# On simple representations of the climate response to external radiative forcing

## Abstract

Global warming in response to external radiative forcing is determined by the feedback of the climate system. Recent studies have suggested that simple mathematical models incorporating a radiative response which is related to upper- and deep-ocean disequilibrium (ocean heat uptake efficacy), inhomogeneous patterns of surface warming and radiative feedbacks (pattern effect), or an explicit dependence of the strength of radiative feedbacks on surface temperature change (feedback temperature dependence) may explain the climate response in atmosphere-ocean coupled general circulation models (AOGCMs) or can be useful for interpreting the instrumental record. We analyze a two-layer model with an ocean heat transport efficacy, a two-region model with region specific heat capacities and radiative responses; a one-layer model with a temperature dependent feedback; and a model which combines elements of the two-layer/region models and the state-dependent feedback parameter. We show that, from the perspective of the globally averaged surface temperature and radiative imbalance, the two-region and two-layer models are equivalent. State-dependence of the feedback parameter introduces a nonlinearity in the system which makes the adjustment timescales forcing-dependent. Neither the linear two-region/layer models, nor the state-dependent feedback model adequately describes the behavior of complex climate models. The model which combines elements of both can adequately describe the response of more comprehensive models but may require more experimental input than is available from single forcing realizations.

## Keywords

Nonconstant global feedback Energy balance models Timescales## 1 Introduction

The need to understand factors influencing climate feedback and sensitivity make simple models indispensable as means to investigate the climate response to changes in the Earth’s energy balance. In this paper we discuss the relationship between the energy balance of the climate system and its temperature response. We focus on the temporal behavior using the conceptual framework of four commonly used conceptual models. We reveal similarity and essential differences among these models, and we provide insight into the climate response in the Max Planck Institute Earth System Model 1.2 (MPI-ESM1.2) as a counterpart to our theoretical considerations. On mathematical grounds, we question the difference between the concept of ocean heat uptake efficacy and a pattern effect due to regional feedbacks weighted by a geographic pattern of surface warming. Further, we challenge the common assumption of these concepts that the characteristic timescales on which the climate system adjusts are independent of the strength of the external perturbation.

*N*, radiative forcing

*F*and surface temperature perturbation

*T*which in turn actuates the radiative feedback parameter \(\lambda\);

*N*and

*T*in experiments of state-of-the-art atmosphere-ocean coupled general circulation models (AOGCMs) in which the preindustrial atmospheric CO\(_2\) concentration is abruptly doubled and then held constant. Details, such as the possibility of a nonlinear dependence on

*T*are omitted from this model by construction. Numerical experiments with more comprehensive models suggest that these details may matter, and that Eq. (1) becomes an increasingly poor description as the forcing is increased to 4 or 8 times preindustrial CO\(_2\). In these cases, there is a tendency for the radiative response in several coupled climate models to decrease over time and with higher temperatures (e.g. Andrews et al. 2015; Bloch-Johnson et al. 2015). If one assumes that Eq. (1) holds at each instance of time, this can be interpreted as nonconstant global feedback. This is modeled by allowing \(\lambda\) to vary over time and rearrange Eq. (1) to obtain

*T*. The outcome of Eq. (2) may, however, also influenced by additional state-variables. It is itself a specific approach to analyze the climate response to external perturbations and provides a simple and vivid description of the radiative response. Unfortunately, to the extent that \(\lambda\) changes with

*t*it does not alone provide a framework for understanding the origin of such changes which becomes evident during the course of this study.

Currently, there are two different branches of literature on conceptual models to explain the more complex climate response in AOGCM experiments. The first branch suggests that on decadal timescales the variation of \(\lambda (t)\) arises from different radiative responses associated with a fast and a slow component of the climate system (two-timescale approaches). The second branch accounts for the possibility that the strength of radiative feedbacks depends explicitly on surface temperature change (feedback temperature dependence).

