m.the radius of flow deformation caused by the gliderโ€™s motion (Fer et al., 2014). We do not install a probe guardin order tominimize thepotential for contaminationof the flow in the immediate vicinity of themeasurement.Besides shear and temperature, the MicroRider also measures pressure, pitch, roll, and transverse accel-erations. Shear, temperature, and acceleration are sampled at 512 Hz, the other channels at 64 Hz. TheMicroRider is produced by Rockland Scientific International (RSI); it is the same model used in the fourglider-microstructure studies referenced in the previous section.2.3. Location, Local Hydrography, and Sampling StrategyThe measurement location was the Amundsen Gulf on the southeastern margin of the Beaufort Sea(Figure 1a). Circulation in the region is complex and highly variable in space and time; it is strongly influ-enced by surface wind stress, complex local bathymetry, submesoscale frontogenesis, and the intermittentpresence of mesoscale eddies (Sรฉvigny et al., 2015; Williams & Carmack, 2008, 2015). Barotropic tidal currentscan be regionally strong but are locally weak where the glider was deployed because of the presence of alocal amphidrome (Kowalik & Matthews, 1982; Kulikov et al., 2004). The presence of sea ice is seasonal; dur-ing our campaign, the southern edge of fragmented sea ice was coincident with themouth of the AmundsenGulf, prohibiting us from guiding the glider north toward the central Beaufort Sea as initially planned. TheAmundsen Gulf was consequently selected for the measurement locale to mimic as closely as possible thehydrographic characteristics of thewider Beaufort Seawhileminimizing the risk of collisionswith sea ice floes.The basin depth in Amundsen Gulf is โˆผ450 m (Figure 1b), well below the typical Beaufort Sea shelf-breakdepth of โˆผ75 m and deep enough to extend across the entire range of the thermocline separating Atlantic-and Pacific-sourced water masses. As a result, the hydrography of the region largely reflects that of thebroader Beaufort Sea (Figures 1cโ€“1e). A 10-m-thick brackish surface lens, resulting from summer sea icemeltand the nearby Mackenzie Riverโ€™s freshwater inflow, caps a near-surface pycnocline that extends to 25-mdepth and has a density anomaly ๐œŽ that ranges between 22 and 24.5 kg/m3. Between โˆผ25 and 200 m, thesignatures of cold Pacific-sourced water dominate the mean temperature profile leading to a temperatureminimum ofโˆ’1.4โˆ˜C at 120-m depth; a complex submesoscale temperature structure is notably visible in thislayer andmodifies themean profile betweenโˆผ40- and 110-m depth. A prominent thermocline characteristicSCHEIFELE ET AL. 3Journal of Geophysical Research: Oceans 10.1029/2017JC013731of the Beaufort Sea extends from 125 m to the temperature maximum associated with the warm core ofAtlantic-sourced water (Rudels, 2015; Williams & Carmack, 2015) at 375-m depth. Stratification is strongthroughout the water column, with buoyancy frequency N of O(10โˆ’2) sโˆ’1 in the near-surface pycnocline andO(10โˆ’3) sโˆ’1 elsewhere.The glider sampled continuously between 25 August and 4 September 2015, following the path outlined inFigure 1. It measured 348 quasi-vertical profiles over a total path length of 186 km, remaining in the deepercentral Amundsen Gulf for the first 4.5 days and partially crossing the continental shelf slope three timesduring the remainder of the mission. The glider dove to a fixed depth of 300 m during the first 3 days; afterthis, it dove to within 15 m of the local bottom.3. Data ProcessingProcessing microstructure measurements from a glider is similar to processing ones from a free-falling pro-filer butwith added complications. Estimating the speed of themicrostructure probes throughwater requiresspecialized procedures, as does screening the data for corruptmeasurements. Becausemeasuring turbulencefrom gliders is still a novel technique, we outline here in detail the steps we take to go from microstruc-ture measurement to dissipation rate estimate, including our procedure for estimating the gliderโ€™s velocityunderwater. Thequality control criteriaweuse toflaganddiscard suspectmeasurements, a comparisonof dis-sipation rate results from upcasts and downcasts, and a brief description of Nasmyth and Batchelor referencespectra are provided in the Appendix.Throughout the text, the symbol ๐œ– is used for the dissipation rate generally, ๐œ–U for dissipation rate estimatesobtained from velocity shearmeasurements, and ๐œ–T for dissipation rate estimates obtained from temperaturemeasurements. All wavenumbers are defined cyclicly, with units cpm. Note that the cyclic wavenumber, k,is related to the radian wavenumber, kฬ‚, which has units mโˆ’1, through the relation k = kฬ‚โˆ•2๐œ‹. The kinematicviscosity, ๐œˆ, of seawater is evaluated locally using TEOS-10 (McDougall & Barker, 2011) because it varies bymore than 20% in our measurements. We use DT = 1.44 ร— 10โˆ’7 m2/s for the molecular diffusion coefficientof temperature and qB = 3.4 for the Batchelor constant; the latter is required when evaluating the Batchelorspectrum (section 3.3), and a sensitivity analysis for this parameter is presented in Appendix C. Measurementsfrom the MicroRiderโ€™s clock, pressure sensor, and temperature probes are prone to low-frequency drift; thus,the low-frequency response from each of these channels is corrected to measurements from the glider. Notethat unless otherwise stated, we average quantities that spanmany orders ofmagnitude using the geometricmean, and we use the term trimmedmean to refer to an average calculated over the central 90% of data.3.1. Glider Velocity EstimatesThe processing of microstructure measurements to obtain dissipation rates is heavily reliant on accurateknowledge of the speed,U, withwhich the probes travel throughwater. Unfortunately, there is no directmea-surement of the gliderโ€™s speed underwater: the glider pitch, ๐œƒ, and rate-of-change of pressure are known, butthis is not enough information to directly obtain U because the glider travels with an unknown and variableangle of attack, ๐›ผ, which in our experience is usually in the range 1โˆ˜ < |๐›ผ| < 10โˆ˜.Studies by Fer et al. (2014), Peterson and Fer (2014), and Palmer et al. (2015) use a hydrodynamic flight modeldeveloped by Merckelbach et al. (2010) to estimate U. The model assumes a steady state balance of drag,buoyancy, and lift forces to optimize estimates of U and a drag coefficient CD0 . The angle of attack is thenobtained numerically from the implicit relation๐›ผ = โˆ’(CD0 + CD1๐›ผ2A tan ๐›พ)(1)where ๐›พ = ๐œƒ โˆ’ ๐›ผ is the glide angle, and CD1 and A are constants optimized for Slocum gliders in Merckelbachet al. (2010).In contrast to this approach, we follow the method of Schultze et al. (2017) and use the steady state modelof Merckelbach et al. (2010) to obtain the angle of attack but then use the measured pitch and pressure toestimate U dynamically usingU = Wsin ๐›พ, (2)SCHEIFELE ET AL. 4Journal of Geophysical Research: Oceans 10.1029/2017JC013731Table 1Meanยฑ 1 Standard Deviation of Glider-Flight Variables From All Upcasts and Downcasts of theMissionProfile Type ๐œƒ (โˆ˜) ๐›ผ (โˆ˜) ๐›พ (โˆ˜) U (cm/s)Upcasts 21.8 ยฑ 1.0 โˆ’4.7 ยฑ 0.2 26.4 ยฑ 0.9 41 ยฑ 4Downcasts โˆ’21.3 ยฑ 0.8 4.8 ยฑ 0.1 โˆ’26.0 ยฑ 0.6 25 ยฑ 4Note. ๐œƒ is the measured pitch, ๐›ผ the estimated angle of attack, ๐›พ the estimated glide angle, and U the estimated speedthrough water. Only data coincident with at least one viable ๐œ– estimate (see Appendix A) are included.whereW is the gliderโ€™s vertical velocity estimated from the measured rate-of-change of pressure. We foundthat this quasi-dynamic estimate of U leads to more consistent results between profiles of ๐œ– from upcastsand downcasts. Note that angles are measured positive upward, ๐œƒ and ๐›พ relative to the horizontal, ๐›ผ relativeto the glide angle ๐›พ .Mean and standard deviation values of selected glider flight characteristics separated by upcasts and down-casts are summarized in Table 1 to enable comparison with previous studies. For the most part, the valuespresented are not remarkable and are similar to ones previously reported, albeit withmarginally larger anglesof attack. One exception is the relatively large discrepancy in U between upcasts and downcasts. The discrep-ancy arises because of the strong near-surface stratification in the Beaufort Sea, resulting in asymmetric diveand climb rates over most of the water column; however, we do not see a significant systematic effect on thedissipation rate results (see Appendix B), and we do not differentiate between upcasts and downcasts fromhere on.3.2. Microstructure ShearThe procedure we use to process the shear measurements uses code provided by RSI and is based onrecommendations outlined in their documentation. We provide an overview here; a comprehensive rationalefor the algorithm and detailed review of recommended procedures are available in