Changes in the Extremes in an Ensemble of Transient Climate Simulations with a Coupled Atmosphere–Ocean GCM

Viatcheslav V. Kharin Canadian Centre for Climate Modelling and Analysis, Victoria, British Columbia, Canada

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Francis W. Zwiers Canadian Centre for Climate Modelling and Analysis, Victoria, British Columbia, Canada

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Abstract

The extremes of surface temperature, precipitation, and wind speed and their changes under projected changes in radiative forcing are examined in an ensemble of three transient climate change simulations for the years 1900–2100 conducted with the global coupled model of the Canadian Centre for Climate Modelling and Analysis. The evolution of the greenhouse gases and aerosols in these simulations is consistent with the Intergovernmental Panel on Climate Change 1992 scenario A. The extremes are analyzed in three 21-yr time periods centered at years 1985, 2050, and 2090.

The model simulates reasonably well the extremes of the contemporary near-surface climate. Changes in extremes of daily maximum and daily minimum temperature are distinctively different and are related to changes in the mean screen temperature, soil moisture, and snow and sea-ice cover. Extreme precipitation increases almost everywhere on the globe. Relative change in extreme precipitation is larger than change in total precipitation. Extreme wind speed in the extratropics changes only modestly. Changes in duration of extended wet and dry periods are consistent with changes in total precipitation. There are temperature-related changes in cooling and heating degree days.

Corresponding author address: Dr. Francis Zwiers, Canadian Centre for Climate Modelling and Analysis, P.O. Box 1700, STN CSC, Victoria, BC V8W2Y2, Canada.

Email: francis.zwiers@ec.gc.ca

Abstract

The extremes of surface temperature, precipitation, and wind speed and their changes under projected changes in radiative forcing are examined in an ensemble of three transient climate change simulations for the years 1900–2100 conducted with the global coupled model of the Canadian Centre for Climate Modelling and Analysis. The evolution of the greenhouse gases and aerosols in these simulations is consistent with the Intergovernmental Panel on Climate Change 1992 scenario A. The extremes are analyzed in three 21-yr time periods centered at years 1985, 2050, and 2090.

The model simulates reasonably well the extremes of the contemporary near-surface climate. Changes in extremes of daily maximum and daily minimum temperature are distinctively different and are related to changes in the mean screen temperature, soil moisture, and snow and sea-ice cover. Extreme precipitation increases almost everywhere on the globe. Relative change in extreme precipitation is larger than change in total precipitation. Extreme wind speed in the extratropics changes only modestly. Changes in duration of extended wet and dry periods are consistent with changes in total precipitation. There are temperature-related changes in cooling and heating degree days.

Corresponding author address: Dr. Francis Zwiers, Canadian Centre for Climate Modelling and Analysis, P.O. Box 1700, STN CSC, Victoria, BC V8W2Y2, Canada.

Email: francis.zwiers@ec.gc.ca

1. Introduction

Extremes are important aspects of climate. Changes in the magnitude and frequency of climatic extremes will have environmental and socioeconomical consequences. It is therefore of great interest to evaluate the extremes as simulated by modern general circulation models (GCMs) and to estimate the changes that may take place under projected anthropogenic forcing.

Many previous studies considered climate changes due to alteration in the chemical composition of the atmosphere as simulated with “equilibrium” climate change simulations. In these experiments the atmosphere’s lower boundary conditions are represented by a relatively simple upper-ocean model that assumes that ocean heat transports remained fixed (e.g., Boer et al. 1992). An equilibrium climate as simulated by a GCM for an anomalous radiative forcing is compared to the corresponding equilibrium climate of the control run performed for standard greenhouse gas and aerosol concentrations. An example of analysis of changes in the extremes in an equilibrium doubled-CO2 experiment conducted with the second generation GCM (CCC GCM2; McFarlane et al. 1992) of the Canadian Centre for Climate Modelling and Analysis (CCCma) is given in Zwiers and Kharin (1998), hereinafter referred to as ZK98.

More recently, modeling groups have simulated the evolution of the climate system under “transient” anthropogenic forcing, such as increasing concentration of greenhouse gases and sulfate aerosols, using global climate models with coupled three-dimensional models of the atmosphere and ocean (Manabe and Stouffer 1994;Mitchell et al. 1995; Haywood et al. 1997; Meehl et al. 2000a; Roeckner et al. 1999; Tett et al. 1999; Boer et al. 2000a,b; Dai et al. 2000). Global three-dimensional coupled models simulate atmosphere–ocean interactions, slow changes in the ocean circulation, and deep ocean heat storage.

CCCma has recently conducted a number of simulations with the first generation Canadian Global Coupled Model (CGCM1). The global coupled model CGCM1 and its control simulation are described in Flato et al. (2000). The experimental design of the transient climate change integrations and analysis of the simulated historical and projected climate changes for two radiative forcing scenarios are given in Boer et al. (2000a,b). The current study examines changes in the extremes of surface temperature, wind speed, and precipitation in an ensemble of the three CGCM1 simulations with increasing greenhouse gas and aerosol concentration.

There have been many studies of weather extremes using a variety of techniques and models, and simulations of various lengths. This is, however, one of the first studies to consider changes in the extremes of basic surface parameters that are simulated by a coupled GCM in a transient climate change simulation. We will not provide an extensive review of the literature in this paper because this is available from a number of other sources such as Kattenberg et al. (1996; see section 6.5) and Meehl et al. (2000b).

The fact that there is considerable variation in the analysis techniques and experimental configurations used in published studies of model simulated extremes makes it difficult to compare specific results. However, the general sense of results from studies on projected changes in the high-frequency variability and extremes of temperature and precipitation is similar to that which we will report below. There is considerably less consensus on projected changes in surface wind variability and extremes.

The changes in simulated extremes under transient climate change that are reported in this paper should certainly not be considered to be reliable projections of future. Rather, they should be considered as scenarios for the future with plausibility limited by the confidence that we have in the model. Confidence is further limited because extremes, by definition, occur at the outer limits of the variability simulated by the model.

In general, confidence in model simulated extremes of precipitation and winds is limited. The subgrid-scale processes that play important roles in producing observed extremes of these variables, such as convection and turbulence in the boundary layer, are highly parameterized. The surface topography, which strongly affects precipitation and channels near-surface winds, is of necessity much smoother in climate models than in the real world. Also, the limited ability of models to simulate some types of tropical variability, such as tropical cyclones and El Niño, reduces the confidence that we have in simulated precipitation and wind speed extremes in some parts of the world. Our confidence in temperature extremes is also limited because these are affected by the representation of the land surface, fluxes involving moisture and other factors, such as the parameterization of surface albedo that affect the conversion of incident solar radiation to latent and sensible heat, and the parameterization of clouds. Nonetheless, we might expect to have somewhat greater confidence in projected changes in temperature extremes than in those of precipitation and near-surface wind speed.

Our ability to validate extremes simulated for the present day climate is limited. This is made difficult on the global scale because observed data gridded on scales comparable to that produced by the model is scarce. In ZK98, the comparison of temperature and wind speed extremes on the global scale was limited to the 1979–95 National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis data (Kalnay et al. 1996). The validation of precipitation extremes was limited to the Canadian region as the NCEP–NCAR reanalysis does not appear to reproduce observed daily precipitation variability very well, particularly in the Tropics. In the current study, we compare simulated precipitation to that in the 1979–93 European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis (Gibson et al. 1997). We also make some comparisons of simulated and reanalyzed precipitation over Canada to estimates derived from 69 Canadian stations (Mekis and Hogg 1999). In addition to the analysis of extreme precipitation, we perform an analysis of extended wet and dry periods and heating and cooling degree days. The NCEP–NCAR reanalysis is used to validate the extremes of surface temperature and wind speed.

The outline of the paper is as follows. First, we give a brief description of the coupled model and the transient runs in section 2. In section 3 we introduce the methodology used to evaluate the extremes in the climate simulations. The temperature, precipitation, and wind speed extremes of the control climate are described in section 4. In section 5 we examine their changes under global warming conditions as simulated by CGCM1. Other impacts such as changes in duration of extended wet and dry periods and changes in heating and cooling degree days are presented in sections 6–7. The results are summarized in section 8.

2. Transient climate change simulations

CGCM1 and its control simulation is described in detail in Flato et al. (2000). We therefore give only a brief summary of the model’s basic features in this section.

The atmospheric component of the coupled model is essentially the same as CCC GCM2 (McFarlane et al. 1992). This model has been used extensively in a number of climate simulations (Boer et al. 1992; Boer 1993;Reader and Boer 1998; Zwiers et al. 2000). It is a spectral model with T32 truncation and 10 vertical levels. The model includes a comprehensive package of physical parameterizations of subgrid-scale processes. The land surface scheme uses a single soil layer with spatially varying moisture field capacity and soil properties. A thermodynamic model is used to simulate sea ice.