Hasselmann (1979) already recognized the multiple timescale structure of the climate response, but at this time the global feedback was assumed to depend on the global mean surface temperature only. Recently, these characteristic adjustment timescales have been explicitly linked to nonconstant global feedback associating them with different state-variables each of which influences the radiative response (e.g. Held et al. 2010; Winton et al. 2010; Geoffroy et al. 2013a, b; Armour et al. 2013; Andrews et al. 2015; Hedemann et al. 2019). Held et al. (2010) and Geoffroy et al. (2013a, b) describe a model based on two distinct timescales to reproduce the transient global mean temperature response of a wide range of AOGCMs. They introduce a globally averaged two-layer ocean model with an efficacy parameter modifying the upper ocean response to heat uptake. Hereafter it is referred to as a two-layer model with efficacy, or just the two-layer model. Two or more adjustment timescales can also be explicitly linked to a one-layer model with two regions. In contrast to Held et al. (2010) and Geoffroy et al. (2013a), Armour et al. (2013) suggest a geographic approach in which constant regional feedbacks are weighted by an evolving pattern of surface warming. This approach is motivated by the idea that the adjustment over parts of the Earth’s surface which are weakly coupled to the state of the deep-ocean is more rapid than over regions which are more strongly coupled to the state of the deep-ocean. Consequently, this can be called as pattern effect (Stevens et al. 2016). In addition to time-varying sensitivity, two-region models have also been used to analyze the effect of perturbation heat transport on stability properties and equilibrium sensitivity (e.g. Bates 2012, 2016) or to explore polar amplification (e.g. Langen and Alexeev 2007). In all of these models, the time-variation of \(\lambda (t)\) does not depend on the strength of the forcing– in this sense these models are linear in forcing. The tendency of such models to have a value of \(\lambda (t)\) that varies with time is thus often paraphrased as a time-dependent feedback though it arises from a dependence of the radiative response on an additional state-variable with its own distinct temporal evolution. Using AOGCM output, e.g. Senior and Mitchell (2000) and Hedemann et al. (2019) discuss time-dependent feedback as a way to understand the radiative response in more comprehensive climate models.

A different approach to representing variation in the radiative response has been to relax the assumption that a temperature perturbation would be proportional to the associated radiative forcing *F* (e.g. Roe and Baker 2007; Zaliapin and Ghil 2010; Roe and Armour 2011; Meraner et al. 2013; Bloch-Johnson et al. 2015). Studies adopting this approach investigate the long-term response on centennial to millennial timescales as well as the nonlinear change in equilibrium sensitivity between different forcing strengths. Conceptual models on nonlinear feedbacks of the climate system can be traced back at least to Budyko (1969) and Sellers (1969). In a recent study, Bloch-Johnson et al. (2015) extend the linear forcing-feedback framework Eq. (1) by a quadratic coefficient *a* representing second-order feedback temperature dependence and solve for stationary solutions, \(-F=\lambda T + a T^2\). They estimate the range of likely feedback temperature dependencies for various GCM simulations and find \(-0.04 \le a \le 0.06\hbox { W m}^{-2}\hbox { K}^{-2}\). This range is in line with Roe and Armour (2011) who find \(-0.058 \le a \le 0.06\hbox { W m}^{-2}\hbox { K}^{-2}\). Positive feedback temperature dependence makes the feedback of the climate system continuously less effective as the surface temperature increases, while the opposite is true for negative feedback temperature dependence. In this connection, the radiative response depends explicitly on the climate state; i.e., feedback temperature dependence gives rise to state-dependent feedback such that the variation of \(\lambda (t)\) is changed as the strength of the forcing is altered.

We formulate the different conceptual models discussed above as simple ordinary differential equations and study their dynamics. To provide a basis, in Sect. 2 we describe the climate response found in an AOGCM. Then, in Sect. 3, we discuss dynamical systems of the two-timescale approaches and demonstrate that the two-layer model with efficacy can be projected onto a two-region model. In Sect. 4 we analyze dynamical systems which incorporate feedback temperature dependence to point out that the temporal behavior of the climate system, or more precisely the temporal behavior of the comprehensive model developed at our institute, is indeed forcing-dependent.

## 2 Noncostant global feedback in MPI-ESM1.2

*N*and

*T*(e.g. Gregory et al. 2004). We briefly illustrate the MPI-ESM1.2 response to an abrupt increase in atmospheric CO\(_2\). A detailed analysis of radiative feedbacks in MPI-ESM1.2 or in its atmospheric component ECHAM6.3 is provided by Mauritsen et al. (2018) and Hedemann et al. (2019). MPI-ESM1.2 is forced by abrupt 2, 4, 8 and 16 times preindustrial CO\(_2\) (284.7 ppm) and is integrated forward in time to 1000 years. In this case, the radiative response is related to radiative feedbacks which emerge on decadal and centennial timescales such as the water vapor, lapse rate, clouds and snow/sea ice albedo feedbacks.

