Thermohaline staircases consist of subsurface mixed layers that are separated by thin interfaces. They are associated with double-diffusive processes, which in turn result from a difference of 2 orders of magnitude between the molecular diffusivity of heat and that of salt . Whenever the vertical gradients of temperature- and salinity-induced stratification have the same sign, these differences in molecular diffusivity can enhance the vertical mixing through double-diffusive convection, leading to effective diffusivities of the order of 10-4 m-2s-1 and references therein.

It is still a topic of discussion how double-diffusive convection leads to the formation of thermohaline staircases in oceanic environments . For example, argued that small-scale mixing processes trigger the formation of internal waves. On the other hand, variations in the turbulent heat and salt fluxes or in the counter-gradient buoyancy fluxes that sharpen density gradients could also lead to the formation of thermohaline staircases. Lastly, subsurface mixed layers can also arise from thermohaline intrusions . Although it remains unclear how these staircases arise, these studies agree that the formation of these subsurface mixed layers are related to double-diffusive processes.

Based on the Turner angle (Tu), which compares the density component of the temperature distribution with the density component of the salinity distribution, two regimes of double diffusion can be distinguished . Waters with -90∘<Tu<-45∘ correspond to a stratification where both temperature and salinity increase with depth and belong to the diffusive-convective regime (DC). Those with 45∘<Tu<90∘ correspond to a stratification where temperature and salinity decrease with depth and belong to the salt-finger regime (SF).

Theoretical and laboratory studies have indicated that diapycnal fluxes of heat and salt in thermohaline staircases are elevated compared to the background turbulence e.g.,. These results were confirmed by a tracer release experiment in the western tropical Atlantic Ocean . Although these enhanced fluxes were observed, the importance of these fluxes for the global mechanical energy budget remain unknown. Moreover, the vertical heat and salt fluxes in thermohaline staircases can also affect water-mass properties. In some regions, persistent thermohaline staircases with layers stretching over a few hundred kilometres have been observed , which could result in significant diapycnal fluxes between water masses. For example, the double-diffusive diapycnal fluxes in the Mediterranean Sea dominate the transport between the deep water masses , and in the Arctic Ocean and Southern Ocean, an upward heat flux has been observed through staircase interfaces .

Modelling studies that incorporated parameterizations of double-diffusive fluxes indicated that the associated double-diffusive diapycnal fluxes can reduce the strength of the global overturning circulation . To be able to study this with observations, we present a global dataset of the occurrence of thermohaline staircases and their properties. The dataset is based on observations from Argo floats and Ice-Tethered Profilers. In the following sections we briefly describe the raw data used to extract the dataset (Sect. ) and the algorithm we designed to detect staircase structures (Sect. ). The sensitivity of this detection algorithm to the chosen input parameters is assessed in Sect. . The dataset is verified in Sect. , followed by some guidelines for the use of the dataset in Sect. .

The dataset contains observations of autonomous Argo floats and autonomous Ice-Tethered Profilers (ITPs). The data of all active and inactive profilers are obtained from http://www.argo.ucsd.edu (last access: 14 May 2020) and http://www.whoi.edu/itp (last access: 14 May 2020) from 13 November 2001 to 14 May 2020. Details on the profilers are described in and for the ITP and in for the Argo floats. First a quality check is performed, where a profile is excluded from analysis if it was taken by an Argo float mentioned on the grey list. This grey list contains floats that may have problems with at least one of the sensors (https://www.nodc.noaa.gov/argo/grey_floats.htm, last access: 14 May 2020). As thermohaline staircases consist of mixed layers with depths of tens of metres, we also require that profiles have continuous data up to 500 dbar with an average resolution finer than 5 dbar. Details on the origin and vertical resolution of the profiles are depicted in Table  and Fig. , in which Fig. b confirms that all profiles have observations deeper than 500 dbar. Furthermore, the average vertical resolution of the profiles indicates the average resolution is well below the 5 dbar that was used as a threshold (Fig. c). After this quality control, 487 493 vertical temperature and salinity profiles remain. Their global distribution is shown in Fig. .

Next, the profiles of the Argo floats and ITP were linearly interpolated to a vertical resolution of 1 dbar from the surface to 2000 dbar so that their data could be analysed in a consistent manner. As a result, the small steps in, for example, Arctic staircases might be missed (see Sect. ). From these interpolated profiles we calculate several variables. Absolute salinity (S) in grammes per kilogrammes and conservative temperature (T) in degrees Celsius are computed with the TEOS-10 software . Note that we use conservative temperature as this is more accurate than potential temperature in computations concerning heat fluxes and heat content . We apply a moving average of 200 dbar (Table ) to obtain the background conservative temperature and absolute salinity profiles of the water column and to compute the thermal expansion coefficient (α, ∘C-1) and the haline contraction coefficient (β, kgg-1). A consequence of the moving average of 200 dbar is that the upper 100 dbar and lower 100 dbar of each profile are omitted in the remainder of the analysis. The Turner angle is computed using profiles that were smoothed with a moving average of 50 dbar instead of 200 dbar, which is similar to , following , from 1Tu=tan⁡-1α∂T∂p-β∂S∂p,α∂T∂p+β∂S∂p, where the vertical gradients are approximated with a central differences scheme.

