1. Introduction
Anthropogenic climate change is characterized by the persistent buildup of heat in the climate system (IPCC 2013) and long-term changes to the hydrological cycle (Durack et al. 2012; Sohail et al. 2022). A vast proportion of excess heat in the climate system is absorbed by the ocean (von Schuckmann et al. 2020), and changes to the water cycle manifest as ocean salinity changes (Pierce et al. 2012). These human-induced changes to ocean heat and salinity occur alongside natural variability in the climate system, driven in part by physical modes of climate variability like El Niño–Southern Oscillation (ENSO) (Trenberth 2020) and the North Atlantic Oscillation (Visbeck et al. 2001). Natural variability in the climate system can obscure forced anthropogenic trends in the ocean, adding “noise” to the signal.
Numerous studies have aimed to tackle the problem of detecting the anthropogenic signal of climate change in observations and climate models. A conventional approach to detecting changes in ocean temperature and/or salinity involves detecting changes to ocean properties at fixed locations on the ocean surface (i.e., in latitude–longitude coordinates, Hawkins and Sutton 2012; Hamlington et al. 2011) or by zonally averaging (i.e., in latitude–depth coordinates, Pierce et al. 2012; Boyer et al. 2005; Swart et al. 2018; Hobbs et al. 2021). In these traditional Eulerian frames of reference, the noise (i.e., natural variability in the climate system) can be reduced by coarsening the grid, filtering out the relevant time scales, taking large ensemble means, and/or by focusing on specific ocean regions that may not be impacted by dominant modes of variability (Hamlington et al. 2011; Penland and Matrosova 2006; Maher et al. 2021; Pierce et al. 2012). In doing so, past research has effectively increased the “signal-to-noise” ratio—allowing for a more robust identification of the long-term climate change-induced trend.
Water-mass-based frameworks have been proposed as an alternative to traditional Eulerian-based methods for tracking ocean change. Tracking changes in ocean properties following isosurfaces of conservative tracers, such as density, temperature and/or salinity, is thought to filter out short time scale, highly variable adiabatic motions, potentially reducing internal variability and noise in the system (Silvy et al. 2020; Palmer et al. 2007; Zika et al. 2015, 2021). In addition, water-mass-based methods can enable a direct attribution of heat or salt content tendencies to surface fluxes and diabatic mixing, as only diabatic flux terms are present in the budget (Walin 1982; Groeskamp et al. 2019; Holmes et al. 2019; Bladwell et al. 2021; Hieronymus et al. 2014).
However, a clean comparison of the internal variability, and thus signal-to-noise ratio, in water-mass-based and Eulerian methods is challenging because the volume bounded by water-mass-based coordinate surfaces can change with time. Thus, a given temperature or salinity surface could expand to fill a large portion of the ocean, while volumes bounded by latitude, longitude and depth surfaces are (by construction) fixed in time. For instance, Palmer et al. (2007) and Palmer and Haines (2009) compared ocean temperature variability above the 14°C isotherm, and the 220 m depth level, which are approximately geographically collocated. While the use of a temperature-based coordinate reduces internal variability, the 14°C isotherm expands over time to accommodate an increasingly warm ocean, while the 220 m depth level remains fixed. This issue exists across work that directly compares isothermal and fixed-depth frameworks (e.g., Weller et al. 2016). Work by Sohail et al. (2021) and Holmes et al. (2022) has avoided this problem by using a percentile-based coordinate system that enables a constant-volume comparison between one-dimensional temperature, depth and latitude coordinate systems. Holmes et al. (2022) showed that internal variability is indeed reduced in one-dimensional temperature coordinates [aligning with findings from Palmer and Haines (2009)], but only for specific time scales and regions of the ocean.
While one-dimensional fixed-depth and fixed-temperature frameworks remain popular choices in assessing ocean heat and salt content (Wolfe et al. 2008; Morrison and Hogg 2013; Sohail et al. 2021, 2022), two-dimensional coordinate systems retain more information and are often used to assess ocean heat and salt content change (e.g., Roemmich et al. 2015; Silvy et al. 2020; Rathore et al. 2020). For instance, in one dimension, “cold” temperature surfaces conflate the ocean interior with surface polar regions, but introducing a second dimension (e.g., salinity) isolates the interior ocean from the polar surface effectively. Variability in two-dimensional water-mass coordinates has been compared to variability in Eulerian coordinates by “re-projecting” diabatic tendencies inferred in water-mass coordinates back onto the geographical coordinates. Evans et al. (2014) inferred seasonal diabatic tendencies in temperature versus salinity (hereafter T–S) coordinates within the Drake Passage and then remapped these onto the average locations of the corresponding T–S classes along a repeat hydrographic section. Similarly, Zika et al. (2021) inferred the diabatic tendencies necessary to explain changes in the global inventories of seawater in T–S coordinates and mapped these onto the 3D geographical distribution of those water masses. In each case, Eulerian changes were larger than the inferred diabatic tendencies. However, these methods have relied on inferring the diabatic tendency from either a numerical model or an inverse model, and the derived solution is not necessarily unique. Thus, a clean, objective comparison assessing whether the projection of internal variability into two-dimensional water-mass frameworks (e.g., T–S coordinates) is reduced compared to Eulerian counterparts (e.g., latitude–longitude, latitude–depth) has not been conducted.
