A review of progress towards understanding the transient global mean surface temperature response to radiative perturbation

Abstract

The correct understanding of the transient response to external radiative perturbation is important for the interpretation of observed climate change, the prediction of near-future climate change, and committed warming under climate stabilization scenarios, as well as the estimation of equilibrium climate sensitivity based on observation data. It has been known for some time that the radiative damping rate per unit of global mean surface temperature increase varies with time, and this inconstancy affects the transient response. Knowledge of the equilibrium response alone is insufficient, but understanding the transient response of the global mean surface temperature has made rapid progress. The recent progress accompanies the relatively new concept of the efficacies of ocean heat uptake and forcing. The ocean heat uptake efficacy associates the temperature response induced by ocean heat uptake with equilibrium temperature response, and the efficacy of forcing compares the temperature response caused by non-CO₂ forcing with that by CO₂ forcing.

In this review article, recent studies on these efficacies are discussed, starting from the classical global feedback framework and basis of the transient response. An attempt is made to structure different studies that emphasize different aspects of the transient response and to stress the relevance of those individual studies. The implications on the definition and computation of forcing and on the estimation of the equilibrium response in climate models are also discussed. Along with these discussions, examples are provided with MIROC climate model multi-millennial simulations.

Introduction

A climate system is said to be in equilibrium when it exhibits a statistically stable state for an extended period of time under a given set of boundary conditions. When external radiative perturbation is added to the system (radiative forcing), it begins to reveal different statistical behavior, which includes the mean state. The statistical behavior evolves as time progresses, and the system eventually reaches a new equilibrium that is constrained by the newly imposed boundary conditions. Historically, the equilibrium response has received much attention; the equilibrium climate sensitivity (ECS) is the most notable example (Knutti and Hegerl 2008; Maslin and Austin 2012). The ECS is defined as the globally and annually averaged value of the equilibrium surface temperature response to a doubling of atmospheric CO₂ concentration. The ECS is an idealized yet useful concept, because it represents the final boundary of the system, and gives a measure of how much warming would occur in the future even if the current CO₂ concentration is sustained and not raised. We note that changes in vegetation and ice-sheet distributions and in atmospheric gas concentrations by a biogeochemical response are excluded from the formal definition of ECS, although the effect of vegetation change is sometimes included in recent model-based estimates. While ice sheets exhibit multiple equilibria (e.g., Abe-Ouchi et al. 2013), the ECS is an inherent property of the climate system, which is independent of the history of radiative forcing.

The transient response, which generally includes the effect of time-varying boundary conditions, is more complicated to characterize and understand than the equilibrium response. In addition, the system does not approach equilibrium at a constant rate even under constant radiative forcing. Nevertheless, the transient response has received more and more attention recently, because it is more relevant to near-future climate change and thus has more practical implications (Allen and Frame 2007). From the viewpoint of climate stabilization scenarios, understanding the transient response is also important for evaluating the magnitude of “committed” warming and the rate of the change accurately (Armour and Roe 2011; Meehl et al. 2005; Plattner et al. 2008; Solomon et al. 2009; Zickfeld et al. 2013). Here, the committed warming is referred to an additional surface warming that is expected to emerge when constant radiative forcing is continuously applied after a stabilization point.

In this review article, we focus on the transient response of the global mean surface temperature to external radiative perturbation, because the magnitudes of many climatic impacts can be scaled proportionally to this global mean variable relatively well (e.g., Ishizaki et al. 2012; Ishizaki et al. 2013; Yoshimori and Abe-Ouchi 2012). One of the motivations for this review arises from the recent, rapid conceptual development on climate feedbacks. Here, climate feedbacks are loosely defined as climate change processes that cause the additional radiative adjustment to the climate system in response to the initial radiative perturbation. For example, previously underappreciated factors such as spatiotemporally varying climate feedbacks related to the ocean heat uptake and the dependence of climate sensitivity on forcing agents have been identified as important. These factors are essential for a quantitative understanding of the transient response. Nevertheless, these factors are rather confusing and not necessarily understood widely by the climate science community. In addition, the recent progress requires further clarification of the definitions of forcing and feedback and discussions on how they are to be computed. It is becoming clear that the omission of such factors would introduce a bias into the estimate of ECS based on observation data. The understanding of the background long-term response to anthropogenic radiative perturbation must also be well established for clarifying observed contemporary climate variations, such as the warming pause or “hiatus” during the early twenty-first century (e.g., Kosaka and Xie 2013; Meehl et al. 2011; Watanabe et al. 2013; Watanabe et al. 2014).

Transient climate response in a global feedback framework

In this section, the transient response of the global mean surface temperature to radiative perturbation is reviewed in a classical global feedback framework. In this framework, the loss of radiative energy from the climate system, i.e., radiative damping, is formulated in proportion to the global mean surface temperature change. In the following, all variables represent global mean values unless noted explicitly. We start with an equilibrium state in which the global mean net radiation at the top of the atmosphere (TOA) is zero.

A classical definition of radiative forcing is instantaneous or stratosphere adjusted forcing in which the radiative perturbation is evaluated at the tropopause instantaneously (on the time scale of radiative transfer) after the perturbation is applied or after the stratospheric temperature responds radiatively to the perturbation. The tropopause is chosen because the troposphere and surface are tightly coupled thermally through radiation and convection. The adjustments of stratospheric temperature are included in the forcing because its response time is on the order of months, and the subsequent forcing thus represents a more effective measure of radiative effect on the troposphere-surface system (Hansen et al. 1997; Stuber et al. 2001). We note that the magnitude of radiative forcing at the TOA is the same as that at the tropopause after the stratospheric adjustments, because any vertical convergence of radiative fluxes in the stratosphere would cause local temperature to adjust (Hansen et al. 1997).

