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Observations of turbulence and mixing in the southeastern Beaufort Sea Scheifele, Benjamin 2020

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Observations of Turbulence and Mixing in the SoutheasternBeaufort SeabyBenjamin ScheifeleM.Sc., The University of British Columbia, 2013B.Sc. Physics, St. Francis Xavier University, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Oceanography)The University of British Columbia(Vancouver)February 2020c© Benjamin Scheifele, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Observations of Turbulence and Mixing in the Southeastern Beaufort Seasubmitted by Benjamin Scheifele in partial fulfillment of the requirements for the de-gree of Doctor of Philosophy in Oceanography.Examining Committee:Stephanie Waterman, University of British ColumbiaSupervisorSusan Allen, University of British ColumbiaSupervisory Committee MemberGreg LawrenceUniversity ExaminerPhil AustinUniversity ExaminerIlker Fer, University of BergenExternal ExaminerAdditional Supervisory Committee Members:Jeff Carpenter, Helmholtz Zentrum GeesthachtSupervisory Committee MemberBernard Laval, University of British ColumbiaSupervisory Committee MemberiiAbstractIn this thesis, I use a novel set of hydrography and turbulence measurements from thesoutheastern Beaufort Sea toi. compare estimates of the turbulent kinetic energy dissipation rate, ε , obtainedindependently from shear and temperature microstructure measurements;ii. characterize turbulence and mixing in the Amundsen Gulf region of the southeast-ern Beaufort Sea; andiii. describe the characteristics of tracer diffusion in an oceanic flow as it transitionsbetween fully turbulent and nearly-laminar.I collected the measurements over 10 days in 2015 using an ocean glider measuringtemperature, conductivity, and pressure on O(10)-cm scales and shear and temperatureon O(1)-mm turbulent scales.The two independent ε estimates agree within a factor of 2 when ε exceeds 3×10−11 Wkg−1, but diverge by up to two orders of magnitude at smaller values. I identify the noisefloor of the shear measurements as the primary reason for this divergence and, therefore,suggest that microstructure temperature measurements are preferable for estimating ε inlow energy environments like the Beaufort Sea.I find that turbulence is typically weak in Amundsen Gulf: ε has a geometric mean valueof 2.8×10−11 W kg−1 and is less than 1×10−10 W kg−1 in 68% of observations. Turbu-lent dissipation varies over five orders of magnitude, is bottom enhanced, and is primar-ily modulated by the M2 tide. Stratification is strong and frequently damps turbulence,inhibiting diapycnal mixing in up to 93% of observations. However, a small numberof strongly turbulent mixing events disproportionately drive net buoyancy fluxes. Heatfluxes are modest and nearly always below 1 W m−2.Finally, I use the turbulence measurements to demonstrate how tracer diffusion in theocean transitions continuously between turbulent diffusion and near-molecular diffu-sion as turbulence weakens and stratification strengthens. I use the buoyancy Reynoldsiiinumber, ReB, to quantify the relative energetic contributions of potential and kinetic en-ergy to the flow dynamics and find that present models for tracer diffusion are accurateto within a factor of 3 when ReB > 10. However, contrary to expectations, I find thatsignificant enhanced tracer diffusivity at turbulent scales remains present when ReB isbelow unity.ivLay SummaryThis thesis outlines my research about ocean turbulence and the impacts it has on theCanadian sector of the Arctic Ocean. Turbulence is a fluid dynamics phenomenon that isubiquitous throughout the world’s oceans and helps to control their ability to support lifeby supplying heat, oxygen, and nutrients to ocean organisms. The distributions of thesecharacteristics are particularly important in the Arctic Ocean because modern climatechange is quickly driving Arctic ecosystems towards states never seen before in humanhistory. Understanding these changes and making predictions about what the regionwill be like in future generations relies on a careful understanding of how turbulencemodulates the ocean environment. This thesis describes original, previously unpublishedresearch on how to measure ocean turbulence, how it impacts the physical environmentin a region of the Beaufort Sea, and on the physical characteristics of turbulent fluid flowin the Arctic Ocean.vPrefaceThis thesis presents original research that was designed and conducted by me. I de-fined the research questions, planned and executed the experiment, analyzed the data,and wrote the thesis and the journal articles mentioned in the next paragraph to reportthe results. The work was, however, also a collaborative effort, and my PhD advisorsDrs. Stephanie Waterman and Jeff Carpenter actively contributed with advice at everystage of the project, from the initial definition of the problem to the final writing of thepapers and thesis. My use of the pronoun “we” throughout the document reflects thiscollaboration.This thesis is written in “paper format”. While I wrote it as a record of a single, co-hesive research project, each science chapter (Chapters 2–4) may also be read as aself-contained standalone study, suitable for journal publication. This results in someredundancies from chapter to chapter, but the overlap is small and, in each case, the in-formation provided is there to serve the purpose of that specific study. Chapter 2 is pub-lished almost verbatim as a peer-reviewed article in Journal of Geophysical Research:Oceans1. A version of Chapter 3 is presently undergoing peer review for publication inan academic journal. In both cases, as with the thesis, the research and manuscript prepa-ration were conducted primarily (to about 90%) by me and supported Drs. StephanieWaterman and Jeff Carpenter. Lucas Merckelbach contributed substantially in the col-lection of the measurements. I led the peer review process for the submission of thematerial in Chapter 2. For the material in Chapter 3, the peer review-driven modifica-tions (i.e. those changes requested by reviewers after the initial submission) are beingcompleted by Jeff Carpenter and Stephanie Waterman; these contributions will amountto about 35% of the material in the final re-submitted document.The project described by this thesis was defined within the broad mandate of the Cana-dian Arctic GEOTRACES Program, and the field work was conducted in collaboration1B. Scheifele, S. N. Waterman, L. Merckelbach, and J. R. Carpenter. Measuring the Dissipation Rate ofTurbulent Kinetic Energy in Strongly Stratified, Low-Energy Environments: A Case Study From the ArcticOcean. Journal of Geophysical Research: Oceans, 19:1817–22, Aug. 2018viwith ArcticNet and the Canadian Coast Guard on the GEOTRACES Cruise Leg 15b. Lu-cas Merckelbach contributed extensively to the design and execution of the field work;he also provided the flight model and scripts to extract the glider data. Rockland Sci-entific Inc. provided scripts to process the microstructure shear data, which I modifiedslightly; Barry Ruddick and Jeff Carpenter provided scripts to process the microstructuretemperature data, which I modified heavily to suit the purposes of this study.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxList of Mathematical Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Project Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Ocean Mixing and Turbulence . . . . . . . . . . . . . . . . . . 41.3.2 Turbulence Measurements from Gliders . . . . . . . . . . . . . 71.3.3 The Dissipation Rate of Turbulent Kinetic Energy . . . . . . . . 81.3.4 Shear Microstructure . . . . . . . . . . . . . . . . . . . . . . . 91.3.5 Temperature Microstructure . . . . . . . . . . . . . . . . . . . 101.3.6 The Osborn Model for Mixing . . . . . . . . . . . . . . . . . . 111.3.7 Amundsen Gulf . . . . . . . . . . . . . . . . . . . . . . . . . . 12viii2 Measuring the Dissipation Rate of Turbulent Kinetic Energy in StronglyStratified, Low Energy Environments . . . . . . . . . . . . . . . . . . . . 142.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Measurement Platform: Slocum Glider . . . . . . . . . . . . . 162.2.2 Shear and Temperature Microstructure . . . . . . . . . . . . . . 162.2.3 Location, Local Hydrography, and Sampling Strategy . . . . . . 172.3 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Glider Velocity Estimates . . . . . . . . . . . . . . . . . . . . 192.3.2 Shear Microstructure . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Temperature Microstructure . . . . . . . . . . . . . . . . . . . 222.4 Comparison of Results from Temperature and Shear Microstructure . . 242.4.1 Spatial Cross Sections . . . . . . . . . . . . . . . . . . . . . . 242.4.2 Mean Vertical Profiles . . . . . . . . . . . . . . . . . . . . . . 262.4.3 Distributions of εU and εT . . . . . . . . . . . . . . . . . . . . 282.4.4 One-to-one comparison of εU and εT . . . . . . . . . . . . . . 302.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5.1 The Effect of Sensor Limitations . . . . . . . . . . . . . . . . . 322.5.2 Turbulence Spectra in Stratified Low Energy Flows . . . . . . . 342.5.3 Understanding Uncertainty for Small ε . . . . . . . . . . . . . 362.5.4 Implications for Interpreting Microstructure Measurements . . . 382.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Turbulence and Mixing in the Arctic Ocean’s Amundsen Gulf . . . . . . 423.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Measurements and Data Processing . . . . . . . . . . . . . . . . . . . 443.2.1 Sampling Strategy . . . . . . . . . . . . . . . . . . . . . . . . 443.2.2 Turbulence Measurements and Data Processing . . . . . . . . . 453.2.3 Arithmetic vs. Geometric Averaging . . . . . . . . . . . . . . . 473.3 Hydrography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Turbulence and Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 493.4.1 Turbulent Dissipation Rates . . . . . . . . . . . . . . . . . . . 493.4.2 The Influence of Stratification . . . . . . . . . . . . . . . . . . 533.4.3 Diffusivity Estimates . . . . . . . . . . . . . . . . . . . . . . . 543.4.4 Vertical Heat Fluxes . . . . . . . . . . . . . . . . . . . . . . . 573.5 Discussion: Mixing Processes . . . . . . . . . . . . . . . . . . . . . . 583.5.1 Tidal Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.5.2 Double Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 613.5.3 Pacific Water Mesoscale and Smaller Features . . . . . . . . . . 623.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64ix4 Enhanced Heat Fluxes in a Marginally Turbulent Flow . . . . . . . . . . 664.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.2.1 Measurements and ε Estimates . . . . . . . . . . . . . . . . . . 684.2.2 Osborn Model . . . . . . . . . . . . . . . . . . . . . . . . . . 704.2.3 Temperature Variance Method: The Osborn-Cox Model . . . . 714.2.4 Idealized Turbulence: Isotropy . . . . . . . . . . . . . . . . . . 724.2.5 Other Diffusivity Models . . . . . . . . . . . . . . . . . . . . . 734.2.6 Mixing Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3.1 Diffusivity Estimates . . . . . . . . . . . . . . . . . . . . . . . 754.3.2 Mixing Efficiency . . . . . . . . . . . . . . . . . . . . . . . . 774.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 794.4.1 Enhanced Heat Fluxes . . . . . . . . . . . . . . . . . . . . . . 794.4.2 Applicability of the Osborn Model . . . . . . . . . . . . . . . . 814.4.3 Mixing Efficiency in High Stratification . . . . . . . . . . . . . 825 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.1 Goals and Representativeness of the Thesis . . . . . . . . . . . . . . . 845.1.1 Observing weak turbulence in strong stratification . . . . . . . 855.1.2 Turbulent mixing in the Arctic Ocean’s Amundsen Gulf . . . . 875.1.3 Enhanced heat fluxes in strongly stratified, weakly turbulent en-vironments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2 The Bigger Picture: Looking Ahead . . . . . . . . . . . . . . . . . . . 89Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.1 Quality Control Measures for Dissipation Rate Estimates . . . . . . . . 102A.2 Comparison of Results from Upcasts and Downcasts . . . . . . . . . . 105A.3 Nasmyth and Batchelor Spectra . . . . . . . . . . . . . . . . . . . . . . 106xList of TablesTable 2.1 Mean ± one standard deviation of glider-flight variables from all up-and downcasts of the mission. θ is the measured pitch, αa the es-timated angle of attack, γ the estimated glide angle, and U the esti-mated speed through water. Only data coincident with at least oneviable ε estimate (see Appendix A.1) are included. . . . . . . . . . . 20Table 2.2 Statistical parameters of the εU and εT distributions shown in Figure2.5. Given, from left to right, are the number, N, of observations;mode; geometric mean; median; first and third quartiles, P25 and P75;arithmetic mean; and geometric standard deviation factor, σg. Thequantities N and σg are dimensionless and unscaled. All other quan-tities are scaled by a factor of 10−11 W kg−1. . . . . . . . . . . . . . 29Table 3.1 Properties of the hydrographic layers. Layers are defined by their ab-solute salinity, SA. Ranges given for depth, conservative temperature,T , density anomaly, σ , and stratification, N2, are for the central 90%of data. The layer labels are SML: Surface Mixed Layer; CH: ColdHalocline; PW: Pacific Water Layer; WH: Warm Halocline; AW: At-lantic Water Layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Table 3.2 Select statistics of ε and Kρ observed in (top) all the data; (middle)all data except that within the turbulent patch; and (bottom) data onlyfrom within the turbulent patch. The turbulent patch is defined as theregion inside the white rectangle in Figures 3.4 and 3.7, between s =52–81 km on the horizontal axis. . . . . . . . . . . . . . . . . . . . 52Table A.1 Quality control parameters, as defined in the text. Percentages arethe fraction of measurements flagged by each condition. . . . . . . . 102xiList of FiguresFigure 1.1 The glider’s 10 day path in the Amundsen Gulf along which it mea-sured 348 microstructure profiles in summer 2015. Shown are thestart and end points/dates, as well as the location of four interme-diate waypoints. The light-blue line indicates approximately thenorth-eastern boundary beyond which it was unsafe to operate theglider because of the possibility of sea ice. . . . . . . . . . . . . . . 2Figure 1.2 (a) Example measured shear spectrum (Φ, black line) with a fit-ted empirical Nasmyth spectrum (blue line). The upper end of theinertial subrange can be seen to the left of the “viscous rolloff“,indicated by the purple shading. Yellow shading indicates the vis-cous subrange where viscosity begins to remove kinetic energy fromthe turbulent flow. The Kolmogorov wavenumber is shown by thevertical dashed line. The dissipation rate estimate εU is also given(the subscript U indicates a velocity shear-derived estimate). (b)Example measured temperature gradient spectrum (Ψ, black line)with a fitted theoretical Batchelor spectrum (blue line). The dissipa-tion rate estimate εT (subscript T indicates a temperature gradient-derived estimate) is indicated. The Batchelor wavenumber is shownby the vertical dashed line. . . . . . . . . . . . . . . . . . . . . . . 10xiiFigure 2.1 (a) The measurement location at the entrance to the Amundsen Gulfin the southeastern Beaufort Sea. The glider path is shown by theline inside the dashed black rectangle. Bathymetry contours aredrawn at 1000 m intervals beginning at 200 m. (b) Enlarged viewof the area inside the dashed rectangle indicated in panel a, showingthe glider path and the local bathymetry. Selected waypoints alongthe path are numbered consecutively and indicated by squares forreference when reading Figures 2.2 and 2.3. Contours are drawn at75 m intervals beginning at 50 m. In both panels, colour indicateswater depth (m); bathymetry data are from IBCAO 3.0 (Jakobssonet al., 2012). Mean vertical profiles of (c) conservative tempera-ture, (d) in-situ density anomaly, and (e) buoyancy frequency arealso shown; these are horizontally averaged over all casts where theglider’s maximum dive depth exceeded 325 m. . . . . . . . . . . . 18Figure 2.2 Cross sections of the turbulent dissipation rate, ε , in log10 space,derived from microstructure measurements of (a) shear and (b) tem-perature. The panels are drawn using the same colour scale. Greyshading indicates the bathymetry, black shading discarded or un-available data (see Section 2.2.3 and the Appendix). Small whitelines along the horizontal axis indicate the locations of individualprofiles. The breaks in the horizontal axis, labelled 1–4, correspondto the waypoints shown in Figure 2.1b. Magenta rectangles withsolid white lines indicate regions of enhanced dissipation discussedin the text. The magenta rectangle with dashed white line in panel(b) indicates the signature of the mesoscale eddy discussed in the text. 25Figure 2.3 Cross section of the ratio εU/εT in log10 space. Shading, panel di-vision, magenta rectangles, and annotations as in Figure 2.2. . . . . 26Figure 2.4 (a) Average vertical profiles of the dissipation rates εU and εT , ob-tained from shear and temperature microstructure and calculated us-ing a trimmed geometric mean in 25 m vertical bins. Shading indi-cates the 95% confidence interval for the mean as indicated by thegeometric standard error. (b) The ratio of the average vertical pro-files of εU and εT , highlighting disagreement by a factor of 5 orgreater between 75–175 m depth. . . . . . . . . . . . . . . . . . . 27xiiiFigure 2.5 Histograms showing the distributions of all (a) εU and (b) εT ob-servations. The interquartile range (IQR) is indicated by the darkershading; the mode, arithmetic and geometric means, and medianare marked in both panels according to the legend in (a). The la-bels N and σg indicate the total number of observations in eachhistogram and the geometric standard deviation factor respectively.Histograms are calculated over 100 logarithmically spaced bins. (c)Quantile-quantile plot demonstrating the goodness of fit of the his-tograms to idealized lognormal distributions. For each set of data,deciles are marked by grey-shaded circles, and the squared linearcorrelation coefficient, R2, is indicated. . . . . . . . . . . . . . . . 28Figure 2.6 Scatter plot comparison of the two coincident dissipation rate esti-mates εU and εT . Identical agreement and agreement within factorsof 2 and 5 are indicated as labelled. Bin averages are calculated per-pendicular to the one-to-one line (see text). Our empirical estimateof the εU noise floor (3× 10−11 W kg−1) is indicated by the hori-zontal dotted line. Purple shading indicates where both estimates ofε simultaneously lie above 3× 10−11 W kg−1 and also delineatesthe region where bin averages agree within a factor of 2. . . . . . . 31Figure 2.7 Sample coincident shear (a-f: Φ) and temperature gradient (g-l: Ψ)spectra (black) for 6 orders of magnitude of ε , as determined by thetemperature measurements. Bold indicates the wavenumbers ex-plicitly included for integration; the remaining variance is estimatedas described in Sections 2.3.2 and 2.3.3. For Ψ, bold also indicatesthe wavenumber range used for the MLE Batchelor fit (see Section2.3.3).Shear spectra have the accelerometer signal removed (Sec-tion 2.3.2) and temperature gradient spectra have the empirically-determined noise spectra removed (Section 2.3.3). Nasmyth (a-f)and Batchelor (g-l) reference spectra (grey) are also drawn. Batche-lor spectra are those determined by the MLE fitting algorithm whichare used to estimate kB (Section 2.3.3). . . . . . . . . . . . . . . . . 33Figure 2.8 Median nondimensionalized shear (a-d) and temperature gradient(e-h) spectra in bold, for regimes of ReB as indicated. Also shownare the 25th and 75 percentile of data (thin solid line) as well asnondimensionalized reference spectra (dashed line): Nasmyth forshear and Batchelor for temperature gradient. The total number ofspectra used in each calculation is indicated by N. Shear spectrawith εU < 10−10 W kg−1 are excluded. . . . . . . . . . . . . . . . 35xivFigure 2.9 Root-mean-square error between Ψ and ΨB, as defined in the text,visualized as a function of (a) buoyancy Reynolds number, and (b)dissipation rate. It quantifies the degree of divergence between ob-served temperature gradient and theoretical Batchelor spectra. Largeopen faced markers are bin averages. Regressions to subsets of thebin averages are shown in each panel; the subsets are those on eitherside of the datum marked by the circle (inclusive). . . . . . . . . . 37Figure 2.10 Histogram of the ratio εU/εT , highlighting the large number of coin-cident measurements where the shear-derived values overestimatedthe temperature-derived ones in our dataset. Agreement by factorsof 5, 10, and 100 is indicated by dashed lines. The histogram iscalculated over 50 logarithmically spaced bins. . . . . . . . . . . . 38Figure 3.1 (a) Map of the southeastern Beaufort Sea, showing the location ofAmundsen Gulf to the east of the Canadian Beaufort Shelf. Theglider path is shown by the thin black line inside the black rectan-gle. (b) Enlarged view of the region given by the black rectangle inpanel (a), showing the path of the glider. The start and end locationsof the track are shown by the large white rectangles; four interme-diate waypoints are shown by the small white rectangles and num-bered consecutively. The color on the glider’s track-line is watertemperature along the 1026.15 kg m−3 isopycnal, using the samecolour scale as shown in Figure 3.11, indicating the location andspatial scale of the warm-core eddy discussed in the text (Section3.5.3). The white circle is the location of ArcticNet mooring CA08.Bathymetry data are from IBCAO 3.0 (Jakobsson et al., 2012). . . 43Figure 3.2 (a) Arithmetic mean profile and spatial cross section of conservativetemperature. (b) Geometric mean profile and spatial cross section ofstratification. For the mean profiles, grey shading indicates the rangeof the central 90% of data; alternating coloured background shadingindicates the approximate depth ranges of the hydrographic layersdefined in the text (PW, WH, and AW are labelled). For the spatialsections, the horizontal axis is broken and consecutively labelled 1–4 at the waypoints marked in Figure 3.1, indicating where the gliderchanged direction. White rectangle in (a) indicates the mesoscaleeddy discussed in the text. . . . . . . . . . . . . . . . . . . . . . . 48xvFigure 3.3 Histograms of (a) the turbulent dissipation rate, ε , and (b) the buoy-ancy Reynolds number, ReB. For each, the number in the top rightindicates the percentage of data that fall within the axis limits; theremaining data are zero-valued and cannot be displayed on a log-arithmic axis. The interquartile range for each set, including zero-valued data, is the span between the two dash-dotted lines. For ε , thegeometric and arithmetic mean values are also indicated (GM andAM, respectively). For ReB, the approximate critical value Re∗B = 10is indicated by the yellow line. . . . . . . . . . . . . . . . . . . . . 50Figure 3.4 Mean vertical profiles and horizontal cross sections of (a) ε , and(b) ReB. Waypoints are indicated as in Figure 3.2. For each, thegeometric mean profile is given in 25 m bins (blue); for ε , the arith-metic mean profile is also given (black). In both cross sections,the white rectangle between waypoints 2 and 3 identifies the patchof enhanced turbulence discussed in the text. In the ReB cross sec-tion, red pixels indicate where a turbulent diapycnal flux is expected;grey pixels indicate a predicted absence of turbulent diapycnal mix-ing. The approximate critical value Re∗B = 10 is indicated in the ReBmean profile by the vertical yellow line. . . . . . . . . . . . . . . . 52Figure 3.5 The three repeat ε transects (left vertical axis) over the continentalshelf slope. The horizontal axis is the distance from Waypoint 3shown in Figure 3.1b. Thick lines are 2.5 km geometric mean bin-averages of ε; coloured markers in the background are individualgeometric mean cast-averages. The quasi-vertical dash-dotted linesconnect peaks and troughs that appear to be stationary between thethree ε transects, as discussed in the text. The bathymetry is shownwith grey shading in the background (right vertical axis) for reference. 53Figure 3.6 Histograms of (a) the diapycnal mixing coefficient, Kρ , of den-sity and (b) the vertical heat flux, FH . Positive Kρ indicate down-gradient density diffusion; negative Kρ indicate up-gradient densitydiffusion. For FH , the green shaded area indicates the region be-tween the 5th and 95th percentiles. . . . . . . . . . . . . . . . . . . 55Figure 3.7 Arithmetic mean vertical profiles, in 25-m bins, and horizontal crosssections of (a) the diapycnal mixing coefficient, Kρ , of density and(b) the vertical heat flux, FH . For the cross sections, the horizontalaxis, waypoint markers, and white rectangle identifying the turbu-lent patch are as in Figure 3.4. The Kρ cross section depicts theabsolute value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56xviFigure 3.8 (a) Power density spectrum of ε , constructed using Welch’s methodand 4 day segments of data. Grey shading indicates the 95% confi-dence interval. The M2 and inertial frequencies are indicated. (b)The ε time series used to construct the power density spectrum. Theseries is made from the geometric cast-averages of ε for all depthsgreater than 100 m and is interpolated to a 15 minute grid. Variabil-ity on scales smaller than 2 hours has been removed. . . . . . . . . 59Figure 3.9 (a) Depth-averaged current velocity components U and V , measuredby ArcticNet mooring CA08 between depths 100–170 m. The greyshading indicates the period of the glider deployment. (b) Powerdensity spectra of the above U and V records, with 95% confidenceintervals. (c) Polar histograms with current speeds of the above Uand V records, decomposed into high frequency and residual com-ponents. High frequencies are defined as those greater than 1.3 cpdand are dominated by the M2 tide. The approximate orientation ofthe Amundsen Gulf’s major axis, azimuth 305◦, is indicated in eachhistogram by the yellow line. The percentage on each histogram’sperimeter is the tick label for the radial axis (Relative Occurrence). . 60Figure 3.10 Geometric mean vertical profile and horizontal cross section of thedensity ratio, Rρ . In the cross section, data are discretized into threeregimes: susceptible to double diffusion (red: Rρ≤7), marginallysusceptible (yellow: 7<Rρ≤10), and not susceptible (purple: Rρ>10).The approximate critical value Rρ=10 is shown in the mean profileby the yellow vertical line. . . . . . . . . . . . . . . . . . . . . . . 62Figure 3.11 (a) An enlarged view of the temperature cross section of the coldhalocline and Pacific Water layers, highlighting the eddy as well assmaller, O(1) km, temperature anomalies. The dashed white linescorrespond, from left to right, to the three T-S lines shown in thelower three panels. (b) T-S diagrams for the three vertical profilesindicated in the upper panel. Grey dots are all the data shown in theupper panel. Dotted lines are density contours. . . . . . . . . . . . 63xviiFigure 4.1 (a) Distribution of ReB from microstructure data collated in Water-house et al. (2014), between the surface mixed layer and 1000 mdepth, for the following experiments: Fieberling, NATRE, BBTRE1996, BBTRE 1997, GRAVILUCK, LADDER, TOTO, DIMES-West, DIMES-DP. (b) Distribution of halocline averaged ReB from afinescale parameterization of ε using CTD data presented in Chanonaet al. (2018). (c) Map showing the locations of the data used in thehistograms; red indicates microstructure data presented in Water-house et al. (2014), and blue indicates finescale data presented inChanona et al. (2018). . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 4.2 Histograms for the measurements used in this study of (a) turbu-lent dissipation rate, ε , (b) squared buoyancy frequency, N2, (c)buoyancy Reynolds number, ReB, and (d) gradient density ratio, Rρ .There are N = 13,190 data. . . . . . . . . . . . . . . . . . . . . . 69Figure 4.3 (a) Scatterplot of the nondimensionalized net temperature diffusiv-ity, KT/κT , as a function of buoyancy Reynolds number, ReB. Whitesymbols indicate mode, arithmetic mean, and geometric mean inlogarithmically spaced ReB bins. Models from Osborn (1980, mod-ified), Shih et al. (2005), and Bouffard and Boegman (2013) areshown for reference, all normalized by κ molT to facilitate compari-son with K∗. The three red data points are select—but in no wayremarkable—points for which the raw temperature microstructurerecords are shown below. The three turbulence regimes proposedby Ivey et al. (2008)—molecular, transitional, and energetic—areindicated by the background shading—purple, white, and orange,respectively. (b) The ratio of the net temperature diffusivity (Equa-tion 4.6) to the net diffusivity calculated from the modified Osbornmodel (Equation 4.5). Symbols as in panel a, excepting the arith-metic mean which is omitted here because it is not informative. (c)The microstructure temperature records used to calculate the threeselect data points (red) in panels a and b. Each record represents 40 sof measurement, spanning an along-path distance ∆x, a vertical dis-tance ∆z, and a temperature difference ∆T between the first and lastmeasurements. For each 40-s segment of measurement, one temper-ature gradient spectrum is calculated by averaging spectra from 19half-overlapping 4-s subsegments (Section 4.2.3). . . . . . . . . . . 76xviiiFigure 4.4 (a) Histogram of flux coefficient estimates for the subset of datawhere ReB > 10. Dash-dotted lines indicate percentiles 5, 25, 75,and 95. The red triangle indicates the canonical value proposed byOsborn (1980). N indicates the number of data points. (b) Fluxcoefficient plotted as a function of ReB. Large open-faced symbolsare the median and geometric mean values in geometrically-spacedbins. Error bars indicate the geometric standard error in the mean,calculated from two geometric standard deviations. (c) As in panela, but for mixing efficiency, R f . (d) As in panel b, but for R f andwith arithmetic mean values and standard errors in place of geomet-ric ones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure A.1 Comparison of results from individual (a,b) shear probes and (c,d)temperature probes. Bin averages are calculated as in Figure 2.6.Agreement within a factor of 5 is indicated in all panels by thedashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure A.2 Overview of select results, separated by upcast and downcast. Foreach of εU and εT , we show the histograms (a,e), averaged verticalprofiles (b,f), and selected spectra (c–d,g–h) separated in this man-ner. The spectra shown are those corresponding to dissipation rateswithin a factor of 1.1 of 10−9 W/kg. Thick black lines depict themedian of the selected spectra at each wavenumber. . . . . . . . . . 