Twenty-first century changes in daily temperature variability in CMIP3 climate models



Changes in the daily mean surface air temperature distribution between the time periods 1981–2000 and 2081–2100 in 15 climate model simulations participating in the Third Coupled Model Intercomparison Project (CMIP3) are analysed. Variability is defined by first removing the annual cycle after which the range and skewness of the temperature distribution are calculated. Present-day biases for the different aspects of the distribution are also studied, using the ERA-Interim reanalysis as the main reference dataset. The variability in the simulations compared to reanalysis is generally smaller over oceans and larger over land areas. For skewness of the temperature distribution, inter-model variability is much larger and systematic model biases are more local. Model performance regarding the mean values is potentially a good indicator of the model performance for range, but for skewness this relation is weaker. For future changes in the range of daily temperature variability, the models agree best over the Northern Hemisphere mid- to high-latitude land areas, where variability is simulated to decrease. Elsewhere, the inter-model differences are generally larger than the multi-model mean changes. Near the sea ice border, significant negative correlations are found between the changes in mean temperature and the temperature range. For changes in skewness, the signal-to-noise ratio is very low over most areas and no firm conclusions can be drawn from the ensemble results. Our results are relatively insensitive to the precise definition of the range and skewness metrics and to internal variability of climate.

1. Introduction

Global climate projections for the 21st century are based on simulations made with coupled atmosphere–ocean global climate models (AOGCMs). In international efforts such as the Third (CMIP3) and the Fifth (CMIP5) Coupled Model Intercomparison Projects (Meehl et al., 2007b; Taylor et al., 2011), large ensembles of AOGCM simulations of future climate have been collected and made available for the research community. One of the most widely used climate variables is the mean surface air temperature (tas in CMIP3 and CMIP5 nomenclature).

Monthly temperature data from AOGCMs have been extensively studied, concentrating on various themes such as spatial distribution either in present-day (Min and Hense, 2006; Knutti and Sedláček, 2013) or in a changing climate (Räisänen, 2001; Meehl et al., 2007a), different sets of metrics for model evaluation (Gleckler et al., 2008; Reichler and Kim, 2008; Räisänen et al., 2010) or low-frequency variability (Räisänen, 2002; Kravtsov and Spannangle, 2007). However, impacts of temperature change are not determined by changes in monthly mean values alone but are also affected by the daily variability of temperatures. Many studies have implicitly assumed unchanged variability, giving a uniform change over the whole temperature distribution (Zhou and Yu, 2006). This is particularly problematic for the extreme values, since the changes taking place in the scale and shape parameters also affect the future probability of present-day extremes (Griffiths et al., 2005). Generally, extreme values are considered to be more sensitive to changes of the scale than the location parameter (Katz and Brown, 1992), which emphasizes the need to study the whole distribution in addition to the mean values (Schaeffer et al., 2005).

Indeed, one common use for the CMIP3 daily temperature data has been analysis of extremes (Tebaldi et al., 2006; Kharin et al., 2007; Orlowski and Seneviratne, 2011; Rummukainen, 2012). Many of these studies use minimum and maximum temperatures and indices derived thereof (Alexander et al., 2006). Applying extreme value theory, Kharin et al. (2007) studied the 20-year return values of both the daily minimum and maximum temperature extremes. They found a larger inter-model variance for the former than for the latter. This also applies to ERA40 and NCEP reanalysis datasets, their differences being the largest over the ice-covered high-latitude areas. Importantly, Kharin et al. (2007) also found the sampling uncertainty in the extreme values to be generally very small compared to the inter-model differences. Furthermore, they found that cold extremes, on global average, are expected to warm 30–40% more than the corresponding warm extremes, though regional differences are substantial: over high latitudes, the amount of warming might be twice as high for the cold extremes. Kharin et al. (2007) also studied the contribution of the annual cycle to the extreme values. This analysis indicated that high extremes of maximum temperature are expected to increase largely in concert with the hottest summer monthly mean temperature, whereas low extremes of minimum temperature are typically projected to warm more than the coldest monthly mean winter temperature, especially over the mid-latitudes near the cryospheric borderline.

Besides extreme values, information about the variability is often very important for impacts. For example, the electricity consumption dependence on temperature is nonlinear (Bessec and Fouquau, 2008) and information about the temperature distribution is crucial when estimating the impacts of the changing climate (Pilli-Sihvola et al., 2010). Changes in the distribution have, however, received much less attention in the literature. These changes can be assessed for example with the help of probability density functions (PDFs), such as in the studies of Dessai et al. (2005), Maxino et al. (2008) and Perkins et al. (2007) for CMIP3 models. Although such a distribution-based approach does not allow a rigorous analysis of extreme values, the variability as a whole is better illustrated.

This article provides a global analysis of the changes in the distribution of daily mean temperatures in the CMIP3 simulations. The subject is motivated in Figure 1, where the multi-model mean (MMM) changes of 5th (left) and 95th (middle) percentile of the December–January–February daily mean temperature distribution between the periods 1981–2000 and 2081–2100 are presented. The right panel is the difference between the two percentile changes (a detailed description of the methods is given in Section 2). The assumption of a uniform change over the whole distribution clearly does not hold in this case as the temperature increase differs between the highest and the lowest percentiles. The difference is negative over the Northern Hemisphere (NH) high latitudes (indicating reduced daily-scale temperature variability) and slightly positive (increased variability) over the tropical land areas. Our aim in this study is to determine how general these features are for different seasons and how robust this information is across the CMIP3 ensemble. In addition to the changes in the width of the distribution, we also study the changes in its skewness. We do not explore extreme values here, but rather aim to determine if the differences between the changes in extreme and mean values, as found by Kharin et al. (2007), are indicative of the changes outside of the immediate tails of the distribution. The changes throughout the distribution axis are described only with scale and shape parameters, so we also explore the robustness of these parameters to their definition.