When radiative forcing F is applied to the TOA, the energy budget equation with the net TOA radiation, called N, may be written in the simplest form as

$$ N=F-\lambda \varDelta T $$

(1)

where λ is termed the climate feedback parameter and represents how much energy is lost to space in accordance with the unit increase of the global mean surface temperature T (e.g., Gregory et al. 2015; Gregory et al. 2004; Knutti and Hegerl 2008; Winton et al. 2010). Symbol ∆ denotes the deviation (or anomaly) from the unperturbed climate. Parameter λ must be positive in order for the system to eventually reach a stable state. We note that a “signed” feedback parameter, e.g., Λ ≡ −λ, which has a positive/negative value for the amplification/suppression of initial perturbation (i.e., positive/negative feedback), is also widely used in other literature (e.g., Armour et al. 2013; Boer and Yu 2003; Winton et al. 2013a). N reflects mostly ocean heat uptake with other forms of energy consumption being latent heat for snow/ice melting, for example (Rhein et al. 2013), and eventually diminishes to zero under constant radiative forcing.

When the feedback is distinguished from the forcing as the surface temperature-mediated response of the climate system as in Eq. (1), it is more consistent to include the so-called rapid tropospheric adjustments in the definition of forcing. The troposphere may respond to forcing directly and rapidly (typically, «1 year) without invoking a surface temperature change. Examples of the tropospheric adjustments include cloud response to lower tropospheric warming due to the increase in atmospheric CO₂ concentration (cloud adjustments, Gregory and Webb 2008), cloud albedo and lifetime changes due to the increase in aerosols (indirect effect of aerosols), and cloud amount changes due to the increase in light-absorbing aerosols (semi-direct effect of aerosols). The resulting forcing is referred to the stratosphere-troposphere-adjusted radiative forcing or effective radiative forcing (ERF). For details on the mechanism of tropospheric adjustments, readers are referred to Andrews et al. (2012a), Kamae et al. (2015), and Sherwood et al. (2015).

It is often assumed that λ is constant, i.e., λ = λ _eq at equilibrium, which leads to the conclusion that the transient response (∆T) to constant radiative forcing is always smaller than the equilibrium response (∆T _eq). Therefore, a rewrite of Eq. (1) indeed gives

$$ \varDelta T=\frac{F-N}{\lambda }<\varDelta {T}_{\mathrm{eq}}=\frac{F}{\lambda_{\mathrm{eq}}} $$

(2)

The time variation of λ is extensively discussed later. The global mean surface temperature change at the time of the doubled atmospheric CO₂ concentration (average of 61 to 80 years in practice) under a 1 % compound annual increase (1pctCO2 experiment of the Coupled Model Intercomparison Project 5 (CMIP5)) is termed transient climate response (TCR, Cubasch et al. 2001). The TCR satisfies

$$ \mathrm{T}\mathrm{C}\mathrm{R}=\frac{F_{2\times }-N}{\lambda }<\mathrm{E}\mathrm{C}\mathrm{S}=\frac{F_{2\times }}{\lambda_{\mathrm{eq}}} $$

(3)

where F _2× denotes the radiative forcing due to a doubling of the atmospheric CO₂ concentration. Figure 1 shows the TCR against ECS for the CMIP5 atmosphere-ocean general circulation models (AOGCMs) (Taylor et al. 2012), in which the TCR is indeed smaller than the ECS and a high correlation exists between the two (Flato et al. 2013).

Fig. 1

Equilibrium climate sensitivity (ECS) and transient climate response (TCR) of 23 CMIP5 models. Redrawn from the data in Table 9.5 and Figure 9.42b in Flato et al. (2013). The linear regression line is also plotted. The number at the lower right corner of the figure indicates the fractional variance shared by ECS and TCR. The right column lists model names

Time scale of the response

In this section, the response time of the climate system is discussed, while still assuming that λ is constant. As stated above, N essentially reflects ocean heat uptake and may thus be expressed as

$$ N=\frac{d\left(C\varDelta T\right)}{dt} $$

(4)

where C = ρc _p h (ρ: density of seawater, c _p : specific heat of seawater at constant pressure, h: effective depth) is the effective heat capacity. Here, the term “effective” is used to represent the fact that the extent of penetration of the heat anomaly with respect to the unperturbed climate varies with time and space. A substitution of Eq. (4) with constant heat capacity into Eq. (1) yields

$$ C\frac{d\left(\varDelta T\right)}{dt}=F-\lambda \varDelta T $$

(5)

At equilibrium, we obtain

$$ \varDelta {T}_{\mathrm{eq}}\equiv \varDelta T\left(t\to \infty \right)=\frac{F}{\lambda }, $$

(6)

and at the transient state, we obtain

$$ \varDelta T=\frac{F}{\lambda}\left(1-{e}^{-t/\tau}\right)\mathrm{or}\frac{\varDelta T}{\varDelta {T}_{\mathrm{eq}}}=1-{e}^{-t/\tau } $$

(7)

where τ ≡ C/λ (Hansen et al. 1981; Hansen et al. 1984; Hansen et al. 1985; Wigley and Schlesinger 1985). Equation (7) means that ΔT reaches 63 % of ΔT _eq at t = τ, and the rate depends on the effective heat capacity and the climate feedback parameter. Substitutions of C = 2.1 × 10⁸ or C = 1.5 × 10¹⁰ J K⁻¹ m⁻², corresponding to a typical ocean mixed-layer depth (50 m) or the average depth of world oceans (3700 m), respectively, and a nominal value of λ = 0.75 W m⁻² K⁻¹ (corresponding to the ECS of about 3 °C), yield τ ≅ 5 or τ ≅ 370 years for the two ocean depths, respectively. The value of τ ≅ 5 does not include the effect of the heat exchange between the mixed layer and the deep ocean, and the τ ≅ 370 value assumes that the heat anomaly is communicated instantly throughout the water column. Both cases are unrealistic but give some insight into the response time. In reality, the heat anomaly spreads gradually from the surface to depth, and thus C varies with time. Schwartz (2007, 2008) applied a single-layer model with a similar formulation to estimate the ECS from the observation data, but their use of constant effective heat capacity for the entire climate system on all time scales of response was criticized (Foster et al. 2008; Knutti et al. 2008; Scafetta 2009). Donohoe et al. (2014) reported that the effective heat capacity is equivalent to about 50-m ocean depth during the first decade (consistent with our estimate of 5 years for the mixed-layer ocean) and equivalent to several hundred meters ocean depth after a century following an instantaneous quadrupling of atmospheric CO₂ concentration (abrupt4xCO2 experiment) for the CMIP5 AOGCMs.

