Distributions of decadal means of temperature and precipitation change under global warming

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jd014502, 2011 Distributions of decadal means of temperature and precipitation change under global warming I. G. Watterson 1 and P. H. Whetton 1 Received 17 May 2010; revised 7 October 2010; accepted 12 January 2011; published 5 April [1] There remains uncertainty in the projected climate change over the 21st century, in part because of the range of responses forced by rising greenhouse gas concentrations among global climate models. This paper applies a method of estimating distributions and probability density functions (PDFs) for forced change, based on the pattern scaling technique and previously used for Australia, to generate changes in temperature and precipitation at locations over the globe, from simulations of 23 CMIP3 models. Changes for 2030 and 2100, under the A1B scenario for concentrations, for both seasonal and annual cases are presented. The PDFs for temperature have a standard deviation that averages 31% of the mean change, and they tend to be positively skewed. The standard deviation for precipitation averages 15% of the base climate mean, leading to five and 95 percentile estimates that are of opposite sign for most of the globe. A further source of uncertainty of change for a particular period of time, such as a decadal average, is the unforced or internal variability of climate. A joint probability distribution approach is used to produce PDFs for decadal means by adding in an estimate of internal variability. In the decade centered on 2030, this broadens the PDFs substantially. The results are related to time series of observations and projections over for the agricultural regions of Iowa and the Murray Darling Basin. For most land areas, warming becomes clearly discernable, allowing for both uncertainties, in the next few decades. Data files of the key results are provided. Citation: Watterson, I. G., and P. H. Whetton (2011), Distributions of decadal means of temperature and precipitation change under global warming, J. Geophys. Res., 116,, doi: /2010jd Introduction [2] In their projection of climate change forced by greenhouse gas (GHG) concentrations following the A1B scenario of the Intergovernmental Panel on Climate Change (IPCC), Meehl et al. [2007a] found that the global mean surface air temperature (T g ) is likely to rise between 1.7 and 4.4 K over the 21st century ( relative to ), with a best (or central) estimate of 2.8 K. Based on the Third Coupled Model Intercomparison Project (CMIP3) [Meehl et al., 2007b] simulations of coupled atmosphereocean climate models, Christensen et al. [2007] found that the warming for most land regions is likely to be at least as large as DT g (where D denotes change). For averages over subcontinental regions, the median warming from the models is large enough to exceed (or be clearly discernable from) the internal (unforced, or natural) variability of 20 year means from the models, within several decades. [3] For small regions, however, it is more difficult to distinguish between forced change and unforced variability. 1 Centre for Australian Weather and Climate Research, Aspendale, Victoria, Australia. Published in 2011 by the American Geophysical Union. This is suggested by the example of Figure 1a, which shows that observed decadal mean temperatures averaged over the state of Iowa have only recently exceeded a peak in the 1930s. Kattsov and Sporyshev [2006] found that such exceedances are often earlier at lower latitudes, as in the case of another important agricultural region, the central part of Australia s Murray Darling Basin (MDB), for which time series are shown in Figure S1 of the auxiliary material. 1 The forced warming as estimated from the median sensitivity of the CMIP3 models (see captions of Figures 1 and S1 for details) is also shown, in each case. This appears consistent with the observed series, given interdecadal variability, but a range of sensitivities would also be. A plausible 90 percentile range for decadal means about the 20th century trend series based on CMIP3 is also shown (following the approach of Watterson [2010]; see below). Extending the forced series through the 21st century under A1B, using these medians, gives temperatures that well exceed this previous range. Note that the two regions are suitably similar in size to the grid squares of most CMIP3 models, but are largely shown here to illustrate the projections. [4] Another key variable, precipitation (P, or rainfall) is expected to increase under global warming at high latitudes 1 Auxiliary material data sets are available at ftp://ftp.agu.org/apend/jd/ 2010jd Other auxiliary material files are in the HTML. 1of13

2 Figure 1. Time series of observed (a) surface air temperature and (b) precipitation over Iowa, United States, averaged over running decades (centered on the end of the year on the time axis and hence ending at 2003, solid line), together with projected values. The central (long dashed) line shows the expected trend forced by global warming, with the average over specified to match the observed mean. Before 2003, the series is based on the median sensitivity of the region (around point 94 W, 43 N) as simulated by the CMIP3 ensemble (+1.42 K for T, +2.8% for P, per degree of global warming; see sections 2 and 4), combined with the CMIP3 multimodel mean global mean warming series [Meehl et al., 2007a, Figure 10.4] for the 20C3M simulations, with smoothing. After 2003, the line represents the expected (or P50) changes for the A1B scenario based on the pattern scaling approach, shown relative to the expected average. The dotted lines show the uncertainty range (P05 and P95 values) for forced change, under the assumption of no uncertainty for (sections 2 and 3). The outer (short dashed) lines over all years are the P05 and P95 values for decadal means, after the inclusion of unforced variability, based on the SD of decadal means simulated by four CMIP3 models (sections 3, 4, and 5). The observed series are averages of data from the 23 Iowa stations in the U.S. Historical Climatology Network [Menne et al., 2009] over , obtained from the Web site www. ncdc.noaa.gov/oa/climate/research/ushcn. and near the equator, and decrease in the subtropics. However, even in subcontinental regions these signals may exceed the internal variability only late in the 21st century [Christensen et al., 