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).





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
2. The North EBM on a sphere
a. Spatial ACFs and temporal PSDs of the North EBM
















































Frequency-dependent spatial ACF of temperature measured vs angular distance θ from any reference point on the uniform unit sphere given by Eq. (10) for (a)
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
(a): Instantaneous spatial covariance,
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1












Log–log plots of the temporal PSD of temperature time series at any point on the uniform unit sphere given by Eq. (9), for
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
b. Relating temporal and spatial correlations

















Log–log plot of power spectral density for the temperature averaged over a fractional area
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
3. Generalizations of the North EBM
a. Transfer function formulation




b. The fractional EBM
















Instantaneous temperature field in a simulation of the fractional EBM on a sphere. The parameters are
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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.
Frequency-dependent spatial ACF of temperature measured vs angular distance θ from any reference point on the uniform unit sphere given by Eqs. (17) and (23) for (a)
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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
Log–log plots of the temporal PSD of temperature time series at any point on the uniform unit sphere given by Eq. (17) for
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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
Log–log plot of power spectral density for the temperature averaged over a fractional area
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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
DFA1 fluctuation functions of the temperature record for central England (blue triangles), Moberg’s reconstruction of the mean surface temperature in the Northern Hemisphere in the last millennium (red crosses), and the instrumental record for global mean temperature (red circles). For the global data the records analyzed are the residuals after the response to the deterministic forcing has been subtracted. The blue dotted line corresponds to
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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
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
(a) The black curve is the global mean land surface temperature. The red solid curve is the mean temperature in the fractional EBM with deterministic forcing and parameters estimated using the least squares method. The estimated β value is 0.61. The red dotted curve is the same as the red solid curve, but in this case the β parameter is estimated from the residual signal obtained by taking the difference between the temperature observations (black curve) and the least squares fit (solid red curve). This β value is 0.28. (b) As in (a), but in this case for the North EBM. The solid blue line shows the response to the deterministic forcing with parameters estimated using the least squares method. This gives the estimate
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
As in Fig. 10, but for sea surface temperatures. (a) The estimated scaling exponent is
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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
The next step is to compute
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
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.
(a) The black circles show the average power spectral density for the temperatures on the Eurasian continent in a control run of the NorESM model. The blue curve is the power spectral density for local temperatures in the North EBM [Eq. (12)] with parameters
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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
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 CO2 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
(a) The temperature response to a step function forcing scenario in the NorESM model (black curve) and a least squares fit of a power-law expression
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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 (



(a): Measured CO2 forcing at the surface measured at the North Slope of Alaska from 2000 throughout 2010. Time resolution is 10−2 yr. The red dashed line is a linear fit to the data. The slope (trend) is 0.2 W m−2 per decade. (b) The power spectral density (PSD) estimated by the periodogram of the time series in (a). The red dashed line has slope
Citation: Journal of Climate 28, 21; 10.1175/JCLI-D-15-0183.1
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





APPENDIX B
The Spatial Covariance






















APPENDIX C
Formulation of the Fractional EBM using Fractional Derivatives







ACFs and PSDs in the generalized, fractional EBM


REFERENCES
Bentsen, M., and Coauthors, 2013: The Norwegian Earth System Model, NorESM1-M—Part 1: Description and basic evaluations of the physical climate. Geosci. Model Dev., 6, 687–720, doi:10.5194/gmd-6-687-2013.
Beran, J., 1994: Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probability Series, Vol. 61, Chapman & Hall/CRC, 315 pp.
Brohan, P., J. J. Kennedy, I. Harris, S. F. B. Tett, and P. D. Jones, 2006: Uncertainty estimates in regional and global observed temperature changes: A new data set from 1850. J. Geophys. Res., 111, D12106, doi:10.1029/2005JD006548.
Budyko, M. I., 1969: The effect of solar radiation variations on the climate of the Earth. Tellus, 21, 611–619, doi:10.1111/j.2153-3490.1969.tb00466.x.
Crowley, T. J., 2000: Causes of climate change over the past 1000 years. Science, 289, 270–277, doi:10.1126/science.289.5477.270.
Feldman, R. D., W. D. Collins, P. J. Gero, M. S. Torn, E. J. Mlawer, and T. R. Shippert, 2015: Observational determination of surface radiative forcing by CO2 from 2000 to 2010. Nature, 519, 339–343, doi:10.1038/nature14240.
Fraedrich, K., and R. Blender, 2003: Scaling of atmosphere and ocean temperature correlations in observations and climate models. Phys. Rev. Lett., 90, 108501, doi:10.1103/PhysRevLett.90.108501.
Geoffroy, O., D. Saint-Martin, D. J. L. Olivié, A. Voldoire, G. Bellon, and S. Tytéca, 2013: Transient climate response in a two-layer energy-balance model. Part I: Analytical solution and parameter calibration using CMIP5 AOGCM experiments. J. Climate, 26, 1841–1857, doi:10.1175/JCLI-D-12-00195.1.
Hansen, J., M. Sato, P. Kharecha, and K. von Scuckmann, 2011: Earth’s energy imbalance and implications. Atmos. Chem. Phys., 11, 13 421–13 449, doi:10.5194/acp-11-13421-2011.
Held, I. M., M. Winton, K. Takahashi, T. Delworth, F. Zeng, and G. K. Vallis, 2010: Probing the fast and slow components of global warming by returning abruptly to preindustrial forcing. J. Climate, 23, 2418–2427, doi:10.1175/2009JCLI3466.1.
Iversen, T., and Coauthors, 2013: The Norwegian Earth System Model, NorESM1-M—Part 2: Climate response and scenario projections. Geosci. Model Dev., 6, 389–415, doi:10.5194/gmd-6-389-2013.
Kantelhardt, J. W., E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, 2001: Detecting long-range correlations with detrended fluctuation analysis. Physica A, 295, 441–454, doi:10.1016/S0378-4371(01)00144-3.
Kim, K.-Y., and G. R. North, 1992: Seasonal cycle and second-moment statistics of a simple coupled climate system. J. Geophys. Res., 97, 20 437–20 448, doi:10.1029/92JD02281.
Kim, K.-Y., G. R. North, and G. C. Hegerl, 1996: Comparisons of the second-moment statistics of climate models. J. Climate, 9, 2204–2221, doi:10.1175/1520-0442(1996)009<2204:COTSMS>2.0.CO;2.
Mann, M. E., 2011: On long-range dependence in global temperature series. Climatic Change, 107, 267–276, doi:10.1007/s10584-010-9998-z.
Moberg, A., D. M. Sonechkin, K. Holmgren, N. M. Datsenko, and W. Karlen, 2005: Highly variable Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data. Nature, 433, 613–617, doi:10.1038/nature03265.
North, G. R., and R. F. Cahalan, 1981: Predictability in a solvable stochastic climate model. J. Atmos. Sci., 38, 504–513, doi:10.1175/1520-0469(1981)038<0504:PIASSC>2.0.CO;2.
North, G. R., J. Wang, and M. G. Genton, 2011: Correlation models for temperature fields. J. Climate, 24, 5850–5862, doi:10.1175/2011JCLI4199.1.
Østvand, L., T. Nilsen, K. Rypdal, D. Divine, and M. Rypdal, 2014: Long-range memory in internal and forced dynamics of millennium-long climate model simulations. Earth Syst. Dyn., 5, 295–308, doi:10.5194/esd-5-295-2014.
Rypdal, K., 2012: Global temperature response to radiative forcing: Solar cycle versus volcanic eruptions. J. Geophys. Res., 117, D06115, doi:10.1029/2011JD017283.
Rypdal, K., and M. Rypdal, 2014: Long-memory effects in linear-response models of Earth’s temperature and implications for future global warming. J. Climate, 27, 5240–5258, doi:10.1175/JCLI-D-13-00296.1.
Rypdal, K., L. Østvand, and M. Rypdal, 2013: Long-range memory in Earth’s surface temperature on time scales from months to centuries. J. Geophys. Res. Atmos., 118, 7046–7062, doi:10.1002/jgrd.50399.
Sellers, W. D., 1969: A global climatic model based on the energy balance of the Earth–atmosphere system. J. Appl. Meteor., 8, 392–400, doi:10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.