a. Observed data

Ishii et al. (2017) and Levitus et al. (2012) datasets are used for this study. Those datasets were constructed using temperature and salinity observations issued from the World Ocean Database. Main differences between both datasets lie in the interpolation techniques and instrumental corrections for biases in expendable and mechanical bathythermograph data. The observed OHC anomalies for the upper 700 m used for this study are the ensemble mean of the two reconstructions.

Reconstructed fields are provided globally and over the full ocean depth, although in situ observations are sparse and unevenly distributed with a bias toward the Northern Hemisphere and the upper ocean (the vast majority of observations are above 700-m depth; Rhein et al. 2013; Abraham et al. 2013). In particular there are very few observations in the Southern Ocean before the year 2000 (Durack et al. 2014). Different techniques are used to fill areas presenting few or no observations. Ishii et al. (2017) and Levitus et al. (2012) use objective interpolation techniques to carry out the infilling, which bias the temperature anomaly toward zero in data-sparse regions (Gregory et al. 2004; Gleckler et al. 2012). This affects the depiction of OHC content changes and variability at the global scale and in regions with sparse data (Gregory et al. 2004; AchutaRao et al. 2007).

To remove this bias and ensure the accuracy of the D&A analysis, we apply an observational mask that restricts the data to areas and periods with a sufficient number of available observations. We build the observational mask using the data distribution field (dd) available in the World Ocean Atlas. For a given grid cell and a given year, the entire 0–700-m water column is considered as observed when at least 10 observations are present along the water column. To avoid seasonal biases, an additional criterion retains a given grid cell and a given year only if observations cover at least two trimesters. We also choose to focus over the period 1971–2005 because the data coverage becomes more global in 1971 (Rhein et al. 2013; Abraham et al. 2013).

Figure 1 shows the total and subsampled yearly mean OHC anomalies. As expected, the subsampling induces a larger variability in particular at the times of the sparsest sampling [i.e., before 1971 and between 1996 and 2003 when the coverage of the global ocean was below 40%; Abraham et al. (2013)]. The subsampled dataset also exhibits a larger trend over the period 1971–2005. This is not surprising as the infilling of nonobserved areas with null trends in Ishii et al. (2017) and Levitus et al. (2012) datasets tends to reduce the global positive trend observed over 1971–2005.

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Yearly mean time series of ocean heat content anomalies for the total reconstructed dataset (continuous line) and for the dataset subsampled with the observational mask (dashed line) as derived from the World Ocean Atlas (see text).

Citation: Journal of Climate 33, 24; 10.1175/JCLI-D-19-0091.1

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b. Climate simulations

In this study, we use various types of CMIP5 numerical experiments: preindustrial control (PIC) and historical (HIST, include all forcing), natural forcing only (NAT), and greenhouse gases forcing only (GHG). The last two sets of simulations are specifically designed for D&A studies, where only a subset of external forcings are varying in time.

PIC runs simulate the climate system under constant forcings, equal to those observed in 1850 or 1860. In particular, greenhouse gas atmospheric concentrations and aerosol rates are kept constant to the level observed in 1850 or 1860. The length of PIC runs ranges from 200 to 1156 years (see second column of Table 1), totaling more than 20 000 years of simulation. Those unforced simulations are used to correct a potential model drift and to estimate the internal variability (Sen Gupta et al. 2013). CMIP5 PIC runs start after the completion of the model spinup. However, the length of this spinup is usually too short to reach the required quasi-equilibrium in the deep ocean, and most PIC and other simulations still exhibit a drift in OHC (in particular in the deep ocean). For each model, and for each grid cell, the drift is computed as the linear trend of the OHC over the full length of the PIC run and is removed from the PIC and other historical simulations following Melet and Meyssignac (2015). Internal variability is estimated from the PIC runs divided into 35-yr segments, taken at 20-yr intervals, matching the length of the period of interest in this study: 1971–2005.

Table 1.

List of models used for this study, length of preindustrial control runs (PIC; in years), and number of realizations available for historical (HIST), natural only (NAT), and greenhouse gases only (GHG) experiments. (Expansions of acronyms are available online at http://www.ametsoc.org/PubsAcronymList.)