2007]. For many smaller midlatitude regions there is a range of results, covering both signs, from the models, and variability is often large, as in the cases of Iowa and MDB shown in Figures 1b and S1b, respectively. There is no clear trend in these observations, and the median forced response to future warming is relatively modest. However, larger changes in rainfall and also temperature are simulated by some models, with potentially significant impacts, for instance on agriculture. Variations on decadal time scales can also have important impacts over both land and ocean [e.g., Woodward et al., 2008; Kravtsov and Spannagle, 2008]. [5] Hawkins and Sutton [2009] recently found that uncertainty in regional and decadal mean T resulting from differences among CMIP3 models and from internal variability exceeds that associated with scenario uncertainty for much of the coming century. Quantification of such uncertainties using continuous distributions in the form of probability density functions (PDFs), rather than discrete values, is increasingly sought and various approaches have been used [e.g., Räisänen et al., 2010; Xu et al., 2010]. Given that regional responses to forcing by GHG are to some extent proportional to DT g [Mitchell, 2003; Meehl et al., 2007a], some authors [e.g., Dessai et al., 2005; Whetton et al., 2005] have combined differing patterns of change with the range of projected DT g. Changes divided, or scaled, by global mean warming are termed standardized changes. Watterson [2008] made this pattern scaling assumption and represented both local standardized change and DT g for a scenario and time period using smooth, but non Gaussian PDFs. Taking the product, using a joint PDF formalism, resulted in a PDF for the actual or net local change (denoted forced change here). Percentiles (denoted Pnn) follow from the corresponding cumulative distribution function (CDF). In the paper, the method was applied to T and P for only a few Australian cases, but results for points over the continent were presented by CSIRO, 2007] (hereafter referred to as C07). [6] There are several caveats to the approach used by C07, a significant one being the simple assumption that the range of uncertainty in the real world future climate should be directly related to the range of CMIP3 model results. This is commonly used [e.g., Hawkins and Sutton, 2009] and received support recently from Annan and Hargreaves [2010]. However, both broader and narrower ranges can result from other methods, as is discussed by Watterson [2008], and the most appropriate approach is yet to be resolved [Knutti et al., 2010]. Murphy et al. [2007] developed a sophisticated approach to generating PDFs that explicitly includes uncertainties in various climate feedbacks, using large ensembles of global model simulations. It also includes a method of downscaling that supported detailed projections for the United Kingdom (denoted UKCP09) presented by Murphy et al. [2009]. While improved projections can be expected to follow from future modeling experiments, Watterson [2008] provides a relatively straightforward approach that produces ranges of change that are broadly consistent with CMIP3 and the IPCC AR4. Furthermore, the use of pattern scaling as part of the approach also makes it easy to replace the global warming PDF used with an alternative one (such as for a different emission scenario, or one based on new scientific understanding). [7] One purpose of the present paper is to apply the method of Watterson [2008] to every grid point of the globe, 2of13

3 and provide percentiles for forced change of T and P in map form, data files (in the auxiliary material, which also includes a table of symbols, Table S1) and time series (as in Figure 1). After a brief description of the method and its caveats, the annual (or ANN) case for temperatures from the full year is then considered in section 2, with a focus on change under A1B at both 2030 and An interesting aspect of the pattern scaling method, which is explored, is its ability to reproduce the asymmetry relative to the mean, or skewness, that is evident in the AR4 range for global mean warming. [8] A second purpose here is to consider the representation of uncertainty in decadal means using PDFs. Such results would be particularly useful in cases where impacts are nonlinear in the climate variable, and probabilities of more extreme values that could occur for a decadal period need to be quantified. A normal (Gaussian) PDF for internal variability about a single base value is trivial to specify given the relevant standard deviation (SD). This has been used in Figure 1, pre 2000, where the SD is determined from long simulations ( runs ) of unforced climate from four CMIP3 models, and the P05 and P95 anomalies are ±1.65 SD. The problem is less simple when the variability needs to be added to a nonnormal change PDF. An extension of the joint PDF approach is used to tackle this and produce percentiles to represent the total of the forced and unforced uncertainty, as described in section 3. While we denote such results as total (decadal) change, the unforced component of this change may be short lived, of course. Both forced and total change results for precipitation are presented in section 4, where further application of the method used by Watterson [2008] to prevent unphysical decreases beyond 100% is made. Seasonal results for both variables are briefly mentioned in section 4.3. Some brief comparisons with UKCP09 are made. Consideration of exceedance times and various other issues concerning the projections is made in section 5. The conclusions follow in section 6. [9] Finally, our approach may be compared to one where the range of future decadal means is formed directly from GCM decadal output [e.g., Christensen et al., 2007]. As well as the advantages of patterns scaling already noted, separate consideration of the signal and variability components (as our approach entails) has the advantage of allowing these components to be more robustly estimated. 