Double-diffusive mixing makes a small contribution to the global ocean circulation

Double-diffusive processes enhance diapycnal mixing of heat and salt in the open ocean. However, observationally based evidence of the effects of double-diffusive mixing on the global ocean circulation is lacking. Here we analyze the occurrence of double-diffusive thermohaline staircases in a dataset containing over 480,000 temperature and salinity profiles from Argo floats and Ice-Tethered Profilers. We show that about 14% of all profiles contains thermohaline staircases that appear clustered in specific regions, with one hitherto unknown cluster overlying the westward flowing waters of the Tasman Leakage. We estimate the combined contribution of double-diffusive fluxes in all thermohaline staircases to the global ocean’s mechanical energy budget as 7.5 GW [0.1 GW; 32.8 GW]. This is small compared to the estimated energy required to maintain the observed ocean stratification of roughly 2 TW. Nevertheless, we suggest that the regional effects, for example near Australia, could be pronounced. The influence of double-diffusive mixing on the global mechanical energy budget is small, according an analysis of in-situ ocean observations, but its regional impact, for example, near Australia, could be important.

D ouble diffusion arises in the ocean when either the temperature-or salinity-induced stratification is statically unstable, while the overall density stratification is statically stable 1 . Two regimes of double diffusion are distinguished: the salt-finger regime characterized by a destabilizing salinity stratification, and the diffusive-convective regime with a destabilizing temperature stratification. The release of potential energy stored in the stratification of the unstable component drives the double-diffusive mixing, resulting in a counter-gradient buoyancy flux that restratifies the water column 2 . Another aspect typical for double-diffusive mixing is an inequality between the density components of the resulting vertical salt and heat fluxes; the density flux ratio γ = F T /F S ≠ 1, where F T is the vertical heat flux and F S is the vertical salt flux 3 .
Ocean general circulation models that incorporate parameterizations of double-diffusive mixing indicate that it induces a weakening of the ocean's meridional overturning circulation [4][5][6] . This decrease could either arise from the counter-gradient diapycnal mixing or from the modification of water masses through the differential vertical fluxes of heat and salt. First, the diapycnal mixing caused by double diffusion contributes to the mechanical energy budget in the deep ocean. In total,~2 TW is required to maintain the abyssal stratification [7][8][9] . Double-diffusive mixing can occur in the open ocean and enhances interior mixing locally 2 . However, the magnitude of its resulting contribution to the global mechanical energy budget is so far unknown.
Second, observations indicate that the double-diffusive vertical fluxes of heat and salt could modify oceanic properties 10,11 . For example, the waters in the southern Indian Ocean became more susceptible to double diffusion over the last decades 10 . Observations 10 indicated that this could lead to stronger double-diffusive fluxes, which in turn provides an explanation for the observed changes in the water masses in this period. Also in the Mediterranean Sea, the vertical transport of heat and salt between the Levantine Intermediate Water and Mediterranean Deep Water seems to be dominated by double-diffusive fluxes 12 . Over the past decades, these fluxes increased the salinity of the Mediterranean Deep Water, which in turn affected the salt and heat input into the Atlantic Ocean 11 . Furthermore, due to the inequality of the strength of the vertical heat and salt fluxes associated with double diffusion, it is thought to be the major consumer of spiciness in the ocean 13,14 .
Although many studies have highlighted the importance of double-diffusive mixing in the ocean, an observationally based analysis of the impact of these processes on the global ocean circulation is lacking. In this study, we analyze the global distribution of thermohaline staircases, which arise from double-diffusive processes. Thermohaline staircases are stepped structures in the temperature and salinity stratification consisting of a sequence of subsurface mixed layers separated by thin interfaces with sharp temperature and salinity gradients. The mixed-layer heights of thermohaline staircases range from several meters in the Arctic Ocean 15,16 to several hundreds of meters in the Tyrrhenian Sea and Black Sea 2 . In contrast to the microstructure of the double-diffusive mixing itself, the vertical length scales of the mixed layers of thermohaline staircases are larger so that they can be captured by Argo floats and Ice-Tethered Profilers 17,18 (see "Methods"). Based on our global distribution of thermohaline staircases 19 , we compute the effective diffusivity of heat and salt in each step of a thermohaline staircase and use that to quantify the total contribution of double-diffusive mixing to the global mechanical energy budget.

