## 1. Introduction

Earth’s oceans play a critical role in regulating the global climate system (Bigg et al. 2003; von Schuckmann et al. 2016) and have been shown to act as a critical sink of excess atmospheric and land-based heat resulting from greenhouse gases (e.g., Barnett et al. 2001, 2005; Pierce et al. 2012; Trenberth et al. 2014). Heat is also redistributed within and released from the oceans, thereby impacting atmospheric temperatures and the global climate system (Bigg et al. 2003). Ocean heat redistribution determines how effectively oceans can store excess heat due to anthropogenic warming, and played a key role in the 1998–2012 global surface warming hiatus (Yan et al. 2016; Liu and Xie 2018). Therefore, a clear understanding of heat transport mechanisms should enable better predictions regarding the extent of global and regional climate change (Keenlyside et al. 2008; Robson et al. 2012; Roberts et al. 2016).

Since heat is conserved, one powerful approach for understanding ocean heat content (OHC) variability is via the ocean heat budget. The budget relates a change in OHC to a variety of driving mechanisms that appear as terms in the heat conservation equation, such as advection, diffusion, and air–sea forcing. A better understanding of which terms in the heat budget are most important can provide a means for interpreting ocean warming patterns and improving predictions of future changes in these patterns. Evaluating ocean heat budgets from direct observations is very difficult, and some unobservable processes must inevitably be inferred from the residual of better-known terms (Roberts et al. 2017). The development of conservative ocean reanalysis products that assimilate observations in a dynamically consistent way—such as the ECCOv4 product used here—offers opportunities to examine the historical ocean heat budget in precise detail, as has been reported previously for specific ocean regions (e.g., Buckley et al. 2014, 2015; Thompson et al. 2016; Piecuch et al. 2017; Desbruyères et al. 2020).

However, a significant downfall of the budget approach is its complexity. Depending on how it is constructed, the budget can contain up to a dozen different terms, which previous studies have simplified either by judiciously combining terms or by focusing only on the main drivers of the budget (Piecuch and Ponte 2012; Buckley et al. 2014, 2015; Piecuch et al. 2017). Furthermore, the budget is evidently quite sensitive to spatial and temporal scale, and different regions of the ocean may have qualitatively different heat budgets (Bishop et al. 2017; Small et al. 2019, 2020). From this complexity, it can be hard to discern general, robust properties of the ocean heat budget.

This paper presents a novel combination of three ocean analysis tools to study the ocean heat budget on a global scale. These methodologies are listed below.

Data-constrained yet dynamically consistent ocean reanalyses, which provide a precise, numerically closed heat budget at each grid point (Forget et al. 2015).

The “covariance ratio” analysis technique, first developed by Doney et al. (2007) and further elaborated by Bishop et al. (2017) and Small et al. (2019, 2020). This method reduces the full time series of heat budget terms at each point in space (or averaged over a region) to a concise set of nondimensional

*O*(1) values characterizing the importance of each term.Unsupervised machine learning, which can help reveal latent patterns in large datasets. The

*k*-means clustering algorithm (Hartigan and Wong 1979; Gong and Richman 1995; Lund and Li 2009) has been successfully applied in oceanography to a wide variety of categorization problems, from identifying regimes of Southern Ocean phytoplankton blooms (Ardyna et al. 2017) to the ocean vorticity budget (Sonnewald et al. 2019). Here we apply clustering to the covariance ratios to identify regions with similar heat content dynamics.

With this analysis, a key question we hope to answer is under what circumstances is OHC variability primarily driven by atmospheric variability versus internal mechanisms? For the internal driving mechanisms, what is the relative importance of advection versus diffusion? And for advection, what is the relative importance of variations in ocean currents versus variations in temperature, and of horizontal versus vertical advective transport?

These are not new questions of course. Many past studies have attempted to understand the drivers of OHC and SST variability in different regions. In a classic pioneering study, Hasselmann (1976) used a stochastic model to describe the temporal relationship between SST and forcing (i.e., the lead–lag correlation between surface heat flux, SST, and its tendency). A series of subsequent studies have suggested that for much of the extratropical regions of the global ocean, SST variability is primarily a function of atmospheric-driven surface heat flux (e.g., von Storch 2000; Wu et al. 2006).

As the spatial resolution of SST and surface heat flux datasets have improved, Bishop et al. (2017) revised the connection between forcing and SST and highlighted regions where ocean dynamics clearly dominate. These regions are delineated by the western boundary currents (WBCs) and the Antarctic Circumpolar Current (ACC). Similarly, Small et al. (2019) showed that latent heat flux is primarily driven by variability in SST (i.e., driven by ocean dynamics) over the eastern tropical Pacific and midlatitude ocean frontal zones (which are associated with WBCs). The above studies described only variability at the sea surface, but similar conclusions can be made for the upper ocean as well, given that SST variability is connected to temperature within the mixed layer (Alexander and Deser 1995). Looking at the upper ocean to full-depth OHC, it is clear that advective heat convergence is a key component. This has been shown by both observation- and model-based studies (Doney et al. 2007; Grist et al. 2010; Buckley et al. 2014, 2015; Piecuch and Ponte 2012; Piecuch et al. 2017; Roberts et al. 2017; Small et al. 2020).

A series of studies have shown that the balance between atmospheric forcing and forcing by ocean dynamics depends on the spatial resolution at which the budget is determined (Kirtman et al. 2012; Bishop et al. 2017; Small et al. 2019, 2020). By using spatial smoothing, Bishop et al. (2017) show that the importance of ocean-driven variability decreases with increasing spatial scale. This suggests that ocean-driven variability is mainly represented by small-scale features such as eddies. The spatial dependence was further confirmed in climate models for the relationship between SST and surface heat fluxes (Small et al. 2019) and for the upper ocean heat budget (Small et al. 2020). Similarly, there is a dependence on the temporal scale. While for monthly to seasonal anomalies atmospheric forcing is the dominant term, ocean dynamics becomes more important in establishing interannual and decadal variations in SST and upper OHC (Buckley et al. 2014, 2015). The time scale at which a switch occurs from an atmospheric- to an oceanic-driven scenario is regionally dependent (Buckley et al. 2015). By using a low-pass filter Bishop et al. (2017) show that importance of ocean-driven variability increases with increasing time scale. Small et al. (2019) expands the time dependency to submonthly variability and show that the ocean-driven signal becomes relevant in the WBCs for time scales longer than 5 days.

