Dynamics of Atmospheres and Oceans

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1 Dynamics of Atmospheres and Oceans (2013) Contents lists availale at SciVerse ScienceDirect Dynamics of Atmospheres and Oceans journal homepage: A diapycnal diffusivity model for stratified environmental flows Damien Bouffard, Leon Boegman Environmental Fluid Dynamics Laoratory, Department of Civil Engineering, Queen s University, Kingston, ON K7L 3N6, Canada a r t i c l e i n f o Article history: Received 19 Septemer 2012 Received in revised form 24 Feruary 2013 Accepted 25 Feruary 2013 Availale online 14 March 2013 Keywords: Vertical diffusivity Mixing efficiency Field parameterization a s t r a c t The vertical diffusivity of density, K, regulates ocean circulation, climate and coastal water quality. K is difficult to measure and model in these stratified turulent flows, resulting in the need for the development of K parameterizations from more readily measurale flow quantities. Typically, K is parameterized from turulent temperature fluctuations using the Osorn Cox model or from the uoyancy frequency, N, kinematic viscosity,, and the rate of dissipation of turulent kinetic energy, ε, using the Osorn model. More recently, Shih et al. (2005, J. Fluid Mech. 525: ) proposed a laoratory scale parameterization for K, at Prandtl numer (ratio of the viscosity over the molecular diffusivity) Pr = 0.7, in terms of the turulence intensity parameter, Re = ε/(n 2 ), which is the ratio etween the destailizing effect of turulence to the stailizing effects of stratification and viscosity. In the present study, we extend the SKIF parameterization, against extensive sets of pulished data, over 0.7 < Pr < 700 and validate it at field scale. Our results show that the SKIF model must e modified to include a new Buoyancy-controlled mixing regime, etween the Molecular and Transitional regimes, where K is captured using the molecular diffusivity and Osorn model, respectively. The Buoyancy-controlled regime occurs over 10 2/3 Pr 1/2 < Re < (3 ln Pr) 2, where K = 0.1/Pr 1/4 Re 3/2 is Pr dependent. This range is shown to e characteristic to lakes and oceans and oth the Osorn and Osorn Cox models systematically underestimate K in this regime Elsevier B.V. All rights reserved. Corresponding author at: Margaretha Kamprad Chair of Aquatic Science and Limnology, IIE-ENAC-EPFL, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. Tel.: ; fax: address: damien.ouffard@a3.epfl.ch (D. Bouffard) /$ see front matter 2013 Elsevier B.V. All rights reserved.

2 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) Introduction Geophysical flows, such as lakes and oceans, are strongly affected y density stratification. One of the consequences is anisotropic turulent mixing, where the vertical diffusivity is limited y the stratification. The vertical diffusivity is a key parameter in stratified flows as it governs a wide range of processes, ranging from local contaminant distriutions to the gloal heat udget of oceans. Since the first asin-scale mixing estimates in the ocean y Munk (1966), the question of how much mixing occurs is still deated (Ivey et al., 2008). The turulent kinetic energy equation leads to a definition of the vertical diffusivity of density K as the ratio etween the uoyancy flux,, and the vertical density gradient (Osorn, 1980) K = N 2 = g w / 0 N 2 (1) where N = ( g/ o )( / z) is the uoyancy frequency, 0 a reference density and and w are the fluctuating components of the turulent density and vertical velocity fields, respectively. The overar indicates a temporal averaging of the turulent quantities. Attempts to directly otain have een made, ut such estimates remain difficult to interpret and require careful examination due to: (i) the non-stationary and intermittent nature of the turulence, and (ii) the difficulty in separating nonreversile from reversile along-gradient mixing (Moum, 1990; Fleury and Lueck, 1994; Yamazaki and Osorn, 1993; Ivey et al., 2008; Saggio and Imerger, 2001). Given the difficulties in directly estimating, as the ratio etween a scalar flux and its gradient, indirect methods have een developed to parameterize K. In contrast, the turulence intensity, often characterized y the rate of dissipation of turulent kinetic energy, ε, has enefited from spectacular improvement in the accuracy with the development of new sensors and instruments, such as temperature and velocity microprofilers or pulse coherent acoustic Doppler profilers (e.g., Ruddick et al., 2000; Lueck et al., 2002; Wiles et al., 2006; Steinuck et al., 2009). The ojective of this paper is to develop a functional parameterization of K that is easy to implement and use with data collected from instruments commonly deployed in the field. More specifically, we need to correctly calculate diffusivities from temperature microstructure to quantify environmental fluxes, such as vertical oxygen transport under low turulence conditions. The parameterizations will e tested against an extensive set of such data. The remainder of the paper is organized as follows. In Section 2, the different parameterizations proposed in the literature are discussed. Given the lack of validated field parameterizations for K, we focus on existing laoratory and numerical work. We then infer a practical parameterization for field measurements from a new analysis of the laoratory and numerical data (Section 3). Lakes represent an ideal field laoratory to test parameterizations at the oceanic scale (Wüest et al., 1996) and this new parameterization is tested against the existing parameterizations using field data from Lake Erie (Section 4). Limitations of the proposed parameterization are presented in Section Existing parameterizations for K Analytical models and experiments suggest that irreversile diascalar mixing depends on the time evolution of the turulence (Ivey and Imerger, 1991; Barry et al., 2001; Smyth et al., 2005; Shih et al., 2005), the stratification (Gargett et al., 1984; Rehmann and Koseff, 2004; Shih et al., 2005), and the molecular diffusivity ( T 10 7 m 2 s 1 and S 10 9 m 2 s 1 for heat and salt, respectively; Stretch et al., 2010; Nash and Moum, 2002). In the following, we discuss the parameterizations resulting from these assumptions Parameterization inferred from the turulent kinetic energy equation The fundamental assumption driving the most commonly used K parameterization is that turulent fluxes are dominated y the motion from the largest eddies and irreversile mixing occurs at a rate consistent with the large-scale turulent production. Developed y Osorn (1980) and extended y

