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Double diffusion in Powell Lake : new insights from a unique case study Scheifele, Benjamin 2013

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Double Diffusion in Powell Lake:New Insights from a Unique Case StudybyBenjamin ScheifeleB.Sc. Physics, St. Francis Xavier University, 2011a thesis submitted in partial fulfillmentof the requirements for the degree ofMaster of Scienceinthe faculty of graduate and postdoctoral studies(Oceanography)the university of british columbia(Vancouver)September 2013c? Benjamin Scheifele, 2013AbstractHigh resolution measurements of temperature and electrical conductivity in Powell Lake,British Columbia provide an extensive set of layer and interface observations of a doublediffusive staircase found between 325?350 m depth. Powell Lake is an ex-fjord with a qui-escent salt layer at thermal steady state in which double diffusion is naturally isolated fromturbulent and advective processes. Layers are coherent on the basin scale and their charac-teristics have a well defined vertical structure. The steady state heat flux is estimated fromthe large-scale temperature profile and agrees with an earlier estimate of the flux in thesediments. These estimates are compared to a 4/3 flux parameterization which agrees withthe steady state flux to within a factor of 2. The discrepancy is explained by testing thescaling underlying the parameterization directly, and it is found that the assumed powerlaw deviates systematically from the observations. Consequently, a different scaling whichbetter describes the observations is presented. The assumption that interfacial fluxes aredominated by molecular diffusion is tested by comparing the interfacial gradient to that ex-pected from the steady state heat flux; at low density ratios, the average interfacial gradientis not sufficiently large to account for transport by molecular diffusion alone, indicating thatdouble diffusive fluxes cannot generally be estimated from bulk interface properties. Salin-ity interfaces are only marginally (9%) smaller than temperature interfaces, and a simplemodel to describe the observed difference is presented and shown to be consistent with theobservations.iiPrefaceThis thesis is authored by me, Benjamin Scheifele, and describes original work carried out byme under the supervision of Dr. Richard Pawlowicz. I independently collected and analyzedthe data described in Chapters 2 and 3 according to a plan jointly developed with Dr.Pawlowicz. Novel theoretical ideas presented in Chapter 4 are mine; however, the discussionsurrounding these ideas was likewise developed in conjunction with Dr. Pawlowicz.Results are unpublished, but are undergoing preparation for submission. Figure 1.4 inChapter 1 is reproduced with permission from Kelley et al. (2003).iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Double Diffusive Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Salt Fingering Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Diffusive Layering Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 The Diffusive Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Modelling Heat and Salt Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Powell Lake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 Research Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Measurements and Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Layer Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 CTD Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Interfacial Temperature Gradient . . . . . . . . . . . . . . . . . . . . . . . . 333.3 Heat Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 TS Diagrams and Layer Coherence . . . . . . . . . . . . . . . . . . . . . . . 39iv4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Large-scale properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Comparison of Layer Characteristics . . . . . . . . . . . . . . . . . . . . . . 434.3 Heat Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 New Scaling Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.5 Interface Characteristics - Temperature Gradient . . . . . . . . . . . . . . . 514.6 Interface Characteristics - Relative Thickness . . . . . . . . . . . . . . . . . 545 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59vList of TablesTable 3.1 Basic layer statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Table 4.1 An overview of double diffusive parameters from five distinct studies . . . . . . . 44viList of FiguresFigure 1.1 A schematic of two double diffusive background stratifications . . . . . . . . . . 2Figure 1.2 A schematic of the salt fingering mode . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.3 The diffusive layering instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.4 Global susceptibility to diffusive layering . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.5 An example of double diffusive layers . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.6 A simplified schematic of two double diffusive layers . . . . . . . . . . . . . . . 8Figure 1.7 Schematic of an interface between two double diffusive cells . . . . . . . . . . . 9Figure 1.8 Maps depicting south-western British Columbia and Powell Lake . . . . . . . . 14Figure 1.9 Schematic cross-section of Powell Lake . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.1 Measurement locations in the South Basin . . . . . . . . . . . . . . . . . . . . . 20Figure 2.2 Our mixed layer and interface detection algorithm . . . . . . . . . . . . . . . . . 23Figure 2.3 Example close-up of a few layers in t and SR . . . . . . . . . . . . . . . . . . . . 23Figure 3.1 Vertical profiles in the South Basin of Powell Lake . . . . . . . . . . . . . . . . 26Figure 3.2 Lateral transect of double diffusive layers in t and SR . . . . . . . . . . . . . . . 27Figure 3.3 Number of double diffusive layers below 324 m . . . . . . . . . . . . . . . . . . . 28Figure 3.4 Histograms of layer/interface characteristics . . . . . . . . . . . . . . . . . . . . 30Figure 3.5 Height of the double-diffusive interface in t and SR . . . . . . . . . . . . . . . . 31Figure 3.6 Layer height and interface height difference . . . . . . . . . . . . . . . . . . . . 31Figure 3.7 Temperature and salinity steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 3.8 Average temperature gradient in the interface between layers . . . . . . . . . . 34Figure 3.9 Parametric plot in depth of ?t against R? . . . . . . . . . . . . . . . . . . . . . 34Figure 3.10 Mean temperature gradient between 310 ? 323 m depth . . . . . . . . . . . . . 36Figure 3.11 Heat fluxes calculated using the 4/3 parameterization . . . . . . . . . . . . . . . 36Figure 3.12 A direct test of the scaling Nu = f1(R?)Ran . . . . . . . . . . . . . . . . . . . . 38Figure 3.13 Temperature-salinity diagram of double diffusive mixed layers . . . . . . . . . . 40Figure 4.1 Parametric plot in depth showing that Nu ? 1/R? . . . . . . . . . . . . . . . . 48Figure 4.2 Schematic depicting the invariance of ?T/H . . . . . . . . . . . . . . . . . . . . 49Figure 4.3 Sixty examples of the double diffusive interface . . . . . . . . . . . . . . . . . . 52Figure 4.4 A simplified model of the temperature interface . . . . . . . . . . . . . . . . . . 55viiAcknowledgmentsThe work you will read about here bears the marks of many who have helped bring thisproject together from conception to conclusion. These include those who were directlyinvolved ?Chris Payne, Lora Pakhomova, Roger Pieters, Greg Lawrence, Kristin Orians,and Bernard Laval? as much as those who shared my life or my office or my living spaceand who have helped make the past two years both fun and memorable. I am grateful toall of you.I also want to extend a special and very heartfelt thank?you to a few people who supportedme the whole way and so have shaped my entire experience at UBC into the meaningfulone it has become:. . . Tara Ivanochko, who has been my mentor, advocate, and friend this year.. . . Rich Pawlowicz, who has encouraged me to explore both oceans and mountains.. . . my parents, Bernd and Josaphine, who love me and put up with me always.And always both last and first, I want to thank God, through whom I do everything that Iaccomplish.viiiChapter 1Introduction1.1 Double Diffusive ConvectionDouble diffusive convection, or simply ?double diffusion?, is a transport process that isobserved over vast regions of the world?s oceans and in numerous lakes around the world(Carpenter and Timmermans, 2012; Kelley et al., 2003). It is comprised of convectivemotions that occur because of instabilities that can form during the simultaneous diffusionof salt and heat. Temperature and salinity determine the relative density of seawater, andwhen these have vertical structures such that one stabilizes the watercolumn while the otheris gravitationally unstable, the much higher diffusion rate of heat relative to that of salt isable to release the stored potential gravitational energy in the unstable density component.Thus even in the absence of mechanical mixing, convective motions can be establishedthrough the creation of an instability which substantially enhance the transports of heatand salt relative to those which would be supported by molecular diffusion alone.To a first approximation, the density ? of a watercolumn can be written as? = ?0 (1 + ?T + ?S) (1.1)where ?0 is a reference density, T the temperature in ?C, and S the salinity in g/kg. Thecoefficient ? is the thermal expansion coefficient which quantifies the density response tochanges in temperature. It is defined as? = ? 1?0???T (1.2)and is usually positive in oceanographic ranges of temperature and salinity. ? is the haline1?Tz?S??T?S?Unstable StableStableUnstable Stable Stable(b)(a)zSalt Fingering Diffusive ConvectionFigure 1.1 ? A schematic of two double diffusive background stratifications, leading to twodistinct modes of double diffusion. (a) The salt fingering mode occurs when the salinitycontribution to the density stratification is unstable, but the temperature stratificationis stable. (b) The diffusive layering mode occurs when the opposite is true and salinityprovides the stabilizing component while temperature stratification is unstable. In bothscenarios, the net density profile is stable.contraction coefficient defined as? = 1?0???S (1.3)which quantifies the density response to changes in salinity and is always positive.Apart from the constant ?0, all variables are in general functions of z, the vertical coor-dinate which we define positive upwards. From Equation 1.1, there are two ways to setup a stratification that has one stable and one unstable component but a net stratificationthat is still stable, and these two scenarios lead to two distinct modes of double diffusion(Figure 1.1). Assuming oceanographic ranges and a net stable stratification, if T and Sboth decrease with depth, temperature must contribute the dominating density effect whilesalinity acts to destabilize the watercolumn ? this leads to the double diffusive mode knownas the salt fingering mode. On the other hand, if T and S both increase with depth thestratification is kept stable by the salinity contribution while the temperature term is un-stable, and this leads to the diffusive layering mode of double diffusion. Except for thefollowing short description of the salt fingering mode, we will restrict ourselves to studyingonly the diffusive layering mode in this thesis.2Figure 1.2 ? A schematic of the salt fingering mode of double diffusion which occurs whenwarm salty water sits above relatively fresh, cold water. Long finger-like vertical intrusionswhich carry salt downwards, and across whose boundaries heat is exchanged by moleculardiffusion, form spontaneously. Salt fingers occurs primarily in subtropical regions whereevaporation creates relatively salty water near the surface.1.2 The Salt Fingering ModeIn both modes of double diffusion, an instability causes convective motions that increasevertical transports relative to molecular diffusion. In the salt fingering case, the instability iseasiest visualized by imagining a long metal conducting tube, open at both ends, extendingdown into a region of the ocean where temperature and salinity both decrease with depth.Initially, the water inside the tube is in thermal and static equilibrium with the surroundingwater. However, if the water inside the tube is displaced upwards, it will be colder thanthe surrounding water and will quickly come to a new thermal equilibrium by thermalconduction through the metal pipe. The displaced water will also be less saline than thewater directly outside the pipe, and since salt cannot diffuse through the metal, it willremain that way. The parcel is now at the same temperature but less salty than thesurrounding water, is therefore lighter than the surrounding water, and will continue torise. As it rises it will obtain more heat via conduction but will not become more salty ?thus, the instability will grow and the parcel will continue rising. This phenomenon hascome to be called the ?perpetual salt fountain? since, once initiated, a tube transportingseawater through a watercolumn in this fashion will continue to do so until the vertical saltand temperature gradients have been equilibrated (Stommel, 1956).In a natural system there is no conducting metal pipe to separate a water parcel from3its surrounding waters. However, the phenomenon can be reproduced without a physicalboundary because the molecular diffusivity of heat in water is about one-hundred timeshigher than that of salt. Thus the difference in the speed at which heat and salt diffuseeffectively takes the place of the conducting pipe, allowing a displaced water parcel to gainheat but not salt from its surroundings. Indeed, lab experiments (Stern, 1960) confirmthat warm salty water above cold fresh water can spontaneously form elongated verticalintrusions, ?salt fingers?, which carry salt downwards and through which heat is exchangedby molecular diffusion (Figure 1.2).The salt-fingering mode of double diffusion was the first to be reproduced in the lab and to betreated analytically (Stern, 1960). In the oceans, it is observed largely in subtropical regionswhere high evaporation tends to increase surface seawater salinity. For an example, see theresults of the Carribean Sheets and Layers Transects (C-SALT) experiments (Schmitt et al.,1987). Historically, the salt-fingering mode has received more attention than the diffusivelayering mode, though this may be more an artifact of convenience and habit rather thanone of scientific motivation (Kelley et al., 2003). Though questions remain, a large body ofresearch exists describing the salt fingering mode and its likely impact on oceanic mixing;for greater detail refer to Kunze (2003), Ruddick and Gargett (2003), and Schmitt (1994,2003).1.3 The Diffusive Layering ModeThe traditional way to conceptualize the instability of the diffusive layering mode is toimagine linear gradients with T and S both increasing with depth, in which a small parcelof water is displaced downwards. As in the salt fingering mode, heat diffuses into the parcelbecause the surrounding water is relatively warmer, while the salt content of the parcelremains virtually unchanged. The parcel is now lighter than its surrounding waters andbegins to rise, overshooting its original position slightly because it has gained a densityanomaly. After it has passed its initial height, it loses the heat it gained, plus a little morebecause its new surroundings are now cooler than they were initially, and again begins tosink ? in this way, an oscillating instability initiates (Figure 1.3).The linear stability analysis for concurrent one-dimensional linearly increasing T and Sgradients bounded above and below is outlined comprehensively by Turner (1973). To givea brief review, the relevant non-dimensional parameters governing the problem areR? =??S??T , Ra =g??TL3??T, and Pr = ??T. (1.4)The first parameter, R?, is the density ratio and gives a measure of the stability of the4Figure 1.3 ? The diffusive layering instability, based on T and S profiles that increaselinearly with depth. Initially, the depicted water parcel is neutrally buoyant on the dashedline. As it is perturbed, the temperature of the parcel will equilibrate with its surroundingsbut the salt content of the parcel will not change much because of the relatively slowermolecular diffusivity of salt. This initiates the growing oscillatory instability, as describedin the text.watercolumn; ?S and ?T are the differences in salinity and temperature between thetop and bottom boundaries. The Rayleigh number Ra carries over directly from single-component, thermally driven convection; it is the ratio of the driving buoyancy forces tothe retarding effects of viscosity and diffusivity. Here, g is the gravitational constant inm/s2, ? the kinematic viscosity and ?T the thermal diffusivity, both in m2/s. L is thevertical length scale of the problem in meters. The last parameter Pr is the Prandtlnumber, the ratio of the kinematic viscosity to the thermal diffusivity, which in the oceanis Pr ? 