105xixList of AcronymsAM Arithmetic meanAW Atlantic Water, as in Atlantic Water layerBB Diffusivity model proposed by Bouffard and Boegman (2013)CA08 ArcticNet mooring in central Amundsen GulfCH Cold haloclineCTD Conductivity-Temperature-Depth profilerFFT Fast Fourier TransformFP07 Fast response microstructure thermistor produced by RSIGM Geometric meanGPS Global Positioning SystemIBCAO International Bathymetric Chart of the Arctic OceanIQR Interquartile rangeITP Ice-Tethered ProfilerM2 Principal lunar semi-diurnal tidal constituentMLE Maximum Likelihood EstimatorPW Pacific Water, as in Pacific Water layerQ-Q Quantile-quantile, as in quantile-quantile plotQC Quality controlRSI Rockland Scientific International Inc.SKIF Diffusivity model proposed by Shih et al. (2005)xxSML Surface mixed layerSPM-38 Microstructure airfoil shear probe produced by RSITEOS-10 Thermodynamic Equation of Seawater - 2010TKE Turbulent Kinetic EnergyxxiList of Mathematical SymbolsUnits used in this thesis are given in brackets. These are common to physical oceanogra-phy and typically, but not always, comply with the International System of Units.Glider variablesαa Angle of attack [◦], measured positive upward relative to the glide angle γA Constant in Merckelbach et al. (2010) hydrodynamic flight modelCD1 Constant in Merckelbach et al. (2010) hydrodynamic flight modelx Distance coordinate along the glider’s path through water [m]CD0 Drag coefficient in Merckelbach et al. (2010) hydrodynamic flight modelγ Estimated glide angle [◦], measured positive upward from the horizontalθ Glider pitch [◦], measured positive upward from the horizontalU Glider speed through water along the direction of travel [m s−1]W Glider vertical speed through water [m s−1]s Horizontal along-track distance coordinate [m]t Time coordinate [s]z Vertical coordinate [m]Data ProcessingqB Batchelor constant in the theoretical Batchelor spectrum [–]kB Batchelor wavenumber [cpm]xxiiΨB Batchelor form for the temperature gradient power spectrum [K2 cpm]sˆ Calibration constant for shear probes [–]χobs Component of χ obtained by integrating Ψ between kl and ku [K2 m−1]Ψns Empirically determined temperature gradient noise spectrum [K2 cpm]ΦN Empirical Nasmyth power spectrum for shear variance [(s2 cpm)−1]f Frequency [Hz]χhw High-wavenumber correction term for χ [K2 m−1]kν Kolmogorov wavenumber [cpm]kl Lower integration limit in k-space [cpm]χlw Low-wavenumber correction term for χ [K2 m−1]Φ∗ Nondimensionalized shear power spectrum [–]Ψ∗ Nondimensionalized temperature gradient power spectrum [–]kˆ Radian wavenumber [m−1], where kˆ = 2pikΨr Raw, i.e. uncorrected, temperature gradient power spectrum [K2 cpm]Φ Shear power spectrum [(s2 cpm)−1]k1 Smallest nonzero wavenumber [cpm]Ψ Temperature gradient power spectrum [K2 cpm]Λ Temperature power spectrum [K2 cpm−1]ku Upper integration limit in k-space [cpm]Ep Voltage across piezoelectric beam in response to trans-axial force [V]k Wavenumber [cpm]Seawater and sea ice propertiesSA Absolute salinity [g kg−1]N Buoyancy frequency [s−1]T Conservative temperature [◦C]xxiiif Coriolis parameter [s−1]ρi Density of sea ice [kg m−3]U Eastward component of current velocity [m s−1]β Haline contraction coefficient of seawater [(g/kg)−1]ν Kinematic viscosity of seawater [m2 s−1]; may also be interpreted as themolecular diffusivity of momentumlo Latent heat of melting sea ice [J kg−1]T AW Mean temperature of Atlantic Water layer [◦C]To Melting temperature of sea ice [◦C]V Northward component of current velocity [m s−1]σ Potential density anomaly [kg m−3], defined ρ−1000 kg m−3ρ Potential density of seawater [kg m−3]Pr Prandtl number (for temperature) [–], equal to ν/κmolTcp Specific heat capacity of seawater [J (kg K)−1]E Thermal energy sequestered in the Atlantic Water layer [J m−2]α Thermal expansion coefficient of seawater [K−1]Statisticsσg Geometric standard deviation factor [–]N Number of data points [–]ξrms Root-mean-square error [–]R2 Squared linear correlation coefficient [–]Turbulence and MixingB Buoyancy flux [kg m−1 s−3]ReB Buoyancy Reynolds number [–], equal to ε/νN2C Cox number [–]xxivRe∗B Critical value for Buoyancy Reynolds number, Re∗B = 10χ Dissipation rate of thermal gradient variance [K2 s−1]ε Dissipation rate of turbulent kinetic energy [W kg−1 or m2 s−3]εT Dissipation rate of turbulent kinetic energy estimate derived fromtemperature microstructure measurements [W kg−1]εU Dissipation rate of turbulent kinetic energy estimate derived from velocityshear microstructure measurements [W kg−1]Γ Flux coefficient [–] in the Osborn (1980) model, defined as R f /(1−R f )Γo Flux coefficient, canonical value, 0.2, from Osborn (1980)R f Flux Richardson number [–]; may also be interpreted as the efficiency ofturbulent mixing [–]R f o Flux Richardson number, canonical value, 0.17, from Osborn (1980)g Gravitational acceleration [m s−2]Rρ Gradient density ratio [–], to quantify susceptibility to double diffusionLK Kolmogorov length scale [m]κ molρ Molecular diffusivity of density in seawater [m2 s−1]κ molS Molecular diffusivity of salinity in seawater [m2 s−1]κ molT Molecular diffusivity of temperature in seawater [m2 s−1]Kρ Net diffusivity of density [m2 s−1]KT Net diffusivity of temperature [m2 s−1]K∗ Nondimensionalized temperature diffusivity [–]LO Ozmidov length scale [m]P Production rate of turbulent kinetic energy [kg m−1 s−3]κ turbρ Turbulent diffusivity of density in seawater [m2 s−1]κ turbT Turbulent diffusivity of temperature in seawater [m2 s−1]u′ Turbulent velocity component perpendicular to the direction of travel [m s−1]FH Vertical heat flux [W m−2]xxvAcknowledgmentsAn honest and heartfelt thank you to my advisors Stephanie Waterman and Jeff Carpen-ter for supporting me, believing in me, and guiding me through my PhD. In addition,Professors Susan Allen, Rich Pawlowicz, and Tara Ivanochko have been important rolemodels who helped instil in me a passion for oceanography and for teaching. Thankyou also to my friends and fellow grad students in the Waterhole; Tara Howatt deservesa special mention for tirelessly listening to me vent about my academic struggles andfailures, and for celebrating with me and sharing in my successes.It goes without saying that I could not be here, at the end of this PhD-road, without thefamily, friends, and colleagues who walked beside me on the journey. Your generosity,patience, and kindness inspire me. You know who you are, and I hope that you eachsee a little bit of yourself reflected in the work presented here. My accomplishments areyours as much as they are mine.xxviFor the preservation of our natural world. May it remain beautiful for generations tocome.Ad maiorem Dei gloriam.xxviiChapter 1Introduction1.1 Project OverviewThis thesis was initially motivated by the central theme of what is now its third chapter,a study of the turbulence and mixing characteristics in Amundsen Gulf, in the southeast-ern Beaufort Sea, using a series of original and tightly resolved ocean microstructuremeasurements from an ocean glider. Direct observations of turbulent mixing in theBeaufort Sea are rare by any measure, but ours are the first to have been collected herewith a robotic platform that allows for the high spatial and temporal resolution requiredto observe the stochastic nature of ocean turbulence. Because turbulence in the oceanis inherently patchy and intermittent, characterizing it from field observations requiresmany densely spaced measurements that can accurately represent the rare but importantenergetic mixing events that dominate tracer fluxes. Proposing to use an autonomousocean glider to carry out these measurements provided us with a way to overcome thischallenge and so, with the prospect of an exciting and novel series of measurements,we set out to plan, organize, and execute the necessary field work. We used the gliderto measure both shear and temperature microstructure in Amundsen Gulf and, in so do-ing, collected what is—to our knowledge—the single largest set of ocean turbulencemeasurements in the Canadian Arctic to date. The field campaign was overwhelm-ingly successful, though as we began to analyze the measurements, it quickly becameclear that the novelty of the data we had collected opened a wholly new set of questions.Largely, these were questions that we hadn’t previously realized needed asking but with-out which we wouldn’t be able to do justice to the original research aims of the project.Those questions could be broadly grouped into two major categories which then becamethe topics of what are now Chapters 2 and 4 of this thesis.The first of these topics—the need for which became painfully obvious during the early1CANADA BASINBeaufort SeaMackenzie River DeltaALASKAYUKONNWTBanksIslandStart25/08End05/091234Figure 1.1 – The glider’s 10 day path in the Amundsen Gulf along which it measured 348microstructure profiles in summer 2015. Shown are the start and end points/dates, as wellas the location of four intermediate waypoints. The light-blue line indicates approximatelythe north-eastern boundary beyond which it was unsafe to operate the glider because of thepossibility of sea ice.stages of the analysis—was understanding why the results we obtained from the shearand temperature measurements were not the same. The results agreed well when tur-bulence was reasonably energetic, but when it became weaker, the dissipation rate esti-mates we derived from the two types of measurement differed by as much as two ordersof magnitude. This difference was not something that could be relegated to the surpris-ingly ubiquitous black box of “uncertainties typical to microstructure measurements”often invoked for ocean turbulence data; rather, it was substantial enough that it had ameaningful impact on the final interpretation of the measurements and, we felt, it couldnot be ignored or averaged away1. It needed to be explained. In addition, the comingdecade is almost certain to see an expansion in the number and scope of Arctic Oceanmicrostructure studies (Carmack et al., 2015), and since the divergence—which we de-termined was largely a result of sensor limitations—was noticeable only because of theBeaufort Sea’s uncharacteristically-weak turbulence, we concluded that sorting out thereasons for the large difference between the shear- and temperature-derived results, andreporting those publicly, was an important and worthwhile endeavour. That endeavourbecame the study that is now Chapter 2, the conclusions of which, we hope, will help in-form the future collection and interpretation of microstructure measurements in weaklyturbulent environments.The second important thing we noticed very early in the analysis was that density strati-1This is not intended to sound irreverent. Consider, as an example, the following assessment by MikeGregg, one of the early pioneers of microstructure measurement: An interesting dichotomy exists betweenthe kinematical models of mixing and the analysis of microstructure data. The kinematical models predictthat all mixing occurs as short-lived overturns formed when the superposition of random internal wavemotions causes the Richardson number to drop below 1/4. On the other hand, measurements of temperaturemicrostructure are interpreted with a model based on the assumption of steady, homogeneous turbulence.Those of us who examine oceanic data have long recognized this incongruity, hoping that it all works outin the averaging. This, however, has yet to be demonstrated. (Gregg, 1987, Section 7)2fication in the environment we were working in was very clearly a dominant contributorto the turbulence dynamics. In one sense this was obviously a problem, and it wouldprove to plague the interpretation of our measurements throughout the project becausethe processing of turbulence measurements relies very explicitly on the assumption thatthe smallest, viscous-scale, turbulent eddies are isotropic and, therefore, unaltered bybuoyancy effects. However, it also led to one of the most fascinating findings of theproject, which is that turbulent-scale variance in the temperature field (and the associ-ated enhanced heat flux) never fully vanished in any of our observations even when mod-els based on laboratory and numerical studies predicted that it should do so given therelatively strong density stratification. We observed a meaningful turbulent heat flux inenvironmental conditions in which present models predict only vanishing tracer fluxes,which was both surprising and exciting. The observation of turbulent heat fluxes insuch weakly turbulent, strongly stratified conditions highlights the limitations of presentturbulence models that are used to interpret microstructure measurements and make pre-dictions about tracer fluxes. Given that much of the global ocean pycnocline is charac-terized by similar strong-stratification-and-weak-turbulence conditions, this observationhas the potential to impact how we conceptualize mixing in the ocean well beyond onlyin the relatively small, localized region of our measurements. The discussion of thisobservation and its potential implications is now presented in Chapter 4.1.2 Research QuestionsGiven the context in Section 1.1, and with the liberty of retroactive motivation, we canusefully define the science topics of this project in the following three subsets, each withits own series of research questions:i. Observing weak turbulence in strong stratification• Estimates of the turbulent kinetic energy (TKE) dissipation rate, ε , are foun-dational to quantifying oceanic diffusivity. However, when turbulence isvery weak, microstructure sensors function at their operational limit. Dosensor limitations hinder the ability to formulate meaningful ε estimates inthese conditions?• If sensor limitations do impact the ability to measure ε when turbulence isweak, as it is in much of the Beaufort Sea, in what conditions and to whatextent do they do so? And, what is the impact of those limitations on theinterpretation of the measurements?• How are uncertainties in ε estimates impacted when turbulence is weak3and stratification is strong and the assumption of isotropic, homogeneous,steadily forced turbulence becomes increasingly intractable?ii. Turbulent mixing in the Arctic Ocean’s Amundsen Gulf• What are the turbulence and mixing characteristics in the Amundsen Gulfregion of the Beaufort Sea? Can we develop statistical metrics of ε andturbulent diffusivity, Kρ , and describe their spatial and temporal variability?• What is the magnitude of vertical heat fluxes associated with turbulent mix-ing in Amundsen Gulf? Is it significant when compared to mean heat budgetestimates of the region and in light of recent increases in sea ice loss?• What physical mechanisms are responsible for the observed turbulence andmixing characteristics in this region?iii. Enhanced heat fluxes in strongly stratified, weakly turbulent environments• Can we observe the transition between turbulent and molecular diffusion inthe real ocean when turbulence weakens and stratification remains strong?• How do predictions of turbulent mixing from models compare to our ob-servations of tracer variance when turbulence is weak and stratification isstrong?• How efficient is turbulent mixing in our observations, and how does thisefficiency compare to the canonical value of 20%?1.3 BackgroundThis section provides a high-level overview of our understanding of ocean turbulence,how we measure it, how it relates to ocean mixing, and why we care about it in Amund-sen Gulf. It is useful for context but, depending on the reader’s background, not essentialfor understanding the main objectives of this thesis and can be skipped if the researchresults are the reader’s primary aim.1.3.1 Ocean Mixing and TurbulenceOcean mixing is, along with advection, one of the primary mechanisms by which tracersare redistributed throughout the world oceans. It is directly linked to global biologicalproduction because it contributes to the availability of heat, nutrients, and oxygen inocean ecosystems (e.g. Sarmiento et al., 2004), and it acts as a control on global climate4patterns because it facilitates the meridional overturning circulation and the maintenanceof the ocean thermocline (Lumpkin and Speer, 2007). As summarized concisely byMunk and Wunsch (1998), “without deep mixing, the ocean would turn, within a fewthousand years, into a stagnant pool of cold salty water”.Away from boundaries like the sea bottom, the continental margins, or the air-sea in-terface, irreversible mixing in the ocean appears to be driven primarily through tur-bulence that is created when internal gravity waves become dynamically unstable andbreak (MacKinnon et al., 2017). These internal waves typically come from one of threesources:i. barotropic ocean tides that are forced over rough or anomalous topography, cre-ating “internal tides” that radiate upwards from the sea floor with a frequencycharacteristic of the barotropic forcing, often that of the dominant diurnal or semi-diurnal tidal constituent (e.g. Garrett and Kunze, 2007);ii. winds that force inertial oscillations in the surface mixed layer, from which “near-inertial” internal waves propagate downwards at frequencies near the Earth’s in-ertial frequency (e.g. Alford, 2003; Plueddemann and Farrar, 2006); oriii. low frequency flows that are forced over rough or anomalous topography, continu-ally creating internal “lee waves” in their wake (e.g. MacKinnon, 2013; Nikurashinand Ferrari, 2013).Modern process studies of ocean mixing tend to focus on the characteristics and effectsof one of these processes, often with the goal of parameterizing the process in globalocean and climate models since the phenomena themselves occur on scales smaller thanmodel grid scales. A comprehensive review of recent advancements in our knowledgeof each process is given by MacKinnon et al. (2017).The most complete collection of ocean mixing rate estimates to date, by Waterhouseet al. (2014), reports globally averaged diapycnal diffusivities of O(10−4) m2 s−1 below1000 m depth and O(10−5) m2 s−1 above 1000 m depth. The variability is large—typical average diffusivities range between O(10−6)–O(10−2) m2 s−1—but the globallyaveraged values are consistent with the estimates put forward by Lumpkin and Speer(2007) required to support the global overturning circulation. Temporal variability inocean mixing averages appears to occur mostly at seasonal and tidal frequencies, con-sistent with variability in generating mechanisms driven by winds and tides (e.g. Whalenet al., 2012; Dosser and Rainville, 2016). The spatial geography of mixing is compli-cated, but there is clear evidence that mixing is substantially increased in regions withcomplex or steep topography, such as along the continental slope margins or over mid-ocean ridges (Polzin et al., 1997; Waterhouse et al., 2014; Rippeth et al., 2015). It is5commonly thought that these regions therefore contribute disproportionately to large-scale water mass transformations, though specific regional sampling is often still toosparse to describe the regional mixing geography in detail. Regional downscaling ofocean mixing observations continues to inform ongoing research questions, includingthose of Chapter 3 of this thesis.Despite the general lack of regional resolution, we have in the last decade accumulatedsufficient measurements to begin to see global patterns in ocean mixing rates (Whalenet al., 2012; Waterhouse et al., 2014), in large part due to the utility of autonomoussampling systems (specifically, the Argo and Ice-Tethered-Profiler systems). However,the vast majority of these diffusivity estimates rely on parameterizations of fine scale (i.e.O(1)–O(10) m) measurements to characterize the effects of centimetre-scale turbulence(Polzin et al., 2014). It is this smaller-scale turbulence that is ultimately responsible forcreating the irreversible mixing that defines water mass transformations, so it is essentialthat the fine scale estimates always be compared to measurements of ocean turbulencewhich can be related directly to the rates of tracer diffusion (as described in Sections1.3.3–1.3.6). This is the primary way in which we validate the effectiveness of the moreeasily employed fine scale parameterizations of turbulent mixing.The fundamental techniques used to measure ocean turbulence were developed in the1970s and early 1980s (see Lueck et al. (2002) for a review) and, though gradually im-proved with newer iterations, have remained largely unchanged since that time. Thetheoretical underpinnings behind the interpretation of those measurements (as outlinedby Osborn and Cox (1972) and Osborn (1980)) have also remained largely the same,with the notable exception that there has been a recent renewed interest in developinga better understanding of the efficiency of turbulent mixing (the proportion of turbulentkinetic energy converted to potential energy through mixing). This efficiency was tra-ditionally considered to be a constant 20% based on theoretical and laboratory resultscompiled by Osborn (1980), but it has now become exceedingly clear that this quan-tity is variable in the real ocean and dependent on the time evolution of the turbulence(Gregg et al., 2018).Traditionally, ocean turbulence measurements have been prohibitively difficult to collecton a large scale because traditional free-falling microstructure profilers are expensiveand tedious to operate and because microstructure probes break easily and, in the caseof shear probes, are prone to contamination from vibrational noise (Lueck et al., 2002).Logistic difficulties are further amplified in the Arctic Ocean because of its remotenessand harsh environmental conditions and by the presence of sea ice. As a result, only arelatively small number of researchers have measured turbulence in the Arctic Ocean—notable studies from the western Arctic include Padman and Dillon (1987); Rainville andWinsor (2008); Bourgault et al. (2011); Shroyer (2012); Shaw and Stanton (2014); Rip-6peth et al. (2015). However, it has recently been demonstrated that autonomous oceangliders are exceptional platforms for microstructure (i.e. turbulence) measurements, onpar with the best free falling profilers (Fer et al., 2014; Peterson and Fer, 2014). Fur-ther, because they do not require continued manual labour or ship time, they are ableto collect turbulence measurements at a much higher spatial density, and in worse seaconditions, than can be practically collected from traditional profilers lowered into theocean from the side of a ship.The high density of measurements available from a glider also naturally addresses con-cerns that arise from two fundamental problems inherent in turbulence measurements:the patchiness and intermittency of turbulent overturns. It has long been established (e.g.Gregg, 1987) that turbulence is a temporally intermittent process that occurs in isolatedpatches in space. As a result, turbulent variables such as the turbulent components ofshear variance (Section 1.3.4) and temperature variance (Section 1.3.5) are lognormallydistributed in the ocean (Gibson, 1987); consequently, turbulent variables need to besampled at a high temporal and spatial resolution if the mean properties of the sampledistributions need to reflect those of the underlying population distributions (as they dowhen, e.g., characterizing mean mixing rates). One important function of this presentthesis is to demonstrate that gliders help to alleviate concerns related to under-samplingbecause, unlike traditional ship-based platforms, gliders can measure turbulence contin-ually over many days or weeks and with a tightly resolved spatial resolution.1.3.2 Turbulence Measurements from GlidersWe opted to use a Slocum G2 ocean glider as the platform for our turbulence measure-ments. Gliders are autonomous underwater vehicles that propel themselves by adjustingtheir density relative to that of the ambient seawater, allowing the positive or negativebuoyancy to accelerate them vertically through the water column. Horizontal motioncomes from hydrodynamic lift that is created by the body and wings of the instrumentas moves it vertically, resulting in a characteristic vertical zig-zag profiling pattern sim-ilar to that of traditional ship-based tow-yo profiles. A nominal glide angle is about 25◦from the horizontal, with a typical profiling speed of about 35 cm s−1. The headingbetween pre-programmed waypoints is maintained with a small digital tail-fin.Because glider propulsion requires no moving parts except at the turnaround points atthe tops and bottoms of profiles, gliders have proven to be ideal platforms for turbu-lence measurements, which tend to be sensitive to mechanical vibrations. Gliders canresolve exceptionally small turbulence signals without vibrationally-induced noise con-tamination, resulting in a measurement quality that is comparable to that from traditionalfree-falling vertical microstructure profilers (Fer et al., 2014). This capability is one of7the reasons we are able to address the first set of research questions of this thesis wherewe compare the very low-end of measurable turbulence signals from shear and tem-perature probes; without a vibration-free platform, the lower limit of the measurementswould not be available to us because it would be masked by vibrational noise.While gliders are becoming standardized technology—they are now well into their sec-ond decade of use for oceanographic science—the collection of microstructure mea-surements from gliders is still relatively novel. For Slocum gliders, the first proof-of-concept for including a self-contained microstructure measurement package, known asa MicroRider and produced by Rockland Scientific International Inc., was published byWolk et al. (2009). The first published field study using a MicroRider and Slocum gliderwas by Fer et al. (2014); since then, the same configuration has been used for variousapplications in studies by Peterson and Fer (2014); Palmer et al. (2015); Schultze et al.(2017); St Laurent and Merrifield (2017), and Merckelbach et al. (2019).1.3.3 The Dissipation Rate of Turbulent Kinetic EnergyThe TKE dissipation rate, ε , is the most common quantitative proxy for “turbulence in-tensity” in oceanographic observations. Formally, it is the rate at which viscous frictionwithin the interior of a fluid removes kinetic energy from a flow. Because kinematicviscosity becomes a dominant force only at small scales, the rate at which it dissipatesenergy is a measure of dissipative-scale fluid motion. That rate is defined asε ≡ 2ν 〈si jsi j〉 , (1.1)written using index notation2 and brackets 〈〉 to denote an ensemble average. The factorν is the kinematic viscosity, and the term si j is the strain-rate tensor:si j =12(∂u′i∂x j+∂u′j∂xi)(1.2)where u′i and xi are components of the 3-dimensional turbulent-velocity and positionvectors, respectively3. In the field, usually only one or two of the nine strain-rate tensorcomponents are measured at any given time, so in practice oceanographers typicallyassume that all perpendicular velocity derivatives in (1.2) are equal—i.e. we assumeisotropy—and the summation in (1.1) collapses to a single term (Taylor, 1935), givingthe simplified relation:ε = (15ν/2)〈(∂u′/∂ z)2〉. (1.3)2sum repeated indices: si jsi j = s11s11 + s12s12 + s13s13 + s21s21 + s22s22+ . . .3for example, s13 = ∂u/∂ z+∂w/∂x8In this example, we are representing ε with the horizontal velocity component u′ and thevertical coordinate z, but since the flow is assumed isotropic, any pair of perpendicularvelocity and space coordinates may be used in their stead. The rate ε has units m2 s−3or W kg−1 and varies in the ocean by over 10 orders of magnitude, typically in the rangeO(10−11) to O(10−2) W kg−1.1.3.4 Shear MicrostructureFrom Equation 1.3, the variance of turbulent-scale velocity shear is linearly proportionalto the TKE dissipation rate, so the problem of estimating ε reduces to measuring ∂u′/∂ z.We do this following the traditional approach of Osborn and Crawford (1980), usingan airfoil shear probe made from a piezoelectric beam that produces a voltage, Ep, inresponse to a small trans-axial force. With knowledge of the profiling speed, U , forcemeasurements are transformed into ∂u′/∂ z estimates according to∂u′∂ z=1sˆ U 2dEpdt, (1.4)where sˆ is a manufacturer-determined calibration constant. A comprehensive review ofthe history and requirements of microstructure probes is given in Lueck et al. (2002).In practice, the calculation of the shear signal variance is always done in Fourier spaceby integrating power density spectra of small subsets of the shear measurements. Per-forming the calculation in Fourier space has two distinct advantages: i. it allows one toexclude components of the signal at wavenumbers contaminated by measurement noise,electronic or otherwise (e.g. Goodman et al., 2006), and ii. it allows comparison to a“universal” shape of turbulence shear spectra, which can provide insight into the natureof the measurement and/or the flow (Figure 1.2a). For example, in reference to thissecond point, shear spectra in fully developed turbulence are anticipated to exhibit aninertial subrange where kinetic energy is passed inviscidly from larger, more energeticeddies to small viscous eddies that dissipate energy (whose wavenumber range is knownas the dissipative subrange). The integration of the power density spectra occurs overa mix of the inertial and dissipative subranges, as available: a larger portion of the in-ertial subrange is typically resolved in energetic turbulence while more of the viscoussubrange is usually resolved in weak turbulence. Details about our variance calculationare given in Section 2.3.2; for greater detail about the subranges of turbulence spectra,the reader is referred to (Shroyer et al., 2017).9Figure 1.2 – (a) Example measured shear spectrum (Φ, black line) with a fitted empirical Nas-myth spectrum (blue line). The upper end of the inertial subrange can be seen to the left ofthe “viscous rolloff“, indicated by the purple shading. Yellow shading indicates the viscoussubrange where viscosity begins to remove kinetic energy from the turbulent flow. The Kol-mogorov wavenumber is shown by the vertical dashed line. The dissipation rate estimate εU isalso given (the subscript U indicates a velocity shear-derived estimate). (b) Example measuredtemperature gradient spectrum (Ψ, black line) with a fitted theoretical Batchelor spectrum (blueline). The dissipation rate estimate εT (subscript T indicates a temperature gradient-derived es-timate) is indicated. The Batchelor wavenumber is shown by the vertical dashed line.1.3.5 Temperature MicrostructureThe dissipation rate ε can also be estimated from microstructure measurements of tem-perature (or any other tracer), though the connection is less direct than it is for the shearmeasurements. Batchelor (1959) used an advection-diffusion balance for temperature toderive a theoretical spectrum for temperature gradients in the vicinity of the dissipativescale,kB =εν(κ molT )2, (1.5)where κ molT is the molecular diffusivity of temperature. The scale kB is known as theBatchelor wavenumber and quantifies the length scale at which the sharpening of tem-perature gradients by turbulent shear is balanced by the softening of those gradientsthrough the molecular diffusion of heat.This spectrum can be written analytically (Appendix A.3) and is a function of ε andthe dissipation rate of temperature variance, χ . Therefore, if we can observe the powerdensity spectrum of temperature gradients, we can determine an estimate of ε by fittingthe theoretical spectrum to the observed spectrum (Figure 1.2b). The fitting procedure iscomputationally expensive and contains a degree of ambiguity because it takes place intwo dimensions (ε and χ) but has been optimized by Ruddick et al. (2000) and appearsto produce accurate estimates of ε .The dissipation rate χ is the rate at which molecular thermal diffusion smoothes micro-scale gradients of temperature and reduces the variance of those gradients. It is linearly10proportional to that variance and is formally defined asχ ≡ 2κ molT〈(∇T ′)2〉. (1.6)In addition, it is possible to derive estimates of the local vertical heat flux from measure-ments of temperature microstructure using the method developed by (Osborn and Cox,1972). In this model, the rate of temperature diffusion is directly related to the variancein the turbulent-scale temperature gradients, and the turbulent temperature diffusivity isgiven byκ turbT = κmolT3〈(∂T ′/∂ z)2〉(∆T/∆z)2, (1.7)assuming isotropy in the temperature gradients at turbulent