I show that the peak-to-peak amplitude of the global mean surface temperature response to the 11-year cyclic total irradiance forcing is an order of magnitude less than the amplitude of a cyclic component roughly in phase with the solar forcing which has been observed in the temperature record in the period 1959–2004. If this cyclic temperature component were a response to the solar forcing, it would imply the existence of strong amplifying feedbacks which operate exclusively for solar forcing, such as top-down mechanisms responding to the large variability in the ultraviolet part of the solar spectrum. I demonstrate, however, that the apparent cyclic component in the temperature record is dominated by the response to five major volcanic eruptions some of which incidentally took place a few years before solar minimum in four consecutive solar cycles, and hence that the correlation with the solar cycle is coincidental. A temperature rise of approximately 0.15 K over the 20th century ascribed to an increasing trend in solar forcing is more than offset by a cooling trend of about 0.3 K due to stratospheric aerosols from volcanic eruptions.
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 Apart from internal variability on scales up to a century, global climate change is considered to be a response to changes in the global energy balance imposed by various forcings. On geological timescales these forcings include orbital variations, continental drift, and long-term changes in atmospheric CO2 resulting from outgassing from the Earth's crust and weathering [Hansen et al., 2011]. In the industrial era anthropogenic forcing is presumed to be very important, adding to negative forcing due to volcanic activity and a positive contribution from increasing solar activity [Hansen et al., 2011; Intergovernmental Panel on Climate Change (IPCC), 2007]. A precise assessment of the anthropogenic contribution depends crucially on correspondingly precise evaluation of the two others, and is currently at the center of the scientific debate on the drivers of global change.
 Although there exists a vast number of reports of regional natural cycles with period close to the 11-year sunspot period [Hoyt and Schatten, 1997], it has been more difficult to detect this cycle in global records like the global mean surface temperature (GMST). However, comprehensive studies, based on spatiotemporal data from observation and climate models [e.g., Stevens and North, 1996, Figure 15], suggest a global temperature response to the solar cycle forcing with peak-to-peak amplitude no more than 0.06 K. Thus a recent paper by Camp and Tung  has stirred some interest, since they have detected a periodic signal with peak-to-peak amplitude near 0.2 K in the GMST-record for the period 1959–2004.This signal is in phase with the solar-cycle signal in the total solar irradiance (TSI), and with roughly the same period, although such a period cannot be determined exactly for a signal covering only four cycles. Such a strong signal is difficult to explain from the standard paradigm for how global surface temperature responds to radiative forcing. Standard in this context means that the forcing is computed from the total solar energy that reaches the Earth's surface (bottom-up mechanisms). Non-standard models may include top-down mechanisms such as absorption of the more variable ultraviolet (UV) part of the solar spectrum and chemistry changes in the middle atmosphere, or cloud formation due to modulations of the galactic cosmic ray flux due to solar-cycle variation of the interplanetary magnetic field [Gray et al., 2010]. If the cyclic signal reported by Camp and Tung  is caused by solar variability, one would have to conclude that the non-standard, and poorly understood, top-down mechanisms are more important than the forcing due to the variations in TSI.
 The goal of this paper is two-fold. The implications of the standard paradigm are explored in terms of very simple energy balance models and data for the different components of the climate forcing and the GMST record for the period 1880–2010. Rather than focusing on advanced statistical analysis, the approach will be to show that any temperature response function that gives a computed temperature record which reflects the main features of the observed GMST record will give a much smaller response to the cyclic TSI signal than reported by Camp and Tung . This is not surprising, and in agreement with the much more sophisticated, but less accessible, work by Stevens and North  and others. The second result, which is a natural by-product of this way to approach the problem, is that stratospheric aerosols due volcanic eruptions have profound effects on global temperature on short (less than a year) as well as longer (several decades) timescales. In particular, the cyclic GMST variations reported by Camp and Tung  is naturally explained as a result of a succession of volcanic eruptions, ending with Mount Pinatubo in 1991.
 This paper does not discuss top-down mechanisms. The focus is on demonstrating that the main features of the GMST record is explicable within the standard paradigm. The scientific debate within this paradigm revolves around the climate response function, in its simplest form parametrized by two constants; the climate sensitivity and the time constant. However, it is generally recognized that there is more than one timescale in the global response, and the scientific debate on this subject is focused around poorly known ocean dynamics; on how, to what extent, and on which timescales heat is redistributed from the well-mixed ocean surface layer to the deep oceans. These issues relate to the question whether it is possible to assign an effective heat capacity to the climate system, which everybody agrees is dominated by the ocean heat uptake, and if possible, how large this heat capacity is. In the present paper we make an attempt to circumvent all questions pertaining to detailed physics of this sort. The idea is to rely only on the equation of radiative balance in its linearized form (without any preconceived or physically motivated idea about values of its coefficients) and then determine these coefficients from the data. The approach is inspired by, and essentially the same as, that of Hansen et al. , but where those authors resort to physically motivated choice of response functions and end up justifying their results with physical arguments, the present paper explores results that follow exclusively from the structure of the energy balance equation, the response function, and the data for forcing and temperature evolution. These differences are discussed in more detail in section 5.
 The structure of the paper is as follows. In section 2 I describe the properties of the energy balance model under the simplifying assumption that one can assign a definite and constant heat capacity C to the climate system, i.e., that the anomaly of the heat content Q of the system can be written in the form ΔQ = CΔT, where ΔT is the global surface temperature anomaly. The resulting equation is a standard linear model equation studied in different areas of elementary college physics, and hence all results described in this section are in principle well known. It may still be useful to present these results in the climate context, and to illustrate how many important aspects of the basic global climate dynamics can be understood from analytical considerations. Section 3 generalizes this to a two-box model, involving the interaction between the mixed ocean surface layer with the deep ocean. This naturally introduces a response function that contains two well-separated timescales and two separate climate sensitivities. Section 4 considers more general response functions and explores two different forms; the instantaneous-plus-exponential response, and the scale-free response, both motivated by observation and/or climate-model experiments. In each section we select model parameters that give good fit of the computed temperature records to the observed GMST time series, and compute the solar-cycle response for these parameters. I also discuss the differences and advantages and disadvantages of the various response models, but emphasize the robustness of the amplitude of the solar cycle response, which seems to be almost the same across all parametrized models. I also point out that the apparent cyclic response over the last few decades remains if I remove the solar forcing, and that it is a result of volcanic forcing. In section 5 I discuss our results in the context of other related work and conclude. In Appendix A I present for completeness a derivation of the energy balance model, with emphasis on explaining how climate feedbacks enter the model parameters.