In this paper, we recast two dimensional coordinate systems, namely, T–S space, latitude–longitude space, and latitude–depth space, onto a constant-volume-based two-dimensional framework using a statistical method called binary space partitioning (BSP). We then track changes to the ocean’s temperature and salinity properties to quantify the internal variability (the noise) with the aim of establishing whether the signal-to-noise ratio of the climate signal increases in water-mass-based frameworks. The coupled climate model data used in this study are described in section 2. We provide details of BSP and its two-dimensional remapping in section 3. Our findings, detailed in section 4, confirm that the median internal variability (the noise) is lowest in T–S and latitude–depth space, and is described by longer time scale processes. We explore the historical “signal” in section 5, and show that signal-to-noise ratio is larger in T–S space in regions of high temperature change compared to its Eulerian counterparts. Conclusions are summarized in section 6.
2. Model data: ACCESS-CM2
In this work, we focus on a number of simulations performed using the ACCESS-CM2 climate model (Bi et al. 2020) as part of the Australian submission to the 6th-generation Climate Model Intercomparison Project (CMIP6) (Eyring et al. 2016). The ocean model component of ACCESS-CM2 is the Modular Ocean Model version 5.1 (Griffies and Greatbatch 2012) and uses Conservative Temperature and Practical Salinity as its standard temperature and salinity variables (McDougall 2003; McDougall and Barker 2011). More details on ACCESS-CM2, the ACCESS-CM2 submission to CMIP6, and in particular, the forcing and spinup of the piControl and historical runs, are provided by Bi et al. (2020) and Mackallah et al. (2022).
We analyze a 500-yr preindustrial control (piControl) simulation, as well as a 165-yr historical simulation (Eyring et al. 2016). In this work, we analyze the model potential temperature, practical salinity, and gridcell volume variables in temperature–salinity, latitude–longitude, and latitude–depth coordinates over the entire preindustrial control period, and the entire historical period, covering 1850–2014. A single ensemble member (r1i1p1f1) is used in this analysis.
The monthly averaged temperature and salinity in the piControl and historical runs are first binned into 2D T–S, latitude–longitude and latitude–depth percentile coordinates using BSP as described in section 3. As in Irving et al. (2020), we find that the preindustrial control simulation has a persistent drift in both temperature and salinity. The globally integrated heat content grows significantly [by O(1024) J] over the 500-yr period of the control run, while the ocean freshwater flux drops by O(1016) kg. To remove these long-term drifts in the preindustrial control run, we de-drift and de-season binned outputs. De-drifting is accomplished by removing a cubic fit of the piControl time series over the relevant overlapping time period, following Irving et al. (2020). The seasonal cycle is removed by subtracting the time-mean seasonal cycle over the entire time period of interest from the monthly time series.
Note that de-drifting and de-seasoning is conducted after aggregating or reorganizing data into their relevant diagnostic. This is primarily because removing the drift from every grid point in the native grid does not guarantee there will be no drift in the aggregated or reorganized data. De-drifting after binning ensures that any drift in the system is removed in the final diagnostic, and thus does not contaminate calculations of variance in this diagnostic.
3. Theory
Typically, water-mass-based analyses involve tracking ocean properties at constant temperature or salinity (Worthington 1981; Walin 1982; Zika et al. 2015, 2018; Holmes et al. 2019). By following constant tracer isosurfaces, the heat and salt budgets contain contributions from diabatic processes only. However, there are still diasurface volume fluxes in these coordinates, which must be accounted for and whose associated tracer flux may be ill-defined (see Holmes et al. 2019 and Bladwell et al. 2021 for details). In addition, as the surface outcrop location of temperature and salinity surfaces may shift over time, it is difficult to link changes at a given tracer isosurface to a specific geographical region in strongly forced ocean models. Thus, cleanly comparing between pure water-mass-based coordinate systems and Eulerian coordinate systems (which track ocean changes at fixed latitude, longitude, or depth) can be difficult, in part because Eulerian coordinate systems are fixed-volume by construction, while the volume of water bounded by temperature or salinity surfaces can change with time.
a. Binary space partitioning
To overcome this issue, we recast all 2D coordinate systems using a statistical method called binary space partitioning (BSP). Originating from computer graphics and image processing fields (e.g., Radha et al. 1996; Thibault and Naylor 1987), BSP is a method for recursively, hierarchically subdividing a distribution using arbitrarily oriented lines. We can use BSP to effectively partition the ocean’s two-dimensional volume distribution into equal weight bins in water-mass and Eulerian space.
To illustrate how BSP works, consider a two-dimensional volume distribution υ(x, y), which is the volume of seawater per unit x and y, where x and y can be coordinates defined by Eulerian space or coordinates defined by time-variable scalars such as T, S, and density. To form a BSP tree, we recursively subdivide the distribution with alternating axis-oriented lines n times, such that the volume of the ocean in each subdivision is 1/2n of the total ocean volume
A simple demonstration of binary space partitioning applied to a generic two-dimensional volume distribution. (a) One slice orthogonal to the y axis at y1 (in red) yields 21 equal-volume bins of υ(x, y). (b) Two additional slices orthogonal to the x axis at x1, x2 (in blue) yield 22 equal-volume bins of υ(x, y).
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The initial slice (Fig. 1a) divides the volume in half along some y-value y1, such that each subdivision contains half of the ocean volume
Once the BSP has been performed for a given choice of x and y coordinates, we can track changes to the mean temperature T and salinity S in each bin over time. This allows us to quantify how variability (noise) behaves in each coordinate system regardless of whether it is Eulerian or water mass based.
In this work, we use BSP to partition the ocean’s volume into 2n equal-volume bins in three 2D coordinate systems: T–S, latitude–longitude, and latitude–depth space. We first illustrate the partitioning of the ocean’s T–S volume distribution in the ACCESS-CM2 piControl run in Fig. 2 for n = 1, 2, 5, and 8.