The ocean model is a version of the Geophysical Fluid Dynamics Laboratory Modular Ocean Model (Pacanowski et al. 1993). The ocean model grid has double the resolution of the atmospheric model in longitude and latitude and 29 vertical levels. The upper four levels are equally spaced at 50 m, with level spacing increasing below 200 m. The atmosphere and the ocean components communicate once per day by exchanging daily average quantities. The model employs flux adjustments for heat and freshwater fluxes. The spinup and coupling procedures are discussed in Flato et al. (2000).

The coupled model has been used to conduct several multicentury integrations under various radiative forcing scenarios. The experimental design of the transient climate change integrations and analysis of the simulated historical and projected climate changes are given in Boer et al. (2000a,b). An ensemble of three independent integrations, discussed in the present study, is performed with an equivalent greenhouse gas forcing corresponding to that observed from 1900 to present, and a forcing corresponding to an equivalent CO2 increase at a rate of 1% yr−1 thereafter until year 2100. The direct effect of sulfate aerosols is also included by increasing the surface albedo as in Reader and Boer (1998). The equivalent CO2 and aerosol evolution is that of Mitchell et al. (1995) and the Intergovernmental Panel on Climate Change 1992 scenario A (Houghton et al. 1992). In the following, we will refer to the simulations as the GHGA (Greenhouse Gases and Aerosol) runs. A subset of the monthly and daily data from several transient simulations is available online at http://www.cccma.bc.ec.gc.ca.

Three 21-yr time periods centered at years 1985, 2050, and 2090 were selected from each integration. The first period for years 1975–95 represents the model climate under the present concentration of greenhouse gases and aerosols. The other two windows in the middle and the end of the twenty-first century correspond roughly to the time of CO2 concentration doubling and tripling, relative to the 1975–95 level. The changes simulated by CGCM1 in 2040–60 and 2080–2100 will be denoted by Δ2050 and Δ2090, respectively.

3. Methodology

For each time period we analyze annual extremes of simulated daily maximum and daily minimum screen (2-m) temperature, denoted as Tmax and Tmin. In mid- and high-latitudes Tmax is typically the warmest summer daytime temperature of the simulated calendar year. Similarly, Tmin is the coldest winter nighttime temperature of the year. We also analyze the annual extremes of 24-h accumulated precipitation P and instantaneous 1000-hPa wind speed S sampled once a day. In addition, we consider extended wet and dry period lengths, denoted as Dwet and Ddry, estimated from the daily precipitation data.

The extreme value analysis approach that we use applies what is essentially a “law of large numbers” to the problem. This is somewhat analogous to the familiar law of large numbers for the sample mean. The latter states that under very general conditions the distribution of the sample mean converges to the Gaussian distribution as the sample becomes large. There is also a law of large numbers that governs the distribution of extremes when samples become large (Gnedenko 1943), even when the daily observations that make up the samples are serially correlated (Leadbetter et al. 1983). This law states that the maximum of a large sample will have one of three extreme value distributions. One of these distributions is the well known Gumbel distribution (Gumbel 1958) that has been used extensively in meteorological and hydrological science. The Generalized Extreme Value (GEV) distribution that we use to analyze annual extremes encompasses all three asymptotic distributions.

The Gumbel distribution is likely the appropriate distribution for the extremes of asymptotically large samples of daily data for the variables considered in this study. However, the samples of 365 daily values from which we extract the annual extremes are not asymptotically large. Moreover, the effective size of the annual sample is reduced by serial correlation and by the effect of the annual cycle, which dictates, for example, that the annual warm temperature extreme at mid- and high-latitudes will happen in summer. The larger family of GEV distributions that we use gives some flexibility that accommodates this discrepancy between the effective size of the annual sample and the asymptotically large sample to which the Gumbel distribution theoretically applies.

In the remainder of this section we describe the standard method of L moments that was used in ZK98 to estimate extreme values. We also discuss statistical goodness-of-fit tests that were performed to assess how well the GEV distribution represents the available samples of annual extremes. Two modifications of the L-moment technique that improve the fit are briefly discussed in this section and in more detail in the appendix.

a. L-moment method

For each time period we estimate return values at every grid point for 10-, 20-, 50-, and 100-yr return periods. For example, for precipitation, we estimate the largest 24-h accumulation that is expected to occur on average once every 10, 20, 50, or 100 yr in a climate with the characteristics of that simulated by the model during the 1975–95, 2040–60, and 2080–2100 periods. These return values are estimated in two steps. First, the GEV distribution is fitted to a sample of annual extremes by the method of L moments (Hosking 1990, 1992). Then, return values are obtained by inverting the fitted GEV distribution.

The GEV distribution:
i1520-0442-13-21-3760-e1
combines the three possible asymptotic extreme value distributions for the maximum of a large sample that are predicted by large sample theory (see, e.g., Leadbetter et al. 1983), that is, EV-I, or the Gumbel distribution, for κ = 0, EV-II for κ < 0, and EV-III for κ > 0. The location parameter ξ, the scale parameter α, and the shape parameter κ are estimated from a sample of annual extremes by the method of L moments (Hosking 1990, 1992). The advantages of the L-moment method over the asymptotically optimal maximum likelihood method, are that it is computationally simpler and that L-moment method parameter estimates have better sampling properties for short samples. The details are given in section a of the appendix.
Having fitted the GEV distribution to a sample of annual extremes the T-year return values XT are estimated from the quantile function as
i1520-0442-13-21-3760-e2

Given the time period of 21 yr and an ensemble of three integrations, the total sample size for each time window is 63, from which we estimate 10-, 20-, 50-, and 100-yr return values. The sampling uncertainty of the estimates is determined by the parametric bootstrap procedure. In this procedure 1000 samples of size 63 are generated from the fitted GEV distribution. A return value is estimated from each generated sample by fitting and inverting a GEV distribution as derived above. The 5th and 95th percentiles of the resulting collection of return value estimates are then used as lower and upper 90% confidence bounds for the true T-yr return value. In the following, the difference between two return value estimates is said to be statistically significant if their 90% confidence intervals do not overlap, which corresponds to the 1% significance level.

Because the precipitation and wind speed extremes show a great deal of spatial noise we apply spatial smoothing to display extremes and their changes for these quantities. First, the “regional” L-moment estimates are obtained by averaging L moments, estimated separately for each individual grid box, over 9 (3 × 3) adjacent grid boxes. These regional L moments are then used to derive the regional parameters of the GEV distribution and the corresponding return value estimates.

This simple approach to smoothing, which was also used by ZK98, is a modified version of a technique proposed by Buishand (1991). It implicitly makes the assumption that the extremes, of say precipitation, at a given grid box have statistical characteristics that are similar to those of the extremes occurring at its nearest neighbors. This seems a reasonable assumption, at least over oceans and in the interior of continents, and applies an amount of smoothing that is commensurate with the smallest spatial scales that are retained in the model’s dynamics. We see very little difference in the large-scale structure of the estimated return values that are obtained with and without smoothing of the L moments. In the case of precipitation we do see a great deal more unorganized noise in the unsmoothed version of the return values.

Confidence intervals are constructed for the return values of P by assuming that the annual maxima of daily precipitation in the adjacent grid boxes are independent of each other. This assumption holds reasonably well for P in tropical and subtropical regions. Annual maxima of wind speed S are more likely to occur on the same day in several adjacent grid points than precipitation extremes. Thus, although the smoothing technique results in better organized patterns, it does not improve the signal-to-noise ratio for S to the same extent as for P.

b. Feasibility of the L-moment estimators

A potential problem with the L-moment method for estimating the parameters of the GEV distribution has been recently raised by Dupuis and Tsao (1998). They pointed out that the method of L moments can produce nonfeasible parameter estimates. That is, the estimated distribution may not contain all of the data from which distribution parameters are estimated. The problem arises because the GEV distribution (1) is bounded above when κ > 0 and below when κ < 0. Sampling variability may lead to parameter estimates that violate these constrains.

Dupuis and Tsao proposed a hybrid estimator that ensures feasibility by adjusting κ̂ when the fitted distribution is nonfeasible. The details of their modification are given in section b of the appendix. We use the hybrid estimators whenever the distribution obtained with the standard method of L moments is nonfeasible.

c. Goodness-of-fit tests

Given the fairly long sample of annual extremes of size 63 available in this study, we conducted goodness-of-fit (GOF) tests to examine whether the GEV distribution fits the annual extremes simulated in CGCM1 satisfactorily. We also performed GOF tests for the Gumbel distribution as a special case of the GEV distribution that occurs when κ = 0. The Gumbel distribution is widely used in the extreme value analysis because it is a limiting distribution of extreme values drawn from many standard distributions, including the normal and exponential distributions.