2. The One-Box Energy Balance Model
 At the core of our thinking about the climate system we find the extremely simple, linear energy balance equation
Here ΔT is the deviation in Earth GMST from a state T0 where the Earth is in radiative balance with an “unperturbed” radiative forcing F0. Thus, F in equation (1) is the deviation from F0. ΔQ is the change in heat content of the climate system when GMST changes by ΔT. Much of the unknown physics of the climate system is associated with the relationship between ΔQ and ΔT. The simplest model is to represent it by ΔQ = CΔT, with C constant in time, representing the effective heat capacity of this system, which is dominated by the heat capacity of the upper few hundred meters of the oceans. The model neglects the heat exchange between this layer and the deep ocean. The term − Seq− 1ΔT represents the change of flux of infrared radiation in response to the temperature change ΔT, and Seq is in first approximation computed from the Stefan-Boltzmann law of radiation. In a second approximation, Seq is corrected for fast feedback processes by a factor f, which is believed to be of the order f ∼ 2 − 3 [Hansen et al., 1985, 1997]. If F changes as a step function F(t) = ΔFH(t), where H(t) is the Heaviside step function, then ΔT will relax to a new equilibrium state,
The parameter Seq measures how much the equilibrium temperature ΔTeq changes in response to a forcing perturbation ΔF, and is therefore called the climate sensitivity.
 For a step-function forcing the time evolution of the relaxation can be written as
where R(t) is the linear response function of the system with time constant τ given by
where and are Fourier transforms of ΔT(t) and F(t), respectively. From these relations one observes that the climate system acts as a low-pass filter for the forcing signal, in the sense that response on timescales longer than the time constant τ is simply obtained by multiplying the forcing with the sensitivity Seq, while response on fast timescales (shorter than τ) is attenuated, and more so the higher the frequency. What this signifies is that for slow forcing components the system manages to stay close to equilibrium and the two terms on the right hand side of equation (1) approximately balance each other, giving ΔT ≈ SeqF. On the other hand, the fast forcing components will have to be balanced by the inertial term on the left, and in the very fast limit ω → ∞ we have ΔT ∼ F/ω → 0. One disturbing consequence is that if the forcing grows rapidly compared to the timescale τ then the radiative loss term will be much smaller than the forcing, and the climate system will accumulate heat. In that case the immediate temperature response on the forcing trend will be weak, but when the growth in the forcing eventually stops the temperature will continue to rise until equilibrium is attained. This delayed warming is what is referred to as “warming in the pipeline” [Hansen et al., 2005, 2008].
 Another observation to make from equation (4) is that frequency components that are faster than τ respond approximately as
indicating that for these high frequencies (ωτ ≫ 1) the response depends only on the ratio Seq/τ = C−1. It is important to have in mind, however, that even if this is valid for fast harmonic components, it does not hold for the response to fast pulses in the forcing, because such pulses also contain slow frequency components (the spectrum of a delta function is flat). The response to such pulses (like volcano forcing) are better studied in the time domain, as shown below by looking in detail at how ΔT(t) responds to a few types of perturbations.
2.1. Step Function and Pulse
 The response to a step in the forcing (like a rapid, but persistent change in infrared opacity due to massive release of chlorofluorocarbon (CFC) gases) is given by the response function R(t) in equation (3), which means that the solution relaxes exponentially toward the new equilibrium with time constant τ. For short times after the step impulse (t ≪ τ) the response is linear in time, R(T) ≈ Seqt/τ, with growth rate Seq/τ. However, as t grows larger than τ, the growth slows and this gives rise to lower frequency components that do not satisfy equation (5), but rather the relation .
 The response to a delta-pulse forcing ΔFδ(t) (in a crude analysis a big volcanic eruption can be described this way) decays exponentially toward the old equilibrium state according to
We observe that the amplitude of the pulse response depends on the ratio Seq/τ, but the duration depends only on τ. Hence the area ∫ΔT(t)dt under the response on the pulse is SeqΔF. Thus the temperature response to the forcing associated with major volcanic eruptions represents a useful tool to determine Seq and τ (and hence also C), provided the one-box model gives an adequate description of the climate response. In principle the area under the response pulse determines Seq, and the amplitude determines Seq/τ = C−1. As will be discussed later, the situation is complicated by factors like uncertainties in the assessment of volcanic aerosol forcing, the cluttering effect of internally generated “climate noise,” and the validity of describing the heat change in the climate system by the simple relation Cd(ΔT)/dt; thus assigning a fixed heat capacity C.
2.2. Linear Trend
 As discussed above it is interesting to know how the temperature responds to a linear trend in the forcing, F(t) = kt, which may be a reasonable model for the anthropogenic forcing through the 20th century. In this case the response is
For τ ≪ t this just gives a delayed response ΔT(t) ≈ Seqk(t − τ) = SeqF(t − τ), i.e., the temperature at time t is close to the equilibrium temperature corresponding to the forcing at time t − τ. In general this means that for a trend in the forcing which has lasted much longer than the time constant (τ ≪ t), the climate system will evolve through states that are close to equilibrium, and hence follow the evolution of the forcing. This will also be the case if the trend is not linear, provided it is approximately linear over the scale τ. In practical terms this means that if the forcing signal is low-pass filtered to yield a signal Ftrend which is smooth on scales τ, then the response to this smooth component will simply be ΔTtrend ≈ SeqFtrend, which is exactly what we observed more generally by looking at the response in the spectral domain.
2.3. Periodic Forcing
 The response to a periodic forcing is interesting for assessing the climate impact of the solar cycle. From equation (1) the reponse to a cyclic forcing F(t) = FAcos(2πt/τc) is
Only if τ ≪ τc/2π will the response have amplitude close to SeqFA, and be in phase with the forcing. If τc is the 11-year sunspot period, this would require a time constant of the order of a year or less. If we consider the example of τ = 10 years the amplitude of the temperature response would be 0.17SeqFA, and for τ = 100 years it is ten times less. In both cases the phase angle θ would be close to −π/2. In the next section we shall see that this means that the climate response to the solar cycle forcing according to the present model should be considerably attenuated compared to the equilibrium response, and hence barely detectable, and the phase should lag the forcing by an angle close to π/2.
