We developed our method using the Biogeochemical Southern Ocean State Estimate (B-SOSE) . B-SOSE is an observationally constrained numerical simulation created using MITgcm (https://mitgcm.org/, last access: 10 August 2021) and a suite of Southern Ocean observations, including Argo float data, ship track data, and satellite data. B-SOSE is part of the Estimating the Circulation and Climate of the Ocean (ECCO) suite of state estimates (https://www.ecco-group.org/, last access: 10 August 2021), which includes a variety of global and regional products covering a range of multi-year to multi-decadal time periods. Examples of other state estimates include the physics-only Southern Ocean State Estimate (SOSE) and the global ECCOv4 state estimate . We chose to develop our method using a state estimate because such products offer (1) uniform coverage in latitude, longitude, and time as well as (2) relatively high fidelity with respect to observations. We chose B-SOSE, in particular, because it represents the Southern Ocean using a spatial resolution of 1/6∘, which is eddy-permitting in the latitude range of the ACC, enabling a realistic representation of mesoscale eddy structure . We expect that training our model on the physics-only SOSE would produce similar results, although we did not attempt that here. In principle, our methods can be readily applied to any gridded temperature and salinity profile dataset. It may be possible to apply these methods to in situ data as well, if the user addresses the problem of non-uniform spatial and temporal sampling. In this paper, we focus only on applications to gridded datasets.

To construct a state estimate, researchers bring a numerical simulation into better consistency with an observational dataset using the 4D-Var method. This method uses adjoint sensitivities to calculate the required changes in the “controls” (e.g. initial conditions, mixing parameters, boundary conditions) needed to improve the agreement between the simulation and the observational dataset .

The B-SOSE domain extends from the Equator to 78∘ S, but we only use data south of 30∘ S to focus on the Southern Ocean and to avoid the model boundary. It uses bathymetry and coastline based on . B-SOSE solves the heat, salt, and momentum equations using a third-order direct space and time advection scheme with a 1 h time step. The time-evolving atmospheric boundary conditions use bulk formulae to solve for fluxes of heat, freshwater, and momentum, with 6-hourly atmospheric state variables as inputs . The state estimation process iteratively adjusts the atmospheric state variables and oceanic initial conditions to improve model–data agreement. B-SOSE uses dynamic sea ice . For vertical mixing, it uses the GLL90 mixed layer parameterisation . It also uses horizontal and vertical viscosity and diffusivity. River runoff comes from the product of augmented with an estimate of Antarctic freshwater input from iceberg and ice sheet melting . It does not include mesoscale eddy parameterisation, as this particular configuration falls into the horizontal resolution range wherein mesoscale parameterisation may actually worsen the representation of the mesoscale . Because we are interested in quantifying physical, large-scale fronts, we only used monthly mean temperature and salinity data. Also, because we are not interested in the surface seasonal cycle at present, we only used temperature and salinity data between 300 and 2000 m, following . We used the whole period of iteration 106 of this state estimate, which covers January 2008 to December 2012. Some key properties of B-SOSE iteration 106 are listed in Table . For further details, see .

Each vertical profile in the full B-SOSE dataset is comprised of temperature and salinity values at multiple depth levels, at every grid cell and every output month. Values close to each other in the water column are correlated to some degree. Therefore, we do not necessarily need values of potential temperature (θ) and salinity (S) at every depth level to capture most of the variability, and reducing the dimensionality of the data can improve the convergence of the training process. One specific dimension reduction technique is principal component analysis (PCA), which identifies the functions that capture most of the variability with depth in the dataset. The result is a representation of the dataset as a linear combination of eigenfunctions (i.e. principal components), sometimes called a principal component expansion or principal component decomposition. Using this procedure, we can describe each profile using a small set of eigenvalues (i.e. coefficients of the principal component expansion) instead of a full set of temperature and salinity values. In addition to improving the speed and efficiency of the GMM algorithm, PCA reveals potential physical structures that may be useful for understanding the stratification of the SO . We choose the number of principal components such that the percentage of variability explained (in a statistical sense) by the PCA expansion is sufficiently high for our purposes.