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HIST runs simulate the climate over the period 1850–2005 in response to all known external forcings, including anthropogenic forcings (from GHGs, aerosols, ozone, and land use when available) and natural forcing (solar and volcanic forcings). NAT runs cover the same period but only the natural forcing varies with time. GHG runs also cover this period, with varying GHG concentrations while other forcings are kept constant. The inclusion of other attribution experiments (e.g., anthropogenic forcings only, aerosol forcing only) was prevented by the limited number of runs available in CMIP5. However, by assuming that the response to a combination of forcings is equal to the sum of the responses to individual forcings, we can infer the response of OHC to other combinations of forcings. The additivity assumption has been verified for the dynamical sea level by Slangen et al. (2015). It holds for OHC as dynamical sea level and OHC are linearly related at global scale. We infer the response of OHC to all anthropogenic forcings (including GHGs and aerosol forcing, referred to as ANT forcing hereafter) by calculating the difference between the response of OHC in HIST versus NAT simulations. We also infer the response of OHC to aerosols (and land use when available, referred to as OA forcing hereafter) as the OHC response to HIST forcing minus NAT forcing minus GHG forcing.

The selection of CMIP5 models was made upon the availability of temperature and salinity fields during the 1971–2005 period, resulting in a total of 39 models for the HIST experiment and 18 models for both NAT and GHG experiments (see Table 1). The HIST, NAT, and GHG columns of Table 1 details the number of realizations used for each model.

c. Calculation of the ocean heat content trends

To compute OHC anomalies, monthly 3D temperature and salinity fields from 0- to 700-m depth are averaged annually and masked over marginal seas and lakes (such as the Mediterranean Sea, Red Sea, Black Sea, Caspian Sea, Baltic Sea, Persian Gulf, Hudson Bay, and Great Lakes) for each dataset. OHC anomalies are computed from the annual mean temperature fields and a climatology of the salinity fields, following the 1980 UNESCO International Equation of State (IES80). The OHC is calculated on each layer and each cell of the reconstruction/model grid and vertically integrated. Anomalies are obtained by subtracting the 1971–2000 climatology for each grid cell. For observations, temperature and salinity fields were first converted into potential temperature fields [using the methodology of Bryden (1973)] before the computation of OHC anomalies.

In a second step, we compute the global (and regional, when necessary) subsampled averages and trends. In this case, a common observational mask, varying in time and space, is applied to all observational datasets and simulations. As a consequence, the spatial coverage might be different from one year to another. The spatial average is computed from subsampled OHC anomalies over the globe (and over different ocean basins when necessary). Finally, linear trends are computed over the period of interest: 1971–2005.

The D&A method is applied to the subsampled OHC anomaly trends. As a consequence, the observational constraint obtained applies to the subsampled OHC trend, which is different from the global OHC trend. This result can be difficult to understand and to be exploited. Therefore, we translate the results obtained on subsampled OHC (sub-OHC) into a result applicable to global OHC (tot-OHC) using a linear regression model between sub-OHC and tot-OHC. For each type of forcing, a linear regression model y = ax + b is fitted from an ensemble of sub-OHC (x) and tot-OHC (y) values issued from available model simulations. An observational constraint on tot-OHC trends, as well as the associated uncertainty, is then derived from this linear regression (see Fig. 2). Uncertainties on the regression line are included in the tot-OHC uncertainties, using a Monte Carlo technique to sample uncertainty on a and b. The regression models assume that, for a given type of experiment, changes over a spatiotemporal subset (as defined by the observational mask) are linearly related to the global changes, and that this ratio is correctly simulated by GCMs. Figure 2 shows the global OHC against the subsampled OHC for the period 1971–2005 for all GCM simulations and for observations [we use here the infilled estimate from Ishii et al. (2017) and Levitus et al. (2012) as a global OHC estimate]. This figure suggest that, for the HIST, NAT, and GHG simulations, the relationship between tot-OHC and sub-OHC is close to linearity. The root-mean-square error (RMSE) estimated as the root-mean-square (RMS) of the residual is below 16% of the RMS of the total signal for the NAT simulations, below 9% for the HIST simulations, and below 7% for the GHG simulations. The relationship between tot-OHC and sub-OHC is also primarily linear for OA and ANT simulations as expected, since these simulations are linear combinations of the HIST, NAT, and GHG simulations. For all simulations that include the GHG forcing (i.e., HIST, GHG, and ANT) the estimated regression coefficients a are similar because the OHC signal is dominated by the response to GHGs (e.g., Gleckler et al. 2012). For the NAT simulation, the internal variability dominates and generates a different ratio between sub-OHC and tot-OHC. For the OA simulations, the ratio between sub-OHC and tot-OHC is significantly different from that found in GHG experiments, because the aerosol forcing is highly variable across space (unlike the GHG forcing). Note that observations lie below the regression line of the HIST simulations. This is because the global observed OHC is biased low due to the infilling technique in Ishii et al. (2017) and Levitus et al. (2012), which tends to bias the temperature anomalies toward 0°C in data-sparse regions such as the Southern Ocean (Durack et al. 2014).