scales.1.3.6 The Osborn Model for MixingFor most applications, the dissipation rate ε is uninteresting by itself; typically, oceanog-raphers care about it because they are interested in the diapycnal diffusivities of density,temperature, or other tracers, since these are the physical quantities needed to modelthe distribution of ocean properties. Osborn (1980) used the turbulent kinetic energyequation to relate the turbulent diffusivity of density, κ turbρ , to ε via the relationκ turbρ =(R f1−R f)εN2. (1.8)Here, N is the buoyancy frequency and R f is the flux Richardson number. To arrive atEquation 1.8, Osborn assumed steady-state and neglected all divergence terms, balanc-ing the production, P , of TKE with a loss to thermal energy by viscous dissipation anda loss to potential energy by a buoyancy flux, B:P =−ρε+B , (1.9)where ρ is the density of seawater. The buoyancy flux is defined B=−g〈u′3ρ ′〉, where gis the gravitational acceleration. Defining R f as the efficiency by which turbulence pro-duces a buoyancy flux, R f = B/P , and using the definitions Kρ ≡ −〈ρ ′u′3〉/(∂ρ/∂ z)and N2 ≡−g(∂ρ/∂ z)/ρ , this balance leads directly to Equation 1.8 without further as-sumptions. Based on theoretical work and then-current laboratory experiments, Osbornrecommended using R f = 0.17 for a mixing efficiency.111.3.7 Amundsen GulfThe Amundsen Gulf is a large embayment in the southeastern Beaufort Sea, directlyadjacent to the Canadian Arctic Archipelago. It is about 250 km east of the MackenzieRiver’s delta and heavily influenced by its outflow in the spring and summer (Carmackand MacDonald, 2002). It is also the site of the Cape Bathurst polynya, the largestrecurring polynya in the Beaufort Sea and part of the larger panarctic circumpolar flawlead system (Barber et al., 2010). These factors make the Amundsen Gulf a regionallyimportant oceanographic environment and endow it with a unique ecological, social, andeconomic significance within the larger Beaufort Sea.Polynyas are notoriously and disproportionately important to the ecology of Arctic re-gions because the perennial open water results in uncharacteristically long growing sea-sons and unusually high primary productivity (Stirling, 1980; Tremblay et al., 2002).The success in primary productivity is passed to higher trophic levels and supports largepopulations of fish, seabirds, and mammals; the Amundsen Gulf in particular is knownto be the site of some of the largest aggregates of animals—polar cod, ringed seals, polarbears, beluga and bowhead whales—found anywhere in the Arctic (e.g. Harwood andStirling, 1992; Stirling, 2002; Asselin et al., 2011; Geoffroy et al., 2011). This ecologicalsuccess also makes the Amundsen Gulf region important to the cultural identities of lo-cal indigenous societies. These have hunted for subsistence along the broader CanadianBeaufort Shelf for nearly a millennium and continue to do so sustainably to the presentday (McGhee, 1988; Harwood et al., 2002), though recent climate change is driving thephysical and ecological environments to a “new normal” state (Serreze and Barry, 2011;Jeffries et al., 2013) and introducing uncertainty about socioeconomic adaptability andthe loss of cultural identity in northern communities (Berkes and Jolly, 2001; Ford et al.,2006, 2007; Post et al., 2009; Adger et al., 2013).Additionally, the decrease in summer sea ice that results from modern climate changebegets an increased interest in the economic role of the Amundsen Gulf region. TheAmundsen Gulf is the Northwest Passage’s western entrance to the Canadian ArcticArchipelago, which became fully navigable for the first time in 2007 (Cressey, 2007)and may become a major future commercial shipping lane as summer sea ice contin-ues to decrease (Prowse et al., 2009; Khon et al., 2010). The first commercial bulkcargo ship transited the Northwest Passage in September 2013 (McGarrity and Gloys-tein, 2013), and it appears likely that shipping traffic throughout the region will expandrapidly in the coming decades (Prowse et al., 2009; Miller and Ruiz, 2014). For compar-ison, the Northeastern Passage along the northern Russian coast, which is losing sea iceand becoming navigable at a quicker rate than its western counterpart, has seen a 20%exponential increase in shipping traffic year-over-year between 2009–2013 (Miller and12Ruiz, 2014); it is likely that the Northwest Passage will see a similar rise in shippingtraffic as it becomes increasingly ice-free.Each of these factors—the ecological, social, and economic reasons for the significanceof the Amundsen Gulf—are inherently dependent on the physical oceanography of theregion. Ocean mixing, in particular, is a primary control mechanism on the biologi-cal production potential of the region because it at least partially determines the rateat which nutrients are supplied to the surface mixed layer from deeper waters (Bour-gault et al., 2011). In order to predict the ecological response of the region to futurechanges in climate forcing, it is important that the science community has a comprehen-sive understanding of the regional geography and intensity of ocean mixing at present.We cannot model potential future changes to the oceanographic environment if we donot understand its current state (Carmack and MacDonald, 2002; Rainville et al., 2011).Likewise, ocean mixing is an important driver of local heat fluxes in the water columnthat directly contribute to the integrated heat budget and rate-of-loss of surface sea icein the Arctic Ocean. If we wish to predict future contributions of ocean heat to regionalsea ice loss, we first need a grounded, quantitative understanding of heat fluxes in theregion presently, without which we cannot predict future changes to the regional sea icepack (Carmack et al., 2015).13Chapter 2Measuring the Dissipation Rate ofTurbulent Kinetic Energy inStrongly Stratified, Low EnergyEnvironments2.1 MotivationThe purpose of this study is to examine the agreement between measures of the turbulentkinetic energy (TKE) dissipation rate, ε , derived from measurements of shear and tem-perature microstructure in a stratified “low energy” environment, i.e. a stratified environ-ment where the amount of turbulent kinetic energy in the flow field is unusually small. Itwas motivated when, in an attempt to quantify turbulent mixing in the Beaufort Sea ther-mocline, we discovered that results from the two measurements diverged strongly at lowε . Where TKE dissipation rate estimates from shear measurements frequently clusterednear a clearly defined lower limit, estimates from temperature measurements distributedto much lower values that were often multiple orders of magnitude smaller. We notedthat this discrepancy may have serious implications for how shear microstructure mea-surements from the Arctic Ocean are interpreted. This study is therefore dedicated todescribing the divergence we observed and discussing its causes with the goal of in-forming the collection and interpretation of microstructure measurements in the ArcticOcean or similar stratified low energy environments.The western Arctic Ocean, where we collected our measurements, is known to be anexceptionally low energy, and highly stratified, ocean environment with some of the14lowest estimates of oceanic turbulence in the world (e.g. Guthrie et al., 2013; Lincolnet al., 2016). Only a relatively small number of microstructure measurements from thisregion exist to date (e.g. Padman and Dillon, 1987; Rainville and Winsor, 2008; Bour-gault et al., 2011; Shroyer, 2012; Shaw and Stanton, 2014; Rippeth et al., 2015), butthis number is certain to increase in the coming years owing to increased interest inconstraining oceanic heat budgets in the Arctic (Carmack et al., 2015). Constrainingthese budgets requires knowledge of turbulent mixing rates in the ocean which are ob-tained most directly from microstructure measurements; we demonstrate here why thisis a challenging endeavour and why special considerations are needed when interpretingthose measurements in stratified low energy environments. Our present study is there-fore timely since turbulent mixing estimates from microstructure measurements havebecome a key component in estimating heat fluxes through the Beaufort Sea thermo-cline.Both shear and temperature microstructure measurements are frequently used to esti-mate the dissipation rate ε , a quantity that characterizes the intensity of turbulent flowsand can range over more than 10 orders of magnitude in the ocean (Gregg, 1999; Luecket al., 2002). Ours is not the first study to compare estimates of ε from coincident shearand temperature microstructure measurements: similar comparisons were performed byOakey (1982), Kocsis et al. (1999), and Peterson and Fer (2014). These three stud-ies all found excellent agreement, generally within a factor of 2, between shear- andtemperature-derived estimates. Our study, however, is distinct because we focus oncomparing ε estimates at the very low end of reported values where shear probes in par-ticular operate at their lower sensitivity limit. The three previous comparative studiesexamined primarily dissipation rates in excess of 10−10 W kg−1. We will demonstratethat it is necessary to resolve dissipation rates lower than this in the Beaufort Sea, thatε estimates from shear and temperature measurements no longer agree at these smallvalues, and that this disagreement can lead to serious biases in the resulting mixingrate estimates in low energy environments. In addition, we will demonstrate the way inwhich turbulence spectra diverge systematically from commonly used reference shapeswhen turbulence becomes weak and stratification becomes strong. We attribute this di-vergence to a breakdown of the assumption that turbulence in the flow is stationary,homogeneous, and isotropic.152.2 Measurements2.2.1 Measurement Platform: Slocum GliderThe platform for our microstructure measurements was the 1000 m-rated Slocum G2ocean glider Comet, one of the gliders also used by Schultze et al. (2017). The glidersamples autonomously in a vertical sawtooth pattern, surfacing at predetermined inter-vals to update its GPS-fix, send low-resolution flight and hydrography data, and receiveupdated sampling instructions from an onshore pilot. For a detailed review of the oper-ation and utility of gliders, see Rudnick (2016).The glider’s onboard sensors include an SBE-41 (pumped) Seabird CTD measuring insitu conductivity, pressure, and temperature; a three-dimensional compass module mea-suring heading, pitch, and roll; and an altimeter measuring height-above-bottom. Theturbulence measurements are taken with a specialized, externally mounted instrument,described in Section 2.2.2. A moveable weight that controls the pitch of the glider wasset to fixed positions and only moved during inflections at the top and bottom of pro-files to avoid mechanical vibrations that affect the quality of turbulence measurementsmid-profile (Fer et al., 2014).The first published use of gliders as a platform for microstructure measurements is in theproof-of-concept study by Wolk et al. (2009). Gliders have since successfully demon-strated their utility for microstructure measurements in studies by Fer et al. (2014), Pe-terson and Fer (2014), Palmer et al. (2015), and Schultze et al. (2017). These have shownthat 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 aspatial and temporal coverage in oceanic microstructure fields that is often unattainablefrom ship-based profiling, especially in inclement weather. The high density and largenumber of measurements obtained from Comet is an important feature for our studybecause it allows us to calculate robust statistical measures of turbulence metrics whichare critical to interpreting microstructure measurements (Gibson, 1987); these have beenlargely unavailable from previous studies in the western Arctic Ocean where microstruc-ture measurements are sparse.2.2.2 Shear and Temperature MicrostructureComet is equipped with an externally mounted turbulence package (“MicroRider”) car-rying two airfoil velocity shear (SPM-38) and two fast-response temperature (FP07)16probes. The shear probes are of the design by Osborn (1974) and sense transverse forcesin a direction perpendicular to the direction of travel (Osborn and Crawford, 1980; Luecket al., 2002). 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 temperatureprobes are sensitive thermistors with response times of ∼10 ms and sensitivities betterthan 0.1 mK (Sommer et al., 2013b). The microstructure probes extend beyond the noseof the glider by∼17 cm, outside of the radius of flow deformation caused by the glider’smotion (Fer et al., 2014). We do not install a probe guard in order to minimize the poten-tial for contamination of the flow in the immediate vicinity of the measurement.Besides shear and temperature, the MicroRider also measures pressure, pitch, roll, andtransverse accelerations. Shear, temperature, and acceleration are sampled at 512 Hz,the other channels at 64 Hz. The MicroRider is produced by Rockland Scientific In-ternational (RSI); it is the same model used in the four glider-microstructure studiesreferenced in the previous section.2.2.3 Location, Local Hydrography, and Sampling StrategyThe measurement location was the Amundsen Gulf on the southeastern margin of theBeaufort Sea (Figure 2.1a). Circulation in the region is complex and highly variable inspace and time; it is strongly influenced by surface wind stress, complex local bathymetry,submesoscale frontogenesis, and the intermittent presence of mesoscale eddies (Williamsand Carmack, 2008, 2015; Se´vigny et al., 2015). Barotropic tidal amplitudes can beregionally large but are locally small where the glider was deployed because of the pres-ence of a local amphidrome (Kowalik and Matthews, 1982; Kulikov et al., 2004). Thepresence of sea ice is seasonal; during our campaign, the southern edge of fragmentedsea ice was coincident with the mouth of the Amundsen Gulf, prohibiting us from guid-ing the glider north towards the central Beaufort Sea as initially planned. The AmundsenGulf was consequently selected for the measurement locale to mimic as closely as pos-sible the hydrographic characteristics of the wider Beaufort Sea while minimizing therisk of collisions with sea ice floes.The basin depth in Amundsen Gulf is ∼450 m (Figure 2.1b), well below the typicalBeaufort Sea shelf-break depth of ∼75 m and deep enough to extend across the entirerange of the thermocline separating Atlantic- and Pacific-sourced water masses. As aresult, the hydrography of the region largely reflects that of the broader Beaufort Sea(Figure 2.1c–2.1e). A 10 m thick brackish surface lens, resulting from summer sea icemelt and the nearby Mackenzie River’s freshwater inflow, caps a near-surface pycnoclinethat extends to 25 m depth and has a potential density anomaly σ that ranges between22–24.5 kg m−3. Between ∼ 25–200 m, the signatures of cold Pacific-sourced water17Figure 2.1 – (a) The measurement location at the entrance to the Amundsen Gulf in the south-eastern Beaufort Sea. The glider path is shown by the line inside the dashed black rectangle.Bathymetry contours are drawn at 1000 m intervals beginning at 200 m. (b) Enlarged view ofthe area inside the dashed rectangle indicated in panel a, showing the glider path and the localbathymetry. Selected waypoints along the path are numbered consecutively and indicated bysquares for reference when reading Figures 2.2 and 2.3. Contours are drawn at 75 m intervalsbeginning at 50 m. In both panels, colour indicates water depth (m); bathymetry data are fromIBCAO 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 horizontallyaveraged over all casts where the glider’s maximum dive depth exceeded 325 m.dominate the mean temperature profile leading to a temperature minimum of -1.4 ◦C at120 m depth; a spatially complex submesoscale temperature structure is notably visiblein this layer and modifies the mean profile between ∼ 40–110 m depth. A prominentthermocline characteristic of the Beaufort Sea extends from 125 m to the temperaturemaximum associated with the warm core of Atlantic-sourced water (Williams and Car-mack, 2015; Rudels, 2015) at 375 m depth. Stratification is strong throughout the watercolumn, with buoyancy frequency N of O(10−2) s−1 in the near-surface pycnocline andO(10−3) s−1 elsewhere.The glider sampled continuously between August 25 – September 4, 2015, followingthe path outlined in Figure 2.1. It measured 348 quasi-vertical profiles over a total pathlength of 186 km, remaining in the deeper central Amundsen Gulf for the first 4.5 daysand partially crossing the continental shelf slope three times during the remainder of the18mission. The glider dove to a fixed depth of 300 m during the first 3 days; after this, itdove to within 15 m of the local bottom.2.3 Data ProcessingProcessing microstructure measurements from a glider is similar to processing ones froma free falling profiler, but with added complications. Estimating the speed of the mi-crostructure probes through water requires specialized procedures, as does screeningthe data for corrupt measurements. Because measuring turbulence from gliders is stilla 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 estimatingthe glider’s velocity underwater. The quality control criteria we use to flag and discardsuspect measurements, a comparison of TKE dissipation rate results from up- and down-casts, and a brief description of Nasmyth and Batchelor reference spectra are providedin the Appendix.Throughout the text, the symbol ε is used for the TKE dissipation rate generally, εU fordissipation rate estimates obtained from velocity shear measurements, and εT for dissi-pation rate estimates obtained from temperature measurements. All wavenumbers aredefined cyclicly, with units cpm. Note the cyclic wavenumber is related to the radianwavenumber kˆ, which has units m−1, through the relation k = kˆ/2pi . The kinematic vis-cosity, ν , of seawater is evaluated locally using TEOS-10 (McDougall and Barker, 2011)because it varies by more than 20% in our measurements. We use κmolT = 1.44× 10−7m2/s for the molecular diffusion coefficient of temperature and qB = 3.4 for the Batch-elor constant; the latter is required when evaluating the Batchelor spectrum (Section2.3.3), and a sensitivity analysis for this parameter is presented in the Appendix Sec-tion A.3. Measurements from the MicroRider’s clock, pressure sensor, and temperatureprobes are prone to low-frequency drift; thus, the low-frequency response from each ofthese channels is corrected to measurements from the glider. Note that unless otherwisestated, we average quantities that span many orders of magnitude using the geometricmean, and we use the term “trimmed mean” to refer to an average calculated over thecentral 90% of data.2.3.1 Glider Velocity EstimatesThe processing of microstructure measurements to obtain dissipation rates is heavilyreliant on accurate knowledge of the speed, U , with which the probes travel through wa-ter. Unfortunately, there is no direct measurement of the glider’s speed underwater: the19θ [◦] αa [◦] γ [◦] U [cm s−1]upcasts 21.8±1.0 −4.7±0.2 26.4±0.9 41±4downcasts −21.3±0.8 4.8±0.1 −26.0±0.6 25±4Table 2.1 – Mean ± one standard deviation of glider-flight variables from all up- and downcasts of themission. θ is the measured pitch, αa the estimated angle of attack, γ the estimated glide angle, and U theestimated speed through water. Only data coincident with at least one viable ε estimate (see AppendixA.1) are included.glider pitch, θ , and rate-of-change of pressure are known, but this is not enough informa-tion to directly obtain U because the glider travels with an unknown and variable angleof attack, αa, which in our experience is usually in the range 1◦ < |αa|< 10◦.Studies by Fer et al. (2014), Peterson and Fer (2014), and Palmer et al. (2015) use ahydrodynamic flight model developed by Merckelbach et al. (2010) to estimate U . Themodel assumes a steady state balance of drag, buoyancy, and lift forces to optimize esti-mates of U and a drag coefficient CD0 . The angle of attack is then obtained numericallyfrom the implicit relationαa =−(CD0 +CD1α2aA tanγ)(2.1)where γ = θ −αa is the glide angle, and CD1 and A are constants optimized for Slocumgliders in Merckelbach et al. (2010).In contrast to this approach, we follow the method of Schultze et al. (2017) and use thesteady state model of Merckelbach et al. (2010) to obtain the angle of attack, but thenuse the measured pitch and pressure to estimate U dynamically usingU = Wsinγ, (2.2)where W is the glider’s vertical velocity estimated from the measured rate-of-changeof pressure. We found that this quasi-dynamic estimate of U leads to more consistentresults between profiles of ε from up- and downcasts. Note that angles are measured pos-itive upwards, θ and γ relative to the horizontal, αa relative to the glide angle γ .Mean and standard deviation values of selected glider flight characteristics separatedby up- and downcasts are summarized in Table 2.1 to enable comparison with previousstudies. For the most part, the values presented are not remarkable and are similar toones previously reported, albeit with marginally larger angles of attack. One exceptionis the relatively large discrepancy in U between up- and downcasts. The discrepancyarises because of the strong near-surface stratification in the Beaufort Sea, resulting inasymmetric dive and climb rates over most of the water column; however, we do not seea significant systematic effect on the dissipation rate results (see Appendix A.2), and we20do not differentiate between up- and downcasts from here on.2.3.2 Shear MicrostructureThe procedure we use to process the shear measurements uses code provided by RSI andis based on recommendations outlined in their documentation. We provide an overviewhere; a comprehensive rationale for the algorithm and detailed review of recommendedprocedures is available in RSI’s Technical Note 028 (Lueck, 2016).We calculate the dissipation rate from the viscosity and the variance of the turbulentvelocity shear according to εU = 7.5ν〈(∂u′/∂x)2〉, assuming isotropic flow. Here, an-gled brackets indicate averaging, x represents the glider’s along-path coordinate, and u′represents either of the two perpendicular turbulent velocity components. As discussedin Section 2.5.3, the isotropy assumption is problematic when energetics are weak andstratification is strong, and it leads to increased uncertainty in the observed dissipationrates, but it is necessary to make the assumption because we measure only two of thenine strain rate tensor components.We estimate the variance from the measured shear record in half-overlapping 40 s seg-ments. Each of these is further subdivided into 19 half-overlapping 4 s subsegmentswhich are detrended, cosine-windowed (in a variance-preserving manner), and trans-formed into shear power spectra using a fast Fourier transform (FFT). These 19 spectraare averaged to create one “observed” shear power spectrum, Φ, for each 40 s segmentof shear measurement. Coherent acceleration signals measured by the MicroRider areremoved from Φ using the algorithm proposed by Goodman et al. (2006). Frequen-cies, f , are transformed to wavenumbers, k, using the glider’s mean speed, U , overthe 40 s and assuming Taylor’s frozen turbulence hypothesis, i.e. Φ(k) = UΦ( f ) andk = f/U .The 4 s length of the subsegments passed to the FFT sets the scale for the largest wave-length (smallest wavenumber) included in the shear spectrum Φ(k); it is identical to theFFT length chosen by Fer et al. (2014) and Schultze et al. (2017). Given the averageglider speeds in Table 2.1, a typical FFT calculation includes along-path wavelengths aslarge as 164 cm on upcasts and 100 cm on downcasts, resolving the low wavenumbertransition between the inertial and viscous ranges of the turbulence spectrum (Luecket al., 2002). The choice of 40 s for the total averaging length, corresponding on averageto 16.4 m on upcasts and 10 m on downcasts, is larger than the 12 s averaging lengthused in the above mentioned studies; it is a heuristic choice and a compromise whichtrades a decrease in the spatial resolution of the observations in favour of an increase inthe statistical confidence of individual εU estimates (Lueck, 2016).21We numerically integrate each Φ(k), calculating εU according toεU = 7.5ν∫ ku0Φ(k)dk. (2.3)Here, ku is an upper integration limit, chosen to exclude large wavenumbers at whichelectronic noise dominates the measurement. To choose ku, we fit a third-order poly-nomial to Φ(k) in order to isolate the location of the spectral minimum which typicallyindicates the onset of noise domination, but we constrain ku to be at least 7 cpm. Notethat in a low-energy environment most of the variance is at low wavenumbers: 90% ofthe variance lies below 7 cpm when ε = 10−10 W kg−1 (Gregg, 1999).To account for unresolved variance, we calculate the fraction, P, of the integral of thenondimensionalized empirical Nasmyth spectrum (Nasmyth, 1970; Oakey, 1982) that isresolved below the nondimensionalized integration limit ku/(εU/ν3)1/4. We then scaleup εU by a factor of 1/P and iterate the correction procedure until the change in εU insuccessive iterations is less than 2%.We further correct for a small integration underestimate that occurs between the originand the first non-zero wavenumber k1. The Nasmyth spectrum rises approximately ask1/3 and so its integral to k1 is proportional to (3/4)k4/31 ; trapezoidal integration betweenk = 0 and k1, however, is proportional only to (1/2)k4/31 . We correct by adding the term7.5ν(1/4)k1ΦN(k1), whereΦN(k1) is the value of the Nasmyth spectrum at k1. Note thatthe two correction procedures described here are both standard features implemented inthe provided RSI codes.There are two distinct shear probes (Section 2.2.2), yielding two independent, simulta-neous εU estimates. When both estimates pass quality control, we average them; whenonly one passes quality control, we use that single estimate for the analysis presentedin Section 2.4. Note that, on average, we do not see a meaningful difference betweenresults from the two shear probes (see Appendix A.1).2.3.3 Temperature MicrostructureThe dissipation rate εT may be estimated from temperature microstructure measure-ments by determining the Batchelor wavenumber, defined kB =(1/2pi)(εT/ν(κmolT )2)1/4,and inverting to yieldεT = ν(κmolT )2(2pikB)4. (2.4)We determine the Batchelor wavenumber by fitting the theoretical Batchelor spectrum(Batchelor, 1959) to observed power spectra of temperature gradients using the pro-cedure outlined below, which is modelled on descriptions by Ruddick et al. (2000),22Steinbuck et al. (2009), and Peterson and Fer (2014).We first calculate a temperature power spectrum, Λ, from the temperature measure-ments for each of the same half-overlapping 40 s segments that we used to calculatethe shear spectra. Each spectrum is again the average of 19 spectra calculated fromhalf-overlapping, detrended and cosine windowed 4 s subsegments. Values of Λ at highfrequencies, where the temperature probes’ temporal response is inadequate, are cor-rected using the transfer function proposed by Sommer et al. (2013b). Like shear spec-tra, temperature spectra are transformed from frequency to wavenumber space using Uand Taylor’s frozen turbulence hypothesis.From each Λ, we next calculate a “raw” one-dimensional temperature gradient powerspectrum, Ψr, using the variance preserving transformationΨr = (2pik)2Λ . (2.5)From each of these we then subtract a probe-specific noise spectrum:Ψ=Ψr−Ψns , (2.6)where Ψns is the noise spectrum, empirically determined for each probe by averagingthe 1% of raw spectra with the least observed variance. We refer to Ψ as the “observed”temperature gradient spectrum. Note the temperature gradients are defined with respectto the along-glider path coordinate.We next estimate the rate, χ , of destruction of temperature gradient variance (Osbornand Cox, 1972) from the observed temperature gradient spectra. Following Steinbucket al. (2009), we iteratively calculateχ = χlw+χobs+χhw = 6κmolT(∫ kl0ΨBdk+∫ kuklΨdk+∫ ∞kuΨBdk), (2.7)on each iteration subsequently fitting the Batchelor spectrum, ΨB, to the observed spec-trum as described below. The term χobs is the component of χ that comes from integrat-ing the observed spectrum between wavenumbers kl and ku. At wavenumbers outsidethis range, the observed spectrum is not reliable and we instead integrate ΨB to obtainthe correction terms χlw and χhw. Note the correction terms are unavailable and thus setto 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 usingthe maximum likelihood procedure described by Ruddick et al. (2000). This procedureminimizes a cost function to choose the best fit from a family of Batchelor curves whichare constructed using constant χ but variable kB. For the upper limit ku we choose the23intersection between Ψr and 2Ψns. The lower limit kl is the smallest available non-zerowavenumber k1 on the first iteration, and on subsequent iterations is the greater of k1and 3k∗, where k∗ = 0.04kB(κmolT /ν)1/2 represents the top of the convective subrange(Luketina and Imberger, 2001). We implement three iterations, enough for kB estimatesto converge (Steinbuck et al., 2009), and then calculate εT from Equation 2.4.There are two distinct thermistors (Section 2.2.2), yielding two independent, simultane-ous εT estimates. As with the shear-derived estimates, when both pass quality control(see Appendix A.1), we average; when only one passes, we use the single estimate forour analysis.2.4 Comparison of Results from Temperature and Shear Mi-crostructureHere we present ε estimates derived from our coincident microstructure measurementsof shear and temperature, demonstrating how the two estimates agree on average withina factor of two when ε > 3× 10−11 W kg−1 but diverge for smaller dissipation rates.We demonstrate that this divergence leads to inconsistencies between statistical metricsthat describe the two sets of observations. Using evidence presented in Sections 2.4.3and 2.4.4, and in anticipation of the discussion presented in Section 2.5.1, we attributedifferences between the εU and εT datasets to the effects of the εU noise floor. Withthis foreknowledge, our description of these differences can be interpreted as a casestudy that demonstrates the degree by which the εU noise floor influences the ability ofthe shear measurements to characterize the dissipation rate in a stratified, low energyenvironment.2.4.1 Spatial Cross SectionsA qualitative comparison of the dissipation rates derived from shear and temperaturemicrostructure measurements is generally favourable. This can be seen in spatial crosssections of εU and εT (Figure 2.2). Both fields exhibit obvious variability over at leastthree orders of magnitude and indicate the same coherent patches of enhanced turbu-lence superimposed on a less turbulent background. In both fields, these patches arecharacterized by dissipation rates O(10−9) W kg−1. They have a spatial coherence onscales O(10)–O(100) m vertically and O(10) km horizontally. Three easily identifiableexamples, seen in both panels, are between approximately (i) 10–20 km at depths 105–305 m, in the central Amundsen Gulf; (ii) 52–81 km at depths 155–400 m, at the edgeof the shelf-slope; and (iii) 161–183 km within a 75 m band above the sea floor, on the24Figure 2.2 – Cross sections of the turbulent dissipation rate, ε , in log10 space, derived frommicrostructure measurements of (a) shear and (b) temperature. The panels are drawn usingthe same colour scale. Grey shading indicates the bathymetry, black shading discarded orunavailable data (see Section 2.2.3 and the Appendix). Small white lines along the horizontalaxis indicate the locations of individual profiles. The breaks