Parsimonious Emulators for the Global Climate Response Across Millennia

Abstract

Parsimonious emulators (PEs) trained on complex climate models (CCMs) are useful when global variables like global mean surface temperature and climate-system energy content are sought. CCM runs over millennia extracted from the LongRunMip repository are used to construct and test PEs for global mean temperature and net incoming radiation flux. For the temperature, the PE is a linear impulse response in the form of a superposition of k decaying exponentials, comprising k weight coefficients and k decay times to be estimated by least-square fitting to the temperature from CCM runs with abrupt step-function forcing. The model fit for $k\ge 3$ is good on all time scales, and the fitted model seems to perform even better for smoother forcing scenarios, suggesting that it reflects essential features of the CCM to which it is fitted. Data for radiation flux are combined with temperature data to produce low-order polynomial fits to Gregory plots and analytic expressions for the evolution of the effective feedback parameter, the radiation fluxes, the evolution of climate-system energy content, and an effective system heat capacity. The analysis reveals four stages of the ocean heat uptake, characterised by increasing effective heat capacity. From these Pes, one can compare the global performance of CCMs under different forcing scenarios, highlighting distinguishing features, such as evolution of albedo feedback and cloud radiative effect. Producing Gregory plots for all-sky and clear-sky outgoing long-wave and short-wave radiation, varying cloud albedo is identified as the main contributor to model spread of equilibrium climate sensitivity.

1. Introduction

Climate models come at all degrees of complexity. The most complex, the Earth System Models (ESMs), incorporates atmospheric dynamics, ocean dynamics, sea ice and ice caps, land surface and land use change, vegetation change and cycle, and chemistry of greenhouse gases. At the core is a coupled atmosphere–ocean general circulation model (AOGCM). In this paper, the term Complex Climate Model (CCM) will be assigned to this broad class of climate models. In the Climate Model Intercomparison Projects (CMIPs) leading up to the periodic Intergovernmental Panel on Climate Change (IPCC) reports, an increasing number of CCMs developed by various research centres are included, and experiments standardised for comparison purposes. Results from phase 6 of this project (CMIP6) were included in the 6th IPCC report (AR6) and CMIP7 is in the pipeline for IPCC AR7 planned for release in 2029. While the increasing complexity of these models yields better projections of climate change on regional and smaller scale and includes an increasing number of climate variables relevant for climate adaptation, it has not led to substantially reduced uncertainty in the estimates of equilibrium climate sensitivity (ECS). Despite impressive advances in climate science over the last 50 years, we do not have much more precise information about the long-term global warming for given emission scenarios [1].

Most climate models in the CMIP protocols are not relaxing to radiative equilibrium during simulations over a few centuries with persistent forcing, and simulations over millennia are not part of the standard protocol. The LongRunMIP protocol [2] was created to fill a gap in the CMIP protocols and collect published data in one location for easy public access. The focus in the present paper is using the LongRunMIP protocol to establish parsimonious emulators (PEs) fitted to CCMs on the millennium time scale and thus incorporating their true equilibrium sensitivities.

The case for parsimonious emulators is obvious for several reasons. Reduced computation cost and time allows for much larger range of experiments and variation of anthropogenic scenarios. The more complex a model, the greater the number of parameters that must be guessed or constrained by observation data. The greater this number, the greater the probability of creating a model that fits the available data, not because the model is a good representation of the system one seeks to model, but because the model’s free parameters give it the necessary degrees of freedom to describe a limited dataset correctly. This caveat of complex models is known as overfitting [1]. Thus, when some of the parameters of a model are not precisely known from basic physics, every reduction of complexity that makes physical sense may have some advantage in certain settings.

An example of the utility of reduced computational cost by model simplicity is found in [3], where a global mean surface temperature (GMST) response of the form

$$T\left(t\right)={\int }_0^{t}G\left(t-s\right) F\left(s\right)ds,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}G\left(t\right)=\sum _{i=1}^3{(S}_i/{\tau }_i) e^{-t/{\tau }_i}$$

(1)

emulates the output of a complex climate model that responds to the total radiative forcing $\mathcal{F}\left(t\right)$. Applied to an ensemble of different emission scenarios representing Shared Socioeconomic Pathways (SSPs) [4], this allows for uncertainty estimates of the remaining carbon budget for specific targets across the CMIP6 ensemble. This allows for uncertainty estimates of the remaining carbon budget for specific targets across the CMIP6 ensemble.

Model parsimony can be very useful in climate attribution studies. One example is found in [5], where a long-memory temperature response PE is applied to produce the footprints of the most important forcing components and modes of internal variability to obtain an accurate model of the historical GMST record.

Avoiding overfitting and forbidding computation cost are not the only rationales for parsimony. Perhaps the most useful feature is the ensuing conceptual simplicity. It is often difficult to identify the physics of CCMs that make them produce divergent projections under the same forcing in very long simulations. The PEs for these CCMs, however, differ only in a few parameters, which may give a clue to the distinguishing physics. Examples will be given in Section 4.

The PEs in this paper are “multi-box” models of the climate system in the form of a superposition of saturating exponentials. They are driven by one scalar input variable; the global mean radiative forcing $\mathcal{F}\left(\mathrm{t}\right)$, and we extract only a few scalar output variables, such as the GMST and the net top-of-the-atmosphere radiation flux density (netTOA) and its decomposition into outgoing longwave radiation (OLR) and incident and outgoing shortwave radiation (ISR/OSR). The PEs are mathematically related to multilayer zero-dimensional energy balance models, which have been studied extensively in the literature [6,7,8,9,10]. The novelty here will be made clear in the structure description of the paper in the next paragraphs and in Section 4.

Section 2 first describes the CCM data used for fitting PEs. In addition to global temperatures, net global top-of-the-atmosphere radiation flux densities (netTOA) and outgoing longwave and shortwave radiation (OLR and OSR) are described, and their relation to the GMST is presented in so-called Gregory plots [11]. The time evolution of the OSR, and hence the Earth’s albedo, turns out to be very different in the models, and it presents itself as the most important explanation of the difference in climate sensitivity. Third-order polynomial fits to Gregory plots are described, and the evolution of an effective feedback parameter ${\mathsf{\lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\mathrm{t}\right)$ is derived from these.

In Section 3, it is shown that the three-box PE with model parameters estimated from the abrupt quadrupling scenario in GISS-E2-R predicts accurately the GMST and netTOA made in the run with 1 percent per year CO₂ increase. In Section 5, this feature is demonstrated in the ECHAM5/MPIOM model as well. This supports the conjecture that once the model parameters have been estimated by fitting to a given CCM run extending over several millennia, the PE will emulate the CCM for any reasonable CO₂ concentration scenario. The combined evolution of the GMST and netTOA is then used to define and compute the evolution of the increased climate system energy content (CSEC), and from this, an effective heat capacity $C_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(t\right)$ of the climate system is defined, and its significance is discussed [12]. It is demonstrated in this section that 150 yr-long CCM simulations (which is the standard in the CMIP protocols) are inadequate for constructing PEs that can predict GMST and netTOA on longer time scales, while three boxes fitted to simulations over several millennia are adequate for smoother forcing scenarios, but four boxes may be necessary to reproduce the first few years after an abrupt 4xCO₂ forcing transition.

Section 4 presents an extensive discussion starting by arguing through illustrative graphs that the four boxes relate to physical stages in the ocean heat uptake. The OLR and OSR are complemented by their clear-sky counterpart, by which the cloud radiative effects are exposed. The section also contains a discussion of some recent related papers. Section 5 concludes and suggests some topics for further work.

Finally, a word about notation. I make a distinction between variables retrieved from the complex CCM and those modelled by the PE. For the former, I use acronyms; for the latter I use a single letter in italic font, often with an explanatory subscript. For instance, the GMST(t) from the CCM contains internal variability. Hence it is not monotonically increasing, even though the forcing $\mathcal{F}\left(t\right)$ considered in this paper is always weakly monotonic, giving rise to strictly monotonic variations of the PE temperature $T\left(t\right)$. Since $T\left(t\right)$ is strictly montonic, there is a one-to-one mapping $t\leftrightarrow T$. Suppose we have a diagnosed physical variable VAR(t) in the CCM, and plot the points (VAR(t), GMST(t)) for all years $t=1,\dots ,L$ in the annual CCM record. If we can regress a simple function $V\left(T\right)$ to these points with a small least-square error, it can be considered a PE for this relationship between T and V. Moreover, we have a PE for the time evolution of this variable; $V(T\left(t\right)$).

Table 1. Variables from CCMs and their corresponding PE variables employed in this paper.

