A scenario of European climate change for the late twenty-first century: seasonal means and interannual variability

Abstract

A scenario of European climate change for the late twenty-first century is described, using a high-resolution state-of-the-art model. A time-slice approach is used, whereby the atmospheric general circulation model, HadAM3P, was integrated for two periods, 1960–1990 and 2070–2100, using the SRES A2 scenario. For the first time an ensemble of such experiments was produced, along with appropriate statistical tests for assessing significance. The focus is on changes to the statistics of seasonal means, and includes analysis of both multi-year means and interannual variance. All four seasons are assessed, and anomalies are mapped for surface air temperature, precipitation and snow mass. Mechanisms are proposed where these are dominated by straightforward local processes. In winter, the largest warming occurs over eastern Europe, up to 7°C, mean snow mass is reduced by at least 80% except over Scandinavia, and precipitation increases over all but the southernmost parts of Europe. In summer, temperatures rise by 6–9°C south of about 50°N, and mean rainfall is substantially reduced over the same area. In spring and autumn, anomalies tend to be weaker, but often display patterns similar to the preceding season, reflecting the inertia of the land surface component of the climate system. Changes in interannual variance are substantial in the solsticial seasons for many regions (note that for precipitation, variance estimates are scaled by the square of the mean). In winter, interannual variability of near-surface air temperature is considerably reduced over much of Europe, and the relative variability of precipitation is reduced north of about 50°N. In summer, the (relative) interannual variance of both variables increases over much of the continent.

Appendices

Appendix A

1.1 Statistical significance of changes in the multi-year means of time-slice ensembles

Here we describe a methodology to assess the significance of differences between multi-year seasonal means of future and control ensembles of HadAM3P. First it is necessary to build a conceptual model to characterise the data in each ensemble. For the control runs this is identical to that used by e.g. Rowell et al. (1995) and Rowell (1998):

$$x_{{ij}} = \mu _{i} + \varepsilon ^{x} _{{ij}} $$

(1)

where x _ij is a simulated seasonal mean quantity in year i=1, ..., N, ensemble member j=1, ..., n _x ; μ _i is the forced component of x _ij in year i (this forcing primarily arises from SST variations, but to a lesser extent also arises from changes in atmospheric composition); and ɛ ^x _ij is the random component of x _ij due to internal atmospheric variations. For the future scenario runs this conceptual model is extended to become:

$$y_{{ij}} = \mu _{i} + g_{i} + m_{j} + i'\tau _{j} + \varepsilon ^{y} _{{ij}} $$

(2)

where y _ij is a simulated future seasonal mean for year i=1, ..., N, ensemble member j=1, ..., n _y ; g _i is the time-evolving component of y _ij due to changes in atmospheric composition and ensemble mean SST in the future minus control scenario; m _j is the component due to the anomalous time-mean SST change in each ensemble member j (and has zero mean over an infinite ensemble); i′=i − i _mid, where i _mid is the central year (or half-year) of the period of interest; τ _j is the component due to the anomalous SST trend in each ensemble member (so that the average of i′ τ _j is zero over the period of interest, and τ _j has zero mean over an infinite ensemble), and ɛ ^y _ij is the random component of y _ij due to internal atmospheric variations.

Now the null hypothesis we wish to test is that the expectations of x and y are equal for the two periods of interest, or in other words that $\bar{g} = 0,$ where $\bar{g}$ is the time-mean of g. This is normally assessed using a t-test, in which a statistic (t) is computed from the x and y data, and then compared against a theoretical t-distribution with a specified number of degrees of freedom. This leads to an estimated probability that the null hypothesis holds true. However, the standard approach cannot be taken here, because the usual assumption required for a t-test, that x and y are independent, is violated by their common component of SST forcing. Thus, the calculation of the t-statistic must be specifically tailored to the character of the data described above.