3 16 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) Ivey and Imerger (1991), K is inferred from the turulent kinetic energy, TKE, equation y assuming a stationary alance, where shear production m = + C- and is parameterized y introducing the flux Richardson numer, a measure of mixing efficiency ( ), as R f = /m into Eq. (1) giving K = R f 1 R f ε N 2 (2) Based on laoratory experiments, Osorn (1980) suggested R f 0.17 and = R f /(1 R f ) 0.2. Although these conditions, with shear production as the sole turulence source and all other terms neglected, are rarely met in the field, it is often assumed R f = 0.17 (Lorke, 2007; MacKinnon and Gregg, 2005); thus potentially grossly overestimating K (Barry et al., 2001; Ivey et al., 2008). Using a comination of laoratory data and scaling arguments, Ivey and Imerger (1991) argue that R f can e characterized y three turulent length scales: the Ozmidov scale, aove which turulent eddies are inhiited y stratification, L O = (ε/n 3 ) 1/2 ; the Kolmogorov scale, which represents the size of the smallest eddies in the momentum field, L k = ( 3 /ε) 1/4 ; and the centred displacement scale, L c, the scale of the most energetic overturns. These length scales may e comined to give two nondimensional numers, the turulent Froude numer, Fr T = (L O /L C ) 2/3 and the turulent Reynolds numer Re T = (L C /L K ) 4/3. Here, R f is non-zero only when L C is located in the andwidth of overturning turulent scales, which is ounded y L O and L k. Beyond this range, where 0 R f 0.2, uoyancy and/or viscosity suppress the turulence. The Ivey and Imerger (1991) parameterization can also e written in terms of a third nondimensional quantity, the turulence intensity parameter Re = ε/(n 2 ) = (L O /L K ) 4/3, which is the ratio etween the destailizing effect of turulence to the stailizing effects of stratification and viscosity or, more importantly, a ratio etween the stratification damping timescale 1/N to the timescale for a turulent event to fully develop ε/ and induce mixing (Ivey et al., 2008). The Re parameter has een used in the literature in various forms. For example, Gargett et al. (1984) and Bluteau et al. (2011) found that the flow is isotropic at high Re. At lower Re, Gison (1980) suggested that 0 when ε N 2. Ivey and Imerger (1991) set this limit, at which uoyancy and viscosity suppress turulence, to Re = 15. This commonly agreed upon value comes from an aritrary average of laoratory estimates, which range etween 8 and 21. However, Huq and Stretch (1995) argued that the vanishing uoyancy flux has no special meaning since oscillations in the uoyancy flux are a natural feature of stratified turulence. Recently Shih et al. (2005, hereafter SKIF) performed direct numerical simulations (DNS) of a homogeneous sheared stratified fluid with Prandtl numer, Pr = 0.7 (air). SKIF confirmed earlier work (Ivey et al., 2000; Barry et al., 2001; Jackson and Rehmann, 2003; Rehmann and Koseff, 2004), that Re is a good parameter to estimate K over its geophysical range where K = C 1 Re n (3) and C 1 and n are constants SKIF distinguished three mixing regimes: a Molecular regime Re < 7, an Intermediate regime 7 < Re < 100, and an Energetic regime Re > 100. The Osorn (1980) model was found to e valid in the Intermediate regime with R f = 0.17 (Fig. 1 in SKIF); however, the model ecomes inappropriate in the Molecular and Energetic regimes, where R f is dependent on Re. SKIF suggested a 0.5 power law (Eq. (3)), as a est fit to the data in the Energetic regime. However, the existence of the regime is controversial as the SKIF DNS were not stationary (Gregg et al., 2012). In the Molecular regime the diffusivity was set to its molecular value. Another issue that arises at the limit etween the Molecular and the Intermediate regimes, as defined in SKIF, is the Pr dependent discontinuity in the parameterization. When applied for temperature in water, K = 0.2Re = = 10 T, resulting in an order of magnitude difference in diffusivity on either side of the transition. This is prolematic when the SKIF parameterization is applied to compute K in weakly turulent thermally stratified geophysical flows.

4 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) K ρ Jackson and Rehmann, 2003, S Jackson and Rehmann, 2003, T Martin and Rehmann, 2006, S Martin and Rehmann, 2006, T Rehmann and Koseff, 2004, S Rehmann and Koseff, 2004, T BIWI, 2001, S Smyth et al., 2005, S Smyth et al., 2005, T Rohr et al., 1985, S Stillinger et al., 1983, S Ivey et al., 1998, S Itsweire et al. 1986, S Re Fig. 1. Total scalar diffusivity of the data used in the present paper. DNS data from SKIF have een omitted here. S and T denote salinity and temperature as the active scalar, respectively. DNS data from Smyth et al. (2005) are an average of the decaying turulence in runs Parameterization inferred from the ratio of the turulent length scales Smyth et al. (2001; hereafter SMC) argue from DNS that the ratio L O /L T is an indicator of the age of the turulence and increases with time following the onset of turulence within a flow. Rather than parameterizing mixing with Re, SMC define the mixing efficiency ( = 0.33 L O L E 3/4 L T 1/4 ) 0.63 ( LO ) L T where L E and L T are the Ellison and Thorpe scales, respectively. SMC found a 60% increase in the diffusivity with Eq. (4) compared to the classic Osorn method (Eq. (1)). Contrary to SKIF, the SMC model suggests the unique existence of a single K regime. (4) 2.3. Parameterization inferred from the variance of the temperature fluctuation equation Similar to Eq. (2), ased on TKE udget, the Osorn Cox model (Osorn and Cox, 1972; hereafter OC) used a scalar udget to relate the vertical diffusivity to the rate of suppression of the scalar variance, expressed for heat under isotropic conditions as T = 6D( T / z) 2, leading to T K = 2( T/ Z) 2 (5) The OC method is widely used in oceanography Parameterization inferred from the scalar properties Jackson and Rehmann (2003, hereafter JR) and Rehmann and Koseff (2004, hereafter RK) conducted laoratory experiments over 10 Re 10 6 for fluids with Pr = 7 and 700 using a towed grid (RK) and an oscillating com (JR). These studies found K to e a function of Re and showed that the traditional Osorn model (Eq. (1), e.g., linear dependence on Re ) was only valid for Re < O(100); aove which an Energetic regime formed where R f decreases and a 0.6 power law (Eq. (3)) is a etter fit to the data (RK). In general, the laoratory and numerical research aove (Section 2.1) show agreement for K p Re n, with n < 1 at high Re and n = 1 at low Re. However, the form of the parameterization for the

