1. Introduction
The Arctic Ocean has a strong halocline and deeper water layers that are warmer than those at the surface in contact with sea ice cover (e.g., Aagaard et al. 1981). Understanding the mechanisms and magnitude of upward fluxes of deep-ocean heat is essential to predictions of Arctic sea ice and climate (e.g., Maykut and Untersteiner 1971; Wettlaufer 1991; Perovich et al. 2008; Carmack et al. 2015; Timmermans 2015). Relatively cold and fresh surface waters, originating from net precipitation, river runoff, inflows from the Pacific Ocean, and seasonal sea ice melt, occupy the upper ~150–200-m water column of the Arctic Ocean’s Canada Basin (e.g., Steele et al. 2008; Timmermans et al. 2014). Below these upper layers lie relatively warm and salty waters associated with Atlantic Water (AW) inflows, centered around 400-m depth in the Canada Basin, with lateral temperature gradients indicating a general cooling moving east from the warm core of the AW layer on the western side of the basin (Fig. 1a).
(a) Map showing locations of ITP 2 profiles over the course of its drift; colors indicate the AW potential temperature maximum (°C). (b) Potential temperature (°C, referenced to the surface) and (c) salinity profiles measured on 22 Aug 2004 (black lines) and 3 Sep 2004 (red lines); locations where both profiles were sampled are marked by squares with corresponding colors on the map in (a). The expanded scales highlight double-diffusive structures. (d) Potential temperature and salinity values measured by ITP 2 between ~200- and ~750-m depth (gray dots, all profiles are shown); black and red lines correspond to the profiles shown in the same colors in (b) and (c). Thin black contours indicate potential density anomaly (kg m−3) referenced to the surface.
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
(a) Map showing locations of ITP 2 profiles over the course of its drift; colors indicate the AW potential temperature maximum (°C). (b) Potential temperature (°C, referenced to the surface) and (c) salinity profiles measured on 22 Aug 2004 (black lines) and 3 Sep 2004 (red lines); locations where both profiles were sampled are marked by squares with corresponding colors on the map in (a). The expanded scales highlight double-diffusive structures. (d) Potential temperature and salinity values measured by ITP 2 between ~200- and ~750-m depth (gray dots, all profiles are shown); black and red lines correspond to the profiles shown in the same colors in (b) and (c). Thin black contours indicate potential density anomaly (kg m−3) referenced to the surface.
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
(a) Map showing locations of ITP 2 profiles over the course of its drift; colors indicate the AW potential temperature maximum (°C). (b) Potential temperature (°C, referenced to the surface) and (c) salinity profiles measured on 22 Aug 2004 (black lines) and 3 Sep 2004 (red lines); locations where both profiles were sampled are marked by squares with corresponding colors on the map in (a). The expanded scales highlight double-diffusive structures. (d) Potential temperature and salinity values measured by ITP 2 between ~200- and ~750-m depth (gray dots, all profiles are shown); black and red lines correspond to the profiles shown in the same colors in (b) and (c). Thin black contours indicate potential density anomaly (kg m−3) referenced to the surface.
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
The basic vertical stratification, in which temperature and salinity both increase with depth, provides conditions amenable to double-diffusive instability, believed to be a key physical process generating thermohaline intrusions and staircases in the Arctic Ocean. Throughout much of the central Arctic Ocean Basins, heat transfer from the AW layer is via double-diffusive convection (e.g., Melling et al. 1984; Padman and Dillon 1987, 1988; Timmermans et al. 2008; Polyakov et al. 2012; Sirevaag and Fer 2012; Guthrie et al. 2015). Two types of double-diffusive convection can arise in a stably stratified ocean: the case when both temperature T and salinity S increase with depth is referred to as diffusive convection [DC; an overview is given by Kelley et al. (2003)], while the case when both temperature and salinity decrease with depth is referred to as the salt-finger (SF) regime [for overviews, see, e.g., Kunze (2003) and Schmitt (2003)]. These stratifications may be characterized by a density ratio, which we define here as 
At the top boundary of the AW layer in the Canada Basin, a prevalent DC staircase is characterized by a sequence of mixed layers of thickness on the order of several meters separated by sharp gradients in temperature and salinity (e.g., Neal et al. 1969; Padman and Dillon 1987; Timmermans et al. 2008; Figs. 1b,c). Thermohaline intrusions, characterized by DC and SF regions alternating in depth, are often found underlying the staircase (e.g., Carmack et al. 1998; Merryfield 2002; Woodgate et al. 2007). Intrusions are believed to be associated with lateral (in addition to vertical) gradients in temperature and salinity and are driven partly by vertical buoyancy flux divergences [overviews of thermohaline intrusions are given by Ruddick and Kerr (2003) and Ruddick and Richards (2003)].
Understanding the observed vertical temperature–salinity structure (whether staircases or intrusions) and associated vertical and lateral heat fluxes from the AW layer requires knowledge of the origins of these features and their relationship to each other (see Kelley 2001). Extensive efforts have been made to explain staircase and intrusion origins and evolution with respect to the SF configuration of double diffusion. There exist around six theories for the origin of SF staircases, as reviewed by Radko (2013). One of these theories is that interleaving motions can develop into a staircase (Merryfield 2000). The idea relies on the presence of lateral temperature and salinity gradients and builds on previous studies that invoke a standard parametric flux model (Walsh and Ruddick 1995, 2000). This model was first introduced by Stern (1967), who delineated three separate scales of motion: small, to describe double-diffusive processes on centimeter to meter scales; medium, to describe the scales of interleaving (order tens of meters vertically and kilometer scales laterally); and large, to characterize the background state (the full thickness of the double-diffusive region in depth and tens to hundreds of kilometers laterally). The main assumption here is that medium-scale dynamics are qualitatively similar to small-scale dynamics and that the effects of double diffusion can be parameterized in terms of eddy diffusivities. Merryfield (2000) applied this formalism in his calculations showing that interleaving motions evolve into SF staircases when the density ratio of the background (SF stratified) state is below a certain value (i.e., there exists a critical density ratio that delineates the boundary between staircase and intrusion formation).
Very little has been done with respect to analysis of staircase origins in the DC-stratified setting. The purpose of this study is to address the relationships between and origins of DC staircases and intrusions, with consideration of those in the Arctic Ocean. Temperature and salinity profiles from the Canada Basin show the presence of both staircases and intrusions shallower than the AW temperature maximum. Often the exact structure differs from region to region; for example, in some regions, only a staircase is observed in the upper part of the AW layer (Figs. 1b,c; black line), while in others, intrusions are observed instead (Figs. 1b,c; red line). Motivated by these observations that show staircases and intrusions coexisting and evolving, we assess the main factors that may determine whether staircases or intrusions will be observed. Perhaps the major limitation on developing a mechanism for staircase formation in the Arctic Ocean has been considered to relate to the fact that the magnitude of the observed vertical density ratio (
The paper is organized as follows: In the next section, we formulate the governing equations (section 2a) and then determine a constraint on the growth rate for growing perturbations to evolve toward a DC staircase (section 2b). Next, in section 2c, we perform a linear stability analysis on the governing Boussinesq equations to determine the fastest-growing mode as a function of vertical and lateral temperature and salinity gradients. Together with the result from section 2b, this allows us to define a critical vertical density ratio 
2. Theory
a. Formulation of the governing equations
Estimates of KT and A
In the staircase region KT may be estimated using a double-diffusive heat flux parameterization based on a 4/3 flux law (Kelley 1990), which uses the temperature difference across an interface between two adjacent mixed layers and the bulk vertical temperature gradient; Guthrie et al. (2015) show the 4/3 flux law to be a reasonable representation of the fluxes. This yields KT = O(10−6) m2 s−1 for typical double-diffusive fluxes in the Canada Basin around 0.1–0.2 W m−2 (Padman and Dillon 1987; Timmermans et al. 2008). This estimate for KT is consistent with microstructure estimates by Guthrie et al. (2013) in the Canada Basin in the region of the staircase, where they find KT = 1–5 × 10−6 m2 s−1.