6.From linear stability analysis, the double-diffusive instability is indeed oscillatory and isbounded by limits on the density ratio, with instability occurring only when1 < R? <Pr + 1Pr + ? (1.5)where ? is the ratio of the diffusivities of salt and heat. That is, ? = ?S/?T ? 0.01 inseawater, suggesting that the instability initiates for 1 < R? < 1.16. However, there isreason to believe that in geophysical examples of double diffusive convection, the analysisof linear T and S profiles is not a useful way to describe the phenomenon.In real geophysical situations, the definition of the density ratio has to be adapted in order tobe relevant to the problem, and so it is standard to define a background density ratioR o? =?(?S/?z)?(?T/?z) (1.6)5Fig. 1. Regions that are susceptible to DL convection. The darker of the two gray tonesindicates regions with the minimum density ratio in the water column lying between and, with the lighter tone indicating the range to . Areas outside this range, and areas inwhich the atlases report no values, are white on this diagram. All other things being equal,the darker tones would indicate greater susceptibility to DL convection. See text for datasources and processing details.have (i.e., highly unstable) through the depth range of m to m. Thesubpolar regions (e.g. the Labrador Sea and the Sea of Okhotsk) have similarly lowvalues in a depth range that extends m closer to the surface. The depth rangeis extended somewhat in the southern polar regions, covering m to m. Inaddition to these high-latitude cases, there is also an DL-susceptible region to theeast of Brazil. This is much deeper ( m to m) than in polar waters, pre-sumably resulting more from juxtaposition of water masses (Bianchi et al., 2002)than from surface forcing. It also has higher density ratios, with . All otherthings being equal, these higher values might imply that DL is less likely to occurin the Brazil region than in other susceptible regions. However, double-diffusionmay be disrupted by turbulence, and so may not be the sole determining factor.For example, the disruptive effect of baroclinic tides is likely to be stronger nearBrazil than in the Arctic. Indeed, it is notable that DDC signatures are prominent inregions which have weak tidal stirring, and thus presumably weak tidal mixing. TheCanada Basin of the Arctic Ocean provides a good example. Melling et al. (1984)have pointed out that the tides in this region are less than m/s. This mayexplain why DL signatures are prominent there, despite the relatively high densityratio of approximately .Thus, while diagrams such as Figure 1 may provide a good indication of suscep-tibility, they should not be taken to indicate either the incidence of DDC or itsimportance in terms of fluxes. Rather, the purpose in constructing susceptibilitymaps is provide a basis for comparison with local observations of DDC signa-tures and fluxes. This might shed light on the interaction of DDC with the environ-5Figure 1.4 ? Global s sceptibility to diffusive layering ? a map depicting regions wherethe large-scale stratification of the watercolumn is susceptible to double diffusion at somedepth. Light shaded regions indicate areas where 3 < R o? < 10 and dark shaded regionsindicate areas with 1 < R o? < 3. Note that this is a susceptibility map based strictly ongradients of T and S and not a map of obs rvation of double diffusion. Reproduced fromKelley et al. (2003) with permission.This definition incorporates the large-scale T and S gradients of the watercolumn, but isotherwise analogous to the single cell representation R?. The background density ratio isthe parameter used to determine the susceptibility to double diffusion of a given T andS stratification in the ocean. Strikingly, the diffusive layering mode is observed in oceansand lakes over the ran e 1 < R o? < 10, and mo t observation tend o be in regions withR o? > 2, leading to t e concl sion that the analysis of linear T and S profiles hich gives theinstability criterium Equation 1.5 is inappropriate for these real situations (Kelley et al.,2003; Padman and Dillon, 1987; Schmid et al., 2010, 2004; Timmermans et al., 2008).Co ditions favouring the diffusive layering mode are often found in high latitude regionsand many of the most promine t examples co e from m asurem nts taken the Arctic(Carmack et al., 2012; J. McDougall, 1983; Padman and Dillon, 1987, 1989; Polyakov et al.,2012; Timmermans et al., 2008). Double diffusive processes are also frequently observed inlakes with warm, subaquatic inflows, and in geothermally heated salt and pit lakes (Osborn,1973; Schmid et al., 2004, 2010). Figure 1.4 highlights those regions of the world oceanswhere the stratification at some depth in the watercolumn is favourable to the diffusivelayering mode of double diffusion.The term layering in the name ?double diffusive layering? refers to the characteristic ther-mohaline staircase structure that frequently accompanies this mode of double diffusion.Observations are usually characterized by a succession of thin layers with homogeneous T6and S properties that are separated from each other by even thinner, high-gradient inter-faces (Figure 1.5). Observed layers tend to be on the order of one to ten meters high inoceanic settings, and interfaces are typically a few tens of centimetres high. Furthermore,Timmermans et al. (2008) have shown that, in the Arctic, individual layers can be tracedhorizontally for hundreds of kilometres, giving them an aspect ratio as high as 106.In a simple conceptual model of the diffusive layering mode, convective motions maintainhomogeneity within each layer, while molecular diffusion transports heat and salt throughthe interfaces that separate layers; hence, the term diffusive layering (Ruddick and Gargett,2003). A schematic of this model is depicted in Figure 1.6. It is not clear from previous workhow the step height, the interface height, or the temperature and salinity differences betweenlayers are determined in a particular situation; a few parameterizations exist (Fedorov,1988; Kelley, 1984), but these seem to give only rough agreement with observed staircasecharacteristics (Kelley et al., 2003; Schmid et al., 2004). Similarly, though a few recentpapers have had some success modelling the initial stages and subsequent evolution of adouble-diffusive thermohaline staircase (Noguchi and Niino, 2010a,b; Radko, 2003), themechanism whereby layers form and are maintained is not evidently clear and has not beenverified observationally.1.4 The Diffusive InterfaceIn the context of a series of well-mixed layers separated by thin, high-gradient interfaces,we can resurrect R? as a governing parameter if we adjust the meaning of ?T and ?Sslightly. Here, these are the differences in the average T and S found in two successivemixed layers, and so R? now quantifies the stability of the interface between those layers.Thus, in real geophysical situations, the two parameters R o? and R? can both be used, but fordifferent purposes: the first to assess the likelihood of double diffusion given a backgroundstratification, the second to characterize the interface between observed double-diffusivelayers.The exact nature of the interface has been considered carefully, and it is of great inter-est to gain an understanding of its dynamics, since the interface is likely the seat of thedouble-diffusive instability which maintains homogeneity within the mixed layers. A recentapproach to studying the interface is described by Carpenter et al. (2012), who model itsstability by using a combination of linear stability analysis and direct numerical simulation.The primary difference to the analysis outlined in the previous section is that this studydoes not use linear profiles, but erf and tanh functions to more realistically model the shapeof the interface.The study by Carpenter et al. (2012) finds that the interface forms a gravitationally un-79.28 9.30327328329330331332333t (?C)p (dbar)16.66 16.67SR (g/kg)12.710 12.714?? kg/m39.28 9.30327328329330331332333t (?C)p (dbar)16.66 16.67SR (g/kg)12.710 12.714?? kg/m39.28 9.30327328329330331332333t (?C)p (dbar)16.66 16.67SR (g/kg)12.710 12.714?? kg/m3Figure 1.5 ? An example of double diffusive layers, measured in Powell Lake. Shownare in-situ temperature t, Reference Salinity SR, and potential density anomaly ??. Smallvariations in SR that are less than 0.0007 g/kg reflect the electronic noise of the conduc-tivity probe, and these are propagated into the calculation of ??. Temperature differencesbetween layers are order 10?3 to 10?2 ?C and salinity differences are order 10?3 to 10?2g/kg. Well-mixed layers with a scale height of 1 to 10 m separated by thin high-gradientregions are characteristic of the diffusive layering mode of double diffusion. Profiles mea-sured at station B03 (Figure 2.1).hConvectionConvectionMolecular DiffusionConvectionConvectionConv.Conv.HT1S1T2S2Figure 1.6 ? A simplified schematic of two double diffusive layers and the interface be-tween them. Convection homogenizes T and S within the layer while molecular diffusionis thought to carry heat and salt through the high-gradient interfaces. However, as dis-cussed in the text, other transport mechanisms may also contribute. It is unknown whatdetermines the cell height H, the interface height h, and the temperature and salinitydifferences between the two layers.8?0?T?0?S?Figure 1.7 ? Schematic of an interface between two double diffusive cells, depicting theunstable boundary layer model described by Linden and Shirtcliffe (1978) and Carpenteret al. (2012). Molecular diffusion thickens the temperature interface relative to the salinityinterface, resulting in gravitationally unstable boundary layers at the edges of the interface(shaded in grey). These eventually break away as plumes which drive the convection inthe double diffusive cells. T and S are normalized to density units.stable boundary layer around a stable core. This occurs because heat diffuses through theinterface faster than salt does, stretching the size of the T interface relative to that of the Sinterface (Figure 1.7). A convective instability initiates within the boundary layer, grows,and eventually forms a plume that breaks away from the core of the interface, adding tothe convective mixing in the layer above it. Defining hT and hS as the heights of the tem-perature and salinity interfaces respectively and by scaling the one dimensional diffusionequation, we can expect the ratio rh = hT /hS to have an upper limit of?1/? ? 10 iftransport through the stable interface core is strictly by molecular diffusion.Where the analysis of linear profiles resulted in the (observationally refuted) instabilitycondition given by Equation 1.5, the analysis performed by Carpenter et al. (2012) allowsfor realistic values of the density ratio R?. While not quoting a stability boundary, theauthors report finding instability over the range 1 < R? < 5 which is in stark contrast tothe stability boundary dictated by the analysis of linear profiles. This is good indicationthat the proposed interface model is a more realistic description of the double diffusiveinstability than the traditional study of linear profiles provided.Despite these advances, there remains the open question whether transport through theinterface is in fact limited by molecular diffusion, or whether a turbulent process is alsoactive. While the former idea of a purely diffusive interface core is certainly the paradigmwhereby double diffusive convection is typically envisioned (Figure 1.6), and while the sim-ulations by Carpenter et al. (2012) support this model, there is also reason to question itscompleteness, and some past results (see below) would tend to support the notion that asecondary transport process may be active in the interface under certain conditions.9The interface model of a laminar, diffusive core surrounded on both sides by growing unsta-ble boundary layers which eventually break away in plumes to support convection withinthe double diffusive cell was first proposed by Linden and Shirtcliffe (1978). However, evenin this early work it was recognized that as R? decreases and approaches unity, the convec-tive motions within a cell may contribute to the vertical fluxes by entraining fluid across theinterface. Marmorino and Caldwell (1976) perform lab experiments where they find thatfor high values of R? the measured heat flux across the interface is that which would beexpected by molecular conduction, but at low values of R? the measured heat flux becomeslarger than the conductive flux. The terms ?high? and ?low? are left somewhat ambiguous,but the experiments cover the approximate range 1.8 < R? < 11 and the authors quote thatat R? = 2 the net vertical heat flux is 2.5 times the conductive flux. Padman and Dillon(1987) also consider the possibility that heat fluxes through the interfaces they observed inthe Arctic may be larger than is supported by molecular diffusion. Two studies of doublediffusion in Lake Kivu by Schmid et al. (Schmid et al., 2004, 2010) do the same, mention-ing that ?there is some additional heat transport through the interfaces besides moleculardiffusion due to intrusions of rising and sinking double-diffusive convective plumes?.1.5 Modelling Heat and Salt FluxesOnce a double diffusive staircase is established it is desirable to quantify the net verticalfluxes of heat and salt through the layers. These are denoted FH and FS and have unitsW/m2 and g/s/m2 respectively. Observations show that double diffusive fluxes are sig-nificantly higher than would be supported by molecular diffusion alone through a similarbackground stratification (Kelley et al., 2003; Ruddick and Gargett, 2003). Measurementsin Lake Kivu, in East Africa?s Rift Valley, indicate they are enhanced relative to molecularfluxes by about an order of magnitude (Schmid et al., 2004, 2010) and the same is true offlux estimates in the Canada Basin of the Arctic Ocean (Padman and Dillon, 1987).Parameterizations for estimating double diffusive fluxes were first developed by Turner(1965) and, though they have been refined, these have remained fundamentally unchangedto date. An overview is given below, with the recommendation to refer to Turner (1973)for greater detail. Double diffusive flux parameterizations are built in analogy to thoseof heat fluxes in single-component thermally driven convection where it assumed that, forsufficiently large Rayleigh number, the flux across a convecting cell is independent of thecell height H. The dimensionless heat flux, or Nusselt number, is given byNu = HFH? cp ?T?T(1.7)10and from dimensionality arguments, has the functional dependenceNu = f(Ra,R?, P r, ?) (1.8)where cp is the specific heat in J/(kg ?C) and H becomes the length scale in Ra.By enforcing the assumption that fluxes are independent of H, choosing Ra as the governingparameter and ignoring any dependence on Pr and ? (which are nearly constant), it ispossible to refine the form of Equation 1.8 to yieldNu = f1(R?)Ra1/3 (1.9)where f1(R?) is an unknown function of the density ratio to be determined experimentally.By substituting the definitions of Nu and Ra, we can retrieve the dimensional equivalentof Equation 1.9:FH = f1(R?) ? cp(?g?TPr)1/3(?T )4/3 (1.10)This expression and similar ones have come to be known as ?4/3 Flux Laws? because ofthe heat flux dependence on (?T )4/3. However, for the purposes of this study we will referto Equation 1.10 more simply as a flux parameterization rather than a law.A similar argument to that outlined above leads to an expression for the dimensionless saltfluxHFS??S?S= f2(R?)Ra1/3 (1.11)where f2 is again an unknown function of the density ratio. The Rayleigh number is chosenas a governing parameter for the salt flux, in analogy to the heat flux, because the convectionthat carries salt across the cell is thermally driven.From Equations 1.9 and 1.11 it is possible to construct a dimensionless flux ratioRF ?cp ?FS?FH= ? f4(R?) (1.12)which, if the preceding arguments are correct, depends only on the density ratio and thephysical properties of seawater. However, regardless of the correctness of the argumentsleading to the right hand side of Equation 1.12, the ratio of salt to heat fluxes is a quantitythat can be measured in a lab experiment. This experiment was carried out by Turner(1965) as well as a few other researchers (Crapper, 1975; Newell, 1984), and these sourcesagree that RF decreases sharply from 1 at low R? and is near constant atRF ? 