1. Introduction
Most of the oceanic heat entering the Arctic Ocean comes through Fram Strait in the Atlantic Water (AW) that then circulates as a warm boundary current along the continental slope of the eastern Arctic shelf seas (e.g., Aagaard 1989; Rudels et al. 1994). The maximum temperature of the AW decreases rapidly along this path, responding initially to direct ocean-to-atmosphere heat loss (e.g., Onarheim et al. 2014; Ivanov et al. 2012, 2016), then to mixing with cooler ambient waters from the continental shelves and in the deep basins (e.g., Walsh et al. 2007). The inferred average heat loss from the AW layer in the Nansen Basin, once it has subducted beneath the cold and relatively fresh Arctic Surface Water, is 4–6 W m−2 (Walsh et al. 2007; Polyakov et al. 2010). This value is large compared with the imbalance of ~1 W m−2 required to explain decadal volume loss of the Arctic sea ice pack (Kwok and Untersteiner 2011; Carmack et al. 2015), suggesting that significant changes in the rate of heat loss from the AW layer could influence evolution of the sea ice.
Carmack et al. (2015) summarized the processes by which AW heat can be transported upward through the pycnocline above the AW layer. Two mechanisms—shear instabilities and double diffusion—are thought to be responsible for most diapycnal fluxes below the direct influence of stress and buoyancy forcing at the ocean surface. Shear instabilities associated with baroclinic tides and other tidally generated internal waves can lead to very large heat fluxes, with measured values sometimes greater than 10 and up to ~50 W m−2 (Padman and Dillon 1991; Rippeth et al. 2015), but these large fluxes appear to be localized near sources of baroclinic tides over the continental slope in the eastern Arctic (Padman 1995; Rippeth et al. 2015). It is not yet known how important shear instabilities are to AW heat transports when averaged over the entire Arctic Ocean. The second mechanism, double diffusion (Turner 1965, 1973), appears above the AW as the diffusive layering (DL) instability, which is known to be widespread throughout the Arctic Ocean (e.g., Timmermans et al. 2008; Rudels et al. 2009; Shibley et al. 2017). Peak measured heat fluxes from the DL instability are much lower than those due to shear instability; however, the basin-averaged contribution of the DL instability to heat loss from the AW core may exceed the contribution from tide-forced internal waves.
In this paper, we focus on the DL process in the eastern Arctic Ocean. The DL instability is described in detail by Kelley et al. (2003) and Radko (2013). When DL is active, vertical profiles of potential temperature θ and salinity S show a series of steps (called a “staircase”) consisting of nearly homogeneous layers separated from each other by interfaces where θ and S change rapidly (Fig. 1b). Fluxes of heat, salt, and buoyancy are driven by the difference between the molecular diffusivities for temperature κT and salinity κS, where κT/κS is of order 100: rapid diffusion of the intrinsically unstable temperature component of total density at the edges of interfaces drives convection in the layers as described by Linden and Shirtcliffe (1978) and illustrated by Padman and Dillon (1987, their Fig. 7). More recent studies indicate that the process is more complex than these one-dimensional interpretations suggest. Carpenter et al. (2012) revealed that the transition to instability of a double-diffusive interface occurs as an oscillating diffusive convection mode at small boundary layer Rayleigh numbers, while nonlinear numerical simulations (Carpenter and Timmermans 2014) suggest that the convective plumes then organize into large-scale circulation cells.
(a) Map showing positions of selected ITP-36 (red/cyan) and ITP-37 (red/red) potential temperature θ profiles. Blue dots show locations of the 44 microstructure profiles made during 2007 and 2008 NABOS cruises and used in the present study. SZ, FJL, and SPB indicate Severnaya Zemlya, Franz Josef Land, and Spitsbergen. (b),(c) Example θ profiles from ITP-37 and ITP-36; 0.1°C offset and color are used to separate these profiles. Note that in (b), ITP-37 profiles contain a set of small steplike structures within the 160–190-m depth range, and thick convective layers separated by HGRs in the lower part of the water column. In (c), the zigzag structure is indicative of intrusions in the ITP-36 profiles. (d),(e) The θ–S (salinity) diagrams for the same profiles. The zigzag structure of the ITP-36 profiles in (c) above the AW core (defined as the maximum) results in peaks in the θ–S diagram, whereas no peaks are found above the AW core in the ITP-37 θ–S diagram.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
Laboratory measurements (e.g., Marmorino and Caldwell 1976; Kelley 1990) and direct numerical simulations (Flanagan et al. 2013) show that the energetics of mixing and the magnitudes of the fluxes are functions of the change in potential temperature Δθ across an interface and the density ratio 
These so-called “flux laws,” applied to hydrographic profiles from the eastern Eurasian Basin (EB), suggest heat fluxes of up to about 10 W m−2 (Lenn et al. 2009; Polyakov et al. 2012), with the largest values for DL steps just above the depth of the AW temperature maximum. These estimated DL heat fluxes are much larger than in the Canada Basin in the western Arctic, where values of order 0.1 W m−2 are typical (Padman and Dillon 1987; Timmermans et al. 2008). The applicability of laboratory-based flux laws to the more complex ocean has, however, been questioned (Padman 1994; Sirevaag and Fer 2012; Guthrie et al. 2015). Given the potential importance of DL heat fluxes to the large-scale heat budgets of the eastern Arctic Ocean above the AW layer, it is important to develop reliable flux parameterizations that are applicable to this oceanic region.
Two distinct classes of DL steps are found in the eastern Arctic: a few large layers, often >10 m high, separated by high-gradient regions (HGRs) with large values of Δθ and ΔS (e.g., between ~190 and 225 m in Fig. 1b); and sequences of several smaller steps with layers of order 1 m high and smaller values of Δθ and ΔS (e.g., above ~190 m in Fig. 1b). The thickness of HGRs can exceed 5 m, with complex internal structure that varies significantly between profiles (Fig. 2). Simultaneous measurements of hydrography and currents from a McLane moored profiler (MMP) record from the eastern Arctic slope (Polyakov et al. 2012) show that the horizontal velocity U(z) varies across large steps, with the largest shear magnitude tending to be coincident with DL interfaces and across HGRs (see example in Fig. 3). These observations are consistent with the interaction of mean, or slowly varying, sheared flow with the DL-driven processes of vertical transport of scalars and momentum described by Padman (1994).