RSIโ€™s Technical Note 028(Lueck, 2016).We calculate the dissipation rate from the viscosity and the variance of the turbulent velocity shear accord-ing to ๐œ–U = 7.5๐œˆโŸจ(๐œ•uโ€ฒโˆ•๐œ•x)2โŸฉ, assuming isotropic flow. Here angled brackets indicate averaging, x representsthe gliderโ€™s along-path coordinate, and uโ€ฒ represents either of the two perpendicular turbulent velocity com-ponents. As discussed in section 5.3, the isotropy assumption is problematic when energetics are weak andstratification is strong, and it leads to increaseduncertainty in theobserveddissipation rates, but it is necessaryto make the assumption because we measure only two of the nine strain rate tensor components.We estimate the variance from the measured shear record in half-overlapping 40-s segments. Each of theseis further subdivided into 19 half-overlapping 4-s subsegments which are detrended, cosine windowed (in avariance-preserving manner), and transformed into shear power spectra using a fast Fourier transform (FFT).These 19 spectra are averaged to create one observed shear power spectrum, ฮฆ, for each 40-s segment ofshearmeasurement. Coherent acceleration signalsmeasuredby theMicroRider are removed fromฮฆusing thealgorithm proposed by Goodman et al. (2006). Frequencies, f , are transformed to wavenumbers, k, using thegliderโ€™smean speed,U, over the 40 s and assuming Taylorโ€™s frozen turbulence hypothesis, that is,ฮฆ(k) = Uฮฆ(f )and k = fโˆ•U.The 4-s length of the subsegments passed to the FFT sets the scale for the largest wavelengths(smallest wavenumbers) included in the shear spectrum ฮฆ(k); it is identical to the FFT length chosen byFer et al. (2014) and Schultze et al. (2017). Given the average glider speeds in Table 1, a typical FFT calcu-lation includes along-path wavelengths as large as 164 cm on upcasts and 100 cm on downcasts, resolv-ing the low-wavenumber transition between the inertial and viscous ranges of the turbulence spectrum(Lueck et al., 2002). The choice of 40 s for the total averaging length, corresponding on average to 16.4 m onupcasts and 10mondowncasts, is larger than the 12-s averaging length used in the abovementioned studies;it is a heuristic choice and a compromise which trades a decrease in the spatial resolution of the observationsin favor of an increase in the statistical confidence of individual ๐œ–U estimates (Lueck, 2016).We numerically integrate eachฮฆ(k), calculating ๐œ–U according to๐œ–U = 7.5๐œˆ โˆซku0ฮฆ(k)dk. (3)SCHEIFELE ET AL. 5Journal of Geophysical Research: Oceans 10.1029/2017JC013731Here, ku is an upper integration limit, chosen to exclude large wavenumbers at which electronic noise domi-nates the measurement. To choose ku, we fit a third-order polynomial toฮฆ(k) in order to isolate the locationof the spectral minimumwhich typically indicates the onset of noise domination, but we constrain ku to be atleast 7 cpm. Note that in a low-energy environment most of the variance is at low wavenumbers: 90% of thevariance lies below 7 cpmwhen ๐œ– = 10โˆ’10 W/kg (Gregg, 1999).To account for unresolved variance, we calculate the fraction, P, of the integral of the nondimensionalizedempirical Nasmyth spectrum (Nasmyth, 1970; Oakey, 1982) that is resolved below the nondimensionalizedintegration limit kuโˆ•(๐œ–Uโˆ•๐œˆ3)1โˆ•4. We then scale up ๐œ–U by a factor of 1โˆ•P and iterate the correction procedureuntil the change in ๐œ–U in successive iterations is less than 2%.We further correct for a small integration underestimate that occurs between the origin and the first nonzerowavenumber, k1. The Nasmyth spectrum rises approximately as k1โˆ•3, and so its integral to k1 is proportionalto (3โˆ•4)k4โˆ•31 ; trapezoidal integration between k = 0 and k1, however, is proportional only to (1โˆ•2)k4โˆ•31 . Wecorrect by adding the term 7.5๐œˆ(1โˆ•4)k1ฮฆN(k1), whereฮฆN(k1) is the value of the Nasmyth spectrum at k1. Notethat the two correction procedures described here are both standard features implemented in the providedRSI codes.There are two distinct shear probes (section 2.2), yielding two independent, simultaneous ๐œ–U estimates.Whenboth estimates pass quality control (see Appendix), we average them; when only one passes quality control,we use that single estimate for the analysis presented in section 4. Note that, on average, we do not see ameaningful difference between results from the two shear probes (see Appendix A).3.3. Microstructure TemperatureThe dissipation rate ๐œ–T may be estimated from temperature microstructure measurements by determiningthe Batchelor wavenumber, defined kB = (1โˆ•2๐œ‹)(๐œ–Tโˆ•๐œˆD2T )1โˆ•4, and inverting to yield๐œ–T = ๐œˆD2T (2๐œ‹kB)4. (4)We determine the Batchelor wavenumber by fitting the theoretical Batchelor spectrum (Batchelor, 1959) toobserved power spectra of temperature gradients using the procedure outlined below, which is modeled ondescriptions by Ruddick et al. (2000), Steinbuck et al. (2009), and Peterson and Fer (2014).Wefirst calculate a temperaturepower spectrum,ฮ›, from the temperaturemeasurements for eachof the samehalf-overlapping 40-s segments that we used to calculate the shear spectra. Each spectrum is again the aver-age of 19 spectra calculated fromhalf-overlapping, detrended and cosinewindowed 4-s subsegments. Valuesofฮ› at high frequencies, where the temperature probesโ€™ temporal response is inadequate, are corrected usingthe transfer function proposed by Sommer, Carpenter, Schmid, Lueck, and Wรผest (2013). Like shear spec-tra, temperature spectra are transformed from frequency to wavenumber space using U and Taylorโ€™s frozenturbulence hypothesis.From each ฮ›, we next calculate a raw one-dimensional temperature-gradient power spectrum,ฮจr, using thevariance-preserving transformationฮจr = (2๐œ‹k)2ฮ› . (5)From each of these we then subtract a probe-specific noise spectrum:ฮจ = ฮจr โˆ’ ฮจns , (6)where ฮจns is the noise spectrum, empirically determined for each probe by averaging the 1% of raw spectrawith the least observed variance. We refer toฮจ as the observed temperature-gradient spectrum. Note that thetemperature gradients are defined with respect to the along-glider path coordinate.We next estimate the rate, ๐œ’ , of destruction of temperature-gradient variance (Osborn & Cox, 1972) from theobserved temperature-gradient spectra. Following Steinbuck et al. (2009), we iteratively calculate๐œ’ = ๐œ’lw + ๐œ’obs + ๐œ’hw = 6DT(โˆซkl0ฮจBdk + โˆซkuklฮจdk + โˆซโˆžkuฮจBdk), (7)SCHEIFELE ET AL. 6Journal of Geophysical Research: Oceans 10.1029/2017JC013731on each iteration subsequently fitting the Batchelor spectrum, ฮจB, to the observed spectrum as describedbelow. The term ๐œ’obs is the component of ๐œ’ that comes from integrating the observed spectrum betweenwavenumbers kl and ku. At wavenumbers outside this range, the observed spectrum is not reliable andwe instead integrate ฮจB to obtain the correction terms ๐œ’lw and ๐œ’hw. Note that the correction terms areunavailable and thus set to zero for the first iteration. The factor of 6 arises from assuming isotropic flow.We fit the Batchelor spectrum between wavenumbers kl and ku on each iteration using the maximum likeli-hood estimation (MLE) procedure described by Ruddick et al. (2000). This procedureminimizes a cost functionto choose the best fit froma family of Batchelor curveswhich are constructed using constant๐œ’ but variable kB.For the upper limit ku we choose the intersection betweenฮจr and 2ฮจns. The lower limit kl is the smallest avail-able nonzero wavenumber k1 on the first iteration, and on subsequent iterations is the greater of k1 and 3kโˆ—,where kโˆ— = 0.04kB(DTโˆ•๐œˆ)1โˆ•2 represents the top of the convective subrange (Luketina & Imberger, 2001). Weimplement three iterations, enough for kB estimates to converge (Steinbuck et al., 2009), and then calculate๐œ–T from equation (4).There are two distinct thermistors (section 2.2), yielding two independent, simultaneous ๐œ–T estimates. Aswith the shear-derived estimates, when both pass quality control (see Appendix), we average; when only onepasses, we use the single estimate for our analysis.4. Comparison of Results From Temperature and Shear MicrostructureHere we present ๐œ– estimates derived from our coincidentmicrostructuremeasurements of shear and temper-ature, demonstratinghow the twoestimates agree on averagewithin a factor of 2when ๐œ– > 3ร—10โˆ’11 W/kgbutdiverge for smaller dissipation rates. We demonstrate that this divergence leads to inconsistencies betweenstatisticalmetrics that describe the two sets of observations. Using evidence presented in sections 4.3 and 4.4,and in anticipation of the discussion presented in section 5.1, we attribute differences between the ๐œ–U and ๐œ–Tdata sets to the effects of the ๐œ–U noise floor. With this foreknowledge, our description of these differences canbe interpreted as a case study that demonstrates the degree by which the ๐œ–U noise floor influences the abilityof the shear