Figure 1.

Multi-model mean daily mean surface temperature changes from 1981–2000 to 2081–2100 in DJF under the A1B scenario in 15 CMIP3 models. Changes in the 5th (left panel) and 95th (middle panel) residual percentiles relative to the annual temperature cycle are added to the mean temperature change of the season. The difference between the residual percentile changes is indicated in the right panel.

Our second aim is to provide information for GCM data users and impact researchers. There are numerous practical issues related to the use of daily resolution data, not least the much larger data volumes compared with monthly mean values. Therefore, it is relevant to ask whether the model-simulated distribution changes are consistent enough across the ensemble to be advantageous to the data user. This is not as obvious as it might seem, because the simulated changes in particular near the tails of the distribution tend to vary more from one model to another than the time mean changes do (Kharin et al., 2007).

This study is organized as follows. Section 2 describes the datasets and methods used. After that, we study how the present-day daily surface temperature distribution is simulated globally in CMIP3 models and different reanalyses (Section 3) and how it is projected to change in the future (Section 4). Discussion and general conclusions are given in Section 5.

2. Data and methods

2.1. Datasets

We include in our study those CMIP3 models for which daily data based on the Special Report on Emissions Scenarios A1B scenario (Nakićenović et al., 2000) were publicly available via the World Climate Research Program (WCRP) CMIP3 database [] in August 2009. As CMIP3 dataset was being used already as a part of the Fourth Assessment Report (AR4) of The intergovernmental Panel on Climate Change (IPCC) (Meehl et al., 2007a), it has not received any remarkable upgrades since 2009 and our results are up-to-date based on this dataset. To minimize the effect of inter-model dependencies in our results, different versions of the same model were omitted so that only the model with the highest resolution or the latest version number was chosen. The total number of models is 15 (Table 1). Only one realization of the simulated climate evolution from each model was used, except for Appendix A2.

Table 1. CMIP3 models used in this study, the responsible institutions and the original calendar used
Model acronymInstitutionOriginal calendar
BCCR-BCM2.0Bjerknes Centre for Climate Research, NorwayGregorian
CCSM3National Center for Atmospheric Research, USA365 day
CCCMA-CGCM3.1 (T63)Canadian Centre for Climate Modelling and Analysis365 day
CSIRO-MK3.0CSIRO Atmospheric Research, Australia365 day
ECHAM5/MPI-OMMax Planck Institute (MPI) for Meteorology, GermanyGregorian
ECHO-GUniversity of Bonn and Model & Data Group, Germany; Korean Meteorological Agency360 day
FGOALS-g1.0Chinese Academy of Sciences365 day
GFDL-CM2.1Geophysical Fluid Dynamics Laboratory, USA365 day
GISS-AOMGoddard Institute for Space Studies, USAGregorian
GISS-ERGoddard Institute for Space Studies, USA365 day
IPSL-CM4Institut Pierre Simon Laplace, France360 day
MIROC3.2 (hires)Center for Climate System Research, National Institute for Environmental Studies and Frontier Research Center for Global Change, JapanGregorian
MRI-CGCM2.3.2Meteorological Research Institute, Japan365 day
PCMNational Center for Atmospheric Research, USA365 day

CMIP3 only offers daily data for the periods 1961–2000, 2046–2065 and 2081–2100. In this study, the simulated and observed present-day climate is described by the period 1981–2000 and the simulated future climate by 2081–2100. Twenty-year periods are not long enough to eliminate the random error caused by chaotic internal climate variability (Macadam et al., 2010), particularly near the tails of the distribution. Nevertheless, as is shown in Appendix A2 by analysing the differences between several simulations made by one of the models, internal variability generally appears to be a smaller source of uncertainty than differences between different models. Moreover, the use of periods with equal length should ensure that sampling problems have no systematic effect on the comparison of the present-day and future climates.

All model data were interpolated from their original resolution to a common 2.5° × 2.5° latitude–longitude grid, using the nearest-neighbour method which avoids the loss of temporal variability associated with other interpolation schemes. As most of the models use the 365-day calendar (Table 1), we removed the leap days from the ones using the Gregorian calendar. For simulations using 360-day calendar, missing days were filled using the previous day.

We used three reanalysis datasets as approximations of observed climate: ERA-Interim (Dee et al., 2011), ERA40 (Uppala et al., 2005) and NCEP-NCAR (National Centers for Environmental Prediction/the National Center for Atmospheric Research) (Kistler et al. 2001). The same time period (1981–2000) was used as for the baseline climate simulations. These data are directly available in the 2.5° × 2.5° grid. Mean temperature was available in 12-h intervals for ERA-Interim and 6-h intervals for ERA40, from which the daily means were averaged.