First we note that under the null hypothesis:

$${\bar{\bar{x}}} - {\bar{\bar{y}}} = {\bar{\bar{\varepsilon}}}^{x} - {\bar{\bar{\varepsilon}}}^{y} - {\bar{m}}$$

(3)

where ${\bar{\bar{a}}}$ is the mean of a over all years and all ensemble members, and ${\bar{m}}$ is the mean of m over all ensemble members. Importantly, the right-hand side of Eq. 3 now satisfies the assumptions required for a t-test (e.g. von Storch and Zwiers 1999), viz. that all realisations of ɛ^x, ɛ^y and m are sampled from independent identical Gaussian distributions, with fixed variances $\sigma_{{\rm INT}x}^{2}, \sigma_{{\rm INT}y}^{2}$ and σ ² _m . Thus, a random variable with an approximate t-distribution can be constructed in the usual way for the case of unequal variances:

$$t = \frac{{\bar{\bar{\varepsilon}}}^{x} - {\bar{\bar{\varepsilon}}}^{y} - {\bar{m}}} {\sqrt{\frac{{\hat{\sigma}}^{2}_{{\rm INT}x}}{Nn_{x}} + \frac{{\hat{\sigma}}^{2}_{{\rm INT}y}}{Nn_{y}} + \frac{{\hat{\sigma}}^{2}_{m}}{n_{y}}}}$$

(4)

where ${\hat{a}}$ represents an estimate of a from sampled data.

Now, in order to compute t from the available model data, we require formulae to estimate ${\hat{\sigma}}^{2}_{{\rm INT}x}, {\hat{\sigma}}^{2}_{{\rm INT}y}$ and ${\hat{\sigma}}^{2}_{m}.$ For the first of these, we follow Rowell et al. (1995):

$${\hat{\sigma}}^{2}_{{\rm INT}x} = \frac{1}{N(n_{x} - 1)} {\sum\limits_{i = 1}^N {{\sum\limits_{j = 1}^{n_{x}} {(x_{{ij}} - {\bar{x}}_{i})^{2}}}}}$$

(5)

Next, to compute the remaining two variances, the following equality can be derived:

$${\hat{\sigma}}^{2}_{{\bar{y}}_{j}} = {\hat{\sigma}}^{2}_{m} + \frac{1}{N}{\hat{\sigma}}^{2}_{{\rm INT}y} $$

(6)

where

$${\hat{\sigma}}^{2}_{{\bar{y}}_{j}} = \frac{1}{n_{y} - 1} {\sum\limits_{j = 1}^{n_{y}} {{\bar{y}}_{j}^{2}}} \quad {\text{and}}\; {\bar{y}}_{j} = \frac{1}{N} {\sum\limits_{i = 1}^N {y_{ij}}}$$

(7)

Thus, by substituting Eqs. 3 and 6 into Eq. 4, we can compute a t-statistic from the model data using:

$$t = \frac{{\bar{\bar{x}} - \bar{\bar{y}}}} {{{\sqrt {\frac{{\hat{\sigma}^{2} _{{\rm INT}x}}}{Nn_{x}} + \frac{{\hat{\sigma}^{2} _{{\bar{y}_{j}}}}}{{n_{y}}}}}}}$$

(8)

along with Eqs. 5 and 7.

Following the usual approach for a t-test with unequal variances (e.g. von Storch and Zwiers 1999), this t-statistic has an approximate t-distribution with ν degrees of freezdom, where:

$$\nu = \frac{{{\left({\frac{{\hat{\sigma}^{2}_{{\rm INT}x}}}{Nn_{x}} + \frac{{\hat{\sigma}^{2}_{{\bar{y}_{j}}}}}{n_{y}}} \right)}^{2}}}{{\frac{{({\hat{\sigma}^{2}_{{\rm INT}x}} \mathord{\left/ {\vphantom {{\hat{\sigma}^{2}_{{\rm INT}x}}{Nn_{x}}}} \right. \kern-\nulldelimiterspace} {Nn_{x}})^{2}}}{{N(n_{x} - 1)}} + \frac{{({\hat{\sigma}^{2} _{{\bar{y}_{j}}}} \mathord{\left/ {\vphantom {{\hat{\sigma}^{2} _{{\bar{y}_{j}}}} {n_{y}}}} \right. \kern-\nulldelimiterspace} {n_{y}})^{2}}}{{n_{y} - 1}}}}$$

(9)