Example profiles of microstructure temperature and vertical shear of horizontal currents containing large steps. (a)–(d) HGRs of Type I include a sequence of continuous small steps with no turbulent patches; (e)–(h) HGRs of Type II incorporate a mix of turbulent “noise” (turbulent patches) and several small steps; (i)–(l) HGRs of Type III lack well-developed small steps. Blue dots indicate ends of convective layers.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
Hydrographic and velocity profiles from a single profile from an MMP over the Laptev Sea slope [see Polyakov et al. (2012) for mooring details]. All data have been filtered to 1.5 m in the vertical. (a) Potential temperature θ; (b) salinity; (c) east u and north υ components of velocity (cm s−1); and (d) shear magnitude (s−1). Horizontal gray lines on each panel mark the locations of three HGRs.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
For typical values of Δθ and Rρ, flux laws suggest heat fluxes of order 10 W m−2 for the larger steps and in the range of 0.1–1 W m−2 for the smaller steps. The focus of this study is on heat fluxes across the larger steps since estimated fluxes and their uncertainties are large, and none of the previous studies of Arctic DL fluxes using microstructure data specifically focused on analysis of these features. Our analyses follow the general approaches applied to smaller steps by Sirevaag and Fer (2012) and Guthrie et al. (2015).
2. Data and methods
a. Summary of microstructure profiler observations
Microstructure observations were made during the 2007 and 2008 summer cruises of the Nansen and Amundsen Basins Observational System (NABOS) research program. About 80 profiles of temperature T, conductivity C, and microscale vertical shear of the horizontal current uz to 600-m depth were collected in three cross-slope transects (Fig. 1a) using a Vertical Microstructure Profiler (VMP) manufactured by Rockland Scientific Instruments. These data have been described previously by Lenn et al. (2009, 2011) and Rippeth et al. (2015). We excluded profiles made in water less than 500 m deep to limit influence of shelf/slope processes. Of the remaining profiles, 44 contained high-quality records of all variables required for our analyses.
The VMP was deployed in tethered, free-fall mode with typical fall speeds of ~0.6 m s−1. Microscale measurements of T, C, and uz were obtained with an FP07 thermistor, SBE7–38 microconductivity, and two SPM38–1 airfoil shear probes, respectively, each recorded at 512 samples per second. A three-axis accelerometer provides high-frequency information on instrument tilt and vibrations that may be excited through the tether. The VMP was also equipped with a Seabird unpumped conductivity–temperature–depth (CTD) system (an SBE-3 thermistor and an SBE-4 conductivity cell), which provided much more accurate values of T and C but at much lower vertical resolution than the microstructure sensors. To identify the data sources for different analyses, we refer to data from the microstructure and Seabird sensors with subscripts “μ” and “SBE,” respectively.
b. Calibration of temperature and salinity
Slow flow rates through the unpumped Seabird conductivity cell degraded the quality of CSBE and SSBE profiles. We therefore calibrated the VMP Seabird data with independent CTD measurements acquired by the Seabird SBE19plus CTD on the main ship system. The VMP casts followed immediately after the ship-based SBE19plus CTD casts were recovered during the 2007 and 2008 cruises, with the VMP casts taking about 30 min each. The VMP data were calibrated to the processed 1-dbar ship CTD data in T–S space to match extrema in water mass properties (i.e., AW intrusions, surface mixed layer temperature minima) while allowing for small isopycnal displacements between the CTD and VMP casts. This calibration process thus preserves the approximately 10-mm vertical sampling of the 64-Hz VMP Seabird data at a free-fall speed of 0.6 m s−1. The CTD operations were based at the stern of the ship, while the VMP operations were based on the bow of the ship about ~150 m forward. We assume that the VMP was free-falling vertically from its drop position. The total separation of the VMP and CTD profiles is, therefore, a combination of the spatial separation of their deployments relative to the ship and the ship drift relative to the underlying water column during the 30 min between profiles. The ship SBE19plus CTD data were processed to reconcile the up- and downcast profiles at each station to produce 1-dbar (~1 m) resolution data. Our calibration process ensures that the properties of the main water masses (i.e., mixed layer, halocline, and AW water) agree in T–S space and the 1-dbar data from the SBE19plus CTD was adequate to the task. Calibrated VMP Seabird data were used for the values of SSBE reported here.
c. Resolution of microscale temperature sensor
The microstructure sensors provide much finer resolution of T and C than the Seabird sensors. For Tμ, the time constant of the FP07 double-pole response is roughly 10 ms (Sommer et al. 2013). At typical VMP fall speeds of ~0.6 m s−1, this response corresponds to the ability to fully resolve scales of ~0.05 m (Gregg and Meagher 1980), much finer than can be resolved in CTD, MMP, and Ice-Tethered Profiler (ITP) profiles (~0.25–1 m) but still insufficient to fully resolve internal structure within small DL interfaces (Padman and Dillon 1987; Sommer et al. 2013; Guthrie et al. 2015).
d. Processing shear measurements and estimating dissipation rate ε
As the first step in processing microscale profiles of shear uz, profiler motion was removed by deconvolving the shear signal with the accelerometer signal. Shear sensors on free-falling microstructure profiles measure uz up to length scales set by the coupling of the profiler motion to the lateral flow: given the length of the VMP (~2 m), we expect that shear estimates between the sensor resolution (~0.03 m) and ~0.5–1 m should be resolvable, provided the shear signal is above the noise level. This band contains most of the shear variance that is expected in spectra for isotropic, fully developed turbulence for typical levels of turbulence in the present dataset. Noise beyond the accurately resolved vertical scales was removed from uz(z) using a 3–60-cm bandpass filter. The upper limit is smaller than the typical thickness of the HGRs between thick layers (Fig. 2); therefore, we cannot measure the large-scale shear magnitude |Uz| across HGRs that is sometimes seen in MMP profiles (Fig. 3).
