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Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[1] This study makes use of calendar month averages of surface temperature taken from the long control runs of 13 coupled atmosphere/ocean general circulation models (GCMs) as well as from the record of observational surface temperature measurements. After aggregating the data into global averages, we examine the mean seasonal cycle as well as the anomaly statistics. We find a large range of different results for the models in the seasonal cycle of anomaly statistics. There appears to be an “empirical” relationship between the published sensitivity of the models to CO2 doubling and the seasonal cycle of the anomaly statistics. We draw an inference about the sensitivity of the real climate based upon this relationship. The relationship appears to be plausible based upon simple considerations of the land–sea distribution and the wintertime variance of climatic noise forcing. This inference leads us to estimate the sensitivity of climate to a doubling of CO2 to be about 2.7°C with the 95% confidence interval (2.3°C, 3.1°C).

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[2] The sensitivity of large-scale climate to anthropogenic or other externally imposed perturbations looms as the greatest problem facing climate change science. The latest survey of climate models [Houghton et al., 2001] suggests a ±50% spread of sensitivities among currently used general circulation models (GCMs) to a doubling of CO2 (equilibrium to equilibrium). This estimate of the range of uncertainty (1.5–4.5°C) has hardly changed over 25 years. Currently, the estimate of uncertainty seems to be mainly based on a comparison of the sensitivities of the currently fashionable models and may not reflect a bias, which could be common to all the models. There are other attempts to estimate climate sensitivity based on comparison of the simulations of a climate model and observations [Forest et al., 2002; Andronova and Schlesinger, 2001]. The 95% confidence interval of climate sensitivity obtained by Forest et al. [2002] is from 1.4°C to 7.7°C, and the 90% confidence interval by Andronova and Schlesinger [2001] is from 1.0°C to 9.3°C. It is very interesting to inquire among the models as to exactly how they differ in certain calculated quantities, especially quantities not normally used in their tuning of parameterizations. It is hoped that such intercomparisons will lead to insights that in turn lead to the improvement of the models and of our understanding of climate change.

[3] The natural variability statistics of climate models possibly provide information that is related to the sensitivity of responses to external forcings. The simplest models based upon damped Brownian Motion type [Hasselmann, 1976] give us a hint as to how this might come about. Consider the very simplified Langevin model of global temperature:

  • equation image

Where T represents the global average temperature and B is a damping coefficient, usually related physically to the infrared radiation to space but possibly including temperature-dependent effects of absorbed solar radiation, and C is an effective heat capacity, related to the mixed layer depth or on longer timescales to the thermocline depth, and N(t) is white noise drawn from a normal distribution with standard deviation σ0, usually attributed to atmospheric weather fluctuations, whose timescales are at most a week or two (compared to the month to years timescale of relaxation of the model (equation image)). One can easily ascertain that the autocorrelation time from solutions to this model is the same as the relaxation time, C/B, but the sensitivity to a steady perturbation to the governing equation's right-hand side leads to a sensitivity of exactly B−1 per unit of heating rate (per unit area) added to the right-hand side. (For example, a doubling of CO2 would lead to ΔQ/B where ΔQ is about 4.3 W m−2). The real story is obviously much more complicated than this simple picture, but it suggests a connection between damping coefficients and sensitivity. One might conjecture that getting the damping coefficients right is a necessary condition for getting the sensitivity right [e.g., Leith, 1976; North et al., 1993]. One might go further and say that getting the autocorrelation statistics right is a necessary condition to getting the time-dependent response to external perturbations right. Complicating the matter is the effective heat capacity which may differ from one climate model to another due to the variety of parameterizations of the air/sea interaction and to the treatments of both atmospheric and oceanic turbulent boundary layers (not to mention the various larger-scale and smaller-scale dynamical means of heat subduction and release in the oceans).

[4] One recent study by Covey et al. [2000] suggested that the mean seasonal cycle of surface temperatures might contain a clue about the sensitivity. Indeed, they found a small correlation between the hemispherically averaged amplitude of the seasonal cycle and the sensitivity of various GCMs to doubling CO2. Lindzen et al. [1995] studied the seasonal cycle of one GCM and found little relationship of the ensemble averaged, global averaged seasonal cycle to the doubling of CO2. However, they did point out the value of using global averages where entire areas are integrated over.