BSP splitting on alternating axes, applied to the time-mean ACCESS-CM2 piControl volumetric distribution in T–S space, with 2n bins, where (a) n = 1, (b) n = 2, (c) n = 5, and (d) n = 8. Note that the salinity axis has three linear scales, delineated by the two horizontal breaks.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
In latitude–longitude and latitude–depth coordinates, we perform BSP on the depth-integrated and zonally integrated time-mean volume distribution, respectively. Figure 3 shows the resulting BSP bins in both Eulerian coordinate systems for n = 8, colored by their mean temperature and salinity. The BSP binning algorithm only “sees” the (depth- or zonally integrated) ocean volume, ignoring any landmasses. The BSP algorithm will thus not abide by continental boundaries and it will form bins that stretch across continents and between ocean basins to meet the equal volume constraint. To limit such interbasin BSP bins, and to account for the periodicity of longitude, we choose to ensure that the Americas and Drake Passage form both the far western and far eastern boundary of the ocean. This is done by slicing the ocean at 70°W longitude from 90°S to 3°N latitude. Then, a diagonal slice is made from 70°W longitude to 100°W between 3°N latitude and 20°N, and the slice continues north from 20° to 90°N along the 100°W longitude. Data points between this line and the Greenwich Meridian, moving east, are labeled with negative longitudes (i.e., are measured west of Greenwich) while the remaining data points to the east of Greenwich are labeled with positive longitudes (i.e., are measured east of Greenwich). This ensures, for example, that data points either side of the Isthmus of Panama do not combine into the same BSP bin (hence the gray, empty cells in Figs. 3a,b)
BSP splitting on alternating axes with 28 bins, applied to the ACCESS-CM2 piControl (a),(c) depth-integrated volumetric distribution in latitude–longitude coordinates and (b),(d) zonally integrated volumetric distribution in latitude–depth coordinates. BSP bins are colored by the time-mean (top) salinity and (bottom) temperature in each bin. Empty cells (where there is no BSP bin) are colored in gray.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The BSP algorithm dynamically adjusts its bin limits to capture equal volumes at all times. In a time-varying vertical grid modeling system (as in ACCESS-CM2, which uses a z* vertical coordinate), this dynamic adjustment, combined with gridcell volume changes in coarse-resolution regions, can lead to an unphysical representation of model properties. Specifically, in the coarsely resolved deep ocean, infinitesimal fluctuations in the gridcell volume due to the movements of the coordinate system surfaces can trigger large changes in BSP bin limits. This means that the deep-ocean variability can appear to be quite large within a given BSP bin, driven primarily by changing bin limits as they accommodate small volumetric changes in sparsely resolved regions of the ocean. The impact of the coarse vertical resolution on the BSP binning algorithm is clear in Figs. 3c,d, where empty regions are scattered through depth among the BSP bins. This is because, as the model grid coarsens vertically, the volume, T and S information becomes aligned along increasingly distant gridcell centers. Hence, the BSP algorithm needs to make a decision about which grid cells to cover to ensure a set of equal-volume bins. This leads to some regions not being covered by any BSP bins, as they do not contain any model information (gray cells in Figs. 3c,d). To minimize the shifting of BSP bin limits in response to minute gridcell volume changes in the deep ocean, we first take the time mean of the gridcell volumes and then use this static field, along with the time-varying T–S properties of the grid cells, to define the BSP bins and the T–S variability within them.
The coarse vertical resolution also impacts the results in T–S space, as regions that are strongly stratified (but coarsely resolved vertically) will have significant gaps in temperature and salinity when re-projected onto T–S space. The impact of coarse vertical resolution in the model can be reduced by linearly interpolating the vertical model grid. In section 5, we show how our results change when the vertical model grid resolution is doubled and quadrupled via linear interpolation.
Note that in BSP, the choice of which axis to cut along, or indeed the angle of the line that makes the cut, is entirely arbitrary. If choosing to cut orthogonal to the distribution axes, there exist 2n combinations of the order of subdivision that are valid. More generally, the choice to slice orthogonally to an axis is also arbitrary, and the BSP algorithm could, for instance, be directed to modify its angle until the volume constraint
b. Visualizing 2D BSP framework
In Eulerian space, the BSP bins generally align with the regular latitude–longitude (and latitude–depth) grid, as demonstrated by the general uniformity in BSP bin size in Fig. 3. However, in T–S space, the ocean’s volume is concentrated over a relatively narrow range of temperatures and salinities (Fig. 2). Thus, the equal-volume binning using BSP leads to a large difference in the temperature and salinity ranges spanned by a given bin in T–S space. Surface waters (which occupy a large range of temperatures and salinities but represent minimal volume) are overrepresented in the visualization, as exhibited in Fig. 4a. Instead, it is advantageous to visualize each bin with an equal area in order to more clearly convey the equal-volume nature of the BSP framework. To achieve this, we make use of the binary tree structure obtained from the BSP. By construction, the corner bins obtained from the BSP (i.e., the top-right, top-left, bottom-right, and bottom-left bins) represent the extremes in T–S space. All other bins are situated relative to these extremes in the BSP tree, and can be remapped relative to these corner bins. Hence, we remap the bins obtained from BSP onto a plot relative to the ocean’s extremes.
