## 1. Calculations

*c*is the heat capacity of the ocean mixed layer,

*T*is the surface temperature anomaly, −

*αT*is a linearized radiative response to temperature,

*N*represents random fluctuations in outgoing energy from the earth,

*f*represents radiative forcings, and

*S*represents internal heating anomalies such as heat exchange with the deep ocean. If the forcings

*f*are accurately known, they can be accounted for before the analysis and do not introduce error into the solution. SB08 show that in the context of this model regressions of outgoing radiation against surface temperature give a slope that differs from

*α*so that the climate sensitivity inferred from the regression differs from the true sensitivity. They go on to use Clouds and the Earth’s Radiant Energy System (CERES) and Tropical Rainfall Measuring Mission (TRMM) data in their model to suggest significant difference between the slope and

*α*. Here we do not discuss the appropriate and inappropriate uses of a simple linear model (Murphy 2010) but instead focus on the choice of parameters.

*α*by the amount:

*b*is the difference between the slope and

_{α}*α*and the primes indicate departures from mean values. Equation (2) differs slightly from SB08, who assume mean

*N*and

*T*values are zero. However, even though

*N*and

*T*nominally have zero mean in their model, in any given simulation statistical noise yields means that are slightly different than zero, and the more correct expression for

*b*is that given here.

_{α}*N*and

*S*terms and then solved the equations numerically. In fact, Eqs. (1) and (2) can be solved analytically for this type of noise and for an infinite time period. Details are given in the appendix. The result is that

*b*can be given by

_{α}*N*

_{0}and

*S*

_{0}are the amplitudes of the Gaussian noise. Note that over an infinite time period the dimensionless difference

*b*/

_{α}*α*is a function only of the relative amplitudes of

*N*and

*S*and is independent of

*α*and

*c*.

We also replicated SB08’s calculations by using Eq. (1) to numerically generate 100-yr temperature time series with random daily noise for *N* and *S*. Differences were then calculated from Eq. (2) over an 80-yr period, as specified in SB08. Seasonal cycles were removed from the model time series for the best comparison with the observed standard deviations, which are computed after removing seasonal cycles. This is a small correction because in Eq. (1) variations with periods that are shorter than *τ* = *c*/*α* are damped and *τ* is greater than 1 yr for all plausible cases. Results shown (Fig. 1) are the average of 4800 box-model runs for each combination of *α* and the ratio *N*/*S*. The numerical results differ from the analytic solution because they are computed over a finite time period. Differences computed over shorter periods are further from the analytic solution than the 80-yr results shown in the figure. As discussed in the next section, subsets of the model runs that match observations are marked with points.

For *α* ≥ 2.5 we were able to replicate the curves in Fig. 2b of SB08 to within the size of points on their figure. For smaller values of *α*, we believe that points in their figure are plotted incorrectly. In particular, given Eq. (3), it is difficult to understand how *b _{α}* can have positive values.

## 2. Scaling to observations

The free parameters in Eq. (1) are *α*, *c*, *N _{o}*, and

*S*

_{0}. SB08 select a single value for

*c*and then scale

*N*

_{0}and

*S*

_{0}to match the standard deviations of CERES and TRMM satellite instruments for the tropical oceans between March 2000 and December 2005. Although purely tropical datasets are not the best choice for understanding global response (e.g., Murphy et al. 2009), for illustration we use the same datasets as SB08 to show how they overestimated the difference between the regression slopes and

*α*.

### a. Heat capacity

For the heat capacity parameter *c* SB08 use the heat capacity of a 50-m ocean mixed layer. This is too shallow to be realistic. Because heat slowly penetrates deeper into the ocean, an appropriate depth for heat capacity depends on the length of the period over which Eq. (1) is being applied (Watterson 2000; Held et al. 2010). For 80-yr global climate model runs, Gregory (2000) derived an optimum mixed layer depth of 150 m. Watterson (2000) found an initial global heat capacity equivalent to a mixed layer of 200 m and larger values for longer simulations. Held et al. (2010) found an initial time constant *τ* = *c*/*α* of about four yr in the Geophysical Fluid Dynamics Laboratory global climate model. Schwartz (2007) used historical data to estimate a globally averaged mixed layer depth of 150 m, or 106 m if the earth were only ocean. He also used observations of historical temperatures to constrain *τ* for use in a single box description of the earth, such as Eq. (1). His revised value (Schwartz 2008) is *τ* = 8.5 ± 2.5 yr, which may still be an underestimate (Scafetta 2008).

