## 1. Introduction

Instrumental measurements and proxy reconstructions of Earth’s surface temperatures show temporal variability on a range of different time scales (Lovejoy 2015; Huybers and Curry 2006). For the global mean surface temperature (GMST), the variability can be parsimoniously described as scale invariant, since the estimated power spectral densities (PSDs) are well approximated by power laws *S*(*f*) ~ 1/*f*^{β} from monthly to centennial scales (Rypdal et al. 2013). The typical scaling exponent is *β* ≈ 1, and the signals are well described as a so-called 1/*f* noise, or pink noise. Some of the low-frequency variability in the temperature records can be accounted for by the variability in the radiative forcing of the planet, but even the residual fluctuations are well described as a scaling stochastic process, with a slightly lower exponent *β*. This suggests that scale-invariant dynamics is an intrinsic property of the climate system, a claim that is supported by the observation of scaling PSDs in unforced control runs of general circulation models (GCMs), on time scales from months to centuries (Fredriksen and Rypdal 2016; Rybski et al. 2008; Fraedrich and Blender 2003).

A signal with power-law PSD can be modeled as a stochastic process with long-range dependence (LRD), and examples of such processes are the fractional Gaussian noises (fGns) and the fractional autoregressive integrated moving average (FARIMA) models. Stochastic processes that exhibit LRD provide more accurate descriptions of the unforced GMST variability compared to the traditional red noise models, such as the Ornstein–Uhlenbeck (OU) processes and the autoregressive processes of order 1 [AR(1)] (Rypdal and Rypdal 2014). The latter are characterized by a single time scale, and are incapable of describing the multiscale nature of the climate fluctuations. Despite this, the LRD processes are largely ignored by many climate scientists, and some consider LRD to be an exotic and redundant notion in climate science (Mann 2011).

One of the aims of this paper is therefore to demystify the notion of LRD in the climate system by demonstrating that the observed phenomena can be produced by simple multibox energy balance models (EBMs). With this, we demonstrate that the exotic physics may be no more than vertical heat conduction in the ocean and that it is reasonable to think of LRD as an approximation to the linear response of EBMs with multiple characteristic time scales. Only a few boxes are needed to obtain power-law PSDs on scales from months to centuries. We also demonstrate how we can construct box models that are consistent with the observation that the exponent *β* is lower for land temperatures than for sea surface temperatures (SSTs) (Fredriksen and Rypdal 2016).

Only a few of the studies that analyze LRD in surface temperatures focus on the mechanisms behind the phenomenon (e.g., Fraedrich 2002; Fraedrich and Blender 2003; Fraedrich et al. 2004; Blender et al. 2006; Franzke et al. 2015). Most treat LRD processes merely as statistical models that fit well with data (Vyushin et al. 2012; Rybski et al. 2006; Franzke 2010). Statistical inference for LRD processes requires special care to avoid the fallacy of circular reasoning: that is, falsely attributing trends in the forced signal to natural variability (Benestad et al. 2016). Incautious trend-significance testing using LRD null models (Cohn and Lins 2005) have led some climate scientists to view LRD processes as exotic mathematical objects that somehow fit with the “climate denier agenda” (Mann 2011; Benestad et al. 2016). This is paradoxical, since climate response models that exhibit LRD actually display more “heating in the pipeline,” and, compared with other response models, they predict that emissions of greenhouse gases must be reduced earlier and more drastically to avoid dangerous anthropogenic influence (K. Rypdal 2016; Rypdal and Rypdal 2014).

Other climate scientists consider scaling to be an inherent property of atmospheric turbulent flows and certain types of regime switching dynamics (Lovejoy and Schertzer 2013; Franzke et al. 2015) and, as such, a signature of the nonlinearity of the underlying dynamics. In fact, Huybers and Curry (2006) hypothesize that the persistent scaling of surface temperatures observed on decadal to multicentennial scales is due to a nonlinear cascade driven by the seasonal forcing. They present a bicoherence spectrum in favor of this hypothesis, but the phase correlations that give rise to high bicoherence do not imply an effective nonlinear energy transfer between the seasonal and the multidecadal scales. We have also had problems in reproducing the bicoherence spectra reported in this paper. In a forthcoming paper, we will examine this hypothesis in depth.

