Interannual variability and expected regional climate change over North America

Abstract

This study aims to analyse the interannual variability simulated by several regional climate models (RCMs), and its potential for disguising the effect of seasonal temperature increases due to greenhouse gases. In order to accomplish this, we used an ensemble of regional climate change projections over North America belonging to the North American Regional Climate Change Program, with an additional pair of 140-year continuous runs from the Canadian RCM. We find that RCM-simulated interannual variability shows important departures from observed one in some cases, and also from the driving models’ variability, while the expected climate change signal coincides with estimations presented in previous studies. The continuous runs from the Canadian RCM were used to illustrate the effect of interannual variability in trend estimation for horizons of a decade or more. As expected, it can contribute to the existence of transitory cooling trends over a few decades, embedded within the expected long-term warming trends. A new index related to signal-to-noise ratio was developed to evaluate the expected number of years it takes for the warming trend to emerge from interannual variability. Our results suggest that detection of the climate change signal is expected to occur earlier in summer than in winter almost everywhere, despite the fact that winter temperature generally has a much stronger climate change signal. In particular, we find that the province of Quebec and northwestern Mexico may possibly feel climate change in winter earlier than elsewhere in North America. Finally, we show that the spatial and temporal scales of interest are fundamental for our capacity of discriminating climate change from interannual variability.

Appendices

Appendix 1: Expected number of years for statistically significant climate change

There are several hypotheses that are considered in the definition of the EYE. First of all that the trend is linear; this seems a reasonable approximation as discussed in Sect. 2.5 and as shown for example by Santer et al. (2011) (who considered that “to first order, the signal is timescale invariant”) and Mahlstein et al. (2011). This is also implicitly assumed when a trend is obtained from the averages of two time windows. However, although linearity is a reasonable approximation for the present century, the choice of the beginning of this trend causes some difficulties. In this work, we do not concern ourselves with this issue, since we are only interested in this measure as a qualitative indicator. Hence, the view taken here is that climate has been stationary until a given time, after which temperature grows linearly (see for example Figs. 10.4 and 10.5 in Meehl et al. 2007b, where results suggest that linear growth is a good approximation after the 1970s and during several decades). In addition, inferences regarding the discrimination of climate change from natural variability will be taken only considering model data and from an arbitrarily determined starting point. This will provide information regarding the expected length of the time series needed to discriminate climate change, but not its actual place in time.

Two possible misunderstandings should be avoided. First, if the chosen starting point for the analysis of the slope were 1990, the number of years N obtained by this method should not be interpreted as meaning that climate change will only be detected in the year 1990 + N. One can choose any starting point as long as the linearity approximation stands. Second, the EYE is by definition an expected value; hence, it is not a prediction and cannot be applied to a single realization. It does not indicate the time that emergence from natural variability will happen, but when it is expected to occur.

Another hypothesis is that seasonal values are interannually decorrelated. Our studies suggest (not shown) that the hypothesis of total decorrelation cannot be rejected with local (grid point) time series of a 100 years.

It is important to keep in mind that the data analyzed from the simulated 140-year time series in Sect. 3.3 may lack a realistic decadal variability as is the case of many climate simulations [see validation analyses of multidecadal variability among CMIP3 models discussed by Kravtsov and Spannagle (2008) and in Santer et al. (2011)]. This is one further reason why these estimations should be considered carefully if applied for policy use.

The test for the slope of a cloud of points based on the Student distribution will be developed. For its comparable performance with respect to other tests in temperature trends see Liebmann et al. (2010) and Mahlstein et al. (2011).

