1Overcoming Chaotic Behavior of Climate Models 2S Fred Singer 3University of Virginia/ Science & Environmental Policy Project, Arlington, VA 22202 singer@sepp.org 5Abstract: A fundamental issue in climate science is Attribution – determining the relative importance of 6human and natural causes The task generally involves comparing temperature trends from observations 7and from greenhouse (GH) models However, a problem arises from the chaotic uncertainties inherent in 8the (non-linear) model calculations Modelers try to overcome this problem by forming an “ensemble9mean” of a number of “simulations” (runs) of the same model Here we conduct a synthetic experiment, 10and use two distinct procedures to demonstrate that no fewer than about 20 runs (of 20-yr length of an 11IPCC Ggeneral-Ccirculation Mmodel) are needed to place useful constraints upon chaos-induced model 12uncertainties 13 14Introduction: 15 16As Lorenz (1963, 1975: Zichichi, 2007, and also Giorgi, 2005) have demonstrated, climate models, using 17non-linear partial differential equations, generate results highly sensitive to the initial conditions To 18reproduce an identical result in successive “simulations” (runs), the parameters describing the model’s 19initial state must be given to a precision that is unattainable in practice The Intergovernmental Panel on 20Climate Change (IPCC-TAR; Meehl et al., 2001) acknowledges that, mathematically speaking, the 21climate is a “complex, non-linear, chaotic object” and that, therefore, “the long-term prediction of future 22climate states is not possible.” Accordingly, any comparison of modeled with observed temperature 23trends cannot be done satisfactorily without understanding the chaotic behavior of a climate model 24One consequence is that successive runs of the same climate model can yield very different values for 25warming trends These trends may vary by an order of magnitude or more, and even their sign may vary 26For example, the Japanese MRI model carried out five runs for IPCC [Santer et al., 2008]: the individual 27trends range from 0.042 to 0.371 K/decade [Fig 1] and this error interval (‘spread’) would have been 28even greater if more runs had been performed 29 30Fig 1: Illustrating the chaotic nature of model trends, using the results of runs (sometimes referred to 31as “realizations” or “simulations”) from 1979 to 1999 of a particular GCM (from Japan’s MRI), as 32presented in figure of Santer et al 2008 The OLS trends of the five runs range from +0.042 to +0.371 33(K per decade) The range of trends would likely to be even larger if more runs had been carried out 34None of the five trends, nor the ‘ensemble-mean’ shown, represents the ‘true’ model trend As discussed 35in the text, one needs to show that the cumulative ensemble-mean approaches an asymptotic value as 36the number of runs increases 37 38Modelers, therefore, carry out several runs (“simulations”) and then publish the ‘model-ensemble-mean,’ 39E, the arithmetic average of the individual trend values generated by the several runs Only rarely we 40learn the results of the individual simulations that are components in E Yet how we know that, say, 41five runs are sufficient to produce a reliable EM to compare with an observed trend? 42The present paper addresses a single but crucial aspect of the impact of chaoticity on the performance of 43general-circulation models: the strong dependence of the error-bars in temperature-trend projections on 44the number of simulations that are run on a particular model We suggest that it may be possible to 45overcome the “chaoticity barrier” by performing a sufficient number of runs 46 47Method: 48The objective of this enquiry was to establish how many simulation runs of a GCM, at minimum, are 49necessary to provide reasonable constraints on the value of E For this investigation, it would have been 50desirable to use climate models which had each been run at least 20 times However, financial and time 51constraints on modelers mean there are no ready examples of such multiple runs Therefore, we 52developed a synthetic approach to the problem 53A single, unforced (‘constant forcing’) control run of 1000 years’ duration was obtained from the Program 54for Climate Model Diagnosis and Intercomparison (PCMDI) at Lawrence Livermore National Laboratory 55 56Fig 2: Temperature values of an unforced 1000-year climate model control run Source: PCMDI 57 58First, the temperatures were plotted against time in years [Fig 2] to check for inevitable drift from what 59should be a straight, zero-slope, horizontal line Next, the temperature series was divided into 25 60segments, each of 40 years’ duration (and also into 50 segments of 20-year runs) For each segment, 61trend values T1 … T25 were determined This procedure is analogous to, and (considering the chaoticity of 62the climate object) equivalent