We apply a standard Kolmogorov–Smirnov GOF test (Stephens 1970) that measures the overall difference between two (cumulative) distribution functions. The Kolmogorov–Smirnov (KS) statistic D is defined as the maximum absolute difference between two distribution functions:
i1520-0442-13-21-3760-e3
where F(x) is the fitted distribution function and SN(x) is an empirical distribution function estimated from a sample of size N as the proportion of data values less than or equal to x. The null hypothesis that the annual extremes are drawn from the distribution F(x) is rejected when D exceeds a certain critical value. This critical value is determined by the parametric bootstrap procedure. In this procedure 1000 samples of size 63 are generated from the fitted GEV or Gumbel distributions, respectively, and the values of D are derived for each generated sample. The 90th percentile of the resulting collections of Ds is then used as the critical value for the rejection of the null hypothesis that the sample originates from the specified distribution at the 10% significance level.

Table 1 shows the percentage of the area of the globe where the null hypothesis (annual extremes of surface temperature, precipitation, and wind speed are drawn from GEV and Gumbel distributions) is rejected in the reference period 1975–95 in the GHGA runs. Similar rejection rates are obtained for the other two time windows. At the 10% significance level we expect to see rejection rates of about 10%, provided the null hypothesis is true. The numbers in brackets indicate the minimum number of spatial degrees of freedom for which the global null hypothesis (the GEV distribution fits at all grid points) will be rejected at the 5% significance level.

The Gumbel distribution does not seem to be a good choice for a distribution function of annual extremes for the analyzed quantities. This is particularly the case for annual extremes of temperature. The parent distribution of daily values is likely to be nearly normal for Tmax and Tmin whereas those for P and S are likely to have exponential-like upper tails. The extremes of asymptotically large samples of daily values drawn from exponential and normal distributions have a Gumbel distribution (e.g., Leadbetter et al. 1983). However, annual extremes, which come from samples of 365 daily values, may have a distribution that has not yet attained the limiting Gumbel form. This is of more concern for near-normal variables than for exponentially distributed variables because convergence to the limiting form is slower for the former (section c of the appendix). Consequently, null hypothesis rejection rates are larger for annual extremes of temperature than for annual extremes of precipitation and wind speed.

The GEV distribution fits the annual extremes of Tmax, P, and S satisfactorily. However for Tmin, the null hypothesis is rejected over more than 20% of the globe and the global null hypothesis is also likely to be rejected at the 5% significance level.

d. Upper tail return value estimates

The GOF tests indicate that the GEV distribution fails to fit the distribution of annual extremes for Tmin in some regions. Figure 1 shows areas where the null hypothesis that the Tmin extremes in 1975–95 in the GHGA runs are drawn from a GEV distribution is rejected at the 10% significance level. The null hypothesis is rejected over most of Australia, South America, southeast of North America, central Europe, and north Africa. The estimated shape parameter κ̂ is negative in roughly the same areas (not shown) while varying between 0.1 and 0.4 in most other regions on the globe. There are also a few small areas, mostly notable in central South America, where the estimated shape parameter is unusually large (κ̂ > 1.0) and the GEV distribution also fails to fit Tmin annual extremes.

Figure 2 shows annual extremes of Tmin simulated by CGCM1 in the reference period 1975–95 at 28°S, 131°E (central Australia) and at 17°S, 60°W (South America), ordered in descending order. The left-hand graph is typical for areas where the estimated shape parameter κ̂ is negative while the right-hand graph is typical for grid points with unusually large κ̂ > 0.5. There is clustering of annual extremes around the freezing point in both cases. This clustering is an artifact of CGCM1’s land surface. CGCM1 represents the land surface as a single layer with properties such as vegetation type, soil type, and water-holding capacity that vary with location. Due to surface energy balance constraints, near-surface temperature does not drop substantially below the freezing point until soil moisture in the whole layer is frozen. In regions with a mild winter climate, near-surface temperatures are not cold enough in some years to freeze all soil moisture resulting in a clustering effect around the freezing point.

Clearly, a GEV distribution is not a proper candidate for approximating the distribution of annual extremes in such situations. We deal with cases in which clustering occurs at the upper tail by obtaining return values from the empirical distribution rather than from the fitted GEV distribution. When clustering occurs at the lower tail, two approaches can be pursued. First, one can increase the time period from which the extremes are drawn from 1 yr to 2, or more, yr. By doing so, the probability of observing an extreme in the vicinity of the clustering point decreases and the distribution function of the extremes converges to the asymptotic GEV distribution. However, the sample size decreases correspondingly so there is less information to reliably estimate the distribution parameters.

An alternative approach is to fit a GEV distribution only to annual extremes in the upper tail. We assume that annual extremes greater than some value x0 can be described by a GEV distribution function and extremes smaller than x0 can be described by some well-defined function. Then a modified GEV distribution function, whose lower tail for x < x0 is replaced by the known function is fitted to the sample.

In this study we replace the extremes that are smaller than x0 with x0. Thus, the modified distribution function (x) that we fit to the transformed sample is given by
i1520-0442-13-21-3760-e4
where F(x) is the GEV distribution function. The L moments of the modified GEV distribution (x) can be derived analytically (see section d in the appendix). Parameter estimates of the modified GEV distribution (x) are obtained by substituting its first three L moments with estimates obtained from the modified sample and solving the resulting system.

To demonstrate the proposed method we fitted standard and modified GEV distributions to the sample shown in Fig. 2a. The analysis of Tmin annual extremes in the transient simulations showed that clustering of annual extremes usually occurs in the temperature range −0.5° to 0.5°C. Thus, we use x0 = −0.5°C. Figure 3 shows quantile–quantile (Q–Q) plots for standard and modified versions of the GEV distribution. In these graphs the sign of annual Tmin extremes is reversed. The annual extreme quantiles estimated from the empirical distribution function are displayed on the x axis. The corresponding quantiles estimated from the fitted distribution function are on the y axis. The closer the points are to the diagonal, the better the fit. The fitted standard GEV distribution disagrees substantially with the sample of the annual extremes, especially at the tails. A better fit is achieved for the modified GEV distribution. The estimated shape parameter κ = 0.7 for the modified GEV distribution is larger than that obtained for the standard GEV fit so that the corresponding return values increase with the return period more slowly.

The sampling properties of return value estimates are evaluated by Monte Carlo simulations that are discussed in section e of the appendix. In these tests, return values are estimated from standard and modified GEV distributions fitted by the method of L moments to 10 000 samples of size 63 generated from a GEV distribution. For positive values of the shape parameter κ, return value estimates derived from a modified GEV distribution fitted to the correspondingly modified samples have total errors that are comparable, or smaller than those obtained by fitting a standard GEV distribution.

4. The extremes simulated during 1975–95

The extremes of Tmax, Tmin, P, and S simulated by CGCM1 in the reference period 1975–95 in the GHGA runs are similar to those produced by CCC GCM2 in the control equilibrium run (ZK98). We give a brief overview of the model’s “extremes” climate description in this section. Globally averaged values for a number of quantities are summarized in Table 2 and land-averaged values are shown in Table 3.

a. Screen temperature

CGCM1 does a credible job of simulating 20-yr return values of daily maximum and minimum temperature (Fig. 4). In comparison with the 1979–95 NCEP–NCAR reanalysis (not shown), temperature extremes over northern land masses are well simulated, while Tmin extremes over western Europe are somewhat too warm. The cold bias reported by ZK98 for temperature extremes simulated by CCC GCM2 over northern Canada, in comparison with Canadian station data, is reduced in CGCM1, mainly because the GHGA runs are slightly warmer during the 1975–95 time period than the control equilibrium run.

b. Precipitation

Estimates derived from 69 Canadian records for the period 1975–95 are used to evaluate the model performance over Canada. The Canadian station records have been adjusted for the known inhomogeneities such as change of observed procedure and instrument deficiencies (Mekis and Hogg 1999). The 1979–93 ECMWF reanalysis (Gibson et al. 1997) is used to validate simulated precipitation extremes on the global scale.

Figure 5 displays annual mean precipitation rate P and 20-yr return values of daily precipitation P20 over Canada as derived from the station records, the ECMWF reanalysis, and the reference period 1975–95 of the transient runs. The latter are very similar to those obtained for the 1 × CO2 equilibrium run in ZK98. Comparison with station data shows that the mean precipitation rate over Canada is well reproduced in the reanalysis. High latitudes north of about 65°N are relatively dry with annual mean precipitation of 1 mm day−1 and less. Precipitation rates increase to 3 mm day−1 and more over southeastern Canada and over the Pacific coast. CGCM1 captures the large-scale precipitation pattern satisfactorily but oversimulates precipitation rates by 0.5–1 mm day−1. The precipitation maximum over western Canada is broader in the coupled model, presumably because of coarse model resolution and smoother model topography.