Following , we only keep values between 300 and 2000 m to exclude most of the surface seasonal variability from the dataset. Because the data are spaced on an irregular grid in the vertical direction, we first interpolate the temperature and salinity profiles onto a regular grid with 10 m cells in the vertical. Following , at each grid cell and time we concatenate the temperature and salinity profiles into a single vector. We normalise each depth level for both temperature and salinity separately: subtracting the mean and dividing by the standard deviation calculated for all time periods on that particular depth level and variable. That is, we standardise the temperature values at each level using the distribution of temperatures at that same depth level, and we standardise the salinity values using the distribution of salinities at that same depth level. This is a slightly different approach from , in which the authors standardise across the entire dataset. We found that, for the work shown in this paper, the choice of normalisation approach does not make a large difference in the results (not shown). After normalisation, we carry out PCA expansion. We keep the first three principal components (PCs), which together statistically explain 98 % of the variability across the thermohaline dataset. For completeness, we show the structure of the principal components in Appendix .

The coefficients associated with PC1 indicate a broad division between polar, high-latitude Southern Ocean waters and the subtropics (Fig. a). The most negative PC1 coefficients are found in the Weddell Gyre, and we also see the imprints of the South Pacific Gyre and the ACC . The coefficients of PC2 bear the imprint of the ACC and of its northward flow along the eastern Pacific Basin (Fig. b). This northward flow is associated with the formation and export of Subantarctic Mode Water and Antarctic Intermediate Water . PC2 also has the imprint of the Agulhas Current around South Africa. Finally, PC3 has strong negative values in the Weddell Gyre and over most of the Pacific, with a band of circumpolar positive values that somewhat mirrors the southward drift of the ACC when considered from west to east. The spatial structure of PC1 and PC2 are largely consistent with those of , but the structure of PC3 is somewhat different from theirs, particularly in the subtropics. These differences are possibly a result of our choice of a different depth range. Given that PC3 explains a small fraction of the variability (7 % of the variance explained), we do not expect these differences to impact our results.

After we perform dimensionality reduction, each monthly output at each model grid cell in latitude and longitude is represented using the first three coefficients of the PC expansion. The three PC values contain combined information about both temperature and salinity, simplifying our analysis. This approach defines an abstract three-dimensional space in which we can perform unsupervised classification. In typical machine learning terminology, this abstract three-dimensional space can be called the “feature space”, in which each PC axis is a “feature”. To be explicit, we can say that each combined temperature–salinity profile in latitude, longitude, and time is represented by a three-dimensional vector of PC values. Each three-dimensional PC vector derived from B-SOSE is an “observation”. In the next section, we use unsupervised classification to identify subpopulations in the three-dimensional distribution of PC values.

Unsupervised classification attempts to identify subpopulations within a data distribution, without the assistance of any predefined labels. In our application, we attempt to identify data subpopulations in the abstract three-dimensional space defined by the PC coefficients (i.e. the “feature space”). Here, we use GMM, an algorithm that attempts to fit a set of multidimensional Gaussian functions to the data by iteratively adjusting the means and covariances of the Gaussians (; see Appendix  for more detail). This method has recently been used to classify Argo temperature profiles in the top 2 km of the North Atlantic Ocean and the Southern Ocean . GMM is well-suited to ocean applications because it offers a probabilistic measure of classification in the form of posterior probabilities, which is useful when working with a highly correlated dataset. Because GMM-derived clusters will likely feature some overlap due to the highly correlated nature of ocean data, such posterior probabilities offer an important complement to the GMM-derived class labels. In this application, we use the posterior probabilities to define coherent thermohaline regimes and their boundaries.

The GMM method attempts to represent the underlying data distribution using a set of K Gaussian functions in D dimensions (in our case D=3): 1Nx;μk,Σk=exp⁡-12x-μkTΣk-1x-μk(2π)DΣk, where x∈RD×1 is a vector in the PC space, μ∈RD×1 is the centre of the Gaussian distribution expressed in vector form, Σk∈RD×D is the covariance matrix, and |Σk| is its determinant. The covariance matrix determines the orientation of the Gaussian ellipsoids in PC space. We model the dataset, in the statistical sense of representing the dataset using a probability distribution, as a weighted sum of Gaussians: 2P(x)≈∑k=1KλkNx;μk,Σk, where λk is the weight associated with the kth Gaussian. The process of fitting the GMM uses expectation maximisation (EM), which consists of iteratively adjusting λk, μk, and Σk to decrease the model–data misfit. For additional details, see Appendix .