Our principal use of microstructure shear profiles is calculation of the dissipation rate of turbulent kinetic energy ε (Gregg 1999). Assuming the turbulence is isotropic, the equation for fully developed isotropic turbulence (Kolmogorov 1941) is ε = 7.5ν〈uz2〉 W kg−1, where ν is the kinematic viscosity of seawater (≈1.8 × 10−6 m2 s−1 at 0°C), and angle brackets denote averaging over some depth interval, typically 0.5 m or greater. In many studies, the averaging depth interval is chosen to be constant. In DL staircases, however, the layers seen in hydrography provide the natural scale of vertical averaging. We use εLAYER to refer to dissipation rates calculated in large layers above and below HGRs and εlayer to refer to rates within single DL layers within HGRs.
We used two methods for estimating 〈uz2〉. First, following Rippeth et al. (2003), we integrated the shear spectra between 2 and 30 cycles per meter (cpm). These integration limits were based on observed characteristics of shear spectra for a wide range of turbulence conditions and are consistent with the instrumentation; measured shear at wavenumbers less than 2 cpm is small because of coupling between the VMP and lateral flow, while the physical size of the shear sensor filters shear at wavenumbers greater than ~30 cpm. Second, we estimated εLAYER as an iterative best fit to the theoretical Panchev–Kesich spectrum of shear (Panchev and Kesich 1969; Gregg 1999) for fully developed turbulence. This approach minimizes issues associated with noise in individual shear spectra, but assumes that the true spectrum of shear at the time of measurement is correctly represented by the Panchev–Kesich spectrum. In our dataset, the values of εLAYER from the two methods are, in general, close to each other and correlated at R = 0.82.
For both approaches to estimating 〈uz2〉, the approximate noise floor for ε is εnoise = 2 × 10−10 W kg−1, consistent with the value cited by Lenn et al. (2009). As discussed in section 3, this noise floor places significant constraints on the accuracy of variables, such as heat flux, derived from single estimates of ε. We estimated the uncertainty in ensemble means of fluxes by analyses of Monte Carlo simulations in which εnoise is treated as a normally distributed random variable with standard deviation of 2 × 10−10 W kg−1 that is added to the set of measured values of ε, with the added condition that ε ≥ 0. For typical sets of measured ε, the resulting uncertainty in ensemble means is primarily a function of the number of values in the ensemble, but is in the approximate range 0.2–0.6 × 10−10 W kg−1 for the ensemble sizes (14–48) considered here.
In further analyses, we used the estimates based on band-passed values. From measurements of εLAYER, the approximate spread of values for the spectral fit for a specific band-passed estimate is about ±1 × 10−10 W kg−1 (i.e., similar to the error associated with noise in shear).
Raw profiles of uz(z) can also be used directly to identify regions of high microscale shear (Padman 1994). However, as noted above, the long-wavelength cutoff caused by VMP coupling to the background velocity profile precludes calculation of background mean shear across thick HGRs.
e. Ancillary datasets
Microstructure observations were complemented by analysis of ship-based (mostly summer) CTD measurements and data from an MMP and drifting ITPs (www.whoi.edu/itp) providing year-round hydrographic measurements in the upper ~750 m. These MMP and ITP measurements have accuracies of 0.002°C for temperature and 0.002 psu for salinity. The CTD SBE19plus V2 sensor has accuracies of 0.005°C for temperature and 0.0005 S m−1 for conductivity.
3. Results
a. Structure of large DL steps in the EB
From the 44 VMP microstructure profiles with good data for all needed variables and in water depths greater than 500 m, we identified 48 “large steps,” defined as depth ranges with two well-mixed several-meters-thick layers separated by an HGR. We evaluated the temperature and salinity change (ΔθHGR and ΔSHGR, respectively) across each HGR from the mean values of TSBE and SSBE in the adjacent two layers. The thickness of the HGR HHGR was evaluated from the clearly defined upper and lower edges of the lower and upper layers (see Fig. 2 for examples). The density ratio (RρHGR) was evaluated using values of α and β based on the central values of T, S, and pressure P for the HGR.
For the 48 large steps, mean values of these variables were 〈ΔθHGR〉 = 0.36°C, 〈ΔSHGR〉 = 0.06, 〈HHGR〉 = 1.8 m, and 〈RρHGR〉 = 2.0, respectively (Fig. 4). However, the probability distribution functions of ΔθHGR, ΔSHGR, and HHGR are highly skewed so that their medians and means differ substantially, and we cannot define confidence intervals for the means. The distribution of RρHGR values is less skewed, and mean and median values are almost identical. The different types of HGR are found across broad ranges of ΔθHGR and ΔSHGR (related through RρHGR), but there is no clear pattern to their relative distributions (Fig. 4e).
