Changes in ocean heat uptake could generate the hiatus period regardless of its origin. Conversely, it has been shown repeatedly that GCMs tend to overestimate the observed ocean heat uptake [IPCC, 2007; Knutti and Tomassini, 2008; Boé et al., 2009]. Therefore, it is paradoxical that models systematically overestimate the SATg trend for 2001–2010. Ocean heat uptake consists of vertical advection, diffusion, and convective mixing, which work differently at different latitudes [Gregory, 2000], and therefore, investigation of individual oceanic processes in observational data is difficult. Instead, we examine the above paradox by means of an approximated form of a two-box energy balance model for surface and deep ocean temperatures [Held et al., 2010; Geoffroy et al., 2012] (see auxiliary material for the derivation).
where N denotes the surface net energy imbalance over oceans, λ is the climate feedback parameter, and κ indicates the heat uptake efficiency coefficient [Gregory and Mitchell, 1997]. The TOA radiative forcing, F, is time-dependent, and Δ indicates changes from the preindustrial climate, defined as the 1851–1900 average. The expression of heat uptake in equation (1) is a posteriori, and will be acceptable only for transient forced response. Moreover, κ has been assumed to be constant for simplicity but varies empirically in time [Raper et al., 2002]. In the literature, another parameter known as the efficacy factor, ε, is sometimes introduced to the energy balance model to represent the dependence of climate feedback on the geographical distribution of temperature increase associated with ocean heat uptake [Held et al., 2010; Winton et al., 2010]. The nonconstant κ is equivalent to introducing ε, but is used here because we do not assume the specific pattern in ΔT necessary for arguing the efficacy change.
 When equation (1) is applied to 16 CMIP5 models, ΔN is shown to be linearly related with ΔSATg but with different regression slopes, namely, κ, for different periods: 1.38 and 0.68 W m−2 K−1 for 1971–2000 and 2001–2030, respectively (Figure 4a). Owing to interannual and decadal fluctuations in ΔN and ΔSATg, we estimate long-term change in κ with a 30 year moving window, the results of which indicate a gradual decrease from 1961 to 2030 in CMIP5 models (squares) and the 11-member MIROC5 ensemble (green circles) in Figure 4b. Although direct measurements of ΔN are not available, equation (1) can be applied to observations partly: F of Hansen et al. , HadCRUT4 SATg, and λ derived from CMIP5 models [Andrews et al., 2012]. The range of λ corresponding to equilibrium climate sensitivities of 2.1–4.7 K (0.63–1.52 W m−2 K−1) gives the uncertainty of our estimate. The result indicates that κ is similar to the direct estimate from GCMs for 1971–2000, but in the observations (triangles) it is clearly strengthened after this time contrasting to GCMs (squares in Figure 4b). This discrepancy is consistent with the systematic warming bias in the SATg trends in GCMs (Figures 1, 2). The larger κ in observations indicates greater mixing of heat into the deeper ocean layers and less surface warming in the hiatus.
Figure 4. Estimates of κ from CMIP5 models and observational data. (a) Scatterplot of ΔN against ΔT in CMIP5 models for 1971–2000 (red) and 2001–2030 (blue). Both variables are smoothed using an 11 year running mean. The error bars and lines indicate ± 1σ and least-squares fits, respectively. Black curve with grey shading represents the evolution in the transient experiments with CO2 increasing at 1% per year and its 95% confidence limit. (b) Estimate of κ with 30 year window from 1961 to 2030 in CMIP5 models (squares), from 1961 to 2020 in MIROC5 (circles), and from 1961 to 2012 in combination of HadCRUT4 SATg, F by Hansen et al. , and a range of λ in CMIP5 models substituted to equation (1) (triangles with vertical lines). For 1971–2000 (red) and 2001–2030 (blue) CMIP5 models, the ensemble-mean values are identical to the regression slopes in Figure 4a, and the estimates from individual models are indicated by “cross” marks (the maximum and minimum are denoted by error bars).
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