Once the weights, means, and covariances are fitted, each data vector x is associated with a posterior probability distribution across all of the K classes. Although we kept the random seed used in the initial guess fixed for this paper, our results are robust to the choice of random seed (not shown). This distribution is the set of likelihoods that the data vector belongs to any particular class, and the probabilities sum to one. GMM assigns each data vector to the class with the maximum posterior probability. We will now use this distribution to define an inter-class metric, which gives us a novel perspective on fronts as transitions in thermohaline structures.

For each combined temperature–salinity profile, GMM returns a probability distribution across all of the K classes. This distribution is called the posterior probability distribution, and it quantifies the probability that a particular profile is in a particular class. If the posterior probability is close to 1.0 for class k and very small for the other classes, then within the context of the Gaussian statistical model (i.e. GMM), the classification of the profile into class k is unambiguous and clear. However, if the posterior probability is close in value for the two classes with highest probabilities, then the classification is ambiguous and less clear. With this in mind, we can use the difference between the highest probability and the second-highest probability to quantify how clearly the profile has been classified. If the classification is unambiguous, then the profile is less likely to be associated with a boundary between coherent thermohaline regimes. If the classification is ambiguous, then the profile is more likely to be associated with a boundary. With this in mind, we propose a probabilistic inter-class comparison metric of the following form: 3Ixn=1-Pc=ckhighest-Pc=clrunner-up, where xn is the nth profile's PC values, and Pc=ckhighest is the highest posterior probability that GMM has assigned the nth profile as belonging to class k. The term Pc=clrunner-up is the second-highest posterior probability belonging to class l. If the difference between the highest and runner-up posterior probabilities is close to one, then I is small. This would indicate that the profile is not likely to be associated with a boundary between thermohaline regimes. If the difference between the highest and runner-up posterior probabilities is small, then I is close to one, indicating that the profile is likely to be associated with a boundary between different thermohaline regimes. The I-metric offers an alternative method for defining boundaries as fuzzy transitions between coherent regimes. In general, some regions will feature sharp transitions across boundaries, whereas other regions will feature more gradual transitions. The relative sharpness of a transition is influenced by the processes that form, mix, and destroy water masses. In contrast with approaches that define fronts as sharp transitions located along property contours or local gradients, the I-metric approach allows for a wider variety of transition types between regimes.

In our I-metric application, GMM clusters the profiles in feature space (Fig. a). The structure of the data shown in PC space is broadly consistent with that found in other studies (e.g. ). The data distribution is reasonably well represented by a linear combination of multidimensional Gaussian functions (Fig. ). The I-metric values indicate transition regions between classes, where the class labelling is relatively ambiguous (Fig. b). We choose K=5 to represent the general, large-scale pattern of the data; we explore the sensitivity of our results to K in Sect. . In the next section, we examine the I-metric and class structure in physical space.

The I-metric viewed in latitude–longitude space illustrates the rich variety of transition types found in the Southern Ocean (Fig. ). In all sectors of the SO, we see sharp transitions where the regions of high I values are narrow and more gradual transitions where the regions of high I values are more spread out. Some features are circumpolar, which is consistent with the view of SO fronts as continuous lines that encircle Antarctica. However, we also see regions where the continuity and circumpolar nature of the fronts is not as clear, suggesting that a broader view may be appropriate . The fronts are not uniformly sharp across all longitudes; for example, the northernmost transition is broad and gradual in the Atlantic sector, sharp in the Indian sector, and relatively broad in the Pacific sector. The southernmost band of high I-metric values is relatively sharp in the Atlantic sector, becoming increasingly broad as we follow it into the Indian and Pacific sectors. In the Pacific sector, it extends into an especially broad region in the Amundsen Sea, which is consistent with the intersection of the classically defined southern boundary (SBdy) with the Antarctic continent . Upstream of Kerguelen Plateau, there is a region where the I-metric is spread-out and diffuse between classes 2 and 3; this region also features a standing meander associated with enhanced eddy kinetic energy . The enhanced mesoscale eddy kinetic energy associated with the meander is consistent with increased lateral mixing and the spread-out pattern in the I-metric found in the same region. Closer to the Antarctic continent, we also see the imprints of both the Weddell Gyre and the Ross Gyre, in regions of coherent structures with low I-metric values, in part enforced by the gyre circulation.