	CCM Variable	PE Variable
Global mean surface air temperature	GMST(t)	T(t)
Net incident top-of-the atmosphere flux density	netTOA(t)	FTOA(T(t))
Outgoing longwave radiation	OLR(t)	FOLR(T(t))
Outgoing shortwave radiation	OSR(t)	FOSR(T(t))
Incident shortwave radiation (from the sun)	ISR(t)	FISR(T(t))
Outgoing longwave feedback radiation	Not directly diagnosed	$F_{\mathrm{O}\mathrm{L}\mathrm{R}}^{\left(\mathrm{f}\mathrm{b}\right)}(T(t))$
Climate system energy content (increase)	CSEC(t)	E(t)

gives an overview of the most important variables.

2. Materials and Methods

2.1. Data Retrieval and Description

Because of the large thermal inertia of the world oceans and the complexity of how heat is distributed from the surface layer into the deep ocean, the rather few millennium-long simulations of CCMs indicate that equilibration occurs on time scales of thousands of years. This paper deals with an abr4x and a 1pct4x 5000 yr simulation of GISS-E2-R, a 5900 yr abr4x simulation of CESM104, and an abr4x 1000 yr and a 1pct4x 6500 yr simulation of ECHAM5/MPIOM. In these simulations, the evolution of the atmospheric CO₂ concentration is prescribed and is the sole radiative forcing. This implies that the CCMs do not include a carbon cycle module and the high constant CO₂ concentration throughout future millennia should not be perceived as a model projection.

From the open access LongRunMip data repository (detailed information is given in [2]), I analyse the monthly sampled gridded data for near-surface air temperature (tas), the net top-of-the-atmosphere incident radiation (netTOA), the TOA incident shortwave radiation from the sun (rsdt), the TOA outgoing longwave radiation (rlut), and the TOA outgoing shortwave radiation (rsut). In Section 4, I also supplement these data with the clear-sky longwave and shortwave radiation (rlutcs and rsutcs, respectively). In climate model archives like LongRunMip, clear-sky variables are computed by taking the exact same atmospheric profiles of humidity, trace gases, and aerosols—and the same surface albedo—and only setting cloud fraction to zero. In the discussion in Section 4, I use the term “surface albedo” as a shorthand for clear-sky albedo, which means that it includes the albedo effect from aerosols.

The area-weighted mean of the temperatures constitutes the GMST in degrees Kelvin. The data for TOA radiation are given as energy flux density in every surface cell, which is multiplied by cell area, summed over cells, and divided by earth surface area to obtain mean global TOA flux densities in Watts per meter square. Note that, since netTOA and ISR are “incident”, and OLR and OSR are “outgoing”, they are connected by the relation

$$\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}\left(t\right)=\mathrm{I}\mathrm{S}\mathrm{R}\left(t\right)-\mathrm{O}\mathrm{L}\mathrm{R}\left(t\right)-\mathrm{O}\mathrm{S}\mathrm{R}\left(t\right).$$

(2)

I make a summation of netTOA(t) over years to obtain the increase in climate system energy content per m² of the earth surface, CSESC(t). Since most of this energy ends up as ocean heat on longer timescales, this quantity multiplied by the area of the earth surface is approximately the increase of the ocean heat content. Sometimes throughout the paper, I refer to CSESC(t) as ocean heat content without mentioning that it should be multiplied by the surface area, and that a small fraction is stored as heat in land mass and as latent heat due to melting of sea ice. Dynamic ice sheets are not modelled in these CCMs.

The simulations start from an initial state ($t=0)$, which is the end state of a millennium-long preindustrial control simulation (pi-control), which ends in the boreal winter 31 December. In Section 2.2, I study how the monthly mean GMST and radiation fluxes behave after abrupt 4xCO₂. In principle, the abrupt change of $F_{\mathrm{TOA}}\left(t\right)$ from the last month of the control to the first month of the forced simulation should represent the forcing ${\mathcal{F}}_{4\mathrm{x}}$ that corresponds to abrupt quadrupling of CO₂ concentration. In practice, however, it is necessary to assess the bias and uncertainty of that estimate due to natural variability in the control.

The analysis of the data in this paper generates more figures than can be presented in the main manuscript. Some figures are only presented for the GISS model, and others only for GISS and CESM. For less essential figures, I refer to Supplements SA and SB.

2.2. Description of the Monthly Data

Figure SB-I (First figure in Supplement SB) shows the incoming radiation flux $F_{\mathrm{ISR}}\left(t\right)$ in the CESM simulations. They are similar, but not completely identical, in the other models. The seasonal variation is due to the eccentricity of the elliptical Earth orbit. This seasonality contributes to seasonal variation of the GMST and the other fluxes, as shown in Figure 1 below. For the GISS model, Figure 1a shows a GMST increase of about 1.5 K during the 1st year after the abrupt 4xCO₂, and then another 1.5 K during the following decade. The netTOA jumps up $\Delta \mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}=$6.8 Wm⁻² from the month of 31 December to 31 January following abrupt 4xCO₂, which could be attributed mainly to the reduction in OLR due to the increased greenhouse forcing. There is a small negative jump in the OSR, but it is not greater than the natural month-to-month variability seen in the control run. The bias and error in the estimated jump in netTOA are assessed by computing the mean and standard deviation of the jump from December to January in the control run. The bias is 0.8 Wm⁻², and the standard deviation is 0.9 Wm⁻². The distribution of the jump is close to Gaussian, so with 0.95% confidence, ${\mathcal{F}}_{4x}=6.0\pm 1.8$ Wm⁻² for the GISS model.

For the CESM model (Figure SB-II), there is a GMST increase like the GISS model during the first decade, and the netTOA jumps up $\Delta \mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}=$7.3 Wm⁻² from the month of December 31st to January 31st following abrupt 4xCO₂, and again, most of it could be attributed mainly to the reduction in OLR. However, the negative jump in the OSR is no longer negligible. This jump in OSR is a reduction in reflected solar radiation (reduced albedo). The bias (mean) and error (standard deviation) in the estimated jump in netTOA pi-control run are 0.3 Wm⁻² and 1.0 Wm⁻², respectively. The 0.95% confidence range is ${\mathcal{F}}_{4x}=7.0\pm 1.8$ Wm⁻² for CESM104.

In Section 2.5 and Section 2.6, alternative estimates for ${\mathcal{F}}_{4x}$ are presented, using the so-called Gregory plots.

2.3. Description of Annual and Decadal Mean Data

Since the seasonal oscillation is quite large, it will clutter graphs of the evolution on longer time scale. For this reason, annual means will be used for plotting and for fitting PEs to the CCM model data. The annual means are computed by averaging over each successive year after $t=0$, and plotted as points with time-coordinates $t=0.5$, 1.5, 2.5 … yr. This puts the annual time average at the midpoint of the calendar year. In most plots, the points are joined by straight lines, as shown in Figure 2. In this figure, annual averages of the GMST anomaly from GISS-E2-R, CESM104, and ECHAM5/MPIOM are plotted for the entire time series. In this paper, when I use the abbreviations ECHAM or ECHAM5, I always mean this coupled atmosphere/ocean model ECHAM5/MPIOM.

In Figure 2a,b,e,f, the evolution for abr4x and 1pct4x both converge to the same path after a few hundred years. In the 1pct4x simulations, the CO₂ concentration grows exponentially up to $t=140$ yr, after which it is kept constant. According to theory, the radiative CO₂ forcing grows approximately as the logarithm of the concentration, so the forcing during those first 140 years grows linearly, and this gives an approximately linearly growing GMST in the 1pct4x simulation. For the CESM model, I only plot the abr4x simulation, as a 1pct4x simulation does not exist for this model. For the ECHAM model, the abr4x simulation is only 1000 yr long, while the 1pct4x run is 6080 yr.

It is apparent from the plots that more action occurs during the first centuries, and this is better resolved by employing a logarithmic time axis, as shown in Figure 2b,d,f. This is a convention that will be used in the remainder of this paper, and it is important to keep this in mind when interpreting the plots: the short time scales are expanded, and the long ones are compressed.

Figure 3 depicts the evolution of the netTOA incident radiation flux from the same simulations shown in Figure 2. The instant rise of netTOA after the abrupt CO₂ rise is due to the resulting reduction of OLR. However, one cannot use the first data point (at approximately 6.5 Wm⁻² for GISS abr4x) as a reliable estimate of ${\mathcal{F}}_{4x}$, because this point measures the average netTOA over the 1st year after the abrupt forcing, and during that year, there is already a substantial surface temperature increase and radiation feedback, resulting in a reduced netTOA. This is clearly demonstrated in Figure 1.