0.15 (1.13)over the range 2 < R? < 8. This result is often used in observational studies as it is usually11difficult to obtain independent estimates of FH and FS in geophysical situations.A considerable amount of work by different authors (Crapper, 1975; Kelley, 1990; Mar-morino and Caldwell, 1976; Turner, 1965) has also gone into determining the factor f1 ofEquation 1.10. We will use the formulation developed by Kelley (1990) because it has beenused in a number of recent studies (Carmack et al., 2012; Polyakov et al., 2012; Schmidet al., 2010; Timmermans et al., 2008) and has met with reasonably good success at esti-mating double diffusive fluxes. Kelley creates an empirical fit to a collection of previouslab-based results, resulting in a factor for Equation 1.10 with the formf1(R?) = 0.0032 exp(4.8/R 0.72?)(1.14)Equations 1.10 and 1.14 together form what we will refer to as the Kelley 4/3 flux param-eterization:FK = 0.0032 exp(4.8/R 0.72? ) ? cp(?g?TPr)1/3(?T )4/3 (1.15)In conjunction with the laboratory derived flux ratio (1.13), this allows for estimates ofdouble diffusive fluxes through a thermohaline staircase based strictly on observations ofT . However, the accuracy of the flux parameterizations has been called into question as anumber of unresolved inconsistencies remain.Firstly, there is theoretical and experimental evidence suggesting that the classical fluiddynamics result for single-component thermally driven (Rayleigh-Be?nard) convectionNu ? Ra1/3 (1.16)may be incorrect or only applicable over certain parameter ranges (Castaing et al., 1989;Kelley, 1990). Indeed, there is a large body of work describing Rayleigh-Be?nard convection,and it is a much more complex field of study than this simple scaling argument suggests.For further reading, see Grossmann and Lohse (2001) and Heslot et al. (1987).The uncertain grounding of Equation 1.16 also casts doubts on the correctness of its deriva-tive Equation 1.9. If the model for the single-component convection parameterization isincorrect or incomplete, we would expect the more complicated double diffusive parame-terization to be likewise. A simple one-cell analytical model developed by Kelley (1990)suggests that the exponent in Equation 1.9 may be variable and near the range 0.27? 0.02.If this is the case, then the traditional 4/3 parameterizations would overestimate fluxes byup to about 30% in oceanographic settings. However, because of a lack of more conclusiveevidence, both exponents 1/3 and 0.27 ? 0.02 continue to be used to estimate heat fluxes.Current work by Sommer and Wu?est (2013) suggests that an exponent that varies with Racould produce more consistent results.12The applicability of double diffusive flux parameterizations remains uncertain in part be-cause of a lack of observational evidence verifying their accuracy. While observations ofdouble diffusive layers are not rare, it is difficult to obtain independent, well-resolved es-timates of vertical heat fluxes by which to calibrate the parameterizations. One commonapproach is to assume pure molecular diffusion through the interfaces and use microstruc-ture measurements to calculate the diffusive flux:FH = ?cp?T (?T/?z) (1.17)The heat flux calculated using this method should be viewed as a lower limit since it isonly accurate if the interface core is strictly laminar. Padman and Dillon (1987) found thetwo methods to agree within about a factor of 4 in the Beaufort Sea; Schmid et al. (2010)observed agreement within a factor of 2 from measurements in Lake Kivu; and Timmermanset al. (2008) also cite agreement within approximately a factor of 2 in the Canada Basin ofthe Arctic.The present study of double diffusion in Powell Lake is unique because it is possible to de-velop and verify an independent measure of the vertical heat flux through the thermohalinestaircase (see Section 1.6 below). As will be shown, this extra condition gives new insightinto the nature of the interface and allows us to make comparisons to the 4/3 flux param-eterizations with much greater precision than has been done before. If the 4/3 exponentis indeed incorrect, it would suggest that the underlying assumption that fluxes are inde-pendent of H is invalid in the context of double diffusion. We present results in this studythat indicate that this may indeed be the case and that it may be possible to develop a newparameterization which more accurately encapsulates the physics of double diffusion.1.6 Powell LakePowell Lake is a glacially formed ex-fjord on the south-west coast of British Columbia,Canada (Figure 1.8). Situated 150 km north of Vancouver near the city of Powell River,the lake is 40 km long, 2-4 km wide, and has a maximum depth of 350 m. It displaysthe characteristics typical of glacial fjords including steep sides and sills separating largelyflat-bottomed basins. Following a bathymetry study by Mathews (1962), the lake can bedivided into six basins: the East and South Basins are permanently stratified and containrelatively warmer relic seawater at depth with maximum salinities of 4.5 and 16.5 g/kgrespectively; the West Basin and the three northern most basins are holomictic and havealmost no measurable salt content (Figure 1.9). The presence of seawater in the lake wasoriginally discovered by Williams et al. (1961) who also noted that the deep waters areanoxic and contain high sulphide and methane contents while having a ?distinct yellow13 36?  32?  124oW  28.00?  24?  20?  55?   50oN   5?  10?  15? 0510 kmStrait of GeorgiaSouth Basin - with relic seawaterEast Basin - with relic seawaterSill at 290 mSill at25 mSill at12 mFour northern basins contain no relic seawaterPowell River:primary freshwater input(b) 126oW  125oW  124oW  123oW   48oN  30?   49oN  30?   50oN  30?   51oN  30? 50 kmPowell LakeVancouverVancouverIslandStrait of Georgia(a)Figure 1.8 ? Maps depicting (a) south-western British Columbia and (b) Powell Lake, aglacially formed ex-fjord adjacent to the Strait of Georgia. The lake was separated fromthe Strait by the emergence of a sill about 10,000-13,000 years ago. The six flat-bottomedbasins of Powell Lake are shaded in dark grey, following the bathymetry study of Mathews(1962). The South and East Basins are permanently stratified and contain relic seawaterand are separated from the rest of the lake by shallow sills.colour which. . . was evidently due to dissolved organic matter?. The supersaturated gasconcentrations, as evidenced by degassing at atmospheric pressure, and distinct colouringare also present in the lake today.Freshwater input to Powell Lake by river run-off is substantial and is largely at the northernhead of the lake. Net flow rates through the dam at the south end of the lake average about3?109 m3/yr (Sanderson et al., 1986). However, because the sills separating the individualbasins are quite shallow, any turbidity currents are likely isolated to the northern end ofthe lake. Flushing by turbidity currents is a likely explanation for the absence of salt waterin the northern basins. At the southern end, the lake is separated from the adjacent Straitof Georgia by a rocky sill on which a hydro-electric dam was built in the early 20th centuryto supply electricity to the local pulp-and-paper mill, stabilizing the current water level ofthe lake to be about 50 m above sea level. Below approximately 130 m depth, the waters inthe South Basin are completely anoxic indicating that there is no seasonal signal and littleturbulent mixing below this depth.14(a)130 m110 m110 mPermanentlyAnoxicRelicSeawater50 m40 kmGeothermalHeat FluxGeothermalHeat FluxFreshwater(holomictic)BasinsTopLayerStrait of GeorgiaMeasured Heat Flux:27 ? 8 mW/m2(Hyndman 1976)(not to scale)(b)South|North5 6 7 8 9 10050100150200250300350T (?C)Depth (m)  0 2 4 6O2 (mL/L)0 5 10 15SR (g/kg)Temp.SalinityOxygenFigure 1.9 ? (a) Schematic cross-section of Powell Lake, not drawn to scale laterally.Anoxia is permanent below 110 m depth indicating limited vertical mixing. Relic seawateris found below 240 m depth and is geothermally warmed by a heat flux of 27? 8 mW/m2.(b) Temperature, Reference Salinity, and dissolved oxygen profiles in the middle of thesouth basin, the deepest part of Powell Lake.Geological considerations suggest that Powell Lake was isolated from the adjacent Strait ofGeorgia 10, 000 ? 13, 000 years ago by isostatic rebound (Mathews et al., 1970). Sander-son et al. (1986) model the subsequent evolution of the lake and are able to successfullyreproduce the general temperature and salinity structure in the South Basin by consideringmolecular diffusion as the dominant process transporting heat and salt from the deep watersto the fresh upper layer; from here, salts are subsequently flushed from the lake. Descendingthrough the watercolumn, temperature decreases quickly with depth to near 5 ?C at 125 m(representing the depth of winter mixing) and then begins to increase quasi-linearly to amaximum of 9.5 ?C at the bottom (Figure 1.9). Salinity is almost zero at the surface andalso begins to increase at about 125 m; there is a very strong halocline between 270 and320 m depth and the maximum salinity occurs at the bottom of the watercolumn. Doublediffusive layers (Figure 1.5) are observed in the lower gradient regions between 301 ? 310 mand 325 ? 350 m and are most clearly defined in the latter.Temperature increases with depth because there is a geothermal heat flux into the bottomof the lake. Hyndman (1976) measured this flux to be 27? 8 mW/m2 from sediment coresamples. This is a number similar to those measured in other nearby inlets of southwestern15British Columbia. Since there are no other sources of heat, the relative linearity of thetemperature profile below the anoxic boundary, especially between 310 and 325 m depth,supports the notion that molecular diffusion is the dominant vertical transport mechanismin these sections of the watercolumn. This can be validified by corroborating the strengthof the observed linear temperature gradient to the measured heat flux (Section 4.3).To our knowledge there are six previous academic studies directly related to Powell Lake.The first is the study by Williams et al. mentioned above which outlines the discovery ofthe salt layer at the bottom of the lake (Williams et al., 1961). This was followed soonafter by the bathymetry study by Mathews (1962) and later by the first (and until now,only) temperature microstructure study in Powell Lake by Osborn (1973). This later studycorrectly hypothesizes that the double diffusive instability may be active in the deepestparts of the lake and presents measurements which show ?characteristic doubly diffusivelayers in the region between 3?5 m above the bottom?. Interestingly, Osborn points outthat there may be a relationship between the strength of the heat flux and the height ofdouble diffusive layers. This publication was followed by the Hyndman (1976) study ofheat flux measurements in two lakes, including Powell Lake, and a number of nearby inlets,and the vertical diffusion study by Sanderson et al. attempting to model the evolution ofthe large-scale temperature and salinity profiles (Sanderson et al., 1986). Finally, the mostrecent study was a chemical study of sulphur speciation and pyrite formation in meromicticPowell and Sakinaw Lakes (Perry and Pedersen, 1993).1.7 MotivationDouble diffusion is a global phenomenon that may have the potential to impact oceancirculation on large spatial scales (Ruddick and Gargett, 2003). As an example, doublediffusive signatures are observed over large portions of the Arctic ocean and are especiallyprominent in the thermocline between the cool, fresh surface waters and the relativelywarmer and saltier Atlantic Water layer that lies between 250 and 800 m depth. Theheat content in this warmer, deeper layer is enough to melt all the Arctic sea ice if itwere transported to the surface (Rudels et al., 2004). Since double diffusive fluxes appearto be the dominant transport mechanism across the central Canada basin thermocline(Timmermans et al., 2008), understanding and quantifying these is of primary importanceto forecasting environmental changes in the Arctic.Some East African rift and crater lakes also exhibit double diffusive signatures, and gain-ing an understanding of these is also critical because a number of African lakes naturallyaccumulate dangerously high concentrations of methane and carbon dioxide gases. Whengas concentrations in the deep waters of these lakes reach supersaturation, a disturbance16in the watercolumn can result in a large uncontrolled gas eruption at the surface. This canhave catastrophic consequences when the released gasses displace the breathable air in thesurrounding communities; in recent history, more than 1700 people died when Cameroo-nian crater lakes Monoun and Nyos erupted large quantities of carbon dioxide gas in the1980s (Sigurdsson et al., 1987; Sigvaldason, 1989). In Lake Nyos, double diffusive steps areobserved in the thermocline that separates the gas-enriched deep waters from the surface,and while it is more probable that the gas eruption was triggered by a localized landslide,double diffusion has been proposed as a possible mechanism responsible for initiating theeruption (Schmid et al., 2004). Since the net effect of double diffusion is to equilibratedensity gradients, it is conceivable that double diffusive effects homogenized sections of thewatercolumn, thus allowing an overturning event to release the gases which had until thenbeen trapped at depth.Lastly, it is worthwhile mentioning that double diffusive phenomena are studied in fieldsoutside of oceanography. Most pointedly perhaps, double diffusion is a current researchtopic in astrophysics (Bruenn and Dineva, 1996; Rosenblum et al., 2011) and researchersmay even straddle fields, applying their knowledge to both the oceanography and the as-trophysics context (Merryfield, 1995, 2000). Indeed, since the phenomenon is the same,much of the formalism and the language are consistent between the two fields, and there isexplicit overlap between their research approaches (e.g. compare Rosenblum et al. (2011)).Applications of double diffusion are also found in chemistry, geology, geophysics, metallurgy,and engineering; an outline of how double diffusion relates to each of these fields can befound in the proceedings of a cross-disciplinary engineering conference dedicated to doublediffusive convection (Chen and Johnson, 1984).Powell Lake presents a unique opportunity to study double diffusive convection because thepresence of warm relic seawater capped by an anoxic boundary at 130 m depth, the linearityof the temperature gradient, the lack of tidal and geostrophic effects, and the absence ofturbidity currents all suggest that the deep waters of Powell Lake are extremely quiescent.The large majority of the double diffusive layers we observe appear undisturbed and sofree from the influence of external mixing processes. In this regard, the deep waters ofPowell Lake are a natural laboratory, and the degree to which the double diffusive processis isolated from other transport mechanism is, to our knowledge, unmatched in any othernatural system that has been studied.Because double diffusion is naturally decoupled from other processes in Powell Lake, itis possible to collect many measurements of extremely well-resolved double diffusive layersover large scales in time and space. This allows us to examine details regarding