Remapping BSP bins from (a),(c) real T–S space to (b),(d) relative T–S space based on a binary tree structure. BSP bins are colored by (a),(b) time-mean salinity and (c),(d) time-mean temperature in the ACCESS-CM2 piControl run. Empty cells (where there is no BSP bin) are colored in gray.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
In Fig. 4, we plot the output of this remapping in T–S space. We plot the mean salinity (Figs. 4a,b) and the mean temperature (Figs. 4c,d) within each BSP bin in T–S and in remapped T–S space. The remapping effectively preserves the fresh-to-salty and hot-to-cold gradient of temperature and salinity in each bin (Figs. 4b,d). The use of the BSP tree structure in the remapping ensures that each bin (representing a single unit of volume) is saltier (fresher) and hotter (colder) than the bin to its left (right) and below (above) it. The fact that we “tag” each BSP bin in this relative space also means that in time series where the overall volume distribution of the ocean changes (for instance, in the historical run), the BSP bins will remain positioned relative to one another, and thus will stay comparable as the xth-percentile warmest (coldest), saltiest (freshest) bin in the model run.
The characteristic salinity and temperature of the global ocean can be seen in the remapped BSP plots in all coordinate systems (Fig. 5). The salty North Atlantic is visible in the top left of Fig. 5b and right side of Fig. 5c, while the relatively fresher Pacific and Southern Oceans are evident in the bottom and right-hand side of Fig. 5b and top left of Fig. 5c, respectively. The clear thermal stratification of the global ocean through depth is also retained in the remapped latitude–depth plots, as shown in Fig. 5f. Overall, the latitude–depth BSP diagnostic aligns well with traditional zonally averaged plots (not shown). However, there are some clear differences between the two diagnostics. Low volume regions are naturally collapsed in the BSP framework and combined with other, adjacent water masses to reach the equal-volume constraint. This is particularly true for the Arctic, which occupies the northern high latitudes but has a low overall volume and thus is collapsed to the rightmost BSP bins in Figs. 5c,f. The (fresh) tropical and (salty) subtropical surface waters are also collapsed to a handful of bins near the surface of the ocean in Fig. 5c.
Time-mean (a)–(c) salinity and (d)–(f) temperature in remapped equal-volume BSP bins, in (a),(d) T–S; (b),(e) latitude–longitude; and (c),(f) latitude–depth coordinates, from the ACCESS-CM2 piControl simulation. The vertical black lines in (b) and (e) roughly delineate the Atlantic (left section), Indian (center section), and Pacific (right section) Ocean basins.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
In this work, we present all results in the form of this remapped BSP visualization, as it provides equal visual weight to each volume of ocean regardless of the space occupied by each bin in its original coordinate system. This remapping also retains the salient features of the different coordinate systems while presenting the data on an equivalent constant-volume metric, enabling a cleaner comparison between different coordinate systems. For ease of interpretation of the BSP remapping and further results in T–S space, we show the broad geographic distribution of the warmest (coldest), freshest (saltiest) 25% volume of the ocean in appendix B (Fig. B1).
c. Signal-to-noise ratio
4. Results
The BSP framework enables an equal-volume comparison between three popular two-dimensional coordinate systems used to assess ocean and climatic changes—the temperature–salinity, latitude–longitude, and latitude–depth coordinate systems. In this section, we explore the internal variability, or noise, in these three coordinate systems.
a. Internal variability
We begin by assessing the internal variability in the mean temperature and salinity of each BSP bin in the three coordinate systems in question. Overall, the T–S coordinate system exhibits a broad range in variance, from low variability in BSP bins corresponding to the ocean interior (bottom-middle bins in Figs. 6a,d), to high variability in BSP bins corresponding to the ocean’s surface (edge and corner bins in Figs. 6a,d). The range in variability between surface and interior BSP bins is also reflected in the latitude–depth plots (Figs. 6c,f), where deep bins have much lower variability than surface bins. Latitude–longitude coordinates (which are depth-integrated) tend to have a smaller range in variability overall (Figs. 6b,e).
Variance in (a)–(c) salinity
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The difference in variability between different BSP bins, and between coordinate systems, can be traced to two possible sources. First, the process of integrating over the ocean volume in different coordinate systems may lead to differing phase-cancellation characteristics of variability that varies in space. For example, any modes of variability that result in warming at one longitude and cooling at another longitude at the same latitude and depth will compensate each other in that given latitude–depth bin, leading to reduced variability in latitude–depth compared to the longitude–latitude coordinate where the two phases of the variability are separated.
Second, water-mass-based coordinates exclude by construction adiabatic processes (associated with, for example, wind-driven circulation changes), which may have a higher amplitude variability. Thus, the difference between variability in T–S space and its Eulerian counterparts may be due to the fact that variability in T–S space is due to diabatic processes, while variability in Eulerian coordinates may be due to both diabatic and adiabatic processes.
The histogram of salinity and temperature variance in each coordinate system (Fig. 7) provides further insight into differences between water-mass-based and Eulerian coordinate systems. T–S coordinates and latitude–depth coordinates have similar median variability, likely for different reasons–T–S coordinates filter out adiabatic processes, resulting in a lower median variability, while latitude–depth coordinates naturally highlight deep-ocean processes separate from the surface ocean, leading to a lower median variance. Latitude–longitude coordinates, on the other hand, have a higher median variance.
Distribution of (a) salinity variance
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
To assess how statistically different these histograms are, we apply the Kolmogorov–Smirnoff (KS) test of “goodness of fit” between histogram pairs. The KS test assesses the probability that a given pair of distributions were randomly sampled from the same data. The T–S and latitude–depth histograms are identified as the only statistically similar pair of histograms, implying that the medians (red and green lines) may not be statistically different. No other distributions in this analysis pass the KS test for statistical similarity.