5 18 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) diffusivity at low turulence intensities, which are aove Molecular yet elow Intermediate regimes, remains unknown for all Pr. The Re parameterization, presented aove (Section 2), is ased on the traditional assumption that at some high Re, the turulent diffusivity is independent of Pr. However, it has long een known (Turner, 1968) that fluids with different Pr have different entrainment or mixing rates, particularly at high gradient Richardson numer (Ri, representing the ratio etween the stailizing effect of the stratification to the destailizing effect of the shear), which suggests a Pr dependence of diffusivity with turulence intensity Re. In oscillating grid experiments Turner showed that at high Ri (or low Re ), the ehaviour of the entrainment coefficient (E) versus Ri follows a 3/2 power law for salinity and an inverse power law for heat. This implies that R f = ERi, is dependent on the fluid properties at high Ri. In the temperature stratified case (Pr = 7), R f would e constant (the Osorn model), ut a slope Ri 1/2 is expected in the salinity stratified case (Pr = 700). Turner s experiment can also e expressed as Re = Re/Ri with 90 < Re < 270 (Nash and Moum, 2002) and therefore suggests that K p varies with Pr. Laoratory experiments and DNS have also shown differential diffusion, or the ratio d = K S /K T, to depend on Re and to e close to unity when Re > O(100) (Jackson and Rehmann, 2003; Martin and Rehmann, 2006; Smyth et al., 2005). Therefore, from a parameterization viewpoint, we infer the existence of a Pr ased threshold, under which a scalar ecomes less efficiently mixed, with eddies ale to re-sort themselves (i.e., reversile stirring). Below this to e determined threshold, there is some evidence of Pr-dependence of K p, and universally Pr independent turulent diffusivity at high Re. This conjecture is supported in the literature. Martin and Rehmann (2006) found a Pr 0.27 dependence of K p in oscillating grid experiments, as did Stretch et al. (2010), who found Pr 0.5 using DNS (Ri = 1000 and Re < 0.1) and showed that the effect of Pr on mixing is negligile for Ri < 10 and Re > 30. A similar result was oserved y Holt et al. (1992); using DNS, where counter gradient fluxes caused K p to e Pr-dependent for large Ri Parameterization inferred from the turulent Prandtl numer The turulent Prandtl numer, defined as the ratio etween the momentum K m and scalar diffusivities, Pr T = K m /K, has een widely used in atmospheric research (e.g., Louis, 1979; Kim and Mahrt, 1992). Similar to Eq. (2), the turulent kinetic energy equation under stationary conditions K = Ri/(Pr T Ri)Re, where Pr T is modelled as a function of Ri. Here, a linear relationship etween Re and K leads to a constant mixing efficiency and corresponds to the Osorn (1980) model, while other parameterizations, such as a power law relationship, suggest that the mixing efficiency will vary with Ri. Elliot and Venayagamoorthy (2011) have recently compared four parameterizations of Pr T as a function of Ri. Although the four parameterizations lead to different results, the study concludes that only the Venayagamoorthy and Stretch (2010) model Pr T = 1.4e 4.28Ri + 7.8Ri is supported y DNS data for homogeneous staly stratified turulent flows. Using the Venayagamoorthy and Stretch (2010) model, Pr T asymptotically approaches a linear relationship for Ri > 1 and therefore confirms that constant at Ri O(1), while will decrease with decreasing Ri for Ri < 1. The parameterization we present in the following, for use with field data, does not depend explicitly on Ri, ut is physically consistent with the Pr T model. This shows that low Ri (or elevated Re ) is associated with decreasing. The existence of this decrease has een confirmed in direct atmospheric flux oservations (Lozovatsky and Fernando, 2012), ut is disputed in indirect oceanic measurements (e.g., Gregg et al., 2012), which we reanalyze elow and compare to our own oservations from Lake Erie. 3. Towards a general K p parameterization from field oservations 3.1. Methodology The underlying assumption of our parameterization is that Re is not only a good indicator of the state of the turulence in stratified fluids, as descried aove, ut is also ale to descrie the turulent diffusivity over its entire range, for all Pr. The distriution of K p with Re from pulished laoratory and numerical research, as descried aove, is presented in Fig. 1. The scatter in the data, etween