The pattern of 20-yr return values of daily precipitation estimated for the ECMWF reanalysis is similar to that derived from the Canadian records. However, the return values are undersimulated by 20 mm day−1 and more in the ECMWF reanalysis compared to the estimates derived from the station data. This is an expected result assuming that reanalyzed precipitation values are representatives for a grid cell while station data are local measurements that are affected by many microclimatological effects (Osborn and Hulme 1997). The CGCM1 estimated 20-yr return values of daily precipitation range roughly between the estimates from the station records and the ECMWF reanalysis.

CGCM1 tends to oversimulate mean precipitation rates in the northern extratropics and in high latitudes (see Fig. 9 in Flato et al. 2000), in comparison with the ECMWF reanalysis. A narrow wet belt along the equator in the eastern Pacific and in the Atlantic is not as pronounced in CGCM1 as it is in the ECMWF reanalysis. Mean precipitation in the Asian summer monsoon outflow area is oversimulated in CGCM1 in comparison with that in the ECMWF reanalysis.

The 20-yr return values of daily precipitation (spatially smoothed) simulated by CGCM1 in 1975–95 and in the ECMWF reanalysis on the globe are displayed in Fig. 6. Return values for the coupled model are somewhat larger than those for the reanalyzed data. The global average of P20 in 1975–95 in the GHGA integrations is 85.3 mm day−1, in comparison with 70.2 mm day−1 in the ECMWF reanalysis. The pattern of extreme precipitation associated with the upward branches of the tropical circulations is broader in the coupled model than that in the ECMWF reanalysis.

c. Wind speed

Extreme wind speed simulated in the reference period 1975–95 of the GHGA integrations is very similar to that in the equilibrium control run (Fig. 8 in ZK98). The 20-yr return values of 1000-hPa wind speed S20 (not shown) are greater than 35 m s−1 in the storm tracks over the North Pacific and Atlantic and over the midlatitude Southern Ocean. Over extratropical land masses, CGCM1 simulated S20 are in the range of 20–30 m s−1. ZK98 indicated that the model produces somewhat stronger extreme winds than the NCEP–NCAR reanalysis.

5. Changes in the extremes

Analysis of various potential climatic consequences of the increasing concentration of atmospheric greenhouse gases and sulfate aerosols in the GHGA simulations is given in Boer et al. (2000b). In this section we examine changes in extreme values for temperature, precipitation, and wind speed.

a. Screen temperature

Changes in 20-yr return values of daily maximum screen temperature in 2040–60 and in 2080–2100 as simulated in the GHGA integrations are shown in Fig. 7. Substantially greater changes occur for daily minimum temperature extremes (Fig. 8). The globally averaged Tmax,20 change is 1.8°C in 2040–60 and 3.9°C in 2080–2100 (Table 2). The corresponding changes for Tmin,20 are 2.4° and 5.2°C. Changes in extreme temperature in the transient runs are statistically significant over most of the globe in both time windows (Table 2). Significance is greater over oceans than over land (Table 3) because natural variability is greater over land. Also, significance decreases as the length of the return period increases, particularly over land.

As in the equilibrium climate change simulations, changes in Tmax,20 and Tmin,20 are remarkably different. Over tropical and temperate oceans changes in both are comparable to those in the mean screen temperature. This is physically reasonable since screen temperature is largely determined by surface temperature over the oceans. Over land masses and polar regions there are changes in both the location and the shape of the screen temperature distribution. The greatest change in return values of daily maximum temperature is found in central and southeast North America, central and southeast Asia, and tropical Africa where there is a substantial decrease in summertime soil moisture content (Fig. 18 in Boer et al. 2000b). Reduced soil moisture means that maximum surface temperatures are less likely to be moderated by evaporative cooling. In contrast, the west of North America is affected by increased precipitation resulting in more soil moisture and a more moderate increase in extreme maximum surface temperature.

Changes in the return values of daily minimum temperature are substantially larger than those of daily maximum temperature over land areas and high-latitude oceans where snow and sea-ice retreat. The amount of solar radiation absorbed at the ground increases in these areas. Some of the absorbed solar radiation warms the overlying air at night and thus raises daily minimum temperatures.

b. Precipitation

The most pronounced changes in mean and extreme precipitation in the transient runs are associated with a warming pattern in the central tropical Pacific that is associated with the model’s “El Niño–like” response in that region. This is a common, but not ubiquitous, feature of transient climate change simulations with coupled climate models (see, e.g., Mitchell et al. 1995; Meehl and Washington 1996; Knutson and Manabe 1998; Noda et al. 1999; Timmerman et al. 1999). Globally averaged, there is little change in the annual mean precipitation in 2040–60 and a modest increase in the end of the next century (Table 2). However, there are substantial shifts in the spatial distribution of mean precipitation (Fig. 17 in Boer et al. 2000b), with drier conditions over Indonesia and the western tropical Pacific and wetter conditions in the central tropical Pacific. Subtropical land masses becomes slightly drier.

Extreme precipitation increases almost everywhere over the globe in the warmer world (Fig. 9). The large increase in 20-yr return values that occurs in the tropical Pacific is statistically significant at the 1% significance level in both time windows. Changes in extremes are generally not as large over land (cf. Tables 2 and 3). While P20 increases are less than 10 mm day−1 over extratropical land masses in 2040–60, large extratropical areas, including eastern North America and Siberia, are affected by larger increases in amplitude of the precipitation extremes in 2080–2100. Globally averaged, P20 increases by 6.9 mm day−1 and 12.2 mm day−1 by the middle and end of the next century, respectively. The increase becomes larger for longer return periods. Changes in extreme precipitation in the transient runs over land masses are generally not statistically significant in 2040–60, but become statistically significant over broad land areas at the end of the integrations.

Changes in return values can also be interpreted in terms of changes in waiting times for 1975–95 size events. For example, the global average of 10-yr return values for daily precipitation simulated in 2080–2100 in the transient runs is 86.8 mm day−1, which is comparable to the global average 85.3 mm day−1 of 20-yr return values in 1975–95 (Table 2). This means that, on average, waiting times for extreme precipitation events, that are expected every 20 yr in the reference period, are reduced to approximately 10 yr at the end of the transient runs.

c. Wind speed

Changes in extreme near-surface wind speed over land masses under increasing radiative forcing as simulated in the transient runs are small. Figure 10 displays change in spatially smoothed 20-yr return values for 1000-hPa wind speed in 2080–2100. There is a slight tendency for weaker wind speed extremes over North America and South America and for stronger extremes over Europe. However, these changes are not statistically significant.

Changes over the oceans are similar in magnitude to those over land (cf. Tables 2 and 3). Weaker wind extremes are found over the tropical Indian Ocean and western Pacific. Extreme wind speed also decreases in the North Pacific and in the southern midlatitudes. A characteristic pattern of positive wind speed anomalies is found in the central tropical Pacific. This shift in extreme wind speeds is associated with changes in the underlying sea surface temperature in the central tropical Pacific simulated in the GHGA runs under increasing greenhouse forcing. These changes are statistically significant. There is also some evidence for stronger wind speed extremes in the Arctic and around Antarctica where sea-ice retreats and the winds blow over open water areas in the warmer world.

6. Wet and dry periods

Changes in duration of extended dry and wet periods under global warming will affect various aspects of the human and natural environment. Here we define dry period length as the time between two consecutive rain days, where a rain day is defined as a day with 1 mm of precipitation or more. Wet period length is defined likewise as the time between two consecutive “dry” days with precipitation less than 1 mm. For convenience, estimates of wet and dry period lengths are determined from daily precipitation data that are interpolated linearly between days. Time points at which the interpolated values cross the rain-day threshold of 1 mm day−1 are used as the starting and ending points of wet and dry periods. Additional computations indicated that the magnitude of the “rain-day” threshold in the range of 0.5–5 mm affects the length of wet and dry periods but has little effect on the spatial pattern of period lengths. Smaller rain-day thresholds result in longer wet periods and shorter dry periods and vice versa.

Extreme precipitation and extended wet and dry period durations characterize different aspects of precipitation. Extreme precipitation is determined by the upper tail of the precipitation distribution. Apart from the accumulation time period, which is 24 h in this study, no other temporal information is involved in estimating precipitation extremes. The length of wet/dry periods is affected by the precipitation distribution and by the temporal properties of the precipitation variations from one day to another. Extremely long wet/dry period lengths can be estimated using the same extreme value technique as described in section 3 by fitting a GEV distribution to a sample of annual longest wet and dry periods at every grid point and estimating return values from the fitted distribution. This is a reasonable approach provided there are many wet/dry periods in a year. This requirement is satisfied reasonably well almost everywhere on the globe.