As discussed in section 1, moving from one-dimensional temperature coordinates to two-dimensional T–S coordinates can enable a cleaner separation of surface and ocean interior water masses due to the addition of the salinity coordinate. The histograms in Fig. 7 indicate that this separation leads to a more skewed distribution of variance, with a large number of weakly varying interior bins and a small handful of surface ocean bins (note the logarithmic x axis). Due to this skewness, the mean variance across the entire distribution (as calculated in the 1D case in Holmes et al. 2022) for our 2D case is strongly impacted by surface bins (which have higher variance). On the other hand, the median variance (vertical lines in Fig. 7) is lower, reflecting the much more numerous interior BSP bins. Moving forward, we opt to compare the median terms of interest, though we do explore the difference between mean and median variance in our spectral analysis below.
The internal variability in Fig. 6 is a consequence of interannual and subdecadal ocean processes, (<10-yr periods, such as El Niño–Southern Oscillation and the North Atlantic Oscillation), and multidecadal and centennial processes (>10-yr periods, such as Atlantic meridional overturning circulation variability). To parse the relative influence of subdecadal processes on internal variability, we present the variability of the 10-yr high-pass-filtered temperature and salinity signal relative to the total temperature and salinity variability, in Fig. 8. A fraction of 1 in Fig. 8 indicates that all of the variability in the given bin may be attributed to subdecadal processes, while a fraction of 0 indicates that all of the variability in the given bin may be attributed to multidecadal processes. Overall, variability in latitude–depth coordinates is influenced most by multidecadal processes (Figs. 8c,f), likely due to the emphasis on deep-ocean processes which change minimally over time in this coordinate system. The bulk of variability in T–S coordinates is also due to multidecadal processes. Surface waters in T–S and latitude–depth space (edge bins in Figs. 8a,c,d,f) have a high proportion of subdecadal variability. Latitude–longitude coordinates have a higher fraction of subdecadal variability overall, particularly in the North Atlantic and equatorial Pacific (possibly due to the influence of ENSO; Figs. 8b,e).
Proportion of variance due to subdecadal processes (a)–(c) in salinity
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The difference between different coordinate systems is highlighted by plotting the distribution of proportion of subdecadal variance (see Fig. 9). In latitude–depth space, approximately 80%–85% of the total variability comes from >10-yr processes (green dashed lines in Fig. 9), again due to the overrepresentation of deep-ocean volumes in this coordinate system. The T–S coordinates also host a high proportion of multidecadal processes, with the overall multidecadal variability representing 77%–80% of the total, suggesting that diabatic processes tend to occur, on average, at multidecadal time scales (red dashed lines in Fig. 9). In contrast, around 50% of the total salinity and temperature variability in latitude–longitude space comes from >10-yr processes (blue dashed lines in Fig. 9). These results are consistent with the one-dimensional analysis of Holmes et al. (2022) who showed that the mean temperature variance in a 1D temperature-based coordinate became comparable to variability in one-dimensional depth and latitude coordinates at decadal to multidecadal time scales, where diabatic processes dominate.
Distribution of proportion of variance due to subdecadal processes (a) in salinity
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The variability fractions presented here are stable across all feasible BSP split combinations (Fig. A1). For all split combinations, a lower fraction of variability comes from subdecadal processes in latitude–depth and T–S coordinates compared with latitude–longitude coordinates.
The variability in all three coordinate systems may be further broken down into characteristic time scales using spectral analysis, as shown in Fig. 10. As highlighted earlier, mean variance is more sensitive to outlier values in the skewed distributions presented. As a consequence, mean power spectra (Figs. 10a,c) are more impacted by outlier (often surface) sources of variability. Our mean results in Fig. 10c compare with the prior one-dimensional analysis of Holmes et al. (2022) (specifically, their Fig. 11a). The mean power spectra of temperature shows a clear peak in the 2–3-yr time period in temperature in all coordinate systems (Fig. 10c), aligning with findings by Holmes et al. (2022), who concluded that this peak is likely due to ENSO. Holmes et al. (2022) found that the mean temperature variability in T space exceeds that in depth space at t > 10 years.
Power spectra of (a),(b) salinity variability and (c),(d) temperature variability in T–S coordinates (red), latitude–longitude coordinates (blue), and latitude–depth coordinates (green). Spectra are presented both as (a),(b) the mean of all BSP bins and (b),(d) the median of all BSP bins. The vertical dashed line shows a period of 10 years, the cutoff used in Fig. 8. Power spectra are calculated from monthly data, using Thompson’s multi-taper method with 19 Slepian tapers.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The median power spectra, that is, the median of all power spectra at each time scale, is a means of comparison between coordinate systems which is more reflective of the more numerous ocean interior bins. The median power spectra highlight the similarity between latitude–depth and T–S coordinates (Figs. 10b,d). Across all time periods, median variance in T–S space is similar (but slightly higher) than that in latitude–depth space. Overall, latitude–longitude coordinates have the highest median variance across most time periods.
b. Modes of variability
The primary modes of variability that drive internal variability in the three coordinate systems may be explored via principal component analysis (PCA), where a principal component (PC) is the eigenvector of the covariance matrix of the distribution. The correlation coefficients obtained from PCA can indicate dominant modes of variability in the time series. PCA yields several PCs which collectively explain the total variance in a time series. We can thus find the number of PCs needed to adequately explain a high proportion of variance in a time series – the lower the number of PCs, the “simpler” the time series can be considered to be. Figure 11 shows the cumulative proportion of variance explained by the PCs obtained from PCA.
Cumulative proportion of total variance captured by principal components in the principal component analysis for monthly (a) salinity and (b) temperature in T–S coordinates (red), latitude–longitude coordinates (blue), and latitude–depth coordinates (green).
