### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Diagnostic model
- 3. Application to IPCC models
- 4. Interpretation
- Acknowledgements
- References

Climate models have traditionally been characterised by their climate sensitivity (equilibrium response to a doubling of CO_{2}) and their ocean heat uptake. Together these determine a third property: the transient climate response to a linear increase in radiative forcing. A fourth property, the *feedback response time* is introduced here and shown to provide a complementary diagnostic of climate model behaviour. In particular, it demonstrates that the discrepancy between recent climate observations and the general circulation models in the ‘IPCC ensemble’ primarily arises because the models are undersampling the range of transient climate responses consistent with recent attributable greenhouse warming. Copyright © 2007 Royal Meteorological Society

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Diagnostic model
- 3. Application to IPCC models
- 4. Interpretation
- Acknowledgements
- References

There is considerable interest in the extent to which current atmosphere-ocean general circulation models (AOGCMs), as used in the latest scientific assessment of the Intergovernmental Panel on Climate Change (IPCC), span the range of responses to external forcing that is consistent with available climate observations. For example, Forest *et al.* (2006) report a clear discrepancy between the parameters of their intermediate-complexity climate model as fitted to the IPCC AOGCMs and the region of parameter space consistent with recent observed transient climate change and ocean heat uptake. This result is important, because any systematic bias in the behaviour of the AOGCMs has implications for climate predictions based upon them. Although Forest *et al.* (2006) discuss some possible explanations, the origin of the discrepancy is not immediately apparent because it is couched in terms of parameters of their intermediate-complexity model.

We show how the broad features of the result from Forest *et al.* (2006) can be reproduced with a simple diagnostic model, allowing us to identify the key properties of the climate system that appear to be inconsistent between some members of the IPCC ensemble and available climate observations. We identify a fourth property of the climate system's response to transient external forcing that complements the familiar properties of effective climate sensitivity (ECS), transient climate response (TCR) and effective heat capacity (EHC or, in Forest *et al.* (2006)'s model, the effective ocean diffusivity). This property, which we call the *feedback response time* (FRT), is the *e*-folding time of adjustment to changes in forcing in the presence of feedback and (more relevant to the interpretation of climate modelling experiments and climate observations) defines the timescale over which surface warming accelerates after the start of a linear increase in forcing. The FRT was discussed implicitly in the context of the ‘cold start problem’ in early AOGCM climate simulations (e.g. Hasselmann *et al.* (1993)) but has not, to our knowledge, been used for comparing climate models with observations.

The FRT complements the TCR just as the EHC complements the ECS: any two of these parameters are sufficient to define a unique version of our simple diagnostic model, but they naturally form these two complementary pairs. We show that the discrepancy between the distribution of the IPCC ensemble and the region of parameter space consistent with recent climate observations can be attributed to a bias in the models' distribution of TCR, with distributions of FRT being broadly consistent between models and observations. This is a matter of concern, because TCR is arguably the most directly observed of all these climate system properties, the warming attributable to increasing greenhouse gases over the past 70 years being very close to the response to a linear forcing increase. TCR is also the key determinant of climate change during the 21st century, so any bias between models and observations in this quantity would have significant implications for climate predictions.

### 2. Diagnostic model

- Top of page
- Abstract
- 1. Introduction
- 2. Diagnostic model
- 3. Application to IPCC models
- 4. Interpretation
- Acknowledgements
- References

A simple energy-balance model of the time-dependent response of the climate system to changes in radiative heating is

- (1)

where *C* is a constant EHC per unit area, *T*(*t*) is the perturbation to the global-mean surface temperature from its value at an initial time *t* = 0, *F*(*t*) is the time-dependent radiative forcing and the constant λ represents feedback processes.

It is important not to over-interpret this model: in particular, the assumptions that λ and *C* are time- and forcing-invariant are over-restrictive. The net strength of radiative feedback λ is assumed to be invariant in many simple and intermediate-complexity models, but has been shown to depend on the forcing history in more complex models (Senior and Mitchell (2000), Boer and Yu (2003), Gregory *et al.* (2004), Stone *et al.* (2007)). The EHC has also been shown to depend on forcing timescales in complex models and apparently in the observed climate as well (Wigley and Raper (1991), Stone and Allen (2005)). Hence if we use Equation (1) for the diagnosis of climate models and observations, we must focus on a single class of forcing profiles: we would not expect a realistic model to display the same EHC and atmospheric feedback in response to a radiative forcing pulse as to a more gradual change. Here we consider the response to a linearly increasing forcing, since this is most relevant to the interpretation of the warming attributable to greenhouse gases over the past century and to the diagnosis of IPCC model ensemble.