The sensitivity to temporal and/or spatial scale has been either focused on particular ocean regions, such as the North Atlantic (Buckley et al. 2014, 2015), or on the global scale for the sea surface using observation-based analyses (Bishop et al. 2017; Small et al. 2019) and subsurface OHC variability based on climate models (Small et al. 2020). In this paper, we use an ocean model that assimilates ocean observations and examine the global distribution of regression coefficients for key drivers of ocean temperature variation. As a key additional step, we allow the data to tell us which regions share common dynamics via a clustering approach.

Our paper is organized as follows: section 2 describes the ocean state estimate and the diagnostics used to describe heat content variability. An anomaly heat budget equation is then derived to describe the temperature tendency anomaly as the sum of distinct variations in ocean heat processes (i.e., forcing, advection, and diffusion). In section 3, we present a local heat budget analysis for the upper ocean as defined by the wintertime MLD, as was first done by Buckley et al. (2014, 2015). The focus here was to evaluate the relative importance of each budget term as a driver of changes in OHC. With this analysis we introduce the covariance ratio, which quantifies the contribution of each budget term to the total variability of temperature. Section 4 presents heat budget variation at different spatial and temporal scales in order to evaluate the contribution of each budget term to the total budget at a range of vertical (i.e., depth) scales and horizontal and temporal (i.e., monthly to decadal) resolutions. In section 5 we introduce an unsupervised machine learning approach to defining ocean regions based on coherent patterns in the local heat budget. The study’s findings are further discussed in section 6, with concluding remarks and suggestions for future work.

## 2. ECCOv4 ocean state estimate and heat budget diagnostics

In this paper, we conduct an investigation of the drivers of variability in OHC using the Estimating the Circulation and Climate of the Ocean (ECCO) consortium state estimate. The third release of version 4 (ECCOv4) was used, which provides a physically consistent ocean state estimate covering the period 1992–2015. Its solution is the output of the Massachusetts Institute of Technology general circulation model (MITgcm) assimilated to available observations for the period 1992 to 2015, which has been thoroughly assessed and found to be a coherent and accurate representation of the ocean state (Forget et al. 2015). In addition to providing closed tracer budgets, ECCOv4 offers detailed diagnostic information about the simulation, making it possible to identify the contributions of specific mechanisms to those budgets. Because of the model’s conservation rules, there are no unidentified sources of heat, which makes ECCOv4 well suited as a reanalysis in order to investigate heat content variability in the ocean over recent decades.

The diagnostic outputs include monthly mean fields from January 1992 to December 2015 for all relevant terms to formulate the heat budget. In addition, diagnostics include monthly snapshots of temperature and sea surface height (taken at the beginning and end of each month). Both the mean and snapshot fields are presented in the Lat–Lon–Cap grid (i.e., LLC90) configuration, which is organized in 12 tiles with each tile including 90 × 90 grid cells (Forget et al. 2015). Horizontal grid spacing is irregular, with an average resolution of 1° × 1°. The grid size in LLC90 ranges from 40 to 50 km at polar to subpolar latitudes, to around 110 km toward the equator. Vertical spacing comprises 50 levels of thickness from 10 m at the surface to 456.5 m for the deepest layer.

### Anomaly heat budget in ECCOv4

*θ*/∂

*t*), the convergence of heat advection [−∇ ⋅ (

*θ*

**u**)], and heat diffusion (−∇ ⋅

**F**

_{diff}), plus downward heat flux from the atmosphere (

*F*

_{forc}). Note that in our definition both latent and sensible air–sea heat fluxes, as well as longwave and shortwave radiation, are contained within

*F*

_{forc}. Shortwave radiation is absorbed throughout the upper 200 m of the water column and is described by an exponentially decaying function (Paulson and Simpson 1977).

*X*are denoted as

*X*′, respectively. The monthly anomaly budget is then derived by subtracting the monthly mean equation from Eq. (1), which removes the mean seasonal cycle and returns the month-to-month interannual variability. The central equation for the budget analysis is thus

The first term on the right-hand side of Eq. (2)

The first two advective terms are horizontal [*R*).

Although the ECCOv4 state estimate allows for the computation of closed tracer budgets that balance to machine precision, our derivation of an anomaly heat budget includes linearized advection terms that are not evaluated online by the ocean model but rather have to be approximated offline (using monthly mean velocity and temperature fields and a linear second-order advection scheme). Furthermore, the offline derivation neglects temporal decomposition of the scaling factor corresponding to the nonlinear free surface in ECCOv4 (Adcroft and Campin 2004; Campin et al. 2004). Thus, we include a residual term in Eq. (2) in order to account for any variability that is ignored in the offline estimation of the advective fluxes. We are able to precisely determine the residual from the full (i.e., non-decomposed) advective heat transport diagnostics. This residual also includes the effects of numerical diffusion, which arise due to the model’s advection scheme (Hill et al. 2012; Megann 2018). The flux due to effective numerical diffusion is present in the model’s diagnostics of the full advective flux, but not in our linearized reconstruction of the flux. As shall be shown, the residual is negligible in virtually all instances.

## 3. Covariance analysis of local heat budget

The ECCOv4 outputs permit calculation of the anomaly budget time series at each point in the global 3D grid. This yields too much information to comprehend or visualize, so to understand which terms drive heat content variability, we consider the correlation between the actual tendency, given by the left-hand side of (2) and denoted *y*, and each individual term on the right-hand side of the equation, denoted *x*. Similar forms of analysis were applied by Small et al. (2020, 2019) and Doney et al. (2007).