Since the ocean has a large heat capacity compared to the atmosphere, the observation that ocean temperatures are more persistent than land temperatures (Fraedrich and Blender 2003; Fredriksen and Rypdal 2016) is an indication that the observed persistence in global temperature to a larger extent must be attributed to ocean heat content and ocean dynamics and to a lesser extent to nonlinear processes in the atmosphere. This hypothesis is further strengthened by the results of Fraedrich and Blender (2003), who find that only models with full ocean circulation show persistence on scales longer than about a decade. In the present paper, we model the slowly responding components of the climate system by including “boxes” that exchange heat with the more rapidly responding mixed layer. This is clearly an oversimplification of the ocean dynamics but reproduces the multiscale characteristics of the surface temperature response.

The paper is structured as follows. Section 2 discusses the construction of multibox EBMs and their corresponding response functions, and in section 3 we demonstrate how the superposition of different response times can be used to approximate an LRD response. Furthermore, we estimate parameters and explore how the response of sea surface temperatures differs from the response of land temperatures. Section 4 presents some concluding discussions.

## 2. Multibox EBMs

*C*denotes the average heat capacity per square meter of the surface, Δ

*T*is the temperature anomaly relative to an equilibrium state,

*S*

_{eq}is the equilibrium climate sensitivity, and Δ

*F*(

*t*) is the forcing (i.e., the perturbation of effective radiative forcing from the initial equilibrium state Δ

*T*= 0). As a response to a constant perturbation Δ

*F*, the temperature will reach a new equilibrium Δ

*T*, and the change in equilibrium temperature relative to the change in radiative forcing is equal to the equilibrium climate sensitivity: that is,

*F*(

*t*), the temperature Δ

*T*(

*t*) is given by a convolution integral

*τ*=

*CS*

_{eq}:

*C*

_{2}. If the energy exchange between the upper and lower box is proportional to the temperature difference between the two boxes, we obtain what is known as the two-box EBM (Geoffroy et al. 2013; Held et al. 2010; Rypdal 2012; Caldeira and Myhrvold 2013):

*κ*

_{1}= 1/

*S*

_{eq}, but it should be noted that the physical meaning of

*κ*

_{1}is different from

*κ*

_{2}. While

*κ*

_{2}is a coefficient of heat transfer between two ocean layers,

*κ*

_{1}is determined by the linearized response of the net outgoing radiation to changes in the surface temperature. It includes all the atmospheric feedbacks and is sometimes referred to as the equilibrium global climate feedback (Armour et al. 2013), or simply the “feedback parameter.”

The natural generalization of the two-box model is to consider *N* vertically distributed boxes. The model is formulated as in Eq. (6), with Δ**T** and Δ**F** being *N* vectors and *N* × *N* matrices. The matrix **F** consists of zeros for all boxes not connected to the surface. We note that, when *N* is large, this model setup can approximate a vertical diffusion model.

*N*-box EBM the

*N*-vector temperature can be written using matrix-exponential notation:

*N*exponentially decaying functions:

*τ*

_{k}= −1/

*λ*

_{k}for

*k*= 1, …,

*N*, where

*λ*

_{k}are the eigenvalues of the matrix

^{−1}

*λ*

_{k}are real and negative.

The model defined by Eq. (6) is meant to describe vertically distributed boxes, but boxes can also be aligned horizontally. This can be useful in order to include the atmosphere over land in the model. In principle, we can have interactions between all boxes, making the matrix *b*_{k} are changed. Several horizontally distributed boxes could also be useful for modeling a space-dependent depth of the mixed layer.