[5] Most GCMs match the mean seasonal cycle of surface temperatures well, and there is little dynamic range in their differences, making a regression on those differences difficult. This mutual agreement among modeled climates for the mean seasonal cycle is not surprising, since seasonal data are readily available and modeling groups use such data in adjusting their parameterizations. This latter concern raises an important point: When a data set is widely available, modeling groups have access to it, and it soon becomes a part of the model through the model's adjustable free parameters. This time-honored process started with mean annual temperature distributions, then the seasonal cycle, variance, and most recently the century-long record of warming. This process makes it difficult if not impossible to test models in any classical sense. This is not a complaint against modeling groups, since the use of every bit of information to reduce the number of or to fix the values of free parameters is a societal obligation. Given that modeling groups use every bit of data available to them, eliminating its availability for testing, it becomes a challenge to observational and diagnostic groups to find innovative data sets and novel ways of analyzing them in such manner as to bring new (and relevant) ways of at least intercomparing the models among themselves and when possible with real data.

[6] In this study we choose the statistics of the calendar month averages of the surface temperature as the basis for our study. Based upon simple model considerations we argue that these kinds of statistics ought to be related to static sensitivity (sensitivity to a steady forcing from one (statistical) equilibrium to another such as the CO2 doubling). We aggregate the calendar month average statistics into global averages. Such aggregation should help in minimizing the many kinds of random error due to sundry causes both in modeling and in observations. The heart of our paper is the analysis of the statistics of the calendar month averages of anomalies (departures from the means). These anomalies can be intercompared among the models and with those in the observed data. Most of the models with larger sensitivities do not agree with the observational record giving a hint that there is a connection between model sensitivity and some of these statistics.

2. Models and Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[7] In this study we used the 140-year record of globally averaged surface temperature from 1856 to 1995 [Jones et al., 1997] (available from http://www.cru.uea.ac.uk/cru/data), and the model climates generated by 13 ocean/atmosphere GCMs. The data streams derived from the models are area-weighted to get the global monthly anomalies from long-term average annual cycle. Some of the model-derived data sets have been made available by the different modeling groups through the Intergovernmental Panel on Climate Change Web site (http://www.dkrz.de/ipcc/ddc/html/dkrzmain.html). We relied exclusively on the long control runs where no external forcings were applied. Table 1 shows the acronyms used for the observations and for the different models. The second column contains the published climate sensitivities to doubling CO2, which were usually estimated from mixed layer models. In column 5 is the number of years available in the control runs of coupled climate models (or record in the case of observations). The lengths of these records are important in establishing confidence intervals for different anomaly statistics. There are also small differences in each model's definition of the surface temperature (see Table 1). We made no attempt to correct these to some common standard, but a few sensitivity studies suggested that it is not important. We emphasize that good estimates of the statistics of calendar month averages is strongly dependent on record length, autocorrelation times, and variance, hundreds of years being desirable. Our choice of models was limited to the availability of output, etc., and the choice does not indicate any desire to prove one model “better” than another. Some of the models had only short control runs available and these often had serious drifts. In these cases, we removed a linear trend before processing. In the case of the observational record, we removed the forced signal components according to a fairly well established procedure. First we used the global and annual averaged data and “detected” the signals due to greenhouse gases, anthropogenic aerosols, volcanic activity, and solar variation over the last 100 years. The method followed was described by North and Wu [2001]. Then we subtracted these “forced response signals” from the original data stream to obtain an “unforced,” “stationary” data record. This procedure seems to be fairly robust and our results do not seem to be sensitive to details in the scheme.

Table 1. Climate Sensitivity and Index of Seasonal Cycle of Global Average Surface Air Temperature (SAT) of the Observational Data and 13 Climate Modelsa
GCMsT2 × imageσ1,227,82SATLengthReferences
  • a

    The first column consists of the acronyms of each model. The second column contains the published sensitivities to doubling CO2 (°C) for the same models. The third column lists the seasonal cycle amplitude. This is the ratio of the variance of January (or February depending on which is larger) SAT to that of July (or August depending on which is smaller). The fourth column reports the definition of SAT for the record and models. In the GFDL and CCSR models, SAT is the temperature of lowest model level. The fifth column shows the length [years] of observational data and model control runs. The sixth column gives the related references.

  • b

    Effective sensitivity.