After the data pre-processing, we apply a detection algorithm that exploits the vertical structure of staircase profiles (Fig. ). The benefit of using the vertical structure, instead of using assumptions based on the Turner angle, is that we can use this angle to verify the results. The detection algorithm consists of five steps. First the algorithm detects all data points that are located in the subsurface mixed layers (ML, green dots in Fig. ) by identifying weak vertical density gradients in conservative temperature and absolute salinity. Next, the properties of any layer lying between the mixed layers (the interfaces, IF, orange dots in Fig. ) are assessed by applying a minimum in temperature and salinity variations. Third, the height of the interface and variations within the interface are limited. The fourth step determines the regime of double diffusion (diffusive convection or salt fingers), and the fifth step is the identification of sequences of interfaces, which eventually characterizes the thermohaline staircases. The different steps of the algorithm applied to three example profiles are shown in Figs. –. In the following subsections, each algorithm step is described in more detail.

The algorithm requires four input parameters: the moving average window, a threshold for the maximum density gradients of the mixed layers, the maximum density difference of the mixed layers, and the maximum height of the interface (Table ). In this section, the sensitivity of the algorithm to each input parameter is assessed (Fig. ).

The moving average window is used by the algorithm to compute the thermal expansion coefficient (α), the haline contraction coefficient (β), and the density ratio (Rρ). We varied the moving average window between 50 and 350 dbar to assess the sensitivity of the outcomes of this choice (Fig. a). We find that the varying moving average window does not result in large variations in detected mixed layers (Fig. a).

In contrast to the moving-average window, the detection algorithm is sensitive to the value set for the density gradient threshold of the mixed layer (Fig. b), which is used to obtain a reference pressure for the sub-surface mixed layers (Sect. ). Not surprisingly, we detect more (fewer) interfaces when we increase (decrease) the allowed threshold density gradient. A small value allows for only the strongest mixed layers to be detected, which are usually referred to as well-defined staircases, while a large density gradient also allows for the detection of rough staircases e.g.,. Although the number of detected interfaces depends on the value set for this density gradient, the detected interfaces remain confined to the two double-diffusive regimes, indicating a robust outcome of the algorithm for the choice of this input parameter.

Similar to the variations in the maximum density gradient, the variation in the maximum density difference allowed within the mixed-layer results in a different number of detected interfaces (Fig. c). The number of detected mixed layers increases when we decrease the maximum density difference allowed within the mixed layer. This effect is mostly visible in the diffusive-convective regime, as we obtained a decrease of 54 % of detected interfaces in the diffusive-convective regime compared to a decrease of 31 % of detected interfaces in the salt-finger regime in the case in which we doubled the density difference in the mixed layer (Δσ1,max=10×10-3kgm-3). This difference between the regimes is due to relatively small interface variations in the diffusive-convective regime compared to the salt-finger regime and can be explained as follows: when a too large density difference is applied, the relatively small density gradients in the interfaces of the diffusive-convective regime are detected as mixed layers by the algorithm. Consequently, multiple mixed layers can be identified as a single mixed layer. However, if the applied density difference is too small, this could result in the detection of multiple mixed layers per staircase step.

The last input parameter of the detection algorithm concerns the interface height (Fig. d). As expected from Fig. b, variations in this input parameter do not result in large differences in the number of detected interfaces. If we omit this input parameter by setting it to infinity, we obtain a total increase in detected interfaces of 17 %.

Overall, the detection algorithm gives robust results as it predominantly detects interfaces within the double-diffusive regime (Fig. ). In line with expectations, the detection algorithm is most sensitive to the threshold value for the maximum density gradient in the mixed layer and the density variations within the mixed layers. The four input variables allow for optimization of the detection algorithm based on the regime and characteristics of the staircases.

The characteristics of thermohaline staircases obtained with the detection algorithm are compared to those obtained from previous observational studies for three major staircase regions: the Canada Basin in the Arctic Ocean, the Mediterranean Sea, and the C-SALT region in the tropical Atlantic Ocean. An overview is given in Tables –.