measurements to characterize the dissipation rate in a stratified, low-energy environment.4.1. Spatial Cross SectionsA qualitative comparison of the dissipation rates derived from shear and temperature microstructure mea-surements is generally favorable. This can be seen in spatial cross sections of ๐œ–U and ๐œ–T (Figure 2). Both fieldsexhibit obvious variability over at least 3 orders of magnitude and indicate the same coherent patches ofenhanced turbulence superimposed on a less turbulent background. In both fields, these patches are charac-terized by dissipation rates O(10โˆ’9) W/kg. They have a spatial coherence on scales O(10)โ€“O(100) m verticallyandO(10) kmhorizontally. Three easily identifiable examples, seen in bothpanels, are between approximately(i) 10โ€“20 km at depths 105โ€“305 m, in the central Amundsen Gulf; (ii) 52โ€“81 km at depths 155โ€“400 m, atthe edge of the shelf slope; and (iii) 161โ€“183 km within a 75-m band above the sea floor, on the shelf slope.These are indicated in the figure with magenta rectangles. The qualitative similarity between the two inde-pendently derived estimates is encouraging and suggests that temperature and shear probes may be usedto qualitatively identify the same regions of enhanced turbulence.Figure 2, however, also reveals an immediate difference between the observed ๐œ–U and ๐œ–T fields: these indicatemarkedlydifferentbackground states, easily seen in thefigurebecause the images in the twopanels aredrawnwith the same color scale. The ๐œ–U field from the shearmeasurements indicates a background ofO(10โˆ’11)W/kg,imaged as a turquoise blue. There appears to be no obvious variability below ๐œ–U โˆผ 5 ร— 10โˆ’11 W/kg. The๐œ–T field from the temperature measurements, on the other hand, suggests a lower background value ofO(10โˆ’12) W/kg, as indicated by the frequent darker blue colors. Additional structure not seen in the ๐œ–U obser-vations is apparent in the ๐œ–T field at dissipation rates between 1 ร— 10โˆ’12 and 5 ร— 10โˆ’11 W/kg. An example ofthis phenomenonmay be seen by looking at the dissipation rate signature of what appears to be amesoscaleeddywhose presence canbe identified in the temperature anddensity fields (not shown) at depths 40โ€“100mbetween 52โ€“87 km: it carries an obvious signature of enhanced dissipation in the ๐œ–T field (Figure 2b, dashedmagenta box), but in the ๐œ–U field (Figure 2a) no such signature can be identified.The magnitude of the discrepancy between the two fields is apparent when visualizing a cross section of theratio ๐œ–Uโˆ•๐œ–T (Figure 3). The discrepancy is largest, at times larger than a factor of 103, in a band approximatelyat 50- to 150-m depth and between 100โ€“186 km. Comparing with Figure 2, this region tends to coincideSCHEIFELE ET AL. 7Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 2. Cross sections of the turbulent dissipation rate, ๐œ–, in log10 space, derived from microstructure measurementsof (a) shear and (b) temperature. The panels are drawn using the same color scale. Gray shading indicates thebathymetry, black shading discarded or unavailable data (see section 2.3 and the Appendix). Small white lines along thehorizontal axis indicate the locations of individual profiles. The breaks in the horizontal axis, labeled 1โ€“4, correspond tothe waypoints shown in Figure 1b. Magenta rectangles with solid white lines indicate regions of enhanced dissipationdiscussed in the text. The magenta rectangle with dashed white line in panel (b) indicates the signature of themesoscale eddy discussed in the text.with the region where ๐œ–T is smallest. The discrepancy is less pronounced, however, in the three patches ofenhanced turbulence identified in Figure 2; here the ratio ๐œ–Uโˆ•๐œ–T tends to unity.4.2. Mean Vertical ProfilesWe find, on average, more than an order of magnitude difference between ๐œ–U and ๐œ–T where dissipation ratesare smallest. This can be seen in Figure 4a, in which the observed dissipation rate fields are horizontallyaveraged in 25-m vertical bins to create mean vertical profiles of ๐œ–U and ๐œ–T . In both profiles, the lowest dis-sipation rates are found between 100- and 125-m depth where, on average, ๐œ–U = 4 ร— 10โˆ’11 W/kg while๐œ–T = 3ร—10โˆ’12 W/kg; there is a factor of 13disagreementbetween the two. This disagreement is statistically sig-nificant: in this depth bin, the respective interquartile range for each data set is ๐œ–U = 2ร—10โˆ’11 โ€“ 8ร—10โˆ’11 W/kgand ๐œ–T = 3 ร— 10โˆ’13 โ€“ 3 ร— 10โˆ’11 W/kg, indicating little overlap between the two measurement distributions.The geometric standard deviation factors ๐œŽg are 2.4 for ๐œ–U and 13.4 for ๐œ–T , reflecting the substantial horizontalvariability seen in Figure 2 and the larger range of ๐œ–T values. Despite the variability, the estimates of themeanvalues are robust: the 95% confidence intervals indicated by the geometric standard error (Kirkwood, 1979)are ๐œ–U = 4 ร— 10โˆ’11 โ€“ 5 ร— 10โˆ’11 W/kg and ๐œ–T = 3 ร— 10โˆ’12 โ€“ 4 ร— 10โˆ’12 W/kg, respectively.Figure 3. Cross section of the ratio ๐œ–Uโˆ•๐œ–T in log10 space. Shading, panel division, magenta rectangles, and annotationsas in Figure 2.SCHEIFELE ET AL. 8Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 4. (a) Average vertical profiles of the dissipation rates ๐œ–U and ๐œ–T , obtained from shear and temperaturemicrostructure and calculated using a trimmed geometric mean in 25-m vertical bins. Shading indicates the 95%confidence interval for the mean as indicated by the geometric standard error. (b) The ratio of the average verticalprofiles of ๐œ–U and ๐œ–T , highlighting disagreement by a factor of 5 or greater between 75- and 175-m depth.The discrepancy between the depth-binned average profiles is less dramatic in the rest of the water column,and the two profiles qualitatively have a similar shape. Both exhibit small dissipation rates below 10โˆ’10 W/kgin the shallowest available bin (25โ€“50m), decreasing further to their distinctminima between 100 and 125mand then gradually increasing with depth to maximum average values near 10โˆ’10 W/kg as they approach theseabed. Disagreement between the geometric mean values, imaged in Figure 4b, is a factor of 5 or greaterbetween75- and175-mdepthand smaller everywhere else.Wheneverbothgeometricmeanvalues are simul-taneously at least 3 ร— 10โˆ’11 W/kg, the agreement between them is better than a factor of 2, highlighting thatthe divergence between ๐œ–U and ๐œ–T occurs only at very low dissipation rates.Note that ourmeasurements tend to exhibit small ๐œ– relative tomeasurements fromother regions of the globalocean, which often exhibit typical averaged values of O(10โˆ’9) W/kg and higher (Waterhouse et al., 2014).This incongruity, however, is not surprising: small dissipation rates are anticipated in thewestern Arctic Oceanwhere turbulence is thought to be exceptionally low because of limited energy input and seasonal sea iceFigure 5. Histograms showing the distributions of all (a) ๐œ–U and (b) ๐œ–T observations. The interquartile range isindicated by the darker shading; the mode, arithmetic and geometric means, and median are marked in both panelsaccording to the legend in (a). The labels N and ๐œŽg indicate the total number of observations in each histogram andthe geometric standard deviation factor, respectively. Histograms are calculated over 100 logarithmically spaced bins.(c) Quantile-quantile plot demonstrating the goodness of fit of the histograms to idealized lognormal distributions. Foreach set of data, deciles are marked by gray-shaded circles, and the squared linear correlation coefficient, R2, is indicated.SCHEIFELE ET AL. 9Journal of Geophysical Research: Oceans 10.1029/2017JC013731Table 2Statistical Parameters of the ๐œ–U and ๐œ–T Distributions Shown in Figure 5Data Set N Mode G. Mean Median P25 P75 A. Mean ๐œŽg๐œ–U 28,575 2.7 6.5 4.6 2.5 13 25 3.7๐œ–T 21,577 1.2 1.6 1.7 0.27 12 61 18.3Note.Given, from left to right, are thenumber,N, of observations;mode; geometricmean;median; first and thirdquartiles,P25 and P75; arithmetic mean; and geometric standard deviation factor, ๐œŽg . The quantities N and ๐œŽg are dimensionlessand unscaled. All other quantities are scaled by a factor of 10โˆ’11 W/kg.cover (Rainville &Woodgate, 2009). Microstructuremeasurements from thewestern Arctic are very sparse butthose that do exist (section 1) have so far indicated typical background dissipation rates of O(10โˆ’10) W/kg.4.3. Distributions of ๐U and ๐TThe histograms of all ๐œ–U and ๐œ–T observations (Figures 5a and 5b) provide further insight into the discrepanciesbetween the two data sets. The histograms have markedly different shapes despite being constructed fromcoincident sets of measurement. Most notably, the distribution of ๐œ–T observations is nearly symmetric withonly small negative skew (skewness, sg = โˆ’0.2) in log10 space, while the ๐œ–U distribution is skewed positiveand more heavily (sg = 1.2). Statistical properties that can