It is apparent from Figure 2 and Figure 3 that the climate of the abr4x and 1pct4x scenarios converge to the same near-equilibrium state after a few centuries. This gives reason to believe that the abrupt CO₂-rise simulations are relevant for prediction of the long-term evolution of the climate also for smoother emission scenarios that end up with permanently elevated CO₂ concentrations.

2.4. Outgoing Longwave and Shortwave Radiation

When dealing with anomalies and annual averages, we have ISR(t) = 0, and Equation (2) implies that the net incoming flux density anomaly is compensated by the sum of outgoing longwave and shortwave flux density anomalies: $\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}\left(t\right)=-\mathrm{O}\mathrm{L}\mathrm{R}\left(t\right)-\mathrm{O}\mathrm{S}\mathrm{R}\left(t\right)$.

Figure 4 shows the evolution of these three fluxes versus time t in annual resolution for the GISS and CESM models after abr4x forcing. However, while netTOA evolves similarly in the two models, the shortwave fluxes of reflected solar radiation behave quite differently. In the GISS model, the reflected radiation increases during the first decade and stabilises at a moderate outgoing flux density anomaly of approximately 1 Wm⁻², whereas in the CESM model, OSR anomaly drops to about $-2$ Wm⁻² during the 1st year and then drops gradually to nearly $-6$ Wm⁻² during the next 6 millennia without completely stabilising. Note, however, that the logarithmic timescale conceals a flattening of the corresponding curve in a linear plot. In both models, the netTOA anomaly decays towards zero after a few millennia, so in this time-asymptotic limit, $\mathrm{O}\mathrm{L}\mathrm{R} \approx -\mathrm{O}\mathrm{S}\mathrm{R}$. This means that the increase in OLR over the entire simulation, after the initial drop following the abrupt forcing, is higher in the CESM ($\approx$10 Wm⁻²) than in GISS ($\approx$5 Wm⁻²). To produce such a large increase in outgoing long-wave radiation, the surface temperature increases more in CESM ($\approx$6.7 K) than in GISS ($\approx$4.8 K).

2.5. Gregory Plots

The outgoing radiation represents feedback to the global temperature change, so it makes sense to plot it as a function of the global temperature anomaly. This is done in the Gregory plots [11], where at each year t in the time series, one plots the point (GMST(t), netTOA(t)). Figure 5c,f show that the absolute value of the slope of the fitted line, known as the feedback parameter, is lower as the temperature approaches equilibrium. In Figure 5, this is done for OLR and OSR as well. In the figure, these are plotted with negative sign, hence the plots represent incoming radiation with positive sign, and the sum of the two curves in the panels to the left and middle equals the netTOA plotted in the rightmost panels. In each panel, a straight line is regressed to the points corresponding to the first 150 years of the record (red points and line), and the grey points and line to the remaining years. For the OLR, the red and grey lines have almost equal slope in both models, but the absolute value of the slope of the grey line for OLR in the CESM model (1.61 Wm⁻²K⁻¹) is greater than that of the grey line in the GISS model (1.15 Wm⁻²K⁻¹). This slope is known as the feedback parameter for the OLR and represents the strength of the response in thermal radiation from the Earth to the warming of its surface. This paper uses the convention of treating the feedback parameter as a positive number (the absolute value of the slope), although it is quite common to assign a negative number to negative feedback.

As shown in Figure 4, the responses in OSR, the changes in reflected radiation, differ a lot between the two models and results in Gregory plots (netTOA versus T) in the rightmost panels, which do not have a distinct slope, although it appears that the curves break around $t=150$ yr, and that meaningful linear fit can be made separately in the time regimes $t<150$ yr and $t>150$ yr.

For GISS, the slope drops from 1.7 Wm⁻²K⁻¹ for $t<150$ yr, to 1.0 Wm⁻²K⁻¹ for $t>150$ yr. For CESM, it drops from 1.17 Wm⁻²K⁻¹ to 0.65 Wm⁻²K⁻¹. This reduction of the radiation feedback parameter when the Earth system approaches its new high-temperature equilibrium state seems to be a universal feature among CCMs, and Figure 5 suggests that the main contribution is increased decline of the albedo with increasing temperature.

2.6. Generalized Gregory Plots and Polynomial Fits

The piecewise linear regression lines may not be the most accurate way to produce regression curves to the Gregory plots. Let us assume that we have established a PE temperature $T\left(t\right)$ by fitting a multi-box model to the GMST time series. For the abr4x simulations, those PEs are established by least-square fitting of the expression given by Equation (8) in Section 3.1 below, determining the parameters $T_i$ and ${\tau }_i$. For the 1pct4x simulation, the impulse response function is established from Equation (1), using the parameters ${ S}_i={ T}_i {\tau }_i$ and ${\tau }_i$ determined from the abr4x simulation of the same model (in this case GISS-E2-R). The PE for the 1pct4x simulation is then established by the convolution integral in Equation (1). For a more detailed explanation of the method, see Supplement SA.

For abr4x, we can formulate a PE for netTOA(t) by utilizing the Gregory plot. The non-constant feedback parameter motivates a search for a non-linear regression curve to the Gregory plot. A simple yet accurate choice is a third-order polynomial fit $N\left(T\right)$. Figure 6a,b show analytical fit $N\left(T\right)$ to the points $\left(\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}\left(t\right), \mathrm{G}\mathrm{M}\mathrm{S}\mathrm{T}\left(t\right)\right), t=1,2,\dots ,L$, where L is the length of the simulation in years. The PE temperature $T\left(t\right),$ regressed to the time series $\mathrm{G}\mathrm{M}\mathrm{S}\mathrm{T}\left(t\right),t=1,2,\dots , L$, then provides a PE for netTOA(t);

$$F_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(t\right)=N\left(T\left(t\right)\right),$$

(3)

The $N\left(T\right)$ obtained from the abr4x simulation is not necessarily applicable to find $F_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(t\right)$ for other forcing scenarios like the 1pct4x, because the feedback in the transient phase may depend on the forcing history $\mathcal{F}\left(t\right)$. This would imply that the feedback parameter ${\mathsf{\lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)$ depends on this forcing history and that we would need data for the specific CCM simulation under the forcing $\mathcal{F}\left(t\right)$ to construct a correct PE for netTOA(t). Suppose we have such data for a forcing $\mathcal{F}\left(t\right)$, which increases monotonically until $t=s$ and stabilizes at $\mathcal{F}\left(s\right)$ thereafter. Let us keep in mind that the interesting feature of the Gregory plot is that it shows the dependence of the feedback radiation on temperature, so that its local slope yields the feedback parameter. We need to generalize $N\left(T\right)$ from the abrupt step forcing to the more general $\mathcal{F}\left(t\right)$ such that this property is preserved.

Since $T\left(t\right)$ is monotonically increasing, there is a one-to-one mapping $t\leftrightarrow T$, such that the function $t\left(T\right)$ is well defined. Recall that $F_{\mathrm{T}\mathrm{O}\mathrm{A}}$ is defined as the net incoming radiation flux, which means that it is positive if it is directed downward. The feedback radiation $F_{\mathrm{f}\mathrm{b}}\left(T\right)$ is normally directed upwards, so we define it as positive if it represents net outgoing radiation caused by increase in surface temperature. The effective feedback parameter is defined as,

$${\mathsf{\lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}(T)={\displaystyle \frac{d{F}_{\mathrm{f}\mathrm{b}}\left(T\right)}{dT}}.$$

(4)

The CO₂ forcing $\mathcal{F}\left(t\right)$ introduces an additional incident radiation such that

$$F_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(T\right)=\mathcal{F}\left(t\left(T\right)\right)-F_{\mathrm{f}\mathrm{b}}\left(T\right)$$

(5)

Now let us define a generalized $N\left(T\right)$ as,

$$N\left(T\right)=\mathcal{F}\left(s\right)-F_{\mathrm{f}\mathrm{b}}\left(T\right)=\mathcal{F}\left(s\right)-\mathcal{F}\left(t\left(T\right)\right)+F_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(T\right),$$

(6)

which has the desired property that ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)=-dN/dT$, and it reduces to the standard $N^{\mathrm{a}\mathrm{b}\mathrm{r}4\mathrm{x}}\left(T\right)=F_{\mathrm{T}\mathrm{O}\mathrm{A}}^{\mathrm{a}\mathrm{b}\mathrm{r}4\mathrm{x}}\left(T\right)$ for abrupt, step-like forcing. In the case of 1pct4x forcing, $\mathcal{F}\left(s\right)={\mathcal{F}}_{4x}$, and $\mathcal{F}\left(t\right)$ increases linearly up to ${\mathcal{F}}_{4x}$ during the first $s=140$ years and is kept constant thereafter;

$${\mathcal{F}}^{1\mathrm{p}\mathrm{c}\mathrm{t}4\mathrm{x}}(t)={\mathcal{F}}_{4x} [(t/s) \theta (s-t)+\theta (t-s)],$$

(7)

where $\theta \left(t\right)$ is the unit step function. The generalized Gregory plot is created by generating the lists $\mathrm{G}\mathrm{M}\mathrm{S}\mathrm{T}\left(t\right)$ and ${N\left(t\right)=\mathcal{F}}_{4x}-\mathcal{F}\left(t\right)+\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}\left(t\right)$ from the CCM records with annual resolution and plotting the points $\left(\mathrm{G}\mathrm{M}\mathrm{S}\mathrm{T}\right(t),N(t\left)\right)$ for $t=1,\dots ,L$. The third-order polynomial fit $N\left(T\right)$ to this plot is shown in Figure 6c for the 1pct4x forcing scenario. The fact that it is not identical to the curve in Figure 6a indicates that the feedback-parameter dependence on T changes as $\mathcal{F}\left(t\right)$ shifts from abr4x to 1pct4x. The definition of $N\left(t\right)$ in Equation (6) implies that $F_{\mathrm{f}\mathrm{b}}\left(T\right)=\mathcal{F}\left(s\right)-N\left(T\right)$, i.e., the outgoing feedback radiation $F_{\mathrm{f}\mathrm{b}}\left(T\right)$ is the distance from the curves in Figure 6a–c to the top of the frame at $N\left(0\right)=\mathcal{F}\left(s\right)$. This distance is smaller in Figure 6c than in Figure 6a, indicating weaker radiation feedback at the same temperatures in the 1pct4x scenario than in the abr4x case. One possible contribution to this could be that there has taken more time for snow and ice to melt in the 1pct4x scenario, and hence a stronger reduction in the surface albedo at a given temperature. This results in weaker albedo feedback in the 1pct4x than in the abr4x scenario.

Figure 6d–f show ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)$ for the three simulations, which are just the derivative of the curves in Figure 6a–c. By inserting the three-box PE result for T(t) obtained for the abrupt4x simulations in GISS and CESM, we find ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(t\right)$ plotted in Figure 6g,h, and by inserting the T(t) for the GISS 1pct4x, we find ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(t\right)$ plotted in Figure 6i. The difference between Figure 6g,i demonstrates that the forcing history during the first 140 years has implications for the evolution of the feedback parameter for this period and the centuries that follow.

One can observe the implications for the construction of a PE for netTOA as follows: By rewriting Equation (6),

$$F_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(T\right)=N\left(T\right)-\mathcal{F}\left(s\right)+\mathcal{F}\left(t\left(T\right)\right),$$

(8)

We have a PE for the net TOA flux density provided we already have established a PE for the temperature $T\left(t\right)$, and this function is monotonic such that the inverse t$\left(T\right)$ is well defined. Unfortunately, use of this equation requires an estimate of $N\left(T\right)$, which makes use of the netTOA(t) data from the $\mathcal{F}\left(t\right)$ run of the CCM we want to emulate, while it would be nice to emulate it by using only data from the abr4x run. One way around this obstacle could be to make the approximation of replacing the true feedback to the $\mathcal{F}\left(t\right)$ forcing by the feedback to the abr4x forcing. Then, Equation (8) reduces to

$$F_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(T\right)=F_{\mathrm{T}\mathrm{O}\mathrm{A}}^{\mathrm{a}\mathrm{b}\mathrm{r}4\mathrm{x}}\left(T\right)-\mathcal{F}(s)+\mathcal{F}(t(T)),$$

(9)

where $T\left(\mathrm{t}\right)\leftrightarrow t\left(T\right)$ has already been established from the abr4x data. Everything on the right-hand side of Equation (9) is now determined from the abr4x simulation, and we have PEs for both GMST and netTOA for any weakly monotonic forcing $\mathcal{F}\left(t\right)$ for which $\mathcal{F}\left(t\right)=\mathcal{F}\left(s\right)$ for $t>s$, which are based only on data from the abr4x simulation. We will show in Section 3 that for netTOA, this PE is inaccurate for $t<s$, but it works well for $t>s.$

The fitted regression curves in Figure 5 and Figure 6 are presented without error bars. This will also be the case for fitted curves in the remainder of the paper. The fitting method is the standard least-square method for which the 95% confidence range is easily estimated under standard assumption of independent and Gaussian internal climate noise. These assumptions are not satisfied; hence, such estimates cannot be trusted. More important is that the visual fit in most cases is so compelling that error estimates are unnecessary.

3. Results

3.1. The Three-Box Model

The conceptual k-box model is illustrated in Figure 7 for the case $k=3$. It models a slab-like water–planet consisting of k ocean layers (boxes) in thermal contact.

The temperature of the upper layer is the surface temperature T(t). The heat capacities of the boxes are $C_i, i=1,\dots ,k$, and the heat conduction coefficient between box i and box i +1 is ${\eta }_i$. The incoming radiation flux, the forcing, is $\mathcal{F}\left(t\right)$, and the outgoing flux is $\mathsf{\lambda }T\left(t\right)$, where l is the feedback parameter. The equations for this idealized system for $k=3$ are:

$$C_1{\displaystyle \frac{d}{dt}}T\left(t\right)=F(t)-\lambda T\left(t\right)-{\mathsf{\eta }}_1\left[{\mathrm{T}}_2\left(\mathrm{t}\right)-\mathrm{T}\left(\mathrm{t}\right)\right],$$

(10)

$${ C}_2{\displaystyle \frac{d}{dt}}{T}_2\left(t\right)={\eta }_1\left[T\left(t\right)-T_2\left(t\right)\right]-{\eta }_2\left[T_2\left(t\right)-T_3\left(t\right)\right],$$

(11)

$$C_3{\displaystyle \frac{d}{dt}}{T}_3\left(t\right)={\eta }_2\left[T_2\left(t\right)-T_3\left(t\right)\right],$$

(12)

with obvious generalization to a system of arbitrary positive integer $k$. It was shown in [13] that this linear system (for arbitrary $k)$ has a solution for the surface temperature on the form Equation (1), provided $\lambda$ is independent of time. For the abrupt 4xCO₂ simulations, the radiative forcing has the form $\mathcal{F}\left(t\right)={\mathcal{F}}_{4\mathrm{x}}\theta \left(t\right)$, and the surface temperature takes the simple form,

$$T\left(t\right)=\sum _{i=1}^3{T}_i{ (1-e}^{-t/{\mathsf{\tau }}_i}),$$

(13)

where $T_i={\mathcal{F}}_{4\mathrm{x}}{S}_i$ and ${\tau }_i$ are six model parameters that are estimated by least-square fitting to the temperature data record from the CCM. Note that we do not need to know the forcing strength ${\mathcal{F}}_{4\mathrm{x}}$ to establish the PE fitted to this CCM run. We only use that the forcing is a step function. However, if we want the sensitivities ${ S}_i=F_{4\mathrm{x}}/T_i$, or the netTOA, we need ${\mathcal{F}}_{4\mathrm{x}}$. For the 1pct4x forcing scenario, it follows from Equation (6) (using that ${\mathcal{F}\left(s\right)=\mathcal{F}}_{4\mathrm{x}}$, $\mathcal{F}\left(0\right)=0,{ F}_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(0\right)=0),$ that ${\mathcal{F}}_{4\mathrm{x}}=N\left(0\right).$ Thus, in all forcing scenarios, ${\mathcal{F}}_{4\mathrm{x}}$ is the ordinate where the generalized Gregory plot intersects the vertical axis.

The forcing ${\mathcal{F}}_{4\mathrm{x}}=N\left(0\right)\approx 10$ Wm⁻² obtained in Figure 6a–c is considerably greater than the estimates ${\mathcal{F}}_{4\mathrm{x}}=6.0\pm 1.8$ Wm⁻² for GISS and ${\mathcal{F}}_{4\mathrm{x}}=7.0\pm 1.8$ Wm⁻² for CESM derived in Section 2.2 from the jump in monthly netTOA at the onset of the abrupt forcing. It is also greater than the values that can be inferred from the linear fits to the first 150 years in the Gregory plots in Figure 5. This inconsistency reflects an inherent ambiguity in how to distinguish between forcing and feedback. By insisting on a constant feedback parameter for the first 150 years, as depicted by the red lines in Figure 5c,f,i, one attributes the positive deviation in observed netTOA from that line to something different from linear feedback. From Figure 5b,e,h, it appears that this “something” arises from changes in albedo that are not linearly dependent on the GMST. On the other hand, our excellent third-order polynomial fits to the Gregory plots demonstrate that the netTOA can be consistently and accurately attributed to the combined effect of a higher forcing parameter ${\mathcal{F}}_{4\mathrm{x}}$ and a temperature-dependent feedback parameter ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)$.