As an intermediate model that ties the two limiting cases of box-ocean models with a mixed layer or with full depth, Gregory (2000) introduced the following two-layer ocean model to investigate the transient temperature response:

$$ C\frac{d\left(\varDelta T\right)}{dt}=F-\lambda \varDelta T-H\ \left(\mathrm{upper}\ \mathrm{ocean}\right) $$

(8)

$$ {C}_D\frac{d\left(\varDelta {T}_D\right)}{dt}=H\ \left(\mathrm{deep}\ \mathrm{ocean}\right) $$

(9)

where the heat exchange between the two layers is assumed to be proportional to their difference in anomalous temperature with respect to the unperturbed climate:

$$ H=\gamma \left(\varDelta T-\varDelta {T}_D\right) $$

(10)

where γ is the proportionality coefficient. The heat capacity of the atmosphere, land, and other surface components of the climate system is small and implicitly included in the heat capacity of the upper ocean (C), and C is much smaller than the heat capacity of the deep ocean (C _D ), i.e., C < <C _D . We note that the addition of Eqs. (8) and (9) yields Eq. (1). That is, the surface temperature change and the ocean heat uptake hold a linear relation under constant radiative forcing in the two-layer model. Geoffroy et al. (2013a) applied this model to analyze the CMIP5 AOGCM simulations and showed that the two-layer models tuned to emulate the global mean surface temperature response of AOGCMs in the abrupt4xCO2 experiment can also emulate the result of the slowly increasing CO₂ (1pctCO2) experiment reasonably well. This supports the assumption that the two-layer model is effective in capturing the minimal physics of the global mean surface temperature transient response.

The upper bound of the ocean response time can be estimated by the following scaling argument based on the fact that the important slow process of heat transport is vertical eddy diffusion:

$$ \rho {c}_p\frac{d\left(\varDelta T\right)}{dt}={\kappa}_T\frac{\partial^2T}{\partial {z}^2}\mathrm{or}\frac{d\left(\varDelta T\right)}{dt}={\overset{\sim }{\kappa}}_T\frac{\partial^2T}{\partial {z}^2} $$

(11)

where ${\kappa}_T$ is the thermal conductivity and ${\tilde{\kappa}}_T={\kappa}_T/\left(\rho {c}_p\right)$ is the thermal diffusivity by eddies. A substitution of the effective value of ${\tilde{\kappa}}_T\sim {10}^{-4}\ {\mathrm{m}}^2\ {\mathrm{s}}^{-1}$ and h = 3700 m yields a time scale of about 4000 years ($\tau \sim {h}^2/{\tilde{\kappa}}_T$). Figure 2 shows the long-term integration of the MIROC3.2 AOGCM (K-1 model developers 2004) for the first 140 years of the 1pctCO2 experiment and the continuous integration under the four times constant CO₂ level (const4xCO2) experiment thereafter. For reference, the two times constant CO₂ level (const2xCO2) experiment, starting from 70 years of the 1pctCO2 experiment, is also plotted. We note that the equilibrium response of these experiments was studied in Yoshimori et al. (2014) and the entire simulations were used to study marine biogeochemical response in Yamamoto et al. (2014). While the surface air temperature responds to the forcing rather quickly, the ocean temperature changes much more slowly. The order of the time scale for the ocean to reach a new steady state in Fig. 2 is loosely consistent with the above estimate based on the scaling argument of 4000 years (i.e., 10³ years). We note that multi-millennial simulations for the increased CO₂ concentration are not available for the CMIPs.

Fig. 2

Global and annual mean temperature changes from the preindustrial control simulation in the MIROC3.2 (medres) AOGCM experiments. Solid lines: surface air temperature; dashed lines: potential temperature averaged over the full ocean depth; black lines: 1 % compound annual increase of CO₂ (1pctCO2, 1–140 years); blue lines: constant 2xCO₂ forcing starting from year 70 of 1pctCO2 (const2xCO2, 71–2699 years); red lines: constant 4xCO₂ forcing starting from year 140 of 1pctCO2 (const4xCO2, 141–3685 years)

Traditionally, one-dimensional box-diffusion ocean models (Hansen et al. 1985; Long and Collins 2013; Siegenthaler and Oeschger 1984; Wigley and Schlesinger 1985) or upwelling-diffusion (UD) ocean models (Andronova and Schlesinger 2001; Baker and Roe 2009; Johansson et al. 2015; Wigley and Raper 2001) have been used. AOGCM studies, however, point out the following limitations of such models: (a) the deep convection at high latitudes identified as an important process in AOGCMs is not properly represented (Gregory 2000); (b) the vertical structure of the heat anomaly stored in the UD ocean models is biased to a shallower depth as compared to an AOGCM (Li et al. 2013); and (c) the total heat anomaly stored in the UD ocean models, which corresponds to the thermosteric sea level rise, is underestimated as compared to an AOGCM (Li et al. 2013).

Kostov et al. (2014) showed that the time scale of the penetration depth of the heat anomaly with respect to the unperturbed climate is positively correlated with the depth and strength of the Atlantic meridional overturning circulation (AMOC) on a century time scale in the eight CMIP5 AOGCMs. This result suggests that the AMOC plays an important role in setting the rate of global ocean heat uptake. In addition to the AMOC, Marshall and Zanna (2014) used a conceptual model to point out the importance of ocean eddy mixing in the Southern Ocean, with larger mixing leading to a faster (yet smaller) global heat uptake response.