First we compare CGCM1 simulated wet and dry period lengths over Canada with those estimated from the ECMWF reanalysis and the Canadian stations. Figure 11 displays mean wet period length Dwet and its 20-yr return values Dwet,20 estimated from the Canadian station records, the 1979–93 ECMWF reanalysis, and the GHGA runs in 1975–95. Annual mean wet period length at the Canadian stations ranges from less than 2 days in northern Canada and the prairies to 3 days in western Canada and more than 4 days at the South Pacific coast. The 20-yr return values for wet period length have a similar pattern with values of about 1 week in northern Canada, 2 weeks in eastern Canada, and up to a month in western Canada. These values are reproduced reasonably well in the ECMWF reanalysis. CGCM1 simulates a similar pattern for Dwet and Dwet,20 but the wet periods are too long, because CGCM1 is wetter in Canada in comparison with the ECMWF reanalysis and station records (cf. Fig. 5).

The patterns of mean dry period length Ddry and its 20-yr return values Ddry,20 (Fig. 12) are largely complementary to those for wet periods. Mean dry periods are longer than a week and 20-yr return values are greater than 2 months in northern Canada. Mean dry period length reduces to 2–3 days and Ddry,20 reduces to about 3 weeks in eastern and western Canada. The ECMWF reanalysis reproduces both patterns reasonably well. CGCM1 simulates dry periods in Canada that are too short.

The global distribution of 20-yr return values for wet period length as simulated by CGCM1 in the GHGA runs in 1975–95 and in the ECMWF reanalysis is shown in Fig. 13. The return values are comparable over extratropical land masses. Over the oceans, especially in the Tropics, the ECMWF reanalysis produces substantially longer wet periods than CGCM1. Globally averaged, Dwet,20 in the reanalysis is almost twice as large as in the GHGA runs. The situation is similar over land (Table 4). The length of dry periods is also undersimulated in comparison with ECMWF analyses. Daily precipitation in CGCM1 is apparently less persistent and more variable than ECMWF reanalyzed precipitation. For example, the globally averaged standard deviation of daily precipitation produced by CGCM1 in 1975–95 is 6.3 mm day−1 in comparison with 5.3 mm day−1 in the reanalysis. This is also consistent with greater return values of P produced by CGCM1 than in the reanalysis. On the other hand, the globally averaged mean precipitation rate is smaller in CGCM1 (2.8 mm day−1) than that in the ECMWF reanalysis (3.0 mm day−1). CGCM1 simulates more precipitation in dry areas such as Antarctica, north Africa, and northern high latitudes than is found in ECMWF reanalysis. This results in longer wet periods in these regions.

The 20-yr return values for the length of dry periods (Fig. 14) are comparable both for CGCM1 and ECMWF reanalysis in areas where dry periods are short. Extended dry periods are shorter over extratropical land masses in the transient runs as CGCM1 is wetter in these areas than ECMWF reanalysis. Dry periods are substantially longer in the reanalysis in areas of descending branches of the large-scale tropical circulations such as north Africa and the Middle East and over Antarctica. These are also the areas where KS GOF test indicates that the GEV distribution does not fit the distribution of longest annual dry period well.

Changes in 20-year return values of wet and dry period length as simulated in the GHGA runs in the end of the next century are shown in Fig. 15 and are summarized for land areas in Table 4. The changes over extratropical land masses are modest and not statistically significant. There is a tendency for longer wet periods in high latitudes and shorter wet periods in moderate latitudes that is associated with changes in precipitation rates in these areas. Greater changes are found in the Tropics and subtropical regions. The increased precipitation rate and the eastward shift of the precipitation pattern in the tropical Pacific under the greenhouse forcing result in longer wet periods in the central and eastern tropical Pacific and shorter wet periods in the western Pacific. More moisture arrives on the Pacific coast of North and South America in the CGCM1 simulated warmer world resulting in shorter dry periods in these regions. Tropical land masses become drier so that extended wet periods become shorter.

7. Heating and cooling degree days

Heating and cooling degree days are integrated measures of temperature that give an indication of the severity of the cold and warm season, respectively. Heating degree days (HDD) in this study are defined as the departures of daily temperature below the reference level of 18°C, totaled over a year. Cooling degree days (CDD) are defined as the temperature departures above 18°C, summed over a year. These measures are commonly used in practical applications such as, for example, estimating weather-related variations in energy consumption.

Annual heating and cooling degree days in Canada derived from station data, the 1979–94 NCEP–NCAR reanalysis, and simulated by CGCM1 in 1975–95 are displayed in Fig. 16. Daily temperature in the NCEP–NCAR reanalysis and the transient runs was estimated as the average of daily maximum and minimum screen temperatures. The station data are the 1961–90 climatological values (EC/AES 1993). The HDD and CDD values in the NCEP–NCAR reanalysis are comparable to those at Canadian stations. Heating degree days simulated in the GHGA runs in 1975–95 are 10%–20% smaller than in the reanalysis, while CGCM1 cooling degree days are somewhat greater than reanalyzed and station data values.

On the global scale, CGCM1 simulates reasonable heating and cooling degree days (Fig. 17, Table 5). As compared with the NCEP–NCAR reanalysis (not shown), HDD are 10%–20% smaller in the northern high latitudes and over North America in the reference period 1975–95 in the transient runs. Cooling degree days are undersimulated by about 10% in the tropical regions but oversimulated by about 40% over North America.

Changes in annual heating and cooling degree days simulated by CGCM1 in the GHGA runs at the end of the next century with respect to the 1975–95 level are displayed in Fig. 18. Heating degree days decrease by 25%–50% over extratropical land masses. The average decrease over North America is 35%. HDD reduction in the tropical regions is greater than 75% but is relatively modest in absolute units. Annual cooling degree days increased by more than 1000 degree days in the Tropics. CDD more than double over wide areas of northern land masses. The average increase over North America is approximately 90%.

8. Summary

The extremes of near-surface climate and their changes under projected anthropogenic forcing as simulated by CGCM1 in an ensemble of three transient integrations are analyzed and discussed. The effective greenhouse gas forcing in these integrations corresponds to that observed from 1900 to present, and effective CO2 increases at a rate of 1% yr−1 thereafter until year 2100. The direct effect of projected sulfate aerosol loadings is also included. Return values of daily maximum and minimum screen temperature, precipitation, wind speed, and wet and dry period lengths for return periods of 10-, 20-, 50-, and 100-yr are estimated from the Generalized Extreme Value (GEV) distribution fitted to samples of annual extremes at every grid point by the method of L moments. The return value XT is the value that will be exceeded, on average, at least once every return period T. Changes in return values in 2040–60 and in 2080–2100 relative to the reference period 1975–95 are determined and discussed.

The standard L-moment method for estimating the parameters of the GEV distribution is modified to address two issues. First, hybrid estimators of Dupuis and Tsao (1998) are used whenever the method of L moments produces nonfeasible estimates. This modification results in only minor changes in estimated return values. Second, a method for estimating the GEV distribution parameters from extreme values in the upper tail is proposed to improve return value estimates in some areas on the globe.

To summarize the model performance in simulating the extremes:

  • CGCM1 does a credible job of simulating 20-yr return values of daily maximum and minimum temperature. Difficulties that were encountered with extreme minimum temperature in some locations were accommodated by modifying the extreme value analysis technique. These problems point to limitations of the land surface parameterization used in CGCM1 and reduce our confidence in projected changes in extreme temperature.

  • When compared with Canadian station data, the model simulates plausible return values for daily precipitation over much of Canada. However, the spatial variation in the observed extremes, some of which is associated with small-scale topography, was not as well reproduced.

  • On the global scale, the pattern of precipitation extremes simulated by CGCM1 resembles that obtained from ECMWF reanalysis. However, the model produces 20-yr return values that are about 20% larger, globally averaged, than those obtained from ECMWF reanalysis. The most prominent difference between model and reanalysis is that the latter confines its large tropical extremes to a narrower equatorial band than does the model.

  • Differences in the means and in the temporal variability of simulated and reanalyzed precipitation result in differences in the length of extended wet and dry periods, with the model tending to have shorter wet and dry periods. This deficiency limits the confidence that we should place in projected changes in the extremes of precipitation and the length of extended wet and dry periods.

  • Heating and cooling degree days are reasonably well simulated by CGCM1 in the reference period 1975–95. We have somewhat greater confidence in these quantities because they integrate information over substantial parts of the simulated and reanalyzed temperature distributions. Over North America the model simulates more cooling degree days than are estimated from reanalysis and fewer heating degree days.

Changes in the extremes as simulated by CGCM1 in the transient runs can be summarized as follows.