Histograms of the (a) potential temperature change ΔθHGR and (b) salinity change ΔSHGR across HGRs, (c) HGR thickness HHGR, and (d) density ratio RρHGR. Mean and median values for each distribution are stated. (e) Relationship between αΔθHGR and βΔSHGR (relative contribution to density change) color coded by interface type. Dotted lines show corresponding values of RρHGR.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
The average thickness of the 48 HGRs of 〈HHGR〉 = 1.8 m is two orders of magnitude larger than the thickness of a few centimeters for typical single DL interfaces (e.g., Padman and Dillon 1987; Sommer et al. 2013, 2014; Guthrie et al. 2015). There are three distinct types of HGR seen in the VMP profiles of Tμ (Fig. 2), each representing about a third of the 48 steps. Fourteen of these HGRs consisted of a set of small and fairly uniform DL steps (Type I: Figs. 2a–d) that, individually, look similar to previously studied DL steps where the interfaces were assumed to be laminar (nonturbulent) with fluxes set by molecular diffusion (e.g., Padman and Dillon 1987; Sirevaag and Fer 2012; Carpenter and Timmermans 2014; Sommer et al. 2014; Guthrie et al. 2015). This type of HGR always contains at least five individual DL interfaces (n ≥ 5). Seventeen HGRs consisted of a few sharp DL interfaces but with more thermal variability in some of the intervening low-gradient regions, and occasional patches with no discernible DL signals (Type II; 1 ≤ n < 5: Figs. 2e–h). These may be similar to Type I steps but with mixing of interfaces caused by shear from convectively driven eddies in the layers (Fernando 1989) or applied shear as reported by Padman and Dillon (1989) and Padman (1994). We refer to these segments of Type II HGRs as turbulent patches. The remaining 17 HGRs had either no small steps or just a hint of emerging small DL steps (Type III; Figs. 2i–l).
The range of HGRs may represent different stages of evolution from a common formation mechanism for large steps (see section 4) or intermittent response to changes in background forcing such as imposed shear (Fig. 3). The different characteristics suggest that the diapycnal heat, salt, and buoyancy fluxes may vary among HGRs that have similar bulk characteristics, as observed in coarsely resolved hydrographic profiles. For example, we expect that fluxes in a Type I HGR are driven entirely by the DL instability, so that the effective diapycnal diffusivity for salt KS is much less than for temperature KT (see, e.g., Kelley 1984). In contrast, these two diffusivities may be similar to each other in a Type III HGR if shear-driven mixing dominates; however, see Gargett (2003) for evidence that KS ≠ KT for some range of stratification and mixing rates when shear instability is the dominant mixing mechanism.
b. Relationship between DL structure and microscale shear
Peaks in profiles of the absolute value of microscale shear |uz| are frequently collocated with interfaces and HGRs [see the example profiles of temperature Tμ and |uz| in Figs. 2a–f and similar analyses by Padman (1994)]. Microscale shear at a single interface or HGR may result from several processes: externally imposed velocity structure (e.g., the large-scale geostrophic shear associated with mesoscale ocean state, and internal tides and other internal waves irradiating the steps; Padman and Dillon 1989; Padman 1994); the abrupt velocity difference associated with the eddy variability of velocities within the bounding quasi-homogeneous layers (Fernando 1987, 1989); and turbulence within the interface or HGR. In the case of externally imposed large-scale shear Uz (e.g., as seen in the MMP velocity profile in Fig. 3), we expect that the shear probes on the VMP would have recorded high microscale shear magnitudes across interfaces because the energetic, convectively driven turbulence in the adjacent layers homogenizes the horizontal velocity in each layer (Padman 1994). The second proposed source of high interfacial shear—eddy velocities associated with the DL-driven turbulence in the quasi-homogeneous layers—has been implicated in the increased diapycnal fluxes associated with values of Rρ less than about 2 (e.g., Fernando 1987, 1989).
We summarize the relationship between |uz| and location relative to HGRs by calculating, for each of the 48 large steps, the value of |uz| as a function of normalized distance z′ from the bottom of the lower layer (z′ = 0) to the top of the upper layer (z′ = 1). The typical thickness of an HGR is about 10% of a complete step, and the two layers have similar thicknesses to each other; therefore, we assign the lower layer, HGR, and upper layer to the ranges 0 < z′ < 0.45, 0.45 < z′ < 0.55, and 0.55 < z′ < 1, respectively. We emphasize that the calculation of |uz|(z′) for each step is applied to microscale shear magnitude without regard to direction changes, and so is not directly comparable with the velocity and associated shear measurements from the MMP current meter profiles illustrated in Fig. 3. The mean profile of |uz|(z′), averaged over all 48 large steps (Fig. 5), shows a clear maximum within the HGR (0.45 < z′ < 0.55). The average value over the HGR is 0.013 s−1 compared with 0.009 s−1 in the upper and deeper convective layers; these estimates of means are statistically different from each other. Typical shear in an HGR is larger for Type II and Type III HGRs than for Type I HGRs (not shown).
Mean magnitude of microscale velocity shear within DL layers and HGRs, averaged over the 48 large steps. For each profile, the depth of each measurement of microscale shear magnitude was normalized to the range 0–1, where 0–0.45 is the lower layer, 0.45–0.55 is the HGR, and 0.55–1 is the upper layer. The blue dotted line is unsmoothed; the blue solid line is smoothed using a running mean of width 0.05 in normalized depth. Red segments show means for lower and upper DL layers and the HGR; the latter differs from layer means at the 95% confidence level.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
c. Heat fluxes through large DL steps in the EB
The heat flux FH through layers and adjacent HGRs in a single microstructure profile will not be the same, since the process of buoyancy production by diffusion at the edges of interfaces is intermittent (see section 1). Over a sufficiently long averaging time, however, vertical fluxes should be approximately continuous over some averaging time scale after lateral heat transport divergence along layers (Hieronymus and Carpenter 2016; Bebieva and Timmermans 2017) is taken into account. The heat flux through each step that we have identified can be estimated in three ways: from dissipation rates in the layers; from laboratory-based DL flux formulas; and from a weighted average of fluxes through small DL interfaces, DL layers, and turbulent patches within HGRs.