Ocean heat uptake efficiency

The ocean heat uptake under the monotonically increasing radiative forcing is often approximated by

$$ N=\kappa \varDelta T $$

(12)

where κ is called the ocean heat uptake efficiency (Gregory and Mitchell 1997; Raper et al. 2002). It is assumed that κ is either constant or slowly varies with ∆T. Physically, Eq. (12) represents that heat exchange between the upper and deep oceans occurs in proportion to the surface temperature change and that the deep ocean is implicitly assumed to have an infinite heat capacity with a constant temperature. This assumption imposes a major limitation on the use of ocean heat uptake efficiency; however, it should be noted that this limitation does not affect the two-layer ocean model introduced earlier. Indeed, Eq. (12) can be derived from Eqs. (8)–(10) by assuming that the total ocean heat uptake is dominated by the deep ocean (N ≈ H) and by replacing γ with κ. Equation (12) is invalid for a transient response immediately after a rapid change in radiative forcing (N ≈ F,ΔT ≈ 0; e.g., abrupt4xCO2 experiment) as it requires an infinite κ. Equation (12) is also invalid on long time scales and in particular near equilibrium because κ approaches zero when the transient response becomes close to the equilibrium (N ≈ 0,ΔT ≠ 0). Figure 3 shows a schematic diagram of the relation between ΔT and N after a step increase in radiative forcing, where κ represents the slope of the line (thin black line) pointing from the origin to a state in the simulation (a point in the thick black curve). An example of a monotonically increasing CO₂ (1pctCO2) experiment is presented by the black crosses in Fig. 4. Approximate linearity is observed except for the initial warming period, thereby supporting the general validity of Eq. (12) in this case.

Fig. 3

Schematic diagram of the summary of concepts developed for understanding of the transient response of the global mean surface temperature. The horizontal and vertical axes are surface temperature and TOA net radiation changes after a step increase in radiative forcing, respectively. Figure A1 of Winton et al. (2013a) is referenced. N: TOA net radiation change; F: effective radiative forcing; ΔT: transient temperature change; λ: (effective) climate feedback parameter; λ _eq: equilibrium climate feedback parameter; κ: ocean heat uptake efficiency; ε: ocean heat uptake efficacy; ΔT _eff: effective climate sensitivity (in the case of 2xCO₂ forcing); ΔT _eq: equilibrium temperature change. See the text for the details

Fig. 4

Relation between global mean surface air temperature and TOA net radiation changes from the preindustrial control simulation in the MIROC3.2 (medres) AOGCM experiments. Black crosses: 1pctCO2 experiment (every year for 1–140 years); blue crosses: abrutp4xCO2 experiment (every year for 1–150 years); red crosses: const4xCO2 experiment (every 20-year average for 141–3680 years); blue line: linear regression for the 150-year abrupt4xCO2 experiment; blue circles: equilibrium temperature response to 4xCO₂ estimated by Gregory’s method for 10 abrupt4xCO2 20-year-long ensemble experiments (that start from different initial conditions in the preindustrial control simulation)

The CMIP5 AOGCM simulations of recent climate change showed that κ decreases with time (Watanabe et al. 2013). This result could reflect less heat uptake in more stratified ocean, i.e., faster warming in the upper ocean that suppresses vertical mixing, or warming of the deep ocean that weakens the effect of mixing. Figure 5a displays the relation between the surface air temperature change and the temperature changes for four oceanic layers of different depths. The figure shows that the upper 300 m of ocean warms at half the speed of the surface air and the ocean below 1000 m warms slightly. Thus, Fig. 5a indicates that the vertical stratification is reduced on a centennial scale. Similarly, Fig. 5b displays the millennial scale of the ocean temperature change with respect to the surface air temperature change. The figure shows that the ocean temperature below 1000 m reaches the upper ocean warming after about a century, although temperature changes never reach the same magnitude. This result suggests that the effect of vertical mixing might become smaller. It also shows that the upper ocean does not warm as much as the surface air at the equilibrium.

Fig. 5

Global mean ocean temperature changes at four different depth layers from the preindustrial control simulation in the MIROC3.2 (medres) AOGCM experiments. The changes are plotted as functions of the surface air temperature change. a 1pctCO2 experiment (every year for 1–140 years). b abrupt4xCO2 experiment (blue, every year for 1–150 years) and const4xCO2 experiment (red: every 20-year average for 141–3680 years)

The ocean heat uptake efficiency is contrasted with the climate feedback parameter. The substitution of Eq. (12) into Eq. (1) yields

$$ F=\rho \varDelta T $$

(13)

where ρ ≡ κ + λ is termed the climate resistance (Gregory and Forster 2008; Gregory et al. 2009, 2010). Here, κ reflects primarily oceanic processes and a larger value indicates that more of the heat anomaly is absorbed in the ocean and hence surface warming is suppressed. On the other hand, λ reflects primarily the atmospheric processes (and sea ice-albedo feedback) and the larger value indicates that excessive energy is more effectively lost to space. The resistance ρ depends on κ (oceanic process) and λ (atmospheric process), and as a result of both processes, the surface temperature change remains small when resistance ρ is large.

By assuming λ = λ _eq, arranging Eq. (2) and Eq. (12) yields

$$ \frac{\varDelta T}{\varDelta {T}_{\mathrm{eq}}}=\frac{1}{1+\kappa /\lambda } $$

(14)

As indicated by Eq. (14), the ratio of transient warming to equilibrium warming is larger when κ is smaller or λ is larger. We note that a case of large λ is consistent with Eq. (7). Raper et al. (2002) found a negative correlation between κ and λ in nine CMIP2 AOGCMs. They noted that larger surface warming caused by the smaller radiative damping rate of λ accompanies stronger ocean stratification and less heat release from the interior of the ocean through convection at high latitudes. This relation corresponds to a larger net heat uptake of the global ocean, i.e., a larger κ In the unperturbed climate, the AMOC transports heat to the northern high latitudes, where heat is released to the atmosphere. The AMOC acting as a heat sink of the ocean was reported by Knutti and Stocker (2000) and Marshall and Zanna (2014). The negative correlation found by Raper et al. (2002) was not verified in later studies with a larger number of models, however (Geoffroy et al. 2013a; Gregory and Forster 2008; Plattner et al. 2008). We note that Kuhlbrodt and Gregory (2012) emphasized the dominant role of the Southern Ocean and found the importance of eddy-induced transport in the model discrepancies of global ocean heat uptake.

Dufresne and Bony (2008) used Eq. (12) to quantify the relative contribution of radiative forcing, ocean heat uptake, and climate feedbacks to the spread of the TCR as well as the ECS in the CMIP3 AOGCMs. They concluded that cloud feedback is the predominant factor for the spread of both quantities. Boe et al. (2010), on the other hand, found a positive correlation in the CMIP3 AOGCMs between present-day ocean mixed-layer depths and future deep ocean warming and concluded that ocean heat uptake plays a major role in the uncertainty of transient warming, given that deep ocean warming is negatively correlated with surface temperature. Their analysis was based on the correlation of a single factor and so does not compare quantitatively with other factors, such as radiative forcing and climate feedbacks.

Effective climate sensitivity and ocean heat uptake efficacy

To discriminate the temperature change estimated by using the feedback parameter during the transient period from the ECS, the term effective climate sensitivity was introduced (Murphy 1995; Senior and Mitchell 2000) and is defined as follows:

$$ \varDelta {T}_{\mathrm{eff}}\equiv \frac{F_{2\times }}{\lambda }=\frac{F_{2\times}\varDelta T}{F-N} $$

(15)

While Watterson (2000) showed that ∆T _eff is stable over time, numerous studies reported that ∆T _eff varies with time or is different from the ECS (e.g., Armour et al. 2013; Bitz et al. 2012; Boer and Yu 2003; Gregory et al. 2004; Kiehl et al. 2006; Li et al. 2013; Long and Collins 2013; Murphy 1995; Senior and Mitchell 2000; Williams et al. 2008; Winton et al. 2010).

In applying Eq. (15), stratosphere adjusted radiative forcing or ERF was used for F. One way to estimate the ERF is the regression method by Gregory et al. (2004), in which the ERF is given by the intercept at ∆T = 0 of the ∆T-N linear regression after a step increase of forcing, if assuming Eq. (1). The blue colors in Fig. 4 represent the result of the abrupt4xCO2 experiment, and the blue crosses represent the global and annual mean anomalies for the initial 150 years. The regression line by Gregory’s method is indicated by a blue line, which gives an ERF of 7.3 W m⁻² at the y-intercept. In Fig. 3, the regression line by Gregory’s method is indicated by a blue dashed line, and the ERF and effective climate sensitivity are denoted by F and ∆T _eff, respectively.

Williams et al. (2008) proposed effective forcing, in which the feedback parameter remains constant in time. Their effective forcing is different from the ERF by Gregory’s method in that Williams applied linear regression for the stabilization period under constant forcing after 70 years of a monotonic CO₂ increase (1pctCO2) experiment and thus included an initial adjustment of the climate system on the decade level. Winton et al. (2010) argued that the decadal scale adjustment clearly contains the oceanic changes, and hence, it is more appropriate to regard the adjustment as a part of the feedback, rather than as a part of the forcing. Winton introduced a new parameter termed ocean heat uptake efficacy (ε), which is described in detail below.

In general, the TOA radiative anomaly induced by a unit surface temperature change during the transient period (λ) is different from that at equilibrium (λ _eq). This led Winton et al. (2010) to decompose the temperature response into equilibrium (∆T _eq) and disequilibrium (or remaining) components, such that ∆T′ ≡ ∆T _eq − ∆T

$$ \varDelta {T}_{\mathrm{eq}}=F/{\lambda}_{\mathrm{eq}} $$

(16)

$$ \varDelta {T}^{\prime }=\varepsilon N/{\lambda}_{\mathrm{eq}} $$

(17)

without using the time-variant λ. Combining Eqs. (16) and (17) yields

$$ \varepsilon N=F-{\lambda}_{\mathrm{eq}}\varDelta T $$

(18)

Therefore, ∆T for a given ocean heat uptake N under a known forcing is determined by a single time-varying parameter ε. We note that Eqs. (16)–(18) correspond to the red dash-dot line in Fig. 3. The feedback parameter is introduced for the ocean heat uptake, λ _o, such that

$$ N\equiv {\lambda}_{\mathrm{o}}\varDelta {T}^{\prime }, $$

(19)

and substitution into Eq. (17) (Rose et al. 2014) yields

$$ \varepsilon ={\lambda}_{\mathrm{eq}}/{\lambda}_{\mathrm{o}} $$

(20)

Here, Eq. (20) means that the ocean heat uptake efficacy parameter represents the ratio between feedbacks operating under radiative forcing and ocean heat uptake alone.

The ocean heat uptake efficacy is analogous to the efficacy of forcing. The efficacy of forcing is defined as the ratio of ∆T caused by a specific forcing agent to that induced by the CO₂ of the equivalent radiative forcing (Hansen et al. 1997; Hansen et al. 2005; Yoshimori and Broccoli 2008):

$$ {\varepsilon}_f\equiv \frac{\varDelta {T}_{\mathrm{non}-{\mathrm{CO}}_2}/{F}_{\mathrm{non}-{\mathrm{CO}}_2}}{\varDelta {T}_{{\mathrm{CO}}_2}/{F}_{{\mathrm{CO}}_2}}=\frac{\lambda_{{\mathrm{CO}}_2}}{\lambda_{\mathrm{non}-{\mathrm{CO}}_2}} $$

(21)

In other words, the efficacy parameter of forcing represents the ratio between feedbacks operating under CO₂ and non-CO₂ forcing alone. To emphasize the similarity to Eqs. (16) and (17), Eq. (21) is written as

$$ \varDelta {T}_{{\mathrm{CO}}_2}={F}_{{\mathrm{CO}}_2}/{\lambda}_{{\mathrm{CO}}_2}, $$

(22)

$$ \varDelta {T}_{\mathrm{non}-{\mathrm{CO}}_2}={\varepsilon}_f{F}_{\mathrm{non}-{\mathrm{CO}}_2}/{\lambda}_{{\mathrm{CO}}_2} $$

(23)

Combining Eqs. (16) and (17), we obtain

$$ \frac{\varDelta T}{\varDelta {T}_{\mathrm{eq}}}=1-\frac{\varepsilon N}{F} $$

(24)

As long as Eq. (12) is valid, Eq. (24) is also expressed as

$$ \frac{\varDelta \mathrm{T}}{\varDelta {T}_{\mathrm{eq}}}={\left(1+\frac{\varepsilon \kappa }{\lambda_{\mathrm{eq}}}\right)}^{-1} $$

(25)

Therefore, Eq. (25) indicates that larger ocean heat uptake efficacy or efficiency leads to a smaller ratio of transient warming to equilibrium warming (Winton et al. 2010).

Geoffroy et al. (2013a) extended the study of Geoffroy et al. (2013b) by applying the following modified version of the two-layer model proposed by Held et al. (2010):

$$ C\frac{d\left(\varDelta T\right)}{dt}=F-{\lambda}_{\mathrm{eq}}\varDelta T-{\varepsilon}_0H\ \left(\mathrm{upper}\ \mathrm{ocean}\right) $$

(26)

$$ {C}_D\frac{d\left(\varDelta {T}_D\right)}{dt}=H\ \left(\mathrm{deep}\ \mathrm{ocean}\right) $$

(27)

where ε ₀ is a constant parameter of ocean heat uptake efficacy. By rewriting, Eq. (26) is more easily compared with Eq. (8)

$$ C\frac{d\left(\varDelta T\right)}{dt}=F-{\lambda}_{\mathrm{eq}}\varDelta T-H+\left(1-{\varepsilon}_0\right)H $$

(28)

The last term on the right side represents the time-varying effect of λ, as explained below.

By combining Eqs. (26) and (27), the ocean heat uptake is written as

$$ N=C\frac{d\left(\varDelta T\right)}{dt}+{C}_D\frac{d\left(\varDelta {T}_D\right)}{dt}=F-{\lambda}_{\mathrm{eq}}\varDelta T-\left({\varepsilon}_0-1\right)H $$

(29)

In Fig. 3, Eq. (29) ties the two limiting cases for the slope α of the ∆T-N relation: Eq. (29) represents α = − λ _eq − ε ₀ γ while the surface temperature is initially adjusting to the imposed forcing (ΔT _D  < < ΔT) and the upper ocean dominates the total ocean heat uptake. Then, α = − λ _eq/ε ₀ while the deep ocean temperature is catching up with the surface warming (ΔT _D  ∝ ΔT) and the deep ocean dominates the total ocean heat uptake. Again, the efficacy parameter (ε ₀) is assumed to be constant. We note the similarity of Eq. (29) to Eq. (18) in Eq. (30):

$$ N=F-{\lambda}_{\mathrm{eq}}\varDelta T-\left(\varepsilon -1\right)N $$

(30)

However, Eqs. (29) and (30) are equivalent (N = H) only when the heat uptake by the upper ocean Eq. (26) is much smaller than that by the deep ocean Eq. (27). We also note that no piecewise behavior is assumed in Eq. (18); the time-varying efficacy parameter (ε) captures the curvature of the slope in Fig. 3. According to Geoffroy et al. (2013a), the model with ocean heat uptake efficacy mimics the CMIP5 AOGCM simulations more closely because it can represent nonlinearity between the surface temperature change and the ocean heat uptake under constant radiative forcing. An example of the nonlinear curvature of the ∆T-N relation in an AOGCM is denoted by the blue and red crosses in Fig. 4. The equilibrium temperature response estimated by Gregory’s method using the initial 20-year (blue circles) or 150-year integration (blue line) in the abrupt4xCO2 experiment tends to be smaller than the “true” equilibrium temperature response obtained from the long millennial integration.

Geoffroy et al. (2012) demonstrated that the spread of the TCR in the CMIP5 AOGCMs is caused by climate feedbacks (λ _eq), radiative forcing (F), and ocean heat uptake efficacy (ε ₀), in the order of large to small contributions. Their result is consistent with that of Dufresne and Bony (2008) and leads to the conclusion that the uncertainty in cloud feedback is a dominant factor to the uncertainty in both ECS and TCR.

Time dependence of global feedbacks and physical meaning of ocean heat uptake efficacy

Armour et al. (2013) emphasized the advantage of a local/regional feedback framework, in which feedbacks are formulated as the TOA radiative anomaly per unit local surface temperature change, rather than per unit global mean surface temperature change (i.e., global feedback framework). In their local feedback framework, radiative damping is approximated by

$$ \lambda (t)\varDelta T(t)=\overline{\lambda \left(r,t\right)\varDelta T\left(r,t\right)}\approx \overline{\lambda (r)\varDelta T\left(r,t\right)}, $$

(31)

where r denotes location and the overbar represents the global average of local variables. The time variation of global mean radiative damping is therefore determined by the time-invariant local feedback parameter λ(r) and the time-varying spatial pattern of surface temperature change ΔT(r,t). If there is no change in the horizontal energy transport of the atmosphere to compensate, the local ocean heat uptake and the corresponding net TOA radiation must be balanced by the local radiative damping to reach equilibrium. Thus, the shift of ocean heat uptake to the region of a less positive local feedback parameter means greater surface warming remains to occur before the equilibrium. In this local feedback framework, an increase of ocean heat uptake efficacy is interpreted as the region of ocean heat uptake shifting to the region of the less positive local feedback parameter, such as that for subpolar oceans. Figure 6 shows the anomalous sea surface temperature (SST) patterns at different times of the increasing CO₂ experiment in (a) and (b) and their difference in (c). The surface warming is more dominant in high latitudes at later times presumably because of the large effective heat capacity in the Southern Ocean and the progress in the Arctic sea ice melting. The difference in the local feedback parameters between low and high latitudes is the time-dependent global mean radiative feedback parameter. As an extension, a larger efficacy of forcing is interpreted as the radiative forcing applied to the region having the less positive local feedback parameter. In an idealized model configuration, Rose et al. (2014), however, showed that local feedback parameters can be sensitive to the latitudinal difference in ocean heat uptake. Because the pattern of ocean heat uptake varies with time, their result challenges the validity of the assumption that local feedback parameters are fixed in time. By conducting atmospheric general circulation model (AGCM) experiments with a fixed SST pattern (scaled by the global mean surface temperature change), Andrews et al. (2015) verified that the time variation of the global feedback parameter develops from the evolving pattern of surface temperature change. They also questioned whether that the local feedback framework explains everything. The dynamical effect of the atmosphere on local feedback parameters, therefore, requires further study (Feldl and Roe 2013; Rose et al. 2014).