6 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) Tale 1 Summary of laoratory experiments and numerical models used in the present parameterization. Note that 0.1 Pr 0.25 = with Pr = 700. Data used for the final est fit of Eq. (3) are in old. Reference Pr Buoyancycontrolled regime Transition regime Energetic regime Fit R 2 Fit R 2 Fit R 2 Rohr et al. (1984) Re 1.42 Itsweire et al. (1986) Re 1.48 Stillinger et al. (1983) Re 1.57 Ivey et al. (1998) Re 1.56 Barry et al. (2001) Re 0.9 Jackson and Rehmann (2003) Rehmann and Koseff (2004) Re Re Re Re Re Re Re Re Re 0.92 Smyth et al. (2005) Re Re Shih et al. (2005) Re 1.0 Martin and Rehmann (2006) Re 1.34 Best fit with fixed exponent 0.02 Re 1.5 Rounded est fit 0.02 Re Re Re Re Re Re Re Re the different studies, potentially results from slight differences in the measured physical properties and hinders any attempt to fit a single parameterization to all of the data. For instance, flux averaging is different in DNS and in the laoratory experiments. The production term could also e a source of error in the laoratory, as it depends on the stirring mechanism: work done y rods or grids. Detailed discussion on these errors can e found in RK, SKIF or in Stretch et al. (2010) where the connection etween DNS and experimental results is investigated. For these reasons, we separately reanalyze each data set. The data are organized into four mixing regimes (Molecular, Buoyancy-controlled, Transitional and Energetic). Three of these regimes (Molecular, Transitional and Energetic) have een defined in previous studies (Barry et al., 2001; SKIF) and our main contriution here is a reanalysis of these regime models for larger data sets, consideration of Pr effects, and the introduction of a new Buoyancy-controlled regime; thus addressing the discontinuous transition etween the Molecular and Transitional regimes in SKIF. Least-square power law fits of Eq. (3) in each regime, were iteratively estimated y moving the regime oundaries in order to optimize R 2, the correlation coefficient. The quality of the data was checked using a ootstrap confidence interval method. Details on the different fits are presented in Tale 1. Although there is a range of uncertainty etween the exponents in the power law fits, the general agreement supports a parameterization ased on the four regimes descried aove (Fig. 2). The isotropic Energetic regime occurs at very high Re and the Transitional regime is at intermediate Re where stratification starts to affect the isotropy. Depending on Pr, a third Buoyancy-controlled regime can occur when the scalar does not entirely diffuse to irreversile mixing, leading to counter gradient fluxes (e.g., Holt et al., 1992). The fourth, Molecular regime occurs once the turulent diascalar flux ecomes zero (fit not shown in Fig. 2) Energetic regime This regime shows a decrease in mixing efficiency, relative to the Osorn (1980) model, for Re > 100. Our results for Pr = 7 indicate that K p follows a 0.5 power law (Eq. (3)) when Re > 100 (Fig. 2) in agreement with SKIF for Pr = 0.7, in which the lower ound was set at Re = 100 to e continuous with the Transitional regime. The ehaviour of K p for Pr = 700 when Re > 100 is less distinct and can e descried using one-fifth to two-third power laws (Tale 1) or, as in the Pr = 7 and 0.7 case, with a square root power law. Interestingly, the more recent data (JR, RK, SKIF) seem to confirm

7 20 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) K ρ /D Re Fig. 2. Scalar diffusivity divided y molecular diffusivity ( S or T). Note that Pr = 50 in Smyth et al. (2005) data. Symols are the same as in Fig. 1 and the SKIF DNS data have een added (small lack dots). Best fits from Tale 1 are: 2 Pr Re 1/2 (solid line, energetic regime), 0.2PrRe (dashed line, Transitional regime), 0.1 Pr 3/4 Re 3/2 (dotted line, Buoyancy-controlled regime), and (dash-dotted line, Molecular regime). the root-square power law (Fig. 3), which satisfies the Batchelor description of turulence where the scalar variance is independent of the scalar at high Re ; a high Re means a large turulence andwidth with a clear inertial surange (Gargett et al., 1984; Brethouwer et al., 2007; Bluteau et al., 2011). To collapse the Pr = 0.7, 7 and 700 data in the Energetic regime, we use a root-square power law with an exponent of 0.5 to get a est fit relation K p = 2.1 Re 0.5 with R 2 = Our results are in agreement with the SKIF parameterization, in this range, when the factor 2.1 is rounded to 2. Though the n < 1 power law, in the Energetic regime, is clearly shown in laoratory and numerical measurements, this decrease in mixing efficiency at elevated Re is disputed y the field-research community (e.g., Gregg et al., 2012). It is argued that the decrease is an artefact of the turulence production eing reduced when the domain is smaller than L O at high Re and/or the way energy is mechanically driven at fixed scale (i.e., the rod or grid scale) in the laoratory experiments. In oth /2 K ρ /D / Re Fig. 3. Diffusivity of salinity scalar divided y molecular diffusivity. The two lines represent the est fits, K S = 0.1/(Pr 0.25 Re 3/2 ) and K S = 2 Re 1/2.