1) Heat flux from layer dissipation rates
The turbulence in the well-mixed layers in DL steps is driven by the buoyancy flux Fb through the diffusive interfaces (e.g., Turner 1965, 1973; Linden and Shirtcliffe 1978; Taylor 1988). This net buoyancy flux is the sum of the destabilizing buoyancy flux due to heat 
In the DL instability, turbulence in a layer is driven by the combined convective forcing of buoyant parcels formed at the upper edge of the lower interface and dense parcels formed at the lower edge of the upper interface. In a series of nonuniform steps, these sources of convection do not contribute equally to the layer turbulence level. This variability can drive a variety of processes including interface migration, layer splitting, and layer merging (Kelley et al. 2003). In our dataset, however, the estimates of εLAYER in the upper and lower layers were well correlated (R = 0.73), and their means were statistically indistinguishable (4.0 ± 0.5 and 4.1 ± 0.6 × 1010 W kg−1). For the present study, we use εLAYER derived from the averages of the layers above and below the HGR.
As Fig. 5 illustrates, the mean magnitude of microstructure shear (denoted 〈|ShμHGR|〉) within HGRs exceeds values in the upper and lower layers. We found that εLAYER is strongly correlated (R = 0.72) with 〈|ShμHGR|〉 within HGRs (Fig. 6), indicating that layers become more turbulent when shear in HGRs increases (note that correlation drops to R = 0.59 if the point with the highest values of εLAYER and 〈|ShμHGR|〉 is removed). This correlation holds when large steps are sorted by type (Fig. 6), with R = 0.70 for Type I, R = 0.65 for Type II, and R = 0.80 for Type III. We do not have the concurrent measurements of large-scale velocity U(z) needed to determine the source of higher 〈|ShμHGR|〉 in HGRs, although other measurements of U(z) through large DL steps (Fig. 3; Polyakov et al. 2012) suggest that layer turbulence may focus background imposed shear at interfaces and HGRs or provide the forcing for development of interfacial turbulence.
Relationship between mean microscale velocity shear magnitude 〈|ShμHGR|〉 and dissipation of turbulent kinetic energy in the thick convective layers εLAYER, sorted by the HGR types. The correlation between εLAYER and 〈|ShμHGR|〉 of R = 0.72 suggests a relationship between turbulence in the well-mixed layers and buoyancy flux through the HGRs resulting from either the DL instability or shear-driven turbulence.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
The associated heat fluxes across large layers 
Heat fluxes (W m−2) for all types of HGRs: shear-based FHLAYER; laboratory-based FHHGR-lab (see text for details); and HGR flux FHHGR, based on weighted fluxes for DL interfaces, DL layers, and turbulent patches. Ratios q1 and q2 compare FHHGR and FHHGR-lab to FHLAYER.
2) Heat flux across HGRs from DL flux laws
Estimates of heat flux for Type I and Type II HGRs (e.g., Figs. 2a–h) were derived from Rρ and Δθ using the Kelley (1990) flux law applied to individual DL steps within the HGR, and to the full temperature step scaled by the number n of individual DL interfaces. The Kelley (1990) flux law is an empirical fit to laboratory measurements of DL fluxes, with the assumption that FH is proportional to Δθ4/3 (e.g., Turner 1965, 1973) based on comparisons with heat transfer at solid conducting planes. However, the “4/3 flux law” assumes that the flux through an interface is independent of the thickness of the adjacent layers (i.e., that the convecting layers are sufficiently thick relative to the thickness of the interfaces; Turner 1965).
Variability in characteristics of individual interfaces within a single HGR leads to convergence or divergence of heat, salt, and buoyancy fluxes, thus altering the values of Δθ and Rρ that would be applicable for future calculation of fluxes through individual interfaces. Given that our primary interest is in the long-term averaged fluxes associated with Arctic HGRs in large DL steps, we assume that the average flux through each interface in an HGR is identical such that diapycnal flux divergence is zero. We cannot, however, estimate whether layer splitting and merging, as discussed by Kelley et al. (2003) and others, might lead to rapid changes in n.
With the assumption of uniform DL fluxes through the individual interfaces in an HGR, we first calculated a heat flux 
For the set of 31 (Types I and II only) HGRs, n varied from 2 to 16, with three HGRs having n ≥ 10. On average, n ≈ 4.8. Estimates of Rρ for individual DL interfaces within each high-gradient region were similar to each other and to the average for the entire large high-gradient region, indicating that the relationship between dS/dz and dθ/dz was fairly constant between the bounding upper and lower large layers. For the following analyses, we used 
The ratio q1 of 
3) Mixing in the stratified HGRs
Each HGR is composed of some combination of small DL interfaces, DL convective layers, and turbulent patches (Figs. 2, 3). Physical mechanisms driving heat transfer across these HGR segments differ, dictating different mathematical expressions for heat fluxes for these three segments of HGRs.
This “mixing efficiency” approach was developed originally for shear-driven turbulence (i.e., mixing created by nonlinear processes such as Kelvin–Helmholtz and Holmboe instabilities; Turner 1973). A typical value of Γ for shear-driven turbulence is ~0.2 (e.g., Gregg 1987). Inoue et al. (2007) proposed that the approach could be extended to cases, such as the DL instability, where buoyancy flux is the driving force. Those authors found that in buoyancy-driven flows, the data fit to Eq. (7) suggested that Γ ≈ 1. This value of Γ was confirmed by Hieronymus and Carpenter [2016, their Eq. (14)] through analyses of the equations of motion with simplifications appropriate to the DL instability.
The choice of Γ ≈ 1 for mixing patches within Type II interfaces is reasonable, given the dominance of the DL instability. For Type III HGRs, we evaluated the optimum value of Γ by comparing 
The rationale for this approach is that we are attempting to estimate the time-averaged fluxes associated with HGRs. For a single microstructure profile, estimates of fluxes for single categories (layer, interface, turbulent patch) may vary substantially from each other even though, in a time-averaged sense, the vertical flux of heat is continuous through the entire HGR (associated lateral flux divergence is small). The weighting in Eq. (9) is, therefore, an attempt to use vertical spatial averaging of instantaneous fluxes to reduce uncertainty in the time-averaged values.