2.2. Methods

The analysis is conducted for all four standard 3-month seasons (December–January–February DJF, March–April–May MAM, June–July–August JJA, September–October–November SON). All data were divided into 99 empirically defined percentiles in the following way: First, the linear 20-year trend of the whole time series was removed. After this, we removed the annual cycle approximated by a fitted third order Fourier series, leaving variability around this annual cycle (denoted as ‘residual’ afterwards). From these n ≈ 1800 daily residual values for each season, percentiles from 1 to 99% were estimated. The mean temperature value of each season was determined by averaging the fitted Fourier series. When studying the future changes, the percentile/mean values in the time periods 2081–2100 and 1981–2000 were simply subtracted from each other for each model, after which the MMM was calculated. To study the robustness of the ensemble signal, the ratio between the ensemble mean and ensemble standard deviation (SD) is used both for the simulations of present-day climate (bias/SD, Section 3) and simulated future changes [signal-to-noise ratio (SNR), Section 4].

For defining the scale parameter of the temperature distribution (hereafter denoted as range), we use the difference between two percentiles, equally positioned around the median.

display math(1)

The shape of the distribution (hereafter skewness) is defined by the Yule–Kendall index

display math(2)

The Yule–Kendall index is a dimensionless measure for skewness, being > 0 ( < 0) for positive (negatively) skewed distributions. Traditionally, x = 25 is used in Equation (2) (Wilks, 2006). In concert with the range, however, we extended the Yule–Kendall index for all percentile intervals of x = [1 , 2, …, 49]. Considering the sensitivity tests described in Appendix A1, our main analysis is done using x = 45 in both Equations (1) and (2). In particular, this provides a higher signal-to-noise ratio in estimating the Yule–Kendall index than lower values of x. We also performed our analysis for another set of residuals for which the annual cycle was estimated by calculating the 20-year mean value for each day of the year independently. For the estimates of residual percentiles 1…99, the difference between the two methods in general is less than 1 °C. For the actual future changes, the effect is smaller, being therefore unimportant for our conclusions. The effect of trend removal on our results is even smaller and neither of these are further analysed.

3. Present-day climate

Comprehensive spatial evaluation of CMIP3 model performance for the mean surface temperature in present-day climate has been made previously. For example, Randall et al. (2007) present the spatial distribution of the annual mean bias, whereas Knutti et al. (2010) present the distribution of ensemble mean bias on seasonal timescales compared to ERA40 reanalysis data. Our analysis mainly uses the ERA-Interim data as the reference dataset, but differences between this dataset and the ERA40 and NCEP-NCAR reanalyses are also briefly described (Figures 3, 4, A2).

Figure 2.

Seasonal mean temperatures in 1981–2000. The different columns correspond to different seasons, different rows to: (a) ERA-Interim reanalysis, (b) MMM, (c) MMM bias compared to ERA-Interim, (d) Standard deviation of individual models (e) MMM bias/SD. Rows 1–4 in unit (°C), row 5 dimensionless. Numerical values above each panel give the global mean (land area mean/sea area mean). Globally, ERA40 is ca. 0.3 °C warmer than ERA-Interim for all seasons. Differences rarely exceed 2 °C. The NCEP reanalysis is colder than ERAInterim (on average 0.3 °C). Differences commonly exceed 2 °C over land areas, especially over Southeastern Asia and high latitudes.

Figure 3.

Difference of 5th and 95th percentiles of residual temperature in 1981–2000. The different columns correspond to different seasons, different rows to: (a) ERA-Interim reanalysis, (b) MMM, (c) MMM bias compared to ERA-Interim, (d) Standard deviation of individual models (e) MMM bias/SD. Rows 1–4 in unit (°C), row 5 dimensionless. Numerical values above each panel give the global mean (land area mean/sea area mean). Differences from ERA-Interim to ERA40 are <2 °C nearly everywhere (globally averaged <0.1 °C), whereas the NCEP/NCAR—reanalysis indicates more variability than ERA-Interim over high latitudes (up to 8 °C), globally averaged differences <0.5 °C in all seasons.

Figure 4.

Yule–Kendall skewness of temperature defined with 5th and 95th residual percentiles in 1981–2000. The different columns correspond to different seasons, different rows to: (a) ERA-Interim reanalysis, (b) MMM, (c) MMM bias compared to ERA-Interim, (d) Standard deviation of individual models (e) MMM bias/SD. All values are dimensionless. Numerical values above each panel give the global mean (land area mean/sea area mean). Over northern South America and parts of central Africa, ERA-Interim has larger skewness in all seasons (by up to 0.3) than ERA40 and NCEP. Otherwise, differences are mainly <0.1 in absolute value.

Figure 2 presents the global distribution of seasonal mean temperatures in 1981–2000. The patterns of MMM (row 2) are smooth compared to the reanalysis dataset (row 1), and there is thus large spatial variability in the MMM bias (row 3). The bias patterns exhibit seasonal dependence, but many of the important characteristics are visible for all of the seasons. The overall tendency towards a slight cold bias in the models, noted by Randall et al. (2007), is replicated here. The most consistent cold biases are seen over NH high-latitude oceans and northern Eurasia. In the Southern Hemisphere (SH), the sign of the bias depends more on the season. Over South America, central Africa and the eastern parts of the tropical Pacific and tropical Atlantic Ocean, the bias is positive in most cases. The bias over the Southern Ocean is positive, whereas that over Antarctica is largely negative with the exception of MAM.