  • Over oceans, temperature extremes increase by an amount that is roughly equal to the change in the mean screen temperature. Over land masses, changes in return values of daily maximum and daily minimum temperatures are very different. Large changes in Tmax,20 occur in regions where soil moisture decreases, such as in central and southeast North America, southeast Asia, and tropical Africa. The larger changes in the return values for daily minimum temperature occur over land areas and high-latitude oceans where snow and ice retreat. While this may, to some extent, be a model artifact, it is broadly consistent with results reported elsewhere [e.g., McGuffie et al. 1999; see also the review by Meehl et al. (2000b)].

  • Globally averaged, the increase in the annual mean precipitation rate in the transient runs is about 1% and 4% in the middle and end of the next century, respectively. There are shifts in the spatial distribution of precipitation, especially in the tropical Pacific. Extreme precipitation increases almost everywhere. Relative changes in extreme precipitation are larger than changes in total precipitation. On the global scale, the 20-yr return values of daily precipitation increase by 8% and 14% in 2040–60 and 2080–2100 respectively. Again, these results are broadly consistent with what we would expect in a climate with a warmer atmosphere that has a greater moisture-holding capacity. They are also consistent with results obtained in other studies including those by Gordon et al. (1992), Gregory et al. (1997), Hennessy et al. (1997), Kothavala (1997), ZK98, and McGuffie et al. (1999).

  • Changes in extreme wind speed are small over extratropical land masses. There is a tendency for stronger extreme winds over high-latitude oceans where sea ice melts in the warmer world and for weaker winds in the southern midlatitudes and in the North Pacific. There is a shift in extreme wind speed pattern in the tropical Pacific associated with the changes in mean circulation as a response to the warmer central tropical Pacific. Results from other studies are mixed (Meehl et al. 2000b). Our confidence in these results is low for reasons including those outlined in the introduction and because this field is not amenable to spatial smoothing for the enhancement of the signal-to-noise ratio of the response to transient forcing.

  • Changes in the length of wet and dry periods are, to the first order, related to the changes in total precipitation. Extended wet periods become longer and dry periods become shorter in the regions with increased precipitation, and vice versa. Other studies have also considered these variables (e.g., Gregory et al. 1997 and McGuffie et al. 1999) but comparison of results is difficult. Our confidence in the projected changes described in this paper is low because the model does not reproduce well the wet and dry period climate that is estimated from reanalyses.

  • Heating degree days, as simulated by CGCM1 in the GHGA runs, decrease by 25%–50% over northern land masses while cooling degree days more than double over wide extratropical land areas at the end of the next century.

Acknowledgments

We would like to thank George Boer and Greg Flato for reading the manuscript and providing valuable comments. We also thank an anonymous Associate Editor and two anonymous reviewers for perceptive comments that lead to an improved paper. The precipitation station data was provided by the Climate Monitoring division of the Atmospheric Environment Service, Environment Canada. Steve Lambert prepared ECMWF reanalyzed precipitation fields.

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APPENDIX

GEV Distribution Parameter Estimation

Method of L moments

The GEV distribution is given by
i1520-0442-13-21-3760-ea1
The first three L moments of a random variable X are defined in terms of its distribution function F(x) as follows (Hosking 1990)
i1520-0442-13-21-3760-ea2
where {X1:r, X2:r, . . . , Xr:r} are the order statistics obtained by sorting the sample of size r in ascending order.

The L moments λ1, λ2, and λ3 are in some ways analogous to the conventional central moments and may be regarded as measures of location, scale, and skewness of the distribution. Clearly, λ1, the mean, is a measure of location. The second L moment λ2 measures the mean distance between two random values and thus may be thought of as measuring the scale or dispersion of the distribution. The third L moment λ3 measures, in a sample of size 3, the mean difference between the distances to the right and to the left of the central value. A positive (negative) difference indicates that the distribution has a wider right- (left) hand tail. Symmetric distributions have λ3 = 0.

For the GEV distribution, κ ≠ 0, the first three L moments are
i1520-0442-13-21-3760-ea3
The corresponding L moments for the Gumbel distribution (κ = 0) are
i1520-0442-13-21-3760-ea4
where γ is Euler’s constant. Unbiased estimates of the L moments are given by
i1520-0442-13-21-3760-ea5
where Cnk = n!/[k!(nk)!]. In the L-moment method, the GEV (Gumbel) distribution parameter estimates are obtained by substituting the first three (two) L moments on the left-hand side of (A3)–(A4) with their estimates from the data and solving the resulting system.
The shape parameter estimate κ̂ is obtained from
τ̂3λ̂3λ̂23κ̂2κ̂
For −0.1 < τ̂3 < 0.5, κ̂ is approximated with accuracy within 4 · 10−4 by a quadratic polynomial
κ̂zz2
where z = 2/(3 + τ̂3) − ln2/ln3. The α and ξ estimates are obtained from
i1520-0442-13-21-3760-ea8
if κ̂ ≠ 0, or from
i1520-0442-13-21-3760-ea9
if κ̂ = 0.

Feasibility of the L-moment estimators

The GEV distribution is bounded above when κ > 0 and below when κ < 0. Sampling variability may lead to parameter estimates that are nonfeasible, that is, estimates that violate one of the following constrains:
i1520-0442-13-21-3760-ea10
where X1:n and Xn:n are the smallest and the largest values in a sample of size n, respectively. Dupuis and Tsao (1998) showed that nonfeasible estimates occur infrequently, one percent of the time or less, when κ ⩽ 0.1, but they occur with increasing frequency when κ is larger. The rate of occurrence of nonfeasible estimates does not decrease with increasing sample size.
Dupuis and Tsao proposed a hybrid estimator that always satisfies (A10) by replacing nonzero κ̂ by κ̃
i1520-0442-13-21-3760-ea11
when the estimate is nonfeasible. They argue that the method of L moments and hybrid estimators are comparable in terms of bias and rms error but the hybrid has the advantage that it is always feasible.

Table A1 shows the number of grid points, at which the method of L moments produced nonfeasible GEV parameter estimates for the reference period 1975–95 in the GHGA runs. Fewer than 1% of estimates is nonfeasible for P and S. The estimated shape parameter κ̂ is close to zero over most of the globe in both cases so that the extreme values are nearly unconstrained. The shape parameter estimated for annual extremes of Tmax and −Tmin is greater than 0.1 over most of the globe so that the extremes are bounded from above. Correspondingly, the percentage of nonfeasible parameter estimates is greater for Tmax and Tmin. We use the hybrid estimators whenever the estimates obtained by the standard method of L moments are nonfeasible.

Convergence to the Gumbel distribution

The rate of convergence of the distribution of extremes drawn from a normal distribution to the asymptotic Gumbel distribution is much slower than that of extremes drawn from an exponential distribution. This is illustrated by Table A2, which shows the parameter estimates of the GEV distribution fitted to maxima of 5000 samples of length L drawn from the standard normal and exponential distributions. Also shown is the rejection rate for the KS test at the 10% significance level of the null hypothesis that an extreme value sample of size 63 is drawn from the Gumbel distribution. The shape parameter κ converges to zero (the value appropriate for the Gumbel distribution) very slowly for the normal distribution resulting in large rejections rates for short samples. For the exponential distribution, κ is nearly zero even for very short samples, and thus the rejections rates are close to the nominal value 10% that is expected by chance under the true null hypothesis.

The parent distribution is likely to be nearly normal for Tmax and Tmin whereas those for P and S are likely to have exponential-like upper tails. Thus, for a given sample size from which extremes are drawn, the distribution of wind and precipitation extremes will be closer to the Gumbel distribution than that of temperature extremes. Also, near-surface air temperature has often a larger decorrelation time than wind speed and daily precipitation. This shortens the effective size of a sample from which annual temperature extremes are drawn.

L moments of a modified GEV distribution

Consider a distribution
i1520-0442-13-21-3760-ea12
where F(x) is a GEV distribution function. Let F0 = F(x0), z0 = −lnF0, γ(1 + κ, z0) = z00ettκ dt, and Ei(1, z0) = 1ez0t/t dt. In these terms, the first three L moments of the modified GEV distribution (x) are given by
i1520-0442-13-21-3760-ea13
for κ ≠ 0, and by
i1520-0442-13-21-3760-ea14
for κ = 0. Systems (A13) and (A14) contain an additional unknown parameter F0, or z0 = −lnF0, respectively. The relation F0 = F(x0) provides the required additional equation:
i1520-0442-13-21-3760-ea15
This equation allows us to eliminate ξ from the expressions for the first L moment λ1 in (A13) and (A14):
i1520-0442-13-21-3760-ea16
For the modified GEV distribution with κ ≠ 0, κ, F0 and z0 = −lnF0 are first determined from
i1520-0442-13-21-3760-ea17
Then α is determined from (A16). Last, ξ is given by (A15). For the modified Gumbel distribution (κ = 0), z0 is determined from
i1520-0442-13-21-3760-ea19
Then α and ξ are derived from (A16) and (A15), respectively.