be used to further compare the distributions aretabulated in Table 2 and indicated in Figures 5a and 5b. Of note are the larger geometric mean and median๐œ–U values, reflecting the relative absence of very small ๐œ–U observations; the separation between the medianand geometric mean of ๐œ–U, reflecting the skewness of that distribution; and the wider interquartile range of๐œ–T , reflecting the larger variability of the ๐œ–T observations.The distributions may be further contrasted using a quantile-quantile (Q-Q) plot (Figure 5c) to quantify howsimilar each distribution is to an idealized lognormal oneโ€”themore linear the plot, the greater the similarity.From this visualization, it is clear that the distribution of ๐œ–T observations can be described as lognormal overall ๐œ– except below the second decile (2 ร— 10โˆ’12 W/kg). In contrast, the distribution of ๐œ–U observations may bedescribed as lognormal only above the seventh decile (1 ร— 10โˆ’10 W/kg). The strong positive curvature in theQ-Q plot for ๐œ–U below the seventh decile indicates that there is substantially less weight on the left side of theobserved distribution relative to an idealized lognormal one. The slight negative curvature in the Q-Q plotfor ๐œ–T below the second decile indicates a small trend in the opposite direction, that is, a marginally heaviertail on left side relative to an idealized lognormal distribution. The squared linear correlation coefficients ofthe Q-Q plots are R2 = 0.917 for ๐œ–U and R2 = 0.995 for ๐œ–T , confirming the qualitative impression that the ๐œ–Uobservations deviate more strongly from an idealized lognormal distribution.We attribute the discrepancy in the shapes of the ๐œ–U and ๐œ–T histograms primarily to the sensitivity limit of theshear probes, which imposes an artificial lower limit (or noise floor) on the ๐œ–U observations. This noise floor willskew the histogram of ๐œ–U observations positive by distributing samples that would otherwise be recorded assmaller values within a narrow range around the lower limit. Given the distinctive peak in the ๐œ–U histogramand the extremely rapid roll-off to the left of the peak, we simply use the mode to approximate the noisefloor as 3 ร— 10โˆ’11 W/kg. The ๐œ–U observations that fall below this estimate of the noise floor are in the range(1 โ‰ค ๐œ–U < 3) ร— 10โˆ’11 W/kg (Figure 5a); we attribute this statistical scatter to errors in the data processingwhichmay, in part, arise because of uncertainty surrounding the characteristics of weakly turbulent, stronglystratified flows (section 5.2). The range of values below the noise floor suggests uncertainty here within afactor of 3, implying that the effects of the noise floor begin to skew the ๐œ–U distribution at โˆผ9 ร— 10โˆ’11 W/kg.This is consistent with the trend in the Q-Q plot (Figure 5c) where the shape of the ๐œ–U histogram begins todiverge from that of the ๐œ–T histogram at dissipation rates below โˆผ1 ร— 10โˆ’10 W/kg.Previous studies using loosely tethered profilers often cite a noise floor of O(10โˆ’10) W/kg for shear-derivedobservations (e.g., Fer, 2014; Gregg, 1999; Lincoln et al., 2016; Shroyer, 2012; Wolk et al., 2002), though twoglider-based studies that incorporated microstructure shear measurements both quote 5 ร— 10โˆ’11 W/kg forthe noise floor (Fer et al., 2014; Wolk et al., 2009). The presence of an ๐œ–U noise floor has practical ramificationsfor the interpretation of microstructure shear measurements; these are particularly important to consider inlow-energy environments, and we discuss them further in section 5.4.SCHEIFELE ET AL. 10Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 6. Scatter plot comparison of the two coincident dissipation rateestimates ๐œ–U and ๐œ–T . Identical agreement and agreement within factorsof 2 and 5 are indicated as labeled. Bin averages are calculatedperpendicular to the one-to-one line (see text). Our empirical estimateof the ๐œ–U noise floor (3 ร— 10โˆ’11 W/kg) is indicated by the horizontaldotted line. Purple shading indicates where both estimates of ๐œ–simultaneously lie above 3 ร— 10โˆ’11 W/kg and also delineates theregion where bin averages agree within a factor of 2.4.4. One-to-One Comparison of ๐U and ๐TA simple scatter plot of the coincident ๐œ–U and ๐œ–T observations (Figure 6)elucidates how the agreement between the two varies over the range ofthe observed dissipation rates. If one considers only observations wherethe two ๐œ– estimates are simultaneously greater thanour empirical estimateof the ๐œ–U noise floor (3ร— 10โˆ’11 W/kg, section 4.3), the agreement betweenthe two sets of observations is encouraging. This subset of data is indicatedin Figure 6 by the purple-shaded region. Here the cloud of individual mea-surements largely scatters around the one-to-one line: 88% of these 8,064observation pairs agree within a factor of 5, and 53% agree within a factorof 2. More importantly, the bin averages that are shown (defined below)always agreewithin a factor of 2. This level of agreement is consistent withthe factor of 2 agreement in themean vertical profiles (Figure 4) wheneverthose averages indicate ๐œ– > 3 ร— 10โˆ’11 W/kg in both estimates. Statisticalagreement within a factor of 2 is comparable to the best agreement seenin other studies (e.g., Kocsis et al., 1999; Peterson & Fer, 2014).When at least one of the ๐œ– estimates is less than 3 ร— 10โˆ’11 W/kg, statis-tical disagreement between the shear- and temperature-derived dissipa-tion rates becomes concerning. Here only 22% of the 16,842 observationpairs agree within a factor of 5, and only 6% agree within a factor of 2.The data diverge systematically from the one-to-one line: as the ๐œ–T esti-mates continue to decrease, the ๐œ–U estimates asymptote to a lower limit ofapproximately 2 ร— 10โˆ’11 W/kg, marginally below but still consistent withour estimate of the ๐œ–U noise floor. The bin averages indicate the same pat-tern as the individual measurements: below ๐œ–T = 1 ร— 10โˆ’11 W/kg they exhibit disagreement greater than afactor of 5 and, even in this averaged sense, suggest disagreement greater than 2 orders of magnitude when๐œ–T is less than 2 ร— 10โˆ’13 W/kg.Note that averaged measures like bin averages are more appropriate than individual measurementswhen evaluating the agreement between ๐œ– estimates because we expect substantial statistical scatter(within about an order of magnitude) in these estimates. This scatter is, in part, attributed to uncertaintiessurrounding the validity of the isotropy, homogeneity, and stationarity assumptions inherent in the data pro-cessing (see section 5.2). The bin averages shown in Figure 6 are averages calculated from the trimmedmeanin logarithmically spaced bins that lie perpendicular to the one-to-one line, that is, in a coordinate systemrotated 45โˆ˜ clockwise from that shown. Defining the bins in this manner helps minimize biases in the aver-age by assuming roughly equal uncertainty in both variables, similar in principle to a bivariate least squaresminimization (Ricker, 1973).5. DiscussionMeasuring turbulence parameters to estimate the turbulent dissipation rate comes with unique challengesin low-energy environments like the Beaufort Sea, and our results in section 4 demonstrate that the twomostcommonmeansof directly estimating thedissipation rate can yield divergent results that disagreebymultipleorders of magnitude at low ๐œ–. Our results suggest that most of this discrepancy can be attributed to thenoise floor of the shear-derived estimates, but fundamental questions about the nature of marginally turbu-lent, strongly stratified flows also introduce uncertainty into the observations. And, more pragmatically, ourresults highlight questions about how to correctly process and interpret shear microstructure measurementsin such environments since it appears that the majority (about 70%, Figure 5c) of ๐œ–U estimates are skewedby the effects of the noise floor. We address these topics in the following discussion: in section 5.1, we lookmore closely at the effect of sensor limitations on themeasurements and the observed spectra; in section 5.2we discuss averaged observed spectral shapes; in section 5.3 we examine uncertainties that arise from the(potentially unjustified) assumptions needed for the processing of microstructure measurements; and insection 5.4 we discuss practical implications, that is, in which circumstances the difference between ๐œ–U and ๐œ–Tmatters and in which circumstances it can be safely ignored.SCHEIFELE ET AL. 11Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 7. Sample coincident shear (aโ€“f: ฮฆ) and temperature-gradient (gโ€“l: ฮจ) spectra (black) for 6 orders of magnitude of ๐œ–, as determined by the temperaturemeasurements. Bold indicates the wavenumbers explicitly included for integration; the remaining variance is estimated as described in sections 3.2 and 3.3.For ฮจ, bold also indicates the wavenumber range used for the