The fourth row of Figure 2 illustrates the inter-model standard deviation. The largest values are seen over the NH high latitudes and especially over the North Atlantic sector, where local temperatures are strongly affected by the simulation of sea ice conditions and meridional heat transport. The fifth row illustrates the extent to which model biases are systematic by the ratio between MMM bias and standard deviation. In areas where the absolute ratio of the value exceeds 1 (blue and red colours), typically five models out of six have the same sign of the bias if model results are approximately normally distributed; for ratios exceeding two in absolute value, generally all 15 models have biases of the same sign. The areas where the ratio exceeds one are very local and seasonally dependent, so over most areas the model simulations and the reanalysis can be assumed to belong to the same statistical population (cf. Annan and Hargreaves, 2010). The most systematic biases are found over northern parts of South America and Africa, over Eurasia in MAM, over the subtropical Atlantic in JJA and over the eastern Atlantic near the equator in JJA.

Figures 3 and 4 summarize the range and the skewness of the distribution of daily mean temperatures, similarly to Figure 2. The geographical distribution of the temperature range (Figure 3) is mostly determined by local surface properties, latitude and season. The temperature variability is larger over land than ocean surfaces. It is greatly enhanced where and when the mean temperature (Figure 2) is well below zero, except for the local summer over Greenland and Antarctica. The variability is the highest in local winter, when the baroclinicity of the atmosphere is the greatest and when the formation of occasional temperature inversions (Connolley, 1996) also tends to accentuate the variability near the surface. Models tend to simulate too little variability over oceans, whereas the bias is mainly positive over land except for North America and northeastern Asia. Due to different land–sea distributions, large inter-model differences extend to lower latitudes in the NH than in the SH. Near the cryospheric borderline the inter-model standard deviation has a sharp gradient. Over ice free Southern Ocean, MMM bias/SD ratio < −1 in all seasons. Otherwise, red and blue colours in the fifth row of Figure 3 are mostly confined to individual seasons, that is, no systematic large biases are found elsewhere.

Physical interpretation of Figure 4 is harder as there is more spatial variability. Nevertheless, a number of general features can be identified: First, skewness has latitudinal dependence in concert with the mean temperature. Near the poles the mean temperature is well below zero and skewness is positive in all seasons. At slightly lower latitudes with higher mean temperatures, negative skewness dominates except during the local summer when positive values in the polar regions extend to lower latitudes. These negative values of skewness are associated with slightly negative mean temperature (Figure 2) combined with large temperature variability (Figure 3). Over snow- or ice-covered surfaces, the highest daily mean air temperatures are often limited by the zero degree threshold, as the energy needed to melt the snow or ice is partially taken from air near the surface. Skewness is the most negative in local winter in concert with the more intense meridional temperature gradient and larger variability. Over sea areas, MMM skewness is typically positive (negative) where and when the mean temperature is below (above) −20 °C. Over land areas, the relation between the mean temperature and the skewness depends more on the season, skewness still being larger for colder mean temperatures (not shown).

Second, negative skewness values are seen in both hemispheres over mid-latitude storm tracks, where strong cold air outbreaks occasionally follow intensifying depressions. Here, the western part of the Atlantic sector storm track is the most visible, caused by the advection of cold continental air masses from North America. Negative skewness is also seen over the northernmost Pacific Ocean south of Alaska in DJF, where the contribution of advection is enhanced by inter-annual variability of sea ice coverage south of Bering Strait. In some years, simulated temperatures can fall well below zero if the area here is ice-covered. The region of El Niño–Southern Oscillation (ENSO) activity can be identified in all the reanalyses with positive skewness over the eastern side of the tropical Pacific. This feature is greatly damped in the MMM maps, indicating much less positive skewness across the ensemble of simulations. Third, the sharp gradient near the coastline caused by the different thermal properties of land and sea is not as characteristic for the skewness as it is for the range of the temperature distribution.

As a measure of mutual dependency between the variables presented in Figures 3, 4, A2, their three inter-model cross-correlation patterns (mean vs range, mean vs skewness and range vs skewness) are summarized in separate rows in Figure 5. For each pair of variables, the linear correlation coefficient between them is calculated separately for each grid point as inferred from the 15 pairs of values provided by the models. Each figure panel also includes the total fraction of the Earth surface where the correlation is significant (at 5% level in a two-sided test, assuming independent and identically distributed samples following normal distribution). These values consistently exceed the fraction of 5% theoretically expected from pure chance, varying from 9% (range vs skewness in MAM) to 22% (range vs mean in SON and DJF). Below, we mainly focus on the statistically significant features, particularly those that can be supported by physical arguments.

Figure 5.

Inter-model correlations for three different pairs of variables in 1981–2000, defined from the 15 models in the ensemble. The different columns correspond to different seasons, different rows to correlations between: (a) Mean temperature and temperature range (b) Mean temperature and skewness (c) Temperature range and skewness. Significant correlations (95% significance level) are coloured in blue and red. Numerical values above each panel give the global fraction of areas with significant correlations (positive/negative).

The strongest correlations are seen between mean temperature and range, especially over the mid- to high-latitude oceans corresponding with 0 °C mean temperature line. Here, the correlation is negative (smaller variability for higher temperatures) and significant for all seasons except for the local summer. This is analogous to the inverse geographical relationship between the mean temperature and temperature range in mid- to high latitudes (first and second rows in Figures 2 and 3), suggesting that most areas of significant correlation in Figure 5 relate to inter-model variations in the extent of snow and ice. However, there are large areas in mid-latitudes (such as North Atlantic and western Europe in MAM) with no sea ice/snow in reality (Figure 2), where this spurious negative correlation is caused by the too cold FGOALS-g1.0 model (Arzel et al., 2006). If this model is omitted, global coverage of areas with significant negative correlation is reduced by 2% in DJF and MAM and 1% in JJA and SON. Omitting this model has much less impact on the correlations shown in the second and third row.