Sampling properties of quantile estimates obtained by the method of L moments for the modified GEV distribution

The sampling properties of the quantile estimates for the modified GEV distributions depends on F0. We conducted a series of Monte Carlo simulations to estimate sampling variability of the return value estimates. For each set of parameters ξ = 0, α = 1, κ = −0.4, . . . , 0.6, and F0 = 0.0, . . . , 0.8 we generated 10 000 samples of size 63 and estimated the 0.95th and 0.98th quantiles, which correspond to 20- and 50-yr return values. The quantiles are estimated from the empirical distribution function and from the fitted by the method of L moments standard and modified GEV distributions. The parameter estimates for the modified GEV distribution are obtained by solving system (A13) by the Newton–Raphson method. In cases when the Newton–Raphson method did not converge after a reasonable number of iterations we used estimates derived from the empirical distribution function.

Table A3 summarizes the results of the Monte Carlo tests. For negative κ, total errors of estimates obtained from the empirical distribution function are the larger than those for all other estimators but become comparable for positive κ. For positive κ, quantile estimates obtained by fitting the modified GEV distribution (F0 > 0) have comparable or even smaller errors than those derived from the fitted standard GEV distribution.

Fig. 1.
Fig. 1.

GOF test for the GEV distribution fitted to annual minima of Tmin in 1975–95 in the GHGA integrations. Shading indicates areas where the null hypothesis that annual Tmin extremes are drawn from the GEV distribution is rejected at the 10% significance level

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 2.
Fig. 2.

Annual Tmin minima simulated in 1975–95 in the GHGA runs at 28°S, 128°E (central Australia) and at 17°S, 60°W (South America) ordered in descending order. The rank of the minima is given at x axis

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 3.
Fig. 3.

Q–Q plots for the (a) standard and (b) modified GEV distribution fitted to the sample of annual Tmin extremes displayed in Fig. 2a. The sign of the extremes is reversed

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 4.
Fig. 4.

The 20-yr return values for (a) daily maximum screen temperature Tmax,20 and (b) daily minimum screen temperature Tmin,20 simulated in 1975–95 in the GHGA runs. (a) Contour interval is 5°C. Light (dark) shading indicates areas where Tmax,20 are larger than 35°C (45°C). (b) Contour interval is 10°C. Light (dark) shading indicates areas where Tmin,20 are smaller than −20°C (−50°C)

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 5.
Fig. 5.

(left) Annual mean precipitation rate P and (right) 20-yr return values of daily precipitation P20 at (top) Canadian stations, (middle) in the 1979–93 ECMWF reanalysis, and (bottom) in 1975–95 in the GHGA integrations. Contour interval is 1 (20) mm day−1 for P (P20). Light (dark) shading indicates areas where P > 2 mm day−1 (>4 mm day−1) and where P20 > 60 mm day−1 (>100 mm day−1). For Canadian stations, numbers display values for individual stations

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 6.
Fig. 6.

(a) Spatially smoothed 20-yr return values for daily precipitation P20 simulated in 1975–95 in the GHGA runs and (b) in the 1979–93 ECMWF reanalysis. Contour interval is 25 mm day−1. Light (dark) shading indicates areas where P20 is greater than 50 (150) mm day−1

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 7.
Fig. 7.

Change in 20-yr return values for daily maximum screen temperature simulated in the GHGA runs (a) in 2040–60 and (b) in 2080–2100 relative to the reference period 1975–95. Contour interval is 4°C. Zero line is omitted. Light, medium, and dark shading indicates areas where changes are greater than 4°, 8°, and 12°C, respectively

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 8.
Fig. 8.

Change in 20-yr return values for daily minimum screen temperature simulated in the GHGA runs (a) in 2040–60 and (b) in 2080–2100 relative to the reference period 1975–95. Contour interval is 4°C. Zero line is omitted. Light, medium, and dark shading indicates areas where changes are greater than 4°, 8°, and 12°C, respectively

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 9.
Fig. 9.

Change in spatially smoothed 20-yr return values for daily precipitation simulated in the GHGA runs (a) in 2040–60 and (b) in 2080–2100 relative to the reference period 1975–95. Contour interval is 10 mm day−1. Zero line is omitted. Light (dark) shading indicates areas where P20 increased by at least 10 (20) mm day−1

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 10.
Fig. 10.

Change in 20-yr return values for 1000-hPa wind speed simulated in the GHGA runs in 2080–2100 relative to the reference period 1975–95. Contour interval is 1 m s−1. Zero line is omitted. Light (dark) shading indicates areas where S20 decreased (increased) by at least 1 m s−1

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 11.
Fig. 11.

(left) Mean wet period length Dwet and (right) 20-yr return values of wet period length Dwet,20 for (top) Canadian stations, (middle) the 1979–93 ECMWF reanalysis, and (bottom) the GHGA integrations in 1975–95. Contour interval is 0.5 (5) days for Dwet (Dwet,20). Light (dark) shading in left panels indicates areas where Dwet < 2 days (>4 days); in right panels it indicates Dwet,20 < 10 days (>20 days). For Canadian stations, numbers display values for individual stations

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 12.
Fig. 12.

(left) Mean dry period length Ddry and (right) 20-yr return values of dry period length Ddry,20 at (top) Canadian stations, (middle) in the 1979–93 ECMWF reanalysis, and (bottom) in 1975–95 in the GHGA integrations. Contour interval is 1 (5) days for Ddry (Ddry,20). Light (dark) shading indicates area where Ddry < 4 days (>8 days) and Ddry,20 < 20 days (>60 days). Numbers display values for individual Canadian stations

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 13.
Fig. 13.

The 20-yr return values of wet period length simulated in the GHGA runs (a) in the reference period 1975–95 and (b) in the 1979–93 ECMWF reanalysis. Rain day is defined as a day with 1-mm precipitation and more. Contour values are 4, 8, 16, . . . , days. Light (dark) shading indicates areas where Dwet,20 < 8 days (>16 days)

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 14.
Fig. 14.

The 20-yr return values of dry period length simulated in the GHGA runs (a) in the reference period 1975–95 and (b) in the 1979–93 ECMWF reanalysis. Dry day is defined as a day with precipitation < 1 mm. Contour values are 4, 8, 16, . . . , days. Light (dark) shading indicates areas where Ddry,20 < 16 days (>128 days)

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 15.
Fig. 15.

Changes in spatially smoothed 20-yr return values for (a) wet period length Dwet,20 and (b) for dry period length Ddry,20 simulated in 2080–2100 in the GHGA runs. (a) Contour interval is 2 days. Contour values are offset by 1 day from zero. Light (dark) shading indicates areas where Dwet,20 decreased (increased) by at least 1 day. (b) Contour interval is 10 days. Contour values are offset by 5 days from zero. Light (dark) shading indicates areas where Ddry,20 decreased (increased) by at least 5 days

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 16.
Fig. 16.

(left) Annual HDD and (right) CDD at (top) Canadian stations, (middle) in the 1979–94 NCEP–NCAR reanalysis, and (bottom) in the GHGA integrations in 1975–95. Contour interval is 1000 (50) degree days for HDD (CDD). Light (dark) shading indicates areas where HDD > 4000 (8000) degree days and CDD > 50 (250) degree days. Numbers display values for individual Canadian stations

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 17.
Fig. 17.

(a) Annual HDD and (b) CDD simulated by CGCM1 in the GHGA runs in 1975–95. (top) Contour interval is 2000 degree days. Light (dark) shading indicates areas where HDD is greater than 2000 (10 000) degree days. (bottom) Contour interval is 500 degree days. Light (dark) shading indicates areas where CDD is greater than 500 (3000) degree days

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Fig. 18.
Fig. 18.

(a) Change in annual HDD and (b) CDD simulated by CGCM1 in 2080–2100 in the GHGA runs. Contour interval is 500 degree days for HDD and 250 degree days for CDD. Light (dark) shading indicates areas where HDD decreased and CDD increased by at least 25% (75%)

Citation: Journal of Climate 13, 21; 10.1175/1520-0442(2000)013<3760:CITEIA>2.0.CO;2

Table 1.

Fraction of the globe where the null hypothesis that the sample of annual extremes in 1975–95 in the transient runs originates from the GEV or Gumbel distribution is rejected at the 10% significance level. The numbers in parentheses indicate the minimal number of spatial degrees of freedom for which the global null hypothesis that the distribution fits at all grid points is rejected at the 5% significance level

Table 1.
Table 2.

Globally averaged annual mean and 10-, 20-, 50-, and 100-yr return values for daily maximum screen temperature Tmax (°C), daily minimum screen temperature Tmin (°C), daily precipitation P (mm day−1), and wind speed at the 1000-hPa level S (m s−1) in the reference period 1975–95 in the GHGA runs (“1985”) and their changes as simulated in the GHGA runs in 2040–60 (“Δ2050”) and in 2080–2100 (“Δ2090”). Percentages indicate the fraction of the globe where the changes in return values are significant at the 1% significance level

Table 2.
Table 3.