2.4. Noisy Forcings
 Some of the forcings of the climate systems do not vary smoothly, but also have irregular variations on all timescales. In particular this is true for the solar forcing which has a strong noisy component. However, this noise is colored, which means that its power spectral density (PSD) depends on frequency like S(ω) ∝ ω1−2H, where H is called the Hurst exponent (for white noise H = 0.5). The autocorrelation function exhibits a power law tail, C(τ) ∝ τ2H−2, and for H > 0.5 the integral ∫ abC(τ)dτ diverges as b → ∞, which is a criterion for long-range persistence [Beran, 1994]. For instance, Rypdal and Rypdal [2012b] demonstrated that the TSI signal is a weakly persistent, non-gaussian noise with Hurst exponent HS = 0.7 ± 0.1. With a forcing where is a colored noise with Hurst exponent HS and unit variance, Fourier transform of equation (1) yields the following relation for the PSD:
For HS = 1/2 (which means that is a white-noise process) equation (1) is the standard Langevin stochastic differential equation and its solution is called the Ornstein-Uhlenbeck stochastic process [Gardiner, 1985]. For HS ≠ 1/2 the corresponding solution is a fractional Ornstein-Uhlenbeck process. Rypdal and Rypdal [2012b] demonstrated how the fractional, non-gaussian noise process can be generated numerically for a discrete-time system and how one can obtain numerical realizations of the discretized version of equation (1).
 Here we are interested in the high-frequencies in the noise (ωτ ≫ 1), for which the PSD in equation (9) takes a power law form . For HS = 0.7 this yields a spectal index β ≡ 2HS + 1 = 2.4. On these short timescales the Sun-driven temperature fluctuation is a persistent fractional motion, since β > 2. Detrended fluctuation analysis of the global temperature records [Lennartz and Bunde, 2009] indicates that land surface temperatures and sea-suface temperatures have different β, with β < 1 (fractional noise) for land and 1 < β < 2 (anti-persistent fractional motion) for sea. The observation that Sun-driven response has β > 2, while the climate noise has β < 2 indicates that the memory effects are generated by internal dynamics in the climate system, and are not imposed by the memory in the solar forcing. Another question is whether the amplitudes of the climate noise are too strong to be explained as a response to the noise in the solar forcing. This question is most easily answered by numerical integration of equation (1) with the appropriate noise added to the forcing, and will be done in the next section. The mismatch in spectral index as well as in amplitude, clearly indicates that the climate noise is internally generated and not a response to fluctuations in solar forcing.
2.5. Fitting the One-Box Model Parameters to Data
 Here and in sections 2.6 and 2.7 I will compute and plot temperature responses to the climate forcing for a selection of simple response models characterized by coefficients representing climate sensitivities and time constants. The “best” choice of these constants will be found by a subjective comparison between the computed temperature evolution and the observed instrumental GMST record. It is certainly possible to infer the sensitivity and the time constant from a nonlinear regression or a maximum likelihood estimate (MLE), but the result will be misleading. The reason for this is the following: the MLE approach assumes that the signal is given by a model. Here, this is the linear response model with Seq and τ as parameters to be determined plus a Gaussian white noise with a variance also to be determined. The noise is supposed to represent the internal fluctuations in the climate system, but these fluctuations is not white noise, as I have discussed in section 2.4. There are, for instance, some multidecadal oscillations which probably are of internal origin, and on shorter timescales the climate noise is not white, but rather pink or red. The measurement bias discussed in the paper by Thompson et al.  gives rise to an anomalous temperature maximum around 1940, and there is at present no authoritative GMST record that is corrected for this bias. This is the likely reason why I never can get a response curve that displays the temperature peak around 1940 with the forcing time series I have used in this paper. Thus, until we have have available a reliable, simple model for the internal fluctuations, and an unbiased data record, an MLE approach is futile.
 A generally accepted data set for GMST is the HadCRUT3gl data [Brohan et al., 2006], which will be used in this paper. (The GMST data can be downloaded from http://www.cru.uea.ac.uk/cru/data/temperature/hadcrut3gl.txt). For the forcing we shall use the data given by Hansen et al. [2011, Figure 1]. This data set is a continuation of the data derived by Hansen et al. . (The forcing data can be downloaded from http://www.columbia.edu/ mhs119/EnergyImbalance/Imbalance.Fig01.txt.) In our Figure 1 I plot the HadCRUT3 GMST data for the time span 1880–2010 along with the total estimated forcing for the same period. Both data sets are given as annual mean values. The original forcing data set is given in units of Wm−2, but in Figure 1 we have multiplied this with a climate sensitivity Seq = 0.75 K/Wm−2, which is the sensitivity held as most probable by the IPCC AR4 [IPCC, 2007]. It thus represents the global temperature we would have observed if the climate responds instantaneously to the forcing. However, it is clear from Figure 1 that the observed temperature response is considerably attenuated compared to the equilibrium response for the small-scale features. The many sharp negative spikes in the forcing signal arise from injection of large amounts of aerosols in the stratosphere from volcanic eruptions. The weak immediate response to these spikes in the forcing indicates that the system time constant must be considerably larger than the duration of the spikes. One plausible explanation of the broad temperature minimum peaking around 1910, but lasting from 1880 till 1930, is that it is a response to the three large volcanic eruptions occurring in that period, in particular the massive 1883 Krakatau event. If this is a correct interpretation the slope of the GMST curve in the period 1960–2010 is probably suppressed due to the large negative forcing from volcanic eruptions during this period, meaning that without these eruptions the anthropogenic warming would have appeared considerably stronger.
 In Figure 2 we plot solutions to equation (1) for Seq = 0.75 K/Wm−2. We have chosen three different time constants, τ = 10, 20, 30 years. Of these the best fit to the GMST curve appears to be the one for τ ≈ 20 years. Recall that equation (6) showed that the amplitude of the response to a very sharp forcing spike is proportional to Seq/τ, while the area under the response curve only depends on Seq. The consequence is that for τ < 20 years the short-time response to negative spikes in the forcing due to volcanic eruptions becomes too large, as shown for the blue curve (τ = 10 years) in the periods 1880–1920 and 1980–2010. The purple curve (τ = 30 years), on the other hand, yields too weak short-time response to volcanic forcing, implying that the responses are smeared out in time, resulting in a too shallow minimum around 1910, too weak volcanic responses in the period 1980–2010 and a too low linear temperature trend in that period.