2. PDFs for Forced Change 2.1. Probabilistic Pattern Scaling Approach [10] In this section we aim to produce a PDF for the GHG forced change v in a variable at a location, following the approach of Watterson [2008]. The pattern scaling approximation is that v is the product of the standardized local change x and DT g ; with the uncertainty in these quantities represented by the probability functions f(x) and g (DT g ), respectively, the joint PDF for the two factors, assuming they are independent, is simply fg. By discretizing the functions, the CDF for v and the corresponding PDF h(v) can be readily calculated, as described more fully by Watterson [2008]. [11] It should be stressed that this approach was not developed from an underlying probabilistic theory, such as those of Rougier [2007] and Tebaldi and Sansó [2009], but evolved from methods of Whetton et al. [2005] and others. In this sense probabilistic is used loosely. As discussed by Watterson [2008] the PDFs calculated are not intended to represent the (hypothetical) range of all possible models, nor to include uncertainties from processes not represented by the CMIP3 GCMs [Stainforth et al., 2007]. The PDFs are, in practice, based on smoothed distributions of CMIP3 results. Furthermore, the local changes primarily reflect the GCMs sensitivity to GHG, and not regional variations that may be associated with changing aerosol loadings, or other effects Calculation of PDFs for Standardized Change [12] For each location the value x i from each of the CMIP3 models used (listed again in Table S2, and denoted by subscript i from 1 to 23) is estimated by regressing the annual (or seasonal) mean from each year of the 21st century under A1B, against the corresponding T g. If multiple runs were available these were averaged to reduce noise, while A2 runs were used in a few cases with no A1B run. The usual uncertainty estimate for the regression coefficient, as x i, denoted s i, is used to specify a normal PDF for the true value for the model, given a hypothetical large ensemble of runs. A combined or Sum PDF is formed from the weighted average of these 23 PDFs. This form is illustrated in Figure 2a by the case of annual temperature at Iowa, discussed shortly. [13] For simplicity, a single set of weights is used for all cases. These are intended to represent the overall ability of a model in simulating patterns of change, and we follow Watterson [2008] in using a quantitative assessment of present climate seasonal means, using global fields here. The weights were determined using a combined skill score statistic ( M of Watterson [1996] and Whetton et al. [2007]) for T, P and sea level pressure from all four seasons; see Text S1 for details. The top four models by this test are the same as those from the more comprehensive assessment of Reichler and Kim [2008], as are the top 11 (common) models. The relative weights (Table S2) range from down to 0.597, so the discrimination is rather small, however. This is because in calculating M errors are scaled relative to the spatial SD of the quantity, which is large over the whole globe. It should be noted that while the weights decrease as GCM errors increase, they are not based on a likelihood function as in a theory like that of Rougier [2007]. In any case, further investigation, described in Text S1, indicates that similar results are obtained if either uniform weighting or more varied weighting is used. The results for Australia that follow are similar to those of C07, which used Australian based weights. [14] The final PDF f is specified to be the (four parameter) beta distribution with the same mean and SD as this Sum PDF, and with the end points matching the range of the x i values (extended by 1 SD uncertainty of each end value). A sharp approach to an end point is avoided by further extensions (see Watterson [2008] for details). The resulting form is bell shaped, like the normal, but has a finite range and can be skewed. We use the normal PDF with the same mean and SD in sensitivity tests. Calculations are performed using values of f at 1000 points in x. These forms for the Iowa case are included in Figure 2a, with results for four 3of13

4 mean warming as 0.9 K, or 0.32 times that for , and the corresponding PDF for that case is also given in Figure 3. The (nondimensional) skewness coefficient (SK) of either beta PDF, calculated as the third moment about the mean divided by the cube of the SD, is As a sensitivity test for the 2100 case, the related normal distribution plotted in Figure 3 is used. In all cases, the DT g PDFs are discretized on 100 values in warming. The PDFs h(v), for the local forced change, are determined on 1000 values. [16] While the range and skewness of the beta PDF is rather consistent with the CMIP3 models, a broader range was not ruled out by Meehl et al. [2007a]. Watterson [2008] provided an example of the use of a wider PDF. In any case, the spread here is already slightly larger than that in UKCP09, with the ratio of P90 to P10 values being 1.8, compared to 1.6 from Murphy et al. [2009], using values given in their Table 4.8. The P50 value for our final period is smaller than theirs (2.8 K compared to 3.4 K). While UKCP09 considers periods centered on 1975 and 2084, rather than 1989 and 2094, this would explain little of the difference. One possibility is that UKCP09 s modeling of carbon cycle feedbacks raises the central estimate of warming beyond that allowed for by the earlier assessment. Figure 2. Distributions for change in temperature at the Iowa grid point (94 W, 43 N): (a) the PDF for change relative to the global warming (f), using three methods, and (b) the CDF for forced change at 2100 under A1B, determined by combining the normal distribution for global warming with the normal PDF in Figure 2a, or the beta PDF for both factors. Values of standardized change from the 23 individual models are marked by crosses in Figure 2a, offset from the x axis for clarity. other locations (discussed shortly) shown in Figure S2. For convenience, SD f denotes the SD of f, which represents an uncertainty in scaled forced response. The effect of uncertainties in the x i values is considered further in section PDFs for Global Mean Temperature [15] The likely range in DT g noted initially represents from 40% to +60% of the multimodel mean warming from CMIP3 (2.8 K), and Meehl et al. [2007a] used this formula for other times and scenarios. While the range took account of other evidence, Watterson [2008] found a similar spread from the individual model warming values for A1B listed (again) in Table S2. The asymmetry in the formula is well represented by a beta distribution, determined as for f and plotted in Figure 3; see Watterson [2008] for the detailed specification. This is the standard PDF g for the 2100 A1B case, as denoted previously, although the time is nominal as the final decade ( ) is intended. It