b. Constraint leading to a staircase
Schematic showing evolution of temperature and salinity according to (16) and (17) (following Merryfield 2002). Dashed lines are constant Rρ contours, bounded by two solid lines where Rρ = 1 and Rρ → ∞. Blue lines show evolution of the initial perturbation from a background state (red dot, characterized by 
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
Schematic showing evolution of temperature and salinity according to (16) and (17) (following Merryfield 2002). Dashed lines are constant Rρ contours, bounded by two solid lines where Rρ = 1 and Rρ → ∞. Blue lines show evolution of the initial perturbation from a background state (red dot, characterized by
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
Schematic showing evolution of temperature and salinity according to (16) and (17) (following Merryfield 2002). Dashed lines are constant Rρ contours, bounded by two solid lines where Rρ = 1 and Rρ → ∞. Blue lines show evolution of the initial perturbation from a background state (red dot, characterized by
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
We next perform a linear stability analysis (see, e.g., Toole and Georgi 1981; Walsh and Ruddick 1995, 2000) on the system (7)–(11) to compute the most unstable mode (i.e., maximal λ and corresponding k and m) for a specified 
c. Critical density ratio
In sum, we take the following steps to compute 
3. Context with ITP observations
As a consistency check, we aim to examine whether the presence of intrusions or staircases in the Arctic Ocean water column is commensurate with the linear theory described in the previous section. Given the significant uncertainty in system parameters, as well as nonlinear effects, we do not assert that the formalism may be used in a predictive capacity. Rather, the motivation for exploring the observations in context with the linear theory is to provide some physical intuition for the physics of staircases and intrusions in the Arctic setting. Water column measurements are from an ITP (Krishfield et al. 2008; Toole et al. 2011) that drifted in the Canada Basin (Fig. 1). ITPs are automated profiling instruments that provide measurements of temperature, salinity, and depth from several meters depth beneath the sea ice, through the Atlantic Water layer to about 750-m depth. The final processed data for ITP system number 2 (ITP 2), operating between August 2004 and September 2004, are analyzed here. Measurements have a vertical resolution of about 25 cm and a horizontal profile spacing of a few kilometers; full processing procedures are given by Krishfield et al. (2008). ITP data have been analyzed in several past studies of double diffusion in the Canada Basin (e.g., Timmermans et al. 2008; Radko et al. 2014; Bebieva and Timmermans 2016; Shibley et al. 2017). We will examine ITP profiles to compare the observed vertical density ratio 
a. Quantifying the temperature and salinity gradients
Before applying the theory, we require some method to assess the bulk temperature and salinity gradients (
1) Vertical gradients
To compute 
2) Horizontal gradients
The drift track of the ITP is approximately from west to east, with some meandering (Fig. 1a); the horizontal distance between profiles varies (between around 2 and 7 km) with variations in sea ice drift speed. All measurements are projected onto 77°N (the x direction) and interpolated to a 4-km horizontal grid. The temperature is taken along isopycnals (effectively parallel to isobars) in the considered depth range (260–360 m), and the lateral background gradient 
b. Comparison with theory
The basic temperature structure sampled by ITP 2 in the considered depth range is as follows: AW temperatures cool from west to east across the basin with stronger lateral gradients in the deeper portion of the depth range compared to the shallower portion (note that the considered depth range includes only the upper part of the AW layer where bulk temperature and salinity gradients are increasing with depth; Fig. 3a). At the depth ranges of interest, the water column structure is effectively unchanging in time over the sampling duration (~40 days) of the ITP as it drifted from west to east (Fig. 1a).