*x*as

*σ*(

*x*,

*y*) is the covariance between

*x*and

*y*and

*σ*(

*y*)

^{2}is the variance of

*y*. In any particular heat budget, the covariance ratio describes the contribution of each budget term to the total temperature tendency. Since the total tendency is the sum of all the budget terms, the sum of the covariance ratios must equal one. This is true regardless of whether or not the terms of the budget are linearly independent (as in fact they are not); this decomposition is a physical, rather than statistical, analysis. A positive covariance ratio implies a positive contribution (and correlation) to the total tendency, and a negative value implies a negative contribution (and an inverse correlation) to the total tendency. Equation (3) is used to separately evaluate the relevance of each term (or sum of terms) in the anomaly heat budget [Eq. (2)].

OHC variability is first investigated for each grid point at the original temporal (monthly) resolution where the anomaly heat budget terms are integrated over the climatological winter mixed layer depth (referred to herein as winter MLD). The MLD in ECCOv4 is defined as the depth at which the potential density *σ* exceeds a variable threshold criterion equivalent to a decrease of 0.8°C (*σ* > *σ*_{surface} + *α*0.8°C, where *α* is the thermal expansion coefficient; Kara et al. 2000; Buckley et al. 2014). From the monthly fields of MLD we then calculate a climatology and choose the annual maximum at each grid point (Fig. S1 in the online supplemental material). The winter MLD defines a bottom boundary of the upper ocean that varies spatially but is fixed in time. This isolates the layer that is in exchange with the atmosphere on an annual time scale, which is most relevant for climate variability (Buckley et al. 2014, 2015). In this layer, we expect only minor influences on the heat budget from vertical mixing and entrainment.

The global distributions of the covariance ratios for the main terms (Figs. 1a,b,f) clearly show that the balance in the anomaly heat budget is largely between anomalous forcing **u***θ*)′. There are distinct global patterns of covariance ratios of the budget terms that are to the first order latitudinal. The covariance ratios for **u***θ*)′ reveals a broad pattern of high covariance ratios in the tropics and much lower covariance ratios in the subtropics and at polar and subpolar latitudes, although −∇ ⋅ (**u***θ*)′ is relevant for most of the extratropical ocean (Fig. 1b).

Figures 1c–e show the temporal decomposition of −∇ ⋅ (**u***θ*)′ into anomalous advection caused by anomalies in circulation

Diffusion (

There is extensive compensation between horizontal and vertical components of **u***θ*) because the mostly positive covariance ratios for −∇ ⋅ (**u***θ*)′ are reflected by

In conclusion, the monthly anomaly heat budget integrated over the winter MLD on the native ECCOv4 grid is largely determined by anomalies in sea surface heat flux and anomalous advection of the mean temperature gradient, while mean advection of temperature anomalies plays a role only in specific regions of relatively strong currents (e.g., western boundary currents). In section 5, we seek further insight into the physics of these patterns by using cluster analysis to identify dynamically similar regions. First, however, we examine the scale sensitivity of these patterns.

## 4. Dependence on spatial and temporal scale

In this section, we explore the sensitivity of the covariance ratio analysis to different choices regarding spatial and temporal aggregation. The point of this is to investigate whether the patterns identified in section 3 and corresponding conclusions about the heat budget are robust over space and time scales, or whether qualitative changes emerge as we consider different scales.

### a. Depth of integration

In contrast to integrating over the winter MLD, we also investigated the balance between budget terms integrated over fixed depths. The aim of this is to understand how the heat budget varies as one considers increasingly deeper portions of the ocean. We know, for example, that all vertical fluxes must eventually vanish as we approach the bottom, but how deep must we go to see this? Small et al. (2019, 2020) focused only on the upper ocean in their analysis, leaving this question open.

The covariance ratios for each term in the heat budget were calculated for a range of depths (i.e., 100, 300, 700, and 2000 m) in order to describe the change in the relative importance of different mechanisms as vertical integration is varied. The principal drivers of the heat budget are consistently **u***θ*)′, but the balance between these mechanisms changes substantially according to the specific depth scale (Fig. 2). As expected, **u***θ*)′ as the dominant factor. Overall the most striking shift in patterns is from 100 to 300 m, while the change in patterns is more subtle when shifting the integration depths from the upper 300 m to deeper layers.

Global distribution of covariance ratios at different depths of integration. Each column represents the main budget terms (from left to right): anomalous forcing

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Global distribution of covariance ratios at different depths of integration. Each column represents the main budget terms (from left to right): anomalous forcing

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Global distribution of covariance ratios at different depths of integration. Each column represents the main budget terms (from left to right): anomalous forcing

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

One might assume that the shift in pattern is related to the vertical extent of shortwave penetration (as it is constrained to the upper 200 m). However, the interannual variability of that term is expected to be very low. Instead, the interannual signal is dominant in the turbulent heat fluxes, which are communicated by diffusive heat fluxes across the mixed layer. At shallow depths (i.e., 100 m) the pattern of covariance ratios for all budget terms closely resembles the covariance pattern of the winter MLD (Fig. 1) in the lower latitudes. In the higher latitudes, the covariance patterns for deeper layers (i.e., >300 m) in Fig. 2 resemble those in Fig. 1. This is mostly due to the spatial pattern of the winter MLD (Fig. S1), which to first order is deeper in the high latitudes (i.e., 200–1000 m) and shallower in the low latitudes (i.e., <200 m).