Results
The global distribution of thermohaline staircases. The global distribution of thermohaline staircases obtained using the methods outlined in the "Methods" section, indicates that thermohaline staircases are formed in specific regions depending on regional water-mass characteristics ( Fig. 1). In total, the global dataset comprises 39,469 profiles with thermohaline staircases in the salt-finger regime (n SF = 8.1% of all 487,493 profiles) and 31,053 profiles with thermohaline staircases in the diffusiveconvective regime (n DC = 6.4% of all profiles).
In general, thermohaline staircases in the diffusive-convective regime occur at high latitudes where fresh and cold surface waters overlie warmer and more saline waters. Especially the Canada Basin that is located within the Arctic Ocean is known for its persistent occurrence of thermohaline staircases 15,16 . There, thermohaline staircases with a high number of steps are observed (dark blue areas in Fig. 1). Previous studies estimated the doublediffusive upward heat transport at 0.004-0.3 W m −2 , which is an order of magnitude smaller than the mean surface mixed-layer heat flux 15,20 . In line with this estimate, we find an upper bound of the average heat fluxes of 0.5 W m −2 (at 135 ∘ W-145 ∘ W, 75 ∘ N-80 ∘ N, Fig. 2a). Besides the Canada Basin, other regions in the Arctic Ocean and Southern Ocean also reveal the presence of thermohaline staircases in the diffusive-convective regime [21][22][23][24] .
At lower latitudes, double diffusion is predominantly in the salt-finger regime (red in Fig. 1). Using the automated detection algorithm, we identify thermohaline staircases in all well-known formation regions: in the western tropical Atlantic Ocean 17,25 , the Caribbean Sea 26,27 , below the Mediterranean outflow 28 , within the Mediterranean Sea 12,29 and along the equator 30 . These  19 . For each profile, the number of steps within thermohaline staircases in the salt-finger regime (red dots) and diffusive-convective regime (blue dots) is plotted. Profiles with the largest numbers of steps are plotted last for clarity. thermohaline staircases have, in general, thicker mixed layers and interfaces than staircases in the diffusive-convective regime 2 , which allows for more accurate estimates of the temperature and salinity steps across interfaces 19 . Using a previous empirical estimate 31 , we obtain an average effective diffusivity of salt of K SF S ¼ 1:92 10 À5 m 2 s −1 [2.5 × 10 −7 m 2 s −1 ; 1.0 × 10 −4 m 2 s −1 ], where the values between brackets correspond to the 2.5-and 97.5-percentile ranges (Fig. 2b).
Besides these well-known regions with thermohaline staircases, the global analysis presented here also reveals a newly discovered staircase region in the Great Australian Bight (Figs. 1 and 3a, b). There, the warm and saline Subtropical Surface Water overlies the cold and fresh Antarctic Intermediate Water (Fig. 3a, b). This interface is susceptible to double-diffusive mixing with Turner angles varying between 45 ∘ < Tu < 90 ∘32 . As expected the mixed layers of the staircases are located at this interface (Fig. 3c). Thermohaline staircases appear abundant in this region (32 ∘ S-42 ∘ S, 125 ∘ E-145 ∘ E): in total, 62% of the 2241 profiles contain staircases (Fig. 3d). The alignment of the temperature and salinity data of each mixed-layer (alignment in Fig. 3e) indicates that the mixed-layer properties and vertical structure of the staircases are similar across multiple profiles. To gain insight in the lateral coherence of these similarities of the properties, we quantify using the aligned data points as an example (red dots in Fig. 3e). We obtain a lateral coherence over a region of several hundreds of kilometers that persisted for almost 2 years, which is similar to what is seen in other major staircase regions 15,25 . The slopes of these aligned points correspond to the density flux ratio and confirm a downward salt and heat transport within the thermohaline staircases (γ SF < 1).
Part of Antarctic Intermediate Water in the Great Australian Bight, known as Tasman Leakage 33 , propagates westward towards the Agulhas region through the southern Indian Ocean 34,35 . Our results show that thermohaline staircases occur over the entire southern Indian Ocean (Fig. 1, red histograms in Fig. 4a) and that the characteristics of the thermohaline staircases change from east to west. In the east, the thermohaline staircases found contain more steps (Fig. 4b). However, the part of the water column that is susceptible to strong salt-fingering (71. 6 ∘ < Tu < 90 ∘ or R ρ > 2, depths between thick white contour in Fig. 4a) 32 is relatively constant from east to west. This illustrates that strong salt fingering most likely occurs along this cross-section in the southern Indian Ocean.