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Linear relation between the subsampled ocean heat content 35-yr trends and the global ocean heat content 35-yr trends for different climate model simulations and for observations. For observations, the 35-yr trend is computed over the period 1971–2005 and indicated with a black dot. HIST refers to historical simulations that include all forcing. NAT refers to simulations with only the natural forcing. ANT refers to simulations with only anthropogenic forcing (including GHG emissions, aerosol forcing, and land use change when available). It is calculated by taking the difference between HIST and NAT simulations. GHG refers to simulations with only the forcing from anthropogenic GHG emissions. OA refers to simulations with only the forcing from anthropogenic activity other than GHG emissions (it includes the forcing from aerosol emissions and land use change when available). It is calculated as HIST simulations minus NAT simulations minus GHG simulations.

Citation: Journal of Climate 33, 24; 10.1175/JCLI-D-19-0091.1

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d. Detection and attribution method

Most D&A methods rely on linear regression models where the observations Y are regressed onto the expected response to forcings i, as simulated by climate models X_i. These methods assume that the spatiotemporal pattern of the response to each forcing given by climate models is correct but that its amplitude is wrong (Hegerl and Zwiers 2011; Bindoff et al. 2013). Their objective is then to find the right amplitude by comparison with observations. The assumption underlying these methods presents two important limitations. First, models do provide information on the amplitude of the response to each forcing. This information is useful and should not be discarded a priori because it can help to constrain the estimate of the real response to each forcing. Second, the spatiotemporal distribution of the response to each external forcing simulated by climate models is not perfectly correct. A careful analysis suggests that there are significant differences among climate models in the spatial response patterns, as shown by the large spread among model patterns (see, e.g., Slangen et al. 2014). So, there are probably some errors in the models’ patterns. This information should be considered through a modeling of the uncertainty in the spatial response pattern.

Recently Ribes et al. (2017) developed a new D&A method to deal with these two limitations. We use this new D&A method, which provides a framework to treat the amplitude and the spatiotemporal pattern of the response to different forcing consistently, without any a priori assumptions on their reliability.

The Ribes et al. (2017) statistical method is based on the additivity assumption: the true and unknown response of the climate system (in terms of OHC here) to all external forcing Y* is assumed to be equal to the sum of the true and unknown responses to each external forcing i taken separately ${\mathbf{X}}_i^{\text{*}}$:

where n_f is the total number of external forcing considered. It is also assumed that the observations of the climate response Y and the multimodel mean response to each forcing i, X_i (i.e., the best available estimate of the expected response), are related to the exact responses Y* and ${\mathbf{X}}_i^{\text{*}}$ with some uncertainty as described in the following equations:

where ε_Y represents the noise in observations Y (including internal variability and observation errors) and ${\mathit{\epsilon }}_{{\mathbf{X}}_i}$ represents the noise in simulated responses (including internal variability, and modeling errors in GCMs). The random terms ε_Y and ${\mathit{\epsilon }}_{{\mathbf{X}}_i}$ are assumed to be independent and to follow Gaussian distributions centered on zero with known variance–covariance matrices Σ_Y and ${\mathbf{\Sigma }}_{{\mathbf{X}}_i}$.

In (1) and (2), Y, X_i, and ε are all vectors made of n variables. The way these vectors are constructed deserves some discussion. Usually in D&A methods, these vectors are spatiotemporal vectors, in which each coordinate corresponds to the mean over a given region and a given period. In this study, we will limit ourselves to the univariate (i.e., with n = 1, using only global averages) or the bivariate (i.e., with two regions and n = 2) cases. We can hardly go further because the small number of independent climate model simulations available in CMIP5 is not large enough to estimate properly each covariance matrix ${\mathbf{\Sigma }}_{{\mathbf{X}}_i}$ when n > 2 (see below for more details). In the univariate case, we only consider the global 1971–2005 OHC trend. In the bivariate case, an extra variable (e.g., the OHC trend over another region and/or period) is considered in addition to the global OHC 1971–2005 trend variable. The idea to use two variables at the same time is to account for some spatial information in the D&A. With this approach we expect to improve the attribution of the forced signal in OHC response and to separate the GHG response from the aerosol response. We run a set of 83 bivariate D&A analysis with a range of 83 different extra variables added to the global OHC 1971–2005 trend. The objective is to test which variable leads to the best separation of the GHG response from the aerosol response.