8 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) Tale 2 Summary of the proposed vertical diffusivity parameterization. The present laoratory/numerical data ased parameterization suggests a transition at Re 100 and a leading factor 2, whereas the field data ased parameterization suggests a transition at Re 400 and a leading factor of 4. * indicates that the value 100 should e replaced with 400 to use the field data-ased parameterization. Although we recommend the present parameterization that includes a decrease in mixing efficiency in the Energetic regime, we include fits for a modified Osorn model where only the Buoyancy-controlled regime is added and the Transitional regime extends through the Energetic regime. Molecular Buoyancy-controlled Transitional Energetic Re < 10 2/3 Pr 1/2 10 2/3 Pr 1/2 < Re < (3 ln Pr) 2 (3 ln Pr) 2 < Re < 100 Re > 100 Range with Pr = 7.0 < < Re < < Re < 100 Re > Modified Osorn Re 3/2 0.2 Re Pr 1/4 0.2 Re Present study (laoratory/numerical data) Present study (field data) 0.1 Re 3/2 Pr 1/4 0.1 Re 3/2 Pr 1/4 0.2 Re 2 Re 1/2 0.2 Re 4 Re 1/2 cases, the outer scale responsile for part of the downward transport (Sun et al., 1996) will not e resolved, causing the mixing efficiency to e strongly underestimated. We find that the SKIF DNS successfully capture L O and L k up to Re O(1000). Similarly, a re-analysis of one run (grid mesh M = 3.81 cm) from Itsweire et al. (1986), using only data with L O lower than 75 percent of the mesh size, also suggests a decrease of mixing efficiency at large Re. We compensate for these restrictions y using DNS data over a limited range of Re. Contrary to these arguments, Turner (1973) reasoned that, in laoratory experiments, stirring mechanisms are primarily driven y internal processes, not y external processes. Moreover, all Pr T ased diffusivity parameterizations, and particularly the most recent one from Venayagamoorthy and Stretch (2010), confirm a decrease of the mixing efficiency at elevated Re (or low Ri). A 0.5 power law at very high Re was also shown y Lozovatsky and Fernando (2012) using direct measurements of fluxes in a stale atmospheric oundary layer. Lozovatsky and Fernando (2012) suggested that the Energetic regime does not support stationary turulence in stratified flows, which would lead to inaccurate estimates when using indirect methods such as OC in this regime. This study theorized that, at elevated Re, the mixing efficiency will decrease as soon as the density gradient has een sufficiently eroded, and the uoyancy flux cannot grow (e.g., ecomes saturated) due to the weak density gradient. The timescale for the uoyancy flux to ecome saturated must e less than that of the large-scale forcing acting to energize the turulence. This condition is satisfied for the field data investigated herein, where progressive internal Poincaré waves provide near-constant aroclinic shear Transitional regime The data in this regime follow the traditional Osorn (1980) model. The K p ehaviour at Re < 100 is dependent on the scalar, where the lower regime limit increases with Pr. Like SKIF (Pr = 0.7), our fit with Pr = 7 also exhiits a linear relation etween Re and K p (Tale 1) and corresponds to a shift from the isotropic Energetic regime towards anisotropy. The lower limit of this regime (where the flow ecomes controlled y uoyancy) was initially defined at Re 15 (Gison, 1980). SKIF found that the relation K = 0.2Re est fit the data (i.e., R f = 0.17, in agreement with Osorn, 1980); however, we find a etter correlation to the data with K = 0.22Re, over 15 < Re < 80 giving R f = 0.18 in this range as a est fit coefficient for the entire data set (Tale 2), which we round to Buoyancy-controlled regime This regime corrects for the discontinuous transition in diffusivity etween the Molecular and Transitional regimes. There is significant spread of data according to Pr over Re < 100 (Fig. 2). The Pr = 700 data suggest an Re 3/2 dependence in this range (Tale 2 and Figs. 2 and 3), where the 3/2 power law means that mixing is less efficient for salinity than for temperature, which still follows the Transitional regime linear law over this range of Re. The limit etween the Buoyancy-controlled and

9 22 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) K ρ /Pr -1/ Re Fig. 4. Diffusivity scaled y Pr 0.25 showing the collapse of the data with different Pr at low Re. Symols are the same as in Fig. 1. Transitional regimes is, therefore, strongly Pr dependent with the Re andwidth of the Transitional regime decreasing significantly for Pr = 700 (Fig. 2). The differing power laws for salinity and heat over Re < 100 are consistent with the development of differential diffusion. Studies on differential diffusion have shown that the ratio d = K S /K T increases with Re up to a limit of d 1 for Re > 100 (Jackson and Rehmann, 2009). Such ehaviour independently confirms the aove choice of a single 0.5 power law (Eq. (3)) in the Energetic regime for all scalars. The question of the transition is addressed y comparing the time scale of decay of a turulent event p O(N 1 ) to the time for a Fourier component to cascade from the Kolmogorov to the Batchelor wavenumer = ln Pr 1/2 1/q(/ε) 1/2 (Nash and Moum, 2002), where the constant q is the nondimensional compressive timescale (Batchelor, 1959). The ratio p / = (q ln Pr 1/2 ) 1 Re 1/2 < 1 can e interpreted as a criterion for differential diffusion to occur. Here, Re (q ln Pr) 2 is the lower limit where the period of a turulent patch, defined as the timescale for energy to cascade to the Batchelor scale, ecomes too short for the scalar to entirely diffuse with respect to the uoyancy period (Nash and Moum, 2002). We apply this criterion to mark the upper ound of the Buoyancy-controlled regime, giving a transition, with Pr = 700, at Re [90 180], where 2.9 < q < 4.1 (Oakey, 1982). Interestingly, for Pr = 7 we find that for Re < [8 16] the stratification and shear timescales ecome important compared to the turulent diffusivity timescale. We adopt q 3 in the following and thus the threshold at Pr = 7 ecomes Re 8.5. The use of q = 5 (Borgas et al., 2004) would shift the oundary to higher Re 25 (Pr = 7) or 250 (Pr = 700), without affecting the validity of the parameterization. The range of q values results from the late alignment of the scalar gradient to the non-constant (oscillating) strain, which affects the efficiency of the compression over the isoscalar surface (Smyth, 1999). Previous work (Martin and Rehmann, 2006; Stretch et al., 2010) has shown a Pr x dependence of the turulent diffusivity at high Ri or low Re. We tested this dependence in the Re 3/2 regime for various Pr and Re < 10; when Pr = 7, Re < 35 when Pr = 50 and Re < 100 when Pr = 700 (Fig. 4). These results suggest the est collapse with 0.11Pr 0.26 Re 3/2. Given the scatter in the data we have rounded the fit to 0.1Pr 0.2 Re 3/2. To verify our parameterization, we plot the parameterized ratio d = (0.1 Re 3/2 )/(0.2 Re ) = 0.5 Pr 1/4 Re 1/2, with Re < 100, and compare it to the d = K S /K T estimates from the literature (Fig. 5). Our parameterization reproduces the experiments with a ratio d = 1 at high Re, a decrease of the ratio with Re up to a plateau, where the two scalars finally follow the same mixing trend. The increase of d at very low Re is not reproduced. The parameterization is also a match (not shown) to the d estimates from Smyth et al. (2005) with Pr = 50.