can also be used as the PDF for the forced global warming in any case with a central estimate of 2.8 K. For the year 2030 (the middle of the decade considered later), C07 estimated the CMIP Temperature at 2100 [17] The uncertainty of DT at a location results from that in both the global warming and the standardized value. The latter can vary substantially, as evident from the field of the sample SD of the 23 model values (interpolated to a common 2 grid) for the ANN case shown in Figure 4a. In the polar regions the SD is often 0.5 or more. In parts of the North Atlantic Ocean it exceeds 1.0, ten times the value at many tropical ocean points. Land values in the midlatitudes are mostly 0.2 to 0.3. The SD at the grid point representing Iowa is 0.27, with individual x i values ranging between 1 and 2, as can be seen in Figure 2a. The resulting Sum PDF, shown there, has a slightly larger SD (which is SD f ) of The Iowa case happens to have a bimodal shape, which might be included if we knew of physical reasons behind this. In this global analysis, we will consider only the fitted beta and normal PDFs, as in Figure 2a. The normal PDF Figure 3. PDFs for global mean warming under the A1B scenario at two times: 2100 and The forms normal and beta have the same mean and SD, with the 2030 values being simply 0.32 times the 2100 values. 4of13

5 from individual models (Figure S2b). In fact, the North Atlantic is one region where the P05 values are negative, the other being in the Southern Ocean, as seen Figure 5a. The P95 value at the North Atlantic point exceeds 5 K, like much of the high latitudes and also the land (Figure 5c). The point Libya (near Tripoli) has a typical mean and median (Figure 5b) warming for land, but rather small SD. The equatorial Borneo point has a rather small mean for land (the mean of f being 0.97). The SD f is among the smallest for any land point but the P05 to P95 uncertainty is still considerable, largely because of DT g. The median field (Figure 5b) has much in common with the multimodel mean field shown by Meehl et al. [2007a, Figure 10.8]. [20] While values from individual grid points vary, particularly between latitudes and from land to sea, it is useful to consider global means of various quantities, including Table 1. Statistics Derived From the PDF for Forced Change of T, Annual Case, at Four Named Locations, at Two Times, and Under Various Cases a Time Type Fit Mean SD SK P05 P50 P95 Figure 4. Variability in decadal mean temperature, quantified by the standard deviation (in K) from two sources: (a) in forced change for a 1 K global warming, with the SD being over the 23 CMIP3 climate models, and (b) in unforced, internal variability, as simulated by four models. In Figure 4b, shading indicates a statistically significant difference to variability in the ERA 40 data set (see text). Heavy (lighter) stippling suggests that the four model variability is larger (smaller) than actual. extends to x values well outside the CMIP3 range, although with small probability. [18] Combining these two PDFs for Iowa with the two global warming PDFs for 2100 produces four CDFs for forced change. Two of the CDFs, using beta or normal for both factors, are shown in Figure 2b, with several statistics given in Table 1, type F (forced). These include SD f and the percentiles 5, 50 and 95 (denoted P05, P50 and P95). In this case, the difference between the beta and normal results is relatively small. The mean of the forced change is given by the product of the f and g means, and hence is the same for both (Table 1, first row case, N normal, and second row, B beta). The P50 or median is not quite the same as the mean, due to asymmetry of the PDFs, but is still similar for beta or normal. The SD is also similar, despite the longer extreme tails of the normal PDFs. The P95 value is slightly larger for beta, as a result of the greater skewness in the PDF (Table 1). These three percentile values are also shown in Figure 1 (the forced change values plotted at year 2094), after the addition of the plotted P50 value for the period [19] Three other locations, with a range of SD f values, feature in Table 1 and Figure S2. The North Atlantic point has the smallest mean for the forced change, and the largest SD, following from both negative and large positive values Iowa, 94 W 43 N, SD f = 0.28, SD I = F N F B T B F B T B North Atlantic, 40 W 49 N, SD f = 0.80, SD I = F N F B T B F B T B Libya, 12 E 31 N, SD f = 0.13, SD I = F N F B T B F B T B Borneo, 112 E 1 N, SD f = 0.11, SD I = F N F B T B F B T B Global Mean of Values, SD f = 0.22, SD I = F R F N F B T B F B T B a Also given are the global means of the results. For each location the longitude, latitude, and standard deviations of the PDF for standardized forced change (SD f, in K per K) and for internal decadal variability (SD I in K) are also given. For Type, F means forced change and T means total of forced and internal variability. Fit is the form of PDF for both global warming and local change per degree, with N meaning normal and B meaning beta. Also given are the global averages for the raw case (Fit R), as explained in the text. The first three statistics are the mean (K), SD (K), and skewness (SK, nondimensional) of the PDF. The others are the change (in K), at the 5, 50, and 95 percentiles, of the corresponding CDF. To clarify differences, SD and SK are given to higher precision. 5of13

6 Figure 5. Percentiles for forced change in temperature (in K) at 2100 under A1B, determined by combining the beta PDF for both global warming and relative change: (a) P05, (b) P50 or median, and (c) P95. those given in Table 1. The global mean of the mean of the standardized PDFs ( f ) is naturally 1.0, while that of the forced change PDFs (h), in Table 1, matches the g mean. Watterson [2008] compared forced changes with the Sum PDF derived at each grid point from the nonscaled or raw data, from the GCMs, with these determined by multiplying each model s standardized change (and its uncertainty) by its own global warming. It is worth giving the global means for this case R in Table 1. That for the PDF mean is again the same. The global mean of SD is also similar to the N and B results. The SDs could differ at grid points, in part because the 23 DT g and local x values may be correlated. Overall, though, the field of correlations averages only 0.058, with a spatial SD of This is quite consistent with sampling uncertainty for 23 uncorrelated