in the horizontal axis, labelled1–4, correspond to the waypoints shown in Figure 2.1b. Magenta rectangles with solid whitelines indicate regions of enhanced dissipation discussed in the text. The magenta rectangle withdashed white line in panel (b) indicates the signature of the mesoscale eddy discussed in thetext.shelf slope. These are indicated in the figure with magenta rectangles. The qualitativesimilarity between the two independently derived estimates is encouraging and suggeststhat temperature and shear probes may be used to qualitatively identify the same regionsof enhanced turbulence.Figure 2.2, however, also reveals an immediate difference between the observed εU andεT fields: these indicate markedly different background states, easily seen in the figurebecause the images in the two panels are drawn with the same colour scale. The εU fieldfrom the shear measurements indicates a background of O(10−11) W kg−1, imaged as aturquoise-blue colour. There appears to be no obvious variability below εU ∼ 5×10−11W kg−1. The εT field from the temperature measurements, on the other hand, suggestsa lower background value of O(10−12) W kg−1, as indicated by the frequent darkerblue colours. Additional structure not seen in the εU observations is apparent in theεT field at dissipation rates between 1× 10−12 and 5× 10−11 W kg−1. An example25Figure 2.3 – Cross section of the ratio εU/εT in log10 space. Shading, panel division, magentarectangles, and annotations as in Figure 2.2.of this phenomenon may be seen by looking at the dissipation rate signature of whatappears to be a mesoscale eddy whose presence can be identified in the temperature anddensity fields (not shown) at depths 40–100 m between 52–87 km: it carries an obvioussignature of enhanced dissipation in the εT field (Figure 2.2b, dashed magenta box), butin the εU field (Figure 2.2a) no such signature can be identified.The magnitude of the discrepancy between the two fields is apparent when visualizing across section of the ratio εU/εT (Figure 2.3). The discrepancy is largest, at times largerthan a factor of 103, in a band approximately at 50–150 m depth and between 100 kmand the end of the glider track. Comparing with Figure 2.2, this region tends to coincidewith the region where εT is smallest. The discrepancy is less pronounced, however, inthe three patches of enhanced turbulence identified in Figure 2.2; here, the ratio εU/εTtends to unity.2.4.2 Mean Vertical ProfilesWe find, on average, more than an order of magnitude difference between εU and εTwhere dissipation rates are smallest. This can be seen in Figure 2.4a, in which the ob-served dissipation rate fields are horizontally averaged in 25 m vertical bins to createmean vertical profiles of εU and εT . In both profiles, the lowest dissipation rates arefound between 100–125 m depth where, on average, εU = 4× 10−11 W kg−1 whileεT = 3× 10−12 W kg−1; there is a factor of 13 disagreement between the two. Thisdisagreement is statistically significant: in this depth bin, the respective interquartilerange (IQR) for each dataset is εU = 2×10−11–8×10−11 W kg−1 and εT = 3×10−13–3×10−11 W kg−1, indicating little overlap between the two measurement distributions.The geometric standard deviation factors σg are 2.4 for εU and 13.4 for εT , reflectingthe substantial horizontal variability seen in Figure 2.2 and the relatively larger range26Figure 2.4 – (a) Average vertical profiles of the dissipation rates εU and εT , obtained fromshear and temperature microstructure and calculated using a trimmed geometric mean in 25 mvertical bins. Shading indicates the 95% confidence interval for the mean as indicated by thegeometric standard error. (b) The ratio of the average vertical profiles of εU and εT , highlightingdisagreement by a factor of 5 or greater between 75–175 m depth.of εT values. Despite the variability, the estimates of the mean values are robust: the95% confidence intervals indicated by the geometric standard error (Kirkwood, 1979)are εU = 4×10−11–5×10−11 W kg−1 and εT = 3×10−12–4×10−12 W kg−1 respec-tively.The discrepancy between the depth-binned mean profiles is less dramatic in the rest ofthe water column, and the two profiles qualitatively have a similar shape. Both exhibitsmall dissipation rates below 10−10 W kg−1 in the shallowest available bin (25–50 m),decreasing further to their distinct minima between 100–125 m, and then gradually in-creasing with depth to maximum mean values near 10−10 W kg−1 as they approach thesea bed. Disagreement between the mean values, imaged in Figure 2.4b, is a factor of 5or greater between 75–175 m depth and smaller everywhere else. Whenever both meanvalues are simultaneously at least 3× 10−11 W kg−1, the agreement between them isbetter than a factor of 2, highlighting that the divergence between εU and εT occurs onlyat very low dissipation rates.Note that our measurements tend to exhibit small ε relative to measurements from otherregions of the global ocean which often exhibit typical averaged values of O(10−9) Wkg−1 and higher (Waterhouse et al., 2014). This incongruity, however, is not surprising:small dissipation rates are anticipated in the western Arctic Ocean where turbulence isthought to be exceptionally low because of limited energy input and seasonal sea icecover (Rainville and Woodgate, 2009). Microstructure measurements from the westernArctic are very sparse, but those that do exist (Section 2.1) have so far indicated typicalbackground dissipation rates of O(10−10) W kg−1.27Figure 2.5 – Histograms showing the distributions of all (a) εU and (b) εT observations. Theinterquartile range (IQR) is indicated by the darker shading; the mode, arithmetic and geomet-ric means, and median are marked in both panels according to the legend in (a). The labels Nand σg indicate the total number of observations in each histogram and the geometric standarddeviation factor respectively. Histograms are calculated over 100 logarithmically spaced bins.(c) Quantile-quantile plot demonstrating the goodness of fit of the histograms to idealized log-normal distributions. For each set of data, deciles are marked by grey-shaded circles, and thesquared linear correlation coefficient, R2, is indicated.2.4.3 Distributions of εU and εTThe histograms of all εU and εT observations (Figure 2.5a,b) provide further insight intothe discrepancies between the two datasets. The histograms have markedly differentshapes despite being constructed from coincident sets of measurement. Most notably,the distribution of εT observations is nearly symmetric with only small negative skew(skewness, sg = −0.2) in log10 space, while the εU distribution is skewed positive andmore heavily (sg = 1.2). Statistical properties that can be used to further compare the dis-tributions are tabulated in Table 2.2 and indicated in Figure 2.5a,b. Of note are the largergeometric mean and median εU values, reflecting the relative absence of very small εUobservations; the separation between the median and geometric mean of εU , reflectingthe skewness of that distribution; and the wider interquartile range of εT , reflecting thelarger variability of the εT observations.The distributions may be further contrasted using a quantile-quantile (Q-Q) plot (Figure2.5c) to quantify how similar each distribution is to an idealized lognormal one — themore linear the plot, the greater the similarity. From this visualization, it is clear that thedistribution of εT observations can be described as lognormal over all ε except belowthe second decile (2× 10−12 W kg−1). In contrast, the distribution of εU observations28Table 2.2 – Statistical parameters of the εU and εT distributions shown in Figure 2.5. Given, from leftto right, are the number, N, of observations; mode; geometric mean; median; first and third quartiles,P25 and P75; arithmetic mean; and geometric standard deviation factor, σg. The quantities N and σg aredimensionless and unscaled. All other quantities are scaled by a factor of 10−11 W kg−1.N Mode G. Mean Median P25 P75 A. Mean σgεU 28575 2.7 6.5 4.6 2.5 13 25 3.7εT 21577 1.2 1.6 1.7 0.27 12 61 18.3may be described as lognormal only above the seventh decile (1×10−10 W kg−1). Thestrong positive curvature in the Q-Q plot for εU below the seventh decile indicates thatthere is substantially less weight on the left side of the observed distribution relative toan idealized lognormal one. The slight negative curvature in the Q-Q plot for εT belowthe second decile indicates a small trend in the opposite direction, i.e. a marginallyheavier tail on left side relative to an idealized lognormal distribution. The squaredlinear correlation coefficients of the Q-Q plots are R2 = 0.917 for εU and R2 = 0.995 forεT , confirming the qualitative impression that the εU observations deviate more stronglyfrom an idealized lognormal distribution.We attribute the discrepancy in the shapes of the εU and εT histograms primarily to thesensitivity limit of the shear probes, which imposes an artificial lower limit (or “noisefloor”) on the εU observations. This noise floor will skew the histogram of εU obser-vations positive by distributing samples that would otherwise be recorded as smallervalues within a narrow range around the lower limit. Given the distinctive peak in theεU histogram and the extremely rapid rolloff to the left of the peak, we simply use themode to approximate the noise floor as 3× 10−11 W kg−1. The εU observations thatfall below this estimate of the noise floor are in the range (1≤ εU < 3)×10−11 W kg−1(Figure 2.5a); we attribute this statistical scatter to errors in the data processing whichmay, in part, arise because of uncertainty surrounding the characteristics of weakly tur-bulent, strongly stratified flows (Section 2.5.2). The range of values below the noisefloor suggests uncertainty here within a factor of 3, implying that the effects of the noisefloor begin to skew the εU distribution at∼9×10−11 W kg−1. This is consistent with thetrend in the Q-Q plot (Figure 2.5c) where the shape of the εU histogram begins to divergefrom that of the εT histogram at dissipation rates below ∼1×10−10 W kg−1.Previous studies using loosely-tethered profilers often cite a noise floor of O(10−10) Wkg−1 for shear-derived observations (e.g. Gregg, 1999; Wolk et al., 2002; Shroyer, 2012;Fer, 2014; Lincoln et al., 2016), though two glider-based studies that incorporated mi-crostructure shear measurements both quote 5×10−11 W kg−1 for the noise floor (Wolket al., 2009; Fer et al., 2014). The presence of an εU noise floor has practical ramifica-tions for the interpretation of microstructure shear measurements; these are particularly29important to consider in low energy environments, and we discuss them further in Sec-tion 2.5.4.2.4.4 One-to-one comparison of εU and εTA simple scatter plot of the coincident εU and εT observations (Figure 2.6) elucidateshow the agreement between the two varies over the range of the observed dissipationrates. If one considers only observations where the two ε estimates are simultaneouslygreater than our empirical estimate of the εU noise floor (3× 10−11 W kg−1, Section2.4.3), the agreement between the two sets of observations is encouraging. This subsetof data is indicated in Figure 2.6 by the purple-shaded region. Here, the “cloud” ofindividual measurements 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 factor of 2. Moreimportantly, the bin averages that are shown (defined below) always agree within a factorof 2. This level of agreement is consistent with the factor of 2 agreement in the meanvertical profiles (Figure 2.4) whenever those averages indicate ε > 3× 10−11 W kg−1in both estimates. Statistical agreement within a factor of 2 is comparable to the bestagreement seen in other studies (e.g. Kocsis et al., 1999; Peterson and Fer, 2014).When at least one of the ε estimates is less than 3×10−11 W kg−1, statistical disagree-ment between the shear- and temperature-derived dissipation rates becomes concerning.Here, only 22% of the 16,842 observation pairs 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: asthe εT estimates continue to decrease, the εU estimates asymptote to a lower limit ofapproximately 2× 10−11 W kg−1, marginally below but still consistent with our esti-mate of the εU noise floor. The bin averages indicate the same pattern as the individualmeasurements: below εT = 1× 10−11 W kg−1 they exhibit disagreement greater thana factor of 5, and, even in this averaged sense, suggest disagreement greater than twoorders of magnitude when εT is less than 2×10−13 W kg−1.Note that averaged measures like bin averages are more appropriate than individual mea-surements when evaluating the agreement between ε estimates because we expect sub-stantial statistical scatter (within about an order of magnitude) in these estimates. Thisscatter is, in part, attributed to uncertainties surrounding the validity of the isotropy,homogeneity, and stationarity assumptions inherent in the data processing (see Section2.5.2). The bin averages shown in Figure 2.6 are averages calculated from the trimmedmean in logarithmically-spaced bins that lie perpendicular to the one-to-one line, i.e. ina coordinate system rotated 45◦ clockwise from that shown. Defining the bins in thismanner helps minimize biases in the average by assuming roughly equal uncertaintyin both variables, similar in principle to a bivariate least-squares minimization (Ricker,30Figure 2.6 – Scatter plot comparison of the two coincident dissipation rate estimates εU andεT . Identical agreement and agreement within factors of 2 and 5 are indicated as labelled. Binaverages are calculated perpendicular to the one-to-one line (see text). Our empirical estimateof the εU noise floor (3× 10−11 W kg−1) is indicated by the horizontal dotted line. Purpleshading indicates where both estimates of ε simultaneously lie above 3× 10−11 W kg−1 andalso delineates the region where bin averages agree within a factor of 2.1973).2.5 DiscussionMeasuring turbulence parameters to estimate the turbulent dissipation rate comes withunique challenges in low energy environments like the Beaufort Sea, and our resultsin Section 2.4 demonstrate that the two most common means of directly estimating thedissipation rate can yield divergent results that disagree by multiple orders of magni-tude at low ε . 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 ofmarginally turbulent, strongly stratified flows also introduce uncertainty into the obser-vations. And, more pragmatically, our results highlight questions about how to correctlyprocess and interpret shear microstructure measurements in such environments since itappears that the majority (about 70%, Figure 2.5c) of εU estimates are skewed by theeffects of the noise floor. We address these topics in the following discussion: in Section2.5.1, we look more closely at the effect of sensor limitations on the measurements andthe observed spectra; in Section 2.5.2 we discuss averaged observed spectral shapes;31in Section 2.5.3 we examine uncertainties that arise from the (potentially unjustified)assumptions needed for the processing of microstructure measurements; and in Section2.5.4 we discuss practical implications, i.e. in which circumstances the difference be-tween εU and εT matters and in which circumstances it can be safely ignored.2.5.1 The Effect of Sensor LimitationsWe propose that the systematic divergence between εU and εT that is obvious in Figures2.5 and 2.6 at small dissipation rates is primarily a result of the effects of the noisefloor of the εU estimates. This low-end divergence is then responsible for the largediscrepancies seen in the spatial cross sections (Figures 2.2 and 2.3) and mean verticalprofiles (Figure 2.4) of εU and εT . This interpretation is consistent with the knownsensitivity limitations of microstructure shear probes (Osborn and Crawford, 1980) andprevious empirical estimates of the εU noise floor (Section 2.4.3).Assuming vibrations from the measurement platform do not contaminate the measuredsignal, the noise floor of an εU estimate is set by the lower limit of a shear probe’sability to detect hydrodynamic transverse forces and distinguish these from electronicmeasurement noise. Hydrodynamic forces from small-scale velocity shear below thisdetection limit may still act on the probe, but the signal is either not recorded or ismasked by the instrument’s electronic noise. As a result (Section 2.3.2), any variance〈(∂u′/∂x)2〉that exists below the probe’s detection limit will yield an εU estimate at(or near) the level of the noise floor, irrespective of what the true dissipation rate at theinstant of the measurement may be. If the true dissipation rate is below the level of thenoise floor in a large proportion of the measurement realizations, this behaviour willlead to an artificially skewed measurement distribution and a “pile-up” of observationsagainst a lower limit, i.e. a distinct peak in the distribution near the noise floor and arapid rolloff towards smaller—unresolved—values, as can be seen in the distribution ofour εU observations (Figure 2.5a).The manifestation of the noise floor can also be seen in the observed shear spectrawhen these are compared to the simultaneously observed temperature gradient spectra.Figure 2.7 depicts six representative pairs of observed shear and temperature gradientspectra, distributed over six consecutive orders of magnitude of ε (as suggested by thetemperature-derived estimates). Each column of panels shows the two coincidently ob-served spectra Φ(k) and Ψ(k), defined in Sections 2.3.2 and 2.3.3. Following panels g–lfrom right to left, the Batchelor fit to the temperature gradient spectra (Section 2.3.3) in-dicates continually decreasing εT , as labelled in each panel. The shear spectra indicate asimilar ε-trend over the four larger orders of magnitude (panels c–f): as anticipated, thepeak of the observed shear spectrum moves downwards and to the left as ε decreases,3210-710-5  [(s2 cpm)-1]O(10-13 ) W/kgU=3e-11(a)101 102k  [cpm]10-710-5  [K2 cpm] T=3e-13(g)O(10-12 ) W/kgU=2e-11(b)101 102k  [cpm]T=3e-12(h)O(10-11 ) W/kgU=2e-11(c)101 102k  [cpm]T=3e-11(i)O(10-10 ) W/kgU=1e-10(d)101 102k  [cpm]T=1e-10(j)O(10-9 ) W/kgU=1e-09(e)101 102k  [cpm]T=2e-09(k)O(10-8 ) W/kgU=1e-08(f)101 102k  [cpm]T=2e-08(l)Figure 2.7 – Sample coincident shear (a-f: Φ) and temperature gradient (g-l: Ψ) spectra (black)for 6 orders of magnitude of ε , as determined by the temperature measurements. Bold indicatesthe wavenumbers explicitly included for integration; the remaining variance is estimated asdescribed in Sections 2.3.2 and 2.3.3. ForΨ, bold also indicates the wavenumber range used forthe MLE Batchelor fit (see Section 2.3.3).Shear spectra have the accelerometer signal removed(Section 2.3.2) and temperature gradient spectra have the empirically-determined noise spectraremoved (Section 2.3.3). Nasmyth (a-f) and Batchelor (g-l) reference spectra (grey) are alsodrawn. Batchelor spectra are those determined by the MLE fitting algorithm which are used toestimate kB (Section 2.3.3).and the integral of the spectrum (Section 2.3.2) indicates decreasing εU , as labelled.However, below O(10−11) W kg−1 (panels a–b) the spectrum runs into a spectral “floor”and does not decrease any further. Here, the integral of Φ(k) no longer reflects the shearvariance or any true physical quantity; instead, it saturates at a lower limit that indicatesthe available precision of εU , which, as anticipated, is in the vicinity of our empiricalestimate of the noise floor 3×10−11 W kg−1 (Section 2.4.3).So far, we have focused our discussion on limitations of the shear measurements. Ofcourse limitations also exist on the measurement of temperature microstructure, butthese are of a different nature than those which affect the shear measurements, andthey tend to be less problematic in our study. Sensitivity limitations are not a concernfor microstructure thermistors in the way they are for shear probes since the FP07 ther-mistors easily respond to within better than 0.1 mK (Sommer et al., 2013b) which isapproximately the smallest temperature scale we need observe (e.g. Figure 2.7g). Therelatively slow time response of thermistors is generally a concern (Gregg, 1999), but atsmall dissipation rates it is possible to adequately account for the slow response usingthe transfer function proposed by Sommer et al. (2013b) or a similar correction method.At rates greater than ∼1× 10−7 W kg−1, the effects of the slow response time can nolonger be adequately corrected and temperature-derived estimates will tend to systemat-33ically underestimate the true dissipation rate (Peterson and Fer, 2014), but this limitationis not a concern in our observations since fewer than 0.1% of our εT estimates are abovethis cutoff value. A more relevant concern for our temperature-derived estimates is thepotential uncertainty that surrounds the characteristics of turbulent eddies and the result-ing turbulence spectra when turbulent energetics are weak and stratification is strong, asis the case in the setting for our measurements. This is the topic of the following twosections.2.5.2 Turbulence Spectra in Stratified Low Energy FlowsOur observations suggest that the shape of shear and temperature gradient spectra de-viate systematically from Nasmyth and Batchelor reference spectra in stratified low en-ergy flows. Fitted Nasmyth and Batchelor spectra are drawn with the selected observedspectra in Figure 2.7 for reference, exemplifying varying levels of agreement; 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 turbulencespectra, we bin all observed spectra by buoyancy Reynolds number, ReB = ε/νN2, andcalculate median temperature gradient and shear spectra in each bin (Figure 2.8). TheReB parameter quantifies the destabilizing effects of turbulent kinetics relative to the sta-bilizing effects of stratification and viscosity. It is proportional to the ratio of the largestvertical (Ozmidov) scale to the smallest isotropic (Kolmogorov) scale of turbulent ed-dies, so when ReB < 1 we anticipate that turbulent eddies of all sizes, including thesmallest ones on dissipative scales, are modified by stratification and exhibit a degree ofanisotropy. Further, modelling results suggest that the characteristics of turbulent struc-tures undergo regime shifts in the vicinity of ReB ∼ 10 and ReB ∼ 100 (Shih et al., 2005;Ivey et al., 2008), and so combining with the above scaling argument, we use 1, 10, and100 to delineate the boundaries of our ReB bins.To calculate median spectra, individual spectra must first be normalized identically sothat the shapes of spectra over varying ε and χ may be compared. To do this, wenondimensionalize shear spectra usingΦ∗ =Φ/(ε3U/ν)1/4 (2.8)and temperature gradient spectra usingΨ∗ =Ψ/(χ√qB/2kBκmolT), (2.9)3410-2 10-10.20.512ShearReB < 1N = 2680(a)10-2 10-1Shear1 < ReB < 10N = 5535(b)10-2 10-1Shear10 < ReB < 100N = 1691(c)10-2 10-1ShearReB > 100N = 274(d)10-1 10010-210-1100Temperature GradientReB < 1N = 31615(e)10-1 100Temperature Gradient1 < ReB < 10N = 6673(f)10-1 100Temperature Gradient10 < ReB < 100N = 2722(g)10-1 100Temperature GradientReB > 100N = 502(h)Figure 2.8 – Median nondimensionalized shear (a-d) and temperature gradient (e-h) spectra inbold, for regimes of ReB as indicated. Also shown are the 25th and 75 percentile of data (thinsolid line) as well as nondimensionalized reference spectra (dashed line): Nasmyth for shearand Batchelor for temperature gradient. The total number of spectra used in each calculation isindicated by N. Shear spectra with εU < 10−10 W kg−1 are excluded.consistent with schemes used by Oakey (1982) and Dillon and Caldwell (1980). Wenondimensionalize wavenumbers using k/kν for shear spectra and k/kB for tempera-ture gradient spectra. The scaling factor kν is the Kolmogorov wavenumber defined(1/2pi)(εU/ν3)1/4; the remaining variable definitions for Equations 2.8 and 2.9 are givenin Section 2.3. For each ReB bin, we then calculate the median spectral height at eachnondimensional wavenumber using all spectra within the bin, creating the median spec-tra shown in Figure 2.8; we also calculate the interquartile range at each nondimensionalwavenumber as a measure of the variability around the median. To exclude artificial ef-fects that may arise because of the εU noise floor (Section 2.5.1), we exclude from thecalculations any shear spectra where εU < 1×10−10 W kg−1.The systematic modification of the temperature gradient spectra with decreasing ReBis clearly visible if one follows panels e–h from right to left: there is a clear trendtowards less curvature and greater low-wavenumber deviation from the theoretical curveas ReB decreases. None of the median temperature gradient spectra exhibit a curvatureas strong as that predicted by the Batchelor spectrum, but the discrepancy increaseswith decreasing ReB, and for the two lowest ReB bins there is no longer a peak androlloff delineating distinctive subranges of the spectrum. This behaviour is similar to35that seen in measurements taken by Dillon and Caldwell (1980) who, as we do, observeddecreasing curvature with smaller turbulence intensities.The median shear spectra likewise vary systematically with ReB, indicating increasingdeviations from the Nasmyth spectrum as ReB becomes small (following panels a–dfrom right to left). The dissipative subrange, to the right of the peak, is not as steep in anyof the median spectra as predicted by the Nasmyth shape, and it becomes increasinglymore 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 allmedian spectra and no longer appears to be well described by a simple power law whenReB < 1. This behaviour is reminiscent of that seen in measurements by Gargett et al.(1984) who found that inertial subranges of shear spectra gradually disappeared whenturbulence became weak and stratification became strong.2.5.3 Understanding Uncertainty for Small εThe above discussion (Section 2.5.2) highlights the manner in which turbulence spectraare systematically modified away from their reference shapes as ReB becomes small. Wepropose that the systematic modification with decreasing ReB occurs as the character-istics of turbulent eddies in strong stratification increasingly depart from the idealizedframework of steady, isotropic turbulence. Evidence for this behaviour can be seen inthe distribution of the ReB parameter which is below unity in 76% of our observations,suggesting that turbulent eddies are frequently anisotropic and modified by the effects ofstratification. Further, the Ozmidov scale, LO = (1/2pi)(ε/N3)1/2, has a median value of0.1 cm, which is exceptionally small and again suggests that even viscous-scale eddiesare squashed by the stratification.These characteristics signal that there is increased uncertainty in the dissipation rateestimates when ε is small and N2 is large. This is especially true for εT estimates wherethe data processing depends on the ability to determine kB from Batchelor spectrum fits(Section 2.3.3). One way to characterize the increased uncertainty is to quantify thedegree by which observed turbulence spectra and idealized reference spectra diverge.We do this here for the temperature measurements and compare temperature gradientspectra to Batchelor spectra by calculating for each spectrum the root-mean-square error,ξrms, of log10(Ψ/ΨB), defined:ξrms =√1nn∑i=1log210(ΨiΨBi)=√1nn∑i=1[log10(Ψi)− log10(ΨBi)]2 . (2.10)The summation index i runs over all wavenumbers included in the Batchelor fitting36Figure 2.9 – Root-mean-square error between Ψ and ΨB, as defined in the text, visualized asa function of (a) buoyancy Reynolds number, and (b) dissipation rate. It quantifies the degreeof divergence between observed temperature gradient and theoretical Batchelor spectra. Largeopen faced markers are bin averages. Regressions to subsets of the bin averages are shown ineach panel; the subsets are those on either side of the datum marked by the circle (inclusive).procedure; n is the number of spectral points included in the fit (Section 2.3.3). Wefind that averages of ξrms increase gradually from 0.3 to 0.5 as ReB estimates decreasefrom O(101) to O(10−2); at smaller ReB, mean ξrms increases rapidly to a maximumvalue of 0.7 (Figure 2.9). A similar, but more pronounced, pattern is visible when ξrmsis visualized as a function of εT . The increase in ξrms is gradual and modest, from 0.3to 0.45, as εT decreases from O(10−7) W kg−1 to O(10−12) W kg−1; this behaviour isfollowed by a sharp increase in ξrms at smaller εT .Further insight into the confidence of the εT values can be gained empirically if wemake the assumption that dissipation rates distribute lognormally in the ocean (Bakerand Gibson, 1987; Gregg, 1987). Under this assumption, we can use the observed distri-bution of εT (Figure 2.5b) to estimate a lower cutoff below which the application of thesteady, isotropic turbulence model becomes problematic and εT estimates become unre-liable. The distribution of εT observations follows the lognormal shape closely over theentire range of data except below the second decile (Figure 2.5c), where the observeddistribution is disproportionately heavy. The distortion in the distribution indicates thatuncertainties in the data processing statistically skew the εT estimates below the seconddecile; our simple statistical model therefore suggests that the εT estimates are reliableand physically meaningful to values as small as εT ≈ 2× 10−12 W kg−1. Below thiscutoff, εT estimates are unreliable and perhaps not meaningful. Note that a lower cutoff37Figure 2.10 – Histogram of the ratio εU/εT , highlighting the large number of coincident mea-surements where the shear-derived values overestimated the temperature-derived ones in ourdataset. Agreement by factors of 5, 10, and 100 is indicated by dashed lines. The histogram iscalculated over 50 logarithmically spaced bins.of 2×10−12 W kg−1 is consistent with the sudden increase in ξrms that occurs in Figure2.9b below εT = O(10−12) W kg−1.2.5.4 Implications for Interpreting Microstructure MeasurementsIn Section 2.4 we demonstrated that there can be a significant difference between dis-sipation rate estimates derived from coincident measurements of shear and temperaturemicrostructure. The εT estimates suggest that the true dissipation rate is below the εUnoise floor of 3× 10−11 W kg−1 in 58% of our observations. A histogram of the ra-tio εU/εT (Figure 2.10) demonstrates the severity and the frequency with which theshear probes may overestimate the dissipation rate in low energy environments like theBeaufort Sea. Using εT as a reference (and acknowledging the associated uncertaintiesdescribed in Section 2.5.3), the shear measurements overestimate the dissipation rate bya factor of at least five in 44% of our measurements, by at least one order of magnitudein 31% of our measurements, and by at least two orders of magnitude in 9% of ourmeasurements. This is a level of error that has the potential to alter the interpretationof the shear measurements, as described in the following paragraph. In contrast, thetemperature measurements overestimate the dissipation rate relative to εU by a factor ofat least five in less than 3% of measurements, a degree of mismatch that could safely beneglected in many applications.The severity with which the bias we found in the shear measurements may alter the in-terpretation of εU estimates depends on the specific goals of a study. If, as in Section382.4, the utility of the measurements is to characterize the variability and the statisticaldistribution of dissipation rates, then the potential for biases greater than an order ofmagnitude cannot be ignored. Without the coincident εT estimates to which to compare,the εU estimates would lead us to misrepresent the degree of spatial variability (Fig-ures 2.2–2.3), the geometric averages representing “typical” values (Figure 