Depth (m)–distance (km) section of (a) potential temperature (°C, referenced to the surface). (b) Turner angle Tu (black contours indicate Tu = 45°); Tu is calculated after first computing a 5-m running average of the full-resolution (~25 cm) T and S profiles. (c) 
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
Depth (m)–distance (km) section of (a) potential temperature (°C, referenced to the surface). (b) Turner angle Tu (black contours indicate Tu = 45°); Tu is calculated after first computing a 5-m running average of the full-resolution (~25 cm) T and S profiles. (c)
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
Depth (m)–distance (km) section of (a) potential temperature (°C, referenced to the surface). (b) Turner angle Tu (black contours indicate Tu = 45°); Tu is calculated after first computing a 5-m running average of the full-resolution (~25 cm) T and S profiles. (c)
Citation: Journal of Physical Oceanography 47, 4; 10.1175/JPO-D-16-0265.1
Detailed examination of the temperature and salinity profiles indicates that some regions exhibit only staircases throughout the considered depth range, while others show either a staircase overlying intrusions or alternating (in depth) staircases and intrusions (Figs. 1b–d). Note that several studies have exploited T–S space in examining these features because the properties of staircase layers and intrusions (Fig. 1d) tend to lie along well-defined regions in T–S space (see, e.g., Timmermans et al. 2008; Walsh and Carmack 2003). However, for our purposes here, we have found that the best metric to characterize the range of double-diffusive structures is the Turner angle Tu = tan−1[(1 + Rρ)/(1 − Rρ)], which indicates the stability of a water column with respect to double-diffusive processes (Ruddick 1983). When a staircase is present, the water column is characterized by −90° < Tu < −45°. When intrusions are present, SF regions (45° < Tu < 90°) alternate in depth with DC regions (−90° < Tu < −45°). A doubly stable water column is characterized by −45° < Tu < 45°. The angle Tu is calculated after first computing a 5-m running average of the full-resolution (~25 cm) T and S profiles; this averaging was chosen as a trade-off between the necessity for fine vertical resolution and elimination of noise in the profiles. Provided the running averaging is over depth intervals smaller than around 5 m, the general pattern of the Tu section, delineating where SF and DC regions are present, is insensitive to the choice of smoothing. The upper part of the AW layer (~260–300 m) consists predominantly of a staircase in the sampled region (Fig. 3b). Deeper than around 320-m depth, SF-unstable regions (indicative of intrusions) may be present. For example, in the west, we observe a staircase over most of the considered depth range; see also the western profile in Figs. 1b and 1c (black lines, with position indicated by the vertical black dashed line in Fig. 3b). In the eastern part of the section, we observe intrusions around 320 m and deeper; see also the eastern profile in Figs. 1b and 1c (red lines, with position indicated by the vertical red dashed line in Fig. 3b).
The quantities 
Our linear stability analysis together with (28) indicates that 
It is instructive to examine the characteristic time and length scales of the most unstable modes leading to either a staircase or intrusions. The horizontal and vertical wavelengths of the most unstable modes are of the order of 1000–10 000 km and 15–50 m, respectively, in the sampled region of the Canada Basin; vertical scales are generally consistent with scales of variability in staircase/intrusion regions of the Canada Basin. Similar scales of the most unstable modes leading to either a staircase or intrusions suggest that these two end states are of the same nature. Of course, the linear theory cannot predict the transient evolution of the linear profile [e.g., how interleaving motions evolve and the associated scale adjustments such as layer splitting/merging (Radko et al. 2014)]; this requires a separate analysis [as performed, e.g., by Walsh and Ruddick (1998) and Li and McDougall (2015) for the SF case]. The time scale of the instability (determined from the fastest-growing mode) ranges from several months to about a year for both intrusions and a staircase and depends on the magnitude of the horizontal temperature gradient (with shorter time scales for larger 
Within our analysis is the implicit assumption that the development of intrusions or a staircase does not affect the magnitudes of the background gradient (i.e., that the background gradients do not evolve on time scales faster than the fastest-growing mode). An order-of-magnitude estimate for the time scale on which double-diffusive fluxes modify the background temperature gradients can be estimated as D2/KT, where D is a characteristic length scale for the background gradient. Taking D ~ 30 m for a vertical scale (a reasonable estimate over which linear gradients remain constant) and eddy diffusivity KT = 10−6 m2 s−1 yields time scales of decades for modification of the vertical background temperature gradient. Similarly, considering a vertical eddy diffusivity for salinity of around 10−7 m2 s−1 (e.g., Bebieva and Timmermans 2016), we find time scales for modification of the vertical background salinity gradient to be an order of magnitude longer than this. Following the same reasoning for the evolution of the lateral gradients [over O(50–100) km horizontal scales] and taking isopycnal diffusivities in the range 5–50 m2 s−1 (Hebert et al. 1990; Walsh and Carmack 2003) also yield time scales of decades for the lateral gradients to evolve. Comparison of these time scale estimates to the maximal growth rates found here (i.e., several months to a year) suggests background gradients do not evolve on time scales shorter than those associated with the development of a perturbation toward either a staircase or intrusions; thus, in this analysis, the background gradients may be approximated as independent of time.
4. Summary and discussion
We have examined a scenario for the origin of double-diffusive staircases and intrusions that are observed to coexist in the Arctic Ocean’s AW. A linear stability analysis of the governing equations was performed to determine the most unstable mode for a given horizontal and vertical linear temperature and salinity stratification that would lead toward either staircases or intrusions. Staircases are the end result of a perturbation if the observed vertical density ratio is below a critical vertical density ratio (
We have shown that the dominant factors that determine the presence of either a staircase or intrusions are 
For the SF configuration, Merryfield (2000) showed that staircases are the end result of a perturbation when the vertical density ratio (defined in the conventional way for the SF configuration, inverse to the definition given here) is small, while intrusions are the end state when that density ratio takes larger values. We have used the same formalism here to demonstrate that an analogous result is applicable to the DC case. That is, a staircase is a possible end state of an interleaving perturbation. Merryfield (2000) showed that a staircase is more likely for smaller values of the eddy diffusivity for salt. By contrast, we have shown that (for the DC configuration) a staircase is more likely for larger values of eddy diffusivity for heat KT (i.e., smaller σ), although the physics of these relationships needs to be explained.
While the linear stability analysis and separation of scales framework described here may be instructive for understanding the general structure of a water column profile (i.e., staircases or intrusions), the Arctic Ocean observations often demonstrate a detailed finestructure that is somewhat more complicated (e.g., Padman and Dillon 1989). Our analysis cannot describe layers that are observed between and within the main staircase mixed layers and intrusive structures. These features, however, may be key to the transition between interleaving structures and staircases—a conjecture that will require further analysis.
Acknowledgments
The Ice-Tethered Profiler data were collected and made available by the Ice-Tethered Profiler Program based at the Woods Hole Oceanographic Institution (Krishfield et al. 2008; Toole et al. 2011); data are available online (at http://www.whoi.edu/itp/data). Funding was provided by the National Science Foundation Division of Polar Programs under Award Number 1350046.
REFERENCES
Aagaard, K., L. Coachman, and E. Carmack, 1981: On the halocline of the Arctic Ocean. Deep-Sea Res., 28A, 529–545, doi:10.1016/0198-0149(81)90115-1.
Bebieva, Y., and M.-L. Timmermans, 2016: An examination of double-diffusive processes in a mesoscale eddy in the Arctic Ocean. J. Geophys. Res. Oceans, 121, 457–475, doi:10.1002/2015JC011105.
Bertsekas, D. P., 2014: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, 344 pp.
