## 1. Introduction

Double-diffusive transport has long been studied by theoretical (Huppert 1971; Turner 1973; Linden and Shirtcliffe 1978; Kelley 1990), laboratory (Turner 1965, 1968; Crapper 1975; Marmorino and Caldwell 1976; Newell 1984), computational (Radko 2003; Noguchi and Niino 2010a,b), and observational (Newman 1976; Padman and Dillon 1989; Schmid et al. 2004; Polyakov et al. 2012; Sirevaag and Fer 2012) approaches. Despite this rich history, many questions regarding the details of double-diffusive transport and layering remain active topics of research, as highlighted by a number of recent studies (Carpenter and Timmermans 2014; Sommer et al. 2013a; Flanagan et al. 2013; Radko et al. 2014).

There are two modes of double diffusion: the salt fingering mode, which may occur when temperature and salinity both decrease with depth, and the diffusive convection mode, which may occur when temperature and salinity both increase with depth. This study focuses solely on the latter, which is observed over vast regions in the thermocline (Padman and Dillon 1987; Timmermans et al. 2008) and in the deep water (Timmermans et al. 2003; Zhou and Lu 2013) of the Arctic Canada basin, may be involved in the formation of Greenland Sea Bottom Water (McDougall 1983), and is actively involved in transports in the Mediterranean and Black Seas (Özsoy et al. 1993; Özsoy and Ünlüata 1997) as well as a number of lakes around the world (Hoare 1968; Schmid et al. 2004, 2010; Wüest et al. 2012).

The diffusive convection mode is often characterized by a thermohaline staircase where smooth large-scale temperature and salinity gradients have broken up into a series of well-mixed, actively convecting layers separated by relatively thin, high-gradient interfaces. Mixed layers tend to have scale heights of about 1 m, and interfaces tend to have scale heights of about 10 cm (Kelley et al. 2003).

One key issue in double-diffusive studies has been to determine the fluxes of heat and salt through staircases and to link these fluxes to the measured staircase characteristics. Early theoretical and laboratory studies resulted in a series of heat flux parameterizations, commonly referred to as the “4/3 flux laws” (Kelley et al. 2003), by which fluxes could be estimated from the temperature difference between consecutive layers. While these parameterizations are generally successful at predicting fluxes to the correct order of magnitude (Padman and Dillon 1987; Schmid et al. 2004; Timmermans et al. 2008), verifying their accuracy to a higher precision has been difficult, and their theoretical basis remains uncertain (Turner 1973; Kelley 1990; Radko 2013).

An alternate method of estimating double-diffusive fluxes, which has been used in a number of recent observational studies (Timmermans et al. 2008; Schmid et al. 2010; Sirevaag and Fer 2012; Sommer et al. 2013a), is to assume that transport through the interface is dominated by molecular diffusion, in which case the fluxes are proportional to the interface gradients. However, it is not obvious that this assumption is always true (Marmorino and Caldwell 1976; Linden and Shirtcliffe 1978), and it has not been verified in a natural setting. In addition, geophysical observations of double-diffusive staircases are often contaminated to an unknown degree by spatial and temporal inhomogeneities, and by shear and lateral advection that may affect the double-diffusive interfaces, further complicating estimates of this “diffusive” flux.

Double-diffusive processes in Powell Lake are largely isolated from other transport mechanisms, and two independent estimates of the steady-state heat flux are possible from measurements of the temperature gradients in the sediments and in a region of the lake where double-diffusive staircases are not present (section 3c). Consequently, Powell Lake provides an opportunity for a case study of naturally occurring double diffusion in which estimates of double-diffusive fluxes are not obscured by complicating turbulent or advective effects, which are often present in open-ocean measurements.

In this study, we describe three sets of observations from this natural laboratory, obtained using both CTD and microstructure measurements, and we characterize the double-diffusive steps in these datasets using two distinct algorithms. Then, using independent heat flux estimates, we test the validity of the assumption that vertical heat fluxes can be estimated by measuring the interfacial gradients in a double-diffusive staircase. We also evaluate the accuracy of the most commonly used 4/3 flux parameterization and objectively compare our measurements to the relationships underlying this parameterization.

## 2. Research methods

### a. Powell Lake

Powell Lake is a meromictic ex-fjord, 50 m above sea level, on the southwest coast of British Columbia, Canada. It consists of six flat-bottomed basins separated by shallower sills (Fig. 1). The southern basin is the deepest of these (maximum depth 350 m), and Williams et al. (1961) discovered that this basin is permanently stratified and contains relic seawater below 150 m, with a maximum salinity of 16.6 g kg^{−1} at maximum depth. The relic seawater is permanently anoxic and from geological considerations is about 11 000 yr old, having been separated from the ocean by continental uplift after the last ice age (Mathews et al. 1970). The basin is geothermally heated (Hyndman 1976), and from early microstructure measurements, Osborn (1973) speculated that the double-diffusive instability may be active in the deep saline layer. However, the details of the temperature and salinity small-scale structure in the lake have not previously been studied. Sanderson et al. (1986) modeled the evolution of the salt and heat budgets of the lake since its formation. While the upper 150 m of the lake were largely flushed in the first 1000 yr, the subsequent flushing of the deep water has been slow (with diffusivities nearly molecular), and a large amount of heat has accumulated in the remaining relic seawater. The model of Sanderson et al. (1986) suggests that relic seawater will continue to remain in the lake for several thousand years.

### b. Measurements

We collected three sets of data for the analysis presented in this paper, consisting of 58 high-quality CTD casts and 32 temperature microstructure profiles (Fig. 1c). The measurements encompass full lateral and longitudinal transects of the flat-bottomed portion of the south basin where double-diffusive layers are observed, with measurement locations no more than 500 m apart. The vast majority of measurement locations were sampled at least twice. There are 1120 individual observations of double-diffusive interfaces from the CTD casts and 779 interface observations from the microstructure casts.

The CTD measurements were collected on 23–25 July 2012 (PL12, 39 casts) and on 3–5 June 2013 (PL13, 18 casts) using a Sea-Bird SBE-25, equipped with an SBE-3F temperature sensor and an SBE-4 conductivity sensor, sampling at 8 Hz. The instrument was lowered through the double-diffusive portion of the water column at a steady speed of 11 cm s^{−1}, yielding a mean vertical sampling resolution of 1.4 cm. All CTD measurements were taken at night when conditions on the lake were calm, and all casts ended within 55 cm of the local sediments.

The finite response time of the sensors results in a small smoothing effect in the temperature and conductivity measurements. The temperature sensor has a response time of 0.10 s, and the salinity cell has a flushing time of 0.16 s; consequently, at a fall rate of 11 cm s^{−1}, instantaneous temperature and salinity changes would be recorded as gradual increases over 1.1 and 1.8 cm, respectively. The observed in situ electronic noise in the CTD temperature measurement is less than 0.5 mK, and the electronic noise in the conductivity cell is less than 0.0005 mS cm^{−1}, resulting in a salinity signal that has an observed in situ electronic noise level of approximately 0.5 mg kg^{−1}. For all measurements, we quote in situ temperature *t* according to the International Temperature Scale of 1990 (ITS-90) and Reference Salinity *S*_{R} according to the International Thermodynamic Equation of Seawater 2010 (TEOS-10) (IOC et al. 2010), assuming a salinity anomaly of zero.