MLE Batchelor fit (see section 3.3). Shear spectra have the accelerometer signal removed(section 3.2) and temperature gradient spectra have the empirically determined noise spectra removed (section 3.3). Nasmyth (aโ€“f ) and Batchelor (gโ€“l)reference spectra (gray) are also drawn. Batchelor spectra are those determined by the MLE fitting algorithm which are used to estimate kB (section 3.3).5.1. The Effect of Sensor LimitationsWe propose that the systematic divergence between ๐œ–U and ๐œ–T that is obvious in Figures 5 and 6 at small dis-sipation rates is primarily a result of the effects of the noise floor of the ๐œ–U estimates. This low-end divergenceis then responsible for the large discrepancies seen in the spatial cross sections (Figures 2 and 3) and meanvertical profiles (Figure 4) of ๐œ–U and ๐œ–T . This interpretation is consistent with the known sensitivity limitationsof microstructure shear probes (Osborn & Crawford, 1980) and previous empirical estimates of the ๐œ–U noisefloor (section 4.3).Assuming vibrations from themeasurement platformdonot contaminate themeasured signal, the noise floorof an ๐œ–U estimate is set by the lower limit of a shear probeโ€™s ability to detect hydrodynamic transverse forcesand distinguish these from electronic measurement noise. Hydrodynamic forces from small-scale velocityshear below this detection limit may still act on the probe, but the signal is either not recorded or is maskedby the instrumentโ€™s electronic noise. As a result (section 3.2), any varianceโŸจ(๐œ•uโ€ฒโˆ•๐œ•x)2โŸฉthat exists below theprobeโ€™s detection limit will yield an ๐œ–U estimate at (or near) the level of the noise floor, irrespective of what thetrue dissipation rate at the instant of themeasurementmay be. If the true dissipation rate is below the level ofthe noise floor in a large proportion of the measurement realizations, this behavior will lead to an artificiallyskewed measurement distribution and a pile-up of observations against a lower limit, that is, a distinct peakin the distribution near the noise floor and a rapid roll-off toward smallerโ€”unresolvedโ€”values, as can beseen in the distribution of our ๐œ–U observations (Figure 5a).The manifestation of the noise floor can also be seen in the observed shear spectra when these are com-pared to the simultaneously observed temperature-gradient spectra. Figure 7 depicts six representative pairsof observed shear and temperature-gradient spectra, distributed over 6 consecutive orders of magnitude of๐œ– (as suggested by the temperature-derived estimates). Each column of panels shows the two coincidentlyobserved spectra ฮฆ(k) and ฮจ(k), defined in sections 3.2 and 3.3. Following panels gโ€“l from right to left, theBatchelor fit to the temperature-gradient spectra (section 3.3) indicates continually decreasing ๐œ–T , as labeledin each panel. The shear spectra indicate a similar ๐œ–-trend over the 4 larger orders of magnitude (panels cโ€“f ):as anticipated, the peak of the observed shear spectrummoves downward and to the left as ๐œ– decreases, andthe integral of the spectrum (section 3.2) indicates decreasing ๐œ–U, as labeled. However, below O(10โˆ’11) W/kg(panels aโ€“b) the spectrum runs into a spectral floor and does not decrease any further. Here the integral ofฮฆ(k) no longer reflects the shear variance or any true physical quantity; instead, it saturates at a lower limitthat indicates the available precision of ๐œ–U, which, as anticipated, is in the vicinity of our empirical estimate ofthe noise floor 3 ร— 10โˆ’11 W/kg (section 4.3).So far, we have focused our discussion on limitations of the shear measurements. Of course limitations alsoexist on the measurement of temperature microstructure, but these are of a different nature than thoseSCHEIFELE ET AL. 12Journal of Geophysical Research: Oceans 10.1029/2017JC013731which affect the shear measurements, and they tend to be less problematic in our study. Sensitivity limita-tions are not a concern for microstructure thermistors in the way they are for shear probes since the FP07thermistors easily respond to within better than 0.1 mK (Sommer, Carpenter, Schmid, Lueck, & Wรผest, 2013),which is approximately the smallest temperature scale we need observe (e.g., Figure 7g). The relatively slowtime response of thermistors is generally a concern (Gregg, 1999), but at small dissipation rates it is possi-ble to adequately account for the slow response using the transfer function proposed by Sommer, Carpenter,Schmid, Lueck, and Wรผest (2013) or a similar correction method. At rates greater than โˆผ1 ร— 10โˆ’7 W/kg, theeffects of the slow response time can no longer be adequately corrected and temperature-derived estimateswill tend to systematically underestimate the true dissipation rate (Peterson & Fer, 2014), but this limitationis not a concern in our observations since fewer than 0.1% of our ๐œ–T estimates are above this cutoff value.A more relevant concern for our temperature-derived estimates is the potential uncertainty that surroundsthe characteristics of turbulent eddies and the resulting turbulence spectra when turbulent energetics areweak and stratification is strong, as is the case in the setting for our measurements. This is the topic of thefollowing two sections.5.2. Turbulence Spectra in Stratified Low-Energy FlowsOur observations suggest that the shape of shear and temperature-gradient spectra deviate systematicallyfrom Nasmyth and Batchelor reference spectra in stratified low energy flows. Fitted Nasmyth and Batchelorspectra are drawn with the selected observed spectra in Figure 7 for reference, exemplifying varying levels ofagreement; however, individual observed spectra have limited utility for providing broader physical insightbecause we anticipate naturally occurring variability in the shapes of individual spectra (e.g., Fer et al., 2014).In order to identify systematic trends in the shapes of turbulence spectra, we bin all observed spectra bybuoyancy Reynolds number, ReB = ๐œ–โˆ•๐œˆN2, and calculate median temperature-gradient and shear spectra ineach bin (Figure 8). The ReB parameter quantifies the destabilizing effects of turbulent kinetics relative to thestabilizing effects of stratification and viscosity. It is proportional to the ratio of the largest vertical (Ozmidov)scale to the smallest isotropic (Kolmogorov) scale of turbulent eddies, so when ReB < 1 we anticipate thatturbulent eddies of all sizes, including the smallest ones on dissipative scales, are modified by stratificationand exhibit a degree of anisotropy. Further, modeling results suggest that the characteristics of turbulentstructures undergo regime shifts in the vicinity of ReB โˆผ 10 and ReB โˆผ 100 (Ivey et al., 2008; Shih et al., 2005),and so combining with the above scaling argument, we use 1, 10, and 100 to delineate the boundaries ofour ReB bins.To calculate median spectra, individual spectra must first be normalized identically so that the shapes ofspectra over varying ๐œ– and ๐œ’ may be compared. To do this, we nondimensionalize shear spectra usingฮฆโˆ— = ฮฆโˆ•(๐œ–3Uโˆ•๐œˆ)1โˆ•4 (8)and temperature-gradient spectra usingฮจโˆ— = ฮจโˆ•(๐œ’โˆšqBโˆ•2kBDT), (9)consistent with schemes used by Oakey (1982) and Dillon and Caldwell (1980). We nondimensionalizewavenumbers using kโˆ•k๐œˆ for shear spectra and kโˆ•kB for temperature-gradient spectra. The scaling factor k๐œˆ isthe Kolmogorov wavenumber defined (1โˆ•2๐œ‹)(๐œ–Uโˆ•๐œˆ3)1โˆ•4; the remaining variable definitions for equations (8)and 9 are given in section 3. For each ReB bin, we then calculate themedian spectral height at each nondimen-sional wavenumber using all spectra within the bin, creating the median spectra shown in Figure 8; we alsocalculate the interquartile range at each nondimensional wavenumber as a measure of the variability aroundthe median. To exclude artificial effects that may arise because of the ๐œ–U noise floor (section 5.1), we excludefrom the calculations any shear spectra where ๐œ–U < 1 ร— 10โˆ’10 W/kg.The systematic modification of the temperature-gradient spectra with decreasing ReB is clearly visible if onefollows panels eโ€“h from right to left: there is a clear trend toward less curvature and greater low-wavenumberdeviation from the theoretical curve as ReB decreases. None of the median temperature-gradient spectraexhibit a curvature as strong as that predicted by the Batchelor spectrum, but the discrepancy increases withdecreasing ReB, and for the two lowest ReB bins there is no longer a peak and roll-off delineating distinctiveSCHEIFELE ET AL. 13Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 8. Median nondimensionalized shear (aโ€“d) and temperature-gradient (e-h) spectra in bold, for regimes of ReB as indicated. Also shown are the 25th and75 percentile of data (thin solid line) as well as nondimensionalized reference spectra (dashed line): Nasmyth for shear and Batchelor for temperature gradient.The total