Contribution to the ocean energy budget. To estimate the combined contribution of double-diffusive fluxes in thermohaline staircases to the global mechanical energy budget, we compute the average effective diffusivity of buoyancy in each detected interface based on the temperature and salinity steps between mixed layers 2,36 . A comparison between the characteristics of temperature and salinity steps found by the algorithm and those found in previous studies on thermohaline staircases indicated that the global dataset contains temperature and salinity steps of the correct magnitude in the salt-finger regime and provides an upper bound for steps in the double-diffusive regime (see "Methods"). As the density flux ratio is different in the two regimes (γ DC > 1 and γ SF < 1), the effective diffusivities and thus their contributions to the global mechanical energy budget are computed separately.
To estimate the contribution of diffusive convection to the global mechanical energy budget, we compute the effective diffusivity of density with flux laws 37 (see "Methods"). This yields an upper bound for the average effective diffusivity of (Fig. 2c). Next, we use this effective diffusivity to compute the dissipation (D DC ) from ref. 8 , using their equation for the vertical fluxes through any depth level in the ocean: where Γ is the mixing efficiency. We use standard values for the gravitational acceleration (g = 9.8 m s −2 ), area of the ocean (A = 3.6 × 10 14 m 2 ) and vertical density difference (Δρ = 1 kg m −3 ) 8 . The mixing efficiency of double-diffusive mixing approaches Γ = −1, because it is driven by the release of potential energy and the production term of the turbulent kinetic energy budget becomes negligible 2,38 . In Eq. (1), ref. 8 assumes that the mixing in the ocean is evenly distributed. To account for the fact that thermohaline staircases do not occur everywhere (Fig. 1), we multiply Eq. (1) with the fraction of the staircase occurrence (n DC ). Moreover, because the depth of thermohaline staircases is variable 19 , Eq. (1) is considered as an upper bound. Using these numbers, we obtain a Our estimate for the total contribution of double diffusion to the global mechanical energy budget by diffusive convection and salt fingers combined thus adds up to D = 7.5 GW [0.1 GW; 32.8 GW]. Owing to the high mixing efficiency of double diffusion (Γ = −1) compared to turbulent mixing (Γ turb = 0.2), double diffusion is able to mix five times more than down-gradient turbulence with the same amount of energy. Notably, the mixing by double diffusion restratifies the water column in contrast to the mixing by down-gradient turbulence. Consequently, doublediffusive mixing contributes to the mechanical energy necessary to maintain the stratification. Depending on its location, the double-diffusive mixing can thus either enhance the downwelling in regions with deep convection in the North Atlantic Ocean or it can prevent the upwelling at lower latitudes 2 . This implies that a part of the double-diffusive mixing in downwelling regions is already contained in the estimates for the amount of abyssal mixing that were previously computed 8 . Therefore, we conclude that the contribution of double-diffusive mixing to the global mechanical energy budget is limited.

Summary and global implications
In this study, we presented a global analysis of thermohaline staircases identified in profiles of Argo floats and Ice-Tethered Profilers. The global distribution of thermohaline staircases shows that thermohaline staircases are confined to specific regions determined by the local water-mass characteristics: thermohaline staircases in the diffusive-convective regime are predominantly found at high latitudes, while staircases in the salt-finger regime dominate at low latitudes. Our analysis revealed a new staircase region in the Great Australian Bight and southern Indian Ocean. As the waters in the southern Indian Ocean are likely to become more susceptible to double-diffusive mixing 10 and previous studies showed that double-diffusive fluxes in thermohaline staircases can modify water-mass characteristics 11,12 , we speculate on the potential implications of this new staircase region.
The thermohaline staircases in the southern Indian Ocean overlie the waters of the Tasman Leakage. As the salt content of the Tasman Leakage waters is considered to affect the stability of the Atlantic Meridional Overturning Circulation (AMOC) 35,41 , the double-diffusive salt fluxes in this region might impact AMOC stability. This impact can be determined qualitatively by using an indicator of AMOC stability, usually referred to M ov , measuring the freshwater transport of the AMOC at 35 ∘ S in the Atlantic [42][43][44][45] . When M ov > 0, the AMOC transports salt out of the Atlantic and it is less sensitive to North Atlantic surface freshwater anomalies [42][43][44][45] . For M ov < 0, the AMOC imports salt and can undergo transitions to a weak AMOC state due to the positive salt advection feedback. Further research is necessary to quantify whether stronger double-diffusive salt fluxes in a future climate 10 can increase the salt content of the Tasman Leakage waters and, consequently, have a destabilizing effect on the AMOC by changing the M ov .