2.4), and theobserved distribution and related statistics (Figure 2.5, Table 2.2) of the turbulent dissi-pation rate here in the Amundsen Gulf. These misrepresentations may then be furtherpropagated into calculations of the mixing rate coefficient Kρ , which typically rely onthe Osborn (1980) model Kρ = 0.2ε/N2, leading to similar misrepresentations of thevariability, the geometric averages representing typical values, and the distribution andrelated statistics of Kρ .Arithmetic mean values of εU , however, are much less sensitive to the bias we describe.This is fortunate, because arithmetic mean values are arguably the appropriate mea-sure to use when estimating bulk buoyancy fluxes and characterizing net water masstransformation from mixing rate estimates (Baker and Gibson, 1987). As noted in Sec-tion 2.4, in this study we have tended to use the geometric mean to average dissipationrates. We do this because the geometric mean effectively characterizes the central ten-dency of lognormally distributed data and more fairly represents “typical” ε realizations(Kirkwood, 1979). In contrast, the arithmetic mean is ineffective at representing typicalvalues of a lognormal-like distribution because it is dominated by a small number ofvery large values at the high end of the distribution. Further, the arithmetic mean tendsto be very sensitive to individual outliers that may exist on the far right-hand-side of thedistribution, but this may be problematic because of the large uncertainty in individualε realizations. However, the disproportionate importance of large values in setting thearithmetic mean also makes it mostly insensitive to errors in small ε estimates. Theeffect can be seen, for example, when comparing the arithmetic mean of the εU and εTdistributions (Figure 2.5, Table 2.2): in contrast to the geometric mean, the median, andthe 25th and 75th percentiles, the arithmetic mean of the εT distribution is greater thanthat of the εU distribution because of a marginally thicker tail on the right-hand-side ofits distribution which more strongly influences its arithmetic mean.A final subtle point remains to be discussed. When carrying out an analysis using mi-crostructure measurements of shear, it is tempting to simply remove observed εU valuesthat sit at or near the estimated noise floor, discarding these as untrustworthy. This ap-proach is viable when only a small number of the observations are near the noise floor;however, in the present study, simply removing data likely to be corrupted by the effectsof the noise floor would only exacerbate the bias evident in the εU observations. For ex-ample, if we remove from the dataset the shear-derived estimates where εU < 5×10−11W kg−1, we increase the positive bias in our sample by removing 53% of the measure-39ments and the entire left half of the εU histogram (Figure 2.5). Rather than helpingto correct biased averages of εU , this change shifts the median from 4.6× 10−11 to1.4×10−10 W kg−1 and the geometric mean from 6.5×10−11 to 1.9×10−10 W kg−1,in both cases increasing the error in these averaging metrics.The best practical way to account for the effects of the noise floor in microstructure shearmeasurements will depend on the goals of each individual study and on the proportion ofthe observations that are in the vicinity of the noise floor. One approach is to set dissipa-tion rates that appear to be near the noise floor to zero (e.g. see Gregg et al., 2012); thisapproach is probably justifiable for arithmetic mean calculations since the averaging isdominated by the large ε values, but it is problematic when describing the variability orwhen calculating a geometric mean to characterize typical ε values. In these situations,it may be more appropriate to fit a lognormal distribution to the part of the observeddistribution that resides above the noise floor, but care is required here also since thetheoretical conditions for expecting lognormality are strict and often not satisfied in aset of field measurements (Yamazaki and Lueck, 1990). In any case, it is clear that theinterpretation of microstructure shear-derived dissipation rate estimates should proceedwith caution if the measurements are from a very low energy environment and it appearsthat a large proportion of the data cluster around a well defined noise floor.2.6 ConclusionsCare must be taken to understand how measurement limitations may bias microstruc-ture measurements in low energy environments like the Beaufort Sea: this is the centraltheme of our study. The results we have presented here suggest that microstructure mea-surements of velocity shear, in particular, are prone to misrepresenting averaged dissi-pation rates—and, consequently, mixing rates—in such environments because the noisefloor of the εU estimates artificially biases the majority of the observations. In addition,our measurements suggest that both shear- and temperature-derived dissipation rate esti-mates may be complicated by further uncertainty when strong stratification modifies thecharacteristics of turbulence in a weakly energetic, strongly stratified flow; this changein the characteristics can be seen in the systematic modification of the shapes of shearand temperature gradient spectra at low ReB.We have documented the discrepancy between the two distinct dissipation rate estimatesεU and εT because we find disagreement large enough to lead to substantial differencesin how the two sets of data would be interpreted independently. The temperature-derivedestimates were able to resolve smaller dissipation rates than the shear-derived estimates:averages of εU began to exhibit biased behaviour below 10−10 W kg−1 and were not able40to resolve rates below 3× 10−11 W kg−1, while averages of εT were reliable to valuesas low as 2× 10−12 W kg−1 and were characterized by unacceptably large uncertaintybelow this. Our experience suggests that caution interpreting shear-derived dissipationrate estimates is warranted if a large number of observations cluster at or near an iden-tifiable εU noise floor, in our case 3× 10−11 W kg−1. In the measurements presentedhere, the temperature-derived estimates suggest that the true dissipation rate lies belowthis noise floor often enough to fundamentally alter the scientific interpretation of themeasurements. Other low energy environments in which the special measurement con-siderations outlined in this study may be applicable include the wider Canada Basin(Rainville and Winsor, 2008), stratified lakes (Sommer et al., 2013a; Scheifele et al.,2014), the central Baltic Sea (Holtermann et al., 2017), and the abyssal global oceanover smooth topography (Waterhouse et al., 2014).41Chapter 3Turbulence and Mixing in theArctic Ocean’s Amundsen Gulf3.1 IntroductionIn this study, we present observations of ocean turbulence and mixing in the BeaufortSea’s Amundsen Gulf (Figure 3.1) from a series of CTD and microstructure measure-ments we collected in summer 2015. Amundsen Gulf is one of the most biologicallyproductive regions of the Arctic Ocean (e.g. Stirling, 1980, 2002; Harwood and Stirling,1992; Dickson and Gilchrist, 2002; Geoffroy et al., 2011), but modern climate changeis rapidly driving the western Arctic to a “new normal” state, and perturbations to re-gional ecosystem dynamics and services are already being observed (Post et al., 2009;Serreze and Barry, 2011; Jeffries et al., 2013). In light of ongoing changes to the broaderphysical environment, it is important that the science community continue developinga detailed understanding of the physical oceanography—and in particular of the mixingcharacteristics—of the western Arctic in order to facilitate studies that will model theenvironmental and ecological responses to future regional climate change (e.g. Carmackand MacDonald, 2002; Rainville et al., 2011; Carmack et al., 2015).The Arctic Ocean is probably the most under-sampled of the major ocean basins with re-spect to mixing (e.g. Waterhouse et al., 2014), and observations of ocean turbulence arenotably scarce in the Beaufort Sea. A number of studies over the previous decade havedeepened our understanding of mixing rates and mechanisms in the broader CanadaBasin (e.g. Rainville and Winsor, 2008; Timmermans et al., 2008a,b; Guthrie et al.,2013; Dosser et al., 2014; Shaw and Stanton, 2014), but, to our knowledge, only fourprevious studies (Padman and Dillon, 1987; Bourgault et al., 2011; Rippeth et al., 2015;Lincoln et al., 2016) have used microstructure measurements to characterize mixing42AlaskaYukonCANADA BASINNunavutBanksIslandCanadian Beaufort  ShelfB e au f or t  Se aNorthwest TerritoriesVictoria IslandAlaskan Beaufort ShelfA m u n d s e n  G u l fCape BathurstCA08Mackenzie Canyon(a)(b)Figure 3.1 – (a) Map of the southeastern Beaufort Sea, showing the location of AmundsenGulf to the east of the Canadian Beaufort Shelf. The glider path is shown by the thin black lineinside the black rectangle. (b) Enlarged view of the region given by the black rectangle in panel(a), showing the path of the glider. The start and end locations of the track are shown by thelarge white rectangles; four intermediate waypoints are shown by the small white rectanglesand numbered consecutively. The color on the glider’s track-line is water temperature alongthe 1026.15 kg m−3 isopycnal, using the same colour scale as shown in Figure 3.11, indicatingthe location and spatial scale of the warm-core eddy discussed in the text (Section 3.5.3). Thewhite circle is the location of ArcticNet mooring CA08. Bathymetry data are from IBCAO 3.0(Jakobsson et al., 2012).43rates in the Beaufort Sea directly. This scarcity of direct observations limits our un-derstanding of the role of turbulent mixing in the Beaufort Sea because a large num-ber of tightly resolved measurements is needed to accurately characterize turbulence,which tends to be described with lognormally distributed variables that are easily under-sampled (Baker and Gibson, 1987; Gregg, 1987). As a result, there remains a pressingneed to continue building a broad record of mixing estimates in order to understand cur-rent and future basin-scale water mass transformations across the western Arctic (Car-mack et al., 2015).With this study, we contribute to a more comprehensive understanding of the physicalenvironment that underpins the ecology of the southeastern Beaufort Sea by providinga detailed description of the turbulence and mixing characteristics of the region. Usingan autonomous ocean glider equipped with a CTD and turbulence sensors, we measurethe hydrography and turbulent dissipation rates of kinetic energy and thermal varianceand use these measurements to statistically characterize diapycnal mixing rates and heatfluxes. We then discuss the relative importance of tidal mixing, double diffusion, andnear-surface mesoscale and smaller processes that may underpin the observed turbulenceenvironment. To our knowledge, this is the first time such a broad characterization ofmixing from direct turbulence measurements has been presented for this region andthe first time an autonomous instrument has been used to characterize the statistics ofturbulence and mixing in the Beaufort Sea.The remainder of the chapter is structured as follows. In Section 3.2, we describe theCTD and microstructure measurements from the glider and briefly outline the data pro-cessing methods. Section 3.3 uses the CTD measurements to describe the relevant hy-drographic context. In Section 3.4, we present the primary results of this study, theturbulence, mixing rate, and heat flux observations. Section 3.5 presents a discussion ofrelevant mixing mechanisms. We synthesize our results in Section 3.6.3.2 Measurements and Data Processing3.2.1 Sampling StrategyWe collected CTD and turbulence measurements in Amundsen Gulf using an autonomous1000-m-rated Teledyne-Webb Slocum G2 ocean glider, fitted with (1) an internallymounted, pumped Seabird CTD measuring conductivity, temperature, and pressure, and(2) an externally mounted turbulence-sensing package measuring shear and temperaturemicrostructure (Section 3.2.2). The measurements used in this study are those first de-scribed in Chapter 2, collected continuously over 11 days during the period 25 August –445 September, 2015.The 186 km horizontal path of the glider, immediately northwest of the basin sill, isshown in Figure 3.1. The glider spent the first 5 days in the central Gulf, where thewater depth exceeds 400 m, and the remaining time on three traverses of the continentalshelf near Banks Island. Along this path, the glider collected 348 discrete quasi-verticalmeasurement profiles, at a nominal glide angle of 26◦ from the horizontal. The first 112profiles, in water ∼410 m deep, extend from the near surface to a fixed depth of 300 m;later profiles typically extend to within 15 m of the local bottom, which ranged between205–430 m depth. The location of each profile is approximated with its mean coordi-nates, neglecting horizontal translation that occurs over the course of one profile. Themean (standard deviation) distance between consecutive profiles is 536 (357) m.3.2.2 Turbulence Measurements and Data ProcessingThe glider carried an externally mounted, self-contained microstructure sensing packageknown as a Microrider, also used in recent studies by Fer et al. (2014), Peterson and Fer(2014), Palmer et al. (2015), and Schultze et al. (2017). The Microrider is manufacturedby Rockland Scientific and is factory-installed on the glider. Our configuration of theMicrorider had two velocity shear probes and two fast response thermistors, each sam-pling at 512 Hz, measuring velocity and temperature gradients on scales smaller than 1cm. All sensors sampled continuously during the deployment, but one of the two shearprobes failed after the first three days of measurement.We derive independent estimates of the turbulent kinetic energy (TKE) dissipation rate,ε , from each of the four microstructure channels. This rate is a measure of how turbu-lent a flow is and is proportional to the rate of diapycnal mixing in the Osborn (1980)model. We briefly outline below our methodologies to derive ε from the shear and tem-perature microstructure measurements; a more detailed description of the methods andtheir limitations is given in Chapter 2.We calculate the TKE dissipation rate from the measured microstructure shear varianceaccording toεU = 7.5ν〈(∂u′∂x)2〉, (3.1)where ∂u′/∂x is a turbulent-scale shear component measured along the glider’s along-path coordinate, x, and ν is the kinematic viscosity of seawater. Angled brackets indicateensemble averaging, and the subscript U indicates that this is a shear-derived dissipationrate estimate. We use half-overlapping 40-s segments of measurement to calculate suc-cessive εU estimates; within each of these segments, we calculate and average 19 shear45power spectra from consecutive half-overlapping 4-s subsegments and integrate to ob-tain the shear variance in the segment. The spatial length encompassed by each 4-ssubsegment depends on the glider’s speed and has a mean (standard deviation) of 163cm (15 cm) for upcasts and 100 cm (15 cm) for downcasts.We calculate the TKE dissipation rate from the temperature microstructure measure-ments using power spectra of temperature gradient variance, calculated over the same40-s segments and 4-s subsegments that we used to calculate the shear spectra. We fita theoretical form for the temperature gradient spectrum—the Batchelor spectrum—tothe observed gradient power spectrum using the maximum likelihood estimator methodproposed by Ruddick et al. (2000) and later modified by Steinbuck et al. (2009). Inthis procedure, ε is a variable fitting parameter that is optimized by minimizing the dif-ference between the observed and theoretical spectra (the full procedure is detailed inSection 2.3.3). We refer to this optimized value as εT , with the subscript T indicating atemperature gradient-derived dissipation rate estimate.Both, εU and εT estimates, are then subjected to a series of quality control criteria thatremove suspect estimates. These routines are designed to flag and remove values where,for example, the glider’s flight was not steady, shear probes contacted small marineorganisms or debris, Taylor’s frozen turbulence hypothesis is violated in the calculationof power spectra, etc. They are detailed in Appendix A.1. Quality control removes 22%of εU and 34% of εT estimates.Finally, dual estimates of ε from each set of probes are arithmetically averaged to ob-tain single εU and εT values for each 40-s segment. The resulting εU and εT esti-mates are combined into a single best ε estimate using the following method. WhenεU ≥ 1× 10−10 W kg−1, we keep only εU because the shear-derived estimate reliesmore directly on the definition of the dissipation rate and is more reliable in energeticconditions (Gregg, 1999). However, if the εU estimate is unavailable because it failedquality control, we keep the coincident εT estimate instead, if this is available. If bothare available, but they differ by more than a factor of 10, both are discarded. Below1× 10−10 W kg−1, εU begins to be statistically biased by the noise floor of the shearmeasurement and is no longer reliable (Chapter 2); for these measurements, we keeponly εT . However, when εT < 2×10−12 W kg−1, this estimate is also no longer reliable(Chapter 2), and we set ε to zero, following the approach used by Gregg et al. (2012).We are left with 22,153 unique ε estimates for the remaining analysis; of these, 4,699(or 21%) are set to zero.463.2.3 Arithmetic vs. Geometric AveragingTurbulence in the ocean is patchy in space and intermittent in time, and the distributionsof dissipation rates and mixing coefficients are typically lognormal-like, spanning manyorders of magnitude (Baker and Gibson, 1987). Arithmetic and geometric mean valuesmay differ by orders of magnitude in such data, so it is important to distinguish betweenthe two and recognize their distinct physical interpretations (Kirkwood, 1979). Geomet-ric averaging characterizes the central tendency of a lognormally distributed variable,giving a measure of a “typical value” of the distribution. Arithmetic averaging charac-terizes the integrated cumulative effect of a turbulent process and is disproportionatelyskewed by a small number of large values on the right-hand side of the distribution. Forexample, while a geometric mean mixing rate represents a “typical” mixing rate in awell-resolved series of observations, the arithmetic mean rate will more accurately char-acterize net buoyancy transformations produced by mixing in those observations. Wepresent both geometric and arithmetic mean values throughout this study, as appropri-ate.3.3 HydrographyIn Figure 3.2, we present conservative temperature, T , and squared buoyancy frequency,N2, fields derived from the CTD measurements; for each field, a mean vertical profileand a spatial cross section are shown. The horizontal coordinate in the cross sectionsis the glider’s along-track distance coordinate, s, measured along the two-dimensionaltrack shown in Figure 3.1b. To guide the eye, each cross section is broken into multiplepanels at waypoints where the glider changed its direction of travel.We identify five distinct hydrographic layers, similar to those used to describe layeringin the Canada Basin. From shallowest to deepest, these are a warm surface mixed layer(SML); a strongly stratified near-surface “cold halocline” (CH); a cold Pacific Water(PW) layer with water sourced at Bering Strait; an intermediate “warm halocline” (WH)where temperature increases with depth; and a warm Atlantic Water (AW) layer withwater sourced from the Atlantic Ocean. We define the boundaries of the layers usingtheir absolute salinity, SA, characteristics, similar to Carmack et al. (1989); the bound-aries and hydrographic characteristics of the layers are summarized in Table 3.1. Thelayering can be seen most easily in the temperature cross section (Figure 3.2a).Three points about the hydrography stand out as noteworthy for the purposes of thisstudy. First, the amount of heat sequestered below the warm halocline in the warm AWlayer is substantial: in the central Gulf, where the water depth is ∼425 m, and the AW47Figure 3.2 – (a) Arithmetic mean profile and spatial cross section of conservative temperature.(b) Geometric mean profile and spatial cross section of stratification. For the mean profiles,grey shading indicates the range of the central 90% of data; alternating coloured backgroundshading indicates the approximate depth ranges of the hydrographic layers defined in the text(PW, WH, and AW are labelled). For the spatial sections, the horizontal axis is broken andconsecutively labelled 1–4 at the waypoints marked in Figure 3.1, indicating where the gliderchanged direction. White rectangle in (a) indicates the mesoscale eddy discussed in the text.Layer SA [g kg−1] Depth [m] T [◦C] σ [kg m−3] N2 [10−4 s−2]SML <28.3 [0 14] [5.9 7.3] [21.2 22.1] —CH 28.3 – 32.0 [12 47] [−1.2 6.6] [22.3 25.8] [0.81 49.9]PW 32.0 – 33.2 [47 122] [−1.4 −0.6] [25.9 27.1] [0.41 2.45]WH 33.2 – 34.8 [126 275] [−1.36 0.07] [27.3 29.1] [0.18 1.46]AW >34.8 [267 —] [0.15 0.36] [29.1 29.8] [0.03 0.31]Table 3.1 – Properties of the hydrographic layers. Layers are defined by their absolute salinity, SA.Ranges given for depth, conservative temperature, T , density anomaly, σ , and stratification, N2, are forthe central 90% of data. The layer labels are SML: Surface Mixed Layer; CH: Cold Halocline; PW:Pacific Water Layer; WH: Warm Halocline; AW: Atlantic Water Layer.48layer is ∆z ≈ 160 m thick and has a mean temperature T AW = 0.30 ◦C, the sequesteredheat, E, is approximatelyE = ρcp(T AW −To)∆z≈ 2×108 J m−2, (3.2)relative to the melting temperature of ice, To = 0 ◦C. The factors ρ and cp are the densityand specific heat capacity of seawater, respectively. If mixed or advected to the surface,this heat could melt Z∗ = 0.66 m of sea ice, where Z∗ = E/ρi lo, ρi = 910 kg m−3 isthe density of sea ice, and lo = 3.3× 105 J kg−1 is the latent heat of melting sea ice.This amount of sea ice loss would be a significant fraction of the Amundsen Gulf’smobile winter ice pack, which is typically 0.6–1.9 m thick in late spring (Peterson et al.,2008).Second, the stratification is strong everywhere in the subsurface relative to that in lower-latitude oceans. Typical values for N2 in the North Atlantic and North Pacific pycn-oclines are O(10−6) s−2 (Emery et al., 1984). This benchmark is comparable to thesmallest N2 values we observe in the AW, but is nearly two orders of magnitude smallerthan N2 in the PW layer and is three orders of magnitude smaller than N2 in the CH. Astudy by Chanona et al. (2018) recently suggested that stratification is a key controllingfeature of the mixing characteristics in many regions of the Beaufort Sea; we will buildon these results in Section 3.4 by combining our stratification observations and directturbulence measurements to demonstrate that density stratification frequently inhibitsturbulent mixing in Amundsen Gulf. Note, the stratification we observe in AmundsenGulf is comparable to that found throughout the Canada Basin, except in the core of theCold Halocline, where we observe marginally stronger stratification (c.f. Chanona et al.,2018).Finally, while most of the subsurface appears to be generally uniform in the horizontal,there is substantial horizontal mesoscale and smaller (O(1) km) temperature variabilityin the PW layer (Figure 3.2a). Most distinctive is the presence of a mesoscale eddy be-tween waypoints 2 and 3. These features, and what they imply for mixing, are discussedfurther in Section 3.5.3.3.4 Turbulence and Mixing3.4.1 Turbulent Dissipation RatesAs is typical for ocean turbulence observations (Gregg, 1987; Lueck et al., 2002), wefind an ε distribution (Figure 3.3a) that spans many orders of magnitude with a relativelysmall number of strongly turbulent events occurring in a less turbulent background flow49Figure 3.3 – Histograms of (a) the turbulent dissipation rate, ε , and (b) the buoyancy Reynoldsnumber, ReB. For each, the number in the top right indicates the percentage of data that fallwithin the axis limits; the remaining data are zero-valued and cannot be displayed on a logarith-mic axis. The interquartile range for each set, including zero-valued data, is the span betweenthe two dash-dotted lines. For ε , the geometric and arithmetic mean values are also indicated(GM and AM, respectively). For ReB, the approximate critical value Re∗B = 10 is indicated bythe yellow line.field. Note that zero-valued ε estimates (21% of the data; Section 3.2.2), represent-ing turbulence too weak for us to observe, are not depicted in Figure 3.3a. Nonzeroε realizations vary over five orders of magnitude, from O(10−12) to O(10−8) W kg−1;however, in 68% of the observations, ε is smaller than 1× 10−10 W kg−1, a commonbenchmark for “low turbulence” open ocean dissipation rates (Gregg, 1999; Lueck et al.,2002).The arithmetic mean dissipation rate in our observations is 4.9× 10−10 W kg−1, andthe geometric mean is 2.8× 10−11 W kg−1. Note that the geometric mean is definedonly for non-zero values, so we set zero-valued estimates to the smallest nonzero value(2.0× 10−12 W kg−1) for this calculation only. The interquartile range, IQR, of our εestimates is (3.0 – 160) ×10−12 W kg−1, and the median value is 2.3×10−11 W kg−1;only 0.4% of the distribution lies above 1× 10−8 W kg−1. For comparison, averagemid-latitude dissipation rates at depths shallower than 1000 m are commonly O(10−10)or O(10−9) W kg−1 (Waterhouse et al., 2014), about one order of magnitude larger thanwe observe.The variability of the ε field has a notable spatial structure that can be identified inthe mean vertical profile and horizontal cross section of the field (Figure 3.4a). In thevertical, there is an ε minimum in the core of the cold PW layer at ∼100 m depth, withlarger average dissipation rates near the sea surface and the seafloor. The geometricaverage of ε is 5×10−10 W kg−1 at 20 m depth, 1×10−11 W kg−1 at 110 m depth, and2×10−10 W kg−1 at 350 m depth. Laterally, the most obvious source of variability is a50prominent near-bottom patch of elevated dissipation at the base of the continental slope,with ε as high as O(10−8) W kg−1. This turbulent patch is found between s = 52–81km and is identified in Figure 3.4a by a white rectangle.Dissipation rates in the turbulent patch are anomalously high relative to the rest of thefield, but modify the statistics of the full data set only marginally (Table 3.2). For exam-ple, the arithmetic mean of ε excluding estimates from within the patch is 4.4×10−10 Wkg−1, only 10% smaller than the estimate from the whole data set. However, the arith-metic mean of data only from within the patch is 11×10−10 W kg−1, an increase by afactor of 2.2 over the mean calculated from the full set of data. For a similar comparisonof the geometric mean, mode, median, and IQR see Table 3.2.Further information about the variability in the ε field is available from the glider’s threerepeat transects over the continental shelf slope. A comparison of the depth-averageddissipation rate estimates along the three transects is shown in Figure 3.5, for each ofwhich ε is plotted as a function of distance from the glider’s eastern-most waypoint,geometrically averaged in 2.5-km bins. This bin-averaged dissipation rate remained ofthe same order of magnitude over the 7 days needed to complete the transects—noticethat the ε axis in Figure 3.5 is linear, not logarithmic—and varied between (1–5)×10−11W kg−1. From the first and last transects, it appears that ε is systematically larger in thecentral Gulf than it is on the shelf slope, but the second transect doesn’t exhibit thispattern; nonetheless, when all transects are averaged together (not shown), the patternof enhanced ε towards the central Gulf remains. The patch of enhanced ε obvious inFigure 3.4 is situated to the immediate left of the leftmost axis limit in Figure 3.5.A notable attribute of the ε transects in Figure 3.5 is that the patterns of local maximaand minima appear quasi-stationary across all three transects. The most obvious featurecorrelating the patterns in the three transects is the peak at 23.8 km, though 7 other peaksor troughs can be traced between transects, indicated in Figure 3.5 by the dash-dottedlines. The stationarity of these features indicates that they may be the result of interac-tions between local topography and the flow field—either tides, internal waves, or thelocal background flow—but we would need higher resolution topography measurementsand more information about the immediate flow to verify this hypothesis. Note that itis impossible to truly decouple time and space variability in measurements taken by aglider; here, we have treated the ε observations primarily as a spatial series in order tohighlight what appear to be primarily geographic features, but we will discuss temporalvariability and its implications in Section 3.5.1.51Figure 3.4 – Mean vertical profiles and horizontal cross sections of (a) ε , and (b) ReB. Way-points are indicated as in Figure 3.2. For each, the geometric mean profile is given in 25 mbins (blue); for ε , the arithmetic mean profile is also given (black). In both cross sections, thewhite rectangle between waypoints 2 and 3 identifies the patch of enhanced turbulence dis-cussed in the text. In the ReB cross section, red pixels indicate where a turbulent diapycnalflux is expected; grey pixels indicate a predicted absence of turbulent diapycnal mixing. Theapproximate critical value Re∗B = 10 is indicated in the ReB mean profile by the vertical yellowline.Arithmetic Mean Geometric Mean Median IQRAll Dataε [10−11 W kg−1] 49 2.8 2.3 0.3 – 16Kρ [10−8 m2 s−1] 450 – −0.31 −1.1 – 0.18Excluding Turbulent Patchε [10−11 W kg−1] 44 2.4 1.9 0.3 – 13Kρ [10−8 m2 s−1] 100 – −0.29 −1.0 – 0.16Turbulent Patch Onlyε [10−11 W kg−1] 110 15 18 2.7 – 79Kρ [10−8 m2 s−1] 4,600 – −0.65 −1.5 – 810Table 3.2 – Select statistics of ε and Kρ observed in (top) all the data; (middle) all data except that withinthe turbulent patch; and (bottom) data only from within the turbulent patch. The turbulent patch is definedas the region inside the white rectangle in Figures 3.4 and 3.7, between s = 52–81 km on the horizontalaxis.52Figure 3.5 – The three repeat ε transects (left vertical axis) over the continental shelf slope.The horizontal axis is the distance from Waypoint 3 shown in Figure 3.1b. Thick lines are 2.5km geometric mean bin-averages of ε; coloured markers in the background are individual geo-metric mean cast-averages. The quasi-vertical dash-dotted lines connect peaks and troughs thatappear to be stationary between the three ε transects, as discussed in the text. The bathymetryis shown with grey shading in the background (right vertical axis) for reference.3.4.2 The Influence of StratificationCombining observations of ε with those of N2 and the kinematic viscosity, ν , we con-struct estimates of the buoyancy Reynolds number, ReB = ε/νN2, which quantify theenergetic capacity of the flow to develop vertical overturns that lead to diapycnal mixing(Figures 3.3b, 3.4b). The ReB parameter is a measure of the relative magnitudes of tur-bulent kinetic energy, which tends to create mixing through vertical density overturns,and potential energy stored in the stratification, which inhibits vertical overturning. Ev-idence from laboratory, numerical, and field studies suggests that the ReB parameter hasa critical value near Re∗B = 10 below which vertical overturns and diapycnal turbulentmixing are unlikely (Stillinger et al., 1983; Shih et al., 2005; Ivey et al., 2008; Bouffardand Boegman, 2013).Imposing this Re∗B criterion separates our data into two regimes, one where turbulentdiapycnal mixing is expected (ReB ≥ 10), and one where it is not (ReB < 10). Doingso, we find