Carmack, E. C., K. Aagaard, J. H. Swift, R. G. Perkin, F. A. McLaughlin, R. W. Macdonald, and E. P. Jones, 1998: Thermohaline transitions. Physical Processes in Lakes and Oceans, J. Imberger, Ed., Coastal and Estuarine Studies, Vol. 54, Amer. Geophys. Union, 179–186.
Carmack, E. C., and Coauthors, 2015: Toward quantifying the increasing role of oceanic heat in sea ice loss in the new Arctic. Bull. Amer. Meteor. Soc., 96, 2079–2105, doi:10.1175/BAMS-D-13-00177.1.
Gregg, M., 1987: Diapycnal mixing in the thermocline: A review. J. Geophys. Res., 92, 5249–5286, doi:10.1029/JC092iC05p05249.
Guthrie, J. D., J. H. Morison, and I. Fer, 2013: Revisiting internal waves and mixing in the Arctic Ocean. J. Geophys. Res. Oceans, 118, 3966–3977, doi:10.1002/jgrc.20294.
Guthrie, J. D., I. Fer, and J. Morison, 2015: Observational validation of the diffusive convection flux laws in the Amundsen Basin, Arctic Ocean. J. Geophys. Res. Oceans, 120, 7880–7896, doi:10.1002/2015JC010884.
Hebert, D., N. Oakey, and B. Ruddick, 1990: Evolution of a Mediterranean salt lens: Scalar properties. J. Phys. Oceanogr., 20, 1468–1483, doi:10.1175/1520-0485(1990)020<1468:EOAMSL>2.0.CO;2.
Kelley, D. E., 1990: Fluxes through diffusive staircases: A new formulation. J. Geophys. Res., 95, 3365–3371, doi:10.1029/JC095iC03p03365.
Kelley, D. E., 2001: Six questions about double-diffusive convection. From Stirring to Mixing in a Stratified Ocean: Proc. ‘Aha Huliko’a Hawaiian Winter Workshop, Honolulu, HI, University of Hawai‘i at Mānoa, 191–198. [Available online at http://www.soest.hawaii.edu/PubServices/2001pdfs/Kelley.pdf.]
Kelley, D. E., H. Fernando, A. Gargett, J. Tanny, and E. Özsoy, 2003: The diffusive regime of double-diffusive convection. Prog. Oceanogr., 56, 461–481, doi:10.1016/S0079-6611(03)00026-0.
Kerr, O. S., and J. Y. Holyer, 1986: The effect of rotation on double-diffusive interleaving. J. Fluid Mech., 162, 23–33, doi:10.1017/S0022112086001908.
Krishfield, R., J. Toole, A. Proshutinsky, and M.-L. Timmermans, 2008: Automated Ice-Tethered Profilers for seawater observations under pack ice in all seasons. J. Atmos. Oceanic Technol., 25, 2091–2105, doi:10.1175/2008JTECHO587.1.
Kunze, E., 2003: A review of oceanic salt-fingering theory. Prog. Oceanogr., 56, 399–417, doi:10.1016/S0079-6611(03)00027-2.
Li, Y., and T. J. McDougall, 2015: Double-diffusive interleaving: Properties of the steady-state solution. J. Phys. Oceanogr., 45, 813–835, doi:10.1175/JPO-D-13-0236.1.
Maykut, G. A., and N. Untersteiner, 1971: Some results from a time-dependent thermodynamic model of sea ice. J. Geophys. Res., 76, 1550–1575, doi:10.1029/JC076i006p01550.
McDougall, T. J., 1985: Double-diffusive interleaving. Part II: Finite amplitude, steady state interleaving. J. Phys. Oceanogr., 15, 1542–1556, doi:10.1175/1520-0485(1985)015<1542:DDIPIF>2.0.CO;2.
Melling, H., R. Lake, D. Topham, and D. Fissel, 1984: Oceanic thermal structure in the western Canadian Arctic. Cont. Shelf Res., 3, 233–258, doi:10.1016/0278-4343(84)90010-4.
