Shifting velocity of temperature extremes under climate change

This work uses high-resolution (0.11°) regional climate models (RCMs) outputs using the RCP 8.5 scenario [38] within the EURO-CORDEX [37] initiative over the period 1951–2100. This long period maximizes the climate evolution and therefore the potential discrepancies between shifting velocities of central trends and extremes. The computational domain covers the following geographical coordinates: (63.55°N, 51.56°W), (63.65°N, 72.56°E), (20.98°N, 14.31°W), and (20.98°N, 72.56°E). The ability of EURO-CORDEX simulations to reproduce present-day temperature extremes has been demonstrated (e.g. [46, 47]) and their outputs have been widely used to analyze projections of extreme temperatures in Europe (e.g. [48]). We use historical runs for the period 1951–2005 and projections for 2006–2100.

Our analysis primarily relies on ALADIN-5.3 (hereafter named ALADIN), a RCM developed by Météo-France [49, 50]. To assess the robustness of our findings against the choice of this particular RCM, we replicate the analysis with three other RCM outputs, namely HIRHAM 5 (HIRAM, DMI [51]), RACMO-22E (RACMO, KNMI [52]), and REMO2009 (REMO, MPI [53]). Unless otherwise specified, the results mentioned below are from ALADIN.

We base the analysis on the screen-level air temperature and consider its daily mean (T2m), minimum (T2m,min), and maximum (T2m,max).

We first compare the evolution of the annual median of T2m (hereafter noted $\overbrace{T2m}$) to that of the cold and hot quantiles (percentiles 1, 2, 5, 10, and 20, and percentiles 80, 90, 95, 98, and 99, respectively) of the annual PDF of T2m. Consistent with the approach of the new normals [45], we consider the PDF as it evolves as a function of time rather than the historical PDF.

In order to assess whether our result depend on the definition of the central trend and the extremes, we also compare the evolution of the yearly averaged T2m (hereafter noted $\overline{T2m}$) with that of the values at ±0.5σ, ±1σ, ±1.5σ, and ±2σ, σ being the yearly standard deviation of the daily T2m values. If T2m were to follow a Gaussian distribution, these thresholds would correspond to percentiles 30 and 70, 16 and 84, 6 and 94, and 2 and 98, respectively. We also consider the number of tropical nights (T2m, min $\geqslant 20$ °C) and the number of frost days (T2m,max $\leqslant 0$ °C) [40].

Finally, as the duration of extreme temperature events is a key aspect of their potential impacts on natural and human systems, we defined a warm days clustering index (WDCI), in an approach consistent with the percentile-based characterization of extremes used in this study. More specifically, we define the WDCI as the total number of days that belong to a sequence of at least three consecutive days with a daily-averaged T2m above the percentile 90 of the local annual T2m PDF. As the WDCI is calculated relative to the annual cycle, it mainly focused on the warm summer episodes. In order to account for the temporal evolution of the PDF with respect to global warming, we detrend the PDF by substracting a 30 year linear regression fit. Tropical nights, frost days, and clustered warm days occur in a limited number each year (e.g. WDCI $\leqslant 35\,{\mathrm{yr}}^{-1}$ according to the above definition). To achieve statistical significance, we bin their yearly counts into 30 years periods (1951–1980, 1981–2010, 2011–2040, 2041–2070, and 2071–2100).

An index of the velocity of temperature change has first been proposed by Loarie et al [18]. Here, we follow the same approach, calculating the shifting velocity of a given value ψ at each grid point as the ratio of its temporal derivative (denoted as temporal gradient by Loarie et al [18]) to its gradient (denoted as spatial gradient by Loarie et al). Although intuitive, this definition relies on several implicit assumptions. We detail further this approach, its mathematical foundations, and we explicitly describe and discuss the underlying assumptions in the Supplementary Material.

At each grid point, the temporal average of the temporal derivative $\overline{\left(\tfrac{\partial \psi }{\partial t}\right)}$ is determined as the slope of a linear regression on the time series over 1951–2100. The gradient is calculated with centered differences, in a way similar as Loarie et al [18]. While the pace of climate change and forcings do not evolve linearly over the 150 years long time span, the consideration of a linear trend over this period provides a first-order assessment with minimal influence of the short-term fluctuations. Furthermore, a longer time period implies wider changes in both the central trends and temperature extremes, maximizing their chances to exhibit different behaviors.

$\overbrace{T2m}$ shifts faster over most marine areas, as expected from previous works [54] (figure 1(b)). This is especially true over the Atlantic Ocean and the Mediterranean Sea. A faster shift is also found across Eastern Europe and Western Russia. Conversely, $\overbrace{T2m}$ shifts slower over mountainous regions such as the Alps, the Atlas, as well as in Scandinavia. These fast (slow) shifting velocities correspond to smoother (steeper) gradients (figure 2(b)), while the temperature change is quite homogeneous over the considered time period (figure 2(e)).

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The shifting velocity of hot and cold extremes, represented by the percentiles 2 and 98 respectively (figures 1(a) and (c)), show spatial patterns and magnitudes similar to those of the median, but differ at specific locations. The cold (hot) extremes shift slower (faster) than the median over the Atlantic Ocean West of Iceland and the Eastern part of the Mediterranean Sea. In contrast, they shift faster (slower) than the median on the East coast of Great Britain and Northern Russia. The similar shifting velocities of the median and the extremes appears to stem from a combination of both similar gradient (figures 2(a)–(c)) and temporal derivatives (figures 2(d)–(f)) on most of the considered region. However, the above-mentioned discrepancies seem to arise either from different gradients (West of Iceland, Eastern Mediterranean Sea, North Sea) or from different temporal derivatives (Northern Russia). Finally, North of Iceland and in Eastern Europe, the dependency of the gradients and temporal derivatives with regard to the quantiles compensate each other, resulting in similar shifting velocities.

The ratio between the shifting velocities of percentiles 2, 50, and 98 (figures 1(d)–(f)) evidence further details pertaining to coastal regions. Off the coasts of Western Europe and in the North-Eastern Mediterranean Sea, both cold and hot extremes shift significantly slower than the median. In contrast, off the North coast of Africa, both in the Atlantic Ocean and the Southern Mediterranean Sea, the cold extremes shift faster than the median, while the hot extremes shift slower.

Figure 3 displays the shifting velocity as a function of the quantile. The rather homogeneous velocities described above translate into very flat dependencies. As all quantiles evolve at the same pace, the histograms of the T2m PDF shift as a whole towards warmer temperature, for each season and on both marine regions and land as shown in figure 4. A faster (slower) heating of the hot (cold) extremes as compared with the median would have resulted in longer and fatter PDF tails at the end of the evolution period, while the opposite would have yielded sharper cutoffs of the tails.

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Three alternative climate models (HIRHAM, RACMO, and REMO) generally yield similar trends as ALADIN in spite of local quantitative differences in the shifting velocities (figure S2 is available online at stacks.iop.org/ERL/15/034027/mmedia). The shifting velocity increases with quantiles in all models in the Mediterranean Sea and the Atlantic Ocean especially West of Iceland. Also, the median shifts faster than both extremes in the North Sea and the Sea of Norway in the four models. In contrast, the decrease of the shifting velocity with the quantile over Russia is much stronger in RACMO than in ALADIN, and smaller in REMO and HIRHAM. As a result, the cold quantiles shift faster over land in RACMO than in the other models (