quantities, supporting the pattern scaling assumption. It is worth noting here that the same holds for precipitation (average, 0.007, SD 0.22). [21] The global mean of the skewness of the grid point PDFs is only 0.06 for (beta) f but rises to 0.45 for h (Table 1, case B). A similar mean skewness occurs in the raw case. The mean skewness for the N case is smaller, at 0.36, but still positive, despite its unskewed f and g components. Skewness develops simply from taking the product of PDFs of quantities that are usually of the same sign. In addition, the skewness field from B is rather uniform, with a spatial SD of The raw field is more erratic, with a spatial SD of Presumably, sampling of only 23 models allows variation in this statistic. These skewness characteristics of the pattern scaling approach seem not to have been analyzed by previous authors. However, the effect is seen also in distributions of DT g generated by combining several factors [e.g., Dessai et al., 2005] Temperature at 2030 [22] With the PDF for DT g in the 2030 case being the same relative shape as that for 2100, the PDF for the forced change at any location also has the same shape. For comparison with the decadal mean case to come, the PDF for Iowa in 2030 is shown in Figure 6. (Computational noisiness in these PDFs has negligible impact on the CDFs and statistics.) Skewness in the PDF is seen, with the coefficient 0.48, again enhanced from that for both g (SK = 0.15) in Figure 3 and f (SK = 0.01) in Figure 2a. Other than SK, the statistics in Table 1, for each location and the globe, are those for 2100 multiplied by the 0.32 global warming factor. [23] The SD of the forced change PDF, as a global field, is shown in Figure 7a. This is closely correlated (r = 0.96) to the SD of the standardized changes (Figure 4a). It averages 31% larger, despite the global mean warming being 0.90 K (10% smaller than 1 K), as a result of the spread of the PDF for DT g. Maps of the P05 and P95 for forced change at 2030 can be seen in Figures S3a and S3b. As for 2100, skewness relates to an asymmetry in the percentiles, with P95 relative to P50 being 37% larger than P50 relative to P05, in the global means. 3. PDFs for Decadal Means 3.1. Decadal Variability [24] The PDFs for forced change for a particular year and scenario are intended to represent uncertainty in the Figure 6. Distributions for change in decadal means of temperature at the Iowa grid point at 2030: the forced change calculated using beta PDFs, and the total change, allowing for internal variability. 6of13

7 Figure 7. Standard Deviation of decadal means of temperature (in K) at 2030 under A1B, from (a) uncertainty in forced change and (b) total change, allowing for internal variability. real world forced response to GHG, relative to some climatological state. With the global warming being assumed relative to , the state can be taken to be a longterm baseline climate average, adjusted somewhat for the warming up to that period. The expected trend curve plotted in Figure 1 attempts to represent this response for the Iowa case. In practice, annual means of local observed values vary from year to year, and even decadal means vary considerably, as seen in Figure 1. [25] Quantifying this internal variability of decadal means is hampered by the limited number of decades of observations. Even with 100 years of reliable data (which excludes much of the globe in standard gridded data sets), the statistical uncertainty of the variance of 10 distinct decades is estimated (by the standard chi squared theory) to have a P05 to P95 range of some 0.53 to 2.70 times this sample variance. These represent a range of SD from 27% less to 64% more. Actual data are also subject to externally forced variations, of course. [26] While we wish to include the real world variability in PDFs for future decades, like Christensen et al. [2007] and Hawkins and Sutton [2009], we consider statistics based on climate models. Watterson [2010] and C07 considered the SD for decadal means from a 1080 year unforced or control run of CSIRO Mk3.5 (see Table S2), but we augment this here with results from three other CMIP3 models GFDL CM2.1 (50 decades), ECHAM5 (50) and HadGEM1 (14). Gleckler et al. [2008] found these (together with the earlier CSIRO Mk3.0) to be among the more skilful models, with respect to interannual variability (from their Figure 9, for the northern hemisphere and tropics). The SDs of decadal means (after small linear trends were removed) tend to be similar at each latitude in these models, as can be seen from the plots of zonal means (of grid point SDs) in Figure S4. For T, the zonal means of SD are also mostly similar to those determined from the global ERA 40 reanalysis archive [Uppala et al., 2005], even though only four decades (not detrended) are used. Variability from one of the less skilful models (as found by Gleckler et al. [2008]) was rather different, even after a substantial trend was removed (Figure S4). [27] The paper is not intended as a study of variability in models, as we only wish to demonstrate the inclusion of a plausible estimate of real world internal variability in PDFs. We specify this variability by the interdecadal SD, denoted SD I, using the simple average of the fields of the four named models. This field is plotted in Figure 4b. As a comparison with the (statistically uncertain) ERA 40 SD field, grid points are shaded when the ERA 40 to model ratio of variances is outside the expected P05 to P95 range based on standard F distribution theory [Hogg and Craig, 1970, p. 205]. (This corresponds to about 0.34 to 1.62 for the ratio of SDs of 4 and 200 or so decades.) The results are compatible, except for a few regions, which may occur by chance. The four model values at Iowa and MDB produce ranges that appear consistent with the observed series in Figures 1 and S1. Typically, there is one excursion outside the range. In some oceanic regions, particularly near sharp gradients in temperature, the model values for decadal T SD vary considerably, for example the Mk3.5 SD is double the average value in the central North Atlantic. Over land, for most regions the variability is fairly consistent. Typically, the SD of the four SD values is 15% of the average in the midlatitudes and 25% in low and high latitudes. [28] As will be discussed in section 5, the SD I field or Figure 4b appears quite similar to the SD of the standardized change from 23 models (which is approximately SD f )of