The main parameters of interest in (1) and (2) are ${\mathbf{X}}_i^{\text{*}}$, which are the responses to each forcing. By estimating these parameters, we detect and attribute the OHC response to each forcing including GHG and aerosols. The maximum likelihood estimate ${\widehat{X}}_i^{\text{*}}$ of ${\mathbf{X}}_i^{\text{*}}$ is given by

where A/B denotes the product A ⋅B⁻¹ and ${\mathbf{\Sigma }}_{\mathbf{X}}={\displaystyle {\sum }_{i=1}^{n_f}{\mathbf{\Sigma }}_{{\mathbf{X}}_i}}$. The distribution of ${\widehat{X}}_i^{\text{*}}$ is a Gaussian given by the following closed-form expression [see Ribes et al. (2017) for more details]:

To estimate ${\mathbf{X}}_i^{\text{*}}$ we need to estimate the variance matrices Σ_Y and ${\mathbf{\Sigma }}_{{\mathbf{X}}_i}$. These matrices represent the noise in respectively the observations of the OHC forced response and the simulations of the OHC forced response. The noise in the observations is generated by two sources of uncertainty: the uncertainty in measurements and the uncertainty induced by the internal variability. The uncertainty induced by the internal variability Σ_υ is estimated from unforced PIC experiments. OHC trends are calculated for 35-yr-long segments, following the same processing as for observations. Segments are taken at 20-yr intervals along the PIC runs (segments can overlap and are thus not totally independent) and Σ_υ is computed as the sample estimate over this ensemble of segments. It leads to an estimate of the uncertainty induced by the internal variability of ±0.48 × 10⁷ J m⁻² yr⁻¹ (see Fig. 3a). The measurement uncertainty is dominated by instrument bias corrections, and the definition of a baseline climatology to estimate the OHC anomalies (see, e.g., Boyer et al. 2016). Boyer et al. (2016) estimated that the uncertainty due to instrument bias corrections and the base line climatology at global scale for the period 1970–2008 ranges between ±0.10 × 10⁷ and ±0.15 × 10⁷ J m⁻² yr⁻¹ (depending on the spatial interpolation scheme used to calculate the global OHC estimate; see their Table 4). Here we consider the period 1971–2005, which is close to the reference period 1970–2008 of Boyer et al. (2016) and we do not interpolate the data to get global OHC estimate. Because we do not interpolate the data, we do not introduce any uncertainty due to the interpolation scheme, so the measurement uncertainty in our case is smaller than the estimate from Boyer et al. (2016). In the worst case, Boyer et al. (2016) estimate the uncertainty due to instrument bias corrections and the base line climatology is ±0.15 × 10⁷ J m⁻² yr⁻¹. This is about 3 times as small as the uncertainty due to the internal variability estimated above. For this reason, we neglect here the uncertainty in measurements and we approximate the uncertainty in the observations by the uncertainty due to the internal variability only (Σ_υ).

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Global 1971–2005 sub-OHC trend values and associated probability distribution functions for (a) PIC simulations (green curve), (b) HIST simulations (purple curve), (c) the combination of NAT (blue curve) and ANT (orange curve) simulations, and (d) the combination of NAT (blue curve), GHG (red curve), and OA (yellow curve) simulations. The trend value obtained by one GCM (averaged over available realizations) is represented by a vertical bar. The observed sub-OHC trend is represented by a black arrow and the sum of the combinated forcings by a pink arrow. The gray error bars and their center represent the 90% confidence interval and the multimodel mean computed from the ensemble of simulations. The colored error bars represent the reduced 90% confidence interval and the new estimation computed from the Ribes et al. (2017) univariate detection and attribution model.

Citation: Journal of Climate 33, 24; 10.1175/JCLI-D-19-0091.1

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The uncertainties in the simulations of the forced response, X_i, are related to both the uncertainty due to internal variability Σ_υ and the uncertainty due to climate modeling errors, Σ_m. The procedure used to estimate each ${\mathbf{\Sigma }}_{{\mathbf{X}}_i}$ is detailed in Ribes et al. (2017). The key step corresponds to the estimation of Σ_m and relies on the paradigm that models are statistically indistinguishable from the truth (Annan and Hargreaves 2010). The resulting Σ_m estimate depends on the spread of responses among models, within the HIST, NAT, and GHG multimodel ensembles.

a. Detection of a signal inconsistent with internal variability alone

The internal variability in OHC is estimated from more than 20 000 years of unforced detrended PIC runs. Global sub-OHC anomalies trends, calculated on 35-yr segments, are shown as vertical bars on Fig. 3a. The data distribution is very similar to a Gaussian distribution of 0 ± 0.58 × 10⁷ J m⁻² yr⁻¹ (90% confidence interval), with the model unforced responses ranging from −1.04 to 1.26 × 10⁷ J m⁻² yr⁻¹. The consistency of the observed trend (black arrows in Fig. 3) with internal variability [see Eq. (18) in Ribes et al. 2017] can be tested statistically, and is rejected with a p value smaller than 10⁻⁴.