10 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) d = K s /K T Re Fig. 5. Diffusivity ratio as a function of the turulence intensity. The upper and lower limits of Turner s experiments (dashed dotted lines) according to Nash and Moum (2002), Jackson and Rehmann (2009) data (diamonds), Martin and Rehmann (2006) data (circles), and present parameterization (solid line). The presence of turulent diffusivity at low Re is also supported y theoretical arguments where energy is injected at small scale, such as the equipartition model (Balmforth et al., 1998; Guyez et al., 2007), internal wave wave interactions (Ivey et al., 2000), or quasi-horizontal vortices induced y Kelvin Helmoltz instailities (Heert and de Bruyn Kops, 2006), which predict mixing at turulence intensities much smaller than those predicted with the classical uoyancy flux model Molecular regime When Re ecomes small, all turulence is damped and the flow in this regime ecomes laminar. Here, R f 0 and the vertical diffusivity is, therefore, set to its molecular value. The upper limit of this regime has not een investigated and is simply deduced as the value of Re where the curves for the Buoyancy-controlled and the Molecular regimes meet: 0.1 Pr 0.25 Re 3/2 =. This limit is thus Pr dependent (Re = 10 2/3 Pr 0.5 ) and was calculated to occur at Re = 5.4, 1.7 and 0.17 for Pr = 0.7, 7 and 700, respectively (Tale 2). In previous parameterizations, the transition to the Molecular regime was assumed to occur at Re = 7 for all Pr, ased on data with Pr = 0.7 (SKIF; Ivey et al., 2008). The present parameterizations are summarized in Tale 2, where all the fits have een rounded. More DNS and laoratory data of the same experiments (and ideally for large0 Pr) are needed to etter adjust the coefficients and ranges. Further adjustment is hopeless with the compilation of data considered herein. Our proposed parameterization is surprisingly analogous to the result from Christodoulou (1986), who compiled pulished entrainment data and distinguished three entrainment rates: at low Ri (high Re ), R f R 1/2 i ; at intermediate Ri, R f R i ; and at high Ri (low Re ), R f R 3/2 i. 4. Application of the parameterization to compute the vertical turulent diffusivity from field data We have proposed a revised parameterization for the vertical turulent diffusivity in stratified geophysical flows, which improves the diffusivity estimate at low Re. In lakes (see elow) and oceans (e.g., Oakey and Greenan, 2004), a non-negligile part of the signal occurs at Re O(10) and, therefore, it is crucial to properly estimate the lower end of the turulence intensity signal, so that the average vertical diffusivity will not e underestimated and processes occurring at low turulence levels, such as layering (Balmforth et al., 1998), quasi horizontal vortices induced y Kelvin Helmoltz instailities (Heert and de Bruyn Kops, 2006), or simply differential diffusion processes (Gargett, 2003) may e

11 24 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) Tale 3 Distriution of microstructure measurements during the summers of 2008 and The numer of consecutives profiles (collected at 10 min intervals) is indicated as is the average wind condition during the measurement day. Station coordinates ate: St. 452 N42 34 W79 55 ; St. 341 N41 47 W82 16 ; St. 357 N41 49 W82 58 ; St N41 48 W82 30 ; St N41 48 W82 21 ; St N41 43 W82 17 ; St N41 47 W82 11 ; St. 84 N42 55 W DOY Station Numer of profiles Average daily wind W 10 (m s 1 ) /1227/1228/ /14/14/ /1227/1228/ /8/8/ /1231 2/ /1227/1228/1229/1230 3/4/4/3/ /1227/1228/ /14/10/ /1227/1228/1231 9/14/10/ /1227/1228/1231 7/13/8/ /341/1227/1233 3/3/3/ /1228/1229/1231/ /6/6/5/ /1229/1231 4/12/ /1231 8/ / / /1227/1228 9/20/ /1227/1228/1231 7/10/7/ /1227/1228/1231 9/9/9/ /1227/1228 7/11/ /452 6/ /1227/1228 7/12/ /1227/1228 7/7/ /1227/1228/1231 5/7/8/7 5.9 modelled. To test the proposed parameterizations, elow we apply them to compute the turulent diffusivity from measured field data Methodology During the summers of , an extensive measurement campaign was conducted in Lake Erie. Special attention was given to investigating the recurring prolem of oxygen depletion in hypolemnic water of the western part of the central asin (Rao et al., 2008) and K estimates were required to compute vertical fluxes of oxygen (Bouffard et al., 2012). Here we focus on the results of this campaign, where more than 600 temperature microstructure profiles were collected using a self-contained autonomous microstructure profiler (SCAMP, Precision Measurement Engineering). The profiles were taken over 26 days in July and August mostly in the western part of the Central Basin (see Tale 3); at each station, the oat was anchored for approximately one hour, allowing for 5 12 microstructure profiles, depending on the water depth. Relatively calm wind conditions were required to successfully perform the casts y hand from the vessels. Wind condition during the measurements is indicated in Tale 3. With the exception of the example profiles in Fig. 6, our collected turulent measurements ( ε,, N 2...) were isopycnaly averaged over each one hour measurement interval. Bouffard et al. (2012) found that the Poincaré wave, with a period of 17 h, was the dominant circulation process in Lake Erie and that the dissipation in the water column resulted from Poincaré wave instaility. Our 600 casts sampled all phases of the Poincaré wave and all stages of instaility growth and decay. Moreover, the 1 h cast averaging is too short to e affected y any mean change in turulence associated with the Poincaré wave activity. The SCAMP was deployed to freefall through the water column at a velocity of 10 cm s 1 and sampled water temperature at 100 Hz with a time response of 7 ms, thus theoretically enaling records of