The microstructure measurements (PLM, 32 casts) were collected on 19–22 June 2013 using a Rockland Scientific International vertical microstructure profiler (VMP). The VMP samples at 512 Hz and freefalls at 38 cm s^{−1}, yielding a vertical resolution of 0.74 mm. The observed high-frequency electronic noise from the temperature probe attached to the VMP is less than 0.1 mK. This probe is very sensitive to high-frequency variability, but tends to be subject to lower-frequency drifts that are corrected using simultaneous CTD measurements. The microstructure measurements extended to within 5 ± 1 cm of the sediment surface. The VMP is described in detail by Sommer et al. (2013b).

The VMP also measures conductivity, but the electronic noise in this measurement was too large to clearly identify the salinity layer/interface boundaries, making it unsuitable for the analysis presented here. This problem has not been reported in microstructure measurements of double-diffusive layers in previous studies because the salinity difference between consecutive layers in Powell Lake is nearly an order of magnitude smaller than in many previously studied locations. For example, the salinity difference between layers in Lake Kivu is about 13 mg kg^{−1} (Sommer et al. 2013a) and in the thermocline of the Canada basin is about 14 mg kg^{−1} (Timmermans et al. 2008); in Powell Lake, the salinity difference between layers is only about 1.9 mg kg^{−1} (Table 1).

Statistics of the double-diffusive parameters in Powell Lake. Shown for each variable are the trimmed mean, median, and IQR (25–75 percentiles). The trimmed mean is the mean of the data between the 5 and 95 percentiles. The last column gives the number of interface observations included in each set.

### c. Layer evaluation

To characterize a double-diffusive staircase, we define *H* as the mixed layer height; *h*_{T} and *h*_{S} as the interface thicknesses in temperature and salinity, respectively; and Δ*t* and Δ*S*_{R} as the differences in the average temperature and salinity between consecutive layers (Fig. 2, insets). We characterize the temperature and salinity interface thicknesses separately because the double boundary layer model of the interface (Linden and Shirtcliffe 1978) suggests that molecular diffusion will thicken the temperature interface faster than the salinity interface such that the ratio *r* = *h*_{T}/*h*_{S} can be expected to lie between 1 < *r* < 10. The difference between the mixed layer heights in *t* and *S*_{R} is small (approximately 2%), and so we do not differentiate between these.

The mixed layer and interface detection algorithm ALG1. The measured (a) temperature (red) and (b) salinity (blue) profiles are each shown by the thick colored line. The mean background signal (dashed line) is derived using a 0.75-m low-pass filter, and the difference (*δt* and *δ**S*_{R}, respectively) between the measured and background profiles is shown by the alternately peaking and troughing gray line. Peaks and troughs in *δt* and *δ**S*_{R}, marked by open squares, correspond to the edges of diffusive interfaces, which are marked by open circles on the profiles. Peaks and troughs are most clearly seen in *δt* but are still unmistakably evident in *δ**S*_{R}. The insets depict our definitions for the basic observable parameters *H*, *h*_{T}, *h*_{S}, Δ*t*, and Δ*S*_{R}. Pressure *p* is converted to depth with a factor of 1.0156 m dbar^{−1}.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

The mixed layer and interface detection algorithm ALG1. The measured (a) temperature (red) and (b) salinity (blue) profiles are each shown by the thick colored line. The mean background signal (dashed line) is derived using a 0.75-m low-pass filter, and the difference (*δt* and *δ**S*_{R}, respectively) between the measured and background profiles is shown by the alternately peaking and troughing gray line. Peaks and troughs in *δt* and *δ**S*_{R}, marked by open squares, correspond to the edges of diffusive interfaces, which are marked by open circles on the profiles. Peaks and troughs are most clearly seen in *δt* but are still unmistakably evident in *δ**S*_{R}. The insets depict our definitions for the basic observable parameters *H*, *h*_{T}, *h*_{S}, Δ*t*, and Δ*S*_{R}. Pressure *p* is converted to depth with a factor of 1.0156 m dbar^{−1}.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

The mixed layer and interface detection algorithm ALG1. The measured (a) temperature (red) and (b) salinity (blue) profiles are each shown by the thick colored line. The mean background signal (dashed line) is derived using a 0.75-m low-pass filter, and the difference (*δt* and *δ**S*_{R}, respectively) between the measured and background profiles is shown by the alternately peaking and troughing gray line. Peaks and troughs in *δt* and *δ**S*_{R}, marked by open squares, correspond to the edges of diffusive interfaces, which are marked by open circles on the profiles. Peaks and troughs are most clearly seen in *δt* but are still unmistakably evident in *δ**S*_{R}. The insets depict our definitions for the basic observable parameters *H*, *h*_{T}, *h*_{S}, Δ*t*, and Δ*S*_{R}. Pressure *p* is converted to depth with a factor of 1.0156 m dbar^{−1}.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

We implement two distinct algorithms to automate interface detection in a double-diffusive staircase. First, an algorithm (ALG1) similar to that described by Polyakov et al. (2012) is implemented on all three datasets (Fig. 2). From a measured staircase, the mean background signal is extracted using a 0.75-m low-pass running-mean filter. The difference (*δt* or *δ**S*_{R}) between the actual and background signal oscillates around zero with sharply defined peaks at the edges between layers and interfaces. Therefore, by identifying the peaks in the difference signal, it is possible to identify the locations of the mixed layers and their corresponding interfaces. For PL12 and PL13, we apply the algorithm to the temperature and salinity profiles independently to derive layer and interface properties in both *t* and *S*_{R}.

ALG1 is reasonably robust to changes in the size of the low-pass filter used to derive the background signals. Implementing the algorithm on a representative profile using a 0.50-m filter results in changes to estimates of *H*, *h*_{T}, and *h*_{S} of less than 12%. Using a 1.00-m filter in place of a 0.75-m filter results in changes of less than 5%. Note that results are more sensitive to the 0.25 m decrease because here the size of the filter is smaller than the mean mixed layer height (Table 1) and does not fully remove the small-scale features from the background profile.

To test the validity of the above algorithm, and to highlight its limitations, it is useful to evaluate the staircase characteristics using a second, distinct algorithm. To do so, we implement a different algorithm, ALG2, described in detail by Sommer et al. (2013b), on the microstructure dataset. This algorithm identifies interfaces using three conditions:

The interface gradient is large compared to the background gradient.

The gradients in the two adjacent mixed layers are small compared to the background gradient.

The measured interface appears undisturbed and is close to linear.

*δt*and

*δ*

*S*

_{R}of a continuous staircase. ALG2 identifies 498 of the 779 interfaces captured by ALG1.