In JJA, the correlation between mean temperature and range is generally positive over NH land areas, although sparsely significant. Over South America, significant positive correlations are found in all seasons. This may at least partially indicate a soil moisture–temperature connection (highest mean temperatures and largest variability in models with drier soils), which is the strongest over areas with limited soil moisture availability (Seneviratne et al., 2010). The significant positive correlations are more localized than negative ones, which makes them traceable to local processes. Over eastern tropical Pacific Ocean, models simulating higher temperatures simulate somewhat less variability in MAM and JJA.

Significant negative correlations between mean temperature and skewness are seen over slightly higher latitudes than the corresponding correlations between mean and range. Here, the MMM mean temperature is substantially below the melting point of 0 °C; therefore, the energy sink associated with snow and ice melt limits the highest daily mean temperatures in model simulations with higher mean temperatures but is less influential in the colder models. Negative correlations are also occasionally significant in dry areas of Sahara and Australia, although the physical mechanism, if any, is to our knowledge unclear. Significant positive correlations are very rare, occurring in less than 2% of the world in all seasons.

The last row in Figure 5 indicates the correlation between range and skewness. The only large area with a significant correlation is seen in austral winter, near the Southern Ocean sea ice border. Here, all three statistics correlate and the freezing point buffer effectively reduces temperature variability and skewness in models with less negative mean temperatures. This mechanism also applies for NH high latitudes, but correlations are generally not significant there.

4. Future climate simulations

MMM changes of the temperature range are presented in Figure 6. Again, the 0 °C line (Figure 2) broadly separates the areas with negative and positive changes. Negative (reduced variability) MMM changes (first row of Figure 6) are mostly confined to areas with negative mean temperatures (Figure 2), being the most negative where the mean temperature is only slightly below zero in present-day climate. For example, the sharp gradient in the change of temperature range near the Southern Ocean sea ice border corresponds very closely with the 0 °C line in present-day climate (Figure 2). Over East Antarctica, where the temperatures are much colder, the range increases. The largest decreases in variability are seen over those ocean areas where sea ice melts and the increased heat flux to the atmosphere prevents very cold daily mean temperatures to be reached. Variability also decreases over NH land areas with negative mean temperatures. Except for Antarctica, variability only increases over land areas with mean temperatures above 0 °C. Between 40°S and 40°N, the changes are predominantly positive in all seasons, and in JJA the increase in variability extends to much higher latitudes over NH.

Figure 6.

Changes (°C) in temperature range (difference of 95th and 5th residual percentiles) between 1981–2000 and 2081–2100. The different columns correspond to different seasons, different rows to: (a) MMM, (b) Standard deviation of individual models (c) SNR. Rows 1–2 in unit (°C), row 3 dimensionless. Numerical values above each panel give the global mean (land area mean/sea area mean).

The inter-model standard deviation for the change in the temperature range (second row in Figure 6) is the largest over high-latitude ice-covered oceans. Since the connection between temperature range and sea ice is strong, this likely reflects a large variance in the reduction of sea ice between the model simulations (probably in part due to differences in present-day ice conditions). The inter-model standard deviation over high latitudes is especially large in local winter.

The ratio between the MMM and the standard deviation (SNR in the last row of Figure 6) indicates the robustness of the simulated changes. The largest negative values of SNR over NH high latitudes are seen over land areas. Over the oceans, SNR values are mostly within −1 to 1, except for the stronger agreement on a decrease in variability over the Arctic Ocean in SON. Areas with SNR > 1 are seen over South Africa and Australia. Here, the present-day soil moisture regime is aridic (Nachtergeale and van Ranst, 2003) and many CMIP3 models simulate drying soil moisture conditions towards the end of the 21st century (Li et al., 2007).

Figure 7 presents the projected MMM changes in the Yule–Kendall skewness. Compared to the range changes, the areas with negative and positive changes show much less spatial coherence, especially over ocean areas. This is because of large inter-model differences (second row), leading to very low SNR over most of the oceans (third row). In addition to high latitudes, the inter-model standard deviation is also high over tropical oceans. Thus, in most areas, the CMIP3 ensemble as a whole allows no conclusions on the future skewness changes. As for the changes in the range, the largest absolute SNR values for skewness changes are seen in the NH high latitudes. Wide areas of negative, even though relatively weak (> −1), values of SNR occur over northern North America and Eurasia in DJF, mainly to the north of the areas with the most negative skewness in the 20th century climate (Figure 4). Thus, along with the warming of climate and reduced snow, the belt of negative skewness is retracting poleward. Another persistent area with negative skewness changes is seen over Australia in all seasons.

Figure 7.

Changes in Yule–Kendall skewness, defined with 5th and 95th residual percentiles, between 1981–2000 and 2081–2100. The different columns correspond to different seasons, different rows to: (a) MMM, (b) Standard deviation of individual models, (c) SNR. All values are dimensionless. Numerical values above each panel give the global mean (land area mean/sea area mean).