the nature ofthe diffusive interface and associated instability which have previously been only modelled.This will help to verify or modify the existing model of a diffusive interface core throughwhich transport is limited by molecular diffusion.17Furthermore, again owing to the relative quiescence of Powell Lake, we are able to gainmultiple separately derived estimates for the heat flux through double diffusive layers towhich we can compare existing flux parameterizations. This allows us to verify the predictiveability of the 4/3 flux parameterizations and see their limitations with a precision that wehave not seen presented before in a geophysical study. We hope that this will enable furtherconversation with the aim of refining parameterizations and addressing some of their currentinconsistencies.1.8 ObjectiveIn light of the motivating features discussed above, the objective of this study is to delineatethe state of double diffusion in the deep basin of Powell Lake in order to provide new insightsinto the nature of double diffusive layers, the interfaces separating layers, and the fluxesof heat and salt transported vertically through the layers. A unique characteristic of thisstudy is that the vertical heat flux through the double diffusive staircase can be estimatedby independent measures. This allows us to test the scaling that underlies current doublediffusive flux parameterizations. Furthermore, an independent estimate of the vertical heatflux provides a comparison by which to confirm whether transport through the interfacebetween mixed layers is indeed controlled by molecular diffusion.In order to accomplish our objective, we carefully measure closely spaced fine-scale temper-ature and salinity structures within the staircase and over the length of the basin. Using asimple but robust algorithm, we detect individual layer/interface boundaries in both tem-perature and salinity from measured profiles, and thereby compile an extensive dataset ofstaircase properties on which to build our conclusions.18Chapter 2Research Methods2.1 Measurements and EquipmentThe data presented here consists of 39 high vertical resolution conductivity-temperature-depth (CTD) measurement casts using a Seabird SBE-25 outfitted with a SBE-3F tem-perature sensor and a SBE-4 conductivity sensor. CTD casts were taken at 21 distinctstations along the length of the South Basin, encompassing nearly a full lateral transect ofthe flat bottomed portion where double-diffusive layers are observed (Figure 2.1). The 21stations are spaced 200?300 m apart. All stations were measured at least once; stationsB01?B06 and B08?B12 were measured twice, and station B07 was measured eight times.Measurements are from July 2012 and were made over a span of two days. Of the 39 CTDcasts, seven extend from the surface to the lake bottom; the rest begin at 250 m depth andextend to the bottom. The instrument was equipped with a landing-device, allowing usto consistently measure up to within 55 cm of the lake sediments. In addition to the highresolution measurements along the length of the lake, we obtained six lower resolution CTDcasts in a line across the South Basin on an earlier trip in June 2012.The SBE-25 sampled at its maximum frequency of 8 Hz. We lowered the instrument throughthe double diffusive portion of the watercolumn at an extremely low average speed of11 cm/s, yielding a mean vertical measurement resolution of 1.4 cm. Compare this to stan-dard CTD sampling procedures which typically measure at 4 Hz and a speed of 100 cm/s,giving a vertical resolution of 25 cm.The resolution of the temperature sensor is 0.0001 ?C and the electronic noise in the temper-ature measurement tends to be less than 0.0005 ?C. The electronic noise in the conductivitycell likewise tends to be less than 0.0005 mS/cm. This results in a salinity signal with aresolution of 0.0001 g/kg and an observed electronic noise level of 0.0007 g/kg. The quotedelectronic noise levels are representative of the in-situ fluctuations of the temperature and19 33?  124oW  32.00?  31?  56?  57?  58?  59?   50oN B01B02B04B05B06B07B08B09B10B11B12B13B14B15B16B17B18B19B20B21300 m150 mFloat7.01.819.26.012.6 (m)Anchor14 x SBE5615 x SBE56 1 x SBE56Figure 2.1 ? Measurement locations in the South Basin, depicting the 21 stations atwhich we measure the double-diffusive staircase (open circles), as well as the locationsof six lower resolution cross-lake CTD casts (x-marks). Depth contours follow Mathews(1962) and are drawn at 150 m and 300 m. The triangle marks the location of an in-situtemperature mooring in the double-diffusive staircase.20conductivity sensors.The quality of our measurements may also be limited by the response time of the tempera-ture and conductivity sensors. The temperature sensor has a known response time of 0.1 s;therefore, at a lowering rate of 11 cm/s, an instantaneous temperature change would berecorded as a gradual increase in temperature over 1.1 cm.The response time of the conductivity cell is slightly more complicated to calculate. Fromthe sensor specifications, the required time to reach 68% of the final conductivity valueafter a step change in salinity is 0.06 s. In practice, the response time is limited by theamount of time required to flush the conductivity cell because the actual conductivitymeasurement is an average over the finite volume of the cell. The conductivity cell has avolume of 2.4 mL, and the measured flow rate of the instrument?s pump which fills the cellis 15 mL/s. Consequently, it takes 0.16 s to flush the interior volume of the conductivitycell. In this time, at a fall rate of 11 cm/s, the instrument covers a vertical distance of1.8 cm. That is, an instantaneous salinity change would be recorded as a gradual increasein conductivity over 1.8 cm.In practice, the finite sensor response times have the effect of smoothing sharp changes intemperature and conductivity measurements over scales of one or two centimetres. We donot correct for the smoothing effect as its length scale is still substantially smaller than theobserved interface height which tends to be between 10 and 20 cm in both temperature andsalinity (Section 3.1). Furthermore, the smoothing effect occurs over approximately thesame distance as the mean vertical measurement resolution of 1.5 cm, meaning it is largelyunnoticeable in our measurements.We additionally find that the quality of our measurements depends on the stability ofthe lowering rate of the instrument; the stability of the lowering rate in turn depends onthe surface motions of the lake because, unlike a true microstructure profiler, the CTD iscoupled to the boat by a physical wire. This means that boat motion due to surface windwaves (or other causes) translates to vertical motions of the instrument which artificallyintroduce localized fluctuations and thereby increase the uncertainty in our measurements.Consequently, we took great care to measure only when conditions on the lake are ideal andthe boat experiences minimal motion. Ideal conditions include low wind speeds (estimatedless than 5 km/h) and no noticeable wind waves; because we found that these conditionstypically occur between sunset and sunrise, we performed our CTD casts at night.Lastly, the conductivity measurement of a parcel of water occurs approximately 0.2 safter the corresponding temperature measurement. This is easily corrected by aligningthe conductivity measurement with the temperature measurement with a fixed time delayof 0.2 s.212.2 Layer EvaluationUnless mentioned otherwise, we quote in-situ temperature t and Reference Salinity SR for allresults. We choose to use in-situ temperature in place of conservative temperature becausethis is the correct temperature variable to use when discussing molecular diffusion. Formally,heat diffuses along gradients of in-situ temperature rather than gradients of conservativetemperature (which is designed to conserve heat content under mixing), though in ourmeasurements the difference is largely negligible. Since the chemical composition of therelic seawater in Powell Lake is similar to that of present-day seawater, we use the measuredconductivity and temperature to compute Reference Salinity according to TEOS-10 (IOCet al., 2010), assuming a salinity anomaly of zero. When plotting depth profiles, we usein-situ pressure p in dbar as a proxy for depth.In order to automate the detection of double diffusive layer and interface boundaries, weimplement an algorithm similar to that described by Polyakov et al. (2012). For the pur-pose of illustration, consider first a temperature profile (Figure 2.2a). From the measuredt staircase, we extract the mean background signal by applying a 0.75 m low-pass runningmean filter. The difference ?t between the actual and background signals oscillates aroundzero with sharply defined peaks at the edges between layers and interfaces. Therefore, byidentifying the peaks in ?t, we can identify the locations of the mixed layers and theircorresponding interfaces. We apply the same algorithm separately to the salinity profiles(Figure 2.2b) to derive independent layer and interface properties in t and SR. Individuallayers are more easily identified in t than in SR because the electronic noise level of theinstrument relative to the signal difference between successive layers is higher for conduc-tivity than it is for temperature; nevertheless, our measurements are precise enough thatwe are able to identify the layer and interface boundaries with confidence in SR as well asin t. In fact, we mark mixed layers only if they are clearly identifiable in both t and SR,and if they have a minimum height of 7 cm.We define ?t and ?SR as the differences in the average in-situ temperature and ReferenceSalinity between consecutive layers (Figure 2.3). The interface heights in t and SR ascalculated by the layer-finding algorithm are given by ht and hS respectively, and the mixedlayer height is labelled H. We use the temperature profiles to determine the layer height,and we do not differentiate between the layer heights in the temperature and salinity profilesbecause the difference between the two relative to their average is small. This is not thecase for the heights of the interfaces, where we expect diffusion to thicken the temperatureinterface more quickly than the salinity interface, motivating us to characterize the twointerfaces separately.Despite the high quality of our measurements, it is still necessary to apply some high-frequency filtering in order to successfully apply the layer-finding algorithm to our data.22?0.002 0 0.002?t (?C)9.29 9.3 9.31 9.32328330332334336338340t (?C)p (dbar)?0.002 0 0.002?SR (g/kg)16.665 16.67 16.675 16.68SR (g/kg)(a) (b)]Figure 2.2 ? Our mixed layer and interface detection algorithm, similar to that describedby Polyakov et al. (2012). The measured profile (t left and SR right) is shown by the thickblack line. The mean background signal (smooth grey line behind the profile) is derivedusing a 0.75 m low-pass filter, and the difference (?t and ?SR for temperature and salinityrespectively) between the measured and background profiles is shown by the alternatelypeaking and troughing grey line. Peaks and troughs in ?t and ?SR, marked by opensquares, correspond to the edges of diffusive interfaces which are marked by open circleson the profiles. Peaks and troughs are most clearly seen in ?t but are still unmistakeablyevident in ?SR. Mixed layers are marked if their height is at least 7 cm. Example profilesshown here are taken at station B18.9.29 9.3 9.31327328329330331332333t (?C)p (dbar)16.665 16.67SR (g/kg)9.29 9.3 9.31327328329330331332333t (?C)p (dbar)16.6 5 16.67SR (g/kg)?t?hT?SR?hSFigure 2.3 ? Example close-up of a few layers in t (left) and SR (right), measured atstation B04. The definitions for the temperature and salinity differences between layers,?t and ?SR respectively, are shown along with those for the mixed layer height H, andthe temperature and salinity interface heights, hT and hS . The layer-interface boundariesas determined by the layer-finding algorithm are depicted by open circles.23To accomplish this, we use a 12.5 cm running median filter (Barner and Arce, 1998) whichpreserves the sharp interface/layer boundaries while successfully removing some of the elec-tronic noise in the temperature and conductivity signals.The only independent variable in calculating the difference profiles ?t and ?SR is the size ofthe low-pass filter used to derive the background signals. We choose the window size 0.75 mbecause this is typical of the observed layer height. In order to test the robustness of ourlayer-finding algorithm, we choose a representative profile and perform the calculation usinga 0.50 m and 1.00 m window. We find that on average H changes by less than 2% for the0.50 m window and less than 1% for the 1.00 m window, hT changes by less than 8% and2% respectively, and hS changes by less than 12% and 2% respectively. These changes aresmall enough to neglect in much of the following analysis, but we must consider them wheninterpreting any observed differences between the temperature and salinity interfaces, aswill be discussed below. Note that the changes in hT and hS are correlated with each otherand anti-correlated with those in H.24Chapter 3Results3.1 CTD MeasurementsIn the South Basin of Powell Lake, temperature and salinity both increase monotonicallywith depth from 250 m to the lake bottom at 348 m (Figure 3.1). Temperature increasesconsistently and quasi-linearly while salinity increases most rapidly from 280?300 m. Bothprofiles have high-gradient regions between 250?270 m, 282?287 m, 293?294 m, 298?301m, and 309?324 m. The intermediate depths between these regions and below the last oneexhibit much lower gradients in t and SR, forming a large-scale step-like structure with ascale height of a few meters. This should not be confused with the double-diffusive layerswe report on in this study, which are found within the larger steps described here; it isunclear how the large-scale steps developed.There are three regions over which double-diffusive layers are observed. All three regionshave background density ratios that reach below 2; however, double-diffusive layering ismost clearly defined and most consistent in the deepest region, between 324 m and thebottom of the watercolumn (Figure 3.2). Here, the density ratio ranges approximatelyover 2 ? R o? < 6, with the lowest values observed near the middle of the staircase. In theshallowest region, between 294?297 m, the layers are observed only near the northern endof the basin; in the second region between 301?309 m, they are observed over most of thelength of the basin, but the double diffusive layer/interface boundaries are not as clear asthey are in the deepest region. For this reason, unless mentioned otherwise, the followinganalysis is restricted to the lower staircase only, which extends to within 1 metre of thebottom of the lake. It is also worth noting that the density ratio between 310 ? 