 Since the ratio Seq/τ is expected to represent the amplitudes of the short-time responses to volcanic forcing it is interesting to investigate how the total response curve changes as we vary climate sensitivity Seq under constant Seq/τ-ratio. In Figure 3 we show the response for Seq = 0.5 K/Wm−2, τ = 13 years (green) and Seq = 1.0 K/Wm−2, τ = 26 years (orange), compared with the red curve from Figure 2 (Seq = 0.75 K/Wm−2, τ = 20 years). We observe that the short-time responses are approximately the same in the three curves, as expected, but for the lower Seq the trend is too low, and for the higher Seq the trend is too high. This supports the IPCC estimate of 0.75 Wm−2 for the climate sensitivity.
2.6. Solar Cycle Temperature Response
 In equation (8) it was shown that with time constant τ = 20 years the solar cycle response is heavily attenuated compared to a response with instantaneous equilibration. In Figure 4 we demonstrate this by solving equation (1) numerically with the solar forcing only (red curve). The purple curve is the solar forcing in Wm−2. The corresponding curve for the instantaneous response (τ = 0) is obtained by multiplying the purple curve by Seq = 0.75 K/Wm−2. Not only is the peak-to-peak amplitude less than 0.02 K, but also the phase is delayed by approximately π/2, both features in accordance with equation (8), and are due to the delayed response from the time constant τ = 20 years. The blue, black, and orange curves in Figure 4 are derived from other response models considered in sections 3 and 4. However, neither of these models yield significantly larger amplitudes of the response, although the instantaneous-plus-exponential response considered in section 4.1 gives almost zero phase delay.
 In Figure 5 the response to the total forcing with the solar forcing subtracted is plotted along with the response to the total forcing for the one-box model. Similar results are obtained from the other models. The difference between the two curves is the response to the solar forcing only. The oscillations on both curves in the period 1959–2004, which were noticed in the observed GMST record by Camp and Tung , could at first sight be interpreted as a solar cycle response, but this cannot be the case since the solar cycle forcing is not present in the blue curve. Nevertheless Camp and Tung  argue that these oscillations with very high statistical confidence are associated with the solar cycle. Their statistical reasoning is based on Monte Carlo simulations of signals with the same PSD but with random phases, noting that the high correlation between the temperature of the solar signal is highly improbable in such an ensemble. This is an example where Bayesian statistical reasoning is highly appropriate. In generating the appropriate statistical ensemble one should take into account all relevant information and not include realizations that are a priori excluded by existing knowledge. The highly relevant existing knowledge in this case is the volcanic forcing, which in this time span was dominated by five eruptions some of which took place few years before solar minimum in four successive solar cycles. These eruptions were Agung (1963), Fernandino (1968), Fuego (1974), El Chichón (1982), and Pinatubo (1991) [Grieser and Schönwiese, 1999]. This becomes clear by looking at Figure 6, where the total forcing, solar forcing, and computed response to the total forcing are plotted together. It is quite apparent that the periodic dips in the time-span 1959–2004 primarily are responses to the negative volcano spikes in the total forcing, and not to the periodic solar forcing. This observation has far-reaching consequences. As shown above a solar cycle response of the magnitude claimed by Camp and Tung  could not be explained by the general variations in total radiative forcing associated with the TSI variation, allowing for the feedback factor that generally applies to radiative forcing (see Appendix A). If the periodic dips were a response to the solar cycle forcing this would have called for strong additional forcing mechanisms associated with the solar cycle, like modulation of cosmic ray flux or selective effects of the UV-part of the solar spectrum, which have much stronger solar-cycle variability than the TSI. The present findings do not exclude that such mechanisms exist, but they make it highly improbable that they are as important as Camp and Tung  claims would imply.
2.7. Internal Dynamics and Climate Noise
 There are some features in the GMST data that are impossible to reproduce with this model and this total forcing; for instance the temperature maximum around 1940 and the following minumum around 1950, and a less signficant multidecadal oscillation around year 1900. Most of the former, in particular the sudden drop in GMST in 1945, has been explained as uncorrected biases in the sea surface temperature records during the Second World War [Thompson et al., 2008]. However, also land temperatures peak, although not as sharply, around 1940, indicating a slow multidecadal oscillation with period around 60 years which is not described by equation (1). Further there are relatively large-amplitude oscillations of periods of 2–5 years usually associated with the El Niño–Southern Oscillation (ENSO) phenomenon and, in monthly GMST records, the climate noise discussed at the end of section 2.4. ENSO is generally recognized as an internal oscillation in the climate system, and the fluctuations on shorter scales are of course not resolved in the description above, since both forcing and response are described with annual resolution. We could, however, assess whether the fluctuations observed on annual to decadal scales in the GMST could possibly arise as a response to fluctuations on corresponding scales in the TSI, by simulating equation (1) with a noise term σFwH(t) with σ given by the standard deviation of the TSI fluctuations on time scales shorter than the sunspot cycle. Such a simulation, with a Gaussian white-noise driving term, yields a response which is an Ornstein-Uhlenbeck process with standard deviation σT ≈ 0.11σF. The TSI forcing standard deviation σF does not exceed the amplitude of the solar cycle TSI forcing, i.e. σF < 0.1 Wm−2, and hence σT < 1 × 10−2 K. This is more than one order of magnitude less than the standard deviation of the annual to decadal fluctuations observed in the GMST, and confirms that the climate noise observed on these timescales arise from volcanoes and internal climate dynamics, and is not driven by noise in the TSI.
3. The Two-Box Energy Balance Model
 It has been known for several decades that atmospheric-ocean general circulation models exhibit climate responses on separated timescales, i.e. there is more than one time constant involved in the response. A simple two-box generalization of equation (1) allows for heat exchange between the upper mixed layer of the ocean and the deep ocean [Held et al., 2010]:
Here ΔT1,2 are the temperatures of the two layers, respectively. C1, 2 are the respective heat capacities, and κ is the coefficient of heat exchange between the two layers. Assuming that the system is in equilibrium at t = 0, i.e. ΔT1(0) = ΔT2(0) = F(0) = 0, the general solution for ΔT1,2 is straightforward but a little bit messy to write out. In the limit C2 ≫ C1, the solution for T1(t) correct to lowest order in the small parameter C1/C2, is very simple and transparent:
with the response function
where we have introduced some new parameters,
These parameters replace the heat capacities C1,2 and the heat coupling constant κ, whose physical meaning is easy to grasp, but hard to measure directly. The meaning of the new parameters is apparent if we consider the response to a step-function forcing F(t) = H(t), for which ΔT1(t) = R(t): since C1/C2 ≪ 1 we have τtr ≪ τeq, and for t ≪ τeq the response is completely dominated by the first term in equation (12), and hence relaxes exponentially with the transient time constant τtr to the new quasi-equilibrium Str, which is denoted the transient climate sensitivity. However, when t approaches τeq the second term comes into play, and there is a new delayed response with time constant τeq giving relaxation to the full radiative equilibrium Seq.