The monthly mean I-metric (Fig. a) also highlights individual ring-like eddies; although these features are not typically considered fronts, they are small-scale transition regions between different hydrographic structures. We do expect the I-metric to be non-zero across these features. The monthly view also features mesoscale meanders, highlighting the detailed structure of the SO, which is partly a result of the energetic mesoscale eddy field. The I-metric does feature some month-to-month variability: in some locations, the fronts meander in their north–south extent, whereas in others, they are relatively stationary, likely due to bathymetric constraints (see animations in , in the “gifs” directory).

By averaging the 4 years worth of monthly means, we obtain a map of the climatological I-metric, which is averaged over many eddy lifetimes (Fig. b). Comparing an example monthly field with the climatological field, we can examine the imprint of eddy spatial variability and the meandering of the fronts on the I-metric pattern. Most of our observations about the metric are unchanged by this averaging; we identify three roughly circumpolar bands of high I-metric values, with significant spatial variability and some overlap. The three bands are fairly distinct in the Atlantic sector, with the northernmost transition being the broadest. Upstream of Kerguelen Plateau, the two northernmost bands become somewhat hard to distinguish. This is possibly a consequence of the eddy mixing and upwelling hotspot in that region, which tends to spread out hydrographic features in latitude–longitude space, increasing the degree of spatial correlation found there. Upstream of Kerguelen Plateau, the Polar Front features strong seasonal variability . Note that the I-metric band aligned roughly with the Polar Front only passes south of the plateau (e.g. south of Heard Island), which is consistent with other studies of the subsurface component of the Polar Front (e.g. ).

The three bands of higher I-metric values are distinct downstream of the Kerguelen Plateau in the Indian sector; notably, the southernmost band features especially high I-metric values in this sector. This pattern is associated with the transition between the Antarctic Circumpolar Current and the Antarctic Slope Current (ASC), which tend to flow in opposite directions . In the Pacific sector, we see the southernmost band turn into the Amundsen Sea and intersect with the Antarctic continental slope, spreading out in a diffuse region that is consistent with the behaviour of the southernmost extent of the ACC, the eastern boundary of the Ross Gyre, and the eastward shelf circulation along the West Antarctic Peninsula . In this same sector, two large regions of low I-metric values spatially correspond to export pathways of Subantarctic Mode Water and Antarctic Intermediate Water . The higher I-metric values delimit the edges of these more coherent thermohaline regimes (Fig. b), which are influenced by basin-scale stratification and the structure of the South Pacific Gyre.

In order to better understand the coherent thermohaline regimes underlying our I-metric results, we examine their lateral extents and their vertical properties. Despite not being given any latitude or longitude information, the underlying GMM captures several coherent, large-scale features of Southern Ocean thermohaline structure (Fig. a). Class 1 contains the coldest waters in the SO, covering both the Weddell and Ross gyres near Antarctica. The mean profile in this class features cold temperatures that are nearly uniform with depth; in general, they are salt stratified in that the near-surface waters are fresher than the subsurface waters, ensuring that the density profile is stable overall (Fig. ). The boundary between class 1 and class 2 broadly lies between the classically defined southern ACC Front (SACCF) and the southern boundary (SBDY) , including the turn of the SBDY towards almost being perpendicular with the Antarctic continent in the Pacific sector of the SO. Class 2 is circumpolar, with excursions into the Amundsen Sea and the area just south of Kerguelen Plateau. It features salt stabilisation, with a fresh layer near 300 m (Fig. ). Class 3 is also circumpolar, with a northward excursion in the Atlantic sector. The boundary between classes 2 and 3 roughly follows the Polar Front (PF), separating the colder, fresher Antarctic waters from the warmer, saltier subtropical waters further north. Finally, there are two subtropical classes: class 4 represents the Atlantic and Indian sectors of the subtropics, and class 5 represents the large-scale South Pacific Gyre. The boundary between classes 3 and 4 roughly aligns with the Subantarctic Front (SAF), particularly over large portions of the Indian and Pacific sectors (Fig. ). The mean of class 5 has a salinity minimum around 700 m, corresponding to the presence of the Antarctic Intermediate Water layer .