SB08 used 80-yr simulations. Their use of *c* from a 50-m ocean mixed layer along with their example value of *α*, 3.5 W m^{−2} K^{−1}, yields *τ* = 2.5 yr, which is well outside the range of previously published results. The value of *c* does not directly change *b _{α}*. However, a more appropriate value of

*c*reduces the temperature fluctuations associated with given values of

*N*

_{0}and

*S*

_{0}. We show below the effect of changing the mixed layer depth from 50 to 110 m.

### b. Standard deviation of outgoing radiation

For the parameter *N*, SB08 use a random daily shortwave flux scaled so that the standard deviation of monthly averages of outgoing radiation (*N* − *αT*) is 1.3 W m^{−2}. They base this on the standard deviation of CERES shortwave data between March 2000 and December 2005 for the oceans between 20°N and 20°S. We have analyzed the same dataset and find that, after the seasonal cycle and slow changes in forcing are removed, the standard deviation of monthly means of the shortwave radiation is 1.24 W m^{−2}, close to the 1.3 W m^{−2} specified by SB08.

However, longwave (infrared) radiation changes the energy budget just as effectively from the earth as shortwave radiation (reflected sunlight). Cloud systems that might induce random fluctuations in reflected sunlight also change outgoing longwave radiation. In addition, the feedback parameter *α* is due to both longwave and shortwave radiation. Modeled total outgoing radiation should therefore be compared with the observed sum of longwave and shortwave outgoing radiation, not just the shortwave component. The standard deviation of the sum of longwave and shortwave radiation in the same CERES dataset is 0.94 W m^{−2}. Even this is an upper limit, since imperfect spatial sampling and instrument noise contribute to the standard deviation. In addition, SB08 set the parameter *f* in Eq. (1) to zero on the premise that known forcings are either constant or can be removed from the data before calculations are made, as in Forster and Gregory (2006). Any variability in *f* would also contribute to the standard deviation of the observed outgoing radiation. We therefore use 0.94 W m^{−2} as an upper limit to the standard deviation of outgoing radiation over the tropical oceans. For comparison, the standard deviation of the *global* CERES outgoing radiation is about 0.55 W m^{−2}.

### c. Standard deviation of temperature

For the parameter *S,* SB08 specify that the amplitude of random fluctuations in *S* was varied until the standard deviation of monthly average temperatures matched 0.134 K. This value was taken from TRMM measurements of sea surface temperature for the same period and latitude band as the CERES data. After removing the seasonal cycle, we also obtain a standard deviation very close to 0.134 K for both the TRMM and Met Office Hadley Centre sea surface temperature records between 20°N and 20°S between March 2000 and December 2005, a 70-month period.

The issue with the standard deviation of the temperature is that Eq. (1) generates a time series similar to a random walk. The standard deviation is a function of the period over which it is computed (Fig. 2). SB08 do not specifically state the period over which they obtained a temperature standard deviation of 0.134 K, but it appears to be 80 yr, the period they state for calculating *b _{α}*. We computed standard deviations of temperature for both 80-yr and successive 70-month series of 31-day averages. We can approximately replicate the marked values of

*N*

_{0}/

*S*

_{0}in their Fig. 2 with the 80-yr standard deviations (Fig. 3a). However, for a proper comparison, the magnitude of

*S*

_{0}should be varied to match the standard deviation of TRMM temperatures over 70-month periods, the same as the observations.

Figure 3 shows *b _{α}* as a function of the box-model feedback parameter

*α*for various combinations of input parameters. The curves show calculations with our best reading of the parameters used by SB08 as well as changing the heat capacity, standard deviation of outgoing radiation, and period for computing standard deviations of temperature. Last, curves are shown for the small effect of computing the

*b*over 80-yr or 70-month intervals. Determining which of these is appropriate depends on the comparison: Forster and Taylor (2006) used regressions from 100-yr model runs, whereas Forster and Gregory (2006) and Murphy et al. (2009) used shorter periods of Earth Radiation Budget Experiment (ERBE) and/or CERES data.

_{α}## 3. Discussion and conclusions

The reason that our values of *b _{α}* are much smaller than those quoted in SB08 can be understood in terms of the dimensionless parameter

*N*

_{0}/

*S*

_{0}, which is the ratio of the amplitude of random fluctuations of outgoing radiation to the fluctuations in internal energy transport. First, the CERES observations suggest a smaller value of

*N*

_{0}than that used by SB08. Second, we use a deeper mixed layer that damps the temperature fluctuations and thus the system requires more internal energy transport

*S*

_{0}to match observed temperature fluctuations. Last,

*S*

_{0}must be increased even further to generate temperature fluctuations on the 70-month period that matches the observations. All three changes individually reduce

*N*

_{0}/

*S*

_{0}so that together there is a substantial reduction in

*N*

_{0}/

*S*

_{0}(Fig. 3a). For small

*N*

_{0}/

*S*

_{0}, the difference

*b*is approximately quadratic in

_{α}*N*

_{0}/

*S*

_{0}[Eq. (3)], so

*b*is reduced by a larger factor than the change in

_{α}*N*

_{0}/

*S*

_{0}.