3.1. Fitting the Two-Box Model Parameters to Data
 This interpretation of the constants Seq, τeq, Str, and τtr as parameters characterizing the transient and equilibrium (fast and slow) responses has given rise to attempts to determine them from climate model experiments. In numerical experiments with the Geophysical Fluid Dynamics Laboratory's Climate Model version 2.1 (CM2.1) Held et al.  find a two-time-scale response to a step-function forcing with τtr ≈ 4 years. The slow time constant is hard to determine in this model and requires runs over several centuries. However, τeq is so large (centuries) in this model that it is unimportant in determining the response to the known forcing in a run spanning the 1870–2000 time period. Hence the one-box model with τ = 4 years and an effective climate sensitivity of S = 0.4 K/Wm−2 gives a good fit to the model temperature evolution in this time period. Of course, approximately the same result would be obtained from a two-box model with τtr = 4 years, Str = 0.4 K/Wm−2, and τeq ∼ centuries. This is not consistent with our results presented in Figures 2 and 3, where the best fit to the observed GMST data was found by choosing τ = 20 years and S = 0.75 K/Wm−2, although Figure 4 showed that we could obtain a reasonable fit by reducing τ and S by the same factor, e.g. for τ ≈ 10 years and S = 0.4 K/Wm−2. The reason for this inconsistency is that we are fitting to the observed GMST, while Held et al. [2010, Figure 3] are fitting to the model results. One major difference between observed and model data is that the model gives a relatively sharp and deep dip in the temperature evolution as a response to the Krakatau eruption in 1883, somewhat similar to our blue curve for τ = 10 years, S = 0.75 K/Wm−2 in Figure 3, but even sharper. This sharp and deep response is absent in the GMST data, which exhibits a broad temperature depression spanning four decades succeeding Krakatau. The only way one can reproduce the observed data from a two-box model (or from any linear response model) is to assume that the slow response is on decadal, not centennial, timescale. In Figure 7 we have plotted the global temperature response for the one-box model with τ = 4 years and S = 0.4 K/Wm−2. This is the same as shown by Held et al. [2010, Figure 3]. The red curve is the global temperature response for the two-box model with τtr = 4 years, τeq = 20 years, Str = 0.3 K/Wm−2, and Seq = 1.0 K/Wm−2, and shows that a better overall fit is produced by adding a slow response on decadal timescale to the fast response with time constant of 4 years. These parameters are well inside the confidence limits established in the statistical study of Padilla et al. . A weakness of that study, however, is that it uses data only from year 1900 onwards, and hence leaves out the information contained in the response to the Krakatau eruption.
 The two-box model response to the solar forcing alone, with the same parameters as used for the red curve in Figure 7 is shown by the blue curve in Figure 4. The peak-to-peak amplitude is approximately the same (≈ 0.02 K) as for the one-box model (red curve) shown in Figure 4, and the phase is still lagged by approximately π/2 with respect to the solar forcing and the cyclic temperature signal found by Camp and Tung .
where is a linear operator. The response function can be defined as the solution to the equation
but another convention is to define it as the solution to
where δ(t) is the Dirac delta function and G(t) is the Greensfunction for the operator . The connection between the two response functions is G(t) = dR(t)/dt. The response to the general forcing F(t) can be written as a convolution integral,
This solution holds even if the GMST response to energy input does not have the simple form Cd(ΔT)/dt, provided the operator is linear. This allows for introduction of other response functions than exponentials, and this is the approach employed by Hansen et al. .
 In this section we examine the potential of such general response models in generating a stronger solar cycle response, under the constraint that the response on the total forcing still fits well to the observed GMST.
4.1. Instantaneous-Plus-Exponential Response
 First we shall idealize the response function to consist of an instantaneous response followed by an exponential relaxation,
where r denotes the fraction of the total response which is instantaneous. Note that this case is equivalent to the two-box model with τtr = 0, Str = rSeq, and τeq = τ. A good fit seems to be obtained for r = 0.15 and τ = 30 years. This solution is shown in Figure 8. In some respects it is a better fit than the one obtained for the one-box model (corresponding to r = 0 and τ = 20) in Figure 2, and perhaps also better than for the two-box model (red curve) in Figure 7, because it spells out more clearly the fast observed volcano responses. Another reason to favor these parameters to those corresponding to the red curve in Figure 7 (for which the transient response parameters were motivated by the climate model results of Held et al. ) could be the short τtr found by Douglass and Knox  in their study of the observed climate response to the Mount Pinatubo eruption in 1991. Their claimed values were τtr ≈ 0.6 years and Str = 0.15 K/Wm−2, although these results have been strongly contested by Wigley et al.  and Robock .
 With the response function given by equation (18) the response to a periodic forcing FAcos(2πt/τc) given in equation (8) will generalize to
where ρ and θ are defined in equation (8). With r = 0.15, τ = 30 years and τc = 11 years the factor , so the signal will be dominated by the first term. Hence, the solar cycle response will be nearly in phase with the forcing with peak-to-peak amplitude approximately 2SeqrFA ≈ 0.016 K, using FA = 0.07 W/m2. Thus, the amplitude of the solar cycle response is approximately the same as we found with the one-box model, but now the temperature signal is dominated by the instantaneous response and hence nearly in phase with the solar cycle forcing. The numerically computed response is given by the black curve in Figure 4. The important message is that all response models considered so far give us a solar-cycle response which is one order of magnitude smaller than the signal claimed by Camp and Tung , and one order of magnitude smaller than the oscillations induced by the volcano forcing.