As expected, the observed global trend (1.38 × 10⁷ J m⁻² yr⁻¹) cannot be explained by the internal variability alone. This finding is consistent with previous studies (Barnett et al. 2001; Pierce et al. 2006; Gleckler et al. 2012, 2016).

b. Role of external forcings

We now compare the observations with historical simulations including all forcings and historical simulations including only a subset of forcings.

First, observations are compared with HIST simulations (see Fig. 3b). The Gaussian distribution describing uncertainty in observations is deduced from the internal variability and centered on the observed global OHC trend. Ensemble means obtained from HIST runs for each model are represented by vertical bars. They range from 0.16 to 2.21 × 10⁷ J m⁻² yr⁻¹, corresponding to X_HIST = 1.43 ± 0.72 × 10⁷ J m⁻² yr⁻¹ assuming a Gaussian distribution. The observed trend is consistent with the simulated response to all forcing, with a p value of 0.94. This shows that models perform well in reproducing the observed trend. By combining the simulated and observed trends, a new estimation of the OHC response to all forcings is computed, using a univariate D&A model (Ribes et al. 2017). This new estimation of the real response (purple error bar) is similar to the multimodel mean (the gray and the purple error bars are both centered on the same value), but with a substantial decrease of the corresponding uncertainty (i.e., the size of the error bars) if compared to the multimodel estimate–error bars are reduced by 37%. The new estimate of the OHC changes in response to all forcings over 1971–2005 is ${\widehat{X}}_{\mathrm{HIST}}^{\text{*}}=1.40\pm 0.45\times {10}^7\mathrm{J}{\mathrm{m}}^{\mathrm{-2}}\mathrm{y}{\mathrm{r}}^{\mathrm{-1}}$.

We now compare the observed trend with the response to different combinations of forcings. Figure 3c shows the response of sub-OHC trends to NAT and ANT forcing. The uncertainty in simulated NAT trends is limited, with a model estimate of X_NAT = −0.18 ± 0.36 × 10⁷ J m⁻² yr⁻¹. The outlier at −1.22 × 10⁷ J m⁻² yr⁻¹ comes from the GFDL-ESM2M simulation, which has only one realization. There are actually some other individual realizations from other models that show similar values but they do not appear on this graph as they are part of a multimember ensemble and thus they are smoothed in the ensemble mean (Fig. 3 only shows, for each model, the ensemble mean across all realizations available). The response to ANT forcing exhibits a larger variance among models (individual model trends ranging from 0.60 to 2.65 × 10⁷ J m⁻² yr⁻¹) than the response to NAT forcing. It shows a large multimodel mean of X_ANT = 1.57 ± 0.94 × 10⁷ J m⁻² yr⁻¹, which dominates the response to all forcing. The observed trend (indicated by a black arrow that is masked by the pink arrow on Fig. 3c) cannot be explained by NAT forcing only. However, it is consistent with the response to ANT forcing only, with a p value of 0.78. It is also consistent with the sum of the NAT and the ANT responses. Indeed, the simulated OHC response to NAT + ANT forcing is almost equal to the observed OHC response (the pink arrow points almost to the same value as the black arrow on Fig. 3c; for this reason the two arrows are separated vertically). We now apply the univariate D&A model to reduce uncertainties and compute a new estimation of the OHC response to each forcing. The D&A analysis yields similar estimates as the multimodel mean in terms of mean OHC response to the NAT forcing and to the ANT forcing. However, it improves the uncertainty around these mean responses. The standard deviation of the uncertainty in the OHC response to the NAT forcing is only improved by 5% (the new estimate is ${\widehat{X}}_{\mathrm{NAT}}^{\text{*}}=-0.18\pm 0.34\times {10}^7\mathrm{J}{\mathrm{m}}^{\mathrm{-2}}\mathrm{y}{\mathrm{r}}^{\mathrm{-1}}$). The improvement is marginal because the uncertainty in the multimodel mean OHC response to the NAT forcing is already small compared to the other uncertainties. For the OHC response to the ANT forcing, the improvement is much clearer, as the uncertainty is reduced by 41% (the new estimate is ${\widehat{X}}_{\mathrm{ANT}}^{\text{*}}=1.57\pm 0.55\times {10}^7\mathrm{J}{\mathrm{m}}^{\mathrm{-2}}\mathrm{y}{\mathrm{r}}^{\mathrm{-1}}$).

We now consider the three-forcing decomposition of the response (NAT + GHG + OA), shown in Fig. 3d. In this analysis, the ANT forcing is decomposed into two distinct forcings: the GHG forcing and the OA forcing (which is largely dominated by the aerosol forcing). As expected, the results for the NAT forcing in this analysis are unchanged compared to the results obtained before. This is because splitting ANT into GHG + OA has no impact on the NAT calculations. Under GHG forcing only, the sub-OHC anomaly trends estimated from the multimodel mean reach the value of X_GHG = 2.41 ± 0.64 × 10⁷ J m⁻² yr⁻¹ (with extreme values at 1.83 and 3.16 × 10 J m⁻² yr⁻¹). Under the OA forcing, the sub-OHC anomaly trends are mainly negative (values ranging from −2.11 to +0.66 × 10⁷ J m⁻² yr⁻¹) with a multimodel mean value of X_OA = −0.84 ± 0.90 × 10⁷ J m⁻² yr⁻¹. This negative response of OHC is due to the cooling effect of aerosols on the ocean (Bindoff et al. 2013). It compensates partly the large positive trend induced by the GHG forcing. On an individual basis, none of the various forcings generate a response in OHC that is consistent with observations. But the sum of the response to all forcings is nearly equal to the observed 1971–2005 trend in OHC (in Fig. 3d the black arrow representing the observed OHC trend is masked below the pink arrow, which represents the combination of the response to all forcings). Among all individual OHC responses, the OHC response to the OA forcing only is the most uncertain. This is because the OA forcing is very different from one model to another (e.g., there is a large spread in the aerosols–cloud interactions across climate models; Boucher et al. 2013). The response of the climate system to the OA forcing is also very different from one model to another. When we apply the D&A procedure, we find no significant shift in the estimates of the OHC response to each forcing compared to the multimodel mean estimate. This means that the multimodel mean estimate agrees with observations within error bars. However, the D&A yields significant improvements on the estimate of the uncertainties. The standard deviations of the uncertainty in OHC response to the NAT, GHG, and OA forcings are reduced respectively by 4%, 13%, and 28%, with new estimates of ${\widehat{X}}_{\mathrm{NAT}}^{\text{*}}=-0.18\pm 0.35\times {10}^7\mathrm{J}{\mathrm{m}}^{\mathrm{-2}}\mathrm{y}{\mathrm{r}}^{\mathrm{-1}}$,${\widehat{X}}_{\mathrm{GHG}}^{\text{*}}=2.41\pm 0.55\times {10}^7\mathrm{J}{\mathrm{m}}^{\mathrm{-2}}\mathrm{y}{\mathrm{r}}^{\mathrm{-1}}$, and ${\widehat{X}}_{\mathrm{OA}}^{\text{*}}=-0.84\pm 0.65\times {10}^7\mathrm{J}{\mathrm{m}}^{\mathrm{-2}}\mathrm{y}{\mathrm{r}}^{\mathrm{-1}}$. These improvements enable a better separation of the OHC response to the different forcings. In particular, the response to GHG and OA forcings is significantly refined. This is an improvement compared to previous studies based on linear regression models that cannot separate effectively the response to GHG from the response to OA (e.g., Barnett et al. 2005; Pierce et al. 2012; Slangen et al. 2014; Tokarska et al. 2019).

Figure 4a shows the contributions of each individual forcing when considering the model ensemble (gray error bars) and after the application of the univariate D&A model (black continuous error bars). The use of the Ribes et al. (2017) D&A model on subsampled OHC trends results in a better determination (i.e., lower uncertainty) of the contributions of individual forcings, especially the most uncertain ones, such as HIST, ANT, and OA.

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Contribution of each forcing to the global 1971–2005 OHC anomaly trends. (a) For the subsampled OHC, uncertainties are computed from the model ensemble (gray error bars) and from the univariate (black continuous error bars) and the bivariate (black dotted error bars) detection and attribution models. (b) The total OHC is inferred from the subsampled bivariate results, using a linear relationship between simulated subsampled and total anomaly trends. The uncertainty is inferred from the subsampled bivariate uncertainty and from the subsampled-to-total transformation uncertainty (black dotted error bars).