*T*

_{O,atmos}is proportional to the temperature in the mixed layer (i.e., Δ

*T*

_{O,atmos}=

*α*Δ

*T*

_{1}), where the factor

*α*> 1 describes the effect of changing sea ice cover (Raper et al. 2001). The parameter

*μ*> 1 quantifies the asymmetry in the heat transport between the atmosphere over the ocean and the atmosphere over land. The parameter

*f*

_{L}= 0.29 is the proportion of Earth’s surface that is covered by land, and

*f*

_{O}= 1 −

*f*

_{L}. The term

*F*

_{N}represents the heat transport into the deep oceans, and

*F*

_{L}and

*F*

_{O}are the forcing terms over land and ocean, respectively. From models in phase 3 of the Coupled Model Intercomparison Project (CMIP3), one finds that the typical values of

*μ*are in the range 1–1.4 (Meinshausen et al. 2011). This implies that, when a new equilibrium is reached after a perturbation of the forcing, the land temperature will have changed more than the SST.

*C*

_{L}→ 0, Eq. (9) becomes

## 3. Approximate scale invariance from aggregation of OU processes

*dB*(

*t*) is the measure of white noise. The equation has a stationary solution of the form

*σ*is called the scale parameter, and

*θ*is the damping rate. Since

*dB*(

*t*) is a white noise, it follows that

*τ*= 1/

*θ*. In the multibox EBM with

*N*vertically distributed boxes, the temperature Δ

*T*

_{1}(

*t*) is given by

^{1}

*N*-box models we consider are linear, the decomposition of the forcing yields a straightforward decomposition of the temperature response:

^{−1}

*τ*

_{k}= −1/

*λ*

_{k}. Taking the Fourier transform of Eq. (14) yields

*T*

_{1}(

*t*) becomes

*T*is the length of the time series Δ

*T*

_{1}(

*t*),

*S*

^{(0)}(

*ω*) is the PSD of an independent superposition of the processes Δ

*T*

_{1,k}(

*t*), and

*S*

^{(cr)}(

*ω*) is the contribution to the PSD from the cross terms, which cannot be neglected since the processes Δ

*T*

_{1,k}(

*t*) are driven by the same forcing Δ

*F*(

*t*). For the stochastic component of the process Eq. (15), we can replace the forcing by a white noise process such that

*ω*.

Before we proceed to estimating parameters from data, we demonstrate that the PSD in Eq. (16) can easily be made to approximate a power law. For instance, can we pick time scales *τ*_{k} such that *τ*_{k+1} = *aτ*_{k} and weights *b*_{k} such that *N* = 5 terms. The sum *S*^{(0)}(*ω*) is shown by the red line in Fig. 1a and the total sum *S*^{(0)}(*ω*) + *S*^{(cr)}(*ω*) by the blue line in Fig. 1b. The idea that a long memory process can be produced by aggregating OU processes is the same as presented by, for example, Granger (1980) and M. Rypdal (2016). In this paper, the time scales *τ*_{k} and their weights *b*_{k} are estimated from data without a priori assumptions of scale invariance.

### a. Example 1: The two-box model

For the classical two-box model, Geoffroy et al. (2013) estimated parameters by fitting the GMST response in the two-box model to the corresponding response in CMIP5 models. The forcing scenario used for fitting was abrupt quadrupling of atmospheric CO_{2} concentration. They find multimodel mean parameter estimates ^{−2} K^{−1}, ^{−2} K^{−1}, ^{−2} K^{−1}, and ^{−2} K^{−1}, which correspond to characteristic time scales of 3.88 and 242 yr. The two-box model provides a good fit to CMIP5 abrupt 4 × CO_{2} experiments and 1% yr^{−1} CO_{2} increase experiments over 140 years, but, if forced with white noise, the PSD is *S*_{1}(*ω*) ∝ *S*^{(0)}(*ω*) + *S*^{(cr)}(*ω*). Using the parameters estimated by Geoffroy et al. (2013) for three different CMIP5 models, we have plotted this expression in Fig. 2a. As seen from Fig. 2, the PSD can be approximated by two different power laws: one in the high-frequency range and another in the low-frequency range. For frequencies corresponding to time scales greater than a few decades, the PSD can be approximated by *S*(*f*) ∝ 1/*f*^{β}, with *β* = 0.3, and for the higher frequencies it can be approximated by *S*(*f*) ∝ 1/*f*^{2}. This result is inconsistent with the PSDs estimated from CMIP5 control run temperatures, which are well approximated by one power law over the entire range of frequencies from months to centuries (Fredriksen and Rypdal 2016).