323 mdepth, where there are no double-diffusive signatures, varies between 4 ? R o? < 12 so thatthere is some overlap in the respective ranges of R o? where double diffusive steps are andare not observed.25t (?C)p (dbar)7 7.5 8 8.5 9 9.52502602702802903003103203303405 10 15SR (g/kg)R?100 101 102N210?5 10?3(a) (b) (c)oFigure 3.1 ? Vertical profiles in the South Basin of Powell Lake. (a) In-situ temperature t(thin line) and Reference Salinity SR (thick line) for the bottom 100 m. Profiles exhibita large-scale step-like structure with double diffusive layers observed in the lowest threesteps, shaded in dark grey. The light shaded region between 310-323 m is the one referencedin Section 3.3 which we use to estimate the steady state vertical heat flux. (b) Backgrounddensity ratio R o? calculated from the t and SR profiles shown in the left panel. (c) Thecorresponding large-scale buoyancy frequency N2. Both b. and c. are smoothed using a1 m running mean low-pass filter. All profiles are from station B06.269.25 9.3 9.35 9.4 9.45 9.5 9.55 9.6325330335340345t (?C)p (dbar)0.015 ?CB01B2116.65 16.7 16.75 16.8 16.85 16.9SR (?C)0.012 g/kg8.48 8.5 8.52 8.54 8.56 8.58 8.6 8.62 8.64303308 p (dbar)0.007 ?C15.52 15.53 15.54 15.55 15.56 15.57 15.58 15.59 15.6 15.61 0.004 g/kg8.22 8.24 8.26 8.28 8.3 8.32 8.34 8.36 8.38294296298 p (dbar)0.007 ?C12.14 12.15 12.16 12.17 12.18 12.19 12.2 12.21 12.22 12.23 0.004 g/kgFigure 3.2 ? Lateral transect of double diffusive layers in t (left) and SR (right) at the three depths at which double diffusion is present. Profilesfrom left to right are from consecutive stations B01 to B21 as in Figure 2.1. The layering structure is least well established in the shallowestsection and most clearly defined in the deepest portion. Each profile is horizontally offset from the previous by the value shown in the bottomleft corner of each panel. The vertical scale is maintained throughout all six panels for easy comparison, and profiles from stations B09 and B10are highlighted in grey. All profiles extend to within 1 m of the lake bottom.270 1 2 3 4 5 651015202530Number of layersDistance along lake (km)  0.70.840.981.121.261.4Layer density (1/m)Layer CountLayer DensityFigure 3.3 ? Number of double diffusive layers below 324 m from south to north and thecorresponding vertical layer density. Both decrease towards the northern end of the basin;the maximum number of layers observed is 27 and the minimum is 9, while the maximumlayer density is 1.30 m?1 and the minimum is 0.81 m?1.In the deepest staircase, double diffusive layering is very well established and though tem-perature and salinity differences between layers are only on the order of one part in ten-thousand, we are able to clearly distinguish between layers over nearly the entire lengthof the basin. The majority of the 21 profiles shown in Figure 3.2 appear undisturbed; theprofiles at stations B09 and B10 are exceptions to the rule as the staircase is not well-definedfor these profiles between 327?335 m. This is a consistent signal over two days of measure-ments (second day of measurements not shown). The bottom bathymetry of the lake can beseen from the depth of the profiles, showing that the lake shallows by about 10 m towardsthe north end of the basin. As the lake shallows northward, there are fewer layers and theobserved layers become about 40 percent larger on average so that the vertical layer densitydecreases from 1.25 to 0.84 m?1 (Figure 3.3). Note that the lake shallows more graduallysouthward and that our profiles do not extend far enough south to show this.From 39 CTD casts, we obtained 756 individual observations of double diffusive layersand their dividing interfaces (Figure 3.4). The median mixed layer height is 0.71 m andthe corresponding interquartile range (IQR) is 0.50?0.89 m; the median interface heightin temperature is 0.18 m with an IQR of 0.13?0.24 m, and the median interface height insalinity is 0.17 m with an IQR of 0.11?0.24 m. On average, the temperature interface is0.015 m or approximately 9% larger than the salinity interface. The median temperaturedifference between consecutive layers is 0.0036 ?C and has an IQR of 0.0029?0.0047 ?C, whilethe median salinity difference is 0.0013 g/kg with an IQR of 0.0011?0.0021 g/kg. We quotethe median value rather than the arithmetic mean because the distributions are log-normal;however we also calculate the mean values (Table 3.1) for reference and intercomparisonswith other studies.The mixed layer and interface heights, and the temperature and salinity steps between28Parameter Median Mean IQRH (m) 0.67 0.71 0.50 - 0.89hT (m) 0.18 0.20 0.13 - 0.24hS (m) 0.17 0.19 0.11 - 0.24?t (?C) 0.0036 0.0043 0.0029 - 0.0047?SR (g/kg) 0.0013 0.0024 0.0011 - 0.0021hT ? hS (m) 0.018 0.015 ?0.02 - 0.05Table 3.1 ? Basic layer statistics, including layer height H, interface heights hT and hS ,temperature and salinity steps ?t and ?SR, and interface size difference hT ? hS . Shownfor each variable are the median, the arithmetic mean, and the interquartile range (IQR).layers all exhibit distinctive vertical structure. To characterize this, we create scatter-plotsin depth (Figures 3.5 ? 3.7) and calculate the average values within 1.75 m vertical bins.We choose this bin size because it is larger than nearly all observed layer heights, but smallenough to provide at least 10 bins over the depth range of the staircase.The interface heights are nearly constant, though they increase slightly from 0.15 m to0.20 m towards the top of the staircase (Figure 3.5). The same is true of the difference?h = hT ? hS whose average remains consistently above zero and increases from approxi-mately 0.010 m to 0.025 m towards the top (Figure 3.6b). There is considerable scatter inour observations of ?h and a number of points fall below zero, indicating that for a portionof our observations the temperature interface is smaller than the salinity interface. This isunlikely as it would imply that the diffusive temperature interface is destroyed by mixingin the adjacent layers while the salinity interface remains intact. It is more probable thatthe scatter in our observations of ?h reflects the limitation of our algorithm to character-ize the boundary between the interface and the mixed layer. This is the case because thenoise-to-signal ratio is quite high in the conductivity measurements. However, we maintainconfidence in our instrument to resolve the vertical height of the interface in both tem-perature and conductivity: if, for example, a large number of salinity interfaces were toosmall for us to resolve, our observations of hS would clump along a lower limit of 1.8 cmas set by the instrument resolution (Section 2.1), forming a dense collection of data pointsalong the left-hand side of Figure 3.5b. Similarly, a large number of measurements belowour detection limit would be reflected in the histograms of Figure 3.4a,b by a large numberof observations at a lower limit. However, since the majority of data lie well above theinstrument?s resolution, it is unlikely that the size of either interface tends to be below ourdetection limit. Therefore, although there is substantial scatter in our observations of ?harising from the inherent difficulties of detecting the layer/interface boundary, we considerthat the trend in our observations (given by the mean ?h ? 1.5 cm) accurately reflects thecharacteristics of the interfaces we observe.The mixed layer height (Figure 3.6a) is distinctly larger towards the middle of the staircase,where H ? 1 m, than it is towards the bottom and the top of the staircase where H ? 0.4 m29hT (m)Occurrence(a)? = 0.18 m0 0.2 0.4 0.6050100hS (m)(b)? = 0.17 m0 0.2 0.4 0.6050100?t (?C)Occurrence(c)? = 0.0036 ?C0 0.005 0.01 0.015050100150?SR (g/kg)(d)? = 0.0013 g/kg0 0.005 0.010100200300H (m)Occurrence(e)? = 0.67 m0 0.5 1 1.5020406080?h (m)(f)? = 0.015 m?0.2 0 0.2050100150Figure 3.4 ? Histograms of layer/interface characteristics, showing (a) temperature inter-face size, (b) salinity interface size, (c) temperature difference between consecutive layers,(d) salinity difference between consecutive layers, (e) layer height, and (f) difference be-tween hT and hS . For each histogram, the lightly shaded area in the background betweenthe dashed lines shows the interquartile range, and the median ? of the observable is given,except in (f) where the mean ? is given.300 0.1 0.2 0.3 0.4 0.5325330335340345hT (m)p (dbar)0 0.1 0.2 0.3 0.4 0.5hS (m)(a) (b)Figure 3.5 ? Height of the double-diffusive interface in t (left) and SR (right). Opengrey circles represent individual interface observations, the dark line is a non-parametricloess curve (Cleveland, 1993) fitted to the observations, and the open squares are verticallybinned averages using 1.75 m bins. Error bars show one standard deviation of the datathat falls within each bin. The symbol and error bar conventions shown here are continuedfor Figures 3.6, 3.7 3.8, and 3.11.?0.15 ?0.1 ?0.05 0 0.05 0.1 0.15?h (m)0 0.5 1 1.5325330335340345H (m)p (dbar)(a) (b)Figure 3.6 ? (a) Layer height and (b) interface height difference. The measured doublediffusive layer height H displays a clear vertical structure with larger values near the middleof the staircase. The difference ?h between interface heights in t and SR is constructeddirectly from Figure 3.5. We observe that hT is about 9% larger than hS throughout mostof the staircase.310 0.005 0.01 0.015325330335340345?t (?C)p (dbar)0 0.004 0.008 0.012?SR (g/kg)(a) (b)Figure 3.7 ? (a) Temperature and (b) salinity steps between consecutive mixed layers,labelled ?t and ?SR respectively. Both remain fairly constant in the lower half of thestaircase but increase strongly near the top. Measurements of ?SR across the two highestinterfaces form distinctive clumps at approximately 0.006 g/kg and 0.011 g/kg which arenot seen in the corresponding measurements of ?t.and H ? 0.6 m respectively. This is a consistent feature at all measurement stations. Thetemperature and salinity differences, ?t and ?SR display a different vertical structurewhich is likewise consistent across the length of the basin. Both become dramatically largertowards the top of the staircase (Figure 3.7). ?t increases approximately from 0.003 ?Cat 345 m, to 0.004 ?C at 330 m, to 0.01 ?C at 325 m depth. Similarly, ?SR increasesapproximately from 0.001 g/kg at 345 m, to 0.002 g/kg at 330 m, to as high as 0.012g/kg at 325 m depth. The shallowest two interfaces span relatively much larger salinitysteps than do the lower ones, resulting in two discrete clumps in the depth profile of ?SR(Figure 3.7b). Also notice that the scatter in observations of both ?t and ?SR is small,indicating that the size of the temperature and salinity steps are laterally consistent acrossthe basin scale.323.2 Interfacial Temperature GradientWe calculate the temperature gradient ?t in the interface using?t = ?t/hT (3.1)We stress that this is the temperature gradient averaged over the interface, which will besystematically lower than the gradient at the centre of an idealized interface because ofthe transition zone at the boundary of the interface and adjacent mixed layers. Similar toother characteristic variables, the interfacial temperature gradient displays a clear verticalstructure that is consistent across the basin (Figure 3.8a). The structure arises primarilybecause hT remains nearly constant while ?t increases strongly towards the top of thestaircase. Near 345 m depth the gradient is approximately 0.02 ?C/m, decreases slightlyaround 340 m, and then begins to increase to a maximum of 0.04 ?C/m at 325 m. Thescatter at a given depth is primarily due to the relatively large scatter in measurements ofhT (Figure 3.5).We calculate the density ratio across each interface usingR? =??SR??t (3.2)where ? and ? are the average values within the interface and are calculated with respect toSR and t. The vertical structure of R? is similar to that of ?t, lowest near the middle, andincreasing gently towards the bottom and more strongly towards the top of the staircase(Figure 3.8b). Values range from 3 at the bottom, to 2 at 340 m depth, to approximately6 at 325 m, and the scatter in the depth-averaged bins of the data is remarkably low. Itis noteworthy that the shape of the vertical structure of ?t reflects that of R?; the linearcorrelation between the two variables is 0.50. Averaging the individual measurements across1.75 m vertical bins yields a linear correlation of 0.95 (Figure 3.9).33p (dbar)?t (?C/m)0 0.02 0.04 0.06325330335340345R?2 3 4 5 6 7 80.047(a) (b)Figure 3.8 ? (a) Average temperature gradient in the interface between layers. Thevertical black line gives the steady state temperature gradient due to molecular diffusionas described in Section 3.3. (b) Density ratio across the interface, which is at its minimumin the middle of the staircase at R? ? 2 and is largest at the top of the staircase at R? ? 6.R = 0.95s = 0.006R??t (?C/m)1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50.0050.010.0150.020.0250.030.0350.040.0450.05Figure 3.9 ? Parametric plot in depth of ?t against R?, constructed from the data inFigure 3.8. At low values of R?, the gradient in the interface is considerably lower than thesteady state gradient (thick black line) we expect from molecular diffusion, while at highervalues of R? it approaches the steady state gradient. Open grey circles are the individualinterface measurements, and open black squares are the 1.75 m vertically binned averages.This convention is continued for Figures 3.12 and 4.1. The depicted regression is thatof the depth-binned data and has a linear correlation coefficient R = 0.95 and a slopes = 0.006.343.3 Heat FluxesIn order to test theories of double diffusive heat transport, independent estimates of thevertical heat flux are useful. Two such estimates are possible: first, Hyndman (1976)estimates the heat flux in the sediments of Powell Lake and finds a value of 27?8 mW/m2.If the heat budget of the saline layer is in a quasi-steady state, then the vertical heatflux through this portion of the watercolumn should be the same as the heat flux in thesediments, and for this reason we refer to it as the steady state heat flux. We consider thequasi-steady state assumption reasonable because the relic seawater in which we conduct ourmeasurements has been quiescent and gradually heated since the most recent deglaciation(Sanderson et al., 1986). While it is true that the large-scale properties of the lake are stillevolving slowly since salt continues to be lost to the surface through molecular diffusion,the rate at which this happens is much lower than that at which heat diffuses through thedouble diffusive region. A simple scaling of the diffusion equation (using t? for time)?T?t? = ?T?2T?z2 (3.3)suggests that the time scale for heat to diffuse through the bottom 50 m of the lake isapproximately 600 years. This is substantially less than the age of the lake, giving thedouble-diffusive region sufficient time to reach a local thermal equilibrium. The quasi-steadystate assumption is supported by a five-year series of CTD measurements (not shown) whichshow no apparent change in the large-scale structure of the temperature and salinity profilesin that time. If the saline layer were not in a quasi-steady state, we would expect to see aslight warming in the lower few meters of the lake, but this is not observed.We gain a second estimate of the heat flux by assuming that vertical transport over thedepths between 310?323 m is dominated by molecular diffusion. Across this depth range, thetemperature profiles are almost perfectly linear and the strength of the temperature gradientis remarkably consistent at all observed stations along and across the lake (Figure 3.10).The gradient varies by less than one percent across all measured locations. Furthermore,we have no evidence to support the presence of significant turbulent energy at this depth.Both observations are consistent with the quiescence condition mentioned above and arediscussed further in Section 4.3.To calculate the mean temperature gradient between 310?323 m, we calculate the averagegradient in thirteen one meter vertical bins at each of the 26 available measurement stationsand then average over all stations. The mean temperature gradient calculated this way is0.0472 ? 0.0003 ?C/m, where the uncertainty is the standard error in the mean based on350 1 2 3 4 5 600.010.020.030.040.050.06Distance along lake (km)?t/?z (?C/m)Mean: 0.047 ?C/m0.2 0.4 0.6 0.8 1Distance across lake (km)Mean: 0.047 ?C/mFigure 3.10 ? Mean temperature gradient between 310 ? 323 m depth, plotted as cross-sections along and across the South Basin. Error bars depict one standard deviation acrossthirteen one meter vertically averaged bins. The measured gradient is in excellent agree-ment with that expected from heat flux measurements by Hyndman (1976). It appears thatvertical transport is dominated by molecular diffusion and that the temperature gradientover this depth is equilibrated to steady state.p (dbar)FK (W/m2)0 0.01 0.02 0.05 0.06325330335340345Figure 3.11 ? Heat fluxes calculated using the 4/3 parameterization from Kelley (1990),given by Equation 1.15. The vertical black line is the steady state vertical heat flux deducedfrom Figure 3.10 and measured by Hyndman (1976). The parameterized heat fluxes appearaccurate to within approximately a factor of two as evidenced by the vertical variationin FK .36one standard deviation of all 26? 13 bins. Using this value and takingcp = 4.1? 103 J/(kg ?C), ? = 1012 kg/m3, ?T = 1.4? 10?7 m2/s (3.4)calculated at t = 10 ?C and SR = 15 g/kg (Sharqawy et al., 2010), we use Equation 1.17 toestimate the vertical steady state heat flux through Powell Lake at 27? 