Fig. 6

SST changes normalized by the global mean SST changes from the control: a at the time of quadrupled CO₂ concentration in the 1pctCO2 experiment (average of 131–150 years); b at equilibrium in the const4xCO2 experiment (average of 3586–3685 years); and c the difference between b and a. The number at the upper right corner of each figure indicates the global mean SST changes from the control in degree Celsius

Winton et al. (2013b) proposed an indirect radiative influence of the AMOC. They argued that weakening of the AMOC under global warming reduces the heat transport to the northern high latitudes and that corresponds to a shift of ocean heat uptake regions from low to high latitudes. This shift results in the increase of the ocean heat uptake efficacy and slows down the global surface temperature rise. As mentioned already, Kostov et al. (2014), on the other hand, proposed that the stronger AMOC enhances the penetration of the heat anomaly into deeper ocean in the perturbed climate, and the resulting increase in the effective heat capacity of the climate system slows down the global surface temperature rise. Therefore, the quantitative contribution of the AMOC still needs to be established.

Meraner et al. (2013) argued another aspect of inconstancy of the global feedback parameter, in which the parameter becomes less positive and hence the ECS becomes larger in a warmer climate through enhanced water vapor feedback. The state dependency of climate feedbacks has been discussed in many previous studies (e.g., Colman and McAvaney 2009; Colman et al. 1997; Feldl and Roe 2013; Hansen et al. 2005; Jonko et al. 2013; Yoshimori et al. 2011), but the result appears to depend on the model and so the effect of the dependency on the transient response still needs to be established. In a similar context, the possibility of inconstancy of the climate resistance parameter (as well as the global feedback parameter) in the CMIP5 AOGCMs was reported by Gregory et al. (2015), although they also listed errors in the estimated radiative forcing as an alternate or additional possibility.

In summary, time-varying global feedbacks or radiative damping may arise from (a) the state dependency of climate feedbacks, that is, the global feedback parameter depends on the background temperature T; (b) changes in the spatial pattern of the surface temperature anomaly ΔT(r,t) associated with ocean heat uptake and dynamics (ocean heat uptake efficacy) or forcing agent (efficacy of forcing); and (c) the non-local effect of atmospheric dynamics resulting from (b) on the local feedback parameter λ(r,t).

Definitions and estimates of forcing

The time variation of the global climate feedback parameter has implications for how forcing and feedback are to be defined and computed. As discussed already, the nonlinearity of the relation between surface temperature and net radiation changes may introduce an error or arbitrariness into the estimate by Gregory’s method. By using Eq. (1), a more general time-varying radiative forcing was estimated by Forster et al. (2013), but the global climate feedback parameter was necessarily assumed to be independent of the forcing agent and also time invariant. In addition to Gregory’s method, the ERF was computed by using the fixed SST method by Hansen et al. (1997) and Hansen et al. (2005). In Hansen’s method, the AGCM was run to equilibrium under constant forcing without allowing the SST to change. Because the typical time scale of adjustments was shorter than a year, Hansen’s method might be applicable to time-varying forcing as well. The notable difference between Gregory’s and Hansen’s methods is that only the global surface temperature change is constrained to zero in Gregory’s method. On the other hand, a local SST change is not allowed in Hansen’s method, although it allows the land surface temperature to respond (Shine et al. (2003) suppressed the land surface response as well as SST change). A high correlation exists between the two methods, but Gregory’s method tends to estimate a smaller ERF than does Hansen’s method (Fig. 7). In both cases, the precise time scale of the adjustments is somewhat ambiguous. While the regression method is useful to investigate the relative spread among models (Andrews et al. 2015; Andrews et al. 2012b), it is expected that magnitude of forcing depends on the choice of period during the transient response, because the ΔT-N relation is nonlinear in general (Andrews et al. 2012b; Chen et al. 2014; Gregory et al. 2004; Winton et al. 2010). We note that the dominant factor of the model spread of the CO₂ ERF is still instantaneous forcing, not tropospheric adjustments (Chung and Soden 2015). As mentioned already, the effective forcing introduced by Williams et al. (2008) precludes the definition of feedback as a surface temperature-mediated response, because the decadal adjustments include the SST change.

Fig. 7

Effective radiative forcing for the doubling of atmospheric CO₂ concentration estimated with Hansen’s method and Gregory’s method for 10 CMIP5 models. See the text for the details. The data taken from Table 9.5 of Flato et al. (2013). The one-to-one line is also plotted. The number at the lower right corner of the figure indicates the fractional variance shared by the two variables. The right column lists the model names with their numbers, which are identical to those in Fig. 1

Implication for the estimate of ECS

As stated in an earlier section, it takes a couple of millennia for a comprehensive climate model or AOGCM to reach a new equilibrium. Therefore, AOGCMs are not routinely run to equilibrium, and the ECS is not obtained precisely by following the definition. Traditionally, atmospheric GCMs coupled to “slab” mixed-layer ocean models (ASGCMs) are used to obtain the ECS. The mixed layer usually has a small yet sufficiently large heat capacity to simulate the annual cycle, and the steady state without considering the change in ocean dynamics is obtained after a few decades of integration. While some ASGCMs exhibit reasonably close ECSs to those obtained by the long integration of AOGCMs, other studies point out the difference (Danabasoglu and Gent 2009; Li et al. 2013; Shell 2013; Stouffer and Manabe 1999; Yokohata et al. 2008). Gregory’s method was adapted by Andrews et al. (2012b) for the CMIP5 AOGCM analysis. It is expected that the linear part of the ΔT-N relation at the later stage of integration (fitting a line to the simulated trajectory closer to the equilibrium point in Figs. 3 and 4) yields an accurate ECS (Armour et al. 2013). Geoffroy et al. (2013a) applied the two-layer energy balance model with the ocean heat uptake efficacy parameter to the CMIP5 AOGCMs and estimated the optimal ECS and other parameters simultaneously. As a consequence, the model captures the nonlinearity of the ΔT-N relations. Although the mean of 16 estimated ECSs differs by only about 8 % compared to that estimated by the ocean heat uptake efficacy parameter of one (Geoffroy et al. 2013b), the difference is more than 20 % for the two AOGCM cases.