that turbulence in the flow is energetic enough to support enhanced diapy-cnal mixing in only 7% of the observations—these are the measurements to the rightof the yellow vertical line in Figure 3.3b. Equivalently, we can say that we do not ex-pect enhanced turbulent mixing in 93% of observations, suggesting that vertical fluxes53of properties like temperature and density are set by molecular diffusion in this largesubset of the data. The ReB distribution, therefore, suggests that stratification plays adominant role in modulating turbulent mixing in Amundsen Gulf, frequently inhibitingthe development of turbulence.Further, it is clear from the mean vertical profile and spatial cross section of the ReB field(Figure 3.4b) that turbulent mixing is not homogeneously distributed in space. Rather,most of the mixing happens within 100 m of the seafloor in the isolated patch of en-hanced ε that we observed at the edge of the shelf slope region (i.e. inside the whiterectangles in Figure 3.4). Only here is ReB commonly of O(10) or larger, with individ-ual values occasionally reaching as large as O(103). The white rectangle representingthe region of enhanced ε in Figure 3.4a encloses only 8% of the observations, but itencloses 41% of the occurrences where ReB ≥ 10 and 64% of those where ReB ≥ 100.Inside the rectangle, 37% of the observations indicate that ReB ≥ 10; in contrast, for allthe data outside the rectangle, only 5% of the observations indicate that ReB≥ 10.3.4.3 Diffusivity EstimatesIn the 7% of observations where ReB ≥ 10 (Section 3.4.2), we expect that turbulencedrives a localized enhanced density flux. For these observations we calculate the rate ofdiapycnal density diffusion, Kρ , using the canonical Osborn (1980) model for turbulentmixing:Kρ = ΓεN2, (3.3)where Γ is a flux coefficient that we take to be 0.2, following Osborn’s original (upperbound) estimate.In the remaining 93% of observations, where ReB < 10 and a turbulent density flux isunlikely, temperature and salinity are expected to diffuse by molecular diffusion. As-suming a linear approximation for the equation of state of seawater, the diffusivity ofdensity in this case is given byKρ =Rρ κ molS −κ molTRρ −1 , (3.4)where κ molT = 1.4×10−7 m2 s−1 and κ molS = 1.0×10−9 m2 s−1 are the molecular diffu-sion rates of temperature and salinity in seawater1. Since κ molT = 140κ molS , this expres-1Equation 3.4 can be verified by substituting the Kρ expression into a Fickian density flux formulation,Fρ = Kρ∂ρ/∂ z, which returns the expected expression Fρ = ρ0(κSβ∂SA/∂ z−κTα∂T/∂ z), where Fρ isthe vertical density flux.54Figure 3.6 – Histograms of (a) the diapycnal mixing coefficient, Kρ , of density and (b) the ver-tical heat flux, FH . Positive Kρ indicate down-gradient density diffusion; negative Kρ indicateup-gradient density diffusion. For FH , the green shaded area indicates the region between the5th and 95th percentiles.sion can be simplified toKρ =(Rρ −140)Rρ −1 κmolS . (3.5)The quantity Rρ is the gradient density ratio, defined asRρ ≡ β (∂SA/∂ z)α (∂T/∂ z) , (3.6)where α and β are the coefficients for thermal expansion and haline contraction of sea-water. Note that Kρ here may be either positive or negative, depending on the sign andmagnitude of Rρ . Specifically, when 1 < Rρ < 140 (which it is in 67% of our obser-vations where ReB < 10), the density flux due to molecular diffusion of temperatureand salinity is downward, in the direction of increasing density. Finally, note also thatthe water column becomes susceptible to double diffusion when 1 < Rρ < 10, a phe-nomenon that we have neglected here; we address possible effects resulting from doublediffusion in Section 3.5.2.A histogram of the diffusivity estimates, separated by up-gradient and down-gradient, isgiven in Figure 3.6a. The discontinuity between 8×10−8 and 3×10−6 m2 s−1 reflectsthe distinction between turbulent and molecular diffusion, with all data to the right of thediscontinuity expected to be turbulent and all data to the left expected to be molecular.The discontinuity is an artifact of the Osborn model’s inability to describe the transitionbetween turbulent and non-turbulent density fluxes (see Chapter 4). However, thoughstriking in the histogram, the discontinuity in the histogram of our Kρ estimates doesnot alter the broader interpretation of the mixing rates because bulk buoyancy transfor-55Figure 3.7 – Arithmetic mean vertical profiles, in 25-m bins, and horizontal cross sectionsof (a) the diapycnal mixing coefficient, Kρ , of density and (b) the vertical heat flux, FH . Forthe cross sections, the horizontal axis, waypoint markers, and white rectangle identifying theturbulent patch are as in Figure 3.4. The Kρ cross section depicts the absolute value.mations are disproportionately driven by the few turbulent, strongly energetic mixingevents described in Section 3.4.2. Arithmetic Kρ averages are largely unaffected byvariability (or inaccuracies) in the smaller-orders of magnitude Kρ estimates. Note that,for the same reason, negative Kρ values are largely immaterial to the arithmetic meandiffusivities shown in Figures 3.6a and 3.7a; though somewhat unusual, the negative Kρestimates are small in magnitude relative to the few large (and positive) Kρ estimatesseen on the right-hand-side of Figure 3.6a which primarily determine the mean diffusiv-ity.The arithmetic mean of all Kρ estimates is 4.5×10−6 m2 s−1, about 32 times larger thanthe molecular diffusivity of temperature and about 4500 times larger than the molecu-lar diffusivity of salinity, highlighting the importance of the relatively small number ofenergetic mixing estimates in setting the mean mixing rate. The arithmetic mean is the94th percentile of data. Note that if we used the Osborn model (Equation 3.3) withoutimposing an Re∗B criterion to separate turbulent and non-turbulent estimates, the arith-metic mean of all Kρ estimates would be 4.8×10−6 m2 s−1, a factor of only 1.1 timeslarger than the Kρ average we present.The disproportionate contribution of a few scattered but strongly turbulent mixing esti-mates is obvious when comparing the arithmetic mean profile of Kρ with its horizontalcross section (Figure 3.7a), especially above 200 m depth. In this part of the water56column, a very small number of turbulent mixing estimates—represented by scatteredgreen and yellow pixels in the section—are superimposed without any recognizable pat-tern overtop of an otherwise non-turbulent background—represented by the dark blue.In the upper 200 m of the water column, only 3% of the observations indicate a turbu-lent density flux. However, the arithmetic mean profile is typically O(10−7) m2 s−1, twoorders of magnitude above the background molecular salinity diffusivity.Below 200 m, the arithmetic average of Kρ increases steadily and reaches a maximum of3.3× 10−5 m2 s−1 between 335–360 m depth. This elevated mean-Kρ signal is mostlydominated by the same energetic patch that we observed in the ε and ReB sections (Fig-ure 3.4). Inside this patch, where 37% of the observations indicate a turbulent signal(Section 3.4.2), the arithmetic mean Kρ value is 4.6×10−5 m2 s−1; in comparison, out-side the patch, where only 5% of the observations have a turbulent diffusivity signal, thearithmetic mean is 1.0×10−6 m2 s−1. Further metrics comparing Kρ inside and outsidethe patch are presented in Table 3.2.3.4.4 Vertical Heat FluxesWe leverage the high resolution of the temperature microstructure measurements to es-timate the diffusivity of temperature using the Osborn-Cox relation (Osborn and Cox,1972):KT = κ molT (C+1) , (3.7)where κ molT = 1.4×10−7 m2 s−1 is the molecular diffusivity of temperature and C is theCox number, definedC ≡3〈(∂T ′/∂x)2〉(∆T/∆z)2, (3.8)which we calculate from the mean vertical background temperature gradient, ∆T/∆z,and the microscale temperature gradient, ∂T ′/∂x, whose ensemble-average is assumedisotropic. Angle brackets indicate ensemble averaging over the same segments and sub-segments used for the ε calculations (Section 3.2.2). The Cox number is a measure of theamount that turbulence has deformed a smooth background temperature structure and isproportional to the turbulent temperature diffusivity (Winters and D’Asaro, 1996). Notethat KT and Kρ are not generally equivalent in our observations because temperature actslargely as a passive tracer.From the KT estimates, the vertical heat flux, FH , is straightforward to obtain using theform for Fickian diffusion:FH =−ρcpKT ∆T∆z . (3.9)57The factor cp = 4.1×103 J kg−1 K−1 is the specific heat capacity of seawater.The heat flux estimates we obtain are almost exclusively small (Figure 3.6b): the abso-lute value of the flux, |FH |, is less than 1 W m−2 in 96% of the observations, and the IQRof |FH | is 0.01–0.08 W m−2. We find |FH | ≥ 10 W m−2 in only 0.6% of observations,and |FH | ≥ 100 W m−2 in only 0.02% of observations. The arithmetic mean heat fluxthrough the warm halocline, separating the warm core of the AW and cold core of thePW, is only 0.03 W m−2. This is small compared to the Arctic Ocean-wide mean heatloss of 6.7 W m−2 out of the AW (Turner, 2010).There is, of course, substantial horizontal and vertical variability (Figure 3.7b), andthe arithmetic mean upward heat flux reaches as high as 0.30 W m−2 in the AW layerbecause of the effects of the locally isolated high-energy patch (Section 3.4.1). However,even inside the patch, heat fluxes are modest: the arithmetic mean value of FH here is0.22 W m−2, and FH exceeds 10 W m−2 in only 0.30% of observations; it never exceeds100 W m−2. The generally-downward heat flux between the surface mixed layer andthe cold core of the PW layer is likewise spatially variable, as seen in the spatial sectionin Figure 3.7b, with interspersed upward and downward fluxes. The arithmetic meandownward flux is −0.34 W m−2 at depths 10–35 m. There are more substantial heatfluxes out of the top and bottom of the warm core eddy which can be as large as O(100)W m−2 in a few isolated observations.The heat fluxes we report here are generally comparable to, or smaller than, those seenin previously published observations for the region. In the central Canada Basin, wheredouble diffusion often dominates the mixing, fluxes are typically observed to be O(0.1)W m−2 (e.g. Timmermans et al., 2008a; Shibley et al., 2017), and on the Beaufort shelfalong the North American continent, heat fluxes are typically O(0.1) or O(1) W m−2(Shaw et al., 2009; Chanona et al., 2018).3.5 Discussion: Mixing Processes3.5.1 Tidal MixingIn addition to geographic variability (Section 3.4.1), the ε field has a systematic temporalsignal that appears to be driven by the M2 tide. A combination of four factors points tothe dominant role of the M2 tide in modulating the temporal turbulence variability: (i)a peak in the ε power spectrum at the M2 tidal frequency; (ii) a substantial tidal signalin the local currents; (iii) a high likelihood of local internal tide generation; and (iv) theknown propensity for localized internal tide dissipation. We briefly outline each of thesehere.58In Figure 3.8a, we have constructed a power density spectrum of the ε observations, ne-glecting spatial variability and treating the glider measurements as a simple time series.The time series, shown in Figure 3.8b, is of geometrically depth-averaged ε observationsdeeper than 100 m, interpolated to a 15-minute grid and filtered to remove temporal vari-ability on scales smaller than 2 hours. The spectrum is constructed with Welch’s methodusing 4-day segments of data, 50% overlapped and Hamming-windowed. The most no-table feature in the ε power spectrum is a rounded peak between frequencies 1.3–2.4cpd, straddling both the M2 tidal frequency, 1.93 cpd, and the local inertial frequency,f= 1.90 cpd. The spectral peak indicates that the dominant mode of temporal variabilityin ε is linked to the M2 tide, inertial forcing, or some combination of both.Figure 3.8 – (a) Power density spectrum of ε , constructed using Welch’s method and 4 daysegments of data. Grey shading indicates the 95% confidence interval. The M2 and inertialfrequencies are indicated. (b) The ε time series used to construct the power density spectrum.The series is made from the geometric cast-averages of ε for all depths greater than 100 m andis interpolated to a 15 minute grid. Variability on scales smaller than 2 hours has been removed.Two lines of reasoning suggest that the tides are more important than the winds in set-ting the ε variability seen in Figure 3.8. First, there is no analogous peak in the εpower spectrum for observations shallower than 100 m (not shown); the signal at theinertial and M2 frequencies is only prevalent in the deeper measurements, suggestingit is unlikely that the forcing originates at the surface (cf. Lincoln et al., 2016). Sec-ond, Acoustic Doppler Current Profiler measurements from a nearby mooring (Arctic-Net, 2018, mooring CA08, Figure 3.1b) indicate that local current variability is stronglytidal. Power density spectra of eastward and northward current velocities, U and V , ex-hibit narrow peaks centred on the M2 frequency (Figure 3.9b), and the slow and steadymodulation of the barotropic velocity amplitude (Figure 3.9a) in the dominant eastwardcomponent suggests that the current variability is indeed predominantly tidal, not wind-59Figure 3.9 – (a) Depth-averaged current velocity components U and V , measured by ArcticNetmooring CA08 between depths 100–170 m. The grey shading indicates the period of the gliderdeployment. (b) Power density spectra of the above U and V records, with 95% confidenceintervals. (c) Polar histograms with current speeds of the above U and V records, decomposedinto high frequency and residual components. High frequencies are defined as those greaterthan 1.3 cpd and are dominated by the M2 tide. The approximate orientation of the AmundsenGulf’s major axis, azimuth 305◦, is indicated in each histogram by the yellow line. The per-centage on each histogram’s perimeter is the tick label for the radial axis (Relative Occurrence).driven.The directionality of the barotropic tide further suggests that tidally forced mixing isan important process. Decomposing the currents into high frequency and residual flowsusing a scale separation of 1.3 cpd, we find that the high frequency flow (dominatedby the tides, and accounting for 23% of the total variance) is predominantly alignedwith the major axis of the Amundsen Gulf (Figure 3.9c). This alignment is significantbecause our measurements were taken near (∼40 km from) the Amundsen Gulf’s sill andthe adjacent complex topography offshore of the southern tip of Banks Island (Figure3.1a). The directionality of the high frequency currents indicates that the barotropic tidemodulates flow over the sill and adjacent topography at the dominant tidal frequency,making this a likely region for internal tide generation (Polzin et al., 1997; MacKinnonet al., 2017). Note that the current speed is strong only once per tidal cycle, not twiceper tidal cycle as would be expected in a tidally dominated region, because the residual60flow is stronger than the tidal flow (Figure 3.9c), preventing the net current vector fromchanging direction on each tidal cycle. Note also that the shelf slope north of nearbyCape Bathurst has previously been identified as a likely region of strong internal tidegeneration (Kulikov et al., 2004) and that observations linking tides and topography tomixing have been recently reported for the broader Arctic Ocean (Rippeth et al., 2015,2017).Finally, an important feature that distinguishes the internal tide here from those in lowerlatitudes is that an M2 internal tide generated in the Beaufort Sea is not expected tobe able to propagate away to the interior of the Canada Basin. Linear wave theorydoes not allow free propagation of the M2 internal tide northward of 74.47◦, and it hasbeen previously suggested that an M2 internal tide generated in this region becomesresonantly trapped between the continent and the critical latitude (Kulikov et al., 2004).Ultimately, internal tides generated in the Beaufort Sea are expected to dissipate neartheir generation site (Morozov and Pisarev, 2002; Kulikov et al., 2010), supporting theidea that the temporal mixing variability we see in Amundsen Gulf is tidally modulatedby a locally generated internal tide. See Kulikov et al. (2004) for an analysis of localbaroclinic tide generation potential.3.5.2 Double DiffusionEven when energetics do not support a turbulent density flux (Section 3.4.2), enhancedvertical mixing can still result from double diffusive convection given the right temper-ature and salinity conditions (Radko, 2013). The susceptibility of a water column todouble diffusion can be characterized by Rρ , the gradient density ratio (Equation 3.6).Empirically, double diffusion is most commonly observed when 1 < Rρ ≤ 7; it is alsosometimes seen when 7 < Rρ ≤ 10, but it is not typically observed when Rρ > 10 (Kel-ley et al., 2003). In the central Canada Basin’s warm halocline, Rρ is typically 6.3±1.4,and coherent double diffusive staircases are observed over horizontal scales exceeding1000 km (Timmermans et al., 2008a; Shibley et al., 2017).We calculate Rρ from our measurements (Figure 3.10) using background gradients fil-tered to exclude vertical scales smaller than 5 m and find that 1 ≤ Rρ < 10 in 21% ofobservations; 19% are in the range 7–10, and 2% are in the range 1–7. Instances whereRρ < 10 are almost exclusively in a band near the top of the Atlantic Water layer: inthe potential density band σ = 28.5–29.5 kg m−3, corresponding approximately to thedepth range ∼200–335 m, 70% of Rρ observations are in the range 1–10. There is alsoa notable number of small Rρ values in the eddy, where 16% of Rρ observations are inthe range 1–10.61Figure 3.10 – Geometric mean vertical profile and horizontal cross section of the density ratio,Rρ . In the cross section, data are discretized into three regimes: susceptible to double diffu-sion (red: Rρ≤7), marginally susceptible (yellow: 7<Rρ≤10), and not susceptible (purple:Rρ>10). The approximate critical value Rρ=10 is shown in the mean profile by the yellowvertical line.Despite conditions near the top of the AW layer that suggest the density structure thereis favourable to double diffusion, we do not find double diffusive staircases like thoseobserved in the central Canada Basin’s thermocline. There are sporadicly dispersed in-dividual temperature steps that are likely related to double diffusive processes, but thereare no pervasive double diffusive features in our observations. It appears, therefore, thatdouble diffusion does not play a substantial role in the broader vertical transport of heator density out of the thermocline in this region. This finding is somewhat surprising be-cause it is often thought that the absence of a double diffusive staircase implies energeticturbulent mixing (e.g. Guthrie et al., 2017; Shibley and Timmermans, 2019); it remainsunclear then why there is no double diffusive staircase in our observations, given thatturbulent mixing estimates from our data set are typically weak.3.5.3 Pacific Water Mesoscale and Smaller FeaturesOne of the most striking features in our observations is the large variability in the temper-ature structure of the Pacific Water layer, visible in an enlarged view of the temperaturecross section (Figure 3.11a). The most obvious feature here is the anticyclonic warm-core mesoscale eddy between s= 52–98 km and depths 40–100 m. In its core, at∼50 mdepth, the maximum temperature is −0.1◦C, about 1.3◦C warmer than the ambient wa-ter. It appears to have at least one outer tendril, transected by the glider twice at s = 104and s = 139 km. Note that the glider needed about 1.5 days to transect the eddy; we donot know how quickly the eddy was moving over ground, but if we assume it was beingadvected by up to 15 cm s−1, it could have translated up to 19 kilometres while beingtransected by the glider, indicating that the eddy’s diameter was 46±19 km.62Ambient  WaterO(1)  km Temperature Anomal iesMesoscale EddyFigure 3.11 – (a) An enlarged view of the temperature cross section of the cold halocline andPacific Water layers, highlighting the eddy as well as smaller, O(1) km, temperature anomalies.The dashed white lines correspond, from left to right, to the three T-S lines shown in the lowerthree panels. (b) T-S diagrams for the three vertical profiles indicated in the upper panel. Greydots are all the data shown in the upper panel. Dotted lines are density contours.The origin of the eddy is unknown, but its T-S characteristics (Figure 3.11b) suggest thatit was not generated locally. Its large temperature anomaly suggests that its origin is ina locale where the Pacific Water layer outcrops to the surface, which occurs periodicallyat Cape Bathurst (Williams and Carmack, 2008; Se´vigny et al., 2015) and at MackenzieCanyon (Williams et al., 2006), and of course at the inflow of Pacific-origin water inthe Chukchi Sea. Fine et al. (2018) estimated the lifespan of a mesoscale eddy on theChukchi shelf to be 1–2 years, indicating that it is possible for any of these three localesto be a source region of the eddy we observe; assuming an eastward advection scale of∼5 cm s−1 along the shelf-break boundary current (Williams and Carmack, 2015), aneddy generated in the Chukchi Sea would reach Amundsen Gulf in ∼250 days.Irrespective of its origin, the eddy likely had a strong, and perhaps dominant, influenceon the local heat budget of the PW layer. An idealized form of the eddy—a cylinderof radius 23 km, height 60 m, and mean temperature −0.65◦C—would carry 310 PJ of63heat relative to the ambient water. The arithmetic mean heat flux we observe out of thetop of the eddy (in the band σ = 25.7–25.8 kg m−3) is 1.6 W m−2, and the arithmeticmean heat flux out of the bottom of the eddy (in the band σ = 26.6–26.7 kg m−3) is−2.9 W m−2, an order of magnitude larger than arithmetic mean fluxes estimated in thePW layer as a whole (Section 3.4.4). We do not have estimates of the lateral heat fluxout of the flanks of the eddy, but Fine et al. (2018) found lateral fluxes due to intrusions400–4000 times larger than vertical fluxes out of the eddy that they observed. If we usethis result as a reference and speculate that the lateral flux out of our eddy is 900 W m−2(i.e. 400 times as large as the mean flux out of the top and bottom), we can integratethe flux estimates over our idealized cylindrical eddy shape. Doing so, we find a totalflux of 7.8 GW out of the sides of the eddy, and 7.5 GW out of the combined top andbottom boundaries of the eddy, resulting in an estimated total heat flux out of the eddyof 15.3 GW. It is important to note this calculation is highly speculative as there wereimportant differences between the eddy we observed and that seen by Fine et al. (2018);nonetheless, using this calculation as a reference and if this flux remained constant, theheat in the eddy would dissipate within 238 days.Equally striking as the eddy is the presence of substantial temperature variability onhorizontal scales of O(1) km seen throughout the PW layer. This variability can be seenin Figure 3.11a as a series of light purple blotches superimposed on the ambient PWoutside of the influence of the eddy. T-S characteristics of the smaller scale structures(Figure 3.11b) are distinct enough from those of the eddy that they are likely distinctfeatures, not tendrils of the eddy. It is unclear how the smaller structures were created,but the presence of excess heat in the anomalies suggests a connection to the warmernear-surface waters. In light of recent results by Se´vigny et al. (2015), who linkedhorizontal temperature structure above 100 m depth in Amundsen Gulf to submesoscalefrontal formation and isopycnal outcropping at Cape Bathurst, it is possible that weare observing remnant features of nearby submesoscale dynamics. Given the otherwiseweak heat fluxes and minimal turbulent mixing in the PW layer, these features are likelyto play a meaningful role in the overall temperature budget of the layer.3.6 ConclusionsCharacterizing turbulent dissipation rates, diapycnal mixing rates, and vertical heat fluxesin Amundsen Gulf, we found that stratification is the dominant modulator of turbulentmixing in the region. Most commonly, the effects of turbulence were weak in relationto the gravitational stability from the density field, precluding the likelihood of turbu-lent diapycnal mixing; as a result, the mean diapycnal diffusivity for density was smallcompared to that typical of lower latitude oceans. However, turbulence appeared to be64energetic enough to drive an enhanced buoyancy flux in a small number (7%) of theobservations, and these had a disproportionate influence on the net mixing rate. Theserelatively few energetic events enhanced the arithmetic mean diffusivity of density byorders of magnitude over that which would result from pure molecular diffusion.We found evidence that much of the overall variability in turbulence below the PWlayer is driven by the M2 tide, adding to the recent understanding that tides appear tobe a dominant forcing mechanism for turbulence and mixing in the Beaufort Sea (e.g.Rippeth et al., 2015). However, the resulting arithmetic mean heat flux from the warmAW layer is small and unlikely to be a leading order contributor to increased future seaice melt (cf. Carmack et al., 2015). In the PW layer, the temperature variability wasdominated by a warm mesoscale eddy which had the largest influence on the localizedheat budget of this layer. This observation supports the notion that eddies are an im-portant, and perhaps a leading order, contributor to the dynamics and heat budget of thenear-surface Beaufort Sea (e.g. Zhao and Timmermans, 2015; Fine et al., 2018).65Chapter 4Enhanced Heat Fluxes in aMarginally Turbulent Flow4.1 IntroductionThe aim of this study is to examine the characteristics of ocean mixing when turbulenceis weak and stratification is strong. It was motivated when, in an attempt to character-ize mixing rates and heat fluxes using a series of microstructure measurements in thesoutheastern Beaufort Sea (Chapter 3), we noticed that models for ocean mixing failedto accurately predict the turbulent-scale tracer variance we observed in our measure-ments. There continued to be notable micro-scale temperature gradients in conditionswhere numerical and laboratory studies previously found that turbulent mixing shouldbe negligible. Finding few previous reports of field-based studies on the characteristicsof turbulence in an analogous locale and mixing regime, and—in light of the increas-ing interest in Arctic Ocean mixing—recognizing the importance of correctly predictingenhanced tracer fluxes in weakly turbulent, strongly stratified environments, we deter-mined to analyze these results in a dedicated study.As outlined in a review by Ivey et al. (2008), enhanced diffusion of ocean tracers dueto turbulent mixing must eventually revert to simple molecular diffusion as turbulenceweakens. For shear-driven turbulence, which is often assumed to drive the majority ofmixing in the ocean’s interior, the buoyancy Reynolds number,ReB =ενN2, (4.1)is used as a parameter to distinguish between molecular and turbulent diffusion regimes.Here, ε is the dissipation rate of turbulent kinetic energy, ν is the kinematic viscosity of66Figure 4.1 – (a) Distribution of ReB from microstructure data collated in Waterhouse et al.(2014), between the surface mixed layer and 1000 m depth, for the following experiments:Fieberling, NATRE, BBTRE 1996, BBTRE 1997, GRAVILUCK, LADDER, TOTO, DIMES-West, DIMES-DP. (b) Distribution of halocline averaged ReB from a finescale parameterizationof ε using CTD data presented in Chanona et al. (2018). (c) Map showing the locations of thedata used in the histograms; red indicates microstructure data presented in Waterhouse et al.