Analogous to Figure 5, Figure 8 presents each of the three pairwise inter-model correlations between the changes in mean temperature, range and skewness. Significant correlations (95% level) are the most common between the mean temperature and range changes in the first row (15–20% of the global area). A significant negative correlation (larger decrease in temperature range for larger warming) in many NH high-latitude areas and near the SH sea ice edge is consistent with the MMM decrease in temperature range (Figure 6) along with the MMM warming in the same areas. Again, the negative correlations around 50°N are caused by sea ice melting simulated by the FGOALS-g1.0 model. Omitting this particular model reduces the global area fraction of significant negative correlation by 1–2% but does not change the results over SH noticeably. In concert with Figure 6, the extent of significant negative correlations is the smallest during the local summer. Over South America and central Africa, larger increases in variability typically occur in models simulating stronger mean warming. Over mid- to high latitudes, areas with significant negative correlations in Figure 8 largely overlap with those in Figure 5. The spatial extent of areas with significant negative correlation is smaller for the changes of than for the present-day values of mean temperature and range, but the sign is typically preserved for the changes. One clear discrepancy is seen over the Eurasian continent in JJA, where correlations are generally positive in present-day climate but negative for the changes. Over low latitudes, similarity between Figures 5 and 8 is weaker. Over South America, however, significant positive correlations are found both in Figures 5 and 8.

Figure 8.

Inter-model correlations for simulated changes from 1981-2000 to 2081–2100, for three different pairs of variables, defined from the 15 models in the ensemble. The different columns correspond to different seasons, different rows to correlations between (a) Changes in mean temperature and temperature range (b) Changes in mean temperature and skewness (c) Changes in temperature range and skewness. Significant correlations (95% significance level) are coloured in blue and red. Numerical values above each panel give the global fraction of areas with significant correlations (positive/negative).

Significantly positive correlations between changes in skewness and mean temperature are very infrequent, but significant negative correlations are somewhat more common (second row in Figure 8). The high-latitude areas with significant negative correlations are largely consistent with the first row of Figure 7: Here, the MMM changes are negative for skewness while positive for mean temperature and, especially in DJF, individual models simulating larger mean warming simulate more negative skewness changes. However, the connection between these two variables is weaker than that between range and mean changes, and the spatial coverage of significant correlations is accordingly smaller in all seasons.

The third row in Figure 8 indicates the correlations between range and skewness changes. Areas with significant correlations are very scattered. However, the significantly positive correlation over the Arctic Ocean in DJF is consistent with the co-occurrence of MMM decreases in both range and skewness in this area (Figures 6 and 7).

The usefulness of deterministic MMM projections to describe the actual climate change depends on the inter-model variability, but also on the metric in question. Because of the higher SNR for range than skewness changes, the MMM spatial patterns are likely to give more useful information of the former, assuming that the dispersion of the CMIP3 results is indicative of the actual uncertainty (Annan and Hargreaves, 2010). In order to further study the robustness of the spatial patterns in the model simulations, we calculated the spatial correlations for each of the individual models with the MMM of the other 14 models (Figure 9). These correlations also reveal which of the individual model simulations are most consistent with the MMM.

Figure 9.

Global spatial correlations for each of the 15 individual models with MMM fields (including the other 14 models) of mean (top), range (middle) and skewness (bottom) change from 1981–2000 to 2081–2100. The last column shows the 15-model mean of the correlations.

The correlations differ between the three metrics, being the highest for mean changes and the lowest for skewness changes. However, correlations for the range changes are often comparable with those for the mean changes. On average, the correlations are >0.6 for mean and range changes, even though models with very low correlations can also be found even for the mean temperature change (FGOALS in JJA). In contrast, the results for skewness changes are quite discouraging: correlations are mostly below 0.2 and never exceed 0.4. Still, the correlation values are positive for all the models except for FGOALS. The correlations for mean changes also show seasonal dependence, being the lowest in JJA and the highest in DJF or SON. This is because the simulated pattern of mean temperature change is typically relatively flat in JJA (so that relatively modest regional differences in warming between the models may reduce the correlation substantially) but shows a strong Arctic amplification in DJF and SON (which is qualitatively robust across the models, resulting in high spatial correlations). For range and skewness changes, the seasonal variation of the correlations is less pronounced and more model-dependent. Characteristics of some of the individual models can be distinguished for all of the three variables. If mean temperature changes in a certain model agree well with the MMM changes, this also indicates a high correlation for range changes. However, the same may not apply for changes in skewness. Correlations for all the three variables are high for MIROC3, ECHAM5 and CCSM3 and low for FGOALS. The results for other models depend more on the variable used.

In all, Figure 9 supports the previous conclusions from Figures 6 and 7: the broad-scale characteristics between the different simulations are more robust for changes in mean temperature and temperature range than for the changes in skewness.

5. Summary and discussion

In this article, we have studied the distribution of daily mean temperatures as simulated by 15 global models of the CMIP3 multi-model ensemble. For each model, we first removed the annual cycle from the temperature time series, after which we calculated the distribution of the residuals for each four 3-month seasons. Using percentiles 5 and 95 of this distribution, we defined its scale (range) and shape (skewness) parameters, both for the present-day climate (1981–2000) and for the end of this century (2081–2100) using simulations based on the A1B emissions scenario. In particular, we studied how robust the changes in the scale and shape of the distribution are to the choice of the model, and whether there are any connections between the inter-model variations of mean temperature, range and skewness, both for the present-day climate and for the simulated changes.