When integrated at 300 m and greater depths, **u***θ*)′ with depth is mostly due to the anomalous circulation term, while the patterns associated with mean circulation of anomalies is relatively insensitive to the depth of integration. The only exception is the relevance of

The effect of

In Fig. 3, we decompose advection into horizontal and vertical components. The compensation (i.e., anticorrelation) between the horizontal and vertical components of advection is particularly prominent at 100 m in the lower latitudes (Fig. 3). Again, there is a stark pattern shift when moving from 100 to 300 m, at which point there is much less compensation in the lower latitudes and more pronounced compensation in the midlatitudes such as in the subtropical gyres. Integrating over deeper layers (i.e., 2000 m) leads to vanishingly small vertical convergences. It is interesting to note that the anticorrelation between horizontal and vertical components only applies to anomalous circulation _{h} ⋅ **u**′ + (∂*w*′/∂*z*) = 0, states that convergence of horizontal transport and vertical transport must be anticorrelated. This anticorrelation logically carries over to the convergence of heat fluxes as well, since interannual variability of velocities drives interannual variability in

Global distribution of the covariance ratios for different depths of integration. Each column represents the following advective terms: anomalous horizontal advection of the mean temperature field, mean horizontal advection of the anomalous temperature field, anomalous vertical advection of the mean temperature field, and mean vertical advection of the anomalous temperature field. Each row represents the depth level over which budget terms are integrated: 100, 300, 700, and 2000 m. The covariance ratios have been evaluated on the original horizontal and temporal resolutions.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Global distribution of the covariance ratios for different depths of integration. Each column represents the following advective terms: anomalous horizontal advection of the mean temperature field, mean horizontal advection of the anomalous temperature field, anomalous vertical advection of the mean temperature field, and mean vertical advection of the anomalous temperature field. Each row represents the depth level over which budget terms are integrated: 100, 300, 700, and 2000 m. The covariance ratios have been evaluated on the original horizontal and temporal resolutions.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Global distribution of the covariance ratios for different depths of integration. Each column represents the following advective terms: anomalous horizontal advection of the mean temperature field, mean horizontal advection of the anomalous temperature field, anomalous vertical advection of the mean temperature field, and mean vertical advection of the anomalous temperature field. Each row represents the depth level over which budget terms are integrated: 100, 300, 700, and 2000 m. The covariance ratios have been evaluated on the original horizontal and temporal resolutions.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Our analysis on the sensitivity of the choice of a fixed depth layer suggests that a fixed depth level of 100 m is a first-order approximation for a global view of the upper ocean that yields a similar balance in the ocean heat budget compared to using the winter MLD (Fig. 1). Furthermore, it should be noted that the full-depth integration yields patterns in covariance ratios that are very similar to the case of the upper 2000 m (bottom row of Figs. 2 and 3). Thus, ocean variability in depths > 2000 m does not substantially contribute to variability in monthly OHC anomalies and is expected to play a role only at much longer time periods.

### b. Temporal scale

The ocean heat anomaly budget up to this point was only evaluated at monthly resolution. Considering the upper ocean (<300 m) and at higher latitudes,

Covariance ratios were averaged into 10° latitude bins to derive zonal means (Fig. 4). These confirm that the balance of the heat budget is dominated by

Zonal means of the covariance ratios for the different budget terms in the upper ocean defined by (top) the winter MLD, (middle) 300 m, and (bottom) 700 m, and for (left) monthly, (center) annual, and (right) pentad temporal averages. Covariance ratios were derived from the original (1 × 1) spatial resolution and averaged into 10° latitude bins.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Zonal means of the covariance ratios for the different budget terms in the upper ocean defined by (top) the winter MLD, (middle) 300 m, and (bottom) 700 m, and for (left) monthly, (center) annual, and (right) pentad temporal averages. Covariance ratios were derived from the original (1 × 1) spatial resolution and averaged into 10° latitude bins.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Zonal means of the covariance ratios for the different budget terms in the upper ocean defined by (top) the winter MLD, (middle) 300 m, and (bottom) 700 m, and for (left) monthly, (center) annual, and (right) pentad temporal averages. Covariance ratios were derived from the original (1 × 1) spatial resolution and averaged into 10° latitude bins.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

We next addressed the following question: How does integration of the heat budget over different depth levels (i.e., the winter MLD versus the upper 300 or 700 m) affect how the budget term balance changes with different time scales? When integrating over 300 m or deeper, there is no apparent compensation (cancelling positive and negative terms) except for the pentad averages. There are multiple terms whose zonal mean of covariance ratios are negative, occurring at latitudes 30°–60°S (corresponding to the Southern Ocean) and at 70°N (corresponding to the Nordic seas). This indicates that in these latitudes there can be strong anticorrelation at pentad time scale for terms that usually contribute to the total tendency (i.e., have positive covariance ratios). At latitude 70°N, the nonlinear advective term shows a strong compensation that is not apparent at higher frequencies (monthly and annual). At 60°S we see that

In summary, advection becomes the principal term with increasing temporal aggregation in the extratropics. In the high latitudes, diffusion increasingly determines variability with increasing temporal aggregation. Section 5 will show specific time scales at which the shifts in relevance occur for particular dynamical regimes.

### c. Horizontal scale

The balance of contributing terms in the heat budget equation varies according to the spatial and temporal scales on which the terms are derived. The remaining question is how the importance of each budget term changes as spatial aggregation changes from the original 1 × 1 grid to increasingly coarse aggregation scales (e.g., 2 × 2, 5 × 5, 10 × 10). The supplemental material provides additional details on how the spatial aggregation is performed. The dependence on horizontal scale has been pointed out by previous studies focusing on the surface ocean (Bishop et al. 2017; Small et al. 2019) and in climate models (Small et al. 2020), which showed that ocean transport is more relevant for higher resolutions. Table 1 lists the global average of covariance ratios of each budget term listed for each spatial aggregation scale, starting with the original resolution (1 × 1) to a maximum binning level of 90 × 90. In general, global mean covariance ratios for the upper ocean are sensitive to spatial scale, changing gradually when spatially aggregating the fields (Table 1).