The standard definition of a Student test can be written as

$$t = \frac{{\hat{\beta }}}{{s_{\beta } }}, $$

(7)

where the numerator is the estimated statistic and the denominator is the estimated standard deviation of the statistic. In the case of the trend of a time series, this expression can be rewritten as

$$t = \frac{{\hat{\beta }}}{{s/\sqrt {SS} }}, $$

(8)

where

$$s^{2} = \frac{1}{n - 2}\sum\limits_{i = 1}^{n} {\left( {y_{i} - \hat{y}} \right)^{2} } , $$

(9)

with i being the year of each value $y_{i}$, with a total of n years, and $\hat{y}$ is the ordinate value obtained by linear regression. This last expression is the detrended variance. The term SS can be expressed as

$$SS = \sum\limits_{i = 1}^{n} {i^{2} } - \frac{1}{n}\left( {\sum\limits_{i = 1}^{n} i } \right)^{2} . $$

(10)

For a thorough derivation of these expressions see Scheaffer and McClave (1990). The term SS can be expressed in a simpler way by rewriting the summations using the properties of finite series of integers as

$$SS = \frac{n}{12}\left( {n^{2} - 1} \right). $$

(11)

Using (11), we can now rewrite expression (8) as

$$t = \frac{{\hat{\beta }}}{s}\sqrt{ \frac{n}{12} \left( {n^{2} - 1} \right)}. $$

(12)

This expression of the statistic t is now function of the estimated slope, the estimated detrended variance of the time series, and the number of years in the sample.

For n reasonably large, t can be described by a Gaussian distribution. By assuming $n^{2} - 1 \approx n^{2}$, we can isolate n for a given level of significance $\alpha$, yielding

$$n_{\alpha } = \root{3} \of {{12\left( {\frac{{t_{\alpha } \sigma }}{{\hat{\beta }}}} \right)^{2} }}, $$

(13)

where $\sigma$ now represents the detrended variance and the trend $\hat{\beta }$ is in degrees per year.

This expression can be interpreted in several ways. In the present work, $\hat{\beta }$ is going to be the expected trend (obtained from a long term climate projection of a single simulation or an ensemble as in Sect. 3.2). At the same time $\sigma$, the estimated interannual variability, will be the average detrended standard deviation (average between past and future) when the climate change trend is estimated by a simple difference between future minus present temperature.

An alternative formulation using two time windows instead of a linear regression was also developed (not shown). It produced a similar functional form.

Appendix 2: Error estimations

2.1 Error in trend and variance

The error in the estimation of the detrended variance can be obtained using the expression from von Storch and Zwiers (1999), but taking into consideration that there are n−2 degrees of freedom,

$$\sigma_{{S_{T}^{2} }}^{2} = 2\frac{{\sigma_{T}^{4} }}{{\left( {n - 2} \right)}}, $$

(14)

where the distribution is assumed to be symmetrical (zero kurtosis), $S_{T}^{2}$ is the estimator of the temperature variance and $\sigma_{T}^{2}$ the population temperature variance. In our case with two time windows of n = 30, the variance considered in Sect. 3.3 is

$$\sigma_{T}^{2} = \frac{1}{2}\left( {\sigma_{{T_{p} }}^{2} + \sigma_{{T_{f} }}^{2} } \right), $$

(15)

where T _p and T _f refer to present and future temperature respectively.

Using properties of the Chi square distribution, and assuming similar variances in both periods, the error of $S_{T}^{2}$ may be written as

$$\sigma_{{S_{T}^{2} }}^{2} = \frac{{\sigma_{T}^{4} }}{{\left( {n - 2} \right)}}, $$

(16)

where the variance over T now considers both present and future temporal windows (the use of a double window explains the decrease in variance estimation error from 14 to 16).

The slope of temperature trends are estimated by using

$$\beta = \frac{1}{{N_{y} }}\left( {\overline{T}_{f} - \overline{T}_{p} } \right), $$

(17)

where N _y is the distance in number of years between the centers of the time windows.