12 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) a) Depth (m) (m) Depth ) T ( o C) or dt/dz ( o C m -1 ) N 2 (s -2 ) L (m) ε (m 2 s -3 ) Re K ρ (m 2 s -1 ) Fig. 6. Examples of temperature microstructure profiles and associated computed data recorded with the SCAMP on 6 August 2009 (a) and 7 August 2008 (). The thermocline region (shaded) exhiits a wide range of Re. Dashed line, dashed dotted line and line are for L k, L O and L c, respectively. temperature microstructure fluctuations as small as 1 mm (2 < L B < 20 mm). The rate of dissipation of turulent kinetic energy was computed y fitting the measured temperature gradient spectra to the theoretical Batchelor spectrum using 25 cm segments. No significant difference was found using larger segments; as already investigated y Steinuck et al. (2009) showing that the enefit from increasing the spectral length on noise reduction and low wave numer contriutions does not vary much for segments larger than 25 cm. The quality of the fit to the theoretical Batchelor spectrum is tested using the maximum likelihood spectral fitting method y Ruddick et al. (2000) and Steinuck et al. (2009). Several strong limitations, however, arise from the use of a temperature microstructure proe. Gregg and Meagher (1980) have shown that a careful caliration and correction of the sensor is needed to use data from the temperature spectrum after 30 Hz. This limitation is severe in oceanic field measurements, were devices are typically free falling or towed at aout U = 1 m s 1, and as a consequence, data are doutful past k = 2fU 1 = 200 cpm leading to a very narrow range where the Batchelor spectrum can accurately e fitted. However, the free falling SCAMP at 0.1 m s 1 allows us to extend this range up to 2000 cpm. We are therefore confident we record dissipation witha10% accuracy (one of our rejection criteria) in k up to ε = m 2 s 3, ut higher dissipation data should e considered with more caution. We have estimated ε within aritrary 25 cm segments, ut the turulent diffusivity parameters were computed over segments of varying size defined y L c (Fig. 6). This insures our segments are of similar size as L O and we do not underestimate the outer scale responsile for part of the downward transport (Sun et al., 1996). We infer the turulent diffusivity using several parameterizations: our parameterization, the SKIF parameterization, the Osorn method (1), the OC method (2), the Re T Fr T diagram from Ivey and Imerger (1991) as parameterized y Ivey et al. (1998) and the ratio L O /L T as parameterized y SMC (5).

13 26 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) Frequency of occurence Re Fig. 7. Distriution (%) of the turulence intensity in summer 2008 and 2009 in Lake Erie. Also shown are distriutions of data collected in the hypolimnion (solid line) and the surface and ottom oundary layers (dashed line) Results Two examples of temperature microstructure profiles recorded with the SCAMP are shown in Fig. 6. The data were collected at the same location on 7 August 2008 and 6 August 2009, in the western part of the Central Basin of Lake Erie. Both profiles show two-layer stratification with a strong seasonal thermocline. Inter-annual climate variaility explains why the thermocline is at 10 m in 2008 and 16 m in 2009, on the same date (e.g., Conroy et al., 2010). The thermocline in Fig. 6 spans from 8 to 13 m depth and turulence in the thermocline is the result of an east wind during the previous days. The wind was weaker on 7 August 2008, progressively veering north. Both temperature and temperature gradient profiles indicate overturns in the surface layer (5 m), at the top of the thermocline (8 m) and in the core of the thermocline (10 m). We estimated the maximum dissipation in thermocline to e m 2 s 3 with Re 60 and K m 2 s 1. Except for this turulent patch, most of the Re values in the thermocline were etween 0.2 and 25 with L k L c. The deep thermocline in 2009 (Fig. 6a) is not distured y the surface layer, as in the previous example, ut rather y the neary ottom oundary; instailities were oserved just eneath the thermocline. In this example, the inner thermocline is less turulent with 0.6 < Re < 15 and thus < K < m 2 s 1. These two examples highlight the strong need for a parameterization that is effective for strongly stratified turulence conditions Turulence distriution A histogram of Re for the entire data set collected over the two summers (Fig. 7) shows the segments to e spread over the four regimes identified in Tale 2. The distriution has a geometric mean at Re = 39 with 28% of the data in the Transitional regime and 14% of the data in the Buoyancy-controlled regime (2< Re <10). In other parameterizations (e.g., SKIF) this range is erroneously assumed to cause mixing at molecular rates. Note that the negative kurtosis in the distriution (road peak) is due to water column averaging, which includes oth turulent (e.g., surface layer and ottom oundary) and non or weakly turulent (e.g., hypolimnion) segments The histogram is consistent with the Re histogram calculated y Steinuck et al. (2010) using 450 profiles collected in the Red Sea during summer We found that the distriution, in Lake Erie, is skewed towards low Re values, as lakes are less turulent than coastal oceans. The Re distriution in Fig. 7 is the superposition of two distriutions, centred around 20 and 90. This is due to segments eing from different regions in the water column with low and high Re in the thermocline and ottom oundary layer, respectively. The average diffusivity over the water