Figure 4a. The global mean values (given in Table 1) are also similar, allowing for the different units. This immediately suggests that uncertainty in decadal means will be comparable to that in forced change for a global warming of about 1 K. While internal variability may change in a warmer world, we follow Hawkins and Sutton [2009] in assuming that this SD field determined from the unforced simulations can be used to represent variability in the coming century. We further assume that decadal anomalies at each grid point associated with internal variability have as their PDF, a normal distribution with this SD and zero mean Method [29] We wish to represent the sum of the forced change and possible decadal anomalies, using a probabilistic approach. As in the case of combining the PDF for DT g with that for local standardized change, we can form a two dimensional joint PDF for the forced change and the internal variability. Making the usual assumption that these are independent [e.g., Hawkins and Sutton, 2009], the joint PDF is simply the product of the individual PDFs, h(v) n(y), where the n represents the variability PDF, in the coordinate y, now. If both these are normal, then the PDF for the sum is, of course, normal, and with variance equal to the sum of the individual variances. [30] From our pattern scaling approach, h is nonnormal, so to gain an accurate PDF for the total change for a decade 7of13

8 given in Table 1, type T. The mean is unchanged from the forced, beta, case, F. The SD is increased only slightly. Even for nonnormal PDFs (such as h), the variance of the sum of independent quantities is still the sum of the variances. And even for the variable North Atlantic point, SD I is not able to greatly boost the square root of the variance at The skewness is always reduced in magnitude through the addition of the nonskewed n. The P95 values are raised slightly, and P05 lowered, by a little more on average. Basic gridded results are available from the auxiliary material (see Table S3) Temperature at 2030 [32] The PDFs for 2030 are intended to represent the uncertainty in the average over the decade around 2030 (say ), as anomalies relative to the expected values during The PDF means are again unchanged by the addition of decadal variability (Table 1, type T relative to type F). The relative effect of the variability on the other statistics is greater than in The enhancement of SD and reduction in skewness is quite evident in Figure 6 for Iowa and in Table 1, for each point. The changes are pronounced in the global mean also, with the mean SD for total change being 24% more than the forced result. As seen from Figure 7b, even for points in the tropical oceans, the SD of the decadal means are usually greater than 0.2 K. Highlatitude points are particularly variable and uncertain. Maps of the P05 and P95 for the total decadal change at 2030 can be seen in Figures S3c and S3d. Negative P05 values occur for some high latitude points, mainly over the oceans but including at the North Atlantic coasts. At most latitudes, P95 exceeds 2 K at some land points. Figure 8. Standard deviation in decadal mean precipitation, as a percentage of base climate mean, in three forms: (a) forced change for a 1 K global warming, using the SD over the 23 CMIP3 models of the standardized change relative to the model mean, (b) unforced, internal variability, as simulated by four CMIP3 models, and (c) in total precipitation change at 2030 under A1B. In Figure 8b, heavy (lighter) stippling suggests the four model variability is larger (smaller) than actual (based on the ERA 40 data set). Note that the SD for the forced change at 2030 is very similar to Figure 8a. an integration of the sum across the joint PDF is required. An illustration of both the joint PDF and the sum, as functions across the domain, for the Iowa, 2030 warming case is provided in Figure S5 (as this has a similar character to Figure 3 of Watterson [2008]). The variability PDF is discretized on 100 points in y, from 3SD I to +3SD I. The total is again determined on 1000 points. The result for Iowa, using our standard h from the beta components, is shown in Figure Temperature at 2100 [31] For the first period considered, the decade up to 2100, global mean and grid point statistics for total change are 4. Application to Precipitation 4.1. PDFs for Forced Change [33] We turn to the other variable considered in our Iowa and MDB time series Figures 1 and S1, precipitation, or P. One difference from T is that the standard pattern scaling method can produce a PDF for DP that includes decreases that are large relative to the present climate mean. In the grid point case examined by Watterson [2008], a large decrease in absolute terms from one model with above average rainfall produced an unrealistic decrease relative to the observational data. This was reduced by using as the variable DP as a percentage of the model mean (over here), or relative P, which is also favored in many climate impact applications. Even using this form, however, there is a considerable range in the standardized changes from the CMIP3 models. The SD field for the annual case, shown in Figure 8a, is often 10% K 1 in subtropical land, and in some low latitude regions double that. The range is small over high latitude oceans, with SD below 2% K 1. The global mean of the field is 4.9% K 1. [34] The individual model values for standardized change in relative P for the MDB point are shown in Figure 9a. Allowing for uncertainty in these regression results, the Sum PDF becomes rather smooth. The beta fit to Sum has a slightly smaller range than Sum and the normal fit (Figure 9a). The median value for the beta PDF leads to a small forced decrease over the past century in the depiction of Figure S1b, while that for Iowa gives a small increase (Figure 1b). The 8of13