The error on beta can be estimated using the standard formula for error in the mean, as well as properties of the variance operator (see von Storch and Zwiers 1999) as

$$\sigma_{\beta }^{2} = \frac{1}{{N_{y}^{2} }}\left( {\frac{{\sigma_{{T_{p} }}^{2} }}{n} + \frac{{\sigma_{{T_{f} }}^{2} }}{n}} \right). $$

(18)

For most gridpoints and NARCCAP models, temperature interannual variances in future and present cannot be said to statistically differ. The exception is the HRM3-HadCM3, which shows a loss of variance in the future during winter over most of Canada, and the CCSM-driven models in the northern tip of Canada during summer, which show an increase of variance (not shown). With this information, the previous expression can confidently be approximated by

$$\sigma_{\beta }^{2} \approx \frac{2}{{N_{y}^{2} }}\frac{{\sigma_{T}^{2} }}{n}, $$

(19)

where $\sigma_{T}^{2}$ represents the mean variance between the present and future climate. This assumption is taken only for the sake of obtaining an estimate of the error bar, and is consistent with results and assumptions used elsewhere (e.g., Hawkins and Sutton 2009).

Under the conditions of the experiment discussed in this paper (n = 30, N _y  = 70, and S of around 3 °C; see Fig. 1), it can be seen that the sampling error in the estimation of the trend in climate change is of around 1 °C/century.

2.2 Error in expected number of years before emergence (EYE)

The other quantity whose error needs to be estimated is the EYE, presented in Sect. 2.5 and derived in Appendix 1. For the sake of completeness, we rewrite below the final expression (13).

$$n_{\alpha } = \root{3} \of {{12\left( {\frac{{t_{\alpha } \sigma }}{\beta }} \right)}}^{2} . $$

(20)

It can be shown that for a function f of two independent variables x and y of the form

$$f = a\left( {\frac{x}{{y^{2} }}} \right)^{1/3} , $$

(21)

its error can be written as a function of those of the independent variables by error propagation as

$$\sigma_{f}^{2} \approx a^{2} \frac{1}{{3^{2} }}\left( {\frac{x}{{y^{2} }}} \right)^{2/3} \left( {\frac{{\sigma_{x}^{2} }}{{x^{2} }} + 4\frac{{\sigma_{y}^{2} }}{{y^{2} }}} \right), $$

(22)

where $\sigma_{f}^{2}$ is the error in function f, and $\sigma_{x}^{2}$ and $\sigma_{y}^{2}$ the error associated to variables x and y. The normalized error can be expressed as

$$\left( {\frac{{\sigma_{f} }}{f} } \right)^{2} \approx \frac{1}{9}\left( {\frac{{\sigma_{x}^{2} }}{{x^{2} }} + 4\frac{{\sigma_{y}^{2} }}{{y^{2} }}} \right). $$

(23)

Using the definition of the EYE from (20) and taking $\sigma^{2}$as x (notice that $\sigma^{2}$ will be estimated by $S_{T}^{2}$), $\hat{\beta }$ as y, $\root{3} \of {{12\left( {t_{\alpha } } \right)^{2} }}$as a, and $n_{\alpha }$ as f, it is easy to see that (23) becomes

$$\left( {\frac{{\sigma_{n_\alpha } }}{{n_{\alpha } }} } \right)^{2} \approx \frac{1}{9}\left( {\frac{{\sigma_{{s_{T}^{2} }}^{2} }}{{S_{T}^{4} }} + 4\frac{{\sigma_{\beta }^{2} }}{{\beta^{2} }}} \right), $$

(24)

Taking advantage of the estimation of errors for each of these variables discussed in (16) and (19), we get

$$\left( {\frac{{\sigma_{n_\alpha } }}{{n_{\alpha } }} } \right)^{2} \approx \frac{1}{9n}\left( {1 + 8\frac{{S_{T}^{2} }}{{N_{y}^{2} \beta^{2} }}} \right). $$

(25)

In most cases this expression can be approximated by neglecting the left hand side of the addition. In per cent units, we obtain

$$\frac{{\sigma_{{n_{\alpha } }} }}{{n_{\alpha } }} \approx 90{\kern 1pt} \% \frac{1}{\sqrt n }\frac{1}{{N_{y} }}\frac{{S_{T} }}{\beta }. $$

(26)

For example, for the typical conditions proposed in this research, with n = 30 and N _y  = 70, a trend of 4 °C/century and an interannual standard deviation of 3 °C gives a relative error of around 20 %. This error increases with larger variability and smaller trend.