Finally, in order to verify that this formulation of the t-statistic provides a reliable test of the null hypothesis, thousands of synthetic ensemble timeseries were created using Eqs. 1 and 2. In each case we specified N=30 and n _x =n _y =3, g _i was specified as a linear trend with $\bar{g} = 0,$ and μ _i , m _j , τ _j , ɛ ^x _ij and ɛ ^y _ij were randomly selected from Gaussian distributions with variances $\sigma_{{\rm SST}}^{2}, \sigma_{m}^{2}, \sigma_{{\tau}}^{2}, \sigma_{{\rm INT}x}^{2}$ and $\sigma_{{\rm INT}y}^{2}$ respectively (where σ ²_SST is the variance of μ _i ). A range of realistic values of the trend of g, and of $\sigma_{{\rm SST}}^{2}, \sigma_{m}^{2}, \sigma_{{\tau}}^{2}, \sigma_{{\rm INT}x}^{2}$ and $\sigma_{{\rm INT}y}^{2}$ were estimated from the HadAM3P data. Then for each of a variety of combinations of these values, 5000 randomly differing ensembles were produced. Next, the proportion of the resultant 5000 t-statistics that fell beyond the 5–95% range of the t-distribution was computed. For every combination of the values referred to above it was found that the rejection rate of the null hypothesis was close to 10%, i.e. exactly that expected from a well-behaved test. Conversely, it was noted that if a conventional t-test were applied by wrongly assuming that x and y derive from independent distributions, then the rejection rate is substantially underestimated, and the null hypothesis accepted far too frequently.

Appendix B

1.1 Estimation and statistical significance of changes in the internal atmospheric variance of seasonal mean data in time-slice ensembles

Here we formulate an estimate for the internal atmospheric interannual variance of seasonal mean data (hereafter, IAIV) for the control and scenario ensembles of HadAM3P, and describe a test to evaluate whether this IAIV changes significantly between the two ensembles. The decision to assess changes in internal variability, rather than total variability, is justified in the introduction to Sect. 3.

First, the estimation of IAIV for the control ensemble is straightforward, and its calculation is given by Eq. 5. This variance estimate has a χ² distribution with N(n _x − 1) degrees of freedom.

Next, the formula to estimate IAIV for the scenario ensemble $(\sigma_{{\rm INT}y}^{2})$ requires a brief derivation. The notation is that of Appendix A. To begin, we note that the variance of the deviations of y _ij from each year’s ensemble mean can be written as follows:

$$\hat{\sigma}^{2}_{y'} = \frac{1}{{N(n_{y} - 1)}}{\sum\limits_{i = 1}^N {{\sum\limits_{j = 1}^{n_{y}} {(y_{{ij}} - \bar{y}_{i})^{2}}}}}$$

(10)

Then, by substituting the conceptual model of Eq. 2 into Eq. 10, and assuming that the time-mean SST anomalies of each ensemble member (m _j ) are uncorrelated with the anomalous SST trends of each ensemble member (τ _j ) (which seems reasonable), it can be shown that:

$${\hat{\sigma}}^{2}_{y'} = {\hat{\sigma}}^{2}_{{\rm INT}y} + \hat{\sigma}^{2}_{m} + \alpha \hat{\sigma}^{2} _{\tau} $$

(11)

where:

$$\alpha = \frac{1}{N}{\sum\limits_{i = 1}^{N} {{i'}^{2}}}$$

(12)

Now, a biased estimate of σ ²_τ can be provided by $\sigma^{2}_{{\hat{\tau}}} ,$ which denotes the variance of the linear trends of y _ij estimated directly from the model data. The bias arises because each estimate of τ _j is ‘contaminated’ by internal atmospheric noise. Nevertheless, this bias can be estimated following Wilks (1995), and hence the following equality can be derived:

$$ \hat{\sigma }^{2}_{\tau } = \hat{\sigma }^{2}_{{\hat{\tau }}} - \frac{{\hat{\sigma }^{2}_{{{\rm INT}y }} }} {{N\alpha }} $$

(13)

Thus, by substituting this into Eq. 11, and also substituting Eq. 6 (which estimates σ ² _m ), it can easily be shown that:

$$\hat{\sigma}^{2} _{{{\rm INT}y}} = \frac{N}{{N - 2}}{\left({\hat{\sigma}^{2} _{{y'}} - \hat{\sigma}^{2} _{{\bar{y}_{j}}} - \alpha \hat{\sigma}^{2} _{{\hat{\tau}}}} \right)}$$

(14)

Importantly, this variance estimate is also close to a χ² distribution, with N(n _y − 1) degrees of freedom, if N is large. This has been verified using synthetic ensemble data with N=30 and n _y =3.