14 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) Tale 4 Average K estimate (m 2 s 1 ) for Lake Erie computed using 5 the different parameterizations. Parameterization Average K estimate (m 2 s 1 ) Osorn (1980) Osorn Cox (1972) Re T Fr T diagram (1991,1998) SMC (2001) SKIF (2005) Present column is m 2 s 1 (Tale 2). This value varies y one order of magnitude, depending on the K parameterization (Tale 4). The reasons for these differences are discussed elow Average turulent diffusivity estimate We compare diffusivity estimates using the present parameterization to those from pulished models in Fig. 8. The lack of a Buoyancy-controlled regime in the SKIF parameterization leads to a 9% difference in average diffusivity compared to our parameterization. This represents the relative importance of low turulence intensities driving weak yet irreversile mixing in lakes. The relationship etween our parameterization and the Osorn model (Eq. (2)) with = 0.20 shows the Osorn model to overestimate the diffusivity in the Energetic and the Buoyancy-controlled regimes. A decrease of the mixing efficiency from = 0.20 to 0.16 and would provide the same average diffusivity and in hindsight could explain the adoption of = 0.15 in many field investigations (Ravens et al., 2000; Wüest et al., 2000; Lorke and Wüest, 2002; Preusse et al., 2010). Given the frequency of Re < 10 and Re > 100 (Fig. 7), universal application of the Osorn model is not recommended. The OC model (Fig. 8) slightly overestimates the mixing efficiency at elevated Re, ut more prolematic is the large scatter (i.e., wide confidence interval) in the OC estimate at low turulence intensity. Uncertainty in estimating K through this method has already een y Van Atta (1999). Wüest et al. (2000) suggested, however, that the OC model is adequate; ut that study only considers segments with Re > 20. As OC is perhaps the most commonly applied method to compute K from microstructure oservations, we investigate the evolution of the mixing efficiency as a function of Re. In Fig. 9, the range defined y the standard deviation of Ɣ OC is shown, using only segments in which ε < m 2 s 3, to insure a 10% accuracy in the k estimate. This constrains and strongly reduces the tendency of the OC method to overestimate mixing efficiency at elevated Re. Our data clearly indicate an increase in mixing efficiency over Re 1 15, with a est fit of OC = (0.028 ± 1.5)Re 0.54± % confidence interval), a plateau until Re 200 with OC = (0.18 ± 0.57)Re 0.02±0.30 (+represents the (i.e., the Osorn (1980) region), followed y a decrease when Re > 400 with a est fit of OC = (6.837 ± 2.3)Re 0.57±0.09 (i.e., a 0.43 power law for the vertical diffusivity). The OC model, applied to our field data, is qualitatively consistent with our suggested parameterization in the 3 regimes. However, the mixing efficiency in the Energetic regime decreases much faster in our parameterization (Tale 1), than in the est fit using the OC method. There has een considerale discussion in the literature (e.g., Kunze, 2011; Gregg et al., 2012) on the accuracy of data purporting to show the existence of decreasing mixing efficiency with increasing Re (i.e., an Energetic regime). We, therefore, also extend the Transitional regime parameterization (i.e., = 0.2) through the Energetic regime (Fig. 9). The corresponding line diverges from the la-scale data used in our parameterization, the est fit to the OC method applied to our field data and oceanic field data from Gregg et al. (2012); clearly indicating that the Osorn (1980) model significantly overestimates mixing efficiency for Re > 400. Recall that we have removed parameterization data where the domain is smaller than the Ozmidov scale and there is potential for grid/ar-scale contamination. Artificial inclusion of red and white noise (from electronics and internal waves, respectively) to our turulent segments decreases OC at low Re ut increases OC at high Re, showing that these trends

15 28 D. Bouffard, L. Boegman / Dynamics of Atmospheres and Oceans (2013) a) ) K ρ (O) K ρ (OC) K ρ (this study) K ρ (this study) c) d) K ρ (IIK) 10 5 K ρ (SMC) K ρ (this study) K ρ (this study) Fig. 8. Comparison of the different pulished parameterizations (O, Osorn (1980); OC, Osorn Cox (1972), IIK, Ivey et al. (1998), and SMC, Smyth et al., 2001) to the present parameterization. Heavy line represents the mean value and shaded area delineates the 95% confidence interval. The thin line shows y = x and the two dashed lines are Re = 10 and 100. White circles in (c) are field data from MacIntyre (1993). Both the Re T Fr T diagram parameterization and the present parameterization are inferred from Tale 1 in MacIntyre (1993). do not result from signal contamination. The decrease in mixing efficiency is less evident when we use all segments (i.e., without the constraint of ε < m 2 s 3 ). In this case the data are more scattered, due to the inaccuracy in dissipation measurements from the sensor limitation in a highly turulent flow and the inaccuracy in the estimate of the temperature gradient required in the OC method in weakly stratified flow, and suggest a wider Transitional regime followed y a steeper decrease of the mixing efficiency. In conclusion the OC method, applied to our data, suggests that the Osorn model 0.2 is valid over a range Re , while lower and higher ranges have smaller mixing efficiencies, as also suggested in the present parameterization. Field measurements suggest a shift in the transition etween the Intermediate and the Energetic regimes from 100 to 400. More work is needed with other instruments (e.g., shear proes) to investigate the case with higher Re and/or stronger turulence levels (oceanic ridges or straits). For this reason, although our data suggest otherwise, we provide oth our parameterization and the Osorn (1980) model for use in the Energetic regime (Tale 2). Our results are also consistent with recent atmospheric data from Lozovatsky and Fernando (2012) who found a 0.5 power law (Eq. (3)) at elevated Re using direct heat flux measurements ut an almost linear