Merryfield, W. J., 2000: Origin of thermohaline staircases. J. Phys. Oceanogr., 30, 1046–1068, doi:10.1175/1520-0485(2000)030<1046:OOTS>2.0.CO;2.
Merryfield, W. J., 2002: Intrusions in double-diffusively stable Arctic waters: Evidence for differential mixing? J. Phys. Oceanogr., 32, 1452–1459, doi:10.1175/1520-0485(2002)032<1452:IIDDSA>2.0.CO;2.
Neal, V. T., S. Neshyba, and W. Denner, 1969: Thermal stratification in the Arctic Ocean. Science, 166, 373–374, doi:10.1126/science.166.3903.373.
Padman, L., 1994: Momentum fluxes through sheared oceanic thermohaline steps. J. Geophys. Res., 99, 22 491–22 499, doi:10.1029/94JC01741.
Padman, L., and T. M. Dillon, 1987: Vertical heat fluxes through the Beaufort Sea thermohaline staircase. J. Geophys. Res., 92, 10 799–10 806, doi:10.1029/JC092iC10p10799.
Padman, L., and T. M. Dillon, 1988: On the horizontal extent of the Canada Basin thermohaline steps. J. Phys. Oceanogr., 18, 1458–1462, doi:10.1175/1520-0485(1988)018<1458:OTHEOT>2.0.CO;2.
Padman, L., and T. M. Dillon, 1989: Thermal microstructure and internal waves in the Canada Basin diffusive staircase. Deep-Sea Res., 36A, 531–542, doi:10.1016/0198-0149(89)90004-6.
Perovich, D. K., J. A. Richter-Menge, K. F. Jones, and B. Light, 2008: Sunlight, water, and ice: Extreme Arctic sea ice melt during the summer of 2007. Geophys. Res. Lett., 35, L11501, doi:10.1029/2008GL034007.
Polyakov, I. V., A. V. Pnyushkov, R. Rember, V. V. Ivanov, Y.-D. Lenn, L. Padman, and E. C. Carmack, 2012: Mooring-based observations of double-diffusive staircases over the Laptev Sea slope. J. Phys. Oceanogr., 42, 95–109, doi:10.1175/2011JPO4606.1.
Radko, T., 2013: Double-Diffusive Convection. Cambridge University Press, 344 pp.
Radko, T., J. Flanagan, S. Stellmach, and M.-L. Timmermans, 2014: Double-diffusive recipes. Part II: Layer-merging events. J. Phys. Oceanogr., 44, 1285–1305, doi:10.1175/JPO-D-13-0156.1.
Reynolds, O., 1894: On the dynamical theory of incompressible viscous fluids and the determination of the criterion. Proc. Roy. Soc. London, 56, 40–45, doi:10.1098/rspl.1894.0075.
Ruddick, B., 1983: A practical indicator of the stability of the water column to double-diffusive activity. Deep-Sea Res., 30A, 1105–1107, doi:10.1016/0198-0149(83)90063-8.
Ruddick, B., and D. Hebert, 1988: The mixing of meddy “Sharon.” Small-Scale Turbulence and Mixing in the Ocean, J. C. J. Nihoul and B. M. Jamart, Eds., Elsevier Oceanography Series, Vol. 46, 249–261, doi:10.1016/S0422-9894(08)70551-8.
Ruddick, B., and O. Kerr, 2003: Oceanic thermohaline intrusions: Theory. Prog. Oceanogr., 56, 483–497, doi:10.1016/S0079-6611(03)00029-6.
Ruddick, B., and K. Richards, 2003: Oceanic thermohaline intrusions: Observations. Prog. Oceanogr., 56, 499–527, doi:10.1016/S0079-6611(03)00028-4.
Schmitt, R. W., 2003: Observational and laboratory insights into salt finger convection. Prog. Oceanogr., 56, 419–433, doi:10.1016/S0079-6611(03)00033-8.