number of spectra used in each calculation is indicated by N. Shear spectra with ๐œ–U < 10โˆ’10 W/kg are excluded.subranges of the spectrum. This behavior is similar to that seen inmeasurements taken byDillon andCaldwell(1980) who, as we do, observed decreasing curvature with smaller turbulence intensities.The median shear spectra likewise vary systematically with ReB, indicating increasing deviations from theNasmyth spectrum as ReB becomes small (following panels aโ€“d from right to left). The dissipative subrange,to the right of the peak, is not as steep in any of the median spectra as predicted by the Nasmyth shape, andit becomes increasingly more shallow with decreasing ReB. In addition, the amplitude of the spectrum in theinertial subrange, left of the peak, is overestimated by the Nasmyth spectrum in all median spectra and nolonger appears to be well described by a simple power lawwhen ReB < 1. This behavior is reminiscent of thatseen in measurements by Gargett et al. (1984) who found that inertial subranges of shear spectra graduallydisappeared when turbulence became weak and stratification strong.5.3. Understanding Uncertainty for Small ๐The above discussion (section 5.2) highlights themanner inwhich turbulence spectra are systematicallymod-ified away from their reference shapes as ReB becomes small. We propose that the systematic modificationwithdecreasingReB occurs as the characteristics of turbulent eddies in strong stratification increasinglydepartfrom the idealized framework of steady, isotropic turbulence. Evidence for this behavior can be seen in thedistribution of the ReB parameter which is below unity in 76% of our observations, suggesting that turbu-lent eddies are frequently anisotropic andmodified by the effects of stratification. Further, the Ozmidov scale,LO = (1โˆ•2๐œ‹)(๐œ–โˆ•N3)1โˆ•2, has a median value of 0.1 cm, which is exceptionally small and again suggests thateven viscous-scale eddies are squashed by the stratification.These characteristics signal that there is increased uncertainty in the dissipation rate estimateswhen ๐œ– is smalland N2 is large. This is especially true for ๐œ–T estimates where the data processing depends on the ability todetermine kB from Batchelor spectrum fits (section 3.3). One way to characterize the increased uncertainty isto quantify the degree by which observed turbulence spectra and idealized reference spectra diverge. We dothis here for the temperaturemeasurements and compare temperature-gradient spectra to Batchelor spectraSCHEIFELE ET AL. 14Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 9. Root-mean-square error between ฮจ and ฮจB, as defined in the text, visualized as a function of(a) buoyancy Reynolds number, and (b) dissipation rate. This metric quantifies the degree of divergence betweenobserved temperature-gradient and theoretical Batchelor spectra. Large open faced markers are bin averages.Regressions to subsets of the bin averages are shown in each panel; the subsets are those on either side of the datummarked by the circle (inclusive).by calculating for each spectrum the root-mean-square error, ๐œ‰rms, of log10(ฮจโˆ•ฮจB), defined:๐œ‰rms =โˆšโˆšโˆšโˆš1nnโˆ‘i=1log210(ฮจiฮจBi)=โˆšโˆšโˆšโˆš1nnโˆ‘i=1[log10(ฮจi) โˆ’ log10(ฮจBi)]2. (10)The summation index, i, runs over all wavenumbers included in the Batchelor fitting procedure; n is the num-ber of spectral points included in the fit (section 3.3). We find that averages of ๐œ‰rms increase gradually from0.3 to 0.5 as ReB estimates decrease from O(101) to O(10โˆ’2); at smaller ReB, mean ๐œ‰rms increases rapidly to amaximum value of 0.7 (Figure 9). A similar, but more pronounced, pattern is visible when ๐œ‰rms is visualized asa function of ๐œ–T . The increase in ๐œ‰rms is gradual and modest, from 0.3 to 0.45, as ๐œ–T decreases from O(10โˆ’7) toO(10โˆ’12) W/kg; this behavior is followed by a sharp increase in ๐œ‰rms at smaller ๐œ–T .Further insight into the confidence of the ๐œ–T values can be gained empirically if we make the assumptionthat dissipation rates distribute lognormally in the ocean (Baker & Gibson, 1987; Gregg, 1987). Under thisassumption, we can use the observed distribution of ๐œ–T (Figure 5b) to estimate a lower cutoff below whichthe application of the steady, isotropic turbulence model becomes problematic and ๐œ–T estimates becomeunreliable. The distribution of ๐œ–T observations follows the lognormal shape closely over the entire range ofdata except below the second decile (Figure 5c), where the observed distribution is disproportionately heavy.The distortion in the distribution indicates that uncertainties in the data processing statistically skew the ๐œ–Testimates below the second decile; our simple statistical model therefore suggests that the ๐œ–T estimates arereliable and physically meaningful to values as small as ๐œ–T โ‰ˆ 2 ร— 10โˆ’12 W/kg. Below this cutoff, ๐œ–T estimatesare unreliable and perhaps notmeaningful. Note that a lower cutoff of 2ร—10โˆ’12 is consistent with the suddenincrease in ๐œ‰rms that occurs in Figure 9b below ๐œ–T = O(10โˆ’12)W/kg.5.4. Implications for Interpreting Microstructure MeasurementsIn section 4 we demonstrated that there can be a significant difference between dissipation rate estimatesderived from coincident measurements of shear and temperature microstructure. The ๐œ–T estimates suggestthat the true dissipation rate is below the ๐œ–U noise floor of 3 ร— 10โˆ’11 W/kg in 58% of our observations.A histogram of the ratio ๐œ–Uโˆ•๐œ–T (Figure 10) demonstrates the severity and the frequency with which the shearprobes may overestimate the dissipation rate in low-energy environments like the Beaufort Sea. Using ๐œ–T asa reference (and acknowledging the associated uncertainties described in section 5.3), the shear measure-ments overestimate the dissipation rate by a factor of at least 5 in 44% of our measurements, by at least 1order of magnitude in 31% of our measurements, and by at least 2 orders of magnitude in 9% of our mea-surements. This is a level of error that has the potential to alter the interpretation of the shear measurements,as described in the following paragraph. In contrast, the temperature measurements overestimate theSCHEIFELE ET AL. 15Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure 10. Histogram of the ratio ๐œ–Uโˆ•๐œ–T , highlighting the large numberof coincident measurements where the shear-derived values overestimatedthe temperature-derived ones in our data set. Agreement by factors of 5,10, and 100 is indicated by dashed lines. The histogram is calculated over 50logarithmically spaced bins.dissipation rate relative to ๐œ–U by a factor of at least 5 in less than 3%ofmea-surements, a degree of mismatch that could safely be neglected in manyapplications.The severity with which the bias we found in the shear measurementsmay alter the interpretation of ๐œ–U estimates depends on the specific goalsof a study. If, as in section 4, the utility of the measurements is to char-acterize the variability and the statistical distribution of dissipation rates,then the potential for biases greater than an order of magnitude cannotbe ignored. Without the coincident ๐œ–T estimates to which to compare, the๐œ–U estimates would lead us to misrepresent the degree of spatial variabil-ity (Figures 2 and 3), the geometric averages representing typical values(Figure 4), and the observed distribution and related statistics (Figure 5and Table 2) of the turbulent dissipation rate here in the Amundsen Gulf.These misrepresentations may then be further propagated into calcula-tions of the mixing rate coefficient K๐œŒ, which typically rely on the Osborn(1980) model K๐œŒ = 0.2๐œ–โˆ•N2, leading to similar misrepresentations of thevariability, the geometric averages representing typical values, and thedistribution and related statistics of K๐œŒ.Arithmetic mean values of ๐œ–U, however, are much less sensitive to thebias we describe. This is fortunate, because arithmetic mean values arearguably the appropriate measure to use when estimating bulk buoyancyfluxes and characterizing net water mass transformation frommixing rateestimates (Baker & Gibson, 1987). As noted in section 4, in this study wehave tended to use the geometricmean to average dissipation rates. We do this because the geometricmeaneffectively characterizes the central tendency of lognormally distributed data andmore fairly represents typi-cal ๐œ– realizations (Kirkwood, 1979). In contrast, the arithmeticmean is ineffective at representing typical valuesof a lognormal-like distribution because it is dominated by a small number of very large values at the high endof the distribution. Further, the arithmeticmean tends to be very sensitive to individual outliers thatmay existon the far right-hand side of the distribution, which may be problematic because of the large uncertainty inindividual ๐œ– realizations. However, the disproportionate importance of large values in setting the