3.2. Three-Box PE Fitted to Millennium-Long abr4x GMST Data

Figure 8a shows the three-box fit to the entire 5000 yr GMST time series from the abrupt 4xCO₂ run in the GISS-E2-R model, and Figure 8b displays the same for the CESM104 model. Figure 8d,e show the fits of the three-box model to the netTOA flux density using Equation (3) established from the third-order polynomial fits to the Gregory plots in Figure 6a,b. Figure 8c presents the fit $T\left(t\right)$ to the 1pct4x run obtained by performing the convolution integral in Equation (1) with the parameters of the integration kernel $G\left(t\right)$ obtained from the abr4x run. The forcing $\mathcal{F}\left(t\right)$ is assumed to be linearly increasing up to ${\mathcal{F}}_{4\mathrm{x}}$ during the first 140 years, and then constant after that time. This is what is expected if CO₂ concentration increases by 1 pct per year, and the forcing depends logarithmically on concentration. It is important to keep in mind that there is no new estimation of the model parameters $T_i,{\mathsf{\tau }}_i$ based on the CCM data for the 1pct4x run. The parameters estimated from the abr4x run are used, and the excellent fit to the 1pct4x run supports the conjecture that the PEM parametrised by one forcing scenario will provide a good description of CCM runs for arbitrary scenarios by application of Equation (1).

For the PE modelling of netTOA, I use Equation (3) with $T\left(t\right)$ estimated as described above and $N\left(T\right)$ from the Gregory plots in Figure 6a–c. The result is shown as blue curves in Figure 8d–f. As mentioned in Section 2, a caveat of creating a PE for the 1pct4x simulation, using the generalized Gregory plot in Figure 6c, is that it uses data from the CCM 1pct4x run. It could be desirable to emulate this run without having to perform another computationally demanding simulation in addition to the abr4x run. The approximate PE for netTOA obtained by substituting the “true” $N\left(T\right)$ shown in Figure 6c by the one obtained from the abr4x simulation shown in Figure 6a is shown as a green curve in Figure 8f. It underestimates the netTOA for $t<s$ because $N\left(T\right)$ obtained from abr4x simulation overestimates the feedback radiation $F_{\mathrm{f}\mathrm{b}}\left(T\right)=\mathcal{F}\left(s\right)-N\left(T\right)$, as explained in Section 2.6 in relation to Figure 6a,c.

3.3. The CCM Data Reveal Three Distinct Timescales

The fact that the three-box PE yields such good fit to the CCM temperature and TOA flux data for millennium-long runs strongly suggests that the global climate system responds on three distinct timescales. These timescales are presented in

Table 2. Response times t_m and new equilibrium temperatures T_m for the three boxes fitted to 5000 abr4x run in GISS-E2-R and 5900 yr abr4x run in CESM104.

	t1 (yr)	t2 (yr)	t3 (yr)	T1 (K)	T2 (K)	T3 (K)
GISS E2 R	1.0	83	800	2.4	1.0	1.4
CESM104	3.5	170	2500	3.4	2.2	1.1

for the GISS-E2-R and CESM104.

The three response times are 2–3 times greater in the CESM model compared to GISS, but for both, the distinct time scales are years, centuries, and millennia, and one observes that the millennium time scale contributes an additional warming of 1.1 K in GISS and 1.4 K in CESM, or an additional contribution to equilibrium climate sensitivity of about 0.5 K and 0.7 K, respectively.

The physical significance of the three timescales is more apparent if one plots the evolution of the increase in the climate system energy content per m² of earth surface, $\mathrm{C}\mathrm{S}\mathrm{E}\mathrm{C}\left(t\right)={\sum }_{i=1}^t\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}\left(i\right),$ as shown in Figure 9.

The grey curves in Figure 9a–c show this evolution for GISS abr4x, GISS 1pct 4x, and CESM abr4x, respectively, and the red curves show the integrated PE fits. The excellent fits demonstrate the goodness of the PEs. Panels (d–f) show the CSEC as a function of T. The grey points are $\left(\mathrm{G}\mathrm{S}\mathrm{M}\mathrm{T}\right(t),\mathrm{C}\mathrm{S}\mathrm{E}\mathrm{C}(t\left)\right)$ for $t=1\dots ,L$. They constitute a kind of Gregory plot for the accumulated TOA flux. The red curves are interpolations of the points ($T\left(t\right),{\int }_0^t{F}_{\mathrm{T}\mathrm{O}\mathrm{A}}\left(t^{\prime }\right)d{t}^{\prime })$ constructed for all years in the data record and represent the fitted PE. The curves in panels (d, e, f) expose three temperature regimes with different slope of the curves, $C_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)=d \mathrm{C}\mathrm{S}\mathrm{E}\mathrm{C}/dT.$ This slope can be perceived as an effective heat capacity of the climate system [12] and describes the amount of energy (predominantly ocean heat) that will be stored in the system if the surface temperature increases by one degree. It is natural to interpret this quantity as measuring the size of the portion of the ocean mass that is effectively being heated by the increased TOA radiation influx, or in other words, the activation of a surface ocean layer, a middle layer, and a deep layer. It is indicated by arrows in panels (d–f) the temperatures and times at which the regime shifts take place. They are also given in

Table 3. The times (t_A, t_B) and temperatures (T_A, T_B) of regime shift in energy uptake, and effective heat capacities C_eff ^(A), C_eff ^(B), and C_eff ^(C) in the corresponding regimes.

	tA (yr)	tB (yr)	TA (K)	TB (K)	Ceff (A) (W yr m−2K−1)	Ceff (B) (W yr m−2K−1)	Ceff (C) (W yr m−2K−1)
GISSabr4x	8	160	2.5	3.5	32	290	780
GISS1pct4x	40	310	3.0	3.7	82	420	750
CESMabr4x	15	500	3.5	5.7	18	230	1570

along with estimates of the slopes $C_{e\mathrm{f}\mathrm{f}}\left(T\right)$ in the three regimes.

3.4. Why Three Boxes and Millennium Timescales Are Necessary

The standard simulation length in the CMIP protocols is 150 years, but the data contained in the LongRunMip protocol demonstrate that CCMs in general need several millennia to equilibrate. For CCM runs of only a few centuries length, a two-box model was shown to work well for a large ensemble of CMIP5 in [8]. Here, the necessity of longer simulations and more boxes is demonstrated by employing two- and three-box PEs fitted to the first 150 yr segment and to the full 5–6 millennia of the CCMs treated above. The goal is to demonstrate why the three-box model fitted to CCM simulations over several millennia seems to be an optimal choice if the focus is on global temperature and equilibrium climate sensitivity.

In Figure 10, data from the long GISS simulation are used to illustrate why two boxes are inadequate to describe the global response on millennium time scale and why 150 years of CCM data do not provide sufficient information for two-box nor three-box models to capture the longer-timescale dynamics of the CCMs. But the figure also shows that the three-box PEM is adequate if the PEM is fitted to the full CCM record. Similar results are shown for CESM and ECHAM in Supplements SA and SB, respectively.

3.5. A Fourth Box Explains the 1st Year in abr4x Runs

For realistic forcing scenarios, there is no need to increase the PE complexity, for instance, by introducing a fourth box. Attempts at doing this do not improve the mean-square error, but they do provide a perfect fit for the 1st year after abrupt forcing by adding a very short response time, as shown in Figure 11.

Table 4. Response times t_i and new equilibrium temperatures T_i for the four boxes fitted to 5000 abr4x run in GISS-E2—and 5900 yr abr4x run in CESM104. The new box with the very short response time has parameters t₀ and T₀.

	t0 (yr)	t1 (yr)	t2 (yr)	t3 (yr)	T0 (K)	T1 (K)	T2 (K)	T3 (K)
GISS-E2-R	0.21	1.7	85	800	0.9	1.6	0.9	1.4
CESM104	0.16	5.4	180	2600	1.0	2.4	2.2	1.1

summarises the parameters of the four-box model for abr4x runs. It should be compared to those of the three-box model in Table 2. I name the addition “box0” because the parameters of box1 is only slightly modified, and those of box2 and box3 are essentially unaltered.

This short response is unimportant in smoother anthropogenic forcing scenarios like the 1pct4x scenario, as shown in Figure 8c. There, the fit T(t) for the three-box model is perfect even in the 1st decade, and there is no need to increase the model complexity by introducing a fourth “box”. It is reassuring that the four-box model improves the fit on the short time scales in the case of abrupt forcing and only makes small modification of the parameters of the other three boxes. Thus, while two-box PEs fitted to the entire temperature record give a poor fit up to 150 years, the three-box model restricts the poor fit to the 1st decade, and introduction of four boxes corrects this mismatch also. This indicates that there is no overfitting; the number of PE parameters (six or eight) is necessary to provide good fit on all relevant time scales. It also makes physical sense, as we shall see in the next section.

4. Discussion

4.1. The Four Timescales in the Temperature Response

In the multi-box PE, the response to an abrupt forcing is modelled as a superposition of exponential relaxations (“boxes”) to a quasi-equilibrium temperature $T_i$ of the form $T_i\left(1-e^{-t/{\tau }_i}\right)$. Since the response scales ${\tau }_i$ are separated by at least one order of magnitude, each box is clearly discernible in the PEM temperature evolution curve, as shown in Figure 12.

The figure shows the parts of the four-box PE solution that appear when summing the contributions from the m first “boxes” (response times), with $m=\mathrm{0,1},\mathrm{2,3},$ respectively. It gives an impression of the contribution to the total temperature change arising from adding more slowly responding layers of the ocean. The blue curve shows that there is a rise that stabilizes at $T_0\approx 1.0$ K during the 1st year in both models. This is the response of the atmosphere and the combined land and ocean. The orange curve shows the result of adding the contribution from the next box, which stabilizes during the 1st decade in GISS at the temperature $T_0+T_1=2.5$ K and somewhat later at $T_0+T_1=3.4$ K in CESM, using the values for $T_0,T_1$ given in Table 3. This can be interpreted as the addition of the delayed response from the ocean mixed layer. The green curves result from adding yet another box, with temperature stabilizing at $T_0+T_1+T_2=3.4$ K after 2–3 centuries in GISS and at $T_0+T_1+T_2=5.6$ K after 1 millennium in CESM. Adding the last box (red curve) gives stabilization at $T_0+T_1+T_2+T_3=$ 4.8 K after 2–3 millennia in GISS, and at 6.7 K after almost 10 millennia in CESM.

The initial box0 response is similar in the two CCMs, but the responses from box1 and box2 are slower and stronger in CESM than in GISS. These are time scales of considerable interest for anthropogenic global warming, from a few years to 1 or 2 centuries in GISS and from a decade to several centuries in CESM. The final box, which determines the time-asymptotic equilibrium temperature, tells us that the succeeding centuries and millennia will add another 1.4 K to the global temperature in GISS and 1.1 K in CESM.

4.2. The Ocean Heat Uptake Is Not Determined by the Surface Temperature

The netTOA multiplied by the area $A_E$ of the earth surface on time scales much longer than annual is essentially the same as the ocean heat uptake (OHU), which means that one can interpret the CSEC multiplied by the area of the earth surface as the anomaly in the ocean heat content (OHC), and the netTOA is a measure of the OHU. Figure 13a shows the full black curves in Figure 6a,b, now in the same panel to display more clearly the difference in OHU(T) between GISS and CESM. There is a growing gap between heat uptakes in the two models as temperature rise increases, and as the GISS temperature stabilises at equilibrium $T_{\mathrm{e}\mathrm{q}}=4.8$ K, $N\left(T_{\mathrm{e}\mathrm{q}}\right)=0,$ the CESM heat uptake is still substantial and leads to a temperature rise, which stabilises at $T_{e\mathrm{q}}=6.7$ K. Figure 13b displays the OHU versus time, $N\left(t\right)=N\left(T\left(t\right)\right)$, but here, there is no growing gap betwen the curves. The heat uptake as a function of time is approximately the same in the two models up to the time when the model with the lower climate sensitivity starts to approach equilibrium.

Figure 14 contains no new information but presents the PE results in a way that corroborates this narrative. Figure 14a shows that T(t) follows roughly the same course in the two models up to 7 years after the abrupt forcing change, when the temperature anomaly in both models has reached 2.5 K. The temperature difference between the two models then increases gradually to 1.6 K after $1000$ years. On the other hand, the anomalies of the accumulated climate system energy content (CSEC) are almost identical up to $1600$ yr, as shown in Figure 14b. This is consistent with Figure 13b, since $\mathrm{C}\mathrm{S}\mathrm{E}\mathrm{C}(t)={\int }_0^t\mathrm{n}\mathrm{e}\mathrm{t}\mathrm{T}\mathrm{O}\mathrm{A}\left(t^{\prime }\right) d{t}^{\prime }$. It suggests that the OHU(t) is determined by internal ocean dynamics triggered by the initial 4xCO₂ forcing, and not by the surface temperature evolution, which is different in the two models.

Inspired by this idea, let us treat the OHU flux as a boundary condition at the bottom of the mixed ocean layer, and the surface temperature change as a rapid response to restore radiative balance of the “box 0+1” comprising the atmosphere, surface, and the mixed layer. Hence, if OHU(t) is “predetermined” by the ocean dynamics, this box must approach a quasi-equilibrium state where the surface temperature is such that the netTOA into the box balances the OHU out of the box, i.e.,

$$A_{\mathrm{E}} {\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}(T){\displaystyle \frac{dT}{dt}}=-{\displaystyle \frac{d\mathrm{O}\mathrm{H}\mathrm{U}\left(t\right)}{dt}}$$

(14)

If ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)$ is given as a third-order polynomial in T from the Gregory plot, and OUH(t) is a function of t, which is common among the CCMs under study, then T(t) is the solution to this nonlinear first-order ordinary differential equation. It shows explicitly a surface temperature rate of change, which is inversely proportional to the effective feedback parameter, which in turn is determined by how longwave and shortwave outgoing fluxes respond to surface temperature change. What distinguishes the two CCMs is then the physics of the atmosphere, land surface, and mixed ocean layer. This will be discussed further in Section 4.3.

The temperature evolution in both models exhibits a plateau in middle time ranges, which starts at point A in Figure 14. The start of these temperature plateaus is found at the break (point A) in the curves at 2.5 K (GISS) and 3.5 K (CESM) in Figure 14c. The plateau corresponds to activation and quasi-equilibration of boxes 0+1 in Figure 12. The CSEC(T) curves in Figure 14c start to bend upwards at point A. The effective climate system heat capacity $C_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)=d\mathrm{C}\mathrm{S}\mathrm{E}\mathrm{C}/dT$ makes an upward jump at point A and indicates that deeper water masses are beginning to take up heat with less effect on surface temperature. The activation of boxes 0+1+2 occurs at the next breaks (point B) in the curves in Figure 14c; around temperature 3.5 K, corresponding to $t=160$ yrs in GISS, and 5.7 K and 500 yrs in CESM. Further on, both temperature and energy content start converging towards the final equilibrium as all four boxes 0+1+2+3 are activated. In this final stage (after point C), $C_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(T\right)$ is approximately constant (the curves in Figure 14c are almost straight), indicating that all heat reservoirs in the climate system have been activated at 10⁴ years.

4.3. Effects of Clouds and Surface Albedo on ECS

Figure 6 gave us a first hint of the main cause of the different climate feedback parameters, hence the climate sensitivities between the GISS and the CESM models. The latter presents a consistently falling OSR throughout the entire simulation due to decreasing albedo. The OLR must increase to compensate decreasing OSR, such that OLR + OSR $\to 0$ as equilibrium is approached after several millennia. These features may be linked to the higher equilibrium temperature and ECS in CESM as compared to GISS, but the linkage is not completely straightforward.

In addition to the all-sky OLR and OSR, the LongRunMip repository contains data for the clear-sky OLR and OSR, defined as the radiation that would take place if all clouds were removed in the instantaneous radiative transfer calculation, with everything else kept constant. Figure 15 presents the evolution in time of these clear-sky variables, OLRcs and OSRcs, along with the previous all-sky, OLR and OSR. The figure also shows the resulting cloud radiative effects, $\mathrm{O}\mathrm{L}\mathrm{R}\mathrm{c}\mathrm{r}\mathrm{e}=\mathrm{O}\mathrm{L}\mathrm{R}-\mathrm{O}\mathrm{L}\mathrm{R}\mathrm{c}\mathrm{s}$ and $\mathrm{O}\mathrm{S}\mathrm{R}\mathrm{c}\mathrm{r}\mathrm{e}=\mathrm{O}\mathrm{S}\mathrm{R}-\mathrm{O}\mathrm{S}\mathrm{R}\mathrm{c}\mathrm{s}$.

In the GISS model, OLRcre is rather small throughout the entire simulation, decaying from about +0.5 to $-0.5$ Wm⁻², indicating that the clouds have a relatively small effect on the longwave radiation. The OSRcre, however, increases by about 1.5 Wm⁻² during the 1st decade, and then by another 1.5 Wm⁻² throughout the next 5 millennia. This increased cloud albedo more than compensates for the drop in surface albedo depicted by the OSRcs curve. The net albedo, shown by the OSR curve, grows to about 0.7 Wm⁻² after the 1st decade, and it remains on that level throughout the remaining simulation, having a moderate cooling effect.

In CESM, the OLRcre is positive and somewhat larger than in GISS, decaying from about 1.2 Wm⁻² to $0.7$ Wm⁻² throughout the 6 millennia. The OSRcre grows from $-1.0$ Wm⁻² to zero during the 1st decade, and then gradually to about 0.7 Wm⁻² at the end of the simulation. The shortwave cloud effect is relatively small, however, compared to the big drop of about 5 Wm⁻² in OSRcs due to a steady loss of surface albedo over the full course of the simulation, as seen in Figure 15d. This drop in OSR must lead to surface warming and increased longwave feedback until radiation balance is restored. Thus, the surface albedo loss is linked to the strong rise in OLR, from $-4$ to $+6$ Wm⁻², as seen in Figure 15c, through a positive-surface albedo feedback loop.

The ECHAM model shows OLR and OLRcs with higher growth rate than in CESM. The higher OLR growth is linked to a faster drop in surface albedo and a negative shortwave cloud radiative effect, which, when combined, yield a faster reduction of OSR (blue curve in panel (f)).

The polynomial fits to the Gregory plots for netTOA shown in Figure 6 are complemented with corresponding plots OLR, OSR, OLRcs, and OLRcs in Figure 16. Analysis of these plots and Figure 15 provide some insight into the overall mechanisms, which cause the large differences between the long-term evolution of CCMs, here exemplified with GISS, CESM, and ECHAM.

Consider first with the red OLRcs curves in the right-hand panels. They depict the OLRcs feedback to T in the three models when the radiative effect of clouds has been removed. We observe that the Gregory curves are quite straight and the slopes in the range $1.5 - 2.0$ Wm⁻². These plots extract the primary greenhouse effect, devoid of surface (including aerosol) albedo feedback and cloud longwave and shortwave feedback. If there were no clouds and no reflection of incoming shortwave radiation from the surface, these would be the effective Gregory plots, and the intersections of the OLRcs curves with the horizontal axis would represent the new equilibrium temperatures. Multiplied by the factor ½, this would yield estimates for the ECS; 2.5 K in GISS, 2.0 K in CESM, and 2.4 K in ECHAM (first column in

Table 5. Shows values of hypothetical ECS = T/2, where T is the point where curves in Figure 16 cross the T-axis. The first column presents the ECS values obtained as solution to OLRcs(T) = 0, which can be perceived as the ECS if no clouds and surface albedo change were present, i.e., the pure greenhouse effect. Second column: ECS given by OLR(T) = 0; no cloud or surface albedo, but effect of clouds on longwave radiation. Third column: ECS given by netTOAcs = 0; clear sky but presence of surface albedo changes. Fourth column: ECS given by netTOA(T) = 0; presence of clouds and surface albedo change, i.e., the actual ECS. The percentages in brackets are percent change of ECS relative to the first column.

	OLRcs = 0: Clear Sky, No Surface Albedo Change	OLR = 0: No Albedo Change, Longwave Cloud Effect	netTOAcs = 0: Clear Sky, Surface Albedo	netTOA = 0: Clouds, Surface Albedo
GISS	2.5	2.7 (+8%)	3.5 (+40%)	2.4 (−4%)
CESM	2.0	1.7 (−15%)	4.0 (+100%)	3.4 (+70%)
ECHAM	2.4	1.9 (−20%)	4.6 (+92%)	6.0 (+150%)

).

The red OLR curves in the left panels include the effects of clouds on the longwave radiation, but still no reflected radiation, i.e., no cloud and surface albedo. The corresponding ECS values are 2.7 K for GISS, 1.7 K for CESM, and 1.9 K for ECHAM (second column in Table 5), hence a rather small influence on ECS from cloud radiative effect on longwave radiation.

The blue OSRcs curves in the right panels show the effect of surface albedo on shortwave radiation (no clouds), and the difference between the black netTOAcs curves and the red OLRcs curves in the right panels show the effect of surface albedo on ECS when the sky is clear. The ECS with surface albedo, but no clouds (third column in Table 5), are 3.5 K for GISS, 4.0 K for CESM, and 4.6 K for ECHAM. The surface albedo change with a clear sky thus leads to 40% increase of ECS for GISS, 100% increase for CESM, and 92% increase for ECHAM (the relative change from the first to the third column).

Introduction of effects of cloud and surface albedo change on shortwave radiation, as well as cloud effects on longwave radiation, leads to the black curves for the actual netTOA in the left panels and to the actual ECS values; 2.4 K in GISS, 3.4 K in CESM, and 6.0 K in ECHAM (fourth column in Table 5). The total effect of clouds when surface albedo change is present is the difference between the black curves in the left and right panels, and it reduces the ECS by 31% for GISS and by 15% for CESM but increases ECS by 30% for ECHAM (the relative change from the third to the fourth column).

4.4. Effect of Pattern Formation in Ocean Heat Uptake on the Feedback Parameter

Pattern formation in ocean heat uptake has been shown to influence the climate feedback and lead to a reduction in the feedback parameter with time. Rugenstein et al. [14] employ the CESM104 model to explore the mechanisms behind the rapid, and then slower, reduction of the effective feedback parameter with time in abrupt 4xCO₂ runs observed in most CMIP5 models (see Figure 6g,i). Their basic idea is that a rapidly developing spatial pattern in the ocean heat uptake evolves over the first few centuries of the simulation, and that the global feedback will depend crucially on this pattern. Their approach is to use the coupled ocean–atmosphere–sea–ice–land CESM104 model to produce the pattern of ocean heat uptake in the form of the geographical distribution of the Q-flux, which is the downward flux at the bottom of the ocean mixed layer. They then use this Q-flux at different times after the abrupt forcing onset as the lower boundary condition on a simulation of a slab-ocean version of the CESM104 model. By running the slab-ocean model to equilibrium, they estimate a feedback parameter associated with the uptake pattern. The patterns at 1, 5, and 20 decades yield progressively lower l, thus explaining (at least qualitatively) the decreasing global feedback parameter during the two first centuries. The authors also decompose the global feedback for these three times into OLRcs, OLRcre, OSRcs, and OSRcre. The result is that OLRcs and OSRcs feedback parameters are the same for all three times, while those for OLRcre and OSRcre decrease by approximately equal amounts (Figure 3c in [14]). Qualitatively, this agrees with Figure 16c,d; the temperature in CESMabr4x is 3.2 K at 1 decade, 3.8 K at 5 decades, and 5.0 K at 20 decades. The slope of the OLRcs and OSRcs curves in this temperature range in Figure 16d does not change noticeably, hence all changes of the slopes in Figure 16c are due to cloud radiative effects.

Huang et al. [15] study the long-term evolution of spatial patterns of GMST and precipitation in CCMs, and among these, GISS-E-R and CESM104. During the 3rd century after abr4x, the spatial GMST patterns in the two models correlate less than during the final century of the millennium simulations, indicating that the transient dynamics is rather different in the two models. The GMST pattern correlation coefficient between a given century and the final century in the same model is lower in GISS than in CESM for the first two millennia, after which both converge towards unity (see Figure 13 in [15]). This indicates that it takes longer for GISS to reach a stable surface temperature pattern. The land–ocean contrast (LOC) is defined as 60° S–60° N mean land surface air temperature change (relative to pi-control) divided by 60° S–60° N mean ocean temperature change. The LOC falls gradually in both models, but more slowly in GISS (from 1.8 to 1.4 during the first two millennia) than in CESM (from 1.5 to 1.3 during the first millennium), which may be the main explanation for the differences in evolution of pattern correlation. Another difference between the models is the evolution of the arctic amplification and the sea–ice cover. In GISS, the arctic amplification falls from 2.5 to 1.8 during the first 2 millennia, while the sea–ice cover rapidly stabilizes around 40%. In CESM, the arctic amplification fluctuates around 2.4 and the sea ice cover falls gradually to a level of 10% at the end of the 6000 yr-long simulation. High arctic amplification is consistent with low sea–ice cover, which may explain the declining OSR observed in CESM and shown in Figure 4b and Figure 15d.