Citation: Journal of Climate 33, 24; 10.1175/JCLI-D-19-0091.1

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c. Exploration of the bivariate case and potential to further reduce uncertainties

In this section, we include an extra variable in the D&A analysis (in addition to the 1971–2005 global mean OHC trend). The objective is to consider more information on the OHC variability to further reduce the uncertainties in the OHC response to the GHG forcing and the OA forcing.

The GHG and OA forcings effects on OHC show distinct spatiotemporal patterns (Slangen et al. 2015). The OA effect is local and its amplitude decreases in the late twentieth century, while the GHG forcing effect is globally quasi-uniform and increases until present (Slangen et al. 2015). Despite these differences, the effects of the GHG and OA forcings are similar on the OHC (with opposite sign) when they are averaged over the ocean and over the period 1971–2005 (Slangen et al. 2014). These similar effects make it difficult to unravel the role of each forcing on OHC when looking only at global OHC. To remove this degeneracy, we use a second variable in the D&A analysis. For a second variable, we take a regional average of the OHC trend. We test different regions for the averaging of the OHC trend and we test also different periods over which the OHC trend is computed. In total, 14 different regions (ranging from global to oceanic basins; see Table S1 in the online supplemental material) and 6 different periods (the periods start after 1957 and end before 2005; see Table S1) were tested leading to a total of 83 different second variables. (That is, 14 regions by 6 periods lead to 14 × 6 = 84 different combinations for the second variable. Within these 84 second variables, we do not consider the combination that corresponds to global OHC over the period 1971–2005 because this variable is already used as primary variable. So, in total it leads to 83 different second variables). Note that when global OHC over a given period is used as primary variable and regional OHC over the same period is used as secondary variable, then the information on the regional OHC is used twice, one time in the primary variable as part of the total information on global OHC and one time in the secondary variable.

Figure 5 summarizes the 83 estimates of the global 1971–2005 sub-OHC trends in response to each forcing and their associated uncertainty. In all tests, the inclusion of the second variable in the D&A analysis does not change substantially the best estimate of the global sub-OHC trend over 1971–2005 (see the x axis in Fig. 5). The best estimates of the global sub-OHC trend over 1971–2005 range between 1.2 and 1.6 × 10⁷ J m⁻² yr⁻¹ for the HIST simulations, between −0.23 and −0.14 × 10⁷ J m⁻² yr⁻¹ for the NAT simulations (excluding the outlier at −0.27 × 10⁷ J m⁻² yr⁻¹), between 2.28 and 2.55 × 10⁷ J m⁻² yr⁻¹ for the GHG simulation, and between −1.05 and −0.9 × 10⁷ J m⁻² yr⁻¹ for the OA simulation. This is consistent with the results of the univariate case. The inclusion of the second variable makes a difference on the uncertainty associated to the best estimate of the global sub-OHC trend over 1971–2005 (see the y axis in Fig. 5). For all tests, the inclusion of the second variable reduces the uncertainty of the sub-OHC response to all forcings compared to the univariate case (see Fig. 5). However, the reduction is not the same for each forcing and for each second variable that is considered. The variables that yield the smallest uncertainty in the sub-OHC response to OA are not the same as the ones that yield the smallest uncertainty in the sub-OHC response to GHG. This suggests that each variable has its own sensitivity to the different forcings.

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The y axis shows the uncertainties of the subsampled global 1971–2005 OHC trends using the bivariate detection analysis (estimated with the standard deviation of the subsampled global 1971–2005 OHC anomalies trends) for (a) HIST, (b) NAT, (c) GHG, and (d) OA. Different regions and periods of trends were used in the second variable of the bivariate case, resulting in different estimators and associated uncertainties in the subsampled global 1971–2005 OHC trends (black dots). Colored symbols indicate particularly interesting cases discussed in main text: blue, 1957–2005; red, 1971–2005; green, 1957–80.

Citation: Journal of Climate 33, 24; 10.1175/JCLI-D-19-0091.1

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We plot in Fig. 5 the detailed results for the cases that lead to the smallest uncertainty in sub-OHC response to GHG (blue plus sign in Fig. 5c, corresponding to the case with the South Pacific sub-OHC trend over 1957–2005 as second variable) and to OA (green square in Fig. 5d, corresponding to the case with the high-latitude sub-OHC trend over 1957–80 as second variable). The pair of variables that leads to the smallest uncertainty in the sub-OHC response to OA (green square in Fig. 5) does not lead to the smallest uncertainty in the sub-OHC response to GHG. The same is true for the pair of variables that leads to the smallest uncertainty in the sub-OHC response to GHG (blue plus sign in Fig. 5). It does not lead to the smallest uncertainty in sub-OHC response to OA. The reason is that for these pairs of variables the regional responses of the sub-OHC to GHG and OA are not still largely anticorrelated such that we are still close to degeneracy (even with two variables).