Figure 1 shows the MPI-ESM1.2 relationship between global and annual mean *N* and *T*. A common feature of these experiments is that there is an abrupt change in this relationship on decadal timescales. The model output for each CO\(_2\) doubling deviates significantly from the linear least-square regression (solid black line) which is computed using only output from year 1 to 25. Extrapolating the initial response (dashed lines), as would be appropriate if the radiative response to temperature were constant, leads to an underestimation of the equilibrium temperature response, increasingly so for larger forcing. In this connection, the initial variation of \(\lambda (t)\) can be explained by distinct timescales on which the climate system adjusts. This is at the heart of the theory of the two-timescale approaches (Sect. 3).

*F*on atmospheric CO\(_2\) concentration as predicted by theory which can be traced back to Arrhenius (1896). That is, the change in the associated radiative forcing

*F*imposed by each doubling cannot explain the nonlinear rise in the equilibrium temperature response: extrapolating the relationship between

*N*and

*T*in MPI-ESM1.2 linearly from year 100 to 1000 gives a change in equilibrium sensitivity from \(2\times \text {CO}_2\) to \(4\times \text {CO}_2\) of about 3.6 K and from the \(4\times \text {CO}_2\) experiment to the \(8\times \text {CO}_2\) of about 4.9 K, while the change from the \(8\times \text {CO}_2\) experiment to the \(16\times \text {CO}_2\) is roughly 9.8 K. As discussed in the introduction, this type of nonlinearity has been attributed to feedback temperature dependence (e.g. Meraner et al. 2013; Bloch-Johnson et al. 2015).

After 1000 years the \(8\times \text {CO}_2\) and \(16\times \text {CO}_2\) experiment are still far from steady state, whereas the \(2\times \text {CO}_2\) experiment is nearly equilibrated. After a forcing is applied, the temperature relaxes to a new equilibrium at specific timescales and the temperature increase actuates radiative feedbacks that amplify or attenuate surface warming. That the adjustment time-scale depends on the forcing is demonstrated by comparing at what time a given fraction of the equilibrium warming is realized as a function of forcing (Fig. 2a). The e-folding time \(\tau _{\mathrm {63}}\) and \(\tau _{\mathrm {80}}\), defined as the time when a fraction of 63 and 80 percent of the equilibrium response is reached, are compared. Evidently, the response times change as the radiative forcing is altered, and these changes occur in a nonlinear way. The change in response times acts on all timescales while the absolute change between different CO\(_2\) doublings increases with higher forcing and higher fractions of the equilibrium response. To highlight this behavior, we plot the difference between specific response times (\(\tau _{\mathrm {80}}\), \(\tau _{\mathrm {75}}\), \(\tau _{\mathrm {70}}\)) to the e-folding time \(\tau _{\mathrm {63}}\) (Fig. 2b). To compute specific response times, the MPI-ESM1.2 temperature output has been smoothed by running mean with a window of thirty years. The difference between these response times increases approximately quadratically as atmospheric CO\(_2\) is progressively doubled. Section 4 explains this phenomenon.

## 3 Two-timescale approaches

*F*which is equivalent to a radiative perturbation imposed by a radiative forcing agent such as CO\(_2\). In the following we explore the temporal dynamics of the energy budget and temperature response of the conceptual models subject to a step (Heaviside) radiative forcing,

*F*. This is conceptually equivalent to the forcing imposed by an abrupt increase in atmospheric CO\(_2\) in a more comprehensive model. The time-dependent unit-area perturbation equation for a zero-dimensional model like Eq. (1) can be generalized by

*N*is the downward energy flux perturbation per unit area at TOA, and

*E*is the system’s enthalpy per unit area. We assume that the atmosphere is always in energetic balance because its heat capacity is much smaller than the oceanic heat capacity. This way, we can parameterize the change in the climate system’s enthalpy by the surface temperature perturbation

*T*multiplied by the effective oceanic heat capacity

*C*such that

*N*. The ocean heat uptake efficacy model is a globally averaged two-layer model (Held et al. 2010; Geoffroy et al. 2013a, b) and based on two state-variables which represent an upper ocean- or mixed-layer temperature (

*T*, comprising atmosphere, land and ocean) and a deep-ocean temperature (\(T_\mathrm {D}\)). As a counterpart, we consider a configuration of the pattern effect which comprises two distinct regions. We demonstrate that, on mathematical grounds, these models can describe the same global temperature evolution and radiative response.