OBS 2.592.0 m140Jones et al. [1997]
PCM2.13.042.0 m300Washington et al. [2000]
CSM2.14.322.0 m300Dai et al. [2001]
EBCM2.22.20Surface10,000Stevens and North [1996]
ECHAM4/OPYC32.6b2.092.0 m240Roeckner et al. [1999]
ECHAM3/LSG3.22.542.0 m1000Voss and Mikolajewicz [2001]
HadCM33.31.841.5 m240Williams et al. [2001]
GFDL_R303.41.67Lowest Level1000Delworth et al. [2001]
CGCM13.51.832.0 m200Boer et al. [2000]
CCSR/NIES3.62.09Lowest Level210Emori et al. [1999]
GFDLml3.71.31Lowest Level1000Stouffer and Manabe [1999]
GFDL_R153.71.47Lowest Level1000Stouffer and Manabe [1999]
HadCM24.11.201.5 m1085Senior and Mitchell [2000]
CSIRO-Mk24.31.702.0 m220Gordon and OFarrell [1997]

3. Anomalies of Calendar Month Averages

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[8] It is in the statistics of the variance of calendar month averages (taken over the ensemble of months available from the control runs) that the models first show remarkable differences in our study. These are shown in Figure 1 along with those of the observations. Note that the errors are not simply in the magnitude of the variances but even in the phase of the annual harmonic. In some cases, a semiannual harmonic is prominent. In constructing these anomalies, it is important to use exactly the same mask for the data since polar regions exhibit large variance.

image

Figure 1. The variances of the anomalies of globally averaged calendar month average temperature for the observed data and 13 climate models. The number in the parenthesis is the sensitivity to CO2 doubling (°C).

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[9] Figure 2 shows a scatter diagram of climate sensitivity versus a measure of the inverse amplitude of the seasonal cycle of variance as indicated by the ratio of July (or August depending on which is smaller) variance to January (or February depending on which is larger) variance. The amplitude of the seasonal cycle of variance represents the largest difference of variance between winter months and summer months. In this case the correlation coefficient is large (0.79) and the shaded vertical bar (obtained by Monte Carlo simulation) indicates the 95% confidence band for the observations based on the 140-year record. The value of the ratio for the observed data is 0.39. The confidence intervals for the model runs are comparable with narrower confidence intervals for the longer runs, etc.

image

Figure 2. Scatter diagram of the sensitivity of models to doubling CO2 versus the ratio of the variance of July (or August depending on which is smaller) global average temperature to that of January (or February depending on which is larger). For the observations, the ratio is about 0.39 and its 95% confidence interval is shaded. The solid lines are values of the EBCM where the value of the damping coefficient B takes on different values. The dashed line represents the relationship between climate sensitivity and inverse amplitude of the seasonal cycle of variance in a climate system with variable damping coefficient B. Its globally averaged temperature satisfies the linear model (2). The effective heat capacity Ce(t) of such system is assumed to have the seasonal shape given by B/(0.46 − 0.25 * cos(2πt/12 + 300)) and the seasonal cycle of σ0(t)/Ce(t) is neglected.

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[10] The solid line on this graph (Figure 2) shows our attempt to indicate the change across a parametric family of Energy Balance Climate Models (EBCMs) (the Stevens [1998] model is summarized in Appendix A of the present paper) [Stevens and North, 1996] by changing their radiative damping coefficient (B). In the EBCM, we did not attempt to “fit” the model to the seasonal cycle mean by adjusting other parameters such as the thermal diffusion coefficient or the effective heat capacity. In one exercise, we tried to fit the EBCM to each of the GCMs by adjusting the available parameters. We found that we could not fit the seasonal cycle within this constraint if we varied B by more than ±15%. While the results do not indicate that the GCMs achieve their results by such a simple “error” as getting the damping wrong, it does give a positively sloped curve indicating that this could be part of the explanation.

[11] We also experimented with the Stevens EBCM [Stevens and North, 1996] by making the entire Northern Hemisphere (NH) land and the Southern Hemisphere (SH) all ocean. Without changing any other parameters or the noise forcing pattern (peaked in winter midlatitudes) we also studied the variance of calendar month averages. Again, we found that a very positively sloped curve resulted for the sensitivity versus ratio of July to January variances. We conclude that the effect is strongly correlated to the land–sea distribution and the pattern of noise forcing being peaked in the winter hemisphere midlatitudes, but also to the damping coefficient which in the EBCM is inversely related to the sensitivity.