Indeed, in Figs. 6b and 6d the semi-major axis of the GHG ellipse is aligned with the semi-major axis of the OA ellipse, which means that the pair of variables have the same response under OA or GHG forcing and thus they do not help in removing the degeneracy in this case.

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Global 1971–2005 sub-OHC trend values vs (a) South Pacific 1971–2005, (b) South Pacific 1957–2005, (c) global 1957–80, and (d) high-latitude 1957–80 trend values for the combination of NAT, GHG, and OA simulations. Trend values obtained by independent GCMs (averaged over available realizations) are represented by a colored cross, observed value by a black cross, and the sum of the combinated forcings by a pink star. The gray error bars and their center represent the 90% confidence interval and the multimodel mean computed from the ensemble of simulations, respectively. The colored error bars represent the reduced 90% confidence interval and the new estimation computed from the Ribes et al. (2017) univariate detection and attribution model.

Citation: Journal of Climate 33, 24; 10.1175/JCLI-D-19-0091.1

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To cope with this issue, and to find cases where the degeneracy is removed, we considered the three cases that lead to the largest uncertainty reduction in sub-OHC responses to GHG and OA simultaneously (the uncertainty of the sub-OHC responses to both GHG and OA is computed as the quadratic sum of the uncertainty in the sub-OHC response to GHG and the uncertainty in the sub-OHC response to OA). These cases are indicated by a red cross (case with the South Pacific sub-OHC trend over 1971–2005 as second variable), a blue cross (case with the South Pacific sub-OHC trend over 1957–2005 as second variable), and a green circle (case with the global sub-OHC trend over 1957–1980 as second variable) in Fig. 5.

Among these three cases, Fig. 6 indicates that two cases (cases with red and blue crosses in Fig. 5) are still very close to degeneracy (see Figs. 6a,b). Indeed, in Figs. 6a and 6b, the semi-major axis of the GHG ellipse is almost aligned with the semi-major axis of the OA ellipse. There is only one case where the semi-major axis of the GHG ellipse is less aligned with the semi-major axis of the OA ellipse. It is the case with the global sub-OHC trend over 1957–80 as second variable (case with the green circle on Fig. 5). For this single case the degeneracy is reduced (see Fig. 6c). The semi-major axes of the GHG and OA ellipses are no longer aligned (although they are still close). This indicates that the ratio of the global sub-OHC 1957–80 trend over the global sub-OHC 1971–2005 trend does not show the same response to GHG forcing and to OA forcing. Thus, the OHC response to GHG and OA cannot compensate anymore with this pair of variables. It means that the combined analysis of the global sub-OHC over the two periods 1957–80 and 1971–2005 enables us to decorrelate (partly) the response to GHG from the response to OA. With the decorrelation, the responses of the sub-OHC to both GHG and OA are independently observable at the same time. It should improve the associated estimate of the global sub-OHC trend response to OA and GHG over 1957–80 and 1971–2005. It should also reduce significantly the associated uncertainty. The resulting estimate of the global sub-OHC trend over 1971–2005 is 1.42 ± 0.18 × 10⁷ J m⁻² yr⁻¹. This trend breaks down into −0.17 ± 0.26 × 10⁷ J m⁻² yr⁻¹ in response to the natural forcing, 2.37 ± 0.34 × 10⁷ J m⁻² yr⁻¹ in response to the GHG forcing, and −0.78 ± 0.37 × 10⁷ J m⁻² yr⁻¹ in response to the OA forcing (Fig. 4). These numbers show that the use of the bivariate case has not changed the best estimate of the OHC response to the different forcing (see Fig. 4). It has improved the uncertainty estimates (by a few percent) but the improvement is marginal compared to the univariate case (see Fig. 4). This is because the decorrelation of the OHC response to OA and GHG even with our best pair of variables is only partial. In all cases analyzed in this study, the bivariate case is our best estimate of OHC response to OA and GHG. When we translate this into tot-OHC trend we find that the global OHC trend over 1971–2005 is 1.22 ± 0.14 × 10⁷ J m⁻² yr⁻¹. This trend breaks down into −0.13 ± 0.09 × 10⁷ J m⁻² yr⁻¹ in response to the natural forcing, 2.12 ± 0.21 × 10⁷ J m⁻² yr⁻¹ in response to the GHG forcing, and −0.84 ± 0.18 × 10⁷ J m⁻² yr⁻¹ in response to the OA forcing (Fig. 4).