Condition iii above is imposed because we use the microstructure measurements to estimate the average heat flux through the staircase by assuming that molecular diffusion dominates transport across the interface (section 3c), but this assumption can only be true if the interface appears laminar and is nonturbulent. While rejecting disturbed interfaces introduces the possibility of artificially biasing the following analysis, the effect is reasonably small. If condition iii is removed, the number of retained interfaces increases from 498 to 561, the mean interface height increases by 8%, and estimates of the mean diffusive flux decrease by 7%. Note that ALG2 is optimized for high-frequency measurements and so can be usefully applied only to the microstructure dataset PLM.

Besides the rejection criterion (condition iii in ALG2), the two algorithms differ most in the following regard: ALG2 uses microstructure measurements to directly estimate the interfacial temperature gradient **∇***t* at the interface center and then calculates the interface height according to *h*_{T} = Δ*t*/**∇***t*. On the other hand, ALG1 uses the bulk staircase properties to first determine layer–interface boundaries and then estimates **∇***t* by effectively averaging across the interface according to **∇***t* = Δ*t*/*h*_{T}. The difference appears subtle but results in substantially different estimates for *h*_{T} and **∇***t* (sections 3b and 4b), highlighting that it is important to consider carefully the correct interpretation when a double-diffusive interface is being evaluated by an automated algorithm.

## 3. Results

### a. Large-scale properties

The south basin of Powell Lake is permanently anoxic below 135 m, which is the maximum depth of convective winter mixing (Fig. 3). The effects of seasonal changes, wind and river forcing, surface currents, and freshwater circulation, which may be seen in the upper layers, are all absent in measurements below this depth, indicating that the deep water is largely decoupled from the overlying freshwater layer. Temperature begins to increase monotonically from 5.1°C at the anoxic boundary to a maximum of 9.3°C at the greatest depth. Salinity begins to increase from near zero at the anoxic boundary to 16.6 g kg^{−1} at maximum depth, with the strongest halocline at depths of 270 to 320 m. These large-scale features of the temperature and salinity structure are uniform across the basin and have remained largely unchanged over the last 50 yr (Fig. 3). The observed lateral and temporal homogeneity is consistent with previous findings that the deep water of Powell Lake is nearly quiescent (Osborn 1973; Sanderson et al. 1986).

Temperature, salinity, and dissolved oxygen profiles from the middle of the south basin at the deepest location in Powell Lake. The basin is permanently anoxic below 135 m. The large-scale properties of all profiles appear homogeneous along the length of the basin, showing little lateral variability, and are consistent over 5 yr of measurements. Lines represent our CTD measurements (July 2012), and symbols represent measurements from Williams et al. (1961). The region in gray highlights the depth-range shown in Fig. 4.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Temperature, salinity, and dissolved oxygen profiles from the middle of the south basin at the deepest location in Powell Lake. The basin is permanently anoxic below 135 m. The large-scale properties of all profiles appear homogeneous along the length of the basin, showing little lateral variability, and are consistent over 5 yr of measurements. Lines represent our CTD measurements (July 2012), and symbols represent measurements from Williams et al. (1961). The region in gray highlights the depth-range shown in Fig. 4.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Temperature, salinity, and dissolved oxygen profiles from the middle of the south basin at the deepest location in Powell Lake. The basin is permanently anoxic below 135 m. The large-scale properties of all profiles appear homogeneous along the length of the basin, showing little lateral variability, and are consistent over 5 yr of measurements. Lines represent our CTD measurements (July 2012), and symbols represent measurements from Williams et al. (1961). The region in gray highlights the depth-range shown in Fig. 4.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Both temperature and salinity exhibit a large-scale steplike structure below 280 m, consisting of five steps with heights of 2–10 m (Fig. 4a). The origin of these large-scale steps is unknown, but they are laterally coherent and their vertical position varies by less than 1 m over 5 yr of CTD measurements (not shown). Note that these large-scale steps should not be confused with the double-diffusive staircases that are found within the larger steps.

(a) Temperature and salinity in the deep layer of relic seawater exhibit a large-scale steplike structure with double-diffusive staircases found within the lowest three large-scale steps (July 2012). The deepest staircase below 330 m is the most well defined and, on close inspection, is just visible in the temperature profile on the scale shown here. The depth range between 315 and 328 m, highlighted in gray, is that used to calculate the temperature gradient in Fig. 8. (b) The background density ratio. Double-diffusive staircases in Powell Lake are observed approximately over the range

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

(a) Temperature and salinity in the deep layer of relic seawater exhibit a large-scale steplike structure with double-diffusive staircases found within the lowest three large-scale steps (July 2012). The deepest staircase below 330 m is the most well defined and, on close inspection, is just visible in the temperature profile on the scale shown here. The depth range between 315 and 328 m, highlighted in gray, is that used to calculate the temperature gradient in Fig. 8. (b) The background density ratio. Double-diffusive staircases in Powell Lake are observed approximately over the range

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

(a) Temperature and salinity in the deep layer of relic seawater exhibit a large-scale steplike structure with double-diffusive staircases found within the lowest three large-scale steps (July 2012). The deepest staircase below 330 m is the most well defined and, on close inspection, is just visible in the temperature profile on the scale shown here. The depth range between 315 and 328 m, highlighted in gray, is that used to calculate the temperature gradient in Fig. 8. (b) The background density ratio. Double-diffusive staircases in Powell Lake are observed approximately over the range

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Double-diffusive staircases are found in three distinct depth ranges: between 298.5 and 303.5 m, 306 and 314 m, and from 330 m to the sediments. The background density ratio *β* ≡ *ρ*^{−1}(∂*ρ*/∂*S*_{R}) = 7.6 × 10^{−4} kg g^{−1} is the haline contraction coefficient, and *α* ≡ −*ρ*^{−1}(∂*ρ*/∂*T*) = 1.3 × 10^{−4} K^{−1} is the thermal expansion coefficient, both calculated at in situ temperature, salinity, and pressure (IOC et al. 2010); *ρ* is the density.

The deepest staircase is very clearly defined and typically extends to within 55 cm of the lake sediments (Fig. 5), indicating that possible boundary effects due to seiching currents are small. The vast majority of profiles exhibit clear, double-diffusive layer–interface boundaries in this staircase, further suggesting that complicating effects of currents or internal waves are small. The shallower staircases appear increasingly more disturbed, making it more difficult to precisely distinguish the layer/interface boundaries. Consequently, the analysis presented in this paper is restricted to the deepest staircase.