(a) Theoretical PSD of the response to white-noise forcing for a two-box model, with three different sets of parameters. The dashed red line shows a power law 1/*f ^{β}*, with

*β*= 0.3 for reference. (b) The response to a unit step forcing for the same parameter choices. The three different sets of parameter choices are those estimated by Geoffroy et al. (2013) from the following models: HadGEM2-ES: (

*κ*

_{1},

*κ*

_{2}) = (0.65, 0.55) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (6.5, 82) W yr m

^{−2}K

^{−1}, IPSL-CM5A-LR: (

*κ*

_{1},

*κ*

_{2}) = (0.79, 0.59) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (7.7, 95) W yr m

^{−2}K

^{−1}, and NorESM1-M: (

*κ*

_{1},

*κ*

_{2}) = (1.11, 0.88) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (8.0, 105) W yr m

^{−2}K

^{−1}.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) Theoretical PSD of the response to white-noise forcing for a two-box model, with three different sets of parameters. The dashed red line shows a power law 1/*f ^{β}*, with

*β*= 0.3 for reference. (b) The response to a unit step forcing for the same parameter choices. The three different sets of parameter choices are those estimated by Geoffroy et al. (2013) from the following models: HadGEM2-ES: (

*κ*

_{1},

*κ*

_{2}) = (0.65, 0.55) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (6.5, 82) W yr m

^{−2}K

^{−1}, IPSL-CM5A-LR: (

*κ*

_{1},

*κ*

_{2}) = (0.79, 0.59) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (7.7, 95) W yr m

^{−2}K

^{−1}, and NorESM1-M: (

*κ*

_{1},

*κ*

_{2}) = (1.11, 0.88) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (8.0, 105) W yr m

^{−2}K

^{−1}.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) Theoretical PSD of the response to white-noise forcing for a two-box model, with three different sets of parameters. The dashed red line shows a power law 1/*f ^{β}*, with

*β*= 0.3 for reference. (b) The response to a unit step forcing for the same parameter choices. The three different sets of parameter choices are those estimated by Geoffroy et al. (2013) from the following models: HadGEM2-ES: (

*κ*

_{1},

*κ*

_{2}) = (0.65, 0.55) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (6.5, 82) W yr m

^{−2}K

^{−1}, IPSL-CM5A-LR: (

*κ*

_{1},

*κ*

_{2}) = (0.79, 0.59) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (7.7, 95) W yr m

^{−2}K

^{−1}, and NorESM1-M: (

*κ*

_{1},

*κ*

_{2}) = (1.11, 0.88) W m

^{−2}K

^{−1}and (

*C*

_{1},

*C*

_{2}) = (8.0, 105) W yr m

^{−2}K

^{−1}.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

The inability of the two-box model to simultaneously describe the average GMST response to certain forcing scenarios in GCMs, as well as the PSD of the background fluctuations, serves as a motivation to analyze more general *N*-box models, and in the next example we consider the EBM with three vertically distributed boxes. We will demonstrate that the three-box model provides accurate descriptions of both the deterministic response to historic radiative forcing and the statistical properties of the response to random forcing.

### b. Example 2: The three-box model

*τ*

_{1},

*τ*

_{2}, and

*τ*

_{3}) and compute the responses

*F*(

*t*), where the integral is estimated by a sum. As in Eq. (13), the GMST response is a linear combination of the responses Δ

*T*

_{1,k}(

*t*):

*b*

_{1},

*b*

_{2}, and

*b*

_{3}from historical data of GMST and forcing. We will subsequently demonstrate that, for the range of time scales we consider in this paper, the results are largely insensitive to the choice of time scales (

*τ*

_{1},

*τ*

_{2}, and

*τ*

_{3}), as long as these are sufficiently separated. We do not only require that the deterministic response to radiative forcing fits well with observations, but also that the PSD of the stochastic component of the response is consistent with the estimated PSD of the residual observational signal (the difference between observed temperature record and the model response to the deterministic forcing). Rypdal and Rypdal (2014) employ a maximum-likelihood method to estimate the variance of the temperature residual

*β*of a long-memory model. This method is inadequate here, since we have more free parameters and the method favors a good fit for the smallest time scales.