### 3.1 Two-layer model

*N*with respect to

*T*is

*T*and \(T_\mathrm {D}\) and it is a monotonically increasing function of time starting from a strong negative feedback. Strictly speaking, the time-variation of \(\lambda (t)\), or time-dependent feedback, does not depend explicitly on time but implictly because it emerges from an additional state-variable which also adjusts on forcing-invariant timescales. In the case of the two-layer ocean model, \(\lambda (t)\) increases or decreases gradually over time because it is relative to the surface temperature perturbation

*T*. This gradual increase or decrease, however, leads to an apparant abrupt change in the relationship between

*N*and

*T*, and the strength of this change is determined by the efficacy factor. The abrupt change in MPI-ESM1.2 on decadal timescales implies an efficacy factor above unity.

Parameters of the analytical solution for the two-layer model

\((b_\mathrm {1},b_\mathrm {2}) =(\frac{\lambda _{\mathrm {eq}} -\varepsilon \eta }{C},\frac{\varepsilon \eta }{C})\) |

\((b_\mathrm {3},b_\mathrm {4}) =(\frac{\eta }{C_\mathrm {D}},-\frac{\eta }{C_\mathrm {D}})\) |

\((\alpha _\mathrm {f},\alpha _\mathrm {s}) = \frac{b_\mathrm {1}+b_\mathrm {4}}{2} (1 \pm \sqrt{1-4 \frac{b_\mathrm {1}b_\mathrm {4}-b_\mathrm {2}b_\mathrm {3}}{({b_\mathrm {1}+b_\mathrm {4})}^2}})\) |

\((\tau _\mathrm {f},\tau _\mathrm {s})=(-\alpha _\mathrm {f}^{-1},-\alpha _\mathrm {s}^{-1})\) |

\((\zeta _\mathrm {f},\zeta _\mathrm {s}) =(\frac{\alpha _\mathrm {f}-b_\mathrm {1}}{b_\mathrm {2}},\frac{\alpha _\mathrm {s}-b_\mathrm {1}}{b_\mathrm {2}})\) |

\((\psi _\mathrm {f},\psi _\mathrm {s}) =(\frac{F/\lambda _{\mathrm {eq}} (1-\zeta _\mathrm {s})}{\zeta _\mathrm {f}-\zeta _\mathrm {s}},\frac{F/\lambda _{\mathrm {eq}} (1-\zeta _\mathrm {f})}{\zeta _\mathrm {s}-\zeta _\mathrm {f}})\) |

### 3.2 Two-region model

*r*denotes (not necessarily contiguous) regions and

*Q*(

*r*) denotes heat transport between these regions (e.g. Armour et al. 2013; Rose and Rayborn 2016). Even though regional feedbacks \(\lambda (r)\) are taken as constant this system may introduce nonconstant global feedback, because regional feedbacks are weighted by a pattern of surface warming. In the following we assume a two-region model in which the climate response in each region is characterized by a specific heat capacity

*C*(

*r*) and a feedback parameter \(\lambda (r)\). In analogy to the two-layer or two-region model we assume Heaviside step forcing input in both regions and five model parameters. The global mean TOA imbalance is given by

*T*and TOA imbalance

*N*of the two-layer model are equal to the global mean temperature perturbation \({\overline{T}}\) and the global mean TOA imbalance \({\overline{N}}\) of the two-region model. We assume that the radiative response of the climate system can be aggregated on these different regions and link the parameters for (regional) feedbacks to the fast component of the climate response and to the slow component of the climate response. The feedback parameters are then given by