Given a linearly increasing forcing *F*(*t*) = α*t* and with *T*(0) = 0, we obtain the temperature perturbation for *t* ≥ 0:

- (2)

cf Hasselmann *et al.* (1993), equation (17). Introducing the *feedback response time* (FRT)

- (3)

(i.e. the *e*-folding adjustment time to forcing in the presence of feedback, cf Hansen *et al.* (1985)) and defining a non-dimensional scaled time τ = *t*/FRT, Equation (2) becomes

The function *G*(τ) shows how *T* evolves in non-dimensional time: see Figure 1. In particular *G*(τ)∼τ^{2}/2 for τ≪ 1 and *G*(τ)∼τ for τ≫ 1, so *T* is approximately quadratic in τ for small τ and approximately linear for large τ.

Now suppose that the forcing is maintained up to some specified time *t* = *t*_{1}: in a conventional CMIP2 experiment (Covey *et al.* (2003)), applying a constant 1% per year increase in CO_{2}, *t*_{1} would be the time of CO_{2}-doubling, but the analysis applies to any linear forcing increase. We define

- (4)

i.e. the temperature perturbation that would occur at *t*_{1} in the absence of feedback processes (λ = 0). In the special case that *F*(*t*_{1}) corresponds to *F*_{2×}, the forcing equivalent to a doubling of CO_{2} from its level at *t* = 0, we can also introduce the *effective climate sensitivity* (ECS):

- (5)

This equals the temperature perturbation that would occur in a steady state if the forcing were maintained at *F*(*t*_{1}) and atmospheric feedback (i.e. λ) were to remain unchanged. The possibility of λ evolving as the system reaches equilibrium has been noted above, so ECS must be distinguished from the *equilibrium* climate sensitivity, which may be significantly higher.

The *transient climate response* (TCR) is defined as the temperature perturbation at time *t*_{1} under this 1%/year increasing CO_{2} scenario, and is given in terms of *T*_{0} and ECS by

- (6)

In Figure 2(a) the solid lines denote contours of the function TCR(*T*_{0}, ECS) as given by Equation (6). Clearly, knowledge of the value of TCR alone does not locate a unique pair of values of *T*_{0} and ECS: to fix both *T*_{0} and ECS we must also have information that is independent of TCR.

This additional information can be provided by another function of *T*_{0} and ECS that specifies each point on any given TCR contour. A simple and convenient choice is FRT, which can be given as

- (7)

using Equations (3), (4) and (5). The contours of FRT are the dashed lines in Figure 2(a).

TCR is the temperature perturbation at time *t*_{1}, whereas FRT indicates how far the evolution has progressed by that time: if *t*_{1} < FRT, the time evolution is still in the quadratic regime at time *t*_{1}, whereas if *t*_{1} > FRT the time evolution has entered the linear regime. Figure 2(b) shows the ‘inverse’ plot of contours of *T*_{0} and ECS as functions of TCR and FRT.

### 3. Application to IPCC models

- Top of page
- Abstract
- 1. Introduction
- 2. Diagnostic model
- 3. Application to IPCC models
- 4. Interpretation
- Acknowledgements
- References

To illustrate the use of various pairs of coordinates for classifying the behaviour of climate models we use data from a set of 18 AOGCMs used in the IPCC Working Group 1 Fourth Assessment Report, Chapter 8 (Randall *et al.* (2007), hereafter IPCC4). Table reproduces data from Tables and S8.1 of IPCC4.

For each model we calculate *T*_{0} from ECS and TCR by inverting the function *G* in Equation (6). We then obtain *C* from Equation (4), using *F*(*t*_{1}) (which differs slightly between models) and *T*_{0} with *t*_{1} = 70 years, and FRT from Equation (7), using ECS, *T*_{0} and *t*_{1}. This gives the data in the last three columns of Table 1.

Table 1. Results from the IPCC4 AOGCMs. Units of ECS, TCR and *T*_{0} are K, of *F*(*t*_{1}) are W m^{−2}, of *C* are GJ K^{−1} m^{−2}, and of FRT are years. The information in the first five columns are from IPCC4 and the data in the last three columns are calculated as described in the textNumber | Model | ECS | TCR | *F*(*t*_{1}) | *T*_{0} | *C* | FRT |
---|