Shibley, N., M.-L. Timmermans, J. Carpenter, and J. Toole, 2017: Spatial variability of the Arctic Ocean’s double-diffusive staircase. J. Geophys. Res. Oceans, 122, 980–994, doi:10.1002/2016JC012419.
Sirevaag, A., and I. Fer, 2012: Vertical heat transfer in the Arctic Ocean: The role of double-diffusive mixing. J. Geophys. Res., 117, C07010, doi:10.1029/2012JC007910.
Steele, M., W. Ermold, and J. Zhang, 2008: Arctic Ocean surface warming trends over the past 100 years. Geophys. Res. Lett., 35, L02614, doi:10.1029/2007GL031651.
Stern, M. E., 1967: Lateral mixing of water masses. Deep-Sea Res. Oceanogr. Abstr., 14, 747–753, doi:10.1016/S0011-7471(67)80011-1.
Timmermans, M.-L., 2015: The impact of stored solar heat on Arctic sea ice growth. Geophys. Res. Lett., 42, 6399–6406, doi:10.1002/2015GL064541.
Timmermans, M.-L., J. Toole, R. Krishfield, and P. Winsor, 2008: Ice-Tethered Profiler observations of the double-diffusive staircase in the Canada Basin thermocline. J. Geophys. Res., 113, C00A02, doi:10.1029/2008JC004829.
Timmermans, M.-L., and Coauthors, 2014: Mechanisms of Pacific summer water variability in the Arctic’s central Canada Basin. J. Geophys. Res. Oceans, 119, 7523–7548, doi:10.1002/2014JC010273.
Toole, J. M., and D. T. Georgi, 1981: On the dynamics and effects of double-diffusively driven intrusions. Prog. Oceanogr., 10, 123–145, doi:10.1016/0079-6611(81)90003-3.
Toole, J. M., R. A. Krishfield, M.-L. Timmermans, and A. Proshutinsky, 2011: The Ice-Tethered Profiler: Argo of the Arctic. Oceanography, 24, 126–135, doi:10.5670/oceanog.2011.64.
Turner, J., 1965: The coupled turbulent transports of salt and heat across a sharp density interface. Int. J. Heat Mass Transfer, 8, 759–767, doi:10.1016/0017-9310(65)90022-0.
Veronis, G., 1965: On finite amplitude instability in thermohaline convection. J. Mar. Res., 23, 1–17.
Walsh, D., and B. Ruddick, 1995: Double-diffusive interleaving: The influence of nonconstant diffusivities. J. Phys. Oceanogr., 25, 348–358, doi:10.1175/1520-0485(1995)025<0348:DDITIO>2.0.CO;2.
Walsh, D., and B. Ruddick, 1998: Nonlinear equilibration of thermohaline intrusions. J. Phys. Oceanogr., 28, 1043–1070, doi:10.1175/1520-0485(1998)028<1043:NEOTI>2.0.CO;2.
Walsh, D., and B. Ruddick, 2000: Double-diffusive interleaving in the presence of turbulence: The effect of a nonconstant flux ratio. J. Phys. Oceanogr., 30, 2231–2245, doi:10.1175/1520-0485(2000)030<2231:DDIITP>2.0.CO;2.
Walsh, D., and E. Carmack, 2003: The nested structure of Arctic thermohaline intrusions. Ocean Modell., 5, 267–289, doi:10.1016/S1463-5003(02)00056-2.
Wettlaufer, J., 1991: Heat flux at the ice-ocean interface. J. Geophys. Res., 96, 7215–7236, doi:10.1029/90JC00081.
Woodgate, R. A., K. Aagaard, J. H. Swift, W. M. Smethie, and K. K. Falkner, 2007: Atlantic water circulation over the Mendeleev Ridge and Chukchi Borderland from thermohaline intrusions and water mass properties. J. Geophys. Res., 112, C02005, doi:10.1029/2005JC003416.