Both range and skewness in the present-day temperature distribution are relatively well simulated in CMIP3 models when compared with ERA-Interim reanalysis. Over open ocean areas and low latitudes, the range is small. Over snow- and ice-covered high-latitude areas it is larger and has a strong seasonal variability, in together with the large inter-model variability. The reanalysis is well within the range of the individual model simulations in most areas, even though the MMM bias is generally negative. The variability in most of the models is too small over the Southern Ocean and too large over tropical land areas.

Spatial patterns of skewness reflect several factors, most importantly the mean temperature, surface type and temperature advection. Because of the smoothing associated with averaging over several models, the MMM skewness field is smoother than the ERA-Interim reanalysis, but manages to capture the most important spatial characteristics of the distribution. A persistent negative bias is seen over the eastern tropical Pacific Ocean, where the models do not capture the positive skewness seen in ERA-Interim. Otherwise the bias patterns are much more scattered for skewness than for range, and systematic seasonal variability in the biases is harder to identify. Errors in the reanalysis might explain some of the apparent bias in the ensemble especially over high latitudes where there are fewer observations available.

Near the sea ice edge, there are significant inter-model correlations between the different characteristics of the temperature distribution. The strongest connection is seen between the mean temperature and range: If a model simulates too cold mean temperatures over the high latitudes, sea ice and snow extend too far to lower latitudes and temperature variability here is increased through surface properties and temperature advection. A similar though somewhat weaker connection is found between mean temperature and skewness, but at slightly higher latitudes. Skewness is very sensitive to the values of mean temperature near 0 °C. For mean temperatures slightly below this value, skewness is the most negative as the highest temperatures are limited by the energy required by melting of snow and ice. At mean temperatures slightly above 0 °C, skewness values are typically less negative. In the coldest regions, with mean temperatures of below −20 °C, skewness is generally positive. Globally, negative skewness values are more common.

The projected changes in temperature range taking place during the 21st century depend strongly on the latitude: At latitudes higher than 40°N/S, the ensemble consistently simulates a narrowing of the daily temperature distribution (excluding Antarctica) while the range increases at lower latitude land areas. An important exception is increased temperature range across Eurasia and North America during JJA. Decreases in sea ice differ in magnitude and location between different models, which causes the inter-model variability to be the largest over the high latitudes. Still, the largest signal-to-noise ratios are also seen over the high latitudes. In Australia and South Africa, the ensemble suggests increasing temperature range. Over sea areas, MMM changes in range are mostly negative.

In light of the CMIP3 results, changes in skewness are much more uncertain than those in the range. Over most regions of the world, inter-model variability prevails over the signal and prevents any firm conclusions to be made. Over Russia, northern Europe and northern North America, skewness in DJF is likely to be reduced, thus becoming either less positive or more negative in the future. Over much of Australia, skewness is likely to decrease in local spring.

Over high latitudes, there are some regions with significant negative correlations between mean temperature changes and skewness changes. However, the inverse relationship between mean temperature change and range change near the sea ice border is stronger. Probabilistic methods for estimating skewness changes would be more appropriate than using only MMM, as global spatial correlations between MMM and individual models never exceed 0.4. Typically this correlation is only very weakly positive (<0.2), indicating a large degree of noise in the fields. The correlations are higher for the changes in range. For some models, high agreement of mean temperature changes with MMM also coincides with higher relative agreement for skewness and range changes.

In conclusion, projected changes in the range of daily mean temperature variability can be inferred much more robustly from the CMIP3 ensemble than the changes in the skewness of the distribution. The areas with the largest signal-to-noise ratio are local, occurring mostly at mid- to high latitudes. Here, the daily temperature range is projected to decrease with increasing temperatures, causing cold (warm) daily mean temperatures to increase more (less) than the mean temperatures.

Our findings are largely in concert with those of Kharin et al. (2007), who compared changes in temperature extremes with those in the climatological seasonal cycle. An exact comparison is, however, not possible, because both the range and skewness metrics are affected by values on both sides of the median. For example, an increase in skewness can result either from low temperatures increasing more than the median or high temperatures increasing more than the median. Range changes, however, seem easier to interpret through seasonal dependence: Range is typically reduced during local winter as cold temperatures increase more than the median, and increased in local summer as the warm temperatures increase more. The areas where the change in 20-year cold extremes in Figure 9 of Kharin et al. (2007) exceeds that expected from the chance in the mean annual cycle co-inside with reduced MMM temperature range in our Figure 6, either in DJF (for the NH) or in JJA (for the SH). On the basis of this similarity and the lack of robustness in the changes in skewness in Figure 7, we would expect the latter to be less important for the findings of Kharin et al. (2007) although this cannot be directly verified by our approach. Importantly, as noted by Kharin et al. (2007), inter-model differences remain the most important component of the total uncertainty in the climate projections. Errors caused by methodology and internal variability of the climate are of secondary importance for our findings (Appendices A1 and A2).

For the end user using GCM data for impact assessments, the findings of our study are relevant if daily scale data is required for the application at hand. If the temperature timeseries is used as such, changes taking place in the temperature distribution are also inherited to the estimates of actual impacts. Estimates of temperature variability can be applied with some confidence over many regions as there are appropriate physical feedbacks to back up our findings (the 0 °C threshold affecting surface properties both over land and sea areas, soil moisture feedback over some dry areas). Our study indicates range change estimates as being relatively robust in many regions and therefore gives confidence in temperature distribution changes and their use in estimating the impacts. Range is a relatively robust metric to estimate from the projections (Appendix A1), which is further emphasized by the stronger connection between range and mean locally: a model simulating cryospheric extent properly in present-day climate is likely to provide more realistic estimates for future range changes as well.