1 mW/m2.The estimate of the vertical heat flux above the staircase agrees with that of the flux inthe sediments below the staircase, and so because of steady state, our best estimate of thetime-averaged vertical flux through the double-diffusive staircase is likewise 27 mW/m2 atall depths within the staircase. We compare this estimate to the 4/3 parameterization ofKelley (1990) given by Equation 1.15 (Figure 3.11). We use values for cp and ?T as above,as well as ? = 1.3?10?6 m2/s also calculated at 10 ?C and 15 g/kg, and g = 9.8 m/s2. Theparameterization calculated across each interface scatters about the steady state heat fluxwithin about a factor of 2 and each 1.75 m bin has a standard deviation of approximately0.01 mW/m2. It has vertical structure resembling that of H (Figure 3.6a), tending tooverestimate near the middle of the staircase (FK ? 33 mW/m2) and underestimate nearthe top (FK ? 22 mW/m2) and bottom (FK ? 15 mW/m2).While our two estimates of the vertical steady state heat flux are in agreement with eachother, the interfacial temperature gradient we estimate using ?t/hT is on average notlarge enough to account for this flux by molecular diffusion alone (Figure 3.8a). UsingEquation 1.17, the estimated heat flux of 27 ? 1 mW/m2 should result in an interfacialtemperature gradient of 0.047? 0.002 ?C/m. The interfacial gradient we observe is lessthan half of this value over much of the staircase and only approaches it towards the top ofthe staircase where the density ratio also becomes large (Figure 3.8). This surprising resultis discussed further in Section 4.5.As outlined in Section 1.5, there has been considerable discussion regarding the correctnessof the assumptions underlying the 4/3 parameterizations, specifically regarding the 1/3exponent in the original scaling. Our measurements allow us to compare Nu and Ra directlyto test for the exponent. In order to do this, we choose a subset of measurements over whichR? is approximately constant, in which case the scaling suggests that Nu ? Ran directly(from Equation 1.9), where n is now a generalized exponent. We include all interfaces acrosswhich the density ratio falls in the range 1.5 < R? < 2.5 and plot log(Nu) against log(Ra)(Figure 3.12); the exponent n is then given by the slope of the linear regression. In total,441 or 58% of the observed interfaces fall within the prescribed range of R?. To calculateNu and Ra, we use the layer height above the interface as a length-scale and the estimatedsteady state heat flux of 27 mW/m2. The logarithms are both calculated at base 10.37log(Ra)log(Nu)n1 = 0.22n2 = 0.431.5 < R? < 2.5  5.5 6 6.5 7 7.5 80.50.7511.251.5 Observations Vertically binned LS Equation 3.5a LS Equation 3.5b Line slope 1/3Figure 3.12 ? A direct test of the scaling Nu = f1(R?)Ran. We use a subset of theavailable data over which R? does not vary much so that the factor f1 is approximatelyconstant. Two linear least squares (LS) regressions incorporating distinct error structures,as described in the text, give that 0.22 ? n ? 0.43. A line with slope 1/3 is shown forcomparison. The vertically averaged bins of the data are connected by a thin line, showingthe progression from deepest (with lowest Ra) to shallowest. Note that the verticallyaveraged bins do not follow an obvious linear trend as would be expected from a scalingof the form Nu ? Ran, but that they deviate systematically from this power law.A standard least squares regression minimizes the sum of the squared measurement errorsalong the vertical axis. It assumes that there is no measurement error along the horizontalaxis. However, in our observations of Nu and Ra, both variables contain an associatedmeasurement error and it is incorrect to associate the entire uncertainty to only one variable.Consequently, we calculate the linear regression to the data shown in Figure 3.12 in twoways. Using the least squares method, we perform a regression assuming each error structureindependently:log(Nu) = n1 (log(Ra) + ? log(Ra)) (3.5a)log(Ra) = 1n2(log(Nu) + ? log(Nu)) (3.5b)where the ? term is the measurement error associated with the respective variable. Thefirst model assumes that all the uncertainty is contained in measurement errors of Ra; thesecond assumes that the entire uncertainity results from measurement errors in Nu. Thetwo methods produce substantially different estimates for the exponent, giving n1 = 0.22and n2 = 0.43, and these values can be taken as lower and upper bounds for an idealregression. If the form of the scaling is correct, then the true exponent lies somewherebetween these two values, and choosing n = 1/3 visually matches the trend in the datareasonably well. The strength of the linear correlation between log(Nu) and log(Ra) forthe range 1.5 < R? < 2.5 is 0.72.38However, notice that the depth-averaged bins of the data do not follow a purely linear trend.The deepest four bins do appear linear with a slope n > 1/3, but the shallowest five bins(with Ra > 107.2) systematically deviate from a linear trend. Taken together, the depth-averaged bins form a convex shape that does not follow a power law of the form Nu ? Ran.This result is surprising because the binning process tends to remove scatter and extractthe underlying trend from the raw observations; the implication is that a scaling of theform Nu ? Ran is fundamentally inconsistent with our observations. This result and itsimplications are discussed further in Sections 4.3 and 4.4.3.4 TS Diagrams and Layer CoherenceThe mixed layers of the double diffusive staircase in Powell Lake group in clusters similar tothose observed in the Canada Basin of the Arctic (Timmermans et al., 2008) when plotted intemperature-salinity space (Figure 3.13). This indicates that individual layers are coherentand can be traced over the scale of the basin. A layer that has a vertical scale heightof 1 m and a lateral scale length of 10 km has an aspect ratio of 104. Furthermore, asour measurements span two days, we can say that layers remain coherent for at least thatperiod. The three shallowest layers are notable exceptions. For these, the temperature-salinity properties appear to split approximately halfway along the basin. The salinityproperties in particular seem to diverge: the upper two layers tend to be less saline in thesouthern portion of the basin and the third layer tends to be more saline in the southernportion of the basin.The other layers appear to maintain relatively similar lateral gradients and are system-atically saltier and cooler, and therefore heavier, towards the northern end of the basin.However, gradients are extremely small: they are order 10?6 ?C/m and 10?5 (g/kg)/m intemperature and salinity respectively. The deepest few layers become difficult to distinguishon the diagram because the salinity difference between layers is no longer larger than thesalinity difference in one layer over the length of the basin. Timmermans et al. define alateral density ratio R?x to characterize the along-layer gradients:R?x =?Sx??x(3.6)which is given by the inverse of the slope of the clusters in Figure 3.13. ? is conservativetemperature, chosen here in place of t because mixed layers are thought to be convective(see Section 2.2). We calculate the slope for seven of the most readily identifiable clustersand so find a lateral density ratio of R?x = ?0.46? 0.18.39Figure 3.13 ? Temperature-salinity diagram of double diffusive mixed layers, not in-cluding measurements of the interfaces. ? and SR are non-dimensionalized by ? and ?respectively. Layers group along distinctive clumps indicating that they are coherent overthe basin scale. They consistently exhibit horizontal gradients in both variables, and themagnitude of the slope for a few layer groupings is shown in the inset.40Chapter 4Discussion4.1 Large-scale propertiesThe gross features of the temperature and salinity structure of the South Basin of PowellLake are described well by a simple one-dimensional diffusion model in which the eddydiffusivity decreases exponentially with depth (Sanderson et al., 1986). The difference inthe shape of the two profiles arises because of the differing boundary conditions in T andS. The temperature profile is continually forced by a geothermal heat flux below whilelosing heat into the mixed fresh layer above; this results in a quasi-steady state in whichthe temperature profile changes only little with time after the first 5000 years. The salinitybudget on the other hand is necessarily negative as salt is continually lost into the mixedlayer, slowly depleting the salt reservoir in the deep water.The large-scale step structure in the temperature and salinity profiles (Figure 3.1) is notreproduced in the one dimensional diffusion model of Sanderson et al. (1986). Similar large-scale steps are observed in Lake Kivu where the steps are thought to be caused by the inflowof subaquatic springs at various depths (Schmid et al., 2010). In Powell Lake there are noobvious indications of subaquatic inflows (for example, consistent mid-depth temperaturemaxima); futhermore, substantial inflows below the halocline would contribute to flushingthe salt from the deep layer more rapidly than is accounted for by the simple diffusionmodel of Sanderson et al. (1986). Since the model is able to account for the bulk of thesalt budget over the last 11,000 years by diffusion, there is strong indication that thereare no substantial inflows below the halocline. Consequently, it appears unlikely that thelarge-scale step structure in Powell Lake results from the effect of subaquatic inflows.Alternatively, it is conceivable that the steps are related to the hypsography of the lake;unfortunately, the only available bathymetry information for Powell Lake is from Mathews(1962) which is too scarce to produce a well-resolved hypsography. A more thorough study41of depth soundings is required to create isobaths at a sufficient resolution to produce aprecise hypsography below 250 m depth. However, echo sounding transects carried outduring the course of numerous trips over five years do not indicate that there are welldefined steps in the bathymetry of the lake (Pawlowicz, personal communication).Lastly, a possibility that has not been explored in previous studies is that the large-scalesteps are themselves related to the double diffusive layers found within them. Perhaps thelow-gradient regions develop because of the formation of double diffusive layers which moreeffectively transport heat and salt through the watercolumn. This idea would explain whydouble diffusive layers are found only in the low-gradient regions of the profiles. It fails toexplain why double diffusive convection becomes active in some regions of the watercolumnbut not in others. In this scenario, the current profiles of R o? and N2 are of little valuein explaining the formation of double diffusive layers as these would have had a differentvertical structure before layering developed than they do currently.The location of double diffusive layering in Powell Lake generally follows the approximatecriterium that 1 < R o? < 6, but this is not sufficient to explain all the observed structure.For example, there is some overlap in the values of R o? that are observed between 310?323 mwhere there is apparently no double diffusion present and below 324 m where double diffusivelayering is well established (Figure 3.1). The overlap exists primarily at the intersection ofthe two regions: approaching the top of the staircase from below, the background densityratio decreases at 324 m to a value of 4, before increasing again to a value of 8 at 320 m. Yet,there appears to be no double diffusive instability over this depth range. It is noteworthythat the large-scale instability N2 does not exhibit a similar overlap in values between thedouble diffusive and the stable regions, suggesting that perhaps double diffusive layeringcan additionally be governed by the large-scale stability.Considering only the background density ratio also does not explain why layering is verywell established below 324 m depth while it appears slightly more disturbed in the higherdouble diffusive region between 301?309 m, and can hardly be seen at all in the shallowestdouble diffusive region between 294?298 m. The most likely explanation for the increasinglydisturbed staircases in the shallower regions is that eddy diffusivities are no longer negli-gibly small. It is likely that some turbulence propagates from the surface to disturb thehigher two staircases. In the modelling scheme of Sanderson et al. (1986), eddy diffusivityis comparable to molecular diffusivity at the bottom but increases exponentially towardsshallower depths.424.2 Comparison of Layer CharacteristicsThe properties of the double diffusive layers in Powell Lake are not unlike those found inother locations (Table 4.1). The layer height in Powell Lake ranges approximately between0.2?1.5 m; in Lake Kivu it ranges between 0.1?0.8 m (Schmid et al., 2010); in Lake Nyosit ranges between 0.2?1.7 m (Schmid et al., 2004); and in the Canada Basin of the Arcticit ranges approximately between 1.0?5.0 m (Timmermans et al., 2008). That is, in allcases the layers tend to have a scale height of about 10 cm to 1 m. It is not clear whatdetermines the layer height in a double diffusive staircase; a parameterization for H doesexist (Kelley, 1984) and is based on the large scale stability N2, but for Powell Lake thisparameterization calculates layer heights that scatter between 1.5?4.0 m which are too largeto accurately describe the observed layer heights.The interface height is remarkably similar in all the cases described above. The temperatureinterface height hT varies approximately between 0.1?0.4 m in Powell Lake, 0.1?0.3 m inLake Kivu, and 0.1?0.3 m in Lake Nyos. No exact interface height information is available forthe Canada Basin staircase; however both Padman and Dillon (1987) and Timmermans et al.(2008) suggest that the interfaces are roughly 0.10 m high. It is qualitatively understoodthat the interface height is likely controlled by (buoyancy driven) separation of the interface?sunstable boundary layer or by convection in the adjacent mixed layers (Carpenter et al.,2012; Linden and Shirtcliffe, 1978; Padman and Dillon, 1987), but no quantitative analysishas successfully predicted the observed interface heights in geophysical situations.Heat fluxes vary by an order of magnitude between the five studies described here. In PowellLake, we estimate that the vertical heat flux is 27 mW/m2 (Section 3.3). From the abovereferences, the vertical heat fluxes in Lake Kivu are approximately 10?100 mW/m2, whilethose in Lake Nyos are roughly 100?600 mW/m2. Padman and Dillon (1987) estimate thevertical heat flux through most of the staircase in the Canada Basin to be about 40?70mW/m2, while Timmermans et al. (2008) estimate the flux to be 50?300 mW/m2. Notethat of the above estimates, only the one in Powell Lake does not rely on either a 4/3flux parameterization or on the assumption that interfaces between layers are controlledby molecular diffusion (though the study in Lake Nyos does make a comparison to anindependent estimate from a heat budget calculation).The observed temperature differences between consecutive layers likewise vary by an orderof magnitude between the above sites. Those in Powell Lake and Lake Kivu are quite similar,with temperature differences that vary between 0.001?0.015 ?C and 0.001?0.018 ?C respec-tively. In Lake Nyos, ?t tends to be higher and is observed in the range 0.015?0.047 ?C.In the Canada Basin, Padman and Dillon (1987) find temperature differences that rangebetween about 0.004?0.012 ?C, while Timmermans et al. (2008) find ?t ? 0.04 ?C. Salinitydifferences are only available for the Canada Basin, where Padman and Dillon (1987) find43R? H (m) hT (m) ?t (?C) ?SR (g/kg) FH (mW/m2)Powell Lake 1.5?8 0.2?1.5 0.1?0.4 0.001?0.015 0.001?0.002 27Lake KivuSchmid et al. (2010) ? 0.1?0.8 0.1?0.3 0.001?0.018 ? 10?100Lake NyosSchmid et al. (2004) ? 0.2?1.7 0.1?0.3 0.015?0.047 ? 100?600CB ? Padmanand Dillon (1987) 4?5 1.2?3.0 ? 0.1 0.004?0.012 0.001?0.005 40?70CB ? Timmer?mans et al. (2008) 2?7 1.0?5.0 ? 0.1 ? 0.04 ? 0.014 50?300Table 4.1 ? An overview of double diffusive parameters from five distinct studies, com-paring Powell Lake, Lake Kivu, Lake Nyos, and the Canada Basin (CB) of the Arctic.?SR ? 