4. An Interpretation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[12] Since the anomalies of month averages generated by the EBCM of Stevens imitate the seasonal cycle rather faithfully, we can assume that the simple physics in that model explains the phenomenon to first order. We now construct a conceptual model that we believe yields the underlying connection of the anomaly statistics to sensitivity.

[13] First note that it is entirely plausible that the maximum in anomaly variance January and February is reasonable based upon the confluence of the two facts: Larger noise forcing in the winter hemisphere and the far larger proportion of landmass in the NH. It should be noted that modeled soil moisture over land can also have a large effect on summer variability. The question is how large should this asymmetry be?

[14] Consider a very simple model for the global average monthly temperature anomaly from long-term average annual cycle, T(t).

  • equation image

where N(t) is Gaussian white noise, equation image and equation image are the simple sinusoids:

  • equation image
  • equation image

[15] First note that in its continuous form the model is analytically solvable by use of an integrating factor, since it is simply a first-order linear ODE. The version with γ = 0 is discussed by Kim et al. [1996].

[16] The effective global heat capacity can be defined:

  • equation image

where r is the spatial coordinate. The time dependence of Ce(t) here comes from the effective heat capacity of the Earth–atmosphere–ocean column being much smaller in NH winter than in NH summer. Note that the effective heat capacity as defined here is very different from the average heat capacity as it involves a harmonic mean.

[17] From the 140-year record of observational data, it was found that B/Ce(t) = 0.46 − 0.25 * cos(2π t/12 + 300), t = 1, …, 12. Consider a climate system whose effective heat capacity Ce(t) has the same seasonal cycle as that in the observational data. Its globally averaged temperature satisfies (2). Also the value of its damping coefficient B can be increased and reduced arbitrarily. Mathematically we can find a relationship between climate sensitivity and the inverse amplitude of the seasonal cycle of variance from the simple linear model solutions, as shown by the dashed line in Figure 2.

5. Discussion and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[18] Climate model solutions for the global average temperature exhibit a seasonal cycle of the anomalies of calendar month averages. This seasonality can be easily traced to the disproportionate landmass in the NH and the fact that variability is strongly concentrated in the winter hemisphere. That this is the case is strongly supported by solutions to Stevens' stochastic EBCM, which exhibits this behavior. As the different AOGCM control solutions are compared, we find that there are considerable differences among the models with regard to this seasonal cycle of variance. Models with small seasonal cycle in the month average anomaly variance tend to have large sensitivities to CO2 doubling. We tested this hypothesis with Stevens' model by varying the model's sensitivity and by attempting to fit the EBCM to the various control runs. We were only successful in fitting the EBCM to the less sensitive models and to the observational data. Finally, the observational seasonal cycle of month average anomaly variances for the observational data can be computed and compared to the AOGCM control runs. The data are consistent with models whose sensitivity to CO2 doubling is around 2.7°C. The 95% confidence interval for this estimate is (2.3°C, 3.1°C).

[19] While we do not claim that our results and accompanying argument are definitive, they raise some interesting questions for AOGCM custodians to consider. We have identified a data set not normally used in tuning the models and it seems to be related to climate sensitivity. It is clearly governed by the hemispherically asymmetric land–sea distribution and the concentration of variability in winter. The land–sea distribution is the same in all models, but the way this is translated into dynamics of the surface temperature fields could be quite different from model to model. In an EBCM, this is governed by the length scale that is the square root of the thermal diffusion coefficient divided by the damping coefficient. Many other factors enter in an AOGCM. The variability of the surface temperature field in the EBCM is caused by white noise forcing that has a seasonal cycle sharply peaked in the midlatitudes and strongest in winter. The EBCM agrees with the observations partly because it was tuned to do so, but this suggests the plausibility of the mechanism. It could be that different AOGCMs supply this variability in drastically different ways. Finally, the models employ very different parameterizations of the ocean heat uptake. This parameterization could be incorrectly forced to make agreement with the mean seasonal cycle with undesirable consequences showing up in the seasonal cycle of month average anomalies.