For comparison with the GMM method, which uses properties of an entire training dataset to detect changes in thermohaline structure, we use a more local front detection method implemented by in the North Atlantic. This method, called the Sobel method, directly examines spatial gradients in the principal component fields using a Sobel operator . To do this, the PCs of each grid point are placed onto a rectangular grid with the same spacing as the data sampling, where points without data are masked. The strength of an edge at a point is found by the two dimensional convolution (represented by *) of the gridded PCs and the following two matrices. In the x direction the Sobel operator is 4Gx=10-120-210-1, and in the y direction the Sobel operator is 5Gy=121000-1-2-1. The effect of this operator is similar to a gradient operator with some smoothing. There is a correlation coefficient of 0.99 between Gy*PC1 and the y gradient, and there is a correlation coefficient of 0.999 between Gx*PC1 and the x gradient of PC1. The motivation for using the Sobel operator rather than the gradient operator is principally that it can reduce the noise in data, as shown by application to photographs . used the magnitude of the x and y Sobel operators, which approximates the magnitude of the gradient, to examine fronts in the Arctic and North Atlantic. They show that the magnitude of the Sobel gradient can be thresholded to highlight features such as the Gulf Stream.

Rather than working with the gradient magnitude, Fig.  shows Gy*PC1, Gy*PC2, and Gy*PC3 alone. This is more interpretable, as the Gy*PC1 component is strongly correlated with the zonal velocity U (r=0.85). use a threshold value to define fronts, but we plot the gradient directly as a colourmap for each PC instead, which is useful as it does not obscure any information about the fronts themselves. Appendix  shows that the correlation between the Sobel Gy gradient of PC1 and the meridional velocity, V, and the correlation between Gx*PC1 and zonal velocity U increase for roughly the first 2 years of B-SOSE iteration 106, suggesting that the model is still spinning up to geostrophic balance.

The GMM and Sobel methods are complementary. GMM reveals the large-scale temperature and salinity structure associated with changes in stratification, which has traditionally been used to define the fronts, whereas edge detection methods like the Sobel method used here reveals the smaller-scale structure of multiple jets, which can merge and separate. As such, both approaches may be useful ways of characterising ocean structure without making ad hoc assumptions related to particular property values or strict requirements that the structures be circumpolar and continuous. The present proliferation of front definition and analysis methods is driven by the need to expand how the oceanographic community deals with ocean structure across a wide variety of spatial and temporal scales .

We chose to use B-SOSE data for this study in order to (1) work with a dataset that features relatively uniform latitude–longitude coverage and (2) to allow us to examine temporal variability as well as spatial variability. B-SOSE is an observationally constrained estimate of the hydrographic structure of the Southern Ocean, so it accurately captures many features of large-scale and mesoscale structure . However, because B-SOSE is a numerical model run, it will no doubt have some biases with respect to observations, particularly on smaller scales. We expect that our results would not change dramatically on basin-wide scales across different state estimate and reanalysis products.

To examine the differences of this bias on the class structure and the structure of the inter-class comparison metric, this study could be repeated with a purely observational dataset such as Argo. One trade-off for such a study would be the fact that observational datasets are relatively sparse in terms of both spatial and temporal coverage relative to a state estimate or other numerical model run. One could attempt to use the same GMM trained on B-SOSE with Argo data, but possible biases between B-SOSE and the Argo dataset could make this challenging. It might be possible to adjust for those biases in the data cleaning and preparation step of the analysis; the standardisation process, which is already a part of the analysis presented here, is a step towards this bias removal and correction that may facilitate comparisons between models and observations. Alternatively, one could attempt to re-train the GMM using Argo data alone. This has been done in other studies, so it should be possible in principle (e.g. ).