4.2. Scale-Free Responses
 It was mentioned in section 2 that global temperature variability on scales up to a decade is dominated by scale-free (self-similar) fluctuations when proper detrending is performed by means of the detrended fluctuation analysis method (DFA) [Lennartz and Bunde, 2009]. These results have been confirmed and the error bars reduced in self-similarity exponent and trend coefficients by means of a maximum likelihood estimation (MLE). Analysis of the northern hemisphere temperature reconstructions of Moberg et al.  (last two millennia) and Mann et al.  (last 1500 years) indicates that this scale-free long-range memory extends to timescales up to several centuries [Rypdal and Rypdal, 2012a]. The interpretation of such analysis is not straightforward, however, since the internal climate variability (for which we want to explore memory properties) is strongly disturbed by the erratic volcano forcing and may also be influenced by the memory in the solar forcing. Nevertheless, since there exist reconstructions of this forcing at least for the last millennium, it is possible to include this forcing in a dynamic-stochastic model whose parameters can be determined by the MLE-method. Memory on centennial timescales obviously involves slow feedback mechanisms such as albedo changes due to melting of ice sheets and CO2 emission from warming oceans. In section 3 I showed that the long time constant of the climate system effectively attenuates the fluctuations of external forcing on sub-decadal timescales like the fluctuations in TSI, and concluded that the observed climate noise must be internally generated. This does not preclude, however, the possibility to conceptually decompose the climate system into (an internally generated) stochastic forcing and a global temperature that responds dynamically to this forcing. If the stochastic forcing is assumed to be uncorrelated (white), the long-range memory of the global temperature may be introduced via the dynamical response, i.e., through the response function.
 One way to generate a long-range correlated self-similar stochastic process is to use the Riemann-Liouville integral;
where α > 0, Γ(α) is the Euler Gamma function, and F(t) is the sum of the external and internal forcing function. The notation Dt−αF(t) refers to a fractional derivative, here of negative order −α and therefore really an integral. For positive order the fractional derivative is defined as
where [α] denotes the smallest integer greater than or equal to α, implying that 0 ≤ [α] − α < 1. If, for instance, 0 < α < 1 equations (20) and (21) yield that we can write the following fractional (long-memory) generalization of equation (1):
If a self-similar process is generated by an integral like equation (20) with F(t) a Gaussian white noise the self-similarity exponent of that process (which is the Hurst exponent of the differentiated process) is H = α − 1/2. To generate a self-similar fractional Brownian motion the forcing F(t) is assumed to be a Gaussian white noise process and the Riemann-Liouville integral has to be slightly generalized to avoid overemphasis of the vicinity of the point t0. This integral has the form: [Γ(H + 1/2)]−1 ∫∞0[(t − t′)H−1/2 − (− t′)H−1/2]F(t′) dt′ + ∫0t (t − t′)H−1/2F(t′)dt′ [Mandelbrot and van Ness, 1968; Embrechts and Maejima, 2002]. It generates a self-similar process only if 0 < H < 1 (i.e. if 1/2 < α < 3/2). The integral in equation (20) converges in the singularity at t′ = t0 only if α > 0. The main purpose of this little excursion into the mathematics of stochastic processes is to motivate exploring the implications of scale-free (power law) response functions of the form
Since R(t) is the response to step-function increase in the forcing at t = 0 we are immediately presented with a paradox, since the response grows without limits as t → ∞. This breaks with the notion that the response to a step in the forcing is a relaxation to a new equilibrium ΔT = SeqΔF. But before dismissing the idea of a scale-free response one should not forget that equation (1) is derived without taking into account slow feedbacks (see Appendix A). Such slow feedbacks can be described in different ways, for instance as a time dependence of the forcings of internal origin induced by external orbital or anthropogenic forcing, or as a time dependence of the feedback parameter. In either case, the effect is that the temperature that was in equilibrium with the forcing a few decades ago is not in equilibrium with the same forcing today. This means that in times of fast climate change, e.g. in periods of glaciation or de-glaciation, the radiative equilibrium may be a moving target and that a measure of climate sensitivity as a single number may make little sense. Like everywhere else in nature where scale-invariance is encountered, it holds only over a certain range of scales. If analysis shows that there is an approximate scale invariance in the internal dynamics of the global temperature fluctuations, and a power law response function proves to be good way to model how today's climate responds to known variations in forcing over the last centuries, then all we need to assume is that this scale invariance holds on these timescales.
 In Figure 9 I have plotted the temperature response to the total forcing for a scale-free response function for α = 0.25, 0.50, and 0.75, corresponding to H = −0.25, 0, and 0.25. Since the climate sensitivity Seq has little meaning in this case, the temperature has been found by convolving the forcing given in units of Wm−2 with G(t) given by equation (23) and then by choosing the coefficient k to give a good fit to the trend of the GMST record during 1970–2010. In Figure 9 we have chosen k = 0.06 for all three values of α. For α = 0.25 (blue) we obtain a temperature record very similar to the two-timescale record with r = 0.15 and τ = 30 years shown in Figure 8, although the spikes in response to the volcanoes are somewhat larger, and clearly more pronounced than in the observed GMST record. The record for α = 0.75 gives a too deep and broad minimum around 1910 and too strong trend since 1960. The one obtained for α = 0.5, however, represents the best fit to the observed record of all the response functions we have considered in this paper. The large-scale features are as good as for the one-timescale exponential response function with τ = 20 years, but the phase and width of the volcano responses show a much better fit, and is also better than the one for the instantaneous-plus-exponential response with r = 0.15 and τ = 30 years.
 The orange curve in Figure 4 shows the scale-free response to the solar forcing only, with the parameters k = 0.06 and α = 0.5 (the parameters corresponding to the red curve in Figure 9). The result is almost identical to what we obtained for the two-box model (the blue curve in Figure 4). Thus the response on time-scales of the solar cycle is essentially the same for the two response models. The difference between exponential and power law relaxation models are apparent only when longer timescales are considered.
 In Figure 10 I show the estimated temperature evolution in the time span 1880–2100 computed from a total forcing which is the historical forcing up to 2010, and which is assumed to stay frozen at the 2010-level for the remainder of the century. Thus the temperature rise after 2010 is to be considered as “warming in the pipeline” [Hansen et al., 2008]. The red curve is computed from the one-box response function that gave the best fit to the observed GMST-curve for 1880–2010 (Seq = 0.75 Wm−2 and τ = 20 years), and yields a saturated growth in temperature of approximately 0.5 K. A similar curve for the evolution for the 21st century is obtained for the instantaneous-plus-exponential response model (black curve). The two-box model (blue) yields a temperature growth of more than 1.0 K, which is due to the higher equilibrium sensitivity (Seq = 1.0 K/Wm−2) we had to use in this model to obtain the best fit to the GMST record. The orange curve is computed from the scale-free response function for k = 0.06 and α = 0.5 and yields an unsaturated increase of 1.5 K before the 21st century is over.