0.001?0.005 g/kg and Timmermans et al. (2008) find ?SR ? 0.014 g/kg. Interfacialdensity ratios are therefore also available only for the Canada Basin where Padman andDillon (1987) find R? ? 4?5 and Timmermans et al. (2008) find an approximate range2 < R? < 7. The latter is very similar to the range of density ratios we observe in PowellLake, which is 1.5 < R? < 8.Lastly, we find that layers in Powell Lake form clumps in T?S space similar to those in theCanada Basin described by Timmermans et al. (2008). In both cases, these clumps crossisopycnals in a systematic manner; however, the lateral density ratio calculated from thesegroups has a larger magnitude and varies less in the Canada Basin (R?x = ?3.7? 0.9) thanit does in Powell Lake (R?x = ?0.46? 0.18).4.3 Heat FluxesOur approach from Section 3.3 of estimating the vertical heat flux by considering thattransport between 310?323 m is dominated by molecular diffusion rests on the assumptionsthat the lake is quiescent at this depth and that the system can be accurately representedby a one-dimensional model. This requires that the eddy diffusivity and lateral divergenceboth be negligibly small. These are strong conditions which are generally not true in theocean, but our observations support the assumption that they do hold in Powell Lake.Firstly, the presence of a permanent anoxic boundary at 130 m depth confirms that thereis no large-scale overturning circulation in the South Basin. This idea is supported by thepresence of the 11,000 year old salt layer which likewise indicates that turbulent mixingacross the halocline is minimal and that eddy diffusivites in the salt layer are extremelysmall. The one-dimensional diffusion model of Sanderson et al. (1986) is largely able to44account for the salt lost from the saline layer over the past 11,000 years by assuming adiffusivity profile in which eddy diffusivity is comparable to molecular diffusivity at depth.Furthermore, it is very unlikely that there are subaquatic inflows below the halocline as thesewould contribute to flushing the salt from the basin. And while there are seiching motions inthe lake which may contribute to enhanced mixing at the boundaries, the observed verticaldisplacement from comparing the 39 CTD casts is small and has a height of approximatelyone meter. It is unlikely that seiching motions produce substantial mixing at the boundariesbecause we do not observe a large mixed boundary layer at the bottom of the lake ? thedouble diffusive staircase is typically undisturbed to within 1 meter of the lake bottom.Perhaps the most compelling evidence in support of the quiescence and one-dimensionalityargument is the observed lateral homogeneity of the large-scale properties, specifically thatof the temperature gradient between 310?323 m. It is unlikely that the temperature gradienthere would be constant within less than 1 percent across the entire basin in all measuredprofiles if localized mixing processes were significant contributors to the vertical transport.The near linearity of the temperature gradient is also consistent with the quasi-steady statecondition and with the physics of molecular diffusion. Lastly, it is remarkable that theflux estimate obtained from this gradient is exactly that estimated by Hyndman (1976) inthe sediments of the lake. This result too is entirely consistent with the quasi-steady statecondition.Implicit in the one-dimensionality argument is the assumption that heating from the basinsides can be neglected in the analysis. The net geothermal heat input to the basin is pro-portional to the total surface area of the bathymetry, and the net integrated heat transportmust be vertically upwards in a symmetric basin. Because we consider only the bottom50 m of the lake and because the lake has a classic fjord-like bathymetry with very steepsides and a relatively flat bottom, we can approximate the longitudinal cross-section below300 m depth as an open rectangle 2 km wide and 50 m high on either side. This impliesthat the side walls of the basin contribute no more than 5% to the total heat budget ofthe lower staircase if the basin sides and bottom are uniformly heated. If we consider shal-lower depths, it may be necessary to include the effects of side-wall heating as the relativeproportion of the influence of the sides of our model box increases.Previous observational studies (Timmermans et al., 2008; Schmid et al., 2010) compare thefluxes calculated by the 4/3 parameterization to that calculated using the gradient in theinterface between layers; this assumes that molecular diffusion is the single dominant trans-port process in the interface and does not account for entrainment that may occur acrossthe interface from rising and falling convective plumes. We are able to obtain two separate(and consistent) measures of the heat flux which rely neither on the assumptions inherentin the flux parameterizations nor on the assumption that interfaces are undisturbed by tur-bulent processes. This allows us to characterize the accuracy of the 4/3 parameterizations45with a confidence that has not been possible in previous studies, and it allows us to test forthe diffusive nature of the interface in a natural system.The instantaneous heat flux from layer-to-layer may not be identical to the large scale tem-porally and spatially averaged flux since laboratory work (Marmorino and Caldwell, 1976)and direct numerical simulation (Carpenter et al., 2012; Noguchi and Niino, 2010b) em-phasize the dynamic nature of double diffusive layering. However, integrated over time orspace, we expect double diffusive fluxes to agree with the steady state heat flux. There-fore, the factor of approximately two that describes the difference in the bin-averages ofthe heat flux parameterization from the steady state flux (Figure 3.11) characterizes theparameterization?s ability to estimate the actual flux. If the quasi-steady state conditionholds, then the vertical structure in the parameterized heat flux reflects a systematic biasfrom the actual flux which could vary with either R? or Ra.This opens the door to a discussion on the correctness of the current double diffusiveflux scaling. There are two possibilities that may be addressed regarding this topic. Thefirst approach assumes the correctness of the scaling Nu = f1(R?)Ran. Originally, n waschosen to be 1/3 because this removes the dependence on H, but more recent work hasshown that other exponents may produce better results (Kelley, 1990). From our observa-tions (Figure 3.12), it is difficult to ascertain whether a different exponent would producebetter results; the exponent may lie anywhere between 0.22 and 0.43, and an exponentof 1/3 would explain the observations as well as any within these limits and over the range1.5 < R? < 2.5.One difficulty of using an exponent other than 1/3 is that this breaks the assumption onwhich the scaling was originally built, that the fluxes should be independent of H. This isan inconsistency which needs to be explained if the scaling continues to be used. However,regardless of which exponent is chosen for the above scaling, a more fundamental question ishighlighted by our observations. Notice that the vertically averaged data in Figure 3.12 donot randomly scatter about a linear regression; rather, they seem to form a convex shape iftraced from deepest to shallowest. This implies that the observations systematically deviatefrom the proposed power law scaling and no constant value of n would be able to capturethis trend. This systematic deviation from a power law is the reason for the observed biasin the vertically averaged heat flux parameterization shown in Figure 3.11. The inability ofthe current scaling to account for this bias indicates that perhaps a new scaling may betterrepresent the physics of the phenomenon. This is the second possibility which should beconsidered when discussing the correctness of double diffusive flux parameterizations.464.4 New Scaling ObservationsWhile we do not fully develop a new theoretical scaling here, we at least suggest that anew scaling may be appropriate. In the original development of the current scaling, Ra waschosen as the governing parameter in Equation 1.8 in analogy to single-component thermalconvection. However, this is partially arbitrary as R? may equally well have been chosen asthe governing parameter (Turner, 1973). In hindsight, it may be more reasonable to chooseR? as the governing parameter because this directly incorporates the stabilizing effect ofthe salt gradient into the scaling; by choosing Ra as the governing parameter, the effect ofthe salt gradient on the heat flux must always be included through an empirical functionof R?. Furthermore, Ra can vary strongly in a particular system while R? typically variesonly over a much narrower range. Even in a simple system such as Powell Lake where thesteady state heat flux can be taken as constant, the Rayleigh number varies by nearly threeorders of magnitude because of the observed variation in H (Figure 3.12).An alternate plausible form for Equation 1.8 with R? as the governing parameter isNu ? R?1? (4.1)and we find that this relation is remarkably successful in describing our observations (Figure4.1). The observed dimensionless proportionality constant is m = 29, so that from ourobservationsNu = 29R?1? (4.2)It is natural to see some scatter in the individual layer/interface measurements (open circles)because we use the steady state flux to calculate Nu even though the individual interfacefluxes are at steady state only when averaged over a number of measurements. However, wedo expect the vertically binned averages (open squares) to reflect the steady state flux, andthese follow Equation 4.2 remarkably well, much better than in the conventional Nu?Rascaling. Compare Figures 3.12 and 4.1.A scaling of the form Nu ? R?1? , if verified by other observations, has a number of theo-retical advantages over the Nu?Ra scaling. Firstly, it directly accounts for the stabilizingeffect of the salt gradient. It is possible to see this more clearly if, following Turner (1973),we define a salinity Rayleigh numberRs = g??SH3??T(4.3)which characterizes the negative buoyancy of the salinity contribution. ThenR? = Rs/Ra (4.4)471/R?NuR = 0.99m = 290.1 0.2 0.3 0.4 0.5 0.60510152025Figure 4.1 ? Parametric plot in depth showing that Nu ? 1/R? in our observations. Therelationship shown here is evidently much stronger than that of the traditional scalingNu ? Ra1/3 which is given in Figure 3.12. Shown also are the correlation R between thedepth averaged bins as well as the slope m of the least squares fit to the binned data.and Equation 4.1 becomesNu ? Ra/Rs (4.5)Incidentally, it is now easy to see that a scaling of this form would not necessarily requirean empirically determined function similar to f1(R?). It already inherently includes twophysically relevant non-dimensional parameters.Secondly, a scaling of the form Nu ? R?1? is independent of variations in the individual layercharacteristics and allows for flux calculations based solely on the large-scale temperatureand salinity gradients. In order to see this, we rewrite Equation 4.2 by substituting thedefinitions of the dimensionless variables and keeping the proportionality constant m in itsgeneralized form:HFH?cp?T?T= m ??T??S (4.6)This can in turn be rewritten asFH?cp?T1(?T/H) = m??(?T/H)(?S/H) (4.7)If the background stratification is fixed, only the ratios ?T/H and ?S/H are importantin determining the shape of the temperature and salinity profiles. However, notice thatdividing smooth temperature and salinity profiles into steps of arbitrary size leaves theseratios unchanged (Figure 4.2). Consequently, variations in individual layer properties donot affect Equation 4.1 as long as the background stratification does not change.48TzH4?T4?T2H2?T1H1?T3H3Figure 4.2 ? Schematic depicting the invariance of ?T/H. Dividing a fixed backgroundtemperature profile (thick black line) into steps of arbitrary size leaves the ratio ?T/Hunchanged. In this example, the value of ?Ti/Hi is the same for i = 1, 2, 3, 4. Thedescription for a fixed salinity profile would be analogous.Isolating FH by rearranging Equation 4.6 givesFH = ?cp?T?THmR?(4.8)Exploiting the demonstrated invariance of Equation 4.1, the discrete ratios become deriva-tives in the limit as H, ?T and ?S become small:?TH ???T?z and R? ?? Ro?Here, the vertical derivative is now that of the large-scale temperature profile. With thisinformation, the expression for the double diffusive heat flux becomes remarkably similarto that of a regular diffusive flux, only scaled by m and R o? :FH = ?cp?T?T?zmR o?(4.9)To make this more clear, we can define an effective diffusion heat flux in analogy to singlecomponent molecular diffusion:FHM = ?cp?T?T?z (4.10)The double diffusive heat flux is then simply given byFH = FHMmR o?(4.11)Notice that Equation 4.9 relies only on the large scale properties of the background strat-ification and not on the individual layer characteristics of the double diffusive staircase.49However, the above analysis leading from Equation 4.1 to Equation 4.11 is only applicablefor the range of R o? over which double diffusion is observed. In the limits R o? ? 0 andR o? ??, the analysis is no longer meaningful because double diffusion is replaced by othertransport processes.One further test of the relation proposed should be possible from our observations by cal-culating the heat flux implied by Equation 4.9 in the higher section of double diffusivelayering between 301?309 m (Figure 3.2). In order to be consistent, the relation would needto calculate a heat flux that is roughly the same as the steady state heat flux estimated inthe lower staircase. However, in practice this is difficult because the double diffusive regionbetween 301-309 m consists of only a small number of layers. Because the vertical heightof the staircase is only a few times larger than the scale-height of a layer, the backgroundtemperature gradient has relatively large transition zones at the edges of the double diffu-sive zone. Furthermore, the layers are not observed consistently across the basin, making itdifficult to determine where the double diffusive instability is present. Using values?T?z = 0.0042?C/m and R o? = 2.75 (4.12)averaged over the staircase between 302?307 m, Equation 4.9 calculates a heat flux of26 mW/m2. While this value is remarkably close to the heat flux estimated in the lowerstaircase, we emphasize that it should be viewed with caution and only as a rough estimatebecause of the inherent difficulty of defining the background temperature gradient.Many open questions remain, but the obvious next step in pursuing this topic would be acomparison with datasets from other locations such as Lake Kivu and the Arctic. If therelationship given by Equation 4.9 holds for observations of double diffusion in these loca-tions, there would be sufficient evidence to encourage further observational and laboratorybased work. The meaning of the proportionality constant m remains a mystery, and it isunclear from one dataset whether it is dependent only on the physical properties of thelake or seawater solution (that is ? and Pr) or also on other parameters. However, ourobservations suggest that m should be independent of R? and Ra.504.5 Interface Characteristics - Temperature GradientThe discrepancy between the interfacial temperature gradient we calculate and the tem-perature gradient that should be expected from molecular diffusion (Section 3.3) can beexplained in two ways. The first explanation is that ?t as we define it does not accuratelyrepresent the maximum gradient in the interface. Because we calculate ?t from the bulkinterface parameters ?t and hT (rather than trying to estimate the maximum gradientsomewhere within the interface directly), we in fact average the gradient over the entireinterface.Using this approach, it is likely that we underestimate the maximum interfacial gradientsince there may be boundary layers at the edges of the interface. However, the boundarylayers at the edges of the interface would need to comprise a sizeable proportion of thetotal interface height in order for us to underestimate the maximum gradient by more thana factor of two. If we take the measured value of ?h to be representative of the size ofthe boundary layer, then on average the two boundary