*T*and

*N*in the two-layer model would be linear with slope \(\lambda _{\mathrm {eq}}\) and no pattern of surface warming and feedback could be resolved. That is, the fast and the slow adjustment timescales must be directly coupled to radiative feedbacks to cause time-variation of \(\lambda (t)\). In Fig. 3b we plot the range of \(\lambda _\mathrm {1}\) and \(\lambda _\mathrm {2}\) for output from the Coupled Model Intercomparison Project (CMIP5) (\(4\times \text {CO}_2\)) using parameter estimates from Geoffroy et al. (2013b) who fit the two-layer model to 16 CMIP5 AOGCMs. In line with the MPI-ESM1.2 experiments, most of these models exhibit an efficacy factor above unity (\(C_\mathrm {1} \ll C_\mathrm {2}\) and \(\lambda _\mathrm {1}<\lambda _2\)). However, these models differ in their temporal adjustment, which is discussed in the following.

According to the projection of the two-layer model onto the two-region model, we plot the fast timescale \(\tau _{\mathrm {f}}\) and the slow timescale \(\tau _s\) against the magnitude of \(\lambda _{\mathrm {1}}\) and \(\lambda _{\mathrm {2}}\) (Fig. 4). Again, we use an idealized configuration of the two-layer model (curves) and vary the efficacy factor to compute regional feedbacks. Furthermore, we plot the pairs of regional feedbacks and timescales for the output from CMIP5 models. The regional approach establishes a simple relationship between regional feedbacks and regional timescales, \(\tau (r)=-C(r)/\lambda (r)\). That is, the regional timescales are inversely proportional to the feedback parameters. The question arises whether CMIP5 models describe a unique relationship or wether the relationship is appropriate to characterize complex climate model behavior. For a given configuration, the relationship between \(\lambda (r)\) and \(\tau (r)\) is independent of the global mean equilibrium feedback parameter: using the idealized case, changing the equilibrium feedback parameter \(\lambda _{\mathrm {eq}}\) changes the regional feedback parameters and timescales in such a way that we move along the same curves to higher or lower values. Using CMIP5 output, the fast timescales are approximately linearly related to the regional feedback parameters in the region of fast adjustment. The effective heat capacity of this region is small and \(\tau _{\mathrm {f}}\) is determined by the magnitude of \(\lambda _{\mathrm {1}}\). This is commonly observed in regions which are weakly coupled to the state of the deep-ocean as found in low latitudes. As far as the slow timescale is concerned, the idealized configuration shows an exponential increase of \(\tau _{\mathrm {s}}\) as \(\lambda _{\mathrm {2}}\) gets more positive. The CMIP5 models deviate from the idealized configuration because they differ in *C*, \(C_{\mathrm {D}}\) and \(\eta\), and these model discrepancies induce changes in the temporal behavior of the climate response. The variation of these intertia parameters would change the relationship between \(\tau (r)\) and \(\lambda (r)\), and this way we would move along a different curve. The model parameters fitted to CMIP5 models, however, do support our general understanding that a more sensitive region adjusts on a longer timescale, and that this timescale changes in a nonlinear way as the magnitude of the radiative feedback is reduced.

How valid is such a regional approach or two-region model conceptually? In their experimental model study based on aquaplanet simulations Haugstad et al. (2017) show that local TOA radiative feedbacks depend on the pattern of the climate forcing modified by ocean heat uptake but the same radiative resonse arises when the surface temperature pattern induced by that forcing is prescribed. In that respect, ocean heat uptake induces a surface temperature pattern but it is secondary how this surface temperature pattern is induced because the same radiative feedbacks govern the relationship between *N* and *T*. We have shown theoretically that, from the perspective of global *N* and global *T*, there is no difference in the radiative response and temperature evolution between the two-layer model and the two-region model. Whereas the two-layer and two-region approaches are mathematically equivalent, the former may be attractive in some cases simply because it does not predicate a fixed spatial distribution of regional feedbacks.

## 4 Feedback temperature dependence

The concepts discussed in the previous section describe a temporal adjustment on forcing-invariant timescales. In this section, following Bloch-Johnson et al. (2015), we add an explicitly state-dependent component and analyze conceptually the implications of feedback temperature dependence for the transient response of the climate system. We first provide insight into the analytical solution of a single mixed-layer to demonstrate that feedback temperature dependence introduces a timescale that depends on the strength of the forcing. Finally, we analyze the interplay between the two-timescale approach and feedback temperature dependence in the case of the two-layer ocean model, since the relationship between *N* and *T* in MPI-ESM1.2 suggests both a time-dependent feedback due to the interaction of different state-variables and state-dependent feedback due to feedback temperature dependence.