3 | CCSM3 | 2.37 | 1.50 | 3.95 | 2.96 | 1.47 | 28.0 |

4 | CGCM3.1(T47) | 3.02 | 1.90 | 3.32 | 3.73 | 0.98 | 28.4 |

7 | CSIRO-Mk3.0 | 2.21 | 1.40 | 3.47 | 2.77 | 1.38 | 27.9 |

8 | ECHAM5/MPI-OM | 3.86 | 2.20 | 4.01 | 3.89 | 1.14 | 34.7 |

9 | ECHO-G | 3.01 | 1.70 | 3.71 | 2.98 | 1.37 | 35.3 |

10 | FGOALS-g1.0 | 1.97 | 1.20 | 3.71 | 2.27 | 1.81 | 30.4 |

11 | GFDL-CM2.0 | 2.35 | 1.60 | 3.50 | 3.49 | 1.11 | 23.5 |

12 | GFDL-CM2.1 | 2.28 | 1.50 | 3.50 | 3.12 | 1.24 | 25.6 |

14 | GISS-EH | 3.04 | 1.60 | 4.06 | 2.65 | 1.69 | 40.2 |

15 | GISS-ER | 2.57 | 1.50 | 4.06 | 2.71 | 1.65 | 33.2 |

16 | INM-CM3.0 | 2.28 | 1.60 | 3.71 | 3.67 | 1.12 | 21.7 |

17 | IPSL-CM4 | 3.83 | 2.10 | 3.48 | 3.59 | 1.07 | 37.3 |

18 | MIROC3.2(hires) | 5.87 | 2.60 | 3.14 | 3.85 | 0.90 | 53.4 |

19 | MIROC3.2(medres) | 3.93 | 2.10 | 3.09 | 3.51 | 0.97 | 39.1 |

20 | MRI-CGCM2.3.2 | 2.97 | 2.20 | 3.47 | 5.60 | 0.68 | 18.6 |

21 | PCM | 1.88 | 1.30 | 3.71 | 2.91 | 1.41 | 22.6 |

22 | UKMO-HadCM3 | 3.06 | 2.00 | 3.81 | 4.12 | 1.02 | 26.0 |

23 | UKMO-HadGEM1 | 2.63 | 1.90 | 3.78 | 4.59 | 0.91 | 20.0 |

We also use an observationally constrained ensemble of 28 800 simple climate models (SCMs) based on Hansen *et al.* (1984), Hansen *et al.* (1985), as used by Frame *et al.* (2005) and Frame *et al.* (2006) (a simplified variant of those used by Forest *et al.* (2000), (2002) and Andronova and Schlesinger (2001)):

- (8)

where *C*_{ml} is the heat capacity per unit area of the ocean mixed layer times the global fractional area of the ocean (∼0.7) and *d*_{ml} is its depth. This is similar to Equation (1) but with an extra term representing the heat flux into a deep diffusive ocean with diffusion coefficient *K*_{v}: this term can be obtained by application of Duhamel's theorem to the expression for the heat flux that applies when *T*(*t*) is a step function (Carslaw and Jaeger (1959), pp. 30, 61). (Note that the EHC diagnosed from these models does not equal *C*_{ml} alone, but also includes a contribution from this heat flux.)

The model is first driven with the radiative forcing due to the observed increase in greenhouse gases over the past 250 years, initialised from equilibrium in 1750, systematically varying λ and *K*_{v} over a broad range. Trends are diagnosed over the 20th century and compared with the warming trends attributable to greenhouse forcing over the same period from a pattern-based detection and attribution analysis (Allen and Stott (2003)) utilising simulations from three coupled AOGCMs (Stott *et al.* (2006)). The analysis separates out the response to greenhouse forcing from the response to other anthropogenic and natural agents. These attributable greenhouse warming trends are directly related to TCR, since the forcing due to the observed increase in greenhouse gases has been nearly linear since the mid-20th century: the fact that the rate of increase in forcing is less than that given by a 1%/year CO_{2} increase is immaterial, since our models are linear. We assign a likelihood to each SCM's 20th century warming trend by equating this with the corresponding likelihood of that magnitude of warming trend in Stott *et al.* (2006)'s analysis.

Since we have two free parameters, an additional constraint is required. Following Forest *et al.* (2002) and Frame *et al.* (2005), we use observations of ocean heat uptake, using Levitus *et al.* (2005)'s estimate of a 14.5( ± 9)× 10^{22} J for the increase in decade-averaged global ocean heat content from 1957–1994, divided by the observed increase in global surface temperature over the same period, to give an estimated likelihood for the EHC, which we then equate to the likelihood of the corresponding quantity diagnosed from the SCM (these are the values used in Frame *et al.* (2006)). This uncertainty in ECS is dominated by Levitus *et al.* (2005)'s estimate of uncertainty in ocean heat content change, with the much smaller fractional uncertainty in surface warming being less important. Any such estimated likelihood function for EHC will be much more contentious than the likelihood function for attributable warming, which emerges directly from the attribution analysis and is based on the real-world response to an increase in forcing very close to the linear ramp used in the idealised CMIP2 experiments. As noted above, both ECS and EHC are likely to depend on the time-history of forcing imposed, so we are definitely stretching the interpretation of our diagnostic model by equating the EHC inferred from the real-world response to a combination of forcings (greenhouse gases, anthropogenic aerosols, solar and volcanic activity) with the EHC appropriate to the response to a linear ramp. We shall see, however, that the main discrepancy between the IPCC ensemble and the observations arises from the better-observed and better-constrained TCR component of the likelihood function. Hence the fact that our estimate of a likelihood for EHC is contentious is unimportant for the key conclusions of this article, although better estimates of the real-world EHC relevant to a secular increase in forcing would be extremely valuable. A likelihood for each member of the SCM ensemble is derived from the product of the likelihoods of that member's warming trend and EHC, while values of *T*_{0} and FRT are derived from that model's ECS and TCR as described above.