Global average covariance ratios for heat budget terms at different spatial aggregations. Monthly heat budget terms were integrated over the wintertime climatological MLD. The aggregation value refers to the level of binning, where *n* × *n* aggregation indicates grouping of *n* grid cells along both *x* and *y* axes in the horizontal space.

There is a notable increase in **u***θ*)′. The relevance of both linear advective terms, **u***θ*)′ is around a 1/4 (25%). The non-continuous increase of

The covariance ratios are again averaged in 10° latitude bins and are plotted against latitude to illustrate the zonal balance between **u***θ*)′ with changing horizontal resolution, from the original 1° × 1° resolution to 30 × 30 aggregation (Fig. 5). Both **u***θ*)′ were determined over the winter MLD and the upper 300- and 700-m depth. Note that the coarsest resolution is set here at 30 × 30, because any coarser resolution fails to retain the latitudinal pattern and there is little change in the global balance of terms beyond that horizontal aggregation scale (Table 1). The zonal means of covariance ratios show similar sensitivity to spatial resolution across all latitudes with only a few exceptions. The significance of **u***θ*)′ are in the midlatitudes, especially in the Southern Hemisphere. The term −∇ ⋅ (**u***θ*)′ remains the main contributor in the low latitudes even at the largest aggregation scales.

Zonal means of the covariance ratios for anomalous forcing **u***θ*)′ (red lines). Lines are shaded by spatial aggregation scale, with darker shades corresponding to coarser aggregations. Covariance ratios were derived from **u***θ*)′ at each aggregation scale and averaged into 10° latitude bins. Zonal means are presented for the upper ocean defined by (top) the winter MLD, (middle) 300 m, and (bottom) 700 m, as well as using (left) monthly, (center) annual, and (right) pentad temporal aggregation.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Zonal means of the covariance ratios for anomalous forcing **u***θ*)′ (red lines). Lines are shaded by spatial aggregation scale, with darker shades corresponding to coarser aggregations. Covariance ratios were derived from **u***θ*)′ at each aggregation scale and averaged into 10° latitude bins. Zonal means are presented for the upper ocean defined by (top) the winter MLD, (middle) 300 m, and (bottom) 700 m, as well as using (left) monthly, (center) annual, and (right) pentad temporal aggregation.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Zonal means of the covariance ratios for anomalous forcing **u***θ*)′ (red lines). Lines are shaded by spatial aggregation scale, with darker shades corresponding to coarser aggregations. Covariance ratios were derived from **u***θ*)′ at each aggregation scale and averaged into 10° latitude bins. Zonal means are presented for the upper ocean defined by (top) the winter MLD, (middle) 300 m, and (bottom) 700 m, as well as using (left) monthly, (center) annual, and (right) pentad temporal aggregation.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Although a clear shift in the covariance ratios is evident, the overall balance across latitude remains the same. Where forcing is dominant (as in the high to mid latitudes) at the native grid resolution (1 × 1), it is still relevant at the coarsest resolutions (30 × 30). This remains true when looking at different temporal scales (i.e., monthly, annual, or pentad averages) as well for different depths of integration (i.e., winter MLD and 300 and 700 m). While the individual terms may shift, there are only a few cases where spatial aggregation causes a change in the overall balance of terms. For the winter MLD (top row), pentad scale includes large compensation between advective convergence term and **u***θ*)′ having covariance much greater than 1.0. Whereas in the upper 300 and 700 m, pentad averages of **u***θ*)′ is affected by spatial aggregation as covariance ratios shift from positive to negative values (Fig. 5).

As the zonal means of covariance ratios in Fig. 4 suggest, the contribution of −∇ ⋅ (**u***θ*)′ [in particular **u***θ*)′ is greater than **u***θ*)′ plays a more important role when looking at OHC variability at longer time scales (e.g., decadal trends).

The varying balance of the budget terms at different integration depths and aggregation scales raise the question of at what spatial scale **u***θ*)′ becomes dominant outside the high latitudes. As we see in Fig. 5, for the upper 300 m (and deeper depths) the contribution of −∇ ⋅ (**u***θ*)′ remains distinctly larger than **u***θ*)′ (and **u***θ*)′ and **u***θ*)′ is still relevant at very coarse resolutions (corresponding to roughly 90° × 90°) and is major at low to middle latitudes when integrating over a fixed depth of >300 m.

The heat budget was also evaluated for three ocean basins (i.e., Pacific, Atlantic, Indian) as a representation of highest spatial aggregation besides the global integral. The spatial masks we use for the ocean basin are provided by the “gcmfaces” toolbox (Forget et al. 2015) and are shown in Fig. S5. The largest contribution to the basin-scale heat budget over the winter MLD is clearly by **u***θ*)′ is also relevant (Table S1). Interestingly, for the Pacific and Atlantic basins, it is mainly the vertical advection, specifically **u***θ*)′. This is consistent with the analysis of vertical heat transport by Liang et al. (2015).

The basinwide heat budgets are further analyzed for different depths and temporal scales for the main terms (Fig. S6). Covariance ratios for **u***θ*)′ does not increase for greater integration depths, but it does become more important at longer time scales, especially in Atlantic and Indian basins. Across the three basins a clear shift occurs at >2 years (Pacific), >3 years (Atlantic), and >2–3 years (Indian). The shift in relevance is due to both greater relevance in

## 5. Classification of dynamical regimes

The balance of terms in the upper ocean heat budget shows clear spatial patterns (Fig. 1) that suggest distinct dynamical regimes, each associated with particular underlying mechanisms controlling heat content variability. Effectively summarizing dynamical regimes relevant to the ocean heat budget on a global scale is challenging given the overwhelming detail necessary to adequately describe each ocean region. Rather than splitting regions based on geographical features, we pursued an unsupervised machine learning technique to assess the global spatial pattern of OHC variability.