Thus, by computing the ratio ${\hat{\sigma }^{2}_{{{\text{INT}}y}} } \mathord{\left/ {\vphantom {{\hat{\sigma }^{2}_{{{\text{INT}}y}} } {\hat{\sigma }^{2}_{{{\text{INT}}x}} }}} \right. \kern-\nulldelimiterspace} {\hat{\sigma}^{2}_{{{\text{INT}}x}} },$ a random variable is constructed which has an F-distribution with N(n _y − 1), N(n _x − 1) degrees of freedom (assuming that the two variance estimates are independent, which is expected). This can then be used to test whether changes in IAIV are statistically significant. Furthermore, since the number of degrees of freedom depends only on the ensemble size, and not in any way on the local climate, the variance ratio at which significance is achieved is spatially homogeneous. For the present study N=30 and n _x =n _y =3, so that variance ratios beyond the range 0.65 to 1.53 are significantly different from unity at the 10% level.

Following the approach of Appendix A, the reliability of this test was verified by computing 5,000 sets of synthetic ensemble data, repeatedly for a range of realistic combinations of $g, \sigma_{{\rm SST}}^{2}, \sigma_{m}^{2}, \sigma_{\tau}^{2}, \sigma_{{\rm INT}x}^{2}$ and $\sigma_{{\rm INT}y}^{2}.$ For each combination, and for $\sigma_{{\rm INT}y}^{2} = \sigma_{{\rm INT}x}^{2} ,$ it was found that the rejection rate of the null hypothesis (F-ratios falling beyond the 5–95% range) was close to 10%, as expected.

Appendix C

1.1 Statistical significance of changes in the coefficient of variation of internal atmospheric variability in time-slice ensembles

For zero-bounded fields, changes in variance (such as those formulated in Appendix B) tend to be similar, to first order, to changes in the mean. This occurs for simple statistical reasons, rather than ‘interesting’ physical reasons. Thus, in this study for precipitation, it is desirable to remove this effect and instead compare the squared coefficient of variation of internal atmospheric interannual variability (hereafter CV²-IAIV). Here a test is described to assess the statistical significance of changes in CV²-IAIV.

Unlike the approach used in Appendices A and B, it is not possible to develop an analytical test to assess changes in CV²-IAIV, because its differences or ratios do not follow a defined distribution. Instead, a test is developed whereby the ratio of the model’s scenario-to-control CV²-IAIV is compared against an estimate of the distribution of CV²-IAIV ratios that would be sampled from an underlying population with similar characteristics, and in which the control and scenario population CV²-IAIVs are identical (i.e. the null hypothesis is satisfied). This distribution is computed from a large set of ensembles of synthetic timeseries sampled from a population that is described by taking estimates $\hat{g}, {\hat{\sigma}^{2} _{{\rm SST}}} \mathord{\left/ {\vphantom {{\hat{\sigma}^{2} _{{\rm SST}}} {\hat{\sigma}^{2} _{{{\rm INT}x}}}}} \right. \kern-\nulldelimiterspace} {\hat{\sigma}^{2} _{{{\rm INT}x}}}, \hat{\sigma}^{2} _{m}$ and $\hat{\sigma}^{2} _{\tau}$ from the model data. The population CV²-IAIV used for this synthetic data is that of a joint estimate computed from both model ensembles. If the model’s actual CV²-IAIV ratio falls beyond the 5–95% range of the distribution of sampled CV²-IAIV ratios then the null hypothesis of an unaltered CV²-IAIV is rejected.

Again, synthetic data (now transformed to a zero-bounded distribution) was used to verify that this test produces the correct rejection rate when the CV²-IAIV of the underlying population is identical in control and scenario data. This verification was successful for precipitation, with rejection rates being only slightly overestimated (typically 10–15% rather than 10%).