arithmeticmean alsomakes it mostly insensitive to errors in small ๐œ– estimates. The effect can be seen, for example, whencomparing the arithmeticmeanof the ๐œ–U and ๐œ–T distributions (Figure 5 and Table 2): in contrast to the geomet-ric mean, the median, and the 25th and 75th percentiles, the arithmetic mean of the ๐œ–T distribution is greaterthan that of the ๐œ–U distribution because of a marginally thicker tail on the right-hand side of its distributionwhich more strongly influences its arithmetic mean.A final subtle point remains to be discussed. When carrying out an analysis using microstructure measure-ments of shear, it is tempting to simply remove observed ๐œ–U values that sit at or near the estimated noisefloor, discarding these as untrustworthy. This approach is viable when only a small number of the observa-tions are near the noise floor; however, in the present study, simply removing data likely to be corrupted bythe effects of the noise floor would only exacerbate the bias evident in the ๐œ–U observations. For example, ifwe remove from the data set the shear-derived estimates where ๐œ–U < 5 ร— 10โˆ’11 W/kg, we increase the pos-itive bias in our sample by removing 53% of the measurements and the entire left half of the ๐œ–U histogram(Figure 5). Rather than helping to correct biased averages of ๐œ–U, this change shifts themedian from 4.6ร—10โˆ’11to 1.4 ร— 10โˆ’10 W/kg and the geometric mean from 6.5 ร— 10โˆ’11 to 1.9 ร— 10โˆ’10 W/kg, in both cases increasingthe error in these averaging metrics.The best practical way to account for the effects of the noise floor in microstructure shear measurementswill depend on the goals of each individual study and on the proportion of the observations that are in thevicinity of the noise floor. One approach is to set dissipation rates that appear to be near the noise floor tozero (e.g., see Gregg et al., 2012); this approach is probably justifiable for arithmetic mean calculations sincethe averaging is dominated by the large ๐œ– values, but it is problematic when describing the variability or whencalculating a geometric mean to characterize typical ๐œ– values. In these situations, it may bemore appropriateto fit a lognormal distribution to the part of the observed distribution that resides above the noise floor, butcare is required here also since the theoretical conditions for expecting lognormality are strict and often notSCHEIFELE ET AL. 16Journal of Geophysical Research: Oceans 10.1029/2017JC013731satisfied in a set of fieldmeasurements (Yamazaki & Lueck, 1990). In any case, it is clear that the interpretationof microstructure shear-derived dissipation rate estimates should proceed with caution if the measurementsare from a very low energy environment and it appears that a large proportion of the data cluster around awell-defined noise floor.6. ConclusionsCare must be taken to understand how measurement limitations may bias microstructure measurements inlow-energy environments like the Beaufort Sea: this is the central theme of our study. The results we havepresented here suggest that microstructure measurements of velocity shear, in particular, are prone to mis-representing averaged dissipation ratesโ€”and, consequently, mixing ratesโ€”in such environments becausethe noise floor of the ๐œ–U estimates artificially biases the majority of the observations. In addition, our mea-surements suggest that both shear- and temperature-derived dissipation rate estimates may be complicatedby further uncertainty when strong stratification modifies the characteristics of turbulence in a weakly ener-getic, strongly stratified flow; this change in the characteristics can be seen in the systematic modification ofthe shapes of shear and temperature-gradient spectra at low ReB.We have documented the discrepancy between the two distinct dissipation rate estimates ๐œ–U and ๐œ–T becausewe find disagreement large enough to lead to substantial differences in how the two sets of data would beinterpreted independently. The temperature-derived estimates were able to resolve smaller dissipation ratesthan the shear-derived estimates: averages of ๐œ–U began to exhibit biased behavior below 10โˆ’10 W/kg andwere not able to resolve rates below 3 ร— 10โˆ’11 W/kg, while averages of ๐œ–T were reliable to values as low as2ร—10โˆ’12 W/kg andwere characterizedby unacceptably large uncertainty below this. Our experience suggeststhat caution interpreting shear-derived dissipation rate estimates is warranted if a large number of observa-tions cluster at or near an identifiable ๐œ–U noise floor, in our case 3ร—10โˆ’11 W/kg. In themeasurements presentedhere, the temperature-derived estimates suggest that the true dissipation rate lies below this noise floor oftenenough to fundamentally alter the scientific interpretation of the measurements. Other low-energy environ-ments in which the special measurement considerations outlined in this study may be applicable includethe wider Canada Basin (Rainville & Winsor, 2008), stratified lakes (Scheifele et al., 2014; Sommer, Carpenter,Schmid, Lueck, Schurter, &Wรผest, 2013), the central Baltic Sea (Holtermann et al., 2017), and the abyssal globalocean over smooth topography (Waterhouse et al., 2014).Appendix A: Quality Control Measures for Dissipation Rate EstimatesHere we describe the quality control procedures we apply to the measurements to identify dissipation rateestimates deemed untrustworthy. These are removed before the analysis described in section 4.Before implementing quantitative quality control measures, we remove any obviously contaminated mea-surements by hand. These include all measurements after 27 August from one of the two shear probesbecause the probe appears to have been damaged at this point and thereafter no longermeasured a sensibleTable A1Quality Control Parameters, as Defined in the TextParameter ๐œ–U ๐œ–TN before QC 45,571 63,507QC1 9.8% 10.3%QC2 13.6% 13.2%QC3 2.7% 17.0%QC4 3.5% 4.1%QC5-U 13.2% โ€”QC6-T โ€” 5.6%QC7-T โ€” 4.6%One or more 22.3% 33.9%N after QC 35,395 41,955Note. Percentages are the fraction of measurements flagged by each condition.SCHEIFELE ET AL. 17Journal of Geophysical Research: Oceans 10.1029/2017JC013731Figure A1. Comparison of results from individual (a,b) shear probes and (c,d) temperature probes. Bin averages arecalculated as in Figure 6. Agreement within a factor of 5 is indicated in all panels by the dashed lines.signal. We then begin systematic quality control measures with 45,571 independent ๐œ–U estimates and 63,507independent ๐œ–T estimates. Note that the estimates from the two distinct shear or two distinct temperatureprobes are not yet averaged at this stage (see sections 3.2 and 3.3).Individual estimates of ๐œ–U and ๐œ–T are flagged untrustworthy and removed if they satisfy one or more of thefollowing conditions:QC1. The magnitude of the gliderโ€™s acceleration |dUโˆ•dt| is above the tenth percentile (|dUโˆ•dt|> 4.6 ร—10โˆ’4 m/s2). This is a heuristic measure, but it satisfactorily isolates measurements where the gliderappears to be changing speed over the span of one ๐œ– estimate. The data processing assumes that U isconstant over the span of an ๐œ– estimate.QC2. The glider is within 15 m of an inflection point. When the glider inflects, the angle of attack and theestimate of U are uncertain and mechanical vibrations from the glider contaminate the measurements.QC3. Estimates from two identical probes differ by greater than a factor of 10. For shear, the higher estimateis removed. For temperature, both estimates are removed.QC4. The ratio Uโˆ•(๐œ–โˆ•N)1โˆ•2 is less than 5. This is the ratio between the gliderโ€™s velocity and an estimate of theturbulent flow velocities (Fer et al., 2014) and may indicate when Taylorโ€™s frozen turbulence hypothesisis violated.In addition, ๐œ–U measurements are flagged and removed ifQC5-U. The maximum of the nondimensionalized shear spectrum ฮฆโˆ—, defined in section 5.2, is greater thantwice the peak of the nondimensionalized Nasmyth spectrum. This isolates shear spectra obviouslycontaminated at low wavenumbers.In addition, ๐œ–T measurements are flagged and removed ifQC6-T . The sum of the correction terms ๐œ’lw and ๐œ’hw (see equation (7)) is greater than the observed term ๐œ’obs.SCHEIFELE ET AL. 18Journal of Geophysical Research: Oceans 10.1029/2017JC013731QC7-T . There are fewer than six distinct wavenumbers available in the closed interval [kl, ku]. This ensures areasonable minimum number of spectral points to which to fit a Batchelor spectrum.Cumulatively, quality assessment conditions flag and remove22.3%of individual ๐œ–U measurements and33.9%of individual ๐œ–T measurements. The percentage of measurements flagged by each individual condition isgiven in Table A1.Beyond these conditions, confidence in the measurements may be indicated by the level of agreementbetweenduplicatemeasurements of ๐œ–U and ๐œ–T . This comparison is favorable: after the quality