The *k*-means clustering algorithm is an efficient tool to reduce the spatial complexity of large datasets (Hartigan and Wong 1979; Gong and Richman 1995; Lund and Li 2009). The *k*-means clustering algorithm works to sort a set of samples (in our case, each geographical point in space) into a fixed number of clusters based on their features (in our case, the covariance ratios) with the aim of minimizing the variance within each cluster. By applying clustering, we can identify regions of shared characteristics in an objective way. A common application of clustering analysis in oceanography is the identification of marine bioregions, which has been done in specific parts of the ocean, such as the northwest Atlantic (Devred et al. 2007) and Southern Ocean (Ardyna et al. 2017), and globally (Sonnewald et al. 2020). A similar approach was used in a recent study in which the mean balance in the barotropic vorticity budget was analyzed (Sonnewald et al. 2019); however, that study focused on classifying the time-mean budgets. Our application of clustering is novel because it is applied to the covariance ratios, rather than the mean budget.

We applied *k*-means clustering to the covariance ratios of the three main heat budget terms *θ***u**), and *O*(1), no extra processing was necessary. The corresponding spatial patterns of the covariance ratios of the three budget terms are shown in Figs. 1a, 1b, and 1f. The dimensions of the parameter space (i.e., features) are defined by the covariance ratios of the three heat budget terms. The optimal number of clusters (*k* = 5) was shown to minimize variation within each cluster and any increase in *k* did not yield further (significant) reduction. Each ocean grid point was assigned to a given cluster based on the proximity to the cluster’s centroids within the parameter space.

The spatial distribution of the five global clusters is shown in Fig. 6a. Having divided the global ocean into these dynamical regions provides the opportunity for a physical interpretation of the drivers of heat content variability (Fig. 6b). The term −∇ ⋅ (**u***θ*)′ clearly dominates the heat budget in regions associated with cluster A. This is mainly because of the presence of strong currents near the equator, the ACC and western boundary currents. In the case of boundary currents and ACC, the relatively high contribution of the advection term is also due to strong spatial gradients in temperature (Bishop et al. 2017).

Classifications of the ocean using *k*-means clustering with five clusters labeled from A to E, representing variation between forcing **u***θ*)′, and diffusion

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Classifications of the ocean using *k*-means clustering with five clusters labeled from A to E, representing variation between forcing **u***θ*)′, and diffusion

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Classifications of the ocean using *k*-means clustering with five clusters labeled from A to E, representing variation between forcing **u***θ*)′, and diffusion

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Cluster B is dominated solely by **u***θ*)′ have major roles. Clusters D and E represent regions where diffusion is relevant, as in the Beaufort Gyre in the Arctic and the Antarctic marginal seas (i.e., the Ross and Weddell Seas). These regions are characterized by a very shallow winter MLD (<100 m; Fig. S1) and can be considered outliers. Dynamical regimes associated with clusters A, B, and C represent the vast majority of the ocean.

We further divided the clusters into basin-specific dynamical regimes (Fig. S9) in order to investigate the heat budget on a regional basis. Particular regions can serve as examples of the key dynamical regimes. In the advection-dominated regions (i.e., cluster A) we identify the Kuroshio and Kuroshio Extension in the North Pacific (Fig. S9; region A1) and the Gulf Stream in the North Atlantic (Fig. S9; region A2). Here the heat budget will be dominated by the western boundary current. The other advection-driven regions are the tropical Indian (region A3), Pacific (region A4), and Atlantic (region A5), as well as the ACC (region A6). The selected forcing-dominated regions (i.e., cluster B) are in the subtropical Atlantic and Pacific (Fig. S9; regions B1–B4). The representative regions for cluster C, where both *F*_{forc} and −∇ ⋅ (**u***θ*)′ are relevant, were chosen from the North Atlantic, North and South Pacific, and Indian basins (Fig. S9; regions C1–C4). The covariance ratios of the integrated heat budget terms corresponding to each of these regions is presented in Table S3 in the supplemental material. The remaining part of this section will focus on a subset of representative regions.

When the heat budget for the Kuroshio is calculated over the winter MLD, **u***θ*)′ each contribute half of the variability (Table S3). Integrating below the winter MLD, −∇ ⋅ (**u***θ*)′ increasingly dominates the variability in the heat budget with increasing depth of integration, and there is a clear shift evident from 100 to 300 m (Fig. 7). The shift toward −∇ ⋅ (**u***θ*)′ is especially notable at longer time scales, at which point it is the main driver with **u***θ*)′ (i.e.,

Covariance ratios for a selection of ocean regions at different integration depths (50, 100, 300, 700, 2000, and 6000 m) and time aggregation scales (1, 3, and 6 months, and 1, 2, 3, 4, 5, and 10 years). Regions represent distinct oceanic regimes and are derived using *k*-means cluster analysis (Fig. 6). Specific locations are shown in Fig. S6. Each column represents the four heat budget terms (forcing, advection, diffusion, residual) for the specified region. Each panel sorts the covariance ratio for each term by integration depth along the vertical axis and time aggregation scale along the horizontal axis.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Covariance ratios for a selection of ocean regions at different integration depths (50, 100, 300, 700, 2000, and 6000 m) and time aggregation scales (1, 3, and 6 months, and 1, 2, 3, 4, 5, and 10 years). Regions represent distinct oceanic regimes and are derived using *k*-means cluster analysis (Fig. 6). Specific locations are shown in Fig. S6. Each column represents the four heat budget terms (forcing, advection, diffusion, residual) for the specified region. Each panel sorts the covariance ratio for each term by integration depth along the vertical axis and time aggregation scale along the horizontal axis.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Covariance ratios for a selection of ocean regions at different integration depths (50, 100, 300, 700, 2000, and 6000 m) and time aggregation scales (1, 3, and 6 months, and 1, 2, 3, 4, 5, and 10 years). Regions represent distinct oceanic regimes and are derived using *k*-means cluster analysis (Fig. 6). Specific locations are shown in Fig. S6. Each column represents the four heat budget terms (forcing, advection, diffusion, residual) for the specified region. Each panel sorts the covariance ratio for each term by integration depth along the vertical axis and time aggregation scale along the horizontal axis.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Covariance ratio for a selection of ocean regions at different integration depths (50, 100, 300, 700, 2000, and 6000 m) and time aggregation scales (1, 3, and 6 months, and 1, 2, 3, 4, 5, and 10 years). Regions represent distinct oceanic regimes and are derived using *k*-means cluster analysis (Fig. 6). Specific locations are shown in Fig. S6. Each column represents the decomposed terms for advection for the specified region. Each panel sorts the covariance ratio for each term by integration depth along the vertical axis and time aggregation scale along the horizontal axis.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Covariance ratio for a selection of ocean regions at different integration depths (50, 100, 300, 700, 2000, and 6000 m) and time aggregation scales (1, 3, and 6 months, and 1, 2, 3, 4, 5, and 10 years). Regions represent distinct oceanic regimes and are derived using *k*-means cluster analysis (Fig. 6). Specific locations are shown in Fig. S6. Each column represents the decomposed terms for advection for the specified region. Each panel sorts the covariance ratio for each term by integration depth along the vertical axis and time aggregation scale along the horizontal axis.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