layers (one on either side of theinterface) together comprise only about 9% of the total interface height. A boundary layerof this size would likewise bias our gradient calculation by approximately 9%. Visuallycomparing the calculated gradient to the observed shape of the interface verifies that theboundary layers are often small, that the layer-finding algorithm correctly identifies thelayer/interface boundaries, and that the gradient calculated from bulk interfacial parametersis largely representative of the actual interfacial gradient (Figure 4.3). While there may yetbe a small portion within the interface in which the gradient is large enough to support thesteady state flux by molecular diffusion (a diffusive ?core?), it would need to be substantiallysmaller than the overall size of the interface to be consistent with our observations.In addition, invoking the argument that our method of calculating ?t dramatically under-estimates the maximum interfacial gradient does not account for the observed correlationbetween the strength of the gradient and the stability of the interface. We find that as theinterface becomes more stable with increasing R?, the strength of the mean temperaturegradient in the interface approaches that which would be expected from molecular diffusion(Figure 3.9). This leads to the second explanation which can be invoked to explain thedifference between the expected and observed interfacial gradients.Lab experiments by Marmorino and Caldwell (1976) find that, while for high values of R?the measured heat flux across a double diffusive interface is that which would be expectedfrom molecular diffusion, at low values of R? the measured heat flux becomes larger thanthe diffusive flux. The authors quote that at R? = 2 the net vertical heat flux is 2.5times the diffusive flux. Likewise, the lower-than-expected interfacial temperature gradientwe calculate in Powell Lake can be explained by allowing for interfacial transport that isnot strictly due to molecular diffusion. Under this interpretation, at low density ratios519.329  338.5  9.344  342.9  9.332  339.4  9.320  335.1  9.337  339.7  9.339  340.9  9.320  335.1  9.336  339.5  9.338  340.5  9.332  339.5  9.334  340.2  9.306  332.0  9.324  337.3  9.327  338.2  9.345  343.6  9.302  331.0  9.319  335.9  9.326  338.0  9.329  338.9  9.324  337.1  9.322  339.2  9.321  337.4  9.327  339.3  9.342  343.1  9.319  337.4  9.329  338.1  9.337  340.6  9.340  341.4  9.298  328.8  9.325  336.6  9.304  330.3  9.332  337.4  9.342  341.1  9.326  336.8  9.337  340.2  9.299  329.8  9.335  340.1  9.337  341.2  9.345  343.3  9.319  335.7  9.331  339.8  9.314  333.8  9.316  334.2  9.336  341.2  9.340  342.0  9.342  342.4  9.314  333.7  9.319  335.4  9.322  336.6  9.307  331.8  9.317  335.0  9.328  336.8  9.349  344.2  9.339  342.1  9.315  334.0  9.334  338.8  9.334  340.5  9.331  342.2  9.316  333.5  9.323  336.0  12 3 4 5 6ABCDEFGHIJFigure 4.3 ? Sixty examples of the double diffusive interface in which the calculatedtemperature gradient (black line) is lower than the expected steady state gradient of0.047 ?C/m (thin grey lines in the background). The actual interface measurements aregiven in blue and the edges of the interface as chosen by our algorithm are shown by opencircles. In Panels A1-I6, the algorithm accurately calculates the observed gradient. InPanels J1-J6, the algorithm underestimates the interfacial gradient. The vertical rangeon all panels is one meter, and the horizontal range on all panels is 0.006 ?C. The lowerbound temperature (?C) and lower bound depth (m) are given in the top right corner ofeach panel.52slightly more than half of the interfacial transport is due to a secondary process active inthe interface. This is evident because the observed gradient is slightly less than half thevalue dictated by molecular diffusion. Molecular diffusion remains important, but at lowdensity ratio, it is no longer the sole mechanism transporting heat from one layer to thenext. Thus our results are in excellent agreement with those of Marmorino and Caldwell(1976), and to our knowledge, there have been no laboratory experiments that contradictthese findings.The above interpretation naturally explains the strong linear relationship between ?t andR? depicted in Figure 3.9. As the interface becomes less stable with decreasing R?, anincreasingly higher proportion of the interfacial transport is due to mixing processes withinthe interface or direct interaction between successive layers. Conversely, as the interfacebecomes more stable with increasing R?, convecting layers cease to interact with each otherdirectly and transport becomes dominated by molecular diffusion. Though this result iscontradictory to the practice of estimating double diffusive fluxes by assuming pure molec-ular diffusion across the interface, it should not be a surprising result that as the interfacebecomes less stable, entrainment from adjacent convecting layers across this relatively thindensity barrier begins to become important and affect the dynamics of the interface. Pre-vious studies have considered that rising and falling convective plumes within mixed layersmay entrain heat from one layer to the next as they push or shear against the interface (Pad-man and Dillon, 1987; Schmid et al., 2004, 2010). However, ours is the first study to provideobservational evidence from a natural setting that this may indeed be the case.Keeping in mind the above discussion, there are cases in which our layer-finding algorithmhas difficulty estimating the interfacial gradient correctly, and in these scenarios it willtend to underestimate rather than overestimate the actual gradient in the interface. A fewexamples of this are given in Figure 4.3, Panels J1?J6. Usually the algorithm will obviouslyunderestimate the actual gradient only if the observed interface is disturbed or asymmetric.We have chosen to include these interfaces in our analysis because there is no basis toconclude that a dynamic interface will always be well described by an idealized functionsuch as a hyperbolic tangent or an error function. In fact, if there is a secondary transportprocess in the interface, we would expect to see interface shapes that appear disturbed andwhich do not follow an idealized shape.While it may not be possible to say definitively which of the two interpretations discussedabove is correct, it is certain from our observations that the mean vertical heat flux cannotin general be calculated accurately from the bulk properties of the interface. This findingshould be incorporated into future studies where double diffusive fluxes are estimated fromdirect interface measurements.534.6 Interface Characteristics - Relative ThicknessMicrostructure observations of the double diffusive interface in salinity are extremely scarceor undocumented, so we are unable to perform comparisons with other studies. However,based on a model in which the diffusion rates of heat and salt determine the relative sizes ofthe temperature and salinity interfaces, it is surprising that the temperature interfaces wemeasure are only marginally larger than the salinity interfaces (Figure 3.6b). If the inter-faces in temperature and salinity grow by molecular diffusion, scaling the one-dimensionaldiffusion equation suggests that the interface heights scale ashT =??T t+ (4.13a)hS =??S t+ (4.13b)where t+ is the time scale of the diffusion. The ratio of the two interface heights is thenindependent of time and given byrh =hThS=??T?S? 10 (4.14)As outlined in Section 1.4, the differing rates at which the two interfaces grow is thoughtto result in unstable boundary layers on either side of the interface which eventually breakaway as convective plumes. The difference in height ?h of the temperature and salinityinterfaces shown Figure 3.6b is then a measure of the boundary layer size. Specifically, it isthe mean combined height of the boundary layers above and below the interface.Using the observed mean interface heights of hT = 18 cm and hT = 17 cm, we find that theinterface height ratio is rh = 1.1 in Powell Lake. This is inconsistent with the predictionfrom Equation 4.14. Though the value rh ? 10 is an upper bound because boundary layersmay be eroded by convection in the adjacent mixed layers, the extremely low value of rhthat we observe may indicate that the size of the interface and adjacent boundary layers isnot controlled solely by molecular diffusion.A possible explanation to resolve the discrepancy between the expected and observed in-terface heights is that there is a semi-permanent background interface of finite thickness ?which is approximately constant (Figure 4.4a). Around this background interface, relativelysmall boundary layers grow because of molecular diffusion, and these small boundary layersbreak away when they become unstable leaving the background interface unchanged. Inthis scenario, the scaling 4.14 for the relative interface heights is replaced by one for therelative boundary layer heights:rb =hT ? ?hS ? ?=??T?S? 10 (4.15)54E ? 14?o cpT h??h/2?h/2??h?h?T?T(a) (b)hThTFigure 4.4 ? (a) A simplified model of the temperature interface based on our observa-tions. There is a semi-permanent background interface of approximately constant height?, shown by the dashed line. Around this background interface, boundary layers grow oneither side, which on average have a height of ?h/2. The average observed interface (thickblack line) then has a height hT = ? + ?h. (b) A typical boundary layer will grow to aheight ?h before an instability sets in and it breaks away from the interface. The thermalenergy ?E per unit area stored in the boundary layer at this time is proportional to thearea of the grey shaded triangle. In both panels the size of the boundary layer relative tothe background interface is exagerated.where rb is the ratio of the diffusive boundary layers in temperature and salinity. In practice,? will be only slightly smaller than hS , and from our measurements the two would beindistinguishable. For example, if hT = 18.0 cm and hS = 17.0 cm, then a value ofrb = 10 would require that ? = 16.8 cm. In the explanation given here, it remains unclearwhat determines the size of ? and this could be the subject of future study, but we willat least show that the model described here is consistent with the estimated heat flux of27 mW/m2.Based on the estimated heat flux and boundary layers that thicken due to molecular dif-fusion, it is possible to infer a scale height for ?h. To begin, consider an interface whoseboundary layers have recently been eroded; the interface now has a height hT = hS = ?.Boundary layers will subsequently begin to form on either side of the interface as the tem-perature interface grows more quickly than the salinity interface. We observe that thecombined height of the two boundary layers is on average ?h; therefore, the temperatureinterface on average has a height hT = ? + ?h. If this is the observed average height, thena typical temperature interface will grow to a size hT = ? + 2 ?h before instability sepa-rates the boundary layer from the interface (Figure 4.4b). The factor of 2 arises becausethe instability should occur at the maximum boundary layer height which will be approxi-mately twice the mean observed boundary layer height averaged over time. Meanwhile, forsimplicity we approximate the salinity interface as constant so that hS = ? always.Now it is possible to estimate the thermal energy ?E accumulating in one of the growing55boundary layers by integrating the temperature profile over the boundary layer. ?E isproportional to the area of the triangle formed in T?z space by the boundary layer (Figure4.4b). This is independent of ?. Approximating the temperature profile in the interface aslinear, the area A of the triangle is 12(?h)(12?T ) and so?E = ?o cpA ?14?o cp?T?h (4.16)The energy accumulated in the interface is also equal to the heat flux into the interfacemultiplied by the time scale t+ of the diffusion:?E ? FH t+ (4.17)Substituting (4.17) into (4.16) and rearranging for ?h gives?h ? 4FHt+?ocp?T(4.18)The time scale, from the one-dimensional diffusion equation, ist+ ? ?h2/kT (4.19)and thus?h ? ?ocp?T?T4FH(4.20)Substituting values from Equation 3.4 along with ?T = 4.0?10?3 ?C and FH = 27 mW/m2yields ?h ? 2 cm. For a simple scaling argument, this is remarkably consistent with ourobservation that ?h = 1.5 cm.As a consequence of the discussion outlined in this Section, we conclude that the overall sizeof the double diffusive interfaces we observe does not appear to be determined by moleculardiffusion alone. This is evidenced by the extremely low observed value of rh. Rather, itcould be that the interface has a semi-permanent background structure of finite size withdiffusive boundary layers that are small compared to the overall interface. It is not evidenthow the size of the background structure of the interface is determined, but it is clear thatscaling the size of the entire interface based solely on the relative diffusion rates of heat andsalt (Equation 4.14) is not the correct approach for describing the interfaces we observe.This conclusion can be taken in conjunction with those of Section 4.5 to further indicatethat the bulk properties of the interface are not necessarily controlled only by moleculardiffusion.56Chapter 5ConclusionThis study reports on observations of double diffusive layering in Powell Lake, BritishColumbia, which contains a 60 m geothermally heated, anoxic layer of relic seawater datingback to the most recent deglaciation. Double diffusion is isolated from turbulent and convec-tive transport mechanisms in the deep, quiescent water, making Powell Lake a unique andideal natural laboratory in which to study the phenomenon. We have obtained basin-widefine-scale measurements of temperature and salinity within the double diffusive staircase,yielding a robust set of layer and interface statistics. Convecting layers tend to be coherenton the basin scale giving them an aspect ratio as large as 104. They tend to have a heightof about 0.5?0.9 m and the dividing interfaces between layers tend to be about 0.1?0.2 mhigh.Powell Lake is a unique system in which to study double diffusion because precise, inde-pendent estimates of the steady state vertical heat flux through the staircase are available.These estimates allow us to test the accuracy of current double diffusive flux parameteriza-tions with unprecidented precision. We find that parameterized heat fluxes are accurate towithin a factor of 2, but deviate from the steady state flux in a systematic manner. We findthat the underlying power law scaling Nu ? Ran does not represent the trend in our ob-servations very well. Consequently, we propose a new scaling of the form Nu ? R?1? whichhas apparently not been examined before, but which much better matches our observations.We also outline that a scaling of this form has theoretical advantages over the traditionalscaling that leads to the current 4/3 flux parameterizations.The combination of fine scale measurements in temperature and conductivity together witha known vertical heat flux allows us to test the properties of the interface between mixeddouble diffusive layers. We find that for low density ratio (approximately R? < 4) theaverage temperature gradient in the interface is less than half that needed to account forthe steady state heat flux by molecular diffusion alone. From our observations it cannot be57ruled out that there is a thin diffusive core somewhere within the interface where moleculardiffusion consistently dominates; however, taking the interface as a whole, it appears thatanother transport mechanism of rougly the same strength as molecular diffusion contributesto the interfacial heat flux. We also find that the strength of the interfacial temperaturegradient correlates to the observed density ratio, which implies that if there is a secondarytransport mechanism through the interface, this mechanism becomes increasingly importantas the density stratification across the interface becomes weaker with decreasing R?.Furthermore, observations of the relative interface thicknesses hT and hS indicate thatmolecular diffusion alone does not set the size of the interface. This is evident becausethe ratio of interface thicknesses rh is close to unity. 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