(2014), and blue indicates finescale data presented in Chanona et al. (2018).seawater, and N is the buoyancy frequency; the nondimensional parameter ReB is there-fore a measure of the competing effects of turbulence (which acts to destabilize a flowand drive enhanced tracer fluxes) and viscosity and stratification (which act to dampenturbulence through friction and buoyancy effects). Larger ReB indicate more energeticturbulence; smaller ReB indicate increasingly damped turbulence. Past numerical andlaboratory experiments suggest that turbulent diffusion ceases when ReB = O(10) (e.g.Stillinger et al., 1983; Itsweire et al., 1993; Shih et al., 2005). What we present in thisstudy are observations of enhanced turbulent-scale tracer variance over a broad range ofReB values, including ones of order unity and smaller. Throughout this study, when wewrite “small ReB”, we mean ReB < 10.The finding from laboratory and numerical studies that turbulent diffusion ceases forReB < 10 is difficult to reconcile with field-based estimates of ocean mixing becausethis regime appears to be characteristic of much of the global ocean pycnocline, indicat-ing that weak turbulent mixing in relatively strong stratification may be of global scalesignificance. Diffusivity estimates from various ocean microstructure experiments inthe Atlantic, Pacific, and Southern Oceans—as collated by Waterhouse et al. (2014)—clearly indicate that ReB is frequently of O(10) or smaller in the pycnocline (Figure4.1a). In this data, 55% of measurements indicate that ReB < 10, and 5% indicate thatReB < 1. Similarly, in the Canadian Arctic, diffusivity estimates presented by Chanonaet al. (2018) indicate that 43% of the western Arctic shelf and shelf-slope waters arecharacterized by ReB < 10, and 3% are characterized by ReB < 1 (Figure 4.1b). Despitethis apparent propensity for small ReB in the global ocean, it remains unclear how to best67characterize water mass transformations and tracer fluxes in these conditions, and inves-tigators employ differing techniques. For example, in Chanona et al. (2018), mixingrates for ReB < 20 are set to molecular values (a similar approach to ours in Chapter 3)to account for the laboratory and numerical results mentioned above, whereas in Water-house et al. (2014), mixing rates are calculated from the Osborn model indiscriminatelyof ReB considerations.In this study, we analyze three aspects of low-ReB ocean mixing using the data set wecollected in the Beaufort Sea, described in Chapters 2 and 3. First, we characterize thedegree of turbulent tracer variance (i.e. the Cox number) as a function of ReB over mul-tiple turbulence regimes, as defined by Ivey et al. (2008); we demonstrate how tracervariance in our measurements decreases, but never vanishes, as ReB becomes small.Second, we compare diffusivity estimates from the Osborn (1980) model to diffusivityestimates obtained from the observed tracer variance, demonstrating how the two di-verge as ReB becomes small, an effect that is at least partially due to the differentialdiffusion of temperature and salinity when turbulence is weak. Finally, we estimate theefficiency of turbulent mixing from our data (where this is justified, i.e. for ReB > 10)and compare these results to the classic Osborn model.4.2 Methods4.2.1 Measurements and ε EstimatesData for this study are derived from a series of ocean hydrography and turbulence mea-surements that we collected in the Amundsen Gulf region of the Beaufort Sea in August2015. They were collected continuously over 10 days from 348 quasi-vertical water col-umn profiles, using an autonomous ocean glider in water 185–425 m deep, and were pre-viously described in Chapters 2 and 3. The glider measures conductivity, temperature,and pressure with a Seabird SBE-41 pumped CTD; it measures shear microstructure andtemperature microstructure with two Rockland Scientific SPM-38 and two RocklandScientific FP07 probes, respectively. It glides through the water column at a nominalangle of 26◦ from the horizontal.We derive estimates of the turbulent kinetic energy (TKE) dissipation rate, ε , in consec-utive half-overlapping 40-s segments of measurement. Each of the four microstructurechannels yields an independent ε estimate: shear-derived ε estimates are calculated byintegrating power spectra of the measured shear variance; temperature-derived ε es-timates are obtained by fitting theoretical Batchelor spectra to observed temperaturegradient variance power spectra. The four independent estimates are subjected to a se-68Figure 4.2 – Histograms for the measurements used in this study of (a) turbulent dissipationrate, ε , (b) squared buoyancy frequency, N2, (c) buoyancy Reynolds number, ReB, and (d)gradient density ratio, Rρ . There are N = 13,190 data.ries of quality control routines to remove untrustworthy values and then combined intoa single best ε estimate using the methodology described in Chapter 3. Details aboutmeasurement limitations, assumptions needed for the ε calculations, Fourier transformparameters, and the Batchelor-fitting algorithm are presented in Chapter 2; details aboutthe quality control conditions are given in Appendix A.1.Data for which no Cox number is available (Section 4.2.3) are discarded; data where ε istoo small to estimate reliably (those set to zero in Chapter 3) are also discarded, leaving13,190 estimates of ε for analysis. Remaining ε estimates vary between O(10−12) andO(10−8) W kg−1 (Figure 4.2a) and have a geometric mean of 5.1× 10−11 W kg−1,about an order of magnitude smaller than commonly observed in lower latitude openocean environments (cf. Waterhouse et al., 2014). Stratification is strong throughoutthe measurements: the squared buoyancy frequency, N2, is typically in the range 10−5to 10−3 s−2 (Figure 4.2b), one to three orders of magnitude larger than is typical inthe North Atlantic and North Pacific pycnoclines (Emery et al., 1984). Mean (standarddeviation) temperature and salinity characteristics were T = −0.68 (0.50) ◦C and S =6933.6 (1.0) g kg−1.Combining estimates of ε and N2, we calculate a buoyancy Reynolds number (Equation4.1) for each data point. The combination of generally small ε and large N2 lead toan ReB distribution (Figure 4.2c) that is almost exclusively “small”: of the 13,190 data,12,396 (i.e. 94%) indicate ReB < 10; only very few (82, or 0.6%) indicate ReB > 100,the regime that Ivey et al. (2008) define as “energetic” turbulence.4.2.2 Osborn ModelIn the model proposed by Osborn (1980), the turbulent diapycnal density diffusivity isrepresented asκ turbρ =(R f1−R f)εN2= ΓεN2= ΓνReB , (4.2)where R f is the efficiency of mixing (formally, the “flux Richardson number”) and Γ isthe dissipation flux coefficient. We evaluate κ turbρ using the traditional upper-bound con-stant values R f o = 0.17 and Γo = 0.2 proposed by Osborn on a theoretical basis and onthe basis of then-current lab experiments. The Osborn model is developed from the tur-bulent kinetic energy equation for a stationary flow with zero flux divergence, assumingkinetic energy production at turbulent scales is balanced by a loss to dissipation (ther-mal energy) and a buoyancy flux (potential energy). In practice, it further depends onthe assumptions of isotropy and homogeneity required to create the ε estimates (Chapter2). The efficiency R f can be interpreted as the fraction of turbulent energy productionconverted to a buoyancy flux.When estimating mixing rates in Chapter 3, we followed the standard observationaloceanography approach and assumed that the net diapycnal diffusivity of density, Kρ ,can be represented purely by the turbulent flux; i.e. we assumed Kρ ≈ κ turbρ . Then, basedon recommendations by Ivey et al. (2008), we set Kρ to the molecular diffusion rate ofdensity—which we modelled asκ molρ =Rρ κ molS −κ molTRρ −1 , (4.3)under a linear equation of state assumption—when ReB was less than 10. The parametersκ molT = 1.4× 10−7 m2 s−1 and κ molS = 1.0× 10−9 m2 s−1 are the molecular diffusionrates of temperature and salinity, respectively. The parameter Rρ is the gradient densityratio,Rρ ≡ β (∂SA/∂ z)α (∂T/∂ z) , (4.4)which quantifies the relative contributions of salinity and temperature to the local den-70sity gradient (Figure 4.2). The factors α and β in Equation 4.4 are the coefficients forthermal expansion and haline contraction of seawater, respectively.However, in the real ocean, the true density diffusivity is more accurately modelled as asuperposition of the molecular and turbulent diffusion rates,Kρ = κ molρ +κturbρ (4.5)over the full ReB parameter space, with a continuous transition between when κ turbρ orκ molρ is dominant. Because one of the goals of this present chapter is to assess thebehaviour of the Osborn model in the small-ReB part of the parameter space, we herecalculate Kρ using Equation 4.5 irrespective of a critical ReB criterion. Fundamentally,this approach is more physical—and, as we will show in Section 4.3, better matches theobservations—but it is encumbered by uncertainties in the Osborn model when ReB issmall, as we will discuss in Section 4.4.2. We will refer to Equation 4.5 as the modifiedOsborn model.4.2.3 Temperature Variance Method: The Osborn-Cox ModelThe net diathermal temperature diffusivity isKT = κ molT +κturbT , (4.6)where κ turbT is the turbulent component of the temperature diffusivity which is a functionof the turbulent-scale temperature gradient variance (Winters and D’Asaro, 1996) andcan be expressed asκ turbT = κmolT C . (4.7)This representation is known as the Osborn-Cox model (Osborn and Cox, 1972). Here,the factor C, the Cox number, is the ratio of turbulent gradient-variance to mean gradient-variance, which we calculate from the microstructure temperature measurements ac-cording toC ≡ 3〈(∂T ′/∂x)2〉(∆T/∆z)2. (4.8)In the limit of no turbulence C→ 0. The factor of 3 assumes isotropic turbulence, an-gle brackets indicate ensemble averaging, x is the glider’s along-path coordinate, andz is the vertical coordinate. We calculate the turbulent gradient-variance〈(∂T ′/∂x)2〉in detrended 40-s segments of measurement by averaging power spectra from 19 half-overlapping 4-s subsegments and integrating over all wavenumbers that have a mea-surable signal. The mean vertical gradient-variance (∆T/∆z)2 is calculated over each40-s segment. The spatial length of the 4-s subsegments effectively defines the largest71turbulent length scale we consider and is dependent on the glider’s instantaneous speedthrough water; the mean (standard deviation) spatial length of the 4-s subsegments is1.19 (0.34) m. Note that the turbulent-scale and mean gradients need not be taken in thesame direction since we assume turbulent gradients are statistically isotropic; the meangradient, however, is not isotropic and must be taken in the vertical since we assume thisthe direction in which the mean gradient is largest. Equation 4.8 is ill-defined when thebackground temperature gradient approaches zero; consequently, we calculate C onlywhen |∆T/∆z| ≥ 2×10−3 ◦C m−1.Unlike ε , the Cox number is a quantity that we can observe directly using the temper-ature measurements. It can be interpreted geometrically as a measure of how much aturbulent flow field has strained a mean background gradient, and it does not rely on thestationarity and non-divergence assumptions required to relate ε to the microstructuremeasurements in the Osborn model (Winters and D’Asaro, 1996). Likewise, Equation4.7 is independent of assumptions about the efficiency of mixing or the dynamics of theflow; it states only that the turbulent diffusion rate of the temperature tracer is propor-tional to the degree of turbulent-scale strain in the temperature field. Consequently, weexpect the theoretical framework behind the temperature variance approach to be physi-cally tractable for all ReB and encumbered only by the assumption of isotropic gradients,which increases the uncertainty in C when turbulence is modified by stratification.It is important to note that temperature is largely passive in the conditions observedhere: mean thermal expansion and haline contraction coefficients for our data are α =3.7×10−5 ◦C−1 and β = 7.8×10−4 (g/kg)−1 respectively, indicating that density is tofirst order set by salinity. As a consequence, temperature and density are not expectedto diffuse at the same rate when turbulence is too weak to fully mask the two-order ofmagnitude difference between the molecular diffusivities of temperature and salinity.Note also that, by substituting Equation 4.7 into Equation 4.6,C+1 =KTκ molT≡ K∗ ; (4.9)that is, C+1 characterizes the net temperature diffusivity (turbulent + molecular) scaledby the molecular diffusivity and can therefore be interpreted as the degree of turbu-lent diffusivity enhancement. We henceforth refer to this nondimensional diffusivity asK∗.4.2.4 Idealized Turbulence: IsotropyThe estimation of ε (and therefore ReB, κ turbρ , and Γ) and the application of the Tem-perature Variance Method (Section 4.2.3) assume spatially homogeneous turbulence72that is statistically stationary in time and spatially isotropic. None of these assump-tions are truly met for measurements in the real ocean (Gregg, 1987, Section 7), addinguncertainty to all oceanic mixing estimates, but the isotropy assumption (which is re-quired because we measure 2 of 9 shear components, and 1 of 3 of temperature gra-dient components) proves to be particularly problematic because it effectively requiresthat dissipation-scale eddies are immune to the buoyancy effects that arise from densitystratification, which becomes increasingly untrue as stratification strengthens and turbu-lence weakens. Field measurements by Gargett et al. (1984) found that buoyancy effectsmodify turbulence at dissipative scales for ReB < 200. For context, 94.1% of data inFigure 4.1a, 90.9% of data in Figure 4.1b, and 99.7% of data in Figure 4.2c are smallerthan this criterion.The degree of uncertainty associated with the departure from isotropy is unclear, butit generally appears to be within the bounds of “typical” uncertainties associated withoceanic turbulence measurements, which are often about a factor of 2–3 for statisticalquantities and a factor of 5–10 for individual measurements. Field measurements ofmultiple shear components by Yamazaki and Osborn (1990) indicate that errors in ε es-timates due to anisotropy are negligible for ReB > 20 and are limited to less than 35%at smaller ReB (the smallest value they report is ReB = 2). Numerical simulations byItsweire et al. (1993) indicate that errors in ε are within a factor of 2–4 in the range50 < ReB < 650, and ones by Smyth and Moum (2000) indicate that anisotropy is negli-gible for ReB > 100. A theoretical analysis by Rehmann and Hwang (2005) suggests thaterrors in ε due to anisotropy when ReB < O(10) are limited to ±33% for unsheared tur-bulence and increases to a factor of 3 for sheared turbulence. In our own measurements(along an angled profile), we find no statistical distinction between the two measuredshear components, indicating that errors due to anisotropy when comparing these twocomponents is smaller than the scatter of the individual data, approximately a factor of5 (Appendix A.1).4.2.5 Other Diffusivity ModelsBesides comparing KT estimates to the modified Osborn model, we further compare withmore recent models by Shih et al. (2005) and Bouffard and Boegman (2013), hereafterSKIF and BB. These are modified versions of the Osborn model that aim to account forvariability in the characteristics of turbulent mixing when turbulence is either very weakor very strong. However, the traditional Osborn model remains the most commonlyemployed method for field measurements (cf. Gregg et al., 2018).In the SKIF model, which is developed from direct numerical simulations, mixing isbroken up into three regimes—molecular, intermediate, and energetic—using ReB as a73criterion to distinguish the regimes. In the intermediate regime, defined for the rangeReB = 7–100, diffusivity is represented by the traditional Osborn model (Equation 4.2),neglecting molecular diffusion. In the energetic regime, defined for ReB > 100, diffu-sivity is represented by Kρ = 2νRe1/2B . In the molecular regime, defined for ReB < 7,diffusivity is represented by molecular diffusion only: Kρ = κ molρ .The BB model, based on an empirical fit to results from multiple laboratory and numeri-cal studies, is identical to the SKIF model but adds a transitional “buoyancy controlled”regime between the molecular and intermediate regimes. This regime is dependent onPrandtl number, Pr, and is bounded by 102/3Pr−1/2 < ReB < (3ln√Pr)2; the purposeof including this regime in the model is to remedy the discontinuity that arises in theSKIF model between the molecular and intermediate regimes, at ReB = 7. The BBmodel’s diffusivity in this regime is represented by Kρ = (0.1ν/Pr1/4)Re3/2B . Note, weuse Pr = ν/κT ≈ 10 for the Prandtl number definition, indicating that the buoyancycontrolled regime is in the range 1.5 < ReB < 11.9.4.2.6 Mixing EfficiencyFor turbulent mixing, the efficiency of mixing, also known as the flux Richardson num-ber, R f , is the ratio of buoyancy flux (i.e. potential energy change) to turbulent kineticenergy production. The efficiency must, by definition, vary between 0 and 1. It is relatedto the flux coefficient in the Osborn model byR f =Γ1+Γ. (4.10)If turbulent mixing is energetic enough that κ turbρ and κ turbT are equal, and if moleculardiffusion can be neglected, we can equate expressions (4.2) and (4.7) and write the fluxcoefficient asΓ=κ molT CN2ε. (4.11)Expression 4.11 is valid if there is no differential diffusion of salinity and temperature(Gargett, 2003). There is no clearly defined regime transition below which differentialdiffusion becomes locally important to mixing, but based on results given in Ivey et al.(2008) and Jackson and Rehmann (2014), we calculate Γ and R f only when ReB >10. Below this cutoff value, it is unlikely that Equation 4.11 is physically meaningfulbecause the effects of molecular temperature and salinity diffusion are expected to be offirst order importance to density transformations. Of our 13,190 data points, 794 satisfythe condition that ReB > 10.744.3 Results4.3.1 Diffusivity EstimatesWe present the nondimensionalized temperature diffusivity estimates derived from theTemperature Variance Method, K∗, as a function of ReB in Figure 4.3a. Mode, arith-metic mean, and geometric mean values are shown in logarithmically-spaced bins alongthe ReB axis. Predictions from Osborn (modified), SKIF, and BB models—all normal-ized by κ molT to facilitate comparison with K∗—are also shown. As anticipated, theprimary trend in the temperature diffusivity observations is that K∗ increases with in-creasing ReB, indicating greater turbulent diffusivity enhancement when turbulence ismore intense and weaker enhancement—trending towards K∗→ 1 as ReB→ 0 from theright—when turbulence is less intense. The modified Osborn model predicts the mea-surements most accurately and agrees with the mode and geometric mean values withina factor of 3 across all ReB bins, tending to underestimate the measurements at ReB ≥ 5and overestimate them at smaller ReB. The mode value tracks the modified Osborn pre-dictions most closely across ReB bins and agrees with the model by better than a factorof 2 in all but three of the bins.The observed trend that K∗ most commonly approaches unity as turbulence weakens isnot surprising; we expect that net diffusion reverts to molecular diffusion in the absenceof turbulence. The more interesting part of Figure 4.3a is the large scatter that exists inK∗, especially in weak turbulence, because it indicates that substantially enhanced heatfluxes are, on average, possible even when turbulence is very weak and stratification isstrong. The scatter in K∗—more specifically, the long tail in the distribution of K∗ atany given ReB—works to drive up the average heat flux from the idealized models byat least an order of magnitude across all observed ReB. Even in the lowest bin, centredon ReB = 1.5× 10−2, the temperature diffusivity averaged across all data in the bin isenhanced by a factor of 11 over the molecular diffusion coefficient κ molT .The net diffusivities of density and temperature, estimated from the modified Osbornmodel (Equation 4.5) and the temperature-variance method (Equation 4.6), respectively,diverge systematically with decreasing ReB. This trend can be seen in Figure 4.3b,where we plot the ratio KT/Kρ as a function of ReB. In the most energetic observations,when ReB > 100, the agreement between the two methods is a factor of 2.2 (geometricmean value), indicating that the diffusivity prediction from the modified Osborn modelmatches the observed temperature diffusion rate reasonably well. The best agreementoccurs at the bin centred on ReB = 7.5, where the two estimates agree within a factorof 1.1 (geometric mean values). However, as ReB decreases further, the disagreementbetween the two diffusivity estimates increases: at ReB = 1, the geometric mean ratio75Figure 4.3 – (a) Scatterplot of the nondimensionalized net temperature diffusivity, KT /κT ,as a function of buoyancy Reynolds number, ReB. White symbols indicate mode, arithmeticmean, and geometric mean in logarithmically spaced ReB bins. Models from Osborn (1980,modified), Shih et al. (2005), and Bouffard and Boegman (2013) are shown for reference, allnormalized by κ molT to facilitate comparison with K∗. The three red data points are select—but in no way remarkable—points for which the raw temperature microstructure records areshown below. The three turbulence regimes proposed by Ivey et al. (2008)—molecular, tran-sitional, and energetic—are indicated by the background shading—purple, white, and orange,respectively. (b) The ratio of the net temperature diffusivity (Equation 4.6) to the net diffusivitycalculated from the modified Osborn model (Equation 4.5). Symbols as in panel a, exceptingthe arithmetic mean which is omitted here because it is not informative. (c) The microstructuretemperature records used to calculate the three select data points (red) in panels a and b. Eachrecord represents 40 s of measurement, spanning an along-path distance ∆x, a vertical distance∆z, and a temperature difference ∆T between the first and last measurements. For each 40-ssegment of measurement, one temperature gradient spectrum is calculated by averaging spectrafrom 19 half-overlapping 4-s subsegments (Section 4.2.3).76between the two is 3.0; at ReB = 0.1, the geometric mean ratio is 9.5; and at ReB = 0.01,the geometric mean ratio is 37. The physical interpretation is that the Osborn modelbecomes increasingly unsuccessful at predicting the observed temperature diffusivity asReB becomes small. That said, it is important to recognize that Kρ and KT should notgenerally be expected to be equal as turbulence weakens because temperature and salin-ity diffuse at differing rates when molecular diffusion is important. The implications ofthis point, together with the appropriate interpretation of the results seen in Figure 4.3b,are discussed further in Section 4.4.2.The red markers in Figures 4.3a and 4.3b are three selected data points for which themeasured 40-s microstructure temperature records are shown (Figure 4.3c). The param-eters ε , ReB, and C are indicated in each respective panel for these three records. Theincrease in temperature gradient variance from left to right is obvious to the eye as C in-creases from O(0.1) to O(1) to O(10), but gradient variance is notably visible in all threepanels. The three examples are representative of nearby data in the ReB–K∗ space and arenot otherwise remarkable. They are presented in order to exemplify what representativetemperature structure looks like in the raw measurements when ReB is O(0.1) or O(0.01)and to highlight that the temperature variance we observe in this small-ReB regime is areal signal, not instrument noise or data processing artifact. Note, the microstructuretemperature probes used for the measurements have a time response of ∼0.01 s and aresensitive to better than 10−4 ◦C, scales much smaller than those relevant to the gradientsin Figure 4.3c.4.3.2 Mixing EfficiencyA histogram of the flux coefficient estimates we infer from the subset of measurementswhere ReB > 10 is shown in Figure 4.4a. These vary between Γ = 0.001–99, though90% are in the range 0.01–0.71; the interquartile range is 0.05–0.19. The geometricmean value is 0.09, and the median value is 0.10, indicating that typical values from ourdata set are about a factor of 2 smaller than the canonical upper bound value Γo = 0.2proposed by Osborn (1980).We observe no statistically significant dependence of Γ on ReB in the range ReB = 10–100, but there is a clear trend towards decreasing Γ in the range ReB = 100–1000.To identify this behaviour, we calculate median and geometric mean values of Γ inlogarithmically-spaced bins of ReB (Figure 4.4b). For ReB < 100, the binned geometricmean values are in the range 0.08–0.09 with no clear trend and geometric standard errorfactors less than 1.5. The two bins centred on ReB = 240 and ReB = 520 indicate geo-metric mean Γ values of 0.06 and 0.02 respectively. The uncertainty in the sample meanincreases with larger ReB because there are fewer data points here, and in the largest bin77Figure 4.4 – (a) Histogram of flux coefficient estimates for the subset of data where ReB > 10.Dash-dotted lines indicate percentiles 5, 25, 75, and 95. The red triangle indicates the canonicalvalue proposed by Osborn (1980). N indicates the number of data points. (b) Flux coefficientplotted as a function of ReB. Large open-faced symbols are the median and geometric meanvalues in geometrically-spaced bins. Error bars indicate the geometric standard error in themean, calculated from two geometric standard deviations. (c) As in panel a, but for mixingefficiency, R f . (d) As in panel b, but for R f and with arithmetic mean values and standarderrors in place of geometric ones.(at ReB = 520), the geometric standard deviation factor is 2.0.Our estimates of the mixing efficiency, R f , vary between 0.00 and 1.00 (Figure 4.4c).The central 90% of data is in the range 0.01–0.42, and the interquartile range is 0.04–0.16. Our estimates are smaller than the canonical value R f o = 0.17 in 77% of the data.The mode and median values are 0.05 and 0.09, respectively, and the arithmetic mean is0.13.As with the Γ estimates, median and mean values of R f in bins of ReB indicate that thereis no statistically significant trend in R f when ReB ≤ 100 (Figure 4.4d); in this subset,the binned mean values of R f are in the narrow range 0.12–0.15, with standard error of780.03 or less. For ReB > 100, there is a statistically significant trend towards less efficientmixing as ReB increases, and in the highest bin (ReB = 520), the mixing efficiency is0.05±0.02.4.4 Summary and DiscussionIn Section 4.3, we presented results from our case study of mixing characteristics in theweakly turbulent, strongly stratified Beaufort Sea. These can be summarized into threekey findings:• We frequently observed substantial turbulent-scale (O(1)–O(100) cm) tempera-ture diffusivity enhancement in a low-ReB regime where models of shear-driventurbulent mixing based on laboratory and numerical studies have previously sug-gested that tracer fluxes should be strictly molecular.• The commonly employed Osborn model, even when modified to account formolecular diffusion, does not accurately predict temperature diffusivities in ourobservations. The degree by which it underestimates KT increases systematicallywith decreasing ReB and is as large as a factor of ∼40 for the smallest observedReB values.• Mixing efficiency estimates from our data in intermediate and energetic regimesare highly variable and span the full range from 0 to 1; however, in general theyappear to be about a factor of 2 smaller than the canonical value R f o = 0.17 when10 < ReB ≤ 100, and they decrease further at larger ReB.In this section, we discuss each of these three findings in turn, addressing implicationsthat arise from our results, uncertainties in the methods, and some recommendations forfuture study.4.4.1 Enhanced Heat FluxesFrom a shear-driven turbulence point-of-view, the turbulent temperature gradient vari-ance we frequently observed at very small ReB is unexpected, as highlighted by the dif-ference between the mean K∗ estimates and the predictions from the Osborn, SKIF, andBB models (Figure 4.3a). The observed micro-scale structure was unexpected not onlybecause laboratory and numerical studies have found that shear-driven turbulent mixingtypically ceases below ReB ≈ 10, but also because ReB values below unity are difficultto interpret dynamically. The length scales of the largest and smallest eddies in stratified79turbulence are given by the Ozmidov and Kolmogorov length scales, LO = (ε/N3)1/2and LK = (ν3/ε)1/4, the ratio of which is proportional to ReB:ReB =ενN2=(LOLK)4/3. (4.12)An ReB value less than unity suggests the absence of a classical cascade of energy fromlarge to small turbulent eddies. Interpreted energetically, it suggests that there is insuf-ficient kinetic energy in the turbulent flow field to create a density overturn through thegiven background stratification. It seems unlikely, then, that the turbulent-scale gradi-ents we observed in this study can be described by the idealized model of shear-driven,isotropic turbulence (Section 4.2.4).A number of mechanisms outside of shear-driven, isotropic turbulence may be responsi-ble for the enhanced temperature gradient variance (and associated heat fluxes) seen inour measurements. The most obvious mechanism that may contribute to the enhancedheat fluxes is double diffusion, since large portions of the Beaufort Sea thermocline arewell known to be susceptible to double diffusion (Timmermans et al., 2008a). We didnot observe an obvious double diffusive staircase in our measurements (Chapter 3), butthe gradient density ratio, Rρ , is conducive to double diffusive convection in at leasta portion of the observations (Figure 4.2). Focusing on a low-ReB subset of the data,we find that 1 < Rρ < 10 in 21% of data for which ReB < 0.1. This characterizationindicates that double diffusion may play an important role in creating the observed gra-dient variance, but it also clearly suggests that double diffusion is not the sole dominantmechanism contributing to the enhanced heat fluxes.Another possibility is that at least a portion of the enhanced gradient variance we observeat small ReB is a remnant feature of some previous, more energetic, turbulence event.A simple scaling argument confirms that the timescale needed for molecular diffusionto naturally smooth the observed gradients is large enough that this could be a realis-tic possibility. For example, in the middle panel of Figure 4.3c, where C = 5 and thedissipation rate of temperature variance is χ = 2κTC (∆T/∆z)2 = 6.5× 10−11 K2 s−1,the timescale T for diffusion to destroy a turbulent temperature anomaly of O(0.01) K isabout T = ∆T 2/χ = 328 days. However, again, it seem unlikely that all the temperaturestructure we observe is a remnant from some previous, more energetic turbulence eventbecause we would not expect that a remnant temperature-gradient signal would decreasesystematically with ReB, as we see it does in Figure 4.3a. The systematic nature of thistrend, rather, is consistent with a dynamic process relating C and ReB, suggesting thatthe creation and sustenance of microstructure gradients becomes less pronounced whenturbulence is weaker.A final possible mechanism that may contribute to the observed enhanced gradient vari-80ance at small ReB is shear-driven turbulence similar to that considered by the Osbornmodel, but not isotropic; the concept is one of anisotropic turbulent eddies that are ac-tively dissipating energy but are vertically squashed by stratification. Such eddies havepreviously been coined “pancake” eddies because buoyancy constraints result in turbu-lence dynamics that are more pronounced in isopycnal directions than in the diapycnalone (e.g. Riley and Lelong, 2000; Lindborg and Fedina, 2009). One problem with thistheory is that we do not actually observe anisotropy from the two perpendicular shearcomponents that we measure (see Appendix A.1, Figure A.1), but this may be becausethe shear measurements begin to lose their reliability when dissipation rates are smallerthan 10−10 W kg−1 (Chapter 2).In reality, it is probably most likely that no single mechanism is responsible for theenhanced heat fluxes at small ReB, but that they are created by some combination of themechanisms described here.4.4.2 Applicability of the Osborn ModelIn Figure 4.3, we found good agreement between the direct KT estimates (Equation 4.6)and the modified Osborn model’s Kρ estimates (Equation 4.5) when ReB > 10. However,as ReB becomes smaller, the modified Osborn model is no longer able to predict KT , andthe Kρ and KT estimates typically diverge by a factor of ∼37 (geometric mean) whenReB = O(0.01). There are at least two distinct reasons for this divergence, one physicaland one an artifact of the Osborn model, the effects of which are convoluted in the finalKT/Kρ results.The first consideration is that, physically, we should not expect KT and Kρ to be equalwhen turbulence is weak. When turbulence is strong, the turbulent components of KTand Kρ are equal and dominant over their molecular analogues, and temperature anddensity diffuse at the same rate. However, in the limit of no turbulence, KT → κ molT andKρ → κ molρ , and (from Equation 4.3) the ratio KT/Kρ approachesκ molTκ molρ=κ molT (Rρ −1)Rρ κ molS −κ molT=140(Rρ −1)Rρ −140 . (4.13)For the Rρ regime represented by our data, the magnitude of κ molT /κ molρ is typically inthe range 5 – 72 (central 90%; median value 13), suggesting that a significant componentof the signal seen in Figure 4.3b at low ReB is caused by the effects of differentialdiffusion of temperature and salinity. Previous studies, from the laboratory and fromnumerical simulations, appear to see the onset of differential diffusion at ReB ≈ 100(Gregg et al., 2018), indicating that nearly all our measurements may be susceptible tothe effects of differential diffusion (Figure 4.2c).81The second consideration is that the physical basis underlying the Osborn model be-comes increasingly intractable as ReB becomes small. This fact is in part due to thebreakdown of isotropy, discussed in Section 4.4.1, and in part due to the assumption thatthe efficiency of diapycnal mixing remains constant in very weak turbulence. This latterpoint especially is concerning because it seems unlikely that a significant fraction ofturbulent kinetic energy is converted to a buoyancy flux when stratification inhibits theoverturning of a dissipation-scale eddy (i.e. when ReB < 1). However, it remains unclearwhat the best representation for the mixing efficiency is in this regime, and there appearsto be no consensus for a justifiable alternative to using R f o = 0.17 when interpreting fieldmeasurements generally (Gregg et al., 2018).Synthesizing these two points, we conclude that the application of the modified Osbornmodel to our data (Figure 4.3b) largely replicates the trend one would expect to see forthe transition between turbulence and laminar flow. In the limit of energetic turbulence(ReB > 100), diffusion rates of temperature and density are equal. In the limit of dampedturbulence (ReB 1), differential diffusion rates of temperature and salinity cause theratio KT/Kρ to diverge systematically. However, the exact behaviour of KT/Kρ in ourdata is also almost certainly influenced by uncertainties in the theoretical framework ofthe Osborn model which, unfortunately, preclude us from suggesting a functional formfor the behaviour of differential diffusion in our data. It is, however, clear that κ turbρ ,as estimated by the Osborn model (Equation 4.2), should not be used to predict heatfluxes or other passive tracer fluxes in low-ReB regimes like the Beaufort Sea withoutaccounting for the potential effects of differential diffusion or comparison to results fromdirectly measured micro-scale temperature gradients. It should be limited to makingpredictions of density diffusivity.4.4.3 Mixing Efficiency in High StratificationEstimating the efficiency of mixing for turbulence in the ocean is precarious business,as highlighted in review articles by Ivey et al. (2008) and, more recently, Gregg et al.