Covariance ratio for a selection of ocean regions at different integration depths (50, 100, 300, 700, 2000, and 6000 m) and time aggregation scales (1, 3, and 6 months, and 1, 2, 3, 4, 5, and 10 years). Regions represent distinct oceanic regimes and are derived using *k*-means cluster analysis (Fig. 6). Specific locations are shown in Fig. S6. Each column represents the decomposed terms for advection for the specified region. Each panel sorts the covariance ratio for each term by integration depth along the vertical axis and time aggregation scale along the horizontal axis.

Citation: Journal of Climate 34, 8; 10.1175/JCLI-D-20-0058.1

As for the Kuroshio and Kuroshio Extension (region A1), the tropical Pacific (region A4) is advection driven, with the distinction that −∇ ⋅ (**u***θ*)′ is much less sensitive to depth of integration and time scale (Fig. 7) and there is no anticorrelation apparent between **u***θ*)′ makes up roughly half of the budget (Fig. 8). In the northeast Pacific, the shift in budget terms with depth is more gradual (Fig. 7). Here the winter MLD is at approximately 150 m, so forcing remains significant at deeper integration depths, but only at shorter time scales (i.e., >2 years). At longer time scales there is a clear shift to advection dominance.

The above comparison of regional budgets for different dynamical regimes shows that the balance of terms in each case is sensitive to the spatial (in this case, depth of integration) and temporal scale. This sensitivity is different for each region, but in most cases there is a clear decrease in the significance of **u***θ*)′ at longer time scales. One exception is the tropical ocean regions where heat content variability is mostly driven by −∇ ⋅ (**u***θ*)′ across all time scales and depth levels. For all regions, diffusion is only relevant within the winter MLD. Although unique to the tropical regions, diffusion is mostly irrelevant even at shallow depths (Fig. 7).

## 6. Conclusions

This study investigated the contribution of individual mechanisms to OHC variability at a range of spatial and temporal scales. By employing ECCOv4, which is constrained by observations in a physically consistent way, the variability investigated in our analysis closely reflects the variability in the observed state of the ocean such as is described by ocean remote sensing and global observational networks (e.g., Argo). The work presented here includes novel approaches in which covariance ratios have been evaluated for a fully closed budget and have been used to define dynamical regimes. These approaches complement previous work describing factors influencing the ocean heat budget.

We have shown here that the balance in the upper ocean heat budget is mainly between anomalous surface forcing **u***θ*)′. Furthermore, the temporal decomposition of mean versus anomalous heat advection provided new insights in the OHC variability. In particular, the decomposition allowed us to see that most of the OHC variability is due to anomalies in circulation

By using the covariance ratio of the main budget terms as the set of features in the clustering algorithm, we defined dynamical regimes such that each would feature similar underlying mechanisms controlling variability in anomalous heat fluxes. Instead of using the mean budget, we focused on variability in the seasonal anomalies to define regions that are in essence dynamically similar. Ocean regions where −∇ ⋅ (**u***θ*)′ dominates the heat budget generally have strong currents and mostly show strong gradients in SST (Bishop et al. 2017). We identified specific areas in the ocean where **u***θ*)′.

Advective convergence due to circulation anomalies is by far the dominant driver of ocean heat change in the tropics, while

By distinguishing between horizontal and vertical components of advection, we have shown that vertical advective flux largely compensates for the horizontal component of the local heat budget. This is observed in the spatial distribution of covariance ratio where the horizontal term is greater than 1.0 while for the vertical term it is negative (Fig. 3). This simply indicates that convergence in the horizontal transport is correlated with divergence in the vertical transport (i.e., volume is conserved). Previous studies (e.g., Abernathey and Wortham 2015; Chemke et al. 2020) have estimated the advective component of the heat budget using just surface data and horizontal velocities, neglecting the vertical component. Our results suggest that surface advection is a useful predictor for the total advective tendency, but it will generally provide an overestimate, due to the compensating nature of the vertical fluxes.