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The T–S and latitude–depth coordinates capture total salinity variance with the fewest PCs, while T–S coordinates are superior in capturing temperature variance with the fewest PCs (cf. green and red lines in Fig. 11). In T–S space, 95% of the total temperature variance is captured in 17 principal components, while in latitude–depth and latitude–longitude coordinates 31 and 104 PCs, respectively, are required to capture 95% of temperature variance.
In the salinity time series (Fig. 11b), 95% of variance can be captured by 26, 25, and 101 PCs in T–S space, latitude–depth space, and latitude–longitude space, respectively. Thus, while T–S coordinates remain the preferred choice to express temperature variability most simply, latitude–depth presents an equivalent alternative for salinity variability.
5. Discussion: Implications for signal-to-noise ratio
Overall, our results so far show that the projection of internal variability, or noise in the global ocean, into latitude–depth and T–S coordinates is roughly equivalent, and is lower than latitude–longitude coordinates. Here we assess the signal, that is, the historical temperature and salinity tendency, in T–S, latitude–depth space, and latitude–longitude space. Figure 12 shows the temperature and salinity tendencies from 1970 to 2014 in the ACCESS-CM2 historical simulation. The salinity tendency (Figs. 12a–c) aligns with previous model and historical estimates of salt content change. In T–S space, salty regions get saltier, and fresh regions get fresher, following a “wet-gets-wetter, dry-gets-drier” pattern (Allan et al. 2020). This is most pronounced in the warmest 50% of the ocean in T–S space, corresponding with the surface ocean that experiences widening salinity contrasts first. The Antarctic Intermediate Water is freshening and subtropical waters are becoming more saline, aligning well with observations of salinity change (see Fig. 12c). Tropical salinity changes are not as obvious in this framework as the tropics constitute a relatively small proportion of the global ocean volume. Overall, the changes in salinity in T–S space and latitude–depth space align with findings by Sohail et al. (2022) and Silvy et al. (2020).
Linear trend in historical (a)–(c) salinity (g kg−1 yr−1) and (d)–(f) temperature (°C yr−1) in (a),(d) T–S; (b),(e) latitude–longitude; and (c),(f) latitude–depth coordinates from 1970 to 2014. The linear trend is calculated by finding the slope of the linear regression on monthly data from January 1970 to December 2014.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
Temperature tendency in a fixed-volume framework is proportional to heat content change, so the temperature tendencies presented in Figs. 12d–f may be thought of as equivalent to the ocean heat content change. In T–S space, there is broad warming over almost all water masses in the 50% warmest BSP bins, save a small water mass in a warm, salty quadrant of the global ocean. Further exploration (not shown) suggested this cooling patch may have originated in the tropical and subtropical Pacific Ocean, though the water masses corresponding to the cooling also exist in the Indian and Atlantic Ocean sectors. This warming profile is consistent, at least in temperature space and depth space, with findings by Sohail et al. (2021). Thus, the BSP remapping captures well previously observed trends in ocean heat and salt content, lending credence to the method as a means to assess changes in historical temperature and salinity, or the climate change signal.
Having quantified both the temperature and salinity signal and noise in the climate system, we proceed to test the signal-to-noise ratio across the three coordinate systems of interest. We focus on the entire historical signal, from 1850 to 2014, as our climate signal (note that this is in contrast to other detection and attribution studies, which look at the recent past, since 1950, e.g., Pierce et al. 2012). We opt to calculate the historical signal over the entire historical period (i.e., 1850–2014) to match the time period over which noise is calculated and to avoid omitting any model data. We follow Eq. (1) to calculate signal-to-noise ratio, and use the linear trend over the historical period (i.e., C in 2014 minus C in 1850), multiplied by the number of years (165) as ΔC in the signal-to-noise ratio calculation. The signal-to-noise ratio in T–S, latitude–depth, and latitude–longitude space is shown for each BSP bin in Fig. 13, for salinity (Figs. 13a–c), and temperature (Figs. 13d–f).
Signal-to-noise ratio of (a)–(c) salinity and (d)–(f) temperature in (a),(d) T–S; (b),(e) latitude–longitude; and (c),(f) latitude–depth coordinates. Stippling shows BSP bins with a signal-to-noise ratio of F/N < 2.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
Latitude–depth coordinates broadly show the highest signal-to-noise ratio, with a large proportion of bins having a signal which exceeds twice the standard deviation of the preindustrial control simulations F/N > 2, particularly in the deep ocean. Latitude–depth F/N is especially high for salinity (Fig. 13c), particularly in the deep ocean. This is in contrast to previous studies that have found the anthropogenic signal to be most pronounced in the surface ocean relative to noise (e.g., Pierce et al. 2012). This is likely because we assess our signal, F, as the linear trend over the entire historical period (1850–2014), rather than the recent past since 1950 considered by other detection and attribution studies. The long-term trend in deep-ocean salinity and temperature is more readily picked up over the entire historical period (not shown), while the noise in the deep ocean is very low (Figs. 6c,f). This leads to a relatively large F/N in the deep ocean compared to the surface. Our analysis also shows a high temperature F/N in water masses corresponding to Subantarctic Mode Waters (Fig. 13f), consistent with past research (Banks et al. 2000, 2002; Swart et al. 2018; Hobbs et al. 2021).
The T–S coordinates also perform relatively well in isolating the forced signal, with the hot spots of F/N broadly distributed across the T–S coordinates in both salinity and temperature. Latitude–longitude coordinates perform the worst in isolating the historical forced signal from internal variability, with the majority of bins having a relatively low signal-to-noise ratio, in both salinity and temperature.