In the Canada Basin (75–80∘ N, 135–145∘ W), the algorithm detects thermohaline staircases in the diffusive-convective regime in 90 % of the profiles (Table ). Both the occurrence and depth range are comparable to what was reported by and , who analysed thermohaline staircases from several Ice-Tethered Profilers, demonstrating that our detection algorithm indeed detects thermohaline staircases at the right location. Microstructure observations suggested that the thermohaline staircases in Canada Basin have interface heights of approximately hIF=0.15m . Due to the vertical resolution of the profiles and the design of the algorithm (recall that the mixed layers are separated from each other by removing the upper and lower data points of the mixed layer, Sect. ), the method is not capable of detecting very thin interfaces (Fig. ). As expected from these limitations for the detection of the interface heights, the algorithm detects conservative temperature and absolute salinity steps (ΔTIF and ΔSIF, respectively) in the interfaces that are in the upper ranges of earlier observations . In the Mediterranean Sea, thermohaline staircases are characterized by relatively thick mixed layers that are separated by thick interfaces of up to 27 m . In this region (30–43∘ N, 0–15∘ E), the detection algorithm detected thermohaline staircases with interfaces up to 21 dbar in 6 % of the profiles, which is comparable to previous observations (Table ). An example of the detection of a Mediterranean staircase is shown in Fig. . We find that the depth at which the thermohaline staircases occur is underestimated by the detection algorithm. This could be explained by the fact that most Mediterranean observations are obtained by the Coriolis DAC (Fig. a). From this DAC, approximately 50 % of the profiles have observations that are deeper than 1000 dbar (Fig. b), which means that the coverage below 1000 dbar is limited in the Mediterranean Sea. Although the Argo floats, and consequently the detection algorithm, do not cover the full extent of the staircases (Fig. ), the conservative temperature and absolute salinity steps that are found are similar to previous observations (Table ). Note that the conservative temperature and absolute salinity steps of the staircases increase with depth , which explains why the conservative temperature and absolute salinity steps detected by the algorithm are slightly smaller than those observed in the deeper observations .

In the C-SALT region in the western tropical North Atlantic Ocean (10–15∘ N, 53–58∘ W), the algorithm detected thermohaline staircases in the salt-finger regime in 60 % of the profiles (Table ). Similar to previous studies , the algorithm detected thermohaline staircases on the main thermocline (see example in Fig. ). Again, the interface height is slightly overestimated by the detection algorithm, but the algorithm obtained conservative temperature and absolute salinity steps comparable to previous studies.

Overall, the comparison between the outcomes of the detection algorithm with previous studies indicates that the detection algorithm performs well. The small overestimation of the interface height can be attributed to the limited vertical resolution and the limitation imposed by the detection algorithm to avoid detection of false positives. Despite this overestimation, the interfaces are detected at the correct depths with conservative temperature and absolute salinity steps within realistic ranges. Therefore, we conclude that the detection algorithm is very suitable for the automated detection of thermohaline staircases in large and quickly growing datasets like the Argo float and Ice-Tethered-Profiler data.

Both algorithm and global dataset are available at DOI: 10.5281/zenodo.4286170 . The algorithm is written in Python 3 and is available under the Creative Commons Attribution 4.0 License. More details on the functions and output of the algorithm are depicted in Tables  and , respectively. The structure of the algorithm is displayed in Fig. . The Argo Program is part of the Global Ocean Observing System .

In this study, we presented an algorithm to automatically detect thermohaline staircases from Argo float profiles and Ice-Tethered Profiles. As these thermohaline staircases have different mixed-layer heights and temperature and salinity steps across the interfaces in different staircase regions, the design of the detection algorithm is based on the typical vertical structure and shape of the staircases (Figs. –). Note that by formulating the algorithm solely on this vertical structure of the staircases, we could use the Turner angle of the detected staircases for verification. Using this Turner angle, we showed that the structures are within the two double-diffusive regimes: the salt-finger regime and the diffusive-convective regime (Fig. ).

We optimized the input of the algorithm such that it provides a global overview and limits the number of detected false positives. As a result, the regional verification in Sect.  indicated that the data pre-processing and data analysis have some limitations. For example, the vertical resolution of 1 dbar in the profiles is too course to capture all staircase steps in the Arctic Ocean. In the Mediterranean, the Argo floats did not dive deep enough to capture the full depth of the staircase region. However, the fact that (i) the algorithm detects thermohaline staircases at realistic depth ranges, with (ii) conservative temperature and absolute salinity steps across the interfaces, and in (iii) the same double-diffusive regime as previous studies (Tables –), indicates that the algorithm itself performs well. Therefore, when considering an individual staircase region, we recommend optimizing the input variables of the algorithm for that specific region and applying the algorithm to additional data, for example high-resolution CTD or microstructure profiles, where available.

A sensitivity analysis to different input parameters showed that the results of the detection algorithm are robust; the detected staircase interfaces are confined to the double-diffusive regimes. Furthermore, the comparison between the detected interface characteristics of thermohaline staircases in three prevailing staircase regions and previous observations suggested that the detection algorithm accurately captures both double-diffusive regimes. The algorithm detected correct magnitudes of the conservative temperature and absolute salinity steps in the interfaces, which allows for adequate estimates of the effective diffusivity in thermohaline staircases.

The global dataset resulting from the detection algorithm contains properties and characteristics of both mixed layers and interfaces. Combined with their locations, these data allow for a statistical analysis of thermohaline staircases on global scales. For example, the global occurrence of thermohaline staircases could give insight into the contribution of double diffusion to the global mechanical energy budget. Moreover, the interface characteristics can be used to validate model and laboratory results on how double-diffusive mixing impacts the regional ocean circulation.