9 Figure 9. Distributions for percentage change in precipitation in the central Murray Darling Basin (148 E, 33 S): (a) the PDF for change relative to the global warming, using three methods, along with individual values (crosses), and (b) the PDFs for 2100 under A1B, for both forced change and total change, allowing for internal variability of decadal means. The beta PDFs for both factors are used, and decreases are calculated using the exponential approach. PDFs for the four other featured grid points, shown in Figure S6, represent values of both signs. [35] Combining the PDF f with the PDF g for forced global warming by 2100 under A1B as for T produces a PDF for forced change at MDB, whose statistics are given in Table 2. The choice of normal PDFs for both factors (Type N, here) can be compared with that for the beta pair (F, Table 2, second row). The means are again the same, while there are only small differences between N and F in the other statistics. [36] While these P05 values for MDB seem plausible, particularly given recent sharp declines (Figure S1b), at some locations this standard method produces P05 values at 2100 that represent unphysical decreases greater than 100%. To avoid this, Watterson [2008] proposed using an exponential approach, whereby 20% decreases, say, for each degree of warming would be compounded rather than summed: the change for 3 K is then 49% rather than 60%. Watterson [2008] described the computational method and gave further justification for this approach, denoted mixed since the usual linear calculations are used for increases. (For simplicity, the x i values are still calculated by linear regression, regardless of sign.) The resulting PDF for MDB (beta pair) is shown in Figure 9b, with statistics given in Table 2 (fourth row). The value of P05 is less negative, while P95 is the same increase. Note, though, that Smith and Chandler [2010] find that decreases over MDB become more likely, under more discriminatory weighting of models. [37] This approach can accommodate large values of standardized change, and movement of rain bands such as the convergence zones can produce these in models. However, some extreme percentage values, particularly in seasonal cases, occur where there is very little rain. To avoid these, values were omitted from the calculations if the regression uncertainty (SD value) was over 40%, while remaining trend values were capped at ±100% per K. Even in the seasonal cases over deserts, there were always values left from more than half the models. [38] The global fields for the percentiles P05, P50 and P95 for change at 2100 determined with this final approach are shown in Figure 10. Naturally, the P50 field has a pattern of sign of change closely matching the multimodel mean shown by Meehl et al. [2007a]. The global mean of the PDF mean is 6%, while that for the PDF SD values is 15%. Only in a few midlatitude regions are the P95 values not positive, while the P05 are decreases at all places outside the high latitudes. The P05 and P95 changes are consistent in sign in for only 18% of the globe, mostly in the high latitudes. For a PDF spanning zero, such as in Figure 9b, there is typically little skewness. However, the global mean of SK is 0.27, as PDFs are positively skewed at both high latitudes, because of the influence of DT g, and in places with very large range, because of the limit on decreases. [39] The results for the smaller warming at 2030, using the beta PDF in Figure 3, are essentially those for 2100 reduced by the same factor (0.32). The negative values are reduced a little less, because of a smaller effect of the exponential approach, as can be seen from the MDB values in Table 2 (sixth row). The P05 and P95 fields for 2030 are plotted in Figures S7a and S7b PDFs for Decadal Means [40] As seen from the time series for Iowa (Figure 1b) and MDB (Figure S1b), decadal means of precipitation can vary substantially relative to the mean. Variability in the CMIP3 models can be comparable, judging by the P05 P95 range Table 2. Statistics, as in Table 1, Derived From the PDF for Percentage Change of ANN Precipitation in Central Murray Darling Basin (148 E, 33 S) at Two Times and for Several Cases a Time Type Mean SD P05 P50 P N F T Exponential Approach 2100 F T F T a Here SD f =6.1%K 1 and SD I =9.9%.For Type, F means forced change, with N denoting the case with normal fit for both global warming and local change per degree. The beta form is used for the other cases. Type T means total of forced and internal variability. In the first three cases the linear approach is used; in the last four the exponential approach is used for decreases. 9of13

10 anomalies close to 100%. The resulting PDF for total change in 2100 at MDB is shown in Figure 9b. The broadening of uncertainty is noticeable, there and in the statistics given in Table 2. Naturally, it is relatively greater for 2030 (Table 2), when the variance is dominated by internal variability. The SD field for the total change at 2030 is included as Figure 8c. This has much in common with the SD I field (Figure 8b), the mean being some 19% larger, and 68% larger than the forced result (not shown). A large broadening of the range is also evident in P05 and P95, from the plots in Figures S7c and S7d. At high latitudes, skewness tends to decrease, on the addition of n, as it did for T. However the restraint on decreases can lead to some PDFs becoming more positively skewed, as in the case of Figure 9b Seasonal Results [42] The Australian projections of C07 included other variables and also seasonal cases. Results for the usual winter/ summer seasons, December February (DJF) and June August (JJA) have been calculated for the globe for this study, also. The basic gridded results for 2030 are included in the auxiliary material, as are maps of P05 and P95 for forced and total change; for T in DJF (Figure S8), JJA (Figure S9), and for P in DJF (Figure S10) and JJA (Figure S11). [43] A few comments and results are warranted here, and we consider temperature also. The main contrast with ANN is due to the greater variability of decadal averages taken over a single season, rather than all four. In the global mean, SD I for T is 43% larger for DJF and 21% larger for JJA than ANN. With some large relative variations of tropical P, the global mean of SD I for either season is double that for ANN. In addition, there is some seasonal variation in the forced change, particularly for P. Figure 10. Percentiles for forced change in precipitation (in %) at 2100 under A1B, determined by combining the beta PDF for both global warming and relative change: (a) P05, (b) P50 or median, and (c) P95. The exponential approach is used for decreases. shown for the past century in Figures 1b and S1b. This range was based on the average of the SD values from the four detrended CMIP3 control runs (as for T), taken as a percentage of the average mean P of the four. Zonal means of the individual SD values (in unit mm d 1 ), shown in Figure S4b, are quite similar, particularly in the midlatitudes. The SD determined from four decades