As in Table 2 except averages are taken over land points only

Table 3.
Table 4.

Land-averaged mean and return values for wet and dry period lengths as estimated from the 1979–93 ECMWF reanalysis and as simulated in the GHGA runs in the reference period 1975–95 (“1985”) and their change in 2040–60 (“Δ2050”) and in 2080–2100 (“Δ2090”). Percentages indicate the fraction of the land area where the changes in return values are significant at the 1% significance level

Table 4.
Table 5.

HDD and CDD averaged over the globe, land surface only, and over North America (NA) between 20° and 60°N in the 1979–94 NCEP–NCAR reanalysis (“NCEP”) and as simulated in the GHGA runs in the reference period 1975–95 (“1985”) and their change in 2040–60 (“Δ2050”) and in 2080–2100 (“Δ2090”)

Table 5.

Table A1. Number and percentage of grid points at which N = 1, 2, . . . annual extremes violate the GEV distribution constraints for the parameter estimates obtained by the L-moment method in the reference period 1975–95 of the GHGA runs. A globally averaged estimate of the shape parameter is also shown

i1520-0442-13-21-3760-t101

Table A2. The GEV parameters estimated from the maxima of 5000 samples of length L drawn from the standard normal and exponential distributions. Also shown rejection rates of the null hypothesis that a sample of the maxima of length 63 originates from the Gumbel distribution at the 10% significance level, based on the KS statistics D.

i1520-0442-13-21-3760-t102

Table A3. Total errors of the 0.95th (Q0.95) and 0.98th quantiles (Q0.98) estimated from 10 000 samples of size 63 generated from the GEV distribution with ξ = 0, α = 1, and κ = −0.4, . . ., 0.6. The estimates are obtained from the empirical distribution and from the fitted by the method of L moments standard and modified GEV distributions. Total errors for a modified GEV distribution (F0 > 0) that are smaller than those for a standard GEV distribution are emphasized in bold

i1520-0442-13-21-3760-t103
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  • Fig. 1.

    GOF test for the GEV distribution fitted to annual minima of Tmin in 1975–95 in the GHGA integrations. Shading indicates areas where the null hypothesis that annual Tmin extremes are drawn from the GEV distribution is rejected at the 10% significance level

  • Fig. 2.

    Annual Tmin minima simulated in 1975–95 in the GHGA runs at 28°S, 128°E (central Australia) and at 17°S, 60°W (South America) ordered in descending order. The rank of the minima is given at x axis

  • Fig. 3.

    Q–Q plots for the (a) standard and (b) modified GEV distribution fitted to the sample of annual Tmin extremes displayed in Fig. 2a. The sign of the extremes is reversed

  • Fig. 4.

    The 20-yr return values for (a) daily maximum screen temperature Tmax,20 and (b) daily minimum screen temperature Tmin,20 simulated in 1975–95 in the GHGA runs. (a) Contour interval is 5°C. Light (dark) shading indicates areas where Tmax,20 are larger than 35°C (45°C). (b) Contour interval is 10°C. Light (dark) shading indicates areas where Tmin,20 are smaller than −20°C (−50°C)

  • Fig. 5.

    (left) Annual mean precipitation rate P and (right) 20-yr return values of daily precipitation P20 at (top) Canadian stations, (middle) in the 1979–93 ECMWF reanalysis, and (bottom) in 1975–95 in the GHGA integrations. Contour interval is 1 (20) mm day−1 for P (P20). Light (dark) shading indicates areas where P > 2 mm day−1 (>4 mm day−1) and where P20 > 60 mm day−1 (>100 mm day−1). For Canadian stations, numbers display values for individual stations

  • Fig. 6.

    (a) Spatially smoothed 20-yr return values for daily precipitation P20 simulated in 1975–95 in the GHGA runs and (b) in the 1979–93 ECMWF reanalysis. Contour interval is 25 mm day−1. Light (dark) shading indicates areas where P20 is greater than 50 (150) mm day−1

  • Fig. 7.

    Change in 20-yr return values for daily maximum screen temperature simulated in the GHGA runs (a) in 2040–60 and (b) in 2080–2100 relative to the reference period 1975–95. Contour interval is 4°C. Zero line is omitted. Light, medium, and dark shading indicates areas where changes are greater than 4°, 8°, and 12°C, respectively

  • Fig. 8.

    Change in 20-yr return values for daily minimum screen temperature simulated in the GHGA runs (a) in 2040–60 and (b) in 2080–2100 relative to the reference period 1975–95. Contour interval is 4°C. Zero line is omitted. Light, medium, and dark shading indicates areas where changes are greater than 4°, 8°, and 12°C, respectively

  • Fig. 9.

    Change in spatially smoothed 20-yr return values for daily precipitation simulated in the GHGA runs (a) in 2040–60 and (b) in 2080–2100 relative to the reference period 1975–95. Contour interval is 10 mm day−1. Zero line is omitted. Light (dark) shading indicates areas where P20 increased by at least 10 (20) mm day−1

  • Fig. 10.

    Change in 20-yr return values for 1000-hPa wind speed simulated in the GHGA runs in 2080–2100 relative to the reference period 1975–95. Contour interval is 1 m s−1. Zero line is omitted. Light (dark) shading indicates areas where S20 decreased (increased) by at least 1 m s−1

  • Fig. 11.

    (left) Mean wet period length Dwet and (right) 20-yr return values of wet period length Dwet,20 for (top) Canadian stations, (middle) the 1979–93 ECMWF reanalysis, and (bottom) the GHGA integrations in 1975–95. Contour interval is 0.5 (5) days for Dwet (Dwet,20). Light (dark) shading in left panels indicates areas where Dwet < 2 days (>4 days); in right panels it indicates Dwet,20 < 10 days (>20 days). For Canadian stations, numbers display values for individual stations

  • Fig. 12.

    (left) Mean dry period length Ddry and (right) 20-yr return values of dry period length Ddry,20 at (top) Canadian stations, (middle) in the 1979–93 ECMWF reanalysis, and (bottom) in 1975–95 in the GHGA integrations. Contour interval is 1 (5) days for Ddry (Ddry,20). Light (dark) shading indicates area where Ddry < 4 days (>8 days) and Ddry,20 < 20 days (>60 days). Numbers display values for individual Canadian stations

  • Fig. 13.

    The 20-yr return values of wet period length simulated in the GHGA runs (a) in the reference period 1975–95 and (b) in the 1979–93 ECMWF reanalysis. Rain day is defined as a day with 1-mm precipitation and more. Contour values are 4, 8, 16, . . . , days. Light (dark) shading indicates areas where Dwet,20 < 8 days (>16 days)

  • Fig. 14.

    The 20-yr return values of dry period length simulated in the GHGA runs (a) in the reference period 1975–95 and (b) in the 1979–93 ECMWF reanalysis. Dry day is defined as a day with precipitation < 1 mm. Contour values are 4, 8, 16, . . . , days. Light (dark) shading indicates areas where Ddry,20 < 16 days (>128 days)

  • Fig. 15.

    Changes in spatially smoothed 20-yr return values for (a) wet period length Dwet,20 and (b) for dry period length Ddry,20 simulated in 2080–2100 in the GHGA runs. (a) Contour interval is 2 days. Contour values are offset by 1 day from zero. Light (dark) shading indicates areas where Dwet,20 decreased (increased) by at least 1 day. (b) Contour interval is 10 days. Contour values are offset by 5 days from zero. Light (dark) shading indicates areas where Ddry,20 decreased (increased) by at least 5 days

  • Fig. 16.

    (left) Annual HDD and (right) CDD at (top) Canadian stations, (middle) in the 1979–94 NCEP–NCAR reanalysis, and (bottom) in the GHGA integrations in 1975–95. Contour interval is 1000 (50) degree days for HDD (CDD). Light (dark) shading indicates areas where HDD > 4000 (8000) degree days and CDD > 50 (250) degree days. Numbers display values for individual Canadian stations

  • Fig. 17.

    (a) Annual HDD and (b) CDD simulated by CGCM1 in the GHGA runs in 1975–95. (top) Contour interval is 2000 degree days. Light (dark) shading indicates areas where HDD is greater than 2000 (10 000) degree days. (bottom) Contour interval is 500 degree days. Light (dark) shading indicates areas where CDD is greater than 500 (3000) degree days

  • Fig. 18.

    (a) Change in annual HDD and (b) CDD simulated by CGCM1 in 2080–2100 in the GHGA runs. Contour interval is 500 degree days for HDD and 250 degree days for CDD. Light (dark) shading indicates areas where HDD decreased and CDD increased by at least 25% (75%)

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