Estimated fluxes were strongest in turbulent patches (mean 3.6 W m−2), somewhat weaker in convective layers (mean 3.5 W m−2), and the weakest across DL interfaces (mean 1.5 W m−2; Fig. 7), although the latter may be underestimated through underresolving temperature gradients with the FP07 thermistor on the VMP (Sommer et al. 2013). Type III HGRs (Fig. 2c) that are composed solely of turbulent patches yield the strongest heat fluxes (Table 1). Recall that for Type III HGRs, flux is linearly dependent on the choice of Γ, so that our choice of Γ = 1 is an upper bound on fluxes.
(a)–(d) Heat fluxes 
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
Uncertainties in estimates for heat fluxes in small DL layers and turbulent patches depend on the noise in measurements of shear that lead to the uncertainties in ε. In contrast, uncertainties in 
Fluxes for individual components in the calculation of net HGR heat flux decline with an increase in the number of small DL steps n, as quantified by negative trends of heat fluxes (Fig. 7). Note that elimination of points associated with the highest n (=15) does not change the result: negative trends still dominate. Although the strongest fluxes are found across the individual turbulent patches, convective DL layers represent the thickest segment of HGRs of Type I and Type II, and as a result, their overall contribution to heat flux across HGRs for these two types is the greatest (Fig. 7). In addition, the computed conductive component of 
The comparison of HGR fluxes from this approach with the concurrent values derived from εLAYER, sorted by the three HGR types shown in Fig. 2, shows that these two fluxes have similar means and are significantly correlated (Fig. 8). The mean ratio of HGR-based to layer-based fluxes for each step type varies by about 10%–20%, with ratios of 0.8 ± 0.3, 1.0 ± 0.1, and 1.1 ± 0.2 for Types I, II, and III, respectively (Table 1). The large errors preclude definitive categorization of ratios by interface type.
Comparison of heat fluxes across DL layers and corresponding HGRs, color coded by HGR type. The fluxes are correlated with mean slope of ~1, and the mean fluxes are similar.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
4. Discussion
The three approaches to calculating heat flux through the large DL steps (section 3) all give similar mean values of ~3–4 W m−2 (Table 1). Given that our estimated fluxes are comparable with the value needed to explain the downstream cooling of the AW layer in the eastern Arctic, we conclude that the DL instability is a major contributor to this regional cooling of the AW layer.
In most profiles, however, the region of large steps is overlaid by a staircase of much smaller steps, as seen in Fig. 1b. Fluxes through these small steps are of order 0.1 W m−2, based on thermal microstructure in interfaces that are assumed to carry fluxes by molecular diffusion (Sirevaag and Fer 2012). Therefore, if no other mechanism exists for increasing these fluxes, most of the heat delivered by the uppermost large step to the base of the staircase of small steps cannot escape farther upward. In this situation, the vertical heat flux convergence may broaden the layer of warm AW to the base of the small steps or initiate lateral heat transports within the EB halocline.
Developing methods to represent the DL-driven fluxes in Arctic Ocean circulation models is, therefore, required to improve predictions of AW layer distributions under projected future climates. Here, we discuss conditions that allow for generation of these large steps.
a. Generation and maintenance of large steps
We consider two possible generation mechanisms for the large steps found just above the AW layer throughout much of the EB: 1) convection driven by diapycnal fluxes immediately above the AW layer and 2) homogenization of intrusions.
1) Diapycnal fluxes
In laboratory experiments, DL layers develop as a salinity-stratified fluid is heated from below. In this case, the heating provides a source of buoyancy that, over time, drives the development of a staircase that can then continue to transport heat through the DL instability even if the original source of heat and buoyancy is removed. A time scale for the development of a layer can be estimated from the heat content change Q required to create a well-mixed layer of height H from an initially linear profile of θ and a temperature change across the resulting HGR after layer homogenization of Δθ. For frequently observed values of H = 25 m and Δθ = 0.36°C, Q = 0.5ρcPHΔθ is ~2 × 107 J m−2. With an estimated heat flux of FH = 3–4 W m−2, it would take of order 1 month to create the first thick layer above the AW core. This time scale is short compared with the ~6–10 months required for AW to be advected in the boundary current along the ~800-km length of the eastern EB slope from St. Anna Trough to the central Laptev Sea at a typical speed of 3–5 cm s−1 (Pnyushkov et al. 2015).
The initiation of a DL layer in this manner requires, however, a source of negative buoyancy flux, not simply a heat flux. If the DL instability is initially inactive, then the only other potential source for a strong heat flux from the AW layer is shear instability, but this would be accompanied by a salinity flux consistent with a flux ratio of RF = Rρ > 1 instead of RF < 1 that is expected for the DL instability (see Kelley 1984, 1990; and section 2). It is possible that instead, the initiation of the DL instability arises through other sources of vertical hydrographic structure and diapycnal divergence of fluxes, including the potential for energetic shear instabilities in regions of strong internal tide generation, within eddies, or as the boundary current of AW interacts with the eastern Arctic continental slope. However, we presently lack the data required to test these speculative hypotheses.
2) Intrusions
Layer heights for the large nearly homogeneous layers studied here are comparable with the vertical length scales of intrusions found in the eastern EB (cf. Figs. 1b,c). Merryfield (2000) proposed that in a region dominated by salt-fingering stratification, heat and salt flux divergences within intrusions could lead to a stable equilibrium staircase with homogeneous layers and sharp interfaces. Bebieva and Timmermans (2017) carried out a similar study for a DL-dominated stratification in the western Arctic Ocean.
There is some evidence that for given background vertical gradients of θ and S, there is an optimum DL layer height (Kelley 1984, his Fig. 1). The thickness of the layers in the large DL steps discussed here greatly exceeds these predicted optimum values. Kelley et al. (2003) propose that layers split and merge depending on the energetics of layer turbulence relative to stabilizing buoyancy gradients at interfaces. If this conceptual model is correct, then the breakdown of large steps into multiple smaller DL steps (e.g., Figs. 2a,b) is a consequence of weakened convective motion in the overly large layers.