of ERA 40 are typically larger, but this seems likely an artefact of ERA 40 [Andersson et al., 2005]. The four model field, used as SD I, is shown in Figure 8b, including the statistical comparison with ERA 40 values. The global mean of SD I is 6.8%, numerically larger than the mean of SD f 5.4% K 1, which suggests a relatively greater effect of decadal uncertainty than for T. The pattern is still similar to that of the SD of the standardized forced changes shown in Figure 8a. [41] Combining the decadal uncertainty PDF n with the PDF for forced change is achieved using the joint PDF approach as for DT, except for decreases. The exponential form is again invoked, to prevent the PDF allowing Table 3. Statistics, as in Table 1, Derived From the PDF for Change of Seasonal Mean Temperature (in K) and Precipitation (in %) for the Decade Centered on 2030 in December February (DJF) and June August (JJA) at Two Locations, Iowa (94 W, 43 N) and Central Murray Darling Basin (MDB, 148 E, 33 S) a Type Mean SD P05 P50 P95 Temperature Iowa, DJF F T Iowa, JJA F T MDB, DJF F T MDB, JJA F T Precipitation Iowa, DJF F T Iowa, JJA F T MDB, DJF F T MDB, JJA F T a For Type, F means forced change and T means total of forced and internal variability. In each case the beta form of PDF is used for both global warming and local change per degree. For precipitation, the exponential approach is used for decreases. 10 of 13

11 Figure 11. Year when the P05 value of total change (for the decade centered on the year and relative to ) first exceeds the P95 value for decadal internal variability for (a) T and (b) P. Grid points are not shaded if exceedance is not reached by [44] There is particular interest in these seasonal aspects with regard to impacts on agriculture, so it is worthwhile including statistics for Iowa and MDB in 2030 as Table 3. The T results for Iowa have a similar mean to ANN, but are broader, particularly for total change. For MDB, there is some seasonal contrast in mean warming, being larger in summer, and little effect of adding variability in JJA. For precipitation, there is some seasonal variation in the mean for both locations. The range is larger than for ANN, and it increases further with decadal variability. [45] Our results over the UK appear to have a similar character to those in UKCP09. As an example, for the central UK region (our grid point 53 N, 2 W), precipitation is more likely to decrease in summer and increase in winter, in both analyses. The annual P50 warming here is a little smaller, and range larger, partly consistent with the different DT g PDFs. 5. Discussion [46] As for 2030, and following C07, projections of forced change for other scenarios or other times during the 21st century can be made by using the central estimate for DT g for the case (relative to 2.8 K) to scale the ( 2100 ) value. The expected or P50 series, and the (inner) P05 and P95 series in Figure 1 and Figure S1 were determined this way, using the CMIP3 multimodel global warming series for A1B. The P05 and P95 range from internal variability prior to 2004 are shown relative to the P50 series. After then, the outer lines represent total uncertainty, relative to the P50 mean. To avoid applying the full method at every year, these percentiles were approximated using a careful interpolation in time between the initial and final values, as described in the caption of Figure S1. [47] All the projections represent change relative to the baseline of the idealized average that would have occurred without variability. In the series plots, the uncertainty in this value is disregarded. With this limitation, Figures 1 and S1 illustrate the spread of both the forced and total change PDFs as determined through our approach. They provide an indication of bounds to the path the future observed decadal means might follow. [48] In the Iowa examples the spread for under the A1B scenario, as a ratio to the internal P05 to P95 range (e.g., at 1989), is 3.4 for T, and 2.1 for P. For MDB the ratio is 4.0 for T and 1.8 for P. The global mean over land of the ratio is for T 4.7, with larger values common in the tropics, and for P 2.5. Clearly, for the later decades, the uncertainty is dominated by that in the forced change, particularly so for temperature. For Iowa, the uncertainty means that not until 2019 (or ) does the total P05 curve for T exceed the 1934 (or ) observed peak. Only by about 2034 does the total P05 change relative to exceed the P95 for internal variability in the decade around For the P case in Figure 1b, no such exceedance time is reached, because of the uncertainty in the sign of the forced change. [49] Indeed, as a test for a change that is discernable over the variability, comparing a future total P05 value to the P95 value in 1989 is rather severe. Variability alone would require the change to be at least 3.3 times the SD for decadal means, or equivalently 2.3 times the SD for the difference of two decadal means. Christensen et al. [2007] compared the median forced change to 2 SD of the difference of two 20 year means over large regions. A comparison of the SDs of 10 and 20 year averages pusing ffiffi the long Mk3.5 simulation suggests that the ratio of 2 expected from independent years is usually a good approximation (within 10% or so) for T and P in the annual case. Thus comparing our P05 and P95 values would require a change at least 1.6 times as large as in the Christensen et al. [2007] test. In addition we use grid point values and we include uncertainty in the forced change, further delaying the exceedance time. It is useful to note here that our 90 percentile range for total change, allowing for the variability of a single 10 year average, can also approximate the range for the difference of two 20 year periods, specifically the future period and Of course, this assumes that the four model variability fields are valid and it ignores anomalies forced by factors other than GHG. [50] Applying our test for a discernable change to the global change fields for 2030, exceedance is achieved over 67% of the globe for T, but nowhere for P. Using the results, the test is passed over 92% of the globe for T, and still only 4% for P. It is worthwhile illustrating the year of exceedance for our test, based on the approximate time series, calculated at each grid point. The field for T, shown in Figure 11a, features years before 2010 for some tropical regions, where there is less uncertainty in the forced change 11 of 13