We employ instead an iterative routine, which in addition to weighting all the time scales in the estimation, allows us to fit to a composite spectrum. We first compute the residual temperature by guessing the model parameters, computing the deterministic responses to the Crowley and Hansen forcing time series, and then subtracting the deterministic responses from the observed Moberg temperature and HadCRUT4 records. The PSD of the residual is estimated using the periodogram and subsequently log binned. The theoretical PSD of the response to white-noise forcing in the three-box model is then fitted to the composite residual PSD derived from the temperatures from Moberg and HadCRUT4 by minimizing the square distance between the theoretical and estimated residual PSDs. From this, we obtain new estimates of the relative size of the weights *b*_{k} at each time scale, and using these we perform a new regression analysis with HadCRUT4 to determine the initial temperature *T*_{0} and the absolute strength of the response. The procedure is repeated with these new model parameters as a starting point. The parameter estimation converges rapidly; only a few iterations are needed to obtain our estimates.

In Figs. 3a,c, we show the deterministic responses to historical forcing with estimated parameters. The three different colors represent different choices of the time scales *τ*_{1}, *τ*_{2}, and *τ*_{3}, and the parameter estimates are presented in Table 1, where we also present the corresponding values of the parameters *κ*_{k} and *C*_{k}, as well as the equilibrium climate sensitivity of the model. We note that the colored curves in Figs. 3a,c are almost indistinguishable, and they all closely follow the HadCRUT4 and Moberg records.

(a) The black curve is the Moberg temperature reconstruction, and the colored curves are the responses to Crowley forcing of the three-box model with estimated parameters. The orange curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 0.5 yr, *τ*_{2} = 5 yr, and *τ*_{3} = 50 yr. The thicker green curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 20 yr, and *τ*_{3} = 400 yr. (c) As in (a), but showing the response to forcing over the instrumental period, and the black curve is the global instrumental temperature from HadCRUT4. (b),(d) The green curves in (a) and (c) decomposed into the responses corresponding to the exponential terms in the response function *R*(*t*). The orange curve corresponds to *τ*_{1} = 1 yr, the red curve corresponds to *τ*_{2} = 10 yr, and the blue curve corresponds to *τ*_{3} = 100 yr.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) The black curve is the Moberg temperature reconstruction, and the colored curves are the responses to Crowley forcing of the three-box model with estimated parameters. The orange curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 0.5 yr, *τ*_{2} = 5 yr, and *τ*_{3} = 50 yr. The thicker green curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 20 yr, and *τ*_{3} = 400 yr. (c) As in (a), but showing the response to forcing over the instrumental period, and the black curve is the global instrumental temperature from HadCRUT4. (b),(d) The green curves in (a) and (c) decomposed into the responses corresponding to the exponential terms in the response function *R*(*t*). The orange curve corresponds to *τ*_{1} = 1 yr, the red curve corresponds to *τ*_{2} = 10 yr, and the blue curve corresponds to *τ*_{3} = 100 yr.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) The black curve is the Moberg temperature reconstruction, and the colored curves are the responses to Crowley forcing of the three-box model with estimated parameters. The orange curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 0.5 yr, *τ*_{2} = 5 yr, and *τ*_{3} = 50 yr. The thicker green curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 20 yr, and *τ*_{3} = 400 yr. (c) As in (a), but showing the response to forcing over the instrumental period, and the black curve is the global instrumental temperature from HadCRUT4. (b),(d) The green curves in (a) and (c) decomposed into the responses corresponding to the exponential terms in the response function *R*(*t*). The orange curve corresponds to *τ*_{1} = 1 yr, the red curve corresponds to *τ*_{2} = 10 yr, and the blue curve corresponds to *τ*_{3} = 100 yr.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

Parameters estimated from data. The time scales *τ*_{k} and their weights *b*_{k} for *k* = 1, 2, and 3 uniquely determine the estimated response functions. The parameter *σ*_{monthly} is the estimated standard deviation of the monthly resolved stochastic forcing. The values for *T*_{0,instr} and *T*_{0,Moberg} are the initial temperature anomalies estimated for global instrumental temperature and the Moberg temperature reconstruction. We note that there is some remnant seasonal variability that we were not able to remove by subtracting a mean seasonal cycle. This is not so apparent in a double-logarithmic plot of the PSD, but it causes our estimates of *σ*_{monthly} to be slightly inconsistent with the actual residual variability. We also point out that our estimates of *σ*_{monthly} are only valid for monthly resolved temperatures, or monthly temperatures sampled at a different time resolution.

Figure 4a shows the theoretical PSD of the stochastic component (the response to white-noise forcing) in the model. The estimated parameters are shown in Table 1, and the choice of time scales are *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The PSD of the model fits well with the PSD estimated from observational data in the time-scale range from months to centuries, and in this range it is close to a power law with an exponent *β* = 0.65.

(a) The blue curve is the estimated PSD of the residual global instrumental temperature after subtracting the estimated deterministic response of a three-box model. The characteristic time scales in the three-box model are chosen to be *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the residual of the Moberg temperature reconstruction. Both curves are normalized by their power on decadal scales. The black curve is the theoretical PSD with the estimated response function, which can be quite well approximated by a power law, shown by the dashed line with slope −*β* = 0.65. (b) The gray curves are the responses to a unit step forcing using two-box model parameters estimated for many climate models by Geoffroy et al. (2013). The colored curves show the three-box responses with the parameters from Table 1. The orange curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 0.5 yr, *τ*_{2} = 5 yr, and *τ*_{3} = 50 yr. The thicker green curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 20 yr, and *τ*_{3} = 400 yr.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) The blue curve is the estimated PSD of the residual global instrumental temperature after subtracting the estimated deterministic response of a three-box model. The characteristic time scales in the three-box model are chosen to be *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the residual of the Moberg temperature reconstruction. Both curves are normalized by their power on decadal scales. The black curve is the theoretical PSD with the estimated response function, which can be quite well approximated by a power law, shown by the dashed line with slope −*β* = 0.65. (b) The gray curves are the responses to a unit step forcing using two-box model parameters estimated for many climate models by Geoffroy et al. (2013). The colored curves show the three-box responses with the parameters from Table 1. The orange curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 0.5 yr, *τ*_{2} = 5 yr, and *τ*_{3} = 50 yr. The thicker green curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 20 yr, and *τ*_{3} = 400 yr.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) The blue curve is the estimated PSD of the residual global instrumental temperature after subtracting the estimated deterministic response of a three-box model. The characteristic time scales in the three-box model are chosen to be *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the residual of the Moberg temperature reconstruction. Both curves are normalized by their power on decadal scales. The black curve is the theoretical PSD with the estimated response function, which can be quite well approximated by a power law, shown by the dashed line with slope −*β* = 0.65. (b) The gray curves are the responses to a unit step forcing using two-box model parameters estimated for many climate models by Geoffroy et al. (2013). The colored curves show the three-box responses with the parameters from Table 1. The orange curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 0.5 yr, *τ*_{2} = 5 yr, and *τ*_{3} = 50 yr. The thicker green curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 10 yr, and *τ*_{3} = 100 yr. The red curve is the response of the three-box model where we have fixed time scales *τ*_{1} = 1 yr, *τ*_{2} = 20 yr, and *τ*_{3} = 400 yr.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

In Fig. 4b, we show the three-box model responses to a unit step-forcing scenario. The different colors correspond to different choices of the time scales *τ*_{1}, *τ*_{2}, and *τ*_{3}, and the gray curves are the corresponding two-box model responses with parameters estimated by Geoffroy et al. (2013) by fitting to abrupt 4 × CO_{2} experiments in CMIP5 models. From the response to a unit step forcing, we can also derive that the equilibrium climate sensitivity for an *N*-box model is given by

The difference we observe in the step-forcing responses is not a result of a difference between the two- and three-box models, but rather a difference in estimation strategy. In our estimation procedure, we use only observational and proxy data and have hence chosen the parameters that best reproduce both the residual spectra and responses to historical forcing. The parameters estimated by this method are not the same as the parameters that best describe the 4 × CO_{2} runs. This discrepancy can have several explanations; perhaps is the random forcing not accurately modeled as a white noise, or the forcing in the 4 × CO_{2} experiments is too strong for a linear approximation to be valid. It is likely that the feedback parameter or parameters related to ocean mixing can change during the strong and abrupt climate change following a quadrupling in CO_{2} concentration. The large differences between the different CMIP5 step responses also reflect the large uncertainty associated with these model runs.

### c. Example 3: Separate boxes for land and ocean

Parameters estimated from global sea and land surface temperatures. The fixed time scales are chosen to be *τ*_{1} = 0.5 yr, *τ*_{2} = 5 yr, and *τ*_{3} = 50 yr.

*σ*

_{L}

*dB*

_{L}(

*t*) is the direct stochastic forcing of the land surface temperature. For the ocean response, we use the previously estimated parameters, given in Table 2. The remaining parameters are chosen such that the deterministic response

*T*

_{G}=

*f*

_{L}Δ

*T*

_{L}+

*f*

_{O}Δ

*T*

_{1}is similar to the three-box temperature response estimated directly from global temperature.

(a) The black curve is the global HadSST3 dataset, and the red curve is the response to forcing on time scales of 0.5, 5, and 50 yr. (b) The blue curve is the spectrum of the residual of HadSST3 after subtracting the red curve in (a), and the black curve is the expected spectrum of the residual. The dashed black line shows a power law 1/*f ^{β}* for reference. (c),(d) As in (a),(b), but for the global CRUTEM4v dataset. In addition to the three time scales, we have an instantaneous response to the forcing for land temperatures.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) The black curve is the global HadSST3 dataset, and the red curve is the response to forcing on time scales of 0.5, 5, and 50 yr. (b) The blue curve is the spectrum of the residual of HadSST3 after subtracting the red curve in (a), and the black curve is the expected spectrum of the residual. The dashed black line shows a power law 1/*f ^{β}* for reference. (c),(d) As in (a),(b), but for the global CRUTEM4v dataset. In addition to the three time scales, we have an instantaneous response to the forcing for land temperatures.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

(a) The black curve is the global HadSST3 dataset, and the red curve is the response to forcing on time scales of 0.5, 5, and 50 yr. (b) The blue curve is the spectrum of the residual of HadSST3 after subtracting the red curve in (a), and the black curve is the expected spectrum of the residual. The dashed black line shows a power law 1/*f ^{β}* for reference. (c),(d) As in (a),(b), but for the global CRUTEM4v dataset. In addition to the three time scales, we have an instantaneous response to the forcing for land temperatures.

Citation: Journal of Climate 30, 18; 10.1175/JCLI-D-16-0877.1

The theoretical PSD of Δ*T*_{1}(*t*) fits well with the estimated PSD of the SST residual, and both are well approximated by a power law with *β*_{O} ≈ 1. In the same way, the theoretical PSD of Δ*T*_{L}(*t*) fits well with the estimated PSD of the LST, and both can be approximated by a power law with *β*_{L} ≈ 0.5. These results are similar to the estimated PSDs of the linearly detrended global LST and global SST analyzed in Fredriksen and Rypdal (2016). With this model, the only reason global LST shows persistence is because of the influence by global SST, but the persistence is weaker for land than for sea as a result of the component responding instantly to forcing.

*r*

_{1}/

*r*

_{2}≪

*S*

_{eq,sea}, there will be a near-constant ratio between land and ocean temperature change, consistent with the findings of Lambert et al. (2011).

## 4. Discussion and conclusions

Simple climate models can be divided in two classes: EBMs and tuned impulse–response (IR) linear statistical models (Good et al. 2011). The power-law response model proposed by Rypdal and Rypdal (2014) is an example of an IR model that reproduces observed temperature variability quite well, but in this mathematical idealization conservation of energy is lost. It may also be unclear what the physical reason for using this model is. In this paper, we demonstrate that such a model is closely approximated by the response of a multibox EBM. This shows that LRD models can be seen as a compact mathematical description of the effect of a range of time scales in the physical response.

Linear EBMs have been studied since Budyko (1969) and Sellers (1969), with various models for how the heat is taken up by the ocean. Many of them include also an additive stochastic forcing, assumed to be generated by a truncation of the dynamics on shorter scales than the response (Hasselmann 1976). Weather systems and day-to-day variations in insolation and outgoing radiation due to variations in cloud cover are likely important parts of the noisy forcing. These are known to be unpredictable on the scales of the climate response, and a Gaussian white noise description in time may be reasonable.

The ocean then integrates the forcing on several time scales, but it is difficult to point out exactly what the time scales are. The reason is that the response appears to be practically indistinguishable from a power law, even if we do not assume this a priori. By picking a set of time scales that is sufficiently separated within the range of the scales where we expect a climate response, we obtain a set of sufficiently different predictors to describe global temperature evolution. We find that three time scales is the least number needed to approximate the temperature response. The resulting description of global temperature in terms of this set of predictors should be seen as a statistical model with a minimal number of parameters to be estimated from data—a model that can also be replaced by a simpler power-law response with even fewer parameters to estimate.

Contrary to the power-law response, the response derived from the EBM has a physical fundament of energy conservation and some description of energy exchange within the system. Even though the model is oversimplified, the power-law-shaped spectrum we obtain for global surface temperature with this simple model is consistent with the slightly more advanced linear diffusion models, where Fraedrich et al. (2004) and Lemke (1977) report 1/*f* noise characteristics, as well as with the power-law spectra observed in complex GCMs (Fredriksen and Rypdal 2016). It is difficult to draw any general conclusions about the physical parameters in the simple model from the large separation of time scales, since each time scale and corresponding response depends on all parameters in the linear model. One general feature for our estimated parameters is that *C*_{1} < *C*_{2} < *C*_{3}, but the large separation of time scales does not necessarily imply that the heat capacities must be well separated.

The EBMs considered in this paper consist only of one or two surface boxes and can therefore only describe the correlation structure of global temperature in time. Several papers consider EBMs extended to describe a horizontal temperature field (North and Cahalan 1981; Kim and North 1991; North et al. 2011). These models were expanded to include a simple model for ocean diffusion and upwelling by Kim and North (1992) and were compared to early versions of GCMs by Kim et al. (1996). More recently, work on two-dimensional stochastic–diffusive EBMs by North et al. (2011) was generalized to include long-memory temporal response by Rypdal et al. (2015). An important result derived from this generalization is power-law spectra for both local and global temperature, and spectral exponent of global temperature, which is twice that of local temperatures, are in good agreement with observations. The spatial model proposed by Rypdal et al. (2015) could be approximated by a sum of spatial fields with AR(1) characteristics in time, similar to our approximation for global temperature in this paper.

As suggested by Ragone et al. (2016), we believe that the global temperature response in the Holocene can be well approximated as linear. Extending the linear response to local and regional temperatures is more problematic, especially for the temperature responses in the Arctic, where strong nonlinear effects such as the sea ice–albedo feedback are present. Despite this, linear responses as in Lucarini et al. (2017) or derived from spatially dependent EBMs have considerable success in describing most local temperatures. The multibox models can also easily be extended to include nonlinear terms: for instance, to describe the rapid sea ice loss in the Arctic. And they can be extended to include different types of tipping points, which allows us to study critical transitions in systems that exhibit LRD. The effect of LRD on the early warning indicators associated with critical transitions in multibox EMBs is a topic that will be pursued in future work.

## Acknowledgments

The authors thank K. Rypdal and O. Løvsletten for useful discussions. The paper was supported by the Norwegian Research Council (KLIMAFORSK Programme) under Grant 229754.

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^{1}

In this paper, we follow Rypdal and Rypdal (2014) and model the random component of the forcing as white noise. However, the models can easily be modified to other stochastic models for the random forcing.