Equation (1) has the form
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. .
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 .
Figure 8. (a) The blue curve is the global temperature response for Seq = 0.75 K/Wm−2 in an instantaneous-plus-exponential model with instantaneous response ratio r = 0.15 and time constant for the slow response τ = 30 years. The red curve is the global temperature response for Seq = 0.75 K/Wm−2 in a one-box model with time constant τ = 20 years. The black curve is the observed GMST. (b) A close-up of Figure 8a for the last 40 years.
Download figure to PowerPoint
 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.
Figure 9. (a) Global temperature records computed from the total forcing with scale-free response functions for α = 0.25 (blue), α = 0.50 (red), and α = 0.75 (green). (b) Close-up of Figure 9a for the last 40 years.
Download figure to PowerPoint
 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.
Figure 10. Global temperature records computed from the total forcing with the forcing frozen at the level of 2010 for the rest of the 21st century. The curves are responses according to the four different response models considered in this paper, for which the response to solar forcing was given in Figure 3. Red curve, one-box model with Seq = 0.75 K/Wm−2 and τ = 20 years. Blue curve, two-box model with Seq = 1.0 K/Wm−2, Str = 0.3 K/Wm−2, τeq = 20 years, and τtr = 4 years. Black curve, instantaneous-plus-exponential response with Seq = 0.75 K/Wm−2, τ = 30 years, and r = 0.15. Orange curve, scale-free model with k = 0.06 and α = 0.5.
Download figure to PowerPoint
 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.