The nonlinearity occurring from the time variation of the global climate feedback parameter or the ocean heat uptake efficacy has a very important implication when the ECS is estimated from observation data. A general tendency of the global climate feedback parameter that is decreasing or the ocean heat uptake efficacy that is larger than one indicates that the ECS based on the energy budget may be underestimated when the transient data are used (Geoffroy et al. 2013a; Winton et al. 2010). In other words, the deviation of the effective climate sensitivity from the equilibrium climate sensitivity illustrated in Fig. 3 is not negligible, as pointed out in many studies (e.g., Armour et al. 2013; Rose et al. 2014; Winton et al. 2010). Because the observed data represent the transient stage of the response under increasing greenhouse gas forcing, this deviation is a concern for the observation-based estimate of the ECS (e.g., Forster and Gregory 2006; Gregory et al. 2002; Lewis and Curry 2015; Otto et al. 2013). The inconstancy of the climate resistance parameter yields similar concerns if the ECS estimate is made based on the assumption of its constancy (Gregory et al. 2015; Gregory and Forster 2008). We note that the efficacy of forcing is another important factor that may distort the estimate of the ECS from observations, as pointed out by Kummer and Dessler (2014), Shindell (2014), and Shindell et al. (2015). Marvel et al. (2015) argued that the best estimate of the TCR moves upward from 1.3 to 1.8 °C and the ECS from 2.0 to 2.9–3.0 °C if the efficacy of forcing based on a single model is taken into account. Therefore, even though the time variation of the climate feedback parameter is taken into account, the lack of the forcing efficacy factor (e.g., Masters 2014) may introduce bias in the ECS estimate. So far, the time variation of the forcing efficacy has not been well established. While climate models of reduced complexity enable us to constrain the ECS and TCR with statistical inference by conducting a large ensemble of parameter perturbations, we must not overlook the fact that the effect of the ocean heat uptake efficacy and the forcing efficacy are usually not taken into account (e.g., Aldrin et al. 2012; Skeie et al. 2014). This is not only limited to the ECS estimate based on historical observations but also is closely related to the ECS estimate based on a volcanic event (Merlis et al. 2014) or a past climate (Yoshimori et al. 2009).

Conclusions

Correct understanding of the transient response to radiative perturbation is important for the interpretation of observed climate change, the prediction of near-future climate change, and committed warming under climate stabilization scenarios, as well as the estimation of ECS based on observation data. It has been known for some time that the radiative damping rate per unit of global mean surface temperature increase (i.e., global climate feedback parameter) varies with time. This inconstancy affects the transient global mean surface temperature change, and different studies have emphasized the various different aspects of this inconstancy.

Recent progress has focused not only on the speed of the ocean heat uptake but also the spatial pattern of the ocean heat uptake. The spatial pattern influences the global radiative damping and thus the global climate feedback parameter. These influences arise from the evolving spatial pattern of the surface temperature anomaly and/or changes in the local feedback parameter. The inconstancy of the global climate feedback may be formulated by a single parameter, called ocean heat uptake efficacy. In addition, the dependency of the global climate feedback parameter on the background climate has also been pointed out. The physical mechanisms behind the behavior of the global climate feedback parameter still need to be established. Detailed analysis of the atmosphere-ocean interaction as well as perturbed physics ensemble experiments including the ocean component might be useful for identifying the physical mechanisms (Collins et al. 2006). Ocean heat uptake processes associated with the AMOC and those operating in the Southern Ocean are of great importance in understanding both speed and spatial pattern of ocean heat uptake and require further study.

The efficacy of forcing is another important factor determining the transient response to radiative perturbation. The current understanding of the forcing efficacy is very limited, and its time variation is particularly not well understood. Nevertheless, its inconstancy is presumed to be in the latitudinal distribution of forcing that varies with time.

All of these studies require the estimate of radiative forcing in a consistent manner. As proposed by Radiative Forcing Model Intercomparison Project,¹ ERF estimated by using AGCMs with a fixed SST, which does not require extrapolation, would greatly advance the study in a more quantitative way. Because the extrapolation using the regression technique in estimating the ECS would introduce ambiguity, we emphasize the usefulness of the integration of AOGCMs to equilibrium under doubled and quadrupled CO₂ forcing as presented in this review article and other studies with different ocean parameterizations (Yamamoto et al. 2015; Yamamoto et al. 2014).

In the fifth intergovernmental panel on climate change (IPCC) assessment report (IPCC, 2013; IPCC-AR5), the lower bound of the ECS estimate was revised from the fourth IPCC assessment report from 2 to 1.5 °C. In addition, the dispersion in the ECS estimate based on the energy budget and the emergent constraint (the empirical relation found in models between the ECS and observable variable) (Fasullo et al. 2015; Klein and Hall 2015) by using observation data precludes the IPCC-AR5 from providing the best estimate. These issues might be resolved at least partially by taking both ocean heat uptake and forcing efficacies into account.

Abbreviation

AGCM, atmospheric GCM; AMOC, Atlantic meridional overturning circulation; AOGCM, atmosphere- ocean GCM; ASGCM, atmosphere-slab ocean GCM; CMIP, coupled model intercomparison project; ECS, equilibrium climate sensitivity; ERF, effective radiative forcing; GCM, general circulation model; IPCC, intergovernmental panel on climate change; IPCC-AR5, the fifth assessment report of the IPCC; UD, upwelling-diffusion; SST, sea surface temperature; TCR, transient climate response; TOA, top of the atmosphere