(left to right) A south-to-north transect depicting the deep double-diffusive staircase in (a) salinity and (b) temperature (locations marked in Fig. 1c). Each consecutive profile is offset from the previous by 0.010 g kg^{−1} in salinity and 0.015°C in temperature. All profiles extend to within 55 cm of the sediments; consequently, the shallowing of the basin toward the south and north can be seen from the maximum depth of the profiles. The staircase becomes less pronounced as the lake shallows southward.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

(left to right) A south-to-north transect depicting the deep double-diffusive staircase in (a) salinity and (b) temperature (locations marked in Fig. 1c). Each consecutive profile is offset from the previous by 0.010 g kg^{−1} in salinity and 0.015°C in temperature. All profiles extend to within 55 cm of the sediments; consequently, the shallowing of the basin toward the south and north can be seen from the maximum depth of the profiles. The staircase becomes less pronounced as the lake shallows southward.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

(left to right) A south-to-north transect depicting the deep double-diffusive staircase in (a) salinity and (b) temperature (locations marked in Fig. 1c). Each consecutive profile is offset from the previous by 0.010 g kg^{−1} in salinity and 0.015°C in temperature. All profiles extend to within 55 cm of the sediments; consequently, the shallowing of the basin toward the south and north can be seen from the maximum depth of the profiles. The staircase becomes less pronounced as the lake shallows southward.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

### b. Staircase characteristics

For all mean values given below, we quote a trimmed mean calculated from the central 90% of the data. In addition, the median value and interquartile range (IQR) for relevant observables are given in Table 1. When quoting the difference between two mean values, we quote the 95% confidence interval of the difference in the population means.

From the 2012 CTD data, the mean of *H* is 69 cm; the means of *h*_{T} and *h*_{S} are 19 and 18 cm, respectively; and the mean of *r* is 1.2. The mean of Δ*t* is 4.0 mK, and the mean of Δ*S*_{R} is 1.9 mg kg^{−1}. The layer-based density ratio *R*_{ρ} = *β*Δ*S*/αΔ*t* characterizes the stability of an interface and has a mean value of 2.7, with 76% of interfaces having *R*_{ρ} < 3.0. The mean of the interfacial temperature gradient **∇***t*, averaged across the interface, is 23 mK m^{−1}.

The staircase extends nearly the entire length of the south basin (Fig. 5), and the mean characteristics of the staircase (Table 1) from the CTD measurements vary by no more than 16% from 2012 to 2013, highlighting a long-term steadiness of the staircase properties. The difference in the mean of *H* from PL12 and PL13 is 11 ± 3 cm. Likewise, the difference in the mean of Δ*t* is 0.3 ± 0.2 mK.

The variability between the microstructure and CTD measurements (in the same year, both evaluated by ALG1) is less than 14%, indicating that the lower vertical resolution of the CTD profiler does not impart a substantial bias to the observations of the bulk staircase properties. The difference in the mean of *H* from PL13 and PLM (ALG1) is −8 ± 4 cm. The difference in the mean of Δ*t* is 0.3 ± 0.2 mK.

The two algorithms ALG1 and ALG2, applied to the same dataset PLM, likewise produce similar statistics for *H* and Δ*t* (Table 1). The difference in the mean of *H* is −4 ± 4 cm, and the difference in the mean of Δ*t* is 0.3 ± 0.2 mK.

As outlined above, the variability of the statistical measures between algorithms and from survey to survey is always on the order of 10%. This variability is of the same order as that resulting from the sensitivity to user-controlled parameters in the evaluating algorithms (section 2c), and so we do not discuss these differences further.

However, there is a noteworthy difference between the results of ALG1 and ALG2 when estimating the interface thicknesses in the microstructure data. The mean value of *h*_{T} calculated by ALG1 (15 cm) is 50% larger than that calculated by ALG2 (10 cm), with a difference of 5 ± 1 cm. A similar discrepancy in the results of the two algorithms can also be seen from estimates of the interfacial temperature gradient **∇***t*, where the mean value from ALG1 is 26 mK m^{−1} (in agreement with the estimates from the CTD measurements), and the mean value from ALG2 is 43 mK m^{−1}. The difference in the mean values is 17 ± 2 mK m^{−1}.

As outlined in section 2c, this discrepancy arises because the two algorithms characterize the interface differently. By using the gradient at the interface center to subsequently estimate *h*_{T}, the measures obtained by ALG2 are more characteristic of the interface core and are more prone to neglecting any boundary layers that may be present at the edges of the interface [as modeled by Linden and Shirtcliffe (1978)]. On the other hand, the measures obtained by ALG1 are more characteristic of the entire bulk interface properties (including potential interface boundary layers), and the temperature gradient obtained from this algorithm is averaged across the entire height of the interface.

As can be seen from the histograms of *h*_{T} and **∇***t* (Fig. 6), ALG2 calculates a larger proportion of thin, high-gradient interfaces than ALG1. Comparing the width of the histograms also reveals that the gradient at the interface core has a much larger natural variability (IQR, 21–63 mK m^{−1}) than the gradient obtained from bulk interface properties (IQR, 16–35 mK m^{−1}).

Histograms of selected interface characteristics from Table 1. (a)–(c) CTD measurements and (d)–(i) microstructure measurements. Each panel is labeled with the total number *N* of interfaces included in the histogram. Comparing the top row and the middle row provides a visual comparison between the CTD and microstructure data. Comparing the middle row and the bottom row provides a visual comparison between the results of the two algorithms ALG1 and ALG2. The vertical axes are maintained across all three rows.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Histograms of selected interface characteristics from Table 1. (a)–(c) CTD measurements and (d)–(i) microstructure measurements. Each panel is labeled with the total number *N* of interfaces included in the histogram. Comparing the top row and the middle row provides a visual comparison between the CTD and microstructure data. Comparing the middle row and the bottom row provides a visual comparison between the results of the two algorithms ALG1 and ALG2. The vertical axes are maintained across all three rows.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Histograms of selected interface characteristics from Table 1. (a)–(c) CTD measurements and (d)–(i) microstructure measurements. Each panel is labeled with the total number *N* of interfaces included in the histogram. Comparing the top row and the middle row provides a visual comparison between the CTD and microstructure data. Comparing the middle row and the bottom row provides a visual comparison between the results of the two algorithms ALG1 and ALG2. The vertical axes are maintained across all three rows.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Individual layers appear to be laterally coherent and clump in *t*–*S*_{R} space (not shown), similar to those described by Timmermans et al. (2008). Staircase properties also exhibit a distinctive vertical structure. This structure is essentially steady over the 2 yr of measurements and can be characterized by creating scatterplots in depth and then calculating the average values within vertical bins of a fixed size (Fig. 7). We choose a bin size of 1.75 m, since this size is larger than nearly all observed layer heights, but small enough to provide at least 10 bins over the depth range of the staircase. To further highlight the vertical structure, a nonparametric, smoothed curve through the data, computed by the loess algorithm (Cleveland 1993), is also shown for each variable.