The same heat balance estimates as for the DL instability can be applied to the transformation of intrusions to well-developed convective layers separated by sharp interfaces; time scales are similar and short compared with advective time scales for AW flow along the EB slope. Therefore, based solely on time scales, we cannot distinguish between these two potential mechanisms for generating the large DL layers discussed in this study. However, intrusion dynamics provide a mechanism for initiating vertical variability in stratification that may then interact with external sources of shear (Fig. 3) to create substantial diapyncal fluxes and heat, salt, and buoyancy flux convergences to initiate large DL steps. We tentatively propose that the large DL steps reported here originate from intrusions, possibly where the two flavors of AW—from the boundary current and via the Barents Sea—meet near St. Anna Trough. In this scenario, the large observed values of Δθ across the HGRs are set by the intrusion characteristics. The formation of multiple internal DL steps (examples in Fig. 2) within each Type I and Type II HGR would then be determined by the layer and interface energetics described previously to explain the rough functional relationship between Rρ and normalized layer heights (Kelley et al. 2003, their Fig. 5) that may be achieved through layer splitting (Kelley 1988).
b. Limitations of the analyses
Based on our analyses of 44 microstructure profiles from near the Laptev Sea continental slope (Fig. 1), the Kelley (1990) laboratory-based DL flux formula provides a reasonable assessment of fluxes measured in large DL steps in the eastern EB, provided the fine structure within each HGR is resolved. If, however, the structure of an HGR is underresolved in measurements (e.g., from older CTD measurements), application of the Kelley (1990) flux model to the resolved characteristics of staircases leads to flux estimates that can be an order of magnitude too large. The flux through large steps appears to be relatively insensitive to the structure of these steps, which fall into the three broad categories illustrated by Fig. 2. This observation suggests that characteristics of HGRs adjust to carry a specified flux that is determined by the hydrographic differences between adjacent thick DL layers, consistent with generalized arguments proposed by Kelley (1984, 1988) and other studies. Our data do not allow us to test whether this flux is consistent across the entire EB; it is plausible that the structure of large interfaces varies across the EB depending on other factors such as proximity to sources of mean and time-dependent shear including internal tides.
The noise floor in our microstructure shear measurements corresponds to a lower bound on heat flux of ~2 W m−2. Comparisons between fluxes through individual DL interfaces evaluated from microscale thermal gradients, with fluxes in the DL layers from microscale shear, indicate that fluxes through Type I and Type II HGRs clearly exceed this lower bound. However, we cannot use shear-based methods for evaluating DL fluxes through the staircase of small steps that lies above the large steps studied here (Fig. 1b). For these steps, the only approach consistent with available microstructure data is evaluation of laminar (Fickian) fluxes through the interfaces (see Sirevaag and Fer 2012). These studies confirm that fluxes through small steps are low, so that the relatively strong heat fluxes delivered to the top of the set of large DL steps cannot continue upward. However, a bulk heat budget analysis for the central EB by Polyakov et al. (2013) found wintertime heat fluxes through the cold halocline above the staircase of about 3–4 W m−2, similar to the flux through the large DL steps just above the AW layer, despite the apparently low fluxes in the staircase of small steps. The Polyakov et al. (2013) study needs to be reconciled with the Sirevaag and Fer (2012) and Kelley (1990) estimates of very low fluxes in the upper thermocline.
One possible mechanism for increasing the flux through the small-steps staircase is the interaction of externally imposed shear with the DL instability (Padman 1994), which relies on shear within the convective layers becoming sufficiently large to create dynamic instabilities of the DL interfaces, so that fluxes through interfaces are turbulent rather than limited to molecular diffusion. The mechanism relies on external shear being significant, but sufficiently low to avoid disrupting the DL layering. Given current limitations on noise levels for shear sensors and resolution of stable microstructure temperature sensors, observing this mechanism and quantifying its impact on net fluxes requires a measurement program that can simultaneously measure fully resolved interfacial gradients of ∂Tμ/∂z and microscale shear magnitude |uz|, and the vertical profile of absolute velocity as a measure of external shear. Multiple profiles would be required, with some profiles at a low fall rate for resolved ∂Tμ/∂z (~0.1–0.2 m s−1; Sommer et al. 2013) and others at a higher fall rate for resolving interfacial shear. Absolute velocity could be obtained from a moored high-frequency ADCP, with the highest-quality data being retrieved from a mooring mounted to sea ice. Alternatively, microconductivity sensors can provide higher vertical resolution of temperature gradients at the present instrument fall speed (Washburn et al. 1996); however, these sensors are sensitive to biofouling and experience large drifts, requiring frequent calibrations to be useful. New microstructure measurements, preferably coincident with observations of external shear, should also cover a larger fraction of the EB since locations of higher flux through the staircase of small steps may not coincide with regions of highest fluxes through the large steps that are the focus of the present study.
5. Conclusions
Using a microstructure dataset from the eastern Arctic Ocean, we have demonstrated that large double-diffusive steps just above the Atlantic Water (AW) layer in the Eurasian Basin (EB) carry substantial vertical heat fluxes, of order 2–8 W m−2 with a mean of ~3–4 W m−2, upward from the core of the AW layer via the diffusive layering (DL) instability. These layers are almost ubiquitous throughout the deep water of the EB (see appendix), so that their integrated contribution to cooling and broadening the AW layer in the Arctic Ocean should be substantial.
Detailed examination of high-gradient regions (HGRs) identified in microstructure profiles shows several types of interface structure, ranging from multiple small DL steps to an HGR lacking the characteristics of the DL instability (Fig. 2). These large, complex HGRs are frequently the sites of large microstructure shear magnitude (Figs. 2, 3, 5). This microscale shear might represent either imposed large-scale shear (e.g., from baroclinic tides), as seen in profiles of finescale velocity from a moored profiler (Polyakov et al. 2012; Fig. 3), or enhanced levels of turbulent mixing. Comparisons between turbulent dissipation rate and inferred heat flux through the layers, and shear within the HGRs (Fig. 6), confirm this relationship. The mean ratio of the two estimates of heat flux—one based on layer-averaged dissipation rate and the other on the Kelley (1990) laboratory-based DL formula (flux law) scaled by the number of sublayers—is close to unity for HGRs with visible steps (Table 1), indicating that the flux law gives a reasonable estimate of heat flux, provided the details of the HGRs are adequately resolved.