We found that the class structure and boundary positions did not feature large temporal variability with respect to the mean state, but much more work could be done to examine this variability and its connection to the processes that determine thermohaline structure (e.g. surface forcing, subsurface mixing, and advection). This is outside the scope of the present study, which is focused on proposing a new metric for identifying and tracking boundaries in Southern Ocean structure.

The Antarctic Slope Current (ASC) that separates warmer open-ocean waters from the colder waters on the Antarctic continental shelf is an important component of heat transport in the Southern Ocean. It acts to control the flow of warm water onto the continental shelf and eventually under the floating ice shelves. In a recent paper, suggest that if the source of the Antarctic Slope Current (ASC) intersects with the ACC in the Bellingshausen Sea, then the ASC source would be considered a major component of the overturning circulation. In our study, we found a diffuse boundary between classes in the Bellingshausen Sea region, which may be relevant for the physical context of the ASC, which is still under investigation (Fig. b).

In this study, we chose K=5 as the number of classes based on sensitivity tests and also based on a priori knowledge. Specifically, previous studies used a front structure with five broad regions, delineated by four fronts, so we might expect a value of around K=5 based on this (e.g. ).

Generally, the choice of the maximum number of classes K can be thought of as a way to select models of varying degrees of complexity. Statistical models with lower K values are potentially easier to interpret, only capturing the most dominant structures in the dataset. For example, the probabilistic boundary between the two classes in a K=2 statistical model roughly separates colder, fresher Antarctic waters from the warmer, saltier subtropical waters (Fig. a). Notably, in this case, the magnitude of the I-metric appears to largely decrease as we follow it from the Atlantic and Indian basins and into the Pacific Basin, indicating that the boundary becomes less sharp with longitude. This possibly reflects the fact that the Pacific Basin hosts some of the dominant northward export pathways of Subantarctic Mode Water and Antarctic Intermediate Water, consistent with a less sharp transition between polar and subtropical waters . A statistical model with K=4 retains most of the features of our analysis with K=5, but the transition region closest to Antarctica in K=5 is no longer present.

The K=5 statistical model that we used in this work captures near-Antarctic and circumpolar structure, as well as some subtropical structure. A more complex statistical model with higher K would capture more of the subtropical structure (not shown). This is consistent with sensitivity studies using temperature-only Argo data, where increasing K added details to the subtropical class structure while leaving the circumpolar class structure largely unchanged . Statistical models with much higher K values may capture more structure in the data, but increasing K also risks overfitting. That is, if we tune the GMM statistical model to match an increasing number of structures in PC space, we risk losing generality; the goal is to represent the dominant structures of the dataset without overfitting every small variation, some of which could represent noise in the data. This has a direct analogue with overfitting in terms of simple statistical models; it is unwise to use a 10th-order polynomial when a quadratic captures the dominant features of the dataset, because the higher-order polynomial is less likely to generalise to other similar datasets. In addition, statistical models with very high K values are increasingly difficult to interpret in terms of our current physical and biogeochemical understanding. Note that regional studies, such as those carried out in specific sectors of the SO, may find it useful to increase K based on local structure (e.g. ). This is consistent with the suggestion by that front definitions may need to be more flexible and region-specific, as opposed to expecting a particular definition to apply globally (or even across a single ocean basin).

The posterior probabilities returned by a Gaussian mixture model are affected by our choice of K. We should be careful not to over-interpret the posterior probabilities as confidences in the correctness of the assigned labels. Notably, GMM does not indicate the probability that a given profile belongs to none of the classes in a given statistical model. With that in mind, we can interpret the posterior probability as a measure of unambiguity in the context of a given statistical model. When one probability is larger than all others with some margin, the profile is unambiguously classified, while probabilities of similar magnitude indicate that the profile cannot be unambiguously classified in the current statistical model with the specified number of classes. In this study, we used the posterior probability distribution to identify boundaries between coherent thermohaline regimes, taking advantage of this property of GMM.