By analyzing the occurrence and properties of the thermohaline staircases, we also estimated the impact of double-diffusive mixing in this study. Of each thermohaline staircase, we estimated the effective diffusivity of buoyancy based on both flux laws (diffusive convection) 37 and empirical estimates (salt fingers) 31 . Although there are some uncertainties regarding the diffusivities obtained from these computations that most likely result in an overestimation of the magnitude of these diffusivities (see "Methods") 46,47 , these computations are necessary to obtain an observationally based estimate of the contribution of double-diffusive mixing to the global mechanical energy , which is relatively small (<1%). The robustness of these results to different input variables of the detection algorithm that was used to obtain the global dataset is confirmed by a sensitivity analysis (Table 1). Hence we conclude that the direct effect of double-diffusive mixing to the maintenance of the abyssal stratification is negligible.
However, the global distribution of thermohaline staircases indicated that double-diffusive mixing is widespread. This implies that in a large part of the ocean the magnitudes of the effective diffusivity of heat and salt differ from each other. By including an inequality of these effective diffusivities in global ocean models based on the characteristics of thermohaline staircases, doublediffusive mixing can in principle be parameterized in ocean models (Fig. 5). Especially in regions with high-staircase occurrence, this is expected to yield more realistic model results on both regional and global scales.

Methods
We use the global dataset of thermohaline staircases that is obtained with an algorithm to automatically detect these structures 19 . In short, the staircase detection algorithm is applied on vertical temperature and salinity profiles obtained from Ice-Tethered Profilers and Argo Floats between 13 November 2001 and 14 May 2020. The data is obtained from http://www.whoi.edu/itp and http://www. argo.ucsd.edu for the Ice-Tethered Profilers and Argo floats, respectively. The average coverage is 1.4 × 10 −3 observations in km −2 (A ocean ≈ 3.6 × 10 8 km −2 ), with the highest observation density of 2.5 km −2 in the Arctic Ocean (83 ∘ N and 99 ∘ W), and smallest observation density in the centers of the subtropical gyres (see Fig. 2 in ref. 19 ). This variation in data coverage results in a (small) overestimate of the occurrence of diffusive-convective staircases as these predominantly occur at high latitudes. Moreover, the Arctic Ice-Tethered Profilers generally follow the ice floe and not the flow at the depth of the staircase, which results in a randomized field of staircase observations in this region. After a quality control, profiles that have an average resolution exceeding 5 dbar and contain observations below 500 dbar are selected, which results in a dataset consisting of 487,493 profiles. Their average vertical resolution is relatively high (finer than 2.5 dbar) in the upper 1000 m of the water column 19 , where most thermohaline staircases are found 2 . Deeper in the water column, the average vertical resolution of the profiles is~2.5 dbar. Afterwards, the profiles are subsequently linearly interpolated from 0 to 2000 dbar with a vertical resolution of 1 dbar. The algorithm itself consists of five steps that detect sequences of interfaces in each temperature and salinity profile: 1. Mixed layers are identified by selecting layers with density gradients below 5 × 10 −4 kg m −3 dbar −1 . Where this criterion is met, the mixed layers are defined as the layer within a density interval of 5 × 10 −3 kg m −3 . 2. The interfaces, defined as the layers between these mixed layers, should have larger temperature, salinity and density variations than those found within the adjacent mixed layers. 3. The interfaces are required to be thinner than the adjacent mixed layers, and the maximum interface thickness is limited to 30 dbar. Furthermore, interfaces are required to contain no temperature or salinity inversion. 4. The double-diffusive regime of each interface is determined: when the temperature and salinity of a mixed-layer below an interface are higher (lower) than the temperature and salinity of the mixed-layer directly above, the regime of the interface is classified as diffusive-convective (salt-finger) regime. 5. Sequences of interfaces that are of the same double-diffusive regime are detected; each sequence of interfaces consisting of more than one step (>2 mixed layers) is classified as a thermohaline staircase.