Our results confirm the previous estimates of DL heat fluxes in the eastern EB based on application of flux laws and show the extent of these features above the AW layer throughout the eastern Arctic. The resulting fluxes in the range 2–8 W m−2 are consistent with the estimates of heat fluxes for the bigger steps reported by Lenn et al. (2009). However, these large DL steps are overlaid by much smaller steps suggesting heat fluxes of order 0.1–1 W m−2. That is, although the large DL steps carry sufficient flux to rapidly cool and freshen the AW layer, over much of the EB, we require another mechanism to move this heat from the top of the uppermost large DL layer to the upper ocean where it could affect sea ice volume. Alternatively, this vertical flux divergence may be balanced by strong lateral AW heat fluxes within the Arctic halocline.
Given that DL fluxes above the AW layer can be large, the steps are spatially extensive, and other mechanisms may be able to transport the AW heat to the surface (e.g., Polyakov et al. 2017), we need to further improve our understanding of how DL fluxes will vary as the Arctic Ocean changes through modifications to AW inflow, surface heat fluxes, and freshwater fluxes from precipitation, river inputs, the sea ice annual cycle, and wind-driven export of ice and liquid freshwater (Carmack et al. 2015). This is a challenging goal that can only be achieved by a combination of additional targeted field work, improved parameterizations of DL fluxes that take into account the existing limitations on resolving structures within HGRs, and numerical models for testing the possible complex response of the Arctic Ocean and its sea ice cover to the parameterized DL fluxes.
Acknowledgments
The ship-based oceanographic observations in the eastern EB and Laptev Sea were conducted under the working frame of the NABOS project with support from NSF (Grants AON-1203473 and AON-1338948) and NERC (NE/D005752/1). Analyses presented in this paper are supported by NSF Grants 1249133 and 1249182 and NERC Grant NE/H016007/1. The Ice-Tethered Profiler data were collected and made available by the Ice-Tethered Profiler Program (Toole et al. 2011; Krishfield et al. 2008) based at the Woods Hole Oceanographic Institution (http://www.whoi.edu/itp). VI acknowledges funding from the Ministry of Science and Higher Education of the Russian Federation (Project RFMEFI61617X0076). We thank Jeff Carpenter and two anonymous reviewers for their valuable comments on previous versions of this manuscript.
APPENDIX
Ubiquity of Large DL Steps in the EB
We illustrate the spatial coherence of large DL steps with three transects (locations shown in Fig. A1d) developed using combinations of CTD and ITP data (Fig. A1). For each temperature profile, we applied the method described by Polyakov et al. (2012) to identify DL steps: calculation of a potential temperature anomaly profile θ′(z) relative to a smoothed profile of θ(z) generated with a 3-m running average. The choice of 3 m for filter length is based on optimizing the identification of HGRs between thick, nearly isothermal layers. Local negative-over-positive anomalies in θ′(z) define the upper and lower boundaries of HGRs.
(a)–(c) Vertical sections (depth; m vs profile number) of potential temperature anomalies (°C) composed from CTD and ITP records. Blue over red stripes identify double-diffusive interfaces or intrusions. Gray vertical segments show missing data. White lines show the position of the isopycnal surface σo = 27.80. Red/green segments at the top of the sections identify DL steps (green) and intrusions (red) at σo = 27.80. (d) Position of sections; FJL and FS are used for Franz Joseph Land and Fram Strait.
Citation: Journal of Physical Oceanography 49, 1; 10.1175/JPO-D-18-0080.1
Section I (Fig. A1a) extends >2000 km along the entire EB from the continental slope in the east to Fram Strait in the west, while section II (Fig. A1b) crosses the eastern EB and section III (Fig. A1c) crosses the central Nansen Basin. These composite cross sections demonstrate strong spatial coherence of DL interfaces throughout the EB. Moreover, they show that the spatial pattern has very distinct features between the eastern and central EB from one side and western EB from the other. For example, section I (Fig. A1a) shows very clear DL interfaces in the eastern and central EB (CTD-09, ITP-36, and ITP-37), whereas the record from the western EB (ITP-38) indicates little spatial coherence. Remarkably, interfaces are always found at the same isopycnal surfaces regardless of specific location or time (e.g., Polyakov et al. 2012). For example, the top of the shallowest thick DL layer in the three transects shown in Fig. A1 is almost always near the depth of the σo = 27.80 isopycnal.
Reduced spatial coherence arises partially from the presence of intrusions [cf. similar measurements from the western Arctic reported by Bebieva and Timmermans (2017)]. The intrusions were distinguished from DL features by checking temperature distributions in both layers above and below an interface (if ∂θ/∂z < 0 is found, then the structure is identified as an intrusion). As an example, ITP-36 profiles 35–50 in transect I (Fig. A1a) show intrusive vertical structure with no well-mixed DL layers. This is distinct from most other eastern EB profiles where the DL layers are present: hundreds of ITP-37 profiles show the classic DL structure with homogeneous convective layers above and below interfaces. Intrusions are often clustered together (Fig. A1). One cluster is near St. Anna Trough, a region that is famous for its active interactions between the Fram Strait and Barents Sea branches of the AW (e.g., Schauer et al. 1997).
The nearly ubiquitous presence of large DL steps above the AW core throughout the EB (Fig. A1) suggests that if heat fluxes associated with these features are large, they would play an important role in the transformation of AW hydrographic properties as it circulates around the eastern Arctic Ocean.
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