assessment pro-cedures, 96% of 6,820 coincident sets of ๐œ–U measurements agree within a factor of 5, and the agreement iscomparable over the full range of ๐œ–U. Note that a factor of 5 is typically considered reasonable agreement forindividual microstructure measurements which often scatter within an order of magnitude and can vary bymore than 10 decades in the ocean. The comparison is slightly more variable but still favorable for ๐œ–T , where87.3% of 20,378 coincident sets agree within a factor of 5 and there is likewise no trend with ๐œ–T .The comparisons are imaged in Figure A1, where subscripts 1 and 2 are arbitrarily designated to measure-ments from distinct probes. Bin averages are calculated as in Figure 6.Following the quality control procedures described here, we average (using an arithmetic mean) results foreach of the two distinct sets of measurements, as described in sections 3.2 and 3.3. This leaves 28,575 ๐œ–U and21,577 ๐œ–T estimates for analysis.Appendix B: Comparison of Results fromUpcasts and DowncastsThe stratification conditions in the Amundsen Gulf resulted in much faster glider speeds on upcasts than ondowncasts (Table 1). Because dissipation rate calculations are very sensitive to the estimated glider speed U(Gregg, 1999; Lueck, 2016; Osborn & Crawford, 1980), this discrepancy can be leveraged to further inform ourconfidence in themeasurements anddata processingmethods. In the absence of systematic errors in the dataprocessing, we would expect to see no systematic difference in the results derived separately from upcastsand downcasts despite the nearly factor of 2 difference in U.We observe little to no difference between dissipation rate estimates when the results are separated byupcasts and downcasts. Figure B1 presents an overview of the ๐œ–U and ๐œ–T results separated in this manner.The two histograms of ๐œ–U (panel a) have nearly indistinguishable characteristics: for example, themedians areFigure B1. Overview of select results, separated by upcasts and downcasts. For each of ๐œ–U and ๐œ–T , we show the histograms (a,e), averaged vertical profiles (b,f ),and selected spectra (cโ€“d,gโ€“h) separated in this manner. The spectra shown are those corresponding to dissipation rates within a factor of 1.1 of 10โˆ’9 W/kg.Thick black lines depict the median of the selected spectra at each wavenumber.SCHEIFELE ET AL. 19Journal of Geophysical Research: Oceans 10.1029/2017JC0137315 ร— 10โˆ’11 and 4 ร— 10โˆ’11 W/kg for upcasts and downcasts, respectively, and the respective geometric stan-dard deviation factors are 3.5 and 3.7. Similarly, the two averaged vertical profiles of ๐œ–U (panel b) are nearlyidentical in magnitude everywhere; they typically agree within a factor of 1.2 and always within a factor of1.8. Shear spectra are likewise similar between upcasts and downcasts, as highlighted in the selected spectrashown in panels c and d. Median spectra are generally alike in shapewith amarginally wider spectral peak forthe upcast median spectrum.There is slightly more discrepancy between upcasts and downcasts in the ๐œ–T results, though the overallagreement is still encouraging and the discrepancy does not impact the results or conclusions of the study.Thehistogramcomparison (panel e) is generally favorable: themedian is 2ร—10โˆ’11 W/kg for bothdistributions,and the geometric standard deviation factors are 23.4 and 16.2 for upcasts and downcasts, respectively.The upcast distribution is wider because of a small unexpected increase in the number of ๐œ–T values below1ร— 10โˆ’13 W/kg. As discussed in section 5.3, there is extensive uncertainty associated with values of ๐œ–T smallerthan 2ร—10โˆ’12 W/kg, and so it is unclear howmuchmeaning can be assigned to this feature of the distribution.Themean profiles (panel f ) demonstrate adequate agreement, typically within a factor of 2 and always withina factor of 3.5, in line with typical uncertainties from microstructure measurements. The shape of the twomedian temperature gradient spectra (panels g and h) compares favorably with only a slightly less roundedroll-off to the Batchelor scale for the upcast spectrum.Appendix C: Nasmyth and Batchelor SpectraThe Nasmyth spectrum,ฮฆN, is an empirically derived form for the one-dimensional power spectrum of veloc-ity shear in an unstratified turbulent flow and is based on measurements collected in a strongly turbulenttidal channel in coastal British Columbia (Nasmyth, 1970). It describes both the inertial subrange of the shearspectrum, predicted by Kolmogorov (1941), and the viscous subrange where viscosity begins to influencethe motion of turbulent eddies. The results of Nasmyth were tabulated by Oakey (1982), and a mathematicalfit was later proposed by Wolk et al. (2002). We use a modified form of that expression, described by Lueck(2016), which can be written nondimensionally as ฮฆโˆ—N = 8.05x1โˆ•3โˆ•(1 + (20.6x)3.715), where x = k(๐œˆ3โˆ•๐œ–)1โˆ•4and the spectrum is nondimensionalized usingฮฆโˆ—N = ฮฆNโˆ•(๐œ–3Uโˆ•๐œˆ)1โˆ•4.The Batchelor spectrum is a theoretical one-dimensional power spectrum describing the wavenumber dis-tribution of a passive tracerโ€™s gradients in an unstratified turbulent flow (Batchelor, 1959); its integral isproportional to the rate, ๐œ’ , at which the tracer gradients are smoothed by molecular diffusion. The spectrumis an analytic solution to the advection-diffusion equation driven by turbulent strain and the large-scale tracergradient. In one dimension, it may be written as follows:ฮจB =๐œ’โˆšqBโˆ•2kBDT(๐›ผ exp(๐›ผ22)โˆ’ ๐›ผ2โˆš๐œ‹2erfc(๐›ผโˆš2))(C1)where๐›ผ = (kโˆ•kB)โˆš2qB . (C2)The factor qB is a dimensionless constant related to the average least principal rate of strain; it representsthe time scale by which compressive strain sharpens scalar gradients (Smyth, 1999). The value of qB is uncer-tain, and experiments by Oakey (1982) suggest that the range 2.2โ€“5.2, though typically qB = 3.4 (e.g.,Ruddick et al., 2000) or qB = 3.7 (e.g., Peterson & Fer, 2014), are used. A percentage error in qB is expected tolead to twice the percentage error in ๐œ–T (Dillon & Caldwell, 1980). We used qB = 3.4 in our analysis but alsoprocessed all the temperature measurements using qB = 3.7, and the difference in results was small: usingqB = 3.7, we found for ๐œ–T amode of 1.5ร— 10โˆ’11 W/kg, a geometric mean of 1.9ร— 10โˆ’11 W/kg, and a geometricstandard deviation factor of 18.3 (compare with Table 2).Note that the Kraichnan spectrum (Kraichnan, 1968) would be an adequate alternative for the fitting pro-cedure described in section 3.3. Peterson and Fer (2014) compared the results of fitting observed temper-ature gradient spectra to Batchelor and Kraichnan spectra and found no significant difference in the final๐œ–T results.SCHEIFELE ET AL. 20Journal of Geophysical Research: Oceans 10.1029/2017JC013731ReferencesBaker, M. A., & Gibson, C. H. (1987). Sampling turbulence in the stratified ocean: Statistical consequences of strong intermittency.Journal of Physical Oceanography, 19, 1817โ€“1836.Batchelor, G. K. (1959). 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Sloan Foundation; the University ofBritish Columbia (UBC); the KillamDoctoral Scholarships program; theVanier Canada Graduate Scholarshipsprogram; the UBC Four Year FellowshipProgram; the Northern ScientificTraining Program; and the NSERCMichael W. Smith Foreign StudySupplement. Logistical support for thefield work was provided by the OceanTracking Network (OTN) and the MarineEnvironmental Observation, Predictionand Response (MEOPAR) Networkwhich are supported by the CanadaFoundation for Innovation (CFI) andNSERC; by ArcticNet; and by theAmundsen Science program which issupported by CFI. ArcticNet andMEOPAR belong to the Networks ofCentres of Excellence Program of theGovernment of Canada. The authorswish to thank the technical staff whooperate the Glider Program of OTN;Keith Lรฉvesque of ArcticNet; Dan Kelleyof Dalhousie University; the crew of theCanadian Coast Guard Ship Amundsen;and three anonymous reviewers of thisarticle. 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The interior shelves of the Arctic Ocean: Physical oceanographic setting, climatology and effects ofsea-ice retreat on cross-shelf exchange. Progress in Oceanography, 139, 24โ€“41.Wolk, F., Yamazaki, H., Seuront, L., & Lueck, R. G. (2002). A new free-fall profiler for measuring biophysical microstructure.Journal of Atmospheric and Oceanic Technology, 19, 780โ€“793.Wolk, F., Lueck, R. G., & Laurent, L. St. (2009). Turbulence measurements from a glider. In 13th Workshop on Physical Processesin Natural Waters Proceedings. Palermo, Italy. Retrieved from https://rocklandscientific.com/wpcontent/uploads/2015/01/Wolk_Lueck_StLaurent_final_PPNW.pdfYamazaki, H., & Lueck, R. (1990). Why oceanic dissipation rates are not lognormal. Journal of Physical Oceanography, 20, 1907โ€“1918.SCHEIFELE ET AL. 22


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