### 4.1 Forcing-dependent timescale

*T*increases, and this temperature dependence can cause a nonlinear temperature response. The first-order feedback framework \({\varDelta }N= \frac{\partial N}{\partial T} {\varDelta }T\) can be extended by retaining terms up to second-order in the Taylor Series expansion of

*N*. This results in nonlinear relationships as

*T*. We call the quadratic coefficient

*feedback temperature dependence*and denote it by

*a*. Using the zero-dimensional model Eq. (1) extended by second-order feedback temperature dependence and solving for the MPI-ESM1.2 equilibrium response for \(2\times \text {CO}_2\), \(4\times \text {CO}_2\) and \(8\times \text {CO}_2\) gives a positive quadratic coefficient

*a*of roughly 0.04 W m\(^{-2}\hbox { K}^{-2}\). In the following we assume positive feedback temperature dependence.

*F*. Adding the quadratic coefficient representing feedback temperature dependence, our problem is described by a first-order nonlinear differential equation with constant coefficients;

*F*and the quadratic coefficient

*a*, and we get a quadratic runaway in the relationship between

*N*and

*T*in the case that \(F>\frac{{\varLambda }^2}{4a}\). Likewise, the characteristic timescale \(\tau\) on which temperature adjusts depends additionally on

*F*and

*a*, and changes in

*F*or

*a*lead to temporal changes in the adjustment of the system. In this connection, the temporal beavior of the climate response is determined by the strength of the forcing.

*a*causes the equilibrium response to be reached far later in time compared to a configuration without feedback temperature dependence. In MPI-ESM1.2 the \(8\times \text {CO}_2\) and \(16\times \text {CO}_2\) experiments are still far from stationarity after 1000 years of integration, while the \(2\times \text {CO}_2\) experiment is nearly equilibrated. Figure 5 shows an example of forcing-induced changes in the normalized timescale of the mixed-layer model with feedback temperature dependence against \(a \rightarrow 0\); i.e.,

In this case, the choice of the effective heat capacity is arbitrary because the timescale \(\tau\) is proportional to *C*. The characteristic timescale increases in a nonlinear way as the forcing is increased, and this nonlinearity is reinforced by a more positive quadratic coefficient. The positive contribution from feedback temperature dependence to the temperature or radiative response is counteracted by the first-order radiative response coefficients comprised in \({\varLambda }\).

### 4.2 Two timescales and feedback temperature dependence

*N*and

*T*is characterized by a continuous function with smoothly changing slope which can describe the long-term response in MPI-ESM1.2 on centennial timescales. The change in the relationship between

*N*and

*T*on a decadal timescale, however, cannot be reproduced due to the missing multiple timescale structure. We combine the two-timescale approach and state-dependent feedback by adding feedback temperature dependence to the mixed-layer energy budget of the two-layer ocean model Eqs. (5, 6). The TOA imbalance

*N*is now given by

*T*into a fast response \(T_\mathrm {f}\) when the deep-ocean is not yet significantly perturbed and a slow component \(T_\mathrm {s}\) when the deep-ocean responds; that is, \(T = T_\mathrm {f}+ T_\mathrm {s}\).

*a*. The effect of feedback temperature dependence on the radiative response is reinforced as the slow component evolves. In contrast, both the fast component and the slow component change significantly as the radiative forcing

*F*is linearly increased (Fig. 6b). The fast response depends sensitively on the magnitude of \({\varLambda }\), and time-dependent feedback dictates to what extent \(T_\mathrm {f}\) and \(T_\mathrm {s}\) behave nonlinearly as we change

*F*. In the case of an efficacy factor below unity, the magnitude of the slow mode \(T_\mathrm {s}\) is reduced compared to an efficacy factor above unity. Likewise, the degree to which the slow mode \(T_\mathrm {s}\) behaves nonlinearly diminishes, because the radiative response is initially suppressed and subsequently enhanced.