 These results should be interpreted with caution, and should not be perceived as an attempt to present “predictions” on the centennial timescale, but rather as an assessment of the uncertainty in the estimated “pipeline warming.” It is worth noticing, however, that the best representation of the temperature evolution for the time span 1880–2010 among the models considered here seems to be obtained by a scale-free response function with α = 0.5. The reason for this is that the singularity of the response function G(T) = ktα−1 describes well the fast response observed after volcanic eruptions, and the long power law tail in the response function provides the necessary long-time responses to provide the large-scale features and trends of the observed GMST record. However, this record would be reproduced well even if the power law tail is truncated after a few decades, but such a truncation would lead to a saturation of the temperature rise on the century timescale and give a future evolution more like the blue curve in Figure 10. Thus, the orange curve should be considered as an upper limit on the warming in the pipeline, which will materialize only if it turns out that the long-range temporal dependence in the climate system extends to centennial timescales. At present, we cannot answer this question with certainty, but careful analysis and stochastic modeling of paleoclimate data could bring us closer. Certainly analysis of results from centennial time-scale simulations of atmospheric-ocean general circulation models, including carbon-cycle and ice sheet dynamics, would be complementary and increase the confidence in such estimates.
5. Discussion and Conclusions
 So far, estimation of climate sensitivity and time constant from physical considerations has proven difficult, and this has given rise to the idea that these parameters could be computed empirically from the GMST time series without any information about the forcing. Schwartz  sets out to determine τ from the autocorrelation function of the linearly detrended GMST time series by invoking the fluctuation-dissipation theorem. In the language of stochastic differential equations this corresponds to representing F in equation (1) as a stochastic Gaussian white-noise force term σw(t), where w(t) has unit variance. Then equation (1) is reduced to the standard Langevin equation which has the Ornstein-Uhlenbeck stochastic process as its solution.
 An interesting question that may be asked in connection with such a model of the detrended GMST is whether F should be interpreted as external forcing, or as internally generated climate noise. While undoubtedly both effects are present, itwas shown in section 2.7 that most of the stochastic fluctuations in the GMST on timescales up to a decade are internally generated, and would be present in the same form even in the case of completely constant external forcing. This assumption would imply that the total heat content of the climate system fluctuates, even in the presence of constant external forcing, and that means that the internal stochastic dynamics of the climate system leads to internally generated fluctuations in the flux of infrared energy radiated to space. Thus internally generated fluctuations do not conserve energy, and this justifies to describe its effect by including a stochastic component in the forcing F(t).
 If equation (1) were correct as it stands, the autocorrelation function would be exponential (r(Δt) ∼ e−Δt/τ) and we would have that Δt/ln r(Δt) = τ = constant. Instead, Schwartz  finds that this quantity grows proportionally to Δt for 0 < Δt < 8 years, before it stabilizes at value around τ ≈ 5 years. He does not consider the possibility that the GMST could be a long-range correlated stochastic process on timescales up to a decade, and that the correlations measured may be associated with the long-range memory in the internally generated fluctuations and thus does not measure the time constant of the climate system's response to external fluctuations. In section 4.2 we showed that such long-range memory could be modeled in terms of fractional derivatives.
 The methods employed in this paper are essentially the same as the Greens-function technique employed by Hansen et al.  and the same data have been used. The difference is in the response functions used. Their starting point is a fit to the response function established from the Russell atmosphere-ocean model [Russell et al., 1995], which they call the slow response function. Then they also investigate an intermediate and a fast response function and argue that they have greater faith in the intermediate version. Their response functions all start at t = 1 year with a response that is 15% of the full equilibration. This should imply that they have an instantaneous response component with r = 0.15, which is exactly the value found in section 4.2 in the present paper to give an optimal fit to the GMST record. The computed temperature record from the intermediate response function model is found as the red curve in Hansen et al. [2011, Figure 7b], and should be compared to the red curves in Figures 7, 8, and 9 in the present paper. They are rather similar, although our curves give better representation of the wide minimum in the period 1880–1930, which is generated as a response to the two large volcanic eruptions in this period. This is due to the relatively slow relaxation on intermediate (decadal) scales in my models. The difference in how the two classes of models treat the long-term response to volcanic eruptions are seen in the red curve of Hansen et al. [2011, Figure 18] and the curves in my Figure 11. In the former, recovery is fast, and the response temperature is nearly constant and close to equilibrium throughout the geologically quiet period 1920–1960, while the traces in my Figure 11 shows an increasing trend due to recovery after the volcanoes in the decades before. More important, however, is that while the rapid recovery in the Hansen et al.  model leaves the total response to the repeated sequence of volcanic eruptions close to zero, our models yield a negative temperature trend of − 0.3 ± 0.1 K from this forcing over the total record. This means that a burst of repeated volcanic eruptions may drive the climate system away from equilibrium and lead to excessive warming in succeeding quiet periods.
Figure 4 shows a temperature increase of approximately 0.15 ± 0.03 K over the entire temperature record deriving from a slow increase in solar forcing over the last century. The results described above indicate that this warming is more than compensated (by a factor 2) by the cooling from volcanic aerosols over the same period. Note also, that those response models with the longer slow response times yield stronger solar warming as well as stronger volcanic cooling.
 Notwithstanding the differences in the response to volcanic forcing on centennial timescale, there is no essential discrepancy between our response models and the model of Hansen et al.  when it comes to the response to solar forcing over the solar cycle. In all the transient response dominates on the timescale of the sunspot period and the peak-to-peak amplitude is of the order 0.02 K. This seems to be a very robust result for all response functions that give reasonable agreement between the computed and observed temperature records. It is also clear that the oscillations in phase with the solar forcing with peak-to-peak amplitude close to 0.2 K observed over the last three solar cycles are due to volcanic forcing, as is apparent from Figures 5 and 6. This result is consistent with the finding of North and Stevens , who find a highly significant GMST signal attributed to volcanic aerosols and a pronounced peak at a period of approximately 10 years in the power spectral density of this forcing over the last century. Thus, the strong correlation between the solar signal and the GMST-signal found by Camp and Tung  should not be interpreted as a causal relationship, and does not lend convincing support to the view that the solar-cycle signal in the GMST record is stronger than what can be explained by the bottom-up TSI mechanism alone.