Our study is the first to globally assess the changes in the distribution skewness in a multi-model framework. Our methodology is easily applicable also to other variables and the analysis would be worthwhile to repeat for the CMIP5 simulations. Our preliminary research on this data shows little difference from our findings presented for the CMIP3 models. We would therefore anticipate the general conclusions of this article to hold also for the CMIP5 ensemble.


We acknowledge the modelling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) and the WCRP's Working Group on Coupled Modelling (WGCM) for their roles in making available the WCRP CMIP3 multimodel dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy. This research has been supported by the Academy of Finland (decisions 127239 and 140801) and Finnish Academy of Science and Letters.

Appendix: A1. Generalization of results for other percentile intervals

To determine how well the findings for a single percentile interval apply for the whole distribution, we performed two separate sensitivity tests. First, we calculated Yule–Kendall index (Wilks, 2006) for all 49 values of x, using various idealized distributions generated by the Pearson system (Johnson et al., 1994) for which the estimated values of central moments (von Storch and Zwiers, 1999) were: mean = 0, standard deviation = 1, skewness = [0, 0.1, …, 1], kurtosis = 0. The uncertainty of the index was quantified analysing 15 distributions each with a sample size of 1800, closely corresponding with the actual datasets used in this study, although these artificial samples lack the serial correlation present in actual time series of temperature. The numerical values of the index increase with larger x and skewness. The variance between the 15 different estimates is insensitive to skewness, but decreases with larger x. Therefore, larger values of x reduce the relative importance of sampling uncertainty. The selected percentile range of 5–95 greatly reduces the sampling noise without focusing on too rare events, even though the selection is otherwise arbitrary. Findings for the range are more robust for all values of x, as the noise is already greatly reduced for values of x > 5.

As the second sensitivity test, we calculated the global spatial correlations between MMM parameter fields obtained with various x (Figure A1). The first two rows present the correlations for present-day climate 1981–2000 and the last two for changes between the time periods 1981–2000 and 2081–2100. Our results for range and its change can be extended to all values of x, as the field correlations are >0.8 for all pairs of percentile intervals. Thus, spatial patterns in Figures 3 and 6 are very robust in all seasons.

Figure A1.

Global field correlation matrices for multi-model mean temperature range and skewness in 1981–2000 (rows 1–2) and changes in range and skewness (rows 3–4) calculated with different values of x in equations (1) and (2). In each panel, the horizontal and vertical axis indicate the percentile ranges over which range and skewness were calculated.

Figure A2.

Standard deviations for temperature range and skewness in 1981–2000 (rows 1–2) and changes in range and skewness (rows 3–4), as derived from five parallel simulations of the CCSM3 model. Parameters are calculated with 5th and 95th residual percentiles. Rows 1 and 3 in unit (°C), rows 2 and 4 dimensionless. Numerical values above each panel give the global mean (land area mean/sea area mean).

Results for skewness depend more on x and the season. The second row emphasizes the relative importance of sampling uncertainty for small values of x: For all seasons, correlations decrease rapidly for values of x < 10. However, all intervals larger than the inter-quartile range (IQR) have cross-correlations >0.9. Over this range of values, spatial patterns of MMM skewness are robust (Figure 4), which allows physical interpretation of the results. However, global field correlations are much weaker for MMM skewness changes (fourth row in Figure A1), because the changes are generally smaller in absolute value than the present-day skewness and have therefore a lower signal-to-noise ratio. The location of the yellow and green areas with absolute SNR > 0.5 of skewness changes in the third row of Figure 7 is relatively stable for values of x > 25, but their global coverage is small. Compared with range changes, skewness changes for the interval 5–95 can only locally be generalized for smaller values of x.

Appendix: A2. Sensitivity of the results to internal variability of climate

As only one realization of climate was used for each model in our study, we briefly estimated also the sensitivity of our results to the internal variability of climate. For this purpose, we used five parallel runs (3, 5, 6, 7 and 8) from 20c3m and A1B simulations as simulated by the CCSM3 model. The selection of this particular model was based only on the data availability.

Figure B1 presents the standard deviation of the 5 CCSM3 ensemble members for the same variables as presented in Figure A1. When comparing the figure with our earlier results (the SD rows in Figures 3, 4, 6 and 7), globally averaged standard deviation is found to be smaller in all the cases, although this difference is smaller for climate change than present-day simulations, and also smaller for skewness than range.

The differences within the CCSM3 ensemble are only due to internal variability, whereas those in the ensemble of 15 CMIP3 models are both due to internal variability and model differences. The variances due to these two contributions are additive in a statistical sense (Räisänen, 2001). Averaging over the global area and the four seasons, the variance of present-day temperature range in the CCSM3 ensemble is only 3% of the variance in the CMIP3 ensemble. If we assume the internal variability in the other models to be similar to CCSM3, this suggests that about 97% of the variance in the latter is due to model differences. The corresponding estimated contribution of internal variability for present-day skewness is 20%, for the change in range 22%, and for the change in skewness 51%. Thus, with the exception of the changes in skewness, model differences appear to be the dominating source of uncertainty.