A detailed description of the algorithm to obtain this dataset and tests of its robustness can be found in ref. 19 . A sensitivity test performed for the chosen input parameters of the detection algorithm shows robust results 19 . As an example, the sensitivity of the occurrence of thermohaline staircases to the value of the mixedlayer criterion (step 1 of the algorithm) is shown in Fig. 6. It clearly shows that while it affects the number of detected steps, the same spatial pattern emerges. Owing to the vertical resolution of the observations and the linear interpolation by the algorithm, it cannot detect very thin interfaces 19 . This particularly plays a role  for staircases in the diffusive-convective regime in the Arctic Ocean, where interfaces can be as thin as 0.1 m 2,48 . The minimum layer height that can be detected by the algorithm is 2 dbar 19 . Consequently, the smallest interfaces are missed by the algorithm and the average temperature and salinity steps in the diffusive-convective interfaces are overestimated 19 . However, the algorithm detects an accurate staircase occurrence for staircases in the diffusive-convective regime (n DC ) in this region 19 . As thermohaline staircases in the salt-finger regime have larger vertical length scales and are, therefore, more easily detected by the algorithm, the staircase occurrence for staircases in the salt-finger regime (n SF ) is also considered reliable. At each staircase interface, the effective diffusivities of heat (K T ), salt (K S ), and buoyancy (K ρ ) are computed. Taking into account that the detection algorithm mainly detects interfaces that arise from double-diffusive mixing 19 , we assume that all detected interfaces result from either diffusive convection (DC) or from salt fingering (SF). To limit detection of thermohaline intrusions, the detection algorithm only detects sequences of interfaces within the same regime (step 5 of the algorithm) as intrusions induce interfaces that alternate in both different regimes 47 . Although it is expected that such intrusions have a limited impact on the results, the detection of thermohaline intrusions could result in an underestimation of the computed fluxes through an interface by 50% 47 .
To compute the effective diffusivities in the diffusive-convective regime, we apply a similar flux law 49 . Although empirical evidence suggests that it is reasonable to extrapolate these flux laws to the oceanic environment, uncertainties do exist about the magnitude of most variables within these flux laws 50 . In general, the flux laws proposed by ref. 37 agree well with observations 15,16,51 ; we therefore choose to apply these in this study. However, it should be noted that there are some indications, mainly from numerical simulations, that fluxes computed with these relations 37 could underestimate observed heat fluxes by 50% 46 . Following ref. 37 , we compute the effective diffusivity of heat: where R ρ is the density ratio (R ρ ¼ α ∂T ∂z À Á β ∂S ∂z À Á À1 , which is limited to 0 < R ρ < 1), κ the molecular diffusivity of heat in m 2 s −1 , α the thermal expansion coefficient in ∘ C −1 , g the gravitational acceleration in m s −2 , Pr the Prantle number and ΔT IF the conservative temperature difference across an interface in ∘ C. The vertical gradients of conservative temperature are computed with a central differences scheme from temperature profiles that are smoothed with a 50 dbar moving average. Recall that the average magnitude of the temperature differences across the interfaces is overestimated by the algorithm 19 . Therefore, the effective diffusivity should be considered as an upper bound. We convert the effective diffusivity of heat to heat fluxes (F H ) as: F H ¼ ρc p K DC T ∂T ∂z , where ρ is a reference density and c p the specific heat of seawater. The effective diffusivity of salt follows from the effective diffusivity of heat in combination with the flux ratio γ: where the density flux ratio (γ) computed following Kelley 37 as: In contrast to the diffusive convection, the flux laws cannot be extrapolated to oceanic environments for thermohaline staircases in the salt-finger regime, as they are known to lead to a significant overestimation of the effective diffusivities 2,39,40,50 . Therefore, we use an empirical estimate 31 to compute the effective diffusivities of salt and heat instead: with a density ratio limit of 1 < R ρ < 10. The effective diffusivity of heat is computed as follows: where γ SF is computed as γ SF ¼ 2:709e À2:513R ρ þ 0:5128 31 . Note that, in contrast to the diffusive-convective regime, the effective diffusivities in the salt-finger regime do not depend on the temperature or salinity jumps across the interfaces. A comparison of our results to tracer-based oceanic observations in the western tropical Atlantic Ocean 17 indicates that we obtained lower effective diffusivities than found in these observations (Fig. 2). However, because these observations most likely also include mixing in the vicinity of topography, these high estimates are most likely a combination of double-diffusive mixing and turbulent mixing. Finally, the effective diffusivity of buoyancy is computed separately for both regimes from the combined effective diffusivities of heat and salt: Data availability Fig. 6 Sensitivity of the number of staircases detected resulting for different input parameters of the staircase detection algorithm 19 . a ∂σ 1 =∂p max = 2.5 x 10 −4 kg m −3 dbar −1 b ∂σ 1 =∂p max = 7.5 x 10 −4 kg m −3 dbar −1 . For each profile, the number of steps per staircase in the salt-finger regime (red dots) and diffusive-convective regime (blue dots) is plotted. Profiles with the largest numbers are plotted last for clarity. Figure 1 shows the same for the standard value of ∂σ 1 =∂p max = 5.0 x 10 −4 kg m −3 dbar −1 .