 Here we present for completeness a derivation of equation equation (1). The derivation follows that of Schwartz  quite closely, but we present some more detail on points of physical interest, elaborating on the nontrivial distinction between climate forcing and feedbacks.
 In the crudest approximation one can consider that most of the heat absorbed by the Earth's surface and atmosphere is instantly deposited uniformly in in a well-mixed layer of the world's oceans. This layer is separated from the cold and non-mixing deep ocean by the thermocline. Supposing the heat capacity per m2 of this mixed layer is CS, the amount of heat deposited per m2s is given by
Here TS is the global mean sea-surface temperature (SST) and TA is the temperature at the top of the atmosphere, more precisely in the lowest atmospheric layer from which infrared radiation can escape to outer space without being reabsorbed. The first term on the right hand side of equation (6) represents the radiative loss per m2s given by Stefan-Boltzmann's law, where σB is the Stefan-Boltzmann constant. One way to understand the greenhouse effect is to realize that a perturbation in the form of increased concentration of greenhouse gases will have to increase the height of this layer. At this increased height the atmospheric temperature is lower, and the radiative loss is reduced. Since in radiative equilibrium the escaping radiative flux must balance the incoming mean flux density I absorbed by the Earth's surface, the temperature TA at the new height will be increased to the previous value at the lower height by heating due to the radiative imbalance. Assuming that the temperature gradient (the lapse rate) in the atmosphere is unchanged after the new equilibrium is attained this also implies an increased temperature T at the surface after equilibration.
 Let us assume that α is the ratio between TA and the global mean surface temperature T, i.e. TA = αT. Note that α depends on the amount of greenhouse gases in the atmosphere. Assume further that β is the ratio between the global mean sea-surface temperature TS and the global mean surface temperature (land+ocean) T, i.e. TS = βT. Finally we denote by γ the ratio between the mean incoming solar radiative flux density J and the absorbed flux density, i.e. I = γJ. About 30% of the incoming radiation is reflected by atmosphere and surface, so the effective coalbedo is γ ≈ 0.7. With these defintions equation (A1) can be written as a differential equation for the global mean surface temperature:
Now denote by T0, J0, α0, β0, γ0 the preindustrial values of these parameters (the values by year 1750), such that the equilibrium equation reduces to
Now, write by Taylor expansion in ΔT, retaining only terms to first order, the perturbed quantities (the present values) in the form
Here ΔT denotes the perturbation of global mean surface temperature since 1880, and JF the corresponding change in solar radiative flux. JF is exclusively due to changes in the solar constant, and therefore represents a climate forcing. In the expressions above α′ is short hand for dα/dT, so the term α′ΔT denotes the change in α due to the change ΔT in global temperature. Since this part of the change in α is a reponse to the change in the climate state it is called a climate feedback effect. The term αF denotes the change in α directly deriving from climate forcing. The same holds for the perturbation terms of β and γ. Let us briefly review the main forcings and feedbacks that contribute to these perturbations. We have already mentioned changes in solar irradiation JF as a primary forcing. The change in α is due to changes in atmospheric composition, such as greenhouse gases. The most logical would be to define the forcing contribution αF as those due to contributions from gases added or subtracted from sources that are unrelated to change in ΔT, e.g. CO2 from human activity or volcanoes. The other contribution α′ΔT would then be due to greenhouse gases emitted from ocean or vegetation cover in response to global warming, or to changed vater vapor content and cloudiness resulting from such warming. However, because it is easy to measure the total CO2 concentration in the atmosphere, but more difficult to assess what fraction of this that originates from human activities and volcanoes, the convention is to consider all increase in atmospheric CO2 as a forcing, and not as a feedback. Land use changes could give rise to forced changes in sea/(land+sea) ratio β, but the main contribution to perturbation of β would be through feedbacks like change in cloudiness in response to sea-surface warming. An important climate forcing is change in coalbedo γF arising from aerosols from human and geological sources (volcanoes), and from land-use change, black soot and change in ice cover unrelated to global temperature change. The total effect of this forcing is at present poorly known. Coalbedo feedbacks γ′ΔT include reduced snow and ice albedo, change in cloud albedo, and change in albedo from change in vegetation cover resulting from global warming.
This allows us to write the climate sensitivity S in the form
is called the climate feedback parameter since it contains all the first order corrections to the coefficients in the energy balance equation that occur in response to the perturbed temperature. Note that since τeq = CSeq the feedbacks also influence the time constant. From equation (A7) we observe that feedbacks occur from change in α (greenhouse gases) and γ (albedo changes). From the expression for the forcing ΔF in equation (A5) it appears that forcing takes place through changes in albedo, solar irradiance and greenhouse gas concentration.
 The feedbacks considered here are so-called fast feedbacks. A more problematic class of feedbacks are those occurring on slower timescales and therefore cannot be accounted for as instantaneous corrections to the coefficients α, β, and γ. When we wrote ΔQ = CSdTs/dt in equation (A1) we assumed implicitly that the sea surface temperature response to the global heat flux is linear. This is reasonable if one assumes that the heat is rapidly distributed in the mixed layer, and there is no heat exchange with the deep ocean. The two-box model allows for such heat exchange, but still leads to a linear response. On centennial timescales, for which ocean heating and ice cap melting could influence the thermohaline circulation, this response is most likely not linear. This is a limitation of the approach presented in this paper when it comes to prediction on those timescales, but does not represent a severe problem for assessing the climate response to the solar cycle. The expansion leading to equation (A4) also represents a linearization which is justified the observation that ΔT/T < 10−2 in the Holocene. Going beyond linear terms in the Taylor expansions would give rise to less than one percent correction in the temperature rate of change, and this would not change the main conclusions of this paper as long as we do not consider timescales much longer than the time constant derived from the linear approach. Forcing due to non-standard mechanisms such as clouds produced by cosmic radiation or change in atmospheric chemistry due to variability of the solar UV spectrum can in principle described by this model by inclusion of additional forcing. However, the main point in this paper is to show that inclusion of such forcings is not necessary to explain the observed GMST record.