4.5. Other Approaches That Use Box Models

In the CCMs, TOA flux density and surface temperature is of course spatially distributed over the earth surface, and the local heat flux density at the ocean surface (the ocean heat uptake) is not necessarily the same as the local TOA flux density. The land surface is also heated by the TOA flux, and heat is transported by atmospheric circulation from land to sea and from low to high latitudes. Moreover, there are evolving patterns of high/low ocean heat uptake, which have been used to explain the evolution of the global netTOA and the changing feedback parameter observed in the Gregory plots. In zero-dimensional energy-balance models like the three-box model Equations (10–12), this and other effects have been modelled by the introduction of an “ocean heat uptake efficacy” factor e [7]. In the three-box model, this parameter changes the heat flux out of box2 towards box3 in Equation (11) by the factor e, i.e., to ${\mathsf{\epsilon }\mathsf{\eta }}_2\left[{\mathrm{T}}_2\left(\mathrm{t}\right)-{\mathrm{T}}_3\left(\mathrm{t}\right)\right]$, and energy conservation then requires, everything else unchanged, that the outgoing TOA flux is changed from $\mathsf{\lambda }\mathrm{T}$ to $\mathsf{\lambda }\mathrm{T}+(\mathsf{\epsilon }-1){\mathsf{\eta }}_2\left[{\mathrm{T}}_2\left(\mathrm{t}\right)-{\mathrm{T}}_3\left(\mathrm{t}\right)\right]$ in Equation (10). The efficacy offers an additional PE parameter, but there is no obvious physical interpretation in a model with more than two boxes.

Cummins et al. [9] fit this model to the combination of GMST(t) and netTOA(t) data of the CCMs, while considering that the forcing may exhibit stochastic components in addition to the deterministic CO₂ quadrupling. The philosophy is the same as in [16], but while the latter assumes a white noise to represent the weather forcing on the ocean and replace the multibox response kernel with a power-law response, Cummins et al. [9] assume a red-noise stochastic TOA-forcing component in addition to an internally generated white noise, with all this applied to a multi-box model with an efficacy factor. Thus, the ambition in that paper is to establish a PE that incorporates internal variability of the global GMST and netTOA. For this purpose, more advanced parameter estimation procedures than the primitive least-square fitting used in the present paper must be employed. They conclude that three or more boxes are needed, which agrees with the results shown in Figure 10 of the present paper.

Wells et al. [10] employ the three-box model Equations (10-12) with the efficacy factor included, but without the stochastic forcing employed in [9]. They fit this model to the combination of GMST(t) and netTOA(t) data from the LongRunMip repository for varying length of the fitting time series. They find that the PE predictions on long time scales improve with increasing fitting interval, even when these exceed 1000 years. This agrees well with the findings in the present paper. The model in [10] exhibits $2k+2$ model parameters; three heat capacities, two heat conductivities, the forcing strength ${\mathcal{F}}_{4x}$, the (time-independent) feedback parameter $\lambda$, and the efficacy factor $\epsilon$, and it requires data for both temperature and net top-of-the-atmosphere flux. The changing slope of the Gregory curve $N\left(t\right)$ is a result of the additional term ${(\epsilon -1) \eta }_2\left[T_2\left(t\right)-T_3\left(t\right)\right]$ in Equation (10), resulting from an efficacy factor $\epsilon \ne 1$.

The emulators described in the present paper exhibit only $2k$ parameters and use only GMST data to establish an emulator for $T\left(t\right)$. To extend the emulation to encompass the energy flux by means of the netTOA data, these are combined with the GMST data through the Gregory plot, and the polynomial fit to this plot allows an analytic emulation of the time-dependent effective feedback parameter ${\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(t\right)$. If a third-order polynomial fit to the Gregory plot is used (a second-order polynomial yields slightly poorer fits), the total number of parameters is $2k+3.$ More important is that for most practical applications, we are only interested in the surface temperature evolution, and then the simple temperature model Equation (13), the GMST time series, and six model parameters (three-box) is all we need.

The direct interpretation of the efficacy factor $\epsilon >1$ employed in [9,10] is that there is a channel by which more heat is lost from the box $k-1$ than received by heat conduction from the box above and lost to the bottom box $k$. Energy conservation requires that his energy is lost to space as outgoing radiation, but the physical mechanism is not clear. While the addition of this parameter obviously can do the job of creating a break in the Gregory curve that provides better fits to the data, the approach employed in the present paper is a more direct construction for this purpose, and the temperature-dependent effective feedback factor can be interpreted by considering the data for OSR and OLR.

5. Conclusions and Outlook

The main objective of this paper has been to establish a method by which one can establish a very simple analytic parsimonious emulator (a PE) for the global temperature evolution over millennia for any reasonable forcing scenario. The model parameters can be estimated by using data from one single millennium-long run of a given complex climate model (CCM). The PE is specific for the given CCM, but the conjecture is that it is valid for any forcing scenario that stabilises at constant level after a transient period of a few centuries. In addition to the GISS model, there is only one other CCM in the LongRunMip repository that contains both abr4x and 1pct4x scenarios. This is the ECHAM5/MPIOM, which contains a 1000 yr abr4x run and a 6080 yr 1pct4x run. I have analysed this model in the same way as the GISS-E2-R and CESM104 models. Figure 17 shows that it supports the conjecture mentioned above.

A remarkable feature observed in Figure 17a is that the three-box GMST fit to the 1000 yr abr4x simulation saturates at almost the same equilibrium temperature as the 6080yr long CCM 1pct4x simulation; the light-green curve covers the black curve after 1000 yr. It suggests that the three-box PE obtained from 1000 yr of CCM data contains the essential information about the upcoming saturation 5000 yrs into the future.

The extension of the PE to encompass the netTOA radiation flux by creating a polynomial fit to a generalised Gregory plot gives rise to some interesting insights into the distinct stages of the global heat uptake, and by studying the OLR, OSR, and the cloud radiative effect, differences in equilibrium climate sensitivity between the GISS, CESM, and ECHAM models are attributed to differences in cloud radiative effects and change in surface albedo. With a clear sky and no change of surface reflectivity, the ECS for the three models would be in the range of 2.0–$2.5$ K. Longwave cloud effects only increase this range to 1.9–$2.7$ K, a small positive change in GISS, and a modest negative change in CESM and ECHAM. Introduction of surface albedo change with a clear sky shifts the ECS-range to 3.5–$4.6$ K, with the greatest increase taking place in CESM and ECHAM. Adding cloud effects to the state with surface albedo change expands the ECS-range to 2.4–$6.0$ K. Here, cloud albedo increases in GISS and CESM, which causes decreased ECS (Figure 15b,d), while decreased cloud albedo gives rise to a high ECS increase in ECHAM (Figure 15f). Thus, spread in cloud albedo among the complex climate models appears to be the most important source of spread in equilibrium climate sensitivity, while spread in surface albedo change and longwave cloud radiation trapping also contribute, but to a lesser extent.

The PE for the GMST is uniquely represented by the six model parameters for T(t). The emulator can be expanded to include variables for radiation fluxes by estimating parameters for the polynomial fits for the radiation feedback for all sky and clear sky. The PEs are constructed as emulators for given CCMs, but in addition to parsimoniously describing the effect of varying forcing profiles, it is of course possible to play with PE parameters to investigate effects of changing global climate variables without rerunning the full CCMs. An application might be to predict global effects of geoengineering measures.

The PE employed in this paper is not meant to model internal variability, which is why a simple mean square fit to the GMST is an adequate procedure. But because of this simplicity, and the large time scales involved, the deviation of CCM data from the corresponding smooth PE curves can serve as a reasonable definition of internal variability. Separating internal variability from forced “trends” has been a notorious subject of controversy in the global warming debate because opponents to the “global warming hypothesis” have argued that most of the recent global warming may be part of the natural internal variability. One line of argument has been based on the hypothesis that natural variability is subject to so-called long-range memory, which implies a power-law spectral density, $S\left(f\right)~f^{-\beta }$, where f is the frequency and $0<\beta <1$ means that the fluctuations constitute a persistent fractional Gaussian noise (see [17] for a review). A salient feature of the long pi-control runs and the several millennia-long abr 4x simulations is that we can study the internal variability of 2 millennium-long near-equilibrium states corresponding pre- and post-industrial global temperatures. This is a natural theme for a forthcoming paper.