*N*and

*T*in the MPI-ESM1.2 on decadal timescales as well as the variation in the \(2\times \text {CO}_2\) experiment. For larger forcing such as \(4\times \text {CO}_2\) and \(8\times \text {CO}_2\), the effect of feedback temperature dependence on the radiative response becomes increasingly important and dictates the long-term response. We could quantify contributions from

*a*and \(\varepsilon\) to the change of

*N*or the variation of \(\lambda (t)\) in the combined run (\(a \ne 0\) and \(\varepsilon \ne 1\)). However, these contributions are relative to the temperature adjustments of the system while the characteristic timescales on which the system adjust themselves depend on

*F*; i.e., these contributions are related to the combined effect of time-dependent and state-dependent feedback. Interpreting time-dependent feedback in AOGCM experiments when both time-dependent feedback and state-dependent feedback are present thus depends sensitively on the conceptual framework chosen and the assumptions made on the temperature-dependent feedback parameter.

To highlight the differences in the temporal adjustment between state-dependent and time-dependent models as well as their combination, we define a timescale \(\tau (t)\) which is allowed to vary over time. We define this timescale as \(\tau (t)= -t/\ln ( \frac{T(t)-T_{\mathrm {eq}}}{-T_{\mathrm {eq}}})\) where *T*(*t*) is the globally averaged surface temperature perturbation and \(T_{\mathrm {eq}}\) is the equilibrium response. That is, we assume a single mode. Figure 8 shows \(\tau (t)\) for purely time-dependent feedback, purely state-dependent feedback and their fitted combination. In general, \(\tau (t)\) increases monotonically from relatively small values which are related to the fast mode of the climate response to relatively high values which are related to amplified warming due to the slow component on a centennial timescale. In the case of purely time-dependent feedback, the variation of \(\tau (t)\) over time is independent of the strength of the radiative focing *F*, since the characteristic timescales of the two-timescale approaches are constant and the amplitudes of the temperature adjustment are linear in forcing. In the case of purely state-dependent feedback due to feedback temperature dependence and unitary efficacy, the variation of \(\tau (t)\) depends additionally on the radiative forcing *F*. Due to the interaction of time-dependent feedback and state-dependent feedback the temporal adjustment is even prolonged and the effective timescales deviate from purely state-dependent feedback despite the same equilibrium response. Understanding the forcing-dependency of the characteristic adjustment and its interaction with time-dependent feedbacks is therefore important to narrow the timing of transient warming on a centennial timescale.

## 5 Conclusion

The two-timescale approaches give rise to time-dependent feedback but we paraphrase time-dependence as implicit because it emerges from the interplay of at least two state-variables and relates to the evolution of the radiative response over time rather than to an explicit time-dependence. The recent conceptual frameworks of the two-timescale approaches, the two-layer model with efficacy and the two-region model, can describe the same global temperature evolution and radiative response. On mathematical grounds, the two-layer ocean model with nonunitary efficacy can be projected onto a two-region model, and we can use the regional model feedbacks and timescales to characterize complex climate model behavior. The mathematical equivalence raises the question which specific mechanism causes time-dependent feedback. We cannot reject one or the other framework given only global mean output for the TOA imbalance *N* and surface temperature change *T*. We suggest that the radiative response of the climate system can be aggregated on different regions such as regions directly coupled to the state of the deep-ocean and not directly coupled to the state of the deep-ocean.

In constrast to time-dependent feedback, state-dependent feedback due to feedback temperature dependence introduces a nonlinearity which makes the timescale on which the climate system adjusts forcing-dependent. The finding that the timescales on which the climate system adjusts depend on the strength of the forcing has not been addressed to date. As we alter the forcing, we change the system’s response times and a certain fraction of the equilibrium response or temperature level is reached earlier or later in time. Further, this forcing-dependency and the combined effect or entanglement of time-dependent feedback and state-dependent feedback poses challenges for determining contributions from different sources such as feedback temperature dependence or a pattern effect. Single forcing realizations may not constrain the model parameters, and this complicates attempts to understand temperature change especially in more strongly forced climates on a centennial timescale but also the details of the instrumental record.

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. This study was supported be the Max-Planck-Gesellschaft (MPG) and TM received funding from the European Research Council (ERC) Consolidator grant 770765. Computational resources were made available by Deutsches Klimarechenzentrum (DKRZ) through support from Bundesministerium fr Bildung und Forschung (BMBF).

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