Figure 3 plots data from Table and contours of likelihood *L* from the SCM ensemble against various pairs of axes. Figure 3(a) gives a ‘standard’ plot, using ECS and TCR as axes: cf Frame *et al.* (2006), Figure 1, Meehl *et al.* (2007), Figure 10.25(a). Figure 3(b) uses heat capacity *C* (derived as described above for the IPCC models and equated to EHC from the SCM ensemble for the *L* contours) and TCR as axes: this is roughly analogous to Figure 1(a,b) of Frame *et al.* (2005). Figure 3(c) uses heat capacity *C* (as in Figure 3(b)) and ECS as axes; this is roughly analogous to Figure 3 of Forest *et al.* (2006) (who, however, use instead of *C*). Note that although the overall appearance of our figure is similar to that of Forest *et al.* (2006), the discrepancy between models and observations is somewhat less severe in our figure. This will be partly due to our use of a different AOGCM ensemble and to differences between the relationships between ECS, TCR and EHC in our SCM and those in the intermediate-complexity model used by Forest *et al.* (2006): see also Stott and Forest (2007). Figure 3(d) uses TCR and our proposed new ‘conjugate’ coordinate FRT as axes. For brevity we omit a plot using *T*_{0} and ECS as axes.

Note that the likelihood for a few IPCC models takes different values in different panels: for example model 20 has *L* > 0.9 in Figure 3(c) but 0.5 < *L* < 0.9 elsewhere. The dependence of EHC on TCR and ECS for the models of the SCM ensemble is not precisely the same as that of *C*(TCR, ECS) in Section 1, because they are not fixed heat capacity models.

### 4. Interpretation

- Top of page
- Abstract
- 1. Introduction
- 2. Diagnostic model
- 3. Application to IPCC models
- 4. Interpretation
- Acknowledgements
- References

Figure 3 illustrates how the coordinate system used to compare models with data can help clarify the origins of model-data discrepancies. Comparing models with observations in ECS–TCR space, Figure 3(a), is difficult to interpret, since these two quantities are so highly correlated. Likewise, the dependence of TCR on EHC complicates the interpretation of Figure 3(b). ECS and EHC might be considered physically more independent quantities so, if these are the quantities of interest, Figure 3(c) is the logical focus, as used in Forest *et al.* (2006). It has, however, been widely noted (e.g. Frame *et al.* (2006)), that ECS is much less relevant than TCR to forecasts of 21st century climate change, so we argue that the most interesting perspective is provided by Figure 3(d), which displays TCR against its logical complementary variable, FRT.

This figure makes clear that the origin of the discrepancy between the IPCC ensemble and the climate system properties consistent with recent climate observations lies in the distribution of TCR: modelled and observed distributions for FRT are more consistent with each other. It might be argued that the model ensemble is under-dispersive in FRT, but there is no obvious bias. There is, however, a clear bias towards lower values of TCR in the IPCC ensemble relative to those values that are consistent with recent observed climate change. The IPCC itself was careful to quote a broader range of uncertainty in TCR than that suggested by the model ensemble, but it is important to continue to bear in mind that any climate forecast based on the assumption that the IPCC models provide a representative sample of responses may be underestimating the likelihood of a ‘high-end’ response.

An alternative explanation of the discrepancy, of course, is that current attribution results may be overestimating this likelihood. Despite the fact that greenhouse gas forcing has been close to a linear ramp over the past 70 years, we cannot observe the response directly because it has been partially obscured by other factors, notably anthropogenic aerosols and natural forcing. The attributable warming trends are derived from a simple linear regression between observations and model-simulated responses to the different forcings, which depends on the assumption that responses superimpose linearly. If further research were to rule out the possibility of anthropogenic aerosol cooling near the upper end of the currently accepted uncertainty range, that would make a high attributable greenhouse warming, and hence a high TCR, correspondingly less likely.