Scatterplots depicting the vertical structure evident in the staircase parameters (a) Δ*S*_{R}; (b) *h*_{S}; (c) *R*_{ρ}; (d) *F*_{K}; (e),(i) Δ*t*; (f),(j) *h*_{T}; (g),(k) **∇***t*; and (h),(l) *H*. Panels (a)–(h) represent the CTD measurements, and (i)–(l) represent the microstructure measurements. Colored open circles represent individual layer–interface measurements. Black squares (ALG1) and black circles (ALG2) represent averages calculated within 1.75-m vertical bins, with error bars depicting the standard error in the mean calculated from two standard deviations. Also shown by the gray-outlined line in each panel is a nonparametric loess curve (Cleveland 1993) fitted to the results of ALG1 to further highlight the vertical structure. The vertical red line in (d) represents the independently estimated steady-state flux, and the red line in (g) and (k) represents the temperature gradient expected from this flux if supported only by molecular diffusion. Note that (e)–(h) and (i)–(l) share respective horizontal axes, allowing a visual comparison between the CTD and microstructure data.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Scatterplots depicting the vertical structure evident in the staircase parameters (a) Δ*S*_{R}; (b) *h*_{S}; (c) *R*_{ρ}; (d) *F*_{K}; (e),(i) Δ*t*; (f),(j) *h*_{T}; (g),(k) **∇***t*; and (h),(l) *H*. Panels (a)–(h) represent the CTD measurements, and (i)–(l) represent the microstructure measurements. Colored open circles represent individual layer–interface measurements. Black squares (ALG1) and black circles (ALG2) represent averages calculated within 1.75-m vertical bins, with error bars depicting the standard error in the mean calculated from two standard deviations. Also shown by the gray-outlined line in each panel is a nonparametric loess curve (Cleveland 1993) fitted to the results of ALG1 to further highlight the vertical structure. The vertical red line in (d) represents the independently estimated steady-state flux, and the red line in (g) and (k) represents the temperature gradient expected from this flux if supported only by molecular diffusion. Note that (e)–(h) and (i)–(l) share respective horizontal axes, allowing a visual comparison between the CTD and microstructure data.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Scatterplots depicting the vertical structure evident in the staircase parameters (a) Δ*S*_{R}; (b) *h*_{S}; (c) *R*_{ρ}; (d) *F*_{K}; (e),(i) Δ*t*; (f),(j) *h*_{T}; (g),(k) **∇***t*; and (h),(l) *H*. Panels (a)–(h) represent the CTD measurements, and (i)–(l) represent the microstructure measurements. Colored open circles represent individual layer–interface measurements. Black squares (ALG1) and black circles (ALG2) represent averages calculated within 1.75-m vertical bins, with error bars depicting the standard error in the mean calculated from two standard deviations. Also shown by the gray-outlined line in each panel is a nonparametric loess curve (Cleveland 1993) fitted to the results of ALG1 to further highlight the vertical structure. The vertical red line in (d) represents the independently estimated steady-state flux, and the red line in (g) and (k) represents the temperature gradient expected from this flux if supported only by molecular diffusion. Note that (e)–(h) and (i)–(l) share respective horizontal axes, allowing a visual comparison between the CTD and microstructure data.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

The interface thicknesses in *t* (Figs. 7f,j) and *S*_{R} (Fig. 7b) are nearly constant in depth, increasing only slightly from 15 to 20 cm toward the top of the staircase. The mixed layer height (Figs. 7h,l) is distinctly larger near 340-m depth, where *H* ≈ 1.0 m, than it is toward the bottom and the top of the staircase where *H* ≈ 0.4 m and *H* ≈ 0.6 m. The temperature and salinity differences Δ*t* and Δ*S*_{R} display a different vertical structure, both increasing strongly toward the top of the staircase. Values of Δ*t* increase approximately from 3 mK at 349 m, to 4 mK at 337 m, to 10 mK at 331-m depth (Figs. 7e,i). Similarly, Δ*S*_{R} increases from approximately 1 mg kg^{−1} at 349 m to 2 mg kg^{−1} at 337 m, to as high as 12 mg kg^{−1} at 331-m depth (Fig. 7a). The layer-based density ratio (Fig. 7c) has lower values (*R*_{ρ} ≈ 2.2) near the middle and higher values (*R*_{ρ} ≈ 6.0) near the top of the staircase. Likewise, the interfacial temperature gradient (calculated using ALG1) increases from **∇***t* ≈ 15 mK m^{−1} in the lower third of the staircase to **∇***t* ≈ 40 mK m^{−1} toward the top of the staircase (Figs. 7g,k). On the other hand, the interfacial gradient calculated from ALG2 does not have a clearly defined vertical structure (Fig. 7k).

### c. Heat fluxes

The 4/3 flux parameterizations allow estimates of vertical heat fluxes through a double-diffusive staircase based on the observable parameters Δ*t* and *R*_{ρ} (Kelley et al. 2003). To assess the accuracy of these parameterizations, independent estimates of the vertical heat flux are necessary; two such estimates are possible in Powell Lake. First, Hyndman (1976) estimated the heat flux into the bottom of the staircase by measuring the temperature gradient and the thermal conductivity in the sediments of the lake. He found that the basin is heated with a geothermal flux of 27 ± 8 mW m^{−2}.

^{−1}(Fig. 8). At this depth, the lake is thought to be nearly quiescent; Sanderson et al. (1986) modeled the evolution of the lake following the last deglaciation and found that the observed large-scale temperature and salinity gradients are consistent with their model only if the vertical diffusivity in the deep water is comparable to the molecular diffusivity. This finding is consistent with the observed smoothness and linearity of the temperature gradient, suggesting that molecular diffusion may dominate transport across the depth range 315–328 m. If this is the case, the vertical heat flux

*F*

_{H}can be estimated from

*c*

_{p}= 4.1 × 10

^{3}J kg

^{−1}°C

^{−1}is the specific heat and

*κ*

_{T}= 1.4 × 10

^{−7}m

^{2}s

^{−1}is the thermal diffusivity of seawater, both at in situ temperature, salinity, and pressure (Caldwell 1974; Sharqawy et al. 2010; IOC et al. 2010). From the observed temperature gradient, Eq. (1) yields a flux of 27 ± 1 mW m

^{−2}, in agreement with the flux estimate from the sediments.

The temperature gradient between 315- and 328-m depth is nearly constant at all measured locations: (a) along the lake and (b) across the lake. For each CTD profile, the gradient between 315 and 328 m is calculated by computing the mean of the average gradient in thirteen 1-m vertical bins. For all symbols, the error bars (standard error in the mean from two standard deviations) are smaller than the symbol size. The mean gradient of all 58 stations is also shown. Symbols as in Fig. 1.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

The temperature gradient between 315- and 328-m depth is nearly constant at all measured locations: (a) along the lake and (b) across the lake. For each CTD profile, the gradient between 315 and 328 m is calculated by computing the mean of the average gradient in thirteen 1-m vertical bins. For all symbols, the error bars (standard error in the mean from two standard deviations) are smaller than the symbol size. The mean gradient of all 58 stations is also shown. Symbols as in Fig. 1.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

The temperature gradient between 315- and 328-m depth is nearly constant at all measured locations: (a) along the lake and (b) across the lake. For each CTD profile, the gradient between 315 and 328 m is calculated by computing the mean of the average gradient in thirteen 1-m vertical bins. For all symbols, the error bars (standard error in the mean from two standard deviations) are smaller than the symbol size. The mean gradient of all 58 stations is also shown. Symbols as in Fig. 1.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

*F*

_{K}from staircase properties, we use the 4/3 parameterization developed by Kelley (1990) that has been used in a number of recent studies (Timmermans et al. 2008; Schmid et al. 2010; Sommer et al. 2013a):

*ν*/

*κ*

_{T}, where

*ν*= 1.4 × 10

^{−6}m

^{2}s

^{−1}is the kinematic viscosity at in situ temperature, salinity, and pressure (Horne and Johnson 1966; Sharqawy et al. 2010).

The mean value of the heat flux calculated using Eq. (2) is 27 mW m^{−2} (IQR, 18–34 mW m^{−2}), which again agrees with the two independent flux estimates. However, there is a clear vertical structure in *F*_{K} introduced by the parameterization (Fig. 7d), similar in shape to the vertical structure seen in *H*, with *F*_{K} ≈ 18 mW m^{−2} near the bottom, *F*_{K} ≈ 38 mW m^{−2} near the middle, and *F*_{K} ≈ 25 mW m^{−2} near the top of the staircase.

A correction factor to Eq. (2) proposed by Sommer et al. (2013a) did not improve estimates of the heat flux when tested (it is only marginally applicable near *R*_{ρ} ≈ 2); it increased heat flux estimates by a factor of 1.5 and did not reduce the vertical variation introduced by the parameterization. Consequently, we do not include this correction factor in the analysis presented here.

Finally, it is possible to estimate the vertical heat flux from measurements of the maximum temperature gradient of undisturbed interfaces, assuming heat through laminar interfaces is transported by molecular diffusion. From ALG2, the mean interfacial temperature gradient is 43 mK m^{−1} (Table 1). Using Eq. (1), the average diffusive flux through undisturbed interfaces is then 25 mW m^{−2} (IQR, 12–36 mW m^{−2}). This estimate again agrees with the two independent estimates of the vertical heat flux. In contrast, the mean interfacial temperature gradient 26 mK m^{−1} calculated from PLM with ALG1 (Fig. 7k) supports a diffusive heat flux of only 15 mW m^{−2} (IQR, 9–20 mW m^{−2}).

## 4. Discussion

### a. A one-dimensional system at steady state?

Powell Lake is about 11 000 yr old, having been formed following the most recent deglaciation. In contrast, the time scale for heat to diffuse through the bottom 50 m of the lake is about 600 yr (one-dimensional diffusion). Consequently, it is reasonable to assume that the system is in a quasi-steady state with an approximately uniform heat flux in the deep water. This does not imply that individual double-diffusive layers are themselves at steady state at any given time, only that the background system remains steady over time scales of a few decades. Furthermore, as outlined in section 3b, the statistical properties of the staircase also remain largely steady across at least 2 yr of measurements (Figs. 7a–h).

Our estimate of the vertical heat flux from the measured temperature gradient above the staircase assumes that molecular diffusion is the dominant transport process between 315 and 328 m. While this estimate is a lower bound and the net heat flux may in fact be larger, the flux estimated in this way varies by less than 1% across the basin (Fig. 8) and is the same as the flux estimated in the sediments of the lake. Consequently, it is likely that 27 mW m^{−2} represents the actual quasi-steady vertical heat flux through the whole staircase.

At the depth range considered in this study, heating from the basin sides can be neglected to a first approximation. Since the lake has steep sides and a nearly flat basin bottom, the longitudinal cross section of the deepest 20 m of the lake can be thought of as an open rectangle 1 km wide, with sides 20 m high. Since the heat flux across a surface is proportional to the surface area, the sides could contribute only about (2 × 20)/(1000 + 2 × 20) ≈ 4% to the heat budget of the lower staircase. Taken together with the lateral homogeneity of both the large-scale properties and the staircase characteristics (Fig. 7), this suggests that the deep saline layer in Powell Lake is well represented as a one-dimensional system.

There is also a possibility that turbulent or advective processes in a thin bottom boundary layer may affect the heat budget of the lake by contributing to an additional flux along the basin sides. However, our measurements give no indication that there is a substantial mixed layer at the basin boundaries nor are there any signs of intruding plumes from the basin sides. Where the staircase does not extend to within 5 cm of the sediments (as measured by the VMP), profiles terminate in high-gradient boundary layers rather than in well-mixed layers indicative of turbulent boundary mixing. Furthermore, while it is difficult to measure closely spaced profiles near the steep basin sides, our attempts to do so did not show regions of uniform properties that could be a sign of increased mixing.

In relation to the oceans, Powell Lake may be most comparable to the deep Canada basin, which likewise is a largely isolated basin, geothermally heated from below and with clear double-diffusive signatures (Timmermans et al. 2003; Zhou and Lu 2013). The geothermal heat flux is of a comparable size (40–60 mW m^{−2}) to that in Powell Lake, and the temperature and salinity steps across interfaces are likewise of comparable magnitude (though the mixed layers are about an order of magnitude larger). However, the two domains also differ in a few important regards; first, the water in the deep Canada basin appears to be gradually warming (Carmack et al. 2012), and this is no longer the case in Powell Lake. Second, interfaces in the deep Canada basin do not appear to be able to support the net geothermal flux, and this suggests that a large proportion of heat may escape along the basin sides (Timmermans et al. 2003). Again, this is in contrast to Powell Lake, where dynamics appear consistently one dimensional.

### b. The interfacial heat flux and temperature gradient

Double-diffusive fluxes are often estimated from measurements of the temperature gradient within interfaces by assuming that steady molecular diffusion dominates transport across the interface (Timmermans et al. 2008; Schmid et al. 2010; Sirevaag and Fer 2012; Sommer et al. 2013a). This approach has also been supported by recent modeling results (Carpenter et al. 2012). However, it is not obvious that the molecular diffusion assumption is always true, and it has been suggested (Linden and Shirtcliffe 1978; Newell 1984; Padman and Dillon 1987; Schmid et al. 2010) that the molecular heat flux may underestimate the actual heat flux through the interface, particularly at low *R*_{ρ} when convective plumes inside mixed layers may partially entrain heat across the interface. Laboratory experiments by Marmorino and Caldwell (1976) found that at *R*_{ρ} = 2 the net vertical heat flux is 2.5 times the molecular flux. Similarly, direct numerical simulations by Carpenter and Timmermans (2014) found that at *R*_{ρ} = 2 the total interfacial heat flux is 2.4 times the molecular value when Coriolis effects are not important.

However, the average temperature gradient calculated at the center of undisturbed interfaces (i.e., using ALG2) closely matches the temperature gradient 47 mK m^{−1} expected from the steady-state heat flux (Table 1; Fig. 7k), providing observational evidence that double-diffusive fluxes can indeed be estimated by measuring the temperature gradient of staircase interfaces. In contrast, the average *bulk* interface gradient estimated from all interfaces (i.e., using ALG1) is on average 40% lower than that expected from molecular diffusion of the steady-state flux. Consequently, it appears that the interfacial flux can be estimated using the molecular diffusion assumption, but that the gradient must be measured at the center of (undisturbed) interfaces. This implies that microstructure measurements are necessary in order to correctly estimate diffusive fluxes through a staircase since a very high vertical resolution is needed to avoid smoothing through the interface properties.

Although the temperature gradient averaged across the interface (from ALG1) substantially underestimates the gradient expected from molecular diffusion, the effect is more pronounced at low density ratios (Figs. 7c,g, 9). At *R*_{ρ} ≈ 2, the gradient averaged across the interface is consistently lower, by a factor of about 2.2, than the expected molecular diffusion gradient. At higher density ratio near *R*_{ρ} ≈ 6, the gradient averaged across the interface approaches the diffusive gradient, perhaps indicating that the whole interface is becoming more stable and is increasingly dominated by molecular diffusion. Our findings are consistent with the laboratory experiments and numerical simulations discussed above, supporting the conclusion that near *R*_{ρ} = 2, the net interfacial heat flux tends to be 2–2.5 times larger than the diffusive flux calculated from the mean bulk gradient in the interface. At low density ratio there may then be, on average, an entrainment flux that is also important in moving heat away from the interface.

Parametric plot of the bulk interface temperature gradient against the density ratio, as calculated by ALG1. The figure is constructed from the data in Figs. 7c and 7g, with square markers showing depth-binned averages. At low *R*_{ρ}, the temperature gradient averaged across the interface, calculated from all available interface observations, is substantially lower than that expected from molecular diffusion of the estimated steady-state heat flux (horizontal red line), indicating that the effects of entrainment into the interface are present at low density ratios. At higher *R*_{ρ}, the interface becomes more stable and the gradient averaged across the interface approaches the anticipated value of the molecular diffusion gradient. The correlation coefficient *R* between the depth-averaged bins is 0.95 with a *p* value less than 0.001. A linear regression to the vertically binned averages is also shown.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Parametric plot of the bulk interface temperature gradient against the density ratio, as calculated by ALG1. The figure is constructed from the data in Figs. 7c and 7g, with square markers showing depth-binned averages. At low *R*_{ρ}, the temperature gradient averaged across the interface, calculated from all available interface observations, is substantially lower than that expected from molecular diffusion of the estimated steady-state heat flux (horizontal red line), indicating that the effects of entrainment into the interface are present at low density ratios. At higher *R*_{ρ}, the interface becomes more stable and the gradient averaged across the interface approaches the anticipated value of the molecular diffusion gradient. The correlation coefficient *R* between the depth-averaged bins is 0.95 with a *p* value less than 0.001. A linear regression to the vertically binned averages is also shown.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Parametric plot of the bulk interface temperature gradient against the density ratio, as calculated by ALG1. The figure is constructed from the data in Figs. 7c and 7g, with square markers showing depth-binned averages. At low *R*_{ρ}, the temperature gradient averaged across the interface, calculated from all available interface observations, is substantially lower than that expected from molecular diffusion of the estimated steady-state heat flux (horizontal red line), indicating that the effects of entrainment into the interface are present at low density ratios. At higher *R*_{ρ}, the interface becomes more stable and the gradient averaged across the interface approaches the anticipated value of the molecular diffusion gradient. The correlation coefficient *R* between the depth-averaged bins is 0.95 with a *p* value less than 0.001. A linear regression to the vertically binned averages is also shown.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

### c. Heat fluxes and the underlying scaling

Finally, we find that the 4/3 parameterization [Eq. (2)] is able to represent the estimated quasi-steady-state flux 27 mW m^{−2} very well on average (Table 1) and within a factor of 2 at any depth (Fig. 7d). However, the parameterization introduces a vertical structure in the fluxes, as highlighted by the depth-averaged bins in Fig. 7d. Furthermore, this structure is very similar to the structure seen in *H* (Figs. 7h,l), which is surprising because the parameterization was initially developed under the assumption that fluxes are independent of *H*. Because there are no lateral divergences in the measured properties of the staircase (section 4a), we anticipate that the true steady-state flux is a uniform 27 mW m^{−2}, consistent with estimates above and below the staircase. Consequently, the vertical structure seen in Fig. 7d is very likely an artifact of the parameterization (discussed in greater detail below).

*HF*

_{H}/

*ρc*

_{p}

*κ*

_{T}Δ

*t*is a function of the density ratio

*R*

_{ρ}and the Rayleigh number Ra =

*gα*Δ

*tH*

^{3}/

*νκ*

_{T}. Turner then suggested a relationship of the form

*η*= ⅓ was chosen to satisfy the assumption that the heat flux across layers is independent of

*H*. The dependence on the stabilizing salinity component was included through

*C*(

*R*

_{ρ}), a then unknown “empirically determined function.” From Kelley (1990), laboratory experiments suggested that

*R*

_{ρ}= 4, and no fit will pass through all the points given by the trend in the depth-binned averages (as in Fig. 7). Consequently, we test the parametric form of Eq. (3) directly. To do this, we assume that Nu is an unknown function of Ra and

*R*

_{ρ}and attempt to deduce the relationship empirically by objectively mapping our data (Fig. 11a). Although much of our raw data are concentrated in a region of low

*R*

_{ρ}and high Ra, there are enough points over the whole domain that the objective map is reasonably robust to changes in the length-scale parameters; here these length scales have been decreased to the point where large-scale features are only lightly smoothed, but small-scale (point to point) variability is largely removed.

Comparison of Kelley’s (1990) scale factor *C*(*R*_{ρ}) with the empirical fit ^{−2} to calculate Nu.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Comparison of Kelley’s (1990) scale factor *C*(*R*_{ρ}) with the empirical fit ^{−2} to calculate Nu.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

Comparison of Kelley’s (1990) scale factor *C*(*R*_{ρ}) with the empirical fit ^{−2} to calculate Nu.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

(a) Objective map depicting contours of log_{10}(Nu) in *R*_{ρ}–Ra space. Red dots represent individual CTD measurements (PL12 and PL13). (b) As in (a), but with contours calculated using Eq. (4) (Kelley 1990). (c) Ra-Nu relationship, using *R*_{ρ} < 3 only. (d) *R*_{ρ}-Nu relationship, using Ra > 10^{6} only. For all panels, depth-binned averages are calculated as in Fig. 7 and sequentially connected by a thin black line. We use the steady-state flux 27 mW m^{−2} to calculate Nu. All axes are log scaled.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

(a) Objective map depicting contours of log_{10}(Nu) in *R*_{ρ}–Ra space. Red dots represent individual CTD measurements (PL12 and PL13). (b) As in (a), but with contours calculated using Eq. (4) (Kelley 1990). (c) Ra-Nu relationship, using *R*_{ρ} < 3 only. (d) *R*_{ρ}-Nu relationship, using Ra > 10^{6} only. For all panels, depth-binned averages are calculated as in Fig. 7 and sequentially connected by a thin black line. We use the steady-state flux 27 mW m^{−2} to calculate Nu. All axes are log scaled.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

(a) Objective map depicting contours of log_{10}(Nu) in *R*_{ρ}–Ra space. Red dots represent individual CTD measurements (PL12 and PL13). (b) As in (a), but with contours calculated using Eq. (4) (Kelley 1990). (c) Ra-Nu relationship, using *R*_{ρ} < 3 only. (d) *R*_{ρ}-Nu relationship, using Ra > 10^{6} only. For all panels, depth-binned averages are calculated as in Fig. 7 and sequentially connected by a thin black line. We use the steady-state flux 27 mW m^{−2} to calculate Nu. All axes are log scaled.

Citation: Journal of Physical Oceanography 44, 11; 10.1175/JPO-D-14-0070.1

The empirically mapped relationship shows that Nu is highest when *R*_{ρ} is low and Ra is high, which is also true of the Kelley (1990) parameterization (Fig. 11b). However, the shape of the empirical surface is somewhat different than the shape of the surface defined by the parameterization, which has contours that go from the lower left to upper right with a downward concavity. The empirical surface, on the other hand, has contours that are strongly curved in the opposite direction, with contours that are close to horizontal on the left side (*R*_{ρ} < 3) and close to vertical on the upper middle/right (*R*_{ρ} > 3 and Ra > 10^{6}) of Fig. 11a.

The objective map implies that, at low *R*_{ρ}, our observations are well described by a relationship with Ra alone (Fig. 11c). Including only those data (76%) with *R*_{ρ} < 3, a functional dependence on Ra^{1/3}, as in Eq. (3), matches the data reasonably well. However, in addition to considering the individual observations, we can also consider the trend given by the depth-binned values (as in Fig. 7) in the same range *R*_{ρ} < 3. At lower Ra, these points lie along the Ra^{1/3} curve, but at higher Ra they no longer follow this line. Kelley (1990) urged caution when applying the 4/3 flux parameterizations and suggested that a different or variable exponent may produce better results. In our measurements, no simple power law will pass through all the points given by the depth-binned values in Fig. 7c, indicating that a Ra^{η} dependence may not always be appropriate, regardless of the value of *η*.

Conversely, the vertical contours of the objective map imply that data at higher Ra can be described by a decreasing function of *R*_{ρ} alone (Fig. 11d). By considering only those data (86%) where Ra > 10^{6}, the individual observations and the depth-binned averages both show a consistent downward trend with increasing *R*_{ρ}. Following Eq. (4), the functional dependence given by Kelley (1990) matches this trend at low *R*_{ρ} and deviates by less than 30% from the trend given by the depth-binned averages at higher values of *R*_{ρ}.

*R*

_{ρ}were possible, it would have the practical advantage that fluxes could be determined from the large-scale background temperature and salinity profiles rather than the individual layer/interface characteristics. With such a parameterization, fluxes would be dependent only on the ratios Δ

*t*/

*H*and Δ

*S*

_{R}/

*H*, and these are fixed for a given set of large-scale background temperature and salinity gradients.

Based on the information presented here, we suggest it may be plausible that double-diffusive fluxes in Powell Lake are governed by two distinct processes. When the stratification across the interface is weak (small *R*_{ρ}), convection in the mixed layers, the strength of which is largely governed by the size of Ra, is able to more easily erode or disrupt the interface and entrain heat and salt between adjacent layers. This is also reflected in the low bulk interface gradient (Fig. 9). However, as *R*_{ρ} becomes large and the density stratification across the interface becomes stronger, bulk interface properties tend to become more laminar (and diffusive), and *R*_{ρ} becomes increasingly dominant in controlling the net vertical flux.

## 5. Conclusions

We present measurements and a preliminary analysis of a double-diffusive staircase that is naturally isolated from outside turbulent effects in a deep, quiescent layer of relic seawater in Powell Lake, British Columbia. Other lakes exist, including Lakes Nyos (Schmid et al. 2004) and Kivu (Sommer et al. 2013a) in eastern Africa, which have similar well-defined double-diffusive staircases; however, in none of these does the phenomenon appear to be as isolated as in Powell Lake, whose observed background stratification is quasi-steady and has remained largely unchanged for at least 50 yr.

The staircase is bounded below by the lake sediments and above by a quiescent layer with high density ratio in which molecular diffusion likely dominates vertical transport. Independent best estimates of the vertical heat flux above and below the staircase are identical (within measurement uncertainty) and in good agreement with estimates of the diffusive flux obtained from microstructure profiling through laminar double-diffusive interfaces. Layers are laterally coherent and appear to have only small boundary effects close to the steep basin sides. This homogeneity suggests that the deep saline layer of Powell Lake can be treated as a one-dimensional system with a known quasi-steady heat flux above and below the staircase.

Microstructure profiling and careful CTD measurements yield similar statistics for the staircase parameters, but the high vertical resolution of the microstructure instrument yields measurements at the interface centers that cannot be obtained from CTD measurements. The CTD, however, was able to sufficiently distinguish the extremely small salinity differences between consecutive layers, which the microstructure probe was unable to do. Staircase statistics remain comparable between 2012 and 2013.

The average interfacial temperature gradient (26 mK m^{−1}) estimated from bulk interface parameters is smaller by approximately a factor of 2 than that (47 mK m^{−1}) expected from molecular diffusion due to the steady-state heat flux. However, the mean gradient (43 mK m^{−1}) at interface centers estimated only from nonturbulent interfaces closely matches that expected from the steady-state flux. This suggests that it is indeed possible to accurately estimate the actual heat flux through a double-diffusive staircase by measuring interface gradients; however, to be correct, only undisturbed interfaces should be used for such an analysis, and, more importantly, the gradient should be measured directly at the interface center rather than estimated by averaging across the bulk features of the interface. It is most important that these conditions are met when the density ratio *R*_{ρ} is low.

Staircase properties derived from bulk parameters are, however, still useful for calculating double-diffusive fluxes. The heat flux calculated from the frequently cited 4/3 parameterization of Kelley (1990) uses bulk parameters and is accurate to within a factor of 2, although it exhibits an artificial depth dependence resembling the depth dependence present in the layer height *H*, suggesting that heat fluxes may not always be independent of *H*, as traditionally assumed. Furthermore, we find that at high Ra, double-diffusive fluxes in Powell Lake can largely be described by a decreasing function of *R*_{ρ} alone, independent of Ra.

Many aspects of double diffusion in Powell Lake remain to be studied. In particular, future work in Powell Lake will concentrate on small spatial and temporal variations in individual layer and interface characteristics, which have not been addressed in this study.

## Acknowledgments

We wish to thank Michael Schurter and Carsten Schubert of Eawag and Roger Pieters, Chris Payne, and Lora Pakhomova of UBC for their assistance in obtaining the measurements described here. The Canadian partners were supported by NSERC under Grant 194270 and a CGS-M scholarship. The Swiss partners were supported by the Swiss National Science Foundation under Grant 200020-140538 (Lake Kivu turbulence and double diffusion in permanent stratification).

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