While latitude–depth coordinates clearly show the greatest F/N across the different coordinate systems assessed here, these bins appear to be isolated to the deep ocean, which does not exhibit a particularly strong climate change signal (as shown in Figs. 12c,f). A coordinate system which has enhanced F/N in regions of high T and S change is of particular utility, as these are the regions of most interest for future studies. To investigate this, we plot the cumulative number of bins with a signal greater than F and an F/N > 2 in each coordinate system in Fig. 14. For salinity (Fig. 14a), latitude–depth coordinates clearly have more BSP bins with F/N > 2 across all signal strengths (but particularly in the high F regime). However, T–S coordinates prove to be superior in isolating high F/N in high temperature change regions. While latitude–depth overall has more BSP bins across all signal strengths for temperature, this advantage is isolated to lower signal regions, and thus may not be as useful. Hence, T–S coordinates are superior to their Eulerian counterparts in capturing the climate change signal in temperature, due to a high F/N in regions of high temperature change F.
Number of BSP bins with signal greater than F for each coordinate system: T–S (red), latitude–longitude (blue), and latitude–depth (green) in (a) salinity and (b) temperature. Only bins with a signal-to-noise ratio of F/N > 2 are accumulated.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The results and discussion have so far focused on analysis of the native ACCESS-CM2 model grid. However, as flagged in section 3, increasing the vertical resolution of the model grid may change the representation of variability in T–S and latitude–depth space, altering the conclusions of this study. In Fig. 15, we show how the median noise and signal-to-noise ratio in T–S and latitude–depth coordinates changes with a doubling and quadrupling of the model vertical resolution. As vertical resolution increases, the median noise in T–S space decreases for both temperature and salinity (Figs. 15a,b). In latitude–depth space, the median variance increases, though not by as much as variance decreases in T–S space, implying that T–S coordinates are more sensitive to vertical model resolution. Therefore, models with a native Eulerian grid will naturally be better represented in Eulerian coordinate systems, but as the vertical grid resolution increases, the representation of T and S is improved, enhancing the utility of T–S coordinates as a diagnostic tool.
The median (a),(b) noise and (c),(d) signal-to-noise ratio of (a),(c) salinity and (b),(d) temperature given a doubling (2×) and quadrupling (4×) of the native model vertical grid via linear interpolation. The first 100 years of the ACCESS-CM2 control simulation are analyzed here. Only T–S (red) and latitude–depth (green) coordinates are compared.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The gap in median signal-to-noise ratio between latitude–depth and T–S narrows for salinity as vertical resolution increases (Fig. 15c). For temperature, the median signal-to-noise ratio in T–S becomes larger than that in latitude–depth upon doubling of the vertical grid resolution (Fig. 15c). Hence, it is essential to use a model which adequately resolves vertical structures of temperature and salinity to unlock the key benefits of water-mass coordinates. The F/N in regions of high T and S change (as shown in Fig. 14) does not change significantly with different vertical resolutions (not shown). Therefore, one of the main conclusions of this study, that is, that water-mass coordinates isolate the historical temperature change signal, is robust regardless of the model vertical resolution.
There are several questions open for further exploration, particularly in terms of the BSP algorithm presented here. In the past, water-mass-based frameworks have been used to develop simple ocean heat and salt content budgets, wherein salt and heat content tendencies can be related solely to diabatic air–sea flux and mixing processes (Holmes et al. 2019; Sohail et al. 2021; Bladwell et al. 2021). In the two-dimensional BSP framework, such a budget is more difficult to formulate, as changes to the properties of a bin can potentially change the BSP bins in adjacent T–S regions. That said, the formulation of a budget in the BSP framework would yield a more process-based understanding of some of the trends and variability seen in this analysis, and is reserved for future work. In addition, while the two-dimensional frameworks assessed here retain regional information, the diagnostics are calculated over the entire global dataset. An analysis that is confined only to certain regions may provide further guidance toward the driving processes in different regions of the ocean. Such a process-based, regional approach may also aid in understanding the tendency results in Fig. 12, including the cooling patch in T–S space. In addition, one-dimensional analyses in temperature space have highlighted the potential benefits of using water-mass-based coordinates to reduce sampling bias arising from adiabatic heave in observations (Palmer et al. 2007; Palmer and Haines 2009). BSP presents an opportunity to extract synthetic profiles from climate model data, following Allison et al. (2019), and assess the influence of two-dimensional coordinate systems on observational sampling biases and observed heat and salt content.
6. Conclusions
Water-mass-based frameworks are becoming popular for capturing changes in ocean heat and salt content, in part because they are believed to reduce internal variability, thus more effectively isolating the historical “signal” of climate change. However, a rigorous comparison between water-mass-based frameworks and Eulerian (latitude–longitude, latitude–depth, etc.) coordinate systems has been difficult due to fundamental differences in the way these coordinate systems are formulated. In this work, we introduce a statistical method, called binary space partitioning (BSP) to recast T–S, latitude–longitude, and latitude–depth coordinate systems onto an equivalent, equal-volume coordinate. Applied to preindustrial control and historical simulations of a state-of-the-art climate model, ACCESS-CM2, BSP enables an apples-to-apples comparison of internal variability between water-mass-based and Eulerian coordinates. We find that T–S and latitude–depth coordinates have equally low global variability, and the majority of this variability can be attributed to multidecadal processes in both coordinate systems. Overall, we find that the historical temperature signal is more effectively isolated in T–S space, with a signal-to-noise ratio that is greater than its Eulerian counterparts in regions of high temperature change. Latitude–depth coordinates, on the other hand, present the best option for isolating the historical salinity change signal, with a signal-to-noise ratio that is greater than T–S and latitude–longitude coordinates in regions of high salinity change. These results present the lower bound of variance and signal-to-noise ratio in T–S coordinates, and are dependent on the model’s vertical grid resolution. Our findings provide a road map for choosing the best two-dimensional coordinate system when analyzing global datasets, suggesting that T–S coordinates are most appropriate for temperature change studies, and latitude–depth coordinates are preferred for salinity change analyses.