(2018). Not only are the many assumptions that lead to Equation 4.11—isotropic, homo-geneous, stationary flow; negligible differential diffusion; all measurements normal tomicrostructure gradients—unrealistically constrained for field measurements, but thereis also a large degree of natural variability in R f depending on the stage of a turbulentbillow and, potentially, the mechanisms driving turbulence. The definitions for “mix-ing efficiency” are not the same across numerical, laboratory, and field studies, makingcomparisons difficult, and there is little convergence between results. Even among ob-servational studies there is little agreement: the 14 previous studies reviewed by Gregget al. (2018) typically found Γ in the range 0.1–0.3, leading the authors to suggest con-82tinued use of Γo = 0.2 until a better consensus emerges.The large variability in our Γ and R f data is likely due to a combination of the naturalvariability and the errors and uncertainties described above. However, the trends seen inFigure 4.4 appear to be robust and there are enough data that the uncertainty in the sam-ple mean in each ReB-bin, at least, is generally small. Our finding that Γ ≈ 0.1 in mostof our data is, therefore, either physical and the result of atypical mixing characteristics,or it is an artifact that arises from a systematic bias in our techniques. The uncertaintiesin estimating Γ certainly allow for the possibility that there is a systematic bias, but wecannot isolate this effect in our measurements, if it is present. We do know, however,that our measurements are distinguished from previous estimates of Γ by locale andstratification conditions. Many previous studies about mixing efficiency do not reportReB—the ones that do mostly report ReB ∼ O(102) or ReB ∼ O(103)—but it appearsthat nearly all previous oceanic estimates of Γ come from mid- or low latitudes, whereN2 tends to be an order of magnitude or more smaller than in the Beaufort Sea. Onenorthern study, by Peterson and Fer (2014), using measurements from Faroe Bank, didreport Γ in similar ReB conditions to those we report here and, unlike us, found largerthan expected mixing efficiencies when ReB was small: the authors reported Γ ≈ 0.6in the range ReB = 5–100. Still, compared to the overwhelming majority of observa-tional stratified turbulence studies, our situation is unique, and so it is important thatfuture studies continue to examine mixing characteristics in Arctic-like conditions be-cause accurate estimates of Arctic Ocean mixing are an important strategic componentfor modelling present and future climate change in northern latitudes (Carmack et al.,2015). Specifically, we recommend future observational studies in the Arctic report R fand Γ alongside ReB, where possible, in order to build a record of mixing efficiencyestimates in similar strongly stratified, weakly turbulent environments.83Chapter 5Conclusion5.1 Goals and Representativeness of the ThesisIn Section 1.2, I defined the objectives of this thesis under three categories:i. measuring turbulence when turbulence is weak and stratification is strong,ii. characterizing turbulent mixing in the Arctic Ocean’s Amundsen Gulf, andiii. understanding enhanced heat fluxes in strongly stratified, weakly turbulent envi-ronments.In the following section, I synthesize the results of my research to respond to the ques-tions I initially posed for each of these categories, in turn.Before doing so, however, it is worth pausing briefly to reflect on how the results ofmy work do—and do not—translate to the broader fields of Arctic oceanography andturbulent ocean mixing. After all, the results presented here stem from measurementstaken in a particular region of the Beaufort Sea, at a particular time of the year, over aspan of only 10 days, and a word on their representativeness is needed.I anticipate that Chapter 2 is generalizable to glider-based microstructure measurementsin all regions where turbulence is weak and stratification dominates the turbulent dy-namics. The methods and insights I propose are in no way unique to the Beaufort Seaand require only that the measurements be taken in a similar parameter space.Chapter 3 is about ocean mixing in Amundsen Gulf in summer; it does not speak toocean mixing in the Beaufort Sea or the wider Arctic Ocean more generally, nor does itaddress seasonal or interannual variability of ocean mixing. That said, I also anticipatethat many of the key features that determine the physical mixing characteristics here areimportant at other times of the year and in other locales. For example, it is unlikely thatthere is substantial seasonal variability in the Atlantic-derived layers since the time scale84for Atlantic Water to reach the Beaufort Sea is on the order of years, and seasonal signalsthat may have existed in this water mass when it was in the North Atlantic have largelydiffused by the time it arrives at Amundsen Gulf (Rudels, 2015). Similarly, the dominantrole of stratification over turbulence in setting vertical mixing translates to much of thewestern Arctic (e.g. Chanona et al., 2018), and recent work indicates that tidal flow is adominant driver for turbulent mixing across the Arctic Ocean generally (Rippeth et al.,2015). Therefore, I think my results here may be viewed as a ”high-resolution snapshot”that cannot generalize in all relevant circumstances but do reflect many relevant featuresof ocean mixing across the western Arctic.The representativeness of Chapter 4 is probably the most challenging to ascertain. Onone hand, the results in this chapter are presented in a way that is decoupled from thelocale of the measurements, and my hope is that they spur further investigation in moregeneralized settings. However, fundamentally, the results we deduced from those mea-surements cannot in good conscience be divorced from the study site in which they weretaken since we do not know what mechanisms created the enhanced temperature vari-ance we observed. In light of that uncertainty, I think it is safest to defer a judgementon the representability of the results of Chapter 4 to other ocean sites until future studiescan reproduce—or provide a rebuttal to—the conclusions I drew here.5.1.1 Observing weak turbulence in strong stratificationa. Do sensor limitations hinder the ability to formulate meaningful ε estimates inthese conditions?Yes, they do, especially for the shear probes, whose measurements are liable tobe influenced by a noise floor below which no meaningful shear signal can bedetected. We found that as much as 70% of the shear-derived ε estimates wereinfluenced by the noise floor of the shear probes. For the temperature measure-ments, we did not find that a sensor limitation hindered our ability to formulatemeaningful ε estimates in weak turbulence, though it is known that in strong tur-bulence the measurement quality suffers from slow response times (e.g. Gregg,1999).85b. In what conditions and to what extent do sensor limitations impact the ability tomeasure ε? And, what is the impact of those limitations on the interpretation ofthe measurements?The quality of our shear measurements was limited by the intensity of the turbu-lence. We found that shear-derived estimates of ε were useful to values as smallas 3×10−11 W kg−1, which we defined as the noise floor of the shear-based mea-surement. In our observations, 58% of the measurements appeared to be belowthis threshold. A more conservative approach to choosing a noise floor would beto consider the threshold where sensor limitations first begin to statistically skewthe distribution of the ε estimates. We found this threshold to be 1× 10−10 Wkg−1; as many as 70% of our measurements were below this threshold.The arithmetic mean of a large sample of shear-based ε estimates from a coastalor otherwise energetic site in the ocean is probably affected only weakly by errorsin small ε values because the arithmetic mean will be dominated by the few largeε values on the right-hand-side of the sample distribution (assuming the distribu-tion spans many orders of magnitude). However, the disproportionate importanceof the large ε values becomes less dominant as ε becomes less variable, and itseems likely that arithmetic mean ε calculations will be skewed for data fromnear quiescent environments like e.g. the abyssal ocean or deep stratified lakes.Geometric average calculations, which characterize the central tendency of a dis-tribution, will be skewed heavily if ε is generally small, e.g. by up to two ordersof magnitude in the measurements presented in this thesis.c. How are uncertainties in ε estimates impacted when turbulence is weak and strat-ification is strong and the assumptions of isotropic, homogeneous, steadily forcedturbulence becomes increasingly intractable?Power spectra of temperature gradients and velocity shear deviate systematicallyfrom their reference shapes with decreasing ReB, presumably in large part due tothe breakdown of the isotropy, homogeneity, and stationarity assumptions. Uncer-tainty in ε estimates increases gradually as result. We determined that our ε es-timates (from temperature measurements) were useful and physically meaningfulfor ε > 2×10−12 W kg−1 which corresponds, approximately, to ReB ≥ O(10−2)in our measurements.865.1.2 Turbulent mixing in the Arctic Ocean’s Amundsen Gulfa. What are the turbulence and mixing characteristics in the Amundsen Gulf regionof the Beaufort Sea? Can we develop statistical metrics of ε and turbulent diffu-sivity, Kρ , and describe their spatial and temporal variability?Turbulence and turbulent mixing in Amundsen Gulf are typically weak and domi-nated by buoyancy effects resulting from stratification. A typical dissipation rate,given by the geometric mean of the whole data set, is ε = 2.8× 10−11 W kg−1.The arithmetic mean diffusivity from all the data is Kρ = 4.5× 10−6 m2 s−1.Buoyancy Reynolds number considerations indicate that a turbulent density fluxis not expected in 93% of the observations because density stratification prohibitsdiapycnal overturning. Variability in ε spans 5 orders of magnitude; variabilityin Kρ spans 3 orders of magnitude, not accounting for the large subset of mea-surements where diffusion is expected to be molecular. Turbulence appears to besurface- and bottom enhanced and varies most strongly at the tidal M2 frequency.b. What is the magnitude of vertical heat fluxes associated with turbulent mixing inAmundsen Gulf? Is it significant when compared to mean heat budget estimatesof the region and in light of recent increases in sea ice loss?Vertical heat fluxes are typically small, especially through the Warm Haloclinelayer overlying the warm Atlantic Water layer, sequestering the heat in that lowerlayer. The mean flux through the Warm Halocline is 0.03 W m−2, negligiblein comparison to the Arctic Ocean-wide mean of 6.7 W m−2. Heat fluxes areenhanced where turbulence is more energetic and in the vicinity of a warm-coremesoscale eddy seen in the observations. The magnitude of the heat flux is greaterthan 10 W m−2 in 0.6% of observations.c. What physical mechanisms are responsible for the observed turbulence and mix-ing characteristics in this region?To first order, it appears that turbulent mixing in the deeper Warm Halocline andAtlantic Water layers is modulated in time by barotropic tidal flow over nearbycomplex topography. Surprisingly, double diffusion does not appear to be a domi-nant contributor to mixing through the Warm Halocline. In the Pacific Water layerand Cold Halocline, the presence of a warm core mesoscale eddy dominated localheat fluxes. Diapycnal density mixing remained generally low in these layers dur-ing our measurements, due primarily to the presence of very strong stratification.875.1.3 Enhanced heat fluxes in strongly stratified, weakly turbulent envi-ronmentsa. Can we observe the transition between turbulent and molecular diffusion in thereal ocean when turbulence weakens and stratification remains strong?It appears that we can. Our measurements indicate a strong relationship betweenthe degree of turbulent diffusivity enhancement, K∗, and the buoyancy Reynoldsnumber, ReB. We observed a continuous transition between dominantly turbu-lent temperature diffusivity (K∗ > 100) and near-molecular temperature diffusiv-ity (K∗ = O(1)) as ReB decreases from O(100), through unity, to O(10−2).b. How do predictions of turbulent mixing from models compare to our observationsof tracer variance when turbulence is weak and stratification is strong?The Osborn, SKIF, and BB models predict the observed diffusivity well when tur-bulence is reasonably energetic, i.e. in the range ReB = 10–100. At lower ReB, theSKIF and BB models predict pure molecular diffusion for parts of the ReB param-eter space where observations indicate a turbulent diffusivity with K∗ = O(10).Part of the reason for this discrepancy is that the SKIF and BB models precludea smooth transition between turbulent and molecular fluxes. The Osborn modelmore accurately predicts this transition and the observed diffusivities at low ReB, ifOsborn’s original formulation for the turbulent flux is superimposed onto a molec-ular component (i.e. as in Equation 4.5). However, predictions of density diffu-sivity from the Osborn model diverge from the observed temperature diffusivityby a factor of ∼ 37 at small ReB; this divergence is most likely due to the onsetof differential diffusion of salinity and temperature, which is not encapsulated bythe Osborn model.c. How efficient is turbulent mixing in our observations, and how does this efficiencycompare to the canonical value of 20%?We were able to estimate the efficiency of mixing when ReB > 10, and foundvalues for R f and Γ that generally were a factor of ∼ 2 smaller than the canonicalvalues proposed by Osborn (1980). Observed median values were R f = 0.09and Γ = 0.10. There was little trend with ReB in the range ReB = 10–100; thereappeared to be a decreasing trend for ReB > 100, but fewer data in this regimeincreased the statistical uncertainty.885.2 The Bigger Picture: Looking AheadIt is my sincere hope that the research I’ve presented in this PhD thesis will one daycontribute to a comprehensive understanding of how ocean mixing in the Beaufort Seacontributes to modulating Arctic climate and ecology, and that that knowledge will beincorporated into governance policies that will preserve the integrity of the region’secosystems for future generations. This aim, the conservation of Arctic ecosystemsand their biodiversity, is the underlying driver that motivated me to do this work andoceanography in the Arctic, more generally. And even though my research is, admit-tedly, a small contribution towards that end goal, I feel it is important to maintain sightof the larger motivation and to propose how my results might inform future research thatwill bring that underlying goal closer to reality. I will try to carry out that final task ofmy thesis here by outlining in what ways I hope my own work will fit into the largerscope of Arctic oceanography in the future.The immediate goal of studying mixing in the Arctic Ocean is to generate an under-standing of how, and by what mechanisms, tracers such as heat, oxygen, and nutri-ents are distributed throughout its basins. The applications of that understanding arefar reaching, even if they are still a work in progress, and they are almost always tiedto predictive capabilities and the effects of modern climate change: predicting whereand how quickly perennial sea ice is lost (Carmack et al., 2015); predicting changesto stratification conditions in the Canada Basin (Davis et al., 2016); predicting changesto community composition and ecological dynamics, especially in light of a changingflaw lead system (Barber et al., 2010). In each case, there is a community-wide goal todevelop models that can describe the present state and make accurate predictions aboutfuture changes, usually with an eye to informing responsible governance.Models of mixing for the Arctic Ocean—present or future—must always be informedby measurements of the environment to remain grounded in reality. This necessity isthe fundamental reason the science community invests in field campaigns in the Arc-tic despite the extreme costs and logistic difficulties of working in this challenging andremote environment. In some ways the investment is a losing battle because there istoo much ocean and there are too few resources to sample sufficiently, but rapid devel-opment of autonomous platforms like gliders, Argo profilers1, or ice-tethered profilers2(ITPs) are closing this gap. This area of progress is one of the most basic, but also mostfundamental, ones in which I hope my work will inspire future Arctic oceanography, bydemonstrating the feasibility and practicality of glider-based turbulence measurements1see http://www.argo.ucsd.edu2see https://www.whoi.edu/website/itp/overview89despite the harshness and remoteness of the environment. On Canada’s eastern coast,the Ocean Tracking Network has within the last decade developed an unprecedentedprogram to continuously sample the Scotian Shelf using a fleet of autonomous gliders,similar to the one used in this study. A similarly ambitious program of autonomoussampling is presently being developed on the west coast of Canada through the PacificRobotic Ocean Observing Facility, again with the help of a new fleet of ocean gliders.In light of these precedents, it is not a stretch, then, to imagine a near future wherea similar fleet of autonomous ocean gliders operates along Canada’s northern coastalocean as well, and if the experience of my work documented here contributes to thatdevelopment, it will be a satisfying and worthwhile contribution indeed.The other area in which I picture my research making an important contribution is withregards to the question of scaling up. Turbulence measurements are important becausethey give us the most direct estimates of mixing we can obtain in the real ocean, and theresults of my thesis should be immediately applicable to future investigators wantingto measure turbulence in the Arctic Ocean. Questions about measurement limitation,regional mixing mechanisms, and stratified mixing models are considerations of firstorder importance for such work, and I hope my research presented here will help toinform many future high latitude turbulence studies. However, notwithstanding theirimportance, turbulence measurements are admittedly cumbersome—even from an au-tonomous platform—because they require sensors that are highly specialized, expensive,subject to low-frequency drift in their calibration, and prone to breaking. In addition, thevery high temporal resolution needed to interpret turbulence measurements are energy-intensive in the field, resulting in reduced operating endurance, and the measurementsproduce vast datasets that are difficult to store and manage3.For these reasons, I imagine that future basin-scale maps of turbulent mixing observa-tions in the Arctic will not be based on direct turbulence measurements, but on param-eterizations using coarser-scale CTD measurements that are collected more easily inlarger number and with larger geographic scope. In many cases the measurements fromwhich mixing rates can be inferred on large scales already exist because of observingprograms like the Woods Hole Oceanographic Institution’s ITP program or ArcticNet’sAmundsen Science Program, and the actual task of parameterizing mixing rates fromthose measurements has already begun. Chanona et al. (2018) recently published astudy using a fine scale parameterization of internal wave signals to estimate mixingrates on the Beaufort Sea’s coastal margin and in the Canadian Arctic Archipelago, andI know that a similar approach is being applied to Arctic Ocean-wide ITP measurements3For example, the 10 days of measurements from a single glider used in this thesis produced 6 GB ofbinary data; once processed and converted to a useable format, the data required about 75 GB of storage.90as I write this. The approach has already been successfully applied to estimate globalpatterns of mixing for the rest of the world ocean by Whalen et al. (2012). Such param-eterizations of turbulence based on coarser scale measurements are useful because theyhelp us to quickly scale up our knowledge of mixing rates and infer large scale patternsof water mass transformations and heat fluxes. They are, however, parameterizations,and they need to be tested and calibrated against more direct estimates of mixing, wherethese are available. This is a topic for which I can see my results, especially those inChapter 3, being especially useful because direct mixing rate estimates with statisticallysignificant metrics are so rare in the Arctic. I hope that the large record of direct turbu-lence measurements I have established here, though regionally isolated, can be used toground-truth parameterizations that can more easily scale up to basin scales and be usedto address questions relevant to the scales of global climate. Answering such questionsis a community effort, of course, and if my work contributes to it, it will be a contributionof which I will be proud.91BibliographyW. N. Adger, J. Barnett, K. Brown, N. Marshall, and K. O’Brien. Cultural dimensionsof climate change impacts and adaptation. Nature Climate Change, 3:112–117, Feb.2013.M. H. Alford. 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These are removed before theanalysis described in Section 2.4.Before implementing quantitative quality control measures, we remove any obviouslycontaminated measurements by hand. These include all measurements after Aug 27from one of the two shear probes because the probe appears to have been damaged atthis point and thereafter no longer measured a sensible signal. We then begin systematicquality control measures with 45,571 independent εU estimates and 63,507 independentεT estimates. Note that the estimates from the two distinct shear or two distinct temper-ature probes are not yet averaged at this stage (see Sections 2.3.2 and 2.3.3).Table A.1 – Quality control parameters, as defined in the text. Percentages are the fraction ofmeasurements flagged by each condition.εU εTN 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% —QC6-T — 5.6%QC7-T — 4.6%One or more 22.3% 33.9%N after QC 35,395 41,955Individual estimates of εU and εT are flagged untrustworthy and removed if they satisfyone or more of the following conditions:102QC1 The magnitude of the glider’s acceleration |dU/dt| is above the tenth percentile(|dU/dt| > 4.6× 10−4 m/s2). This is a heuristic measure, but it satisfactorilyisolates measurements where the glider appears to be changing speed over thespan of one ε estimate. The data processing assumes U is constant over the spanof an ε estimate.QC2 The glider is within 15 m of an inflection point. When the glider inflects, the angleof attack and the estimate of U are uncertain and mechanical vibrations from theglider contaminate measurements.QC3 Estimates from two identical probes differ by greater than a factor of 10. For shear,the higher estimate is removed. For temperature, both estimates are removed.QC4 The ratio U/(ε/N)1/2 is less than 5. This is the ratio between the glider’s velocityand an estimate of the turbulent flow velocities (Fer et al., 2014) and may indicatewhen Taylor’s frozen turbulence hypothesis is violated.In addition, εU measurements are flagged and removed ifQC5-U The maximum of the nondimensionalized shear spectrum Φ∗, defined in Sec-tion 2.5.2, is greater than twice the peak of the nondimensionalized Nasmyth spec-trum. This isolates shear spectra obviously contaminated at low wavenumbers.In addition, εT measurements are flagged and removed ifQC6-T The sum of the correction terms χlw and χhw (see Equation 2.7) is greater thanthe “observed” term χobs.QC7-T There are fewer than 6 distinct wavenumbers available in the closed interval[kl,ku]. This ensures a reasonable minimum number of spectral points to which tofit a Batchelor spectrum.Cumulatively, quality assessment conditions flag and remove 22.3% of individual εUmeasurements and 33.9% of individual εT measurements. The percentage of measure-ments flagged by each individual condition is given in Table A.1.Beyond these conditions, confidence in the measurements may be indicated by thelevel of agreement between duplicate measurements of εU and εT . This comparisonis favourable: after the quality assessment procedures, 96% of 6,820 coincident sets ofεU measurements agree within a factor of 5, and the agreement is comparable over thefull range of εU . Note that a factor of 5 is typically considered reasonable agreementfor individual microstructure measurements which often scatter within an order of mag-nitude and can vary by more than 10 decades in the ocean. The comparison is slightlymore variable but still favourable for εT , where 87.3% of 20,378 coincident sets agree103Figure A.1 – Comparison of results from individual (a,b) shear probes and (c,d) temperatureprobes. Bin averages are calculated as in Figure 2.6. Agreement within a factor of 5 is indicatedin all panels by the dashed lines.within a factor of 5 and there is likewise no trend with εT .The comparisons are imaged in Figure A.1, where subscripts 1 and 2 are arbitrarilydesignated to measurements from distinct probes. Bin averages are calculated as inFigure 2.6.Following the quality control procedures described here, we average (using an arithmeticmean) results for each of the two distinct sets of measurements, as described in Sections2.3.2 and 2.3.3. This leaves 28,575 εU and 21,577 εT estimates for analysis.104Figure A.2 – Overview of select results, separated by upcast and downcast. For each of εUand εT , we show the histograms (a,e), averaged vertical profiles (b,f), and selected spectra (c–d,g–h) separated in this manner. The spectra shown are those corresponding to dissipation rateswithin a factor of 1.1 of 10−9 W/kg. Thick black lines depict the median of the selected spectraat each wavenumber.A.2 Comparison of Results from Upcasts and DowncastsThe stratification conditions in the Amundsen Gulf resulted in much faster glider speedson upcasts than on downcasts (Table 2.1). Because dissipation rate calculations are verysensitive to the estimated glider speed U (Osborn and Crawford, 1980; Gregg, 1999;Lueck, 2016), this discrepancy can be leveraged to further inform our confidence inthe measurements and data processing methods. In the absence of systematic errorsin the data processing, we would expect to see no systematic difference in the resultsderived separately from upcasts and downcasts despite the nearly factor of 2 differencein U .We observe little to no difference between dissipation rate estimates when the resultsare separated by upcast and downcast. Figure A.2 presents an overview of the εU andεT results separated in this manner. The two histograms of εU (panel a) have nearlyindistinguishable characteristics: for example, the medians are 5×10−11 W/kg and 4×10−11 W/kg for up- and downcasts respectively, and the respective geometric standarddeviation factors are 3.5 and 3.7. Similarly, the two averaged vertical profiles of εU(panel b) are nearly identical in magnitude everywhere; they typically agree within afactor of 1.2 and always within a factor of 1.8. Shear spectra are likewise similar between105up- and downcasts, as highlighted in the selected spectra shown in panels (c,d). Medianspectra are generally alike in shape with a marginally wider spectral peak for the upcastmedian spectrum.There is slightly more discrepancy between up- and downcasts in the εT results, thoughthe overall agreement is still encouraging and the discrepancy does not impact the resultsor conclusions of the study. The histogram comparison (panel e) is generally favourable:the median is 2×10−11 W/kg for both distributions, and the geometric standard devia-tion factors are 23.4 and 16.2 for upcasts and downcasts respectively. The upcast distri-bution is wider because of a small unexpected increase in the number of εT values below1×10−13 W/kg. As discussed in Section 2.5.2, there is extensive uncertainty associatedwith values of εT smaller than 2×10−12 W/kg, and so it is unclear how much meaningcan be assigned to this feature of the distribution. The mean profiles (panel f) demon-strate adequate agreement, typically within a factor of 2 and always within a factor of3.5, in line with typical uncertainties from microstructure measurements. The shape ofthe two median temperature gradient spectra (panels g,h) compare favourably with onlya slightly less rounded rolloff to the Batchelor scale for the upcast spectrum.A.3 Nasmyth and Batchelor SpectraThe Nasmyth spectrum, ΦN , is an empirically derived form for the one-dimensionalpower spectrum of velocity shear in an unstratified turbulent flow and is based on mea-surements collected in a strongly turbulent tidal channel in coastal British Columbia(Nasmyth, 1970). It describes both the inertial subrange of the shear spectrum, predictedby Kolmogorov (1941), and the viscous subrange where viscosity begins to influence themotion of turbulent eddies. The results of Nasmyth were tabulated by Oakey (1982), anda mathematical fit was later proposed by Wolk et al. (2002). We use a modified formof that expression, described by Lueck (2016), which can be written nondimensionallyas Φ∗N = 8.05k˜1/3/(1+(20.6k˜)3.715), where k˜ = k(ν3/ε)1/4 and the spectrum is nondi-mensionalized using Φ∗N =ΦN/(ε3U/ν)1/4.The Batchelor spectrum is a theoretical one-dimensional power spectrum describingthe wavenumber distribution of a passive tracer’s gradients in an unstratified turbulentflow (Batchelor, 1959); its integral is proportional to the rate, χ , at which the tracergradients are smoothed by molecular diffusion. The spectrum is an analytic solutionto the advection-diffusion equation driven by turbulent strain and the large-scale tracergradient. In one dimension, it may be written as:ΨB =χ√qB/2kBκ molT(aexp(a22)−a2√pi2erfc(a√2))(A.1)106wherea = (k/kB)√2qB . (A.2)The factor qB is a dimensionless constant related to the average least principal rate ofstrain; it represents the timescale by which compressive strain sharpens scalar gradients(Smyth, 1999). The value of qB is uncertain, and experiments by Oakey (1982) suggestthe range 2.2–5.2, though typically qB = 3.4 (e.g. Ruddick et al., 2000) or qB = 3.7 (e.g.Peterson and Fer, 2014) are used. A percentage error in qB is expected to lead to twicethe percentage error in εT (Dillon and Caldwell, 1980). We used qB = 3.4 in our analysisbut also processed all the temperature measurements using qB = 3.7, and the differencein results was small: using qB = 3.7, we found for εT a mode of 1.5× 10−11 W/kg, ageometric mean of 1.9×10−11 W/kg, and a geometric standard deviation factor of 18.3(compare with Table 2.2).Note that the Kraichnan spectrum (Kraichnan, 1968) would be an adequate alternativefor the fitting procedure described in Section 2.3.3. Peterson and Fer (2014) comparedthe results of fitting observed temperature gradient spectra to Batchelor and Kraichnanspectra and found no significant difference in the final εT results.107

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