Appendix A:: Stevens' EBCM

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[20] For completeness, we review here the EBCM devised by Stevens [1998] in his dissertation. EBCMs have been constructed over the years for two different purposes. The first is to clarify basic conceptual ideas about the climate system. Examples of this are ice cap models [North, 1990], studies of the seasonal cycle for paleoclimate purposes [North et al., 1983], small icecap instability [North, 1984; Mengel et al., 1988], idealized models of predictability in stochastic versions of the model [North and Cahalan, 1981] or in the demonstration that simple stochastic models can be used to simulate natural variability [Kim and North, 1991]. A second reason for running EBCMs is to provide simulation models of natural variability for signal processing or other related purposes [e.g., Stevens and North, 1996; North and Stevens, 1998] and in studies of error assessment in observing systems [Shen et al., 1994]. In the latter class of cases, one fits as closely as possible to the real data using more than a bare minimum of adjustable phenomenological coefficients. Here the idea is similar to the use of autoregressive models in time series analysis: one fits the data to a simplified model whose analytical or numerical properties are easy to study. The result is smooth features such as spectra, etc. The model by Stevens falls in this second category. It is forced as much as possible through parameter adjustment to imitate the present climate. An example of its utility is that it is entirely feasible to make a 10,000-year run on a workstation is a very short time.

[21] The surface temperature is governed by the energy balance equation:

  • equation image

where Independent variables: θ and ϕ are latitude and longitude; t is time. Dependent variable: T(θ,ϕ,t) is surface temperature. C(θ,ϕ) is effective heat capacity per unit area. A is a radiation parameter, 210.3 W m−2. B is the radiative damping coefficient (2.15 W m−2 K−1), C/B is a relaxation time which takes on different values depending on surface type. Over land: 31 days; over ocean 4.3 years; over sea ice 40 days; over snow 26 days. Q is the solar constant 1371.75 W m−2 ÷ 4. D(θ,ϕ) is a macroscopic thermal conductivity coefficient: Over land it has a maximum value of 0.6 at the Equator and tapers toward the poles, going to 0.3 in the NH and 0.2 in the SH. Over ocean, it is also 0.6 at the Equator, tapering toward both poles to a value of 0.4. The asymmetry between hemispheres over land is attributed to the elevation over Antarctica. α(θ,t) is set to the observed zonal average value through the seasonal cycle. It is fixed, so there is no snow/ice albedo feedback in this version of the model. N(θ,ϕ,t) is the noise driving force. It is white noise in space and time modulated by two factors: a constant plus sinusoid differing in each hemisphere, being positive in the NH during NH winter and peaking around 1 February, being small in NH summer. The SH factor is peaked in SH winter at around 1 September. There is also a spatial factor that is the magnitude of the horizontal gradient of the mean seasonal variation of temperature, taken from observations (or the seasonal model data). The three factors make for a noise that is peaked in winter and midlatitudes over land. However, as pointed out by Kim and North [1991], the effect gives qualitatively correct answers for the spatial dependence of variance in response if only the white noise factor is retained. The seasonal variation must come from the other factors. H(θ,ϕ,t) is a heat flux which serves as a “flux correction.” it is proportional to (T(θ,ϕ,t) − T0(θ,ϕ,t)) where the second term is the long-term mean for that position and time of year. The proportionality factor differs between the hemispheres. It leads to relaxation times of 44 days in the NH and 55 days in the SH. This term can also be thought of as the horizontal transport by the deeper ocean. This latter is not allowed to change in our simulations. Note that the retoring term does not contribute to static sensitivity (e.g., doubling CO2).

[22] The model is solved by a relaxation method based upon the study of Bowman and Huang [1991] but speeded up by Stevens [1998].

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

[23] We are grateful to the National Oceanographic and Atmospheric Administration for its support through a grant from its Office of Global Programs. In addition, some support came from the Office of the Vice President for Research at Texas A&M University. We thank the AOGCM groups for making their data available. We have also enjoyed may helpful discussions with Mark Stevens. Several comments by John Mitchell and two anonymous reviewers have helped improve the manuscript.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Models and Data
  5. 3. Anomalies of Calendar Month Averages
  6. 4. An Interpretation
  7. 5. Discussion and Conclusions
  8. Appendix A:: Stevens' EBCM
  9. Acknowledgments
  10. References
  11. Supporting Information

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.