Acknowledgments.
We acknowledge the World Climate Research Programme (WCRP), the CMIP6 climate modeling groups for producing and making available their model output, and the Earth System Grid Federation (ESGF) as well as the funding agencies that support the WCRP, CMIP6, and ESGF. All data analysis was conducted at facilities which form part of the National Computational Infrastructure (NCI), which is supported by the Commonwealth of Australia. The authors are supported by the Australian Research Council (ARC) Centre of Excellence for Climate Extremes (CLEX), the Australian Centre for Excellence in Antarctic Science (ACEAS), and the ARC Discovery Project Scheme DP190101173. We acknowledge support from ARC Award DE21010004. We thank Dr. John A. Church for assistance in the interpretation of results, and the Climate Modelling Support (CMS) team at CLEX for assistance in developing the BSP binning algorithm. We thank reviewers Yona Silvy and Jonas Nycander for their thorough comments and feedback.
Data availability statement.
All data used in this work are publicly available via ESGF: https://esgf-node.llnl.gov/search/cmip6/.
APPENDIX A
Variability across 2n Combinations of Axis Subdivisions
In this study, we opt to subdivide alternating axes (starting with the y axis) eight times, to yield 28 = 256 bins. However, as mentioned in section 3, there are 256 possible combinations of axis subdivisions that may have been chosen: xxxxxxxx, xxxxxxy, yyyyyyyx, etc. In this appendix, we explore the influence of choosing some of these other combinations of axis subdivisions on our results.
When assessing internal variability in two-dimensional tracer space, an ideal coordinate system would equally represent changes in both the x and y axes. For instance, in some climate model grids latitude and longitude have roughly equivalent resolutions as variability in the latitudinal and longitudinal directions is roughly similar. Of course, for the sake of reducing computational complexity, dimensions which are known a priori to exhibit characteristically lower variability may have reduced resolution–for instance, ocean model grids typically have lower depth resolution than latitude or longitude. Without such a priori knowledge of variability in a given dimension, and in an attempt to create a like-for-like coordinate system, we argue that the most appropriate BSP split combinations would be ones that preserve the aspect ratio of bins. Thus, we propose that the most physically plausible BSP split combinations are combinations of xy and yx. Always splitting in axis pairs ensures that no long, thin bins are created which span a large range in one dimension but a small range in another dimension.
For n = 8, there are 16 yx and xy combinations that preserve the BSP bin aspect ratio. As the variance distributions (Fig. 7) are highly skewed, we examine how the median (rather than mean) variance changes across the 16 BSP split combinations in Figs. A1a and A1c. For salinity, T–S, and latitude–depth coordinates have extremely similar median variances across all split combinations. For temperature on the other hand, latitude–depth coordinates have consistently lower median variance than T–S, and this gap changes based on the specific split combination used. That said, the median variance of T–S and latitude–depth remains quite close relative to the latitude–longitude coordinate system, which has a median variance that is approximately one order of magnitude larger. Across all BSP split combinations, T–S and latitude–depth coordinates are dominated by multidecadal processes, while latitude–longitude coordinates have a roughly equal split between subdecadal and multidecadal processes (see Figs. A1b,d). Our exploration of alternative BSP split combinations further solidifies our findings, showing that our results are insensitive to the order of (physically constrained) BSP splitting used.
(a),(c) Median variance and (b),(d) median fraction of subdecadal variance in T–S (red), latitude–longitude (blue), and latitude–depth (green) coordinates across 16 plausible BSP split combinations.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
APPENDIX B
Geographic Location of Water Masses in T–S Space
It is difficult to conceptualize changes in water-mass space in terms of the geographic location of said water masses. In an attempt to aid in interpretation of the results we show the volume fraction in each latitude–longitude and latitude–depth grid cell that corresponds to the warmest (coldest) and freshest (saltiest) 25% volume of the ocean, in Fig. B1. The 25% coldest and freshest ocean by volume is predominantly located in the Southern Ocean and surface Arctic Ocean (Figs. B1a–c). Antarctic Bottom Water and Pacific subsurface waters are captured in this quadrant.
The volume fraction, in latitude–longitude and latitude–depth coordinates, occupied by four water masses in T–S space: (a)–(c) the coldest, freshest 25% of the ocean; (d)–(f) the coldest, saltiest 25% of the ocean; (g)–(i) the warmest, freshest 25% of the ocean; and (j)–(l) the warmest, saltiest 25% of the ocean. Note that these quadrants are the same as those presented in Fig. 2 for 22 bins, and their temperature and salinity limits are denoted in gray. Landmasses are colored in dark gray.
Citation: Journal of Climate 36, 9; 10.1175/JCLI-D-22-0363.1
The 25% coldest and saltiest ocean is much more broadly distributed–and largely corresponds to the deepest ocean water (Figs. B1d–f). The North Atlantic Deep Water and North Atlantic overturning are captured in this quadrant. The 25% warmest and freshest ocean is largely isolated to the surface Pacific Ocean, as well as the Antarctic Intermediate Water, but excludes the Pacific subpolar gyres (Figs. B1g–i). The 25% warmest and saltiest ocean, on the other hand, is almost exclusively isolated to the Indian and Atlantic oceans (excluding the Indo-Pacific warm pool), and includes the Pacific subpolar gyres (Figs. B1j–l).
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