## 1. Introduction

The enormous complexity of present-day general circulation climate models implies that model data can only be interpreted through advanced data analysis. Implicit in such data analysis is always the application of concepts based on simpler dynamic, stochastic–dynamic, or statistical model frameworks. Such models will in some form describe the correlation structure of Earth’s climatic fields. Zero-dimensional energy balance models (EBMs) only describe correlation structure in time—in simplest form, as an exponential relaxation time with a time constant of a few years determined by the heat capacity of the ocean mixed layer. Zero-dimensional two-layer models include the energy exchange between the mixed layer and the deep ocean, which introduces another and larger time constant of the order of a century (Held et al. 2010; Geoffroy et al. 2013). An alternative generalization of the one-layer model, which yields a power-law impulse response, was introduced by Rypdal (2012) and, by further introducing stochastic forcing, in Rypdal and Rypdal (2014). These generalizations thus provide a phenomenological stochastic–dynamical model describing the long-range temporal correlation structure on time scales from years to centuries observed in global temperature data as well as in millennium-long climate model simulations (Østvand et al. 2014).

One-dimensional EBMs describe meridional energy transport in addition to the vertical radiation balance (Budyko 1969; Sellers 1969), and two-dimensional models include also zonal transport. There is a plethora of papers on such models, many of which include a stochastic forcing. One of the earlier studies of such stochastic-diffusive EBMs was made by North and Cahalan (1981). This model was expanded to include a simple model for ocean diffusion and upwelling by Kim and North (1992), and was compared to early versions of atmospheric–ocean general circulation models (AOGCMs) by Kim et al. (1996). The latter study revealed power-law temporal spectra corresponding to strong long-range temporal correlation structure in global temperature on time scales up to a decade in the AOGCMs, but a loss of correlations on longer time scales. The stochastic–diffusive EBM showed a more gradual transition toward uncorrelated noise on longer time scales. Their spectra of instrumental global temperature also shows the transition to uncorrelated noise on time scales longer than a decade, which is now known not to reflect the true correlation structure on time scales from decades to centuries revealed in millennium-long temperature reconstructions (Rypdal et al. 2013) and AOGCM simulations (Østvand et al. 2014). The explanation of the observed flattening of the spectra on long time scales may be the particular spectral analysis method used in Kim et al. (1996), which is based on fitting a high-order autoregressive (AR) process to the data. AR processes cannot model long-range dependence in data (Beran 1994).

*t*is time,

It is shown in North et al. (2011) that the stationary random field resulting from Eq. (1) exhibits an exponentially decaying, frequency-dependent, spatial autocorrelation function (ACF). They express this as a frequency-dependent spatial autocovariance

North et al. (2011) do not compute explicit instantaneous spatial ACFs or temporal power spectral densities (PSDs). From a data-analysis viewpoint temporal spectra are of great interest because of the published evidence that such spectra exhibit power-law scaling of the form *β* measures the degree of persistence in the record, and in local records it is close to unity over oceans, and close to zero over land (Fraedrich and Blender 2003). It also appears that *β* is larger in records averaged over large areas, like global and hemispheric averages, than in local time series. As we will demonstrate in section 2, many of these features are described by the stochastic-diffusive EBM of North et al. (2011) (in the following called the North EBM) for time scales up to the relaxation time *β* than observed in the control simulations. This motivates a generalization of the fractional EBM to include persistent, stochastic forcing in section 5. In section 6 we summarize and conclude, and in the appendixes we elaborate on some mathematical technicalities.

## 2. The North EBM on a sphere

### a. Spatial ACFs and temporal PSDs of the North EBM

*θ*is the polar angle, and

*ϕ*is the azimuthal angle (longitude). This equation is now formulated on the rectangle

*K*and

*N*, and since all spatiotemporal scales are represented by equal power in the random field the total power diverges as

*α*such that

*n*th-order Legendre polynomial, we arrive at the frequency-dependent ACFwhereThe frequency-dependent ACF

*K*(i.e., if we assume that the forcing field is smooth on time scales shorter than

*λ*. Note that since we operate on the unit sphere

*λ*is measured in radians. For time scales longer than the relaxation time

### b. Relating temporal and spatial correlations

*θ*between two points on the sphere. If

*β*for local and global temperature observed in Fig. 4. Equation (14) shows that in general, if

*τ*, then local and global temperature should scale the same way for all time scales greater than

*τ*. In the NorESM data we are not able to identify such a time scale

*τ*(i.e., if it exists it must be more than several centuries). This is a major motivation for searching for a generalization of the North EBM, which does not exhibit the same scaling for local and global time series at low frequencies.

## 3. Generalizations of the North EBM

### a. Transfer function formulation

### b. The fractional EBM

*η*is a constant parameter of dimension time that characterizes the strength of the response. The unit constant

^{−1}. The Fourier transform of

*c*has dimension (time)

^{β/2}, hence

### c. Spatial and temporal correlations in the fractional EBM

The frequency-dependent ACF is plotted in Fig. 6 for two values of *λ* and is the analog of Fig. 1 generalized to the fractional EBM. We have the same tendency toward loss of long-range spatial correlation at high frequencies, but in the zero frequency limit the correlation function is uniform; that is, the fluctuations are dominated by spatially uniform (global) variations. Such behavior does not appear in the conventional EBM since that model lacks a long-range global response reflecting the slow response of the deep ocean.

In Fig. 7 we plot the PSD of the local temperature for a number of different *λ* values. This figure is the fractional analog of Fig. 3. For small and large *λ* these spectra are perfect power laws over most of the frequency range. For small *λ* (i.e., in the regime relevant for the Earth climate system) the spectral exponent is *β*. This result is obvious, since in the large *λ* limit the field will be spatially uniform and the model reduces to the zero-dimensional one. For intermediate values of *λ* there is a break in the scaling from exponent *β* at low frequencies (dominated by global fluctuations) to

We can also compute the PSD for the temperature averaged over a fraction *ξ* of the globe surface, as we did in Fig. 4 for the conventional EBM. The result is shown in Fig. 8. It may not come as a surprise that the spectra are power laws, and that local spectra have exponent *β* with increasing degree of spatial averaging up to global.

In Fig. 9 we show that this feature is also reflected in observation data. The figure shows the fluctuation functions of a first-order detrended analysis (DFA1) (Kantelhardt et al. 2001) of the central England instrumental temperature record and two globally averaged records. DFA1 does not eliminate linear trends, but this trend is small in the 350-yr-long central England record. The averaged records are derived from the 160-yr-long HadCRUT3 global instrumental surface temperature (Brohan et al. 2006) and the 2000-yr-long Northern Hemisphere reconstruction of Moberg et al. (2005). The scaling properties of these records may be strongly influenced by the global radiative forcing and hence not representative of the internal (unforced) variability. However, by using a reconstruction of the forcing (Hansen et al. 2011; Crowley 2000) and a model for the global temperature response we can compute the temperature response to this deterministic forcing. This was done by Rypdal and Rypdal (2014), who also demonstrated that a simple zero-dimensional, fractional EBM response model yields a deterministic response almost indistinguishable from the mean response in ensembles of CMIP5 model runs. The residual noise obtained by subtracting the deterministic response from the observed/reconstructed record represents the internal variability, and it is these records that have been subject to analysis in Fig. 9. The slope *α* of the fluctuation-function curve is related to the spectral exponent *β* via *β* is twice the local *β*.

## 4. Comparison to NorESM data

The simple version of the North EBM and the fractional EBM we have explored here assumes a uniform Earth surface. Since the global spatial average of these models are identical to the zero-dimensional “exponential” and “scale-invariant” response models studied in Rypdal and Rypdal (2014), we can use their methods for estimating the model parameters *β* and *η* (for the fractional EBM). The temperature data records to use as input for such estimates can be the instrumental record for global land temperature or sea surface temperature, depending on whether we want to study spatiotemporal persistence over continental interiors or over oceans.

As discussed in the previous section, the *β* for unforced dynamics should be estimated from the residual obtained by subtracting the deterministic response from the observed record. In Figs. 10a and 11a the red full curve is the deterministic solution to the fractional EBM when *β* of that fGn that minimizes the mean square error. If we subtract the red full curve from the observed record and estimate *β* again from the residual record, we find *β* is the dotted red curve, which is almost the same. Hence in the fractional EBM the response to the deterministic forcing is not very sensitive to *β*, and hence that *τ*, but the random fluctuations requires a shorter response time, and suggests that the response is characterized by more than one time constant, as in the two-layer model considered by Geoffroy et al. (2013). Similar observations are made for the ocean temperatures as shown in Figs. 11a and 11b.

The diffusion parameter *λ* cannot be estimated from such globally averaged records, so these must be obtained from spatiotemporal observation data, reanalysis data, or data from AOGCMs. The latter provide the best data coverage in space and longer temporal records, which make it possible to “calibrate” the EBMs to general circulation models. For this purpose we use the frequency-dependent ACF ^{−1} and the other frequencies are

The next step is to compute *β* and *η* estimated from the observed global land temperature and different *λ* and select the value of *λ* that minimizes the mean square error between the theoretical and observed ACFs in the range

In Fig. 10d we show similar results for the North EBM, where the time constant

Corresponding results for ocean temperatures are shown in Fig. 11. To avoid a strong influence from El Niño–Southern Oscillation in the ACF on interannual time scales we study only the oceans south of 20°S in the NorESM control run. We obtain _{2} forcing.

The performance of the two EBM models with respect to reproducing the observed global-scale PSDs is shown in Fig. 12. The black curves are the spatially averaged spectra of local temperatures over the Eurasian continent (Fig. 12a) and over the Southern Ocean (Fig. 12b). The red curves are the theoretical spectra from the fractional EBM and the blue curves for the North EBM, demonstrating very clearly the superiority of the fractional model.

## 5. Fractional forcing noise

From Figs. 10c and 11c we have observed that the fractional EBM provides a less accurate description of the correlation structure for sea surface temperatures as compared to the land temperatures. However, the comparison with the NorESM model improves significantly if we allow the stochastic forcing *F* to be an fGn with an exponent *β*, and this EBM is driven by an fGn with spectral exponent

Such a generalization of the fractional EBM would be rather ad hoc if it was not directly suggested by other climate model simulations. Geoffroy et al. (2013) studied a large number of such runs of CMIP5 models, with results that were all similar to the black curve in Fig. 13a, which is the global temperature following sudden quadrupling of atmospheric CO_{2} concentration in the NorESM model. Geoffroy et al. (2013) find good fits to these curves with a function that is a linear combination of two exponential functions with one small time constant of the order of a few years and one larger of the order of a century. In Fig. 13a the red curve is a fit of a power-law function *β* for the response is roughly half the value of *β* from the transient evolution shown in Fig. 13a. Hence we have *local* spectrum for the generalized, fractional EBM has exponent *local* spectra of the SO. Here the red line is the power-law spectrum with exponent

## 6. Conclusions

In an editorial comment, Mann (2011) asserted that the scaling behavior in instrumental and long-term proxy temperature reconstructions appears consistent with the results of a standard, zero-dimensional EBM forced by estimated natural and anthropogenic radiative forcing changes, and subject to white-noise stochastic weather forcing. According to this author, “nothing more exotic than the physics of such a simple model is necessary to explain the apparent scaling behavior in observed surface temperatures.” This conclusion was drawn from application of a number of standard estimation techniques for *β* to realizations of the purely stochastically forced, and stochastic plus radiatively forced, EBM. These estimates were compared to results from the same techniques applied to observation data. Such comparisons show overlap of the distributions of *β* estimates for the model realizations and the observation records, which lead the author to conclude that the scaling properties of the observation data are consistent with this simple model.

The results derived in the present paper demonstrate that there is a clear discrepancy between the scaling properties of the North EBM and data derived from observations and climate models. The conclusions of Mann (2011) arise from uncritical application of estimation techniques for the scaling exponent to time series that do not exhibit scaling. For instance, the stochastically forced model signal is an AR(1) process, which scales like a Wiener process (

*β*for the response than observed in the control runs. The consistency is restored by assuming that the stochastic forcing is a fractional noise with

_{2}greenhouse effect, also has a noisy component that may exhibit long-range persistence. This assertion is supported by recent direct observations of CO

_{2}forcing at Earth’s surface. Feldman et al. (2015) measured the clear-sky radiative CO

_{2}surface forcing and obtained time series as shown in Fig. 14a. The PSD of this time series has a spectral exponent

_{2}forcing, and radiative forcing in general exhibits a noisy component that is persistent, and not white. The length of the observation record is too short to claim statistical significance of this persistence on time scales longer than a month, but the noisy CO

_{2}forcing record illustrates that radiative forcing has a noisy component, and there are good reasons to believe that this noise exhibits persistent scaling properties.

## Acknowledgments

This paper was supported by the Norwegian Research Council, project 229754. The authors acknowledge Dr. Odd Helge Otterå for providing data from the NorESM simulations.

## APPENDIX A

### Expansion Coefficients for a Random Field

*σ*is a constant indicating the strength of the field. By substituting Eq. (A3) into Eq. (A2) we findand henceThus, we have proven the validity of Eq. (4) for a random field.

## APPENDIX B

### The Spatial Covariance

*η*and

*n*is given by

*n*is a nonnegative integer;

*A*being a constant and

## APPENDIX C

### Formulation of the Fractional EBM using Fractional Derivatives

#### ACFs and PSDs in the generalized, fractional EBM

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