This study has also clearly demonstrated the importance of the depth of integration chosen to define the upper ocean. Previous studies have used the wintertime climatological MLD as the bottom boundary (Buckley et al. 2014, 2015) while other studies used a fixed depth horizon (e.g., Doney et al. 2007; Grist et al. 2010; Piecuch et al. 2017; Small et al. 2020). As we have shown, there are substantial differences in the spatial patterns of the covariance ratios between the heat budget terms when determined by integrating over a fixed depth versus when determined by integrating over the winter MLD. A striking example is given by the forcing-dominated subtropical regions (regions B2–B4 in Fig. S9). This is consistent with previous studies that showed that

For shallower layers (i.e., upper 100 m) the balance between **u***θ*)′ is comparable to the one determined over the winter MLD. With increasing depths of integration, the balance between **u***θ*)′ shifts toward higher contribution of the advective terms. The contribution of **u***θ*)′ becomes more important and forcing diminishes in the lower latitudes. When integrating over the upper 2000 m (or the entire water column),

The heat budget is also sensitive to the temporal scale. Averaging over longer time intervals (i.e., varying the temporal mean from monthly to decadal), results in a decrease in **u***θ*)′. This suggests that heat content variability is largely forcing driven on shorter time scales, while advective processes are increasingly important at longer time scales. Such time scale dependencies have been reported for the North Atlantic by Buckley et al. (2014, 2015) and for the WBCs and ACC by Bishop et al. (2017). We have shown here that this transition from forcing to advection-driven budgets as temporal aggregation increases is common in most dynamical regimes. Thus, for future studies, it is important to clearly define at what temporal scales heat content variability is analyzed.

Interestingly, it is mostly

Consistent with recent studies by Bishop et al. (2017) and Small et al. (2019, 2020), we find that spatial aggregation of the gridded ECCOv4 fields to coarser resolutions changes the balance between forcing and advection. However, in our case the overall patterns remain the same when aggregating the grid from the original resolution of 1° × 1° up to a factor of 90. This low sensitivity of the heat budget to aggregation scale is surprising, as our initial expectation was that the balance of mechanisms in the budget would shift strongly toward

Similarly, the contribution by −∇ ⋅ (**u***θ*)′ decreases only slightly with coarsening, mostly in the high latitudes. Advection remains the main contributor in the low latitudes, even at the largest aggregation scale (i.e., 90 × 90). In contrast to our results, previous studies using similar methods (Bishop et al. 2017; Small et al. 2019, 2020) instead found that spatial aggregation changed the heat budget qualitatively. The cause for this discrepancy is probably that the spatial resolution of the ECCOv4 state estimate is already too coarse to resolve mesoscale dynamics. The only possible exception is for the tropical oceans, where the eddy-like motions (e.g., tropical instability waves) occur on such large scales that they can be partially resolved in ECCOv4.

The highest *n* value (*n* = 90) corresponds to approximately 90° × 90°, which can be considered a basinwide scale. Any coarser aggregation would lead to summing over different ocean basins (across continents), which would yield ambivalent results in terms of potential underlying mechanisms. Coarsening of the grid beyond 90° × 90° was addressed by evaluating the heat budget for the three major ocean basins (Pacific, Atlantic, Indian). This tells us that heat transport by the basin-scale circulation (e.g., large-scale gyres and overturning circulations) is playing a dominant role in the basin-scale heat budget. Thus, it is not possible to determine a specific resolution scale at which −∇ ⋅ (**u***θ*)′ will become zero, other than the entire global ocean.

We note certain caveats associated with our study. First and foremost, ECCOv4 is a relatively coarse-resolution model and therefore unable to resolve mesoscale ocean processes. Similar to the work presented here, Small et al. (2020) evaluated ocean heat budgets over the upper 50 and 400 m, using both a high- and low-resolution setup. An important insight regarding the impact of resolution arose when performing spatial smoothing with their high-resolution model output to determine at what scale the high-resolution model results reflects the low-resolution results. They found that for most regions this occurs when averaging over a box of 3°–5° for the 50-m budget and 5°–7° for the 400-m budget. As most of the sensitivity to spatial resolution lies below 1° (Bishop et al. 2017; Small et al. 2020), it makes sense that the spatial aggregation with ECCOv4 did not lead to large differences globally, as the spatial resolution of ECCOv4 is around 1°

While higher spatial resolution is important in capturing ocean dynamics relevant to heat content variability, it is currently not feasible in a reanalysis framework to present estimates at resolutions below 1° and still ensure constraining them to available observations. Despite these limitations, ECCOv4 presents a distinct advantage in that it is a physically consistent estimate of the observed ocean state. It accurately reflects the ocean variability over larger regions, though it must be recognized that once the spatial resolution is increased, variability in mesoscale ocean dynamics will likely play a more important role in characterizing overall variability.

Another caveat of our approach is that only 24 years of data are available, limiting our capability to analyze the budget on a decadal time scale. The issue that arises is that at longer temporal aggregation scales, the time series have fewer and fewer points and so the correlations become more noisy. Thus, with the pentad averages the number of data points may be too small to yield robust results. On the other hand, our results are consistent with the findings of other studies (Buckley et al. 2014, 2015; Bishop et al. 2017). By using multiple temporal aggregations we were able to reveal a clear shift toward advective-driven heat budgets which often occurs at particular time scales. For most dynamic regions this was shown with averaging beyond a 2-yr time scale. We encourage the application of our time aggregation methodology to longer dataset runs (e.g., hindcast simulations or coupled climate models), in order to provide an independent and more robust way to identify important time scales at which shifts in the heat budget balance can be expected.

## Acknowledgments

JET acknowledges funding from NASA’s Goddard Space Flight Center (Award NNX15AN27H). RPA acknowledges support from NSF Award OCE-1553593 and a Sloan Fellowship in Ocean Sciences. Computational tools for performing this research were provided by Pangeo, supported by NSF EarthCube award OCE-1740648. The authors thank Spencer Jones for providing helpful comments. We wish to thank Martha Buckley and two anonymous reviewers for their careful assessment of the manuscript and for their helpful suggestions on improving it.

## Data availability statement

All results of this study are based on ECCO Version 4, Release 3 (ECCOv4r3) for which standard output and documentation can be obtained at https://ecco.jpl.nasa.gov/drive/files/Version4/Release3/. We reproduced the ECCOv4r3 ocean state estimate with a custom set of diagnostics that are available as a dataset on Pangeo (https://catalog.pangeo.io/browse/master/ocean/ECCOv4r3/) or can be requested from the corresponding author.

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