to, 25 separate runs of a GCM over a single 40-year time-interval Another 63advantage of using an unforced model is that the true trend is known in advance – namely, zero (except 64for drift effects) 65 66First procedure: the cumulative ensemble mean Ecum In the first of two procedures, a cumulative 67ensemble mean Ecum was derived by adding the trend value of an additional run to the previous values, Eq 68(1), so as to determine a new value of E Finally, the cumulative trend was plotted as a function of n, the 69number of trend values used [Fig 3]: 70 71 72 n EMcum = 1/n Σ Ti i=1 , (1) 73 74It was then possible to observe where this cumulative ensemble mean, Ecum, approaches an asymptotic 75value that may be termed the ‘true’ trend Results [Fig 3] indicate that about 10 runs of the model seem 76to be sufficient for 40-yr runs (and 20 runs of 20-yr length) 77 78 79 • 80 • 81 • 82 • 83 84 85 86 87 88 89 90 91 92 93Fig 3: Procedure #1: Cumulative ensemble-means of trend values as a function of n, the number of runs 94(of length 20 or 40 years) The cumulative ensemble mean, Ecum, is seen to reach an asymptotic value 95close to zero as the number of runs exceeds about 20 (for a run-length of 20 years) – and about 10 (for a 96run-length of 40 years) In the absence of model drift, the asymptotic value would presumably be zero 97 98Control experiments for the first procedure 99 100 101 102 103 • To investigate the influence of drift, which is seen to exist in the 1000-yr model run [Fig 2], we have also carried out the same procedure for two additional time periods (Years 200-1000 and 400-1000), for which the drift appears to be more uniform – or at least does not change its sign For 40-yr runs, the asymptote of the cumulative ensemble mean Ecum is reached again after at least 10 runs 104 105 106 107 108 • In addition to the 25 time-segments, each of 40 years, starting in years 1, 41, 81, … , a further 24 trend values may be determined by starting the 40-year time-segments in years 11, 51, 91, … , with still further trends determinable by starting in years 21, 61, 101, … , and at 31, 71, 111, …, for a total of 97 trend values All of these were found on examination to behave in the same way Lagged auto-correlation does not seem to be significant 109 110 • We also checked against a possible influence of contiguity by selecting non-contiguous segments to form cumulative ensemble-means 111 112 113 • Finally, by starting in each year on the interval [1, 961], it is possible to obtain 961 (partly overlapping) segments of 40-year length When all 961 trend values are plotted, they are found to form a Gaussian distribution 114 115Second procedure: constraining the error-interval 116In the second of two procedures, the interval on which the values of E fall (‘spread’), was investigated for 117(assumed) three synthetic models as a function of the number of runs The result is shown in Fig.4, with 118the spread plotted against n, the number of simulation runs For a run length of 40 years, the trend interval 119is seen to approach zero for n > 10 (Fig 4) 120 121Fig 4: Procedure #2: Error interval (“spread” = Tmax - Tmin) of ensemble-means of trend values of 122(synthetic) models as a function of n, the number of runs (All trend values shown on the y-axis were 123multiplied by 1000.) For a run length of 40 years, the spread is seen to collapse towards zero as the 124number of runs exceeds 10 Similar results are obtained for the cases of models and models The 125dashed lines result from a different method of selecting trend values (see text) 126 127Details for the second procedure 128 • The time-series is truncated by removing years 1-200, to minimize possible effects of drift 129 130 • The remaining 800 years were divided into 40 segments of length 20 years, whereupon 40 trend values T1 … T40 were determined 131 132 • We assume we have models, each with the same number of runs n We therefore assign n trend values, arbitrarily selected from the 40 values of T, to each of the three models 133 • For n=1, the error interval among the trend values is simply (Tmax – Tmin) 134 135 136 • For n=2, a series of ensemble-mean trend values was constructed thus: T΄1 = ½(T1+T2); T΄2 = ½(T3+T4); T΄3 = ½(T5+T6) The trend interval among the values was then determined as (T΄max– T΄min) 137 • The procedure was repeated for n = 4, 5, 8, 10, and 13, respectively Results are plotted in Fig 138 139 • To test sensitivity, the procedure was next repeated for an assumed models (permitting n-values up to 10), and then for models Similar results were obtained 140 141 142 • As a further check, we repeated all procedures by starting the selection of T-values with T40 (instead of T1) and proceeding backwards Those max and trends are indicated by dashed lines in Fig 143 144Discussion: 145We suggest here that it may be possible to overcome the “chaoticity barrier” of climate models identified 146by IPCC in the 20CEN intercomparison 147We have demonstrated that the ensemble-mean (E) trend obtained from a multi-run model is more reliable 148than the trend obtained from a model that is run only once In general, ten or more 40-yr runs may be 149necessary to form a reliable E 150Sensitivity Analysis of Segment Length: Our initial choice of a segment length of 40 years was 151arbitrary We now investigate the effects of segment length on the convergence of the ensemble mean 152value We find convergence after runs for model runs of length 80 years and after 20 runs (see Fig 3) 153for a segment length of 20 years (which is typical of the models in the IPCC compilation; see Fig 1) 154Empirically, it appears that convergence is achieved in 400 run-years – i.e., (20 x 20), (10 x 40), and (5 x 15580) We have not discovered a theoretical explanation for this useful result 156Discussion: Few modelers have the resources to carry out ten or more runs of a particular general157circulation model They frequently report a temperature trend based on only a single run For example, 158the IPCC’s compilation of 22 ‘20CEN’ models has five models with just one run, five with runs, and 159only seven with or more runs The run lengths varied between 20 and 24 years Modeler should be 160encouraged to report not only the details of the forcings and parameterizations used in their particular 161models but also the results of each run and its length (in years) 162Most investigators when considering a group of models compound the problem by simply using the 163average of the ensemble-mean trends of the group to compare with an observed trend This procedure, 164however, is defective in that it gives equal weight to single-run models and multi-run models, and leads to 165greatly enhanced uncertainty of modeled trends Yet we demonstrate here that the ensemble-mean trend 166obtained from a multi-run model is more reliable than the trend obtained from a model that is run only 167once 168The “spread” in trend values trend values discussed in Procedure #2 is akin to the “range’ of extreme 169values of a Gaussian distribution But “range” is an improper statistical metric.; it inceases as the number 170of independent data points increases –while the Standard Deviation of their distribution decreases For 171example, it can be shown that the wide model uncertainty displayed as a grey region in Fig of Santer et 172al [2008] and labeled as a “2-sigma Standard Deviation” is actually an artifact, caused by the presence of 173five single-run models in the IPCC compilation of models A compilation comprising only multi-run 174models would, therefore, help to constrain chaotic uncertainty, and would provide a more reliable means 175of comparing the consistency of modeled with observed trends 176 177Acknowledgements 178We are grateful to Roger Cohen, Curtis Covey, Robert Levine, Craig Loehle, Christopher Monckton, and 179Ronald Stouffer for useful discussion and to Garrett Harmon and Will McBride for technical assistance in 180preparing the figures 181References 182Giorgi, F 2005 Climate Change Prediction Climatic Change 73: 239-265: DOI: 10.1007/s10584-0051836857-4 184 185Lorenz, Edward N 1963 Deterministic nonperiodic flow Journal of the Atmospheric Sciences, 20: 130186141 187 188Lorenz, E N 1975 The physical bases of climate and climate modeling, in Climate predictability, #16 in 189GARP Publication Series, pp 132-136, World Meteorological Organization 190 191Meehl, G., et al 2001 Climate Change: The Scientific Basis, in Fourth Assessment Report of the 192Intergovernmental Panel on Climate Change, Cambridge University Press, London 193Santer, B.D., Thorne, P.W., Haimberger, L., Taylor, K.E., Wigley, T.M.L., Lanzante, J.R., Solomon, S., 194Free, M., Gleckler, P.J., Jones, P.D., Karl, T.R., Klein, S.A., Mears, C., Nychka, D., Schmidt, G.A., 195Sherwood, S.C., Wentz, F.J 2008 Consistency of modeled and observed temperature trends in the 196tropical troposphere Int J Climatol.: doi:1002/joc.1756 197 198Zichichi, A Meteorology and climate: Problems and expectations Pontifical Council for Justice and 199Peace, The Vatican, 26-27 April 2007 200******************************************************** ... crucial aspect of the impact of chaoticity on the performance of 43general-circulation models: the strong dependence of the error-bars in temperature-trend projections on 44the number of simulations... ensemble-means of trend values of 122(synthetic) models as a function of n, the number of runs (All trend values shown on the y-axis were 123multiplied by 1000.) For a run length of 40 years, the... caused by the presence of 173five single-run models in the IPCC compilation of models A compilation comprising only multi-run 17 4models would, therefore, help to constrain chaotic uncertainty,