Although highly simplified, a single box model of the earth has some pedagogic value. One must remember that the heat capacity *c* and feedback parameter *α* are not really constants, since heat penetrates more deeply into the ocean on long time scales and there are fast and slow climate feedbacks (Knutti et al. 2008). It is tempting to add a few more boxes to account for land, ocean, different latitudes, and so forth. Adding more boxes to an energy balance model can be problematic because one must ensure that the boxes are connected in a physically consistent way. A good option is to instead consider a global climate model that has many boxes connected in a physically consistent manner.

The Hadley Centre Slab Climate Model, version 3 (HadSM3) is such a model. It has a 50-m slab ocean that can be compared with the 50-m mixed layer in SB08. Gregory et al. (2004) compared the feedback parameter *α* derived from a 20-yr regression of the global radiation balance against global surface temperature with that derived from the equilibrium temperature change, reached in about 30 yr with a shallow slab ocean. The global values—0.99 ± 0.07 and 1.04 ± 0.01 W m^{−2} K^{−1}, respectively—are not statistically different from each other. The small difference between these two numbers is in contrast to the bias of approximately −0.4 W m^{−2} K^{−1} that would be predicted by applying the results from SB08. A caveat to this model regression comparison is that internal variability in a slab ocean model is not directly comparable to the internal variability in either a coupled climate model or the simple SB08 model. Nevertheless, we discuss it here to show that regression remains a useful and accurate tool for these kinds of global analyses, in contrast to the conclusion of SB08.

Global climate models with coupled oceans often show that correlations between surface temperature and outgoing radiation over a few decades underestimate the equilibrium temperature change (Gregory et al. 2004; Williams et al. 2008; Held et al. 2010). The combination of these slow climate responses and the small values of *b _{α}* found here suggests that observed regressions between temperature and radiative imbalance underestimate long-term temperature change rather than overestimate it, as inferred by SB08. For example, the short-term regression derived from ERBE data in Forster and Gregory (2006) yields a smaller climate sensitivity than employing other methods (Hegerl et al. 2007).

Our analysis shows that in each case, SB08 chose an input parameter that yielded a larger estimate of *b _{α}* than supported by our analysis of the same observational datasets. Accurate values of the model parameters show that the difference identified by SB08 is a minor correction to certain studies of outgoing radiation. It is also important to note that this argument only applies to regressions of outgoing radiation against surface temperature. Many estimates of climate sensitivity do not depend on such regressions.

## Acknowledgments

Researcher T. Wong provided CERES data for the tropical ocean regions. This work was supported by NOAA base and climate change funding.

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## APPENDIX

### Analytical Solution

Equation (1) is a linear first-order differential equation that can be readily solved for sinusoidal inputs. It is analogous to a resistor–capacitor circuit in electrical engineering. One feature of such linear equations is that the output has the same frequency as the input. This is why it makes sense to remove seasonal cycles from the data and model: in the context of Eq. (1), seasonal cycles cannot induce any variations on longer time scales.

*ω*is the angular frequency and

*ϕ*is the phase difference between

*N*and

*S*. In general

*N*

_{0}is complex, but for the derivation here it is sufficient to use real

*N*

_{0}and

*S*

_{0}and only carry the phase difference between

*N*and

*S*. Substitution into Eq. (1) yields

*τ*=

*c*/

*α*. The expression for

*b*can be expressed in terms of the Fourier series of

_{α}*N*and

*T*:

*ω*

_{1}and

*ω*

_{2}. Because of the orthogonal properties of Fourier series, the only terms that will be nonzero when summed over time are those with

*ω*

_{1}=

*ω*

_{2}. Therefore, the two sums over

*ω*can be replaced by a single sum,

*N*and

*T*and with some algebra for finding the real portions of the products, the sums can be expressed as

*ϕ*between

*N*and

*S*is random. For an independent pair of infinitely long random series,

*ϕ*will take on all values from 0 to 2

*π*over every interval of

*ω*. Any term proportional to cos

*ϕ*or sin

*ϕ*will therefore average to zero when summed over

*ω*. For an infinite series, the sums will also be replaced with integrals. Last, if both

*N*and

*S*are white noise, as assumed by SB08, then the scaling factors

*N*

_{0}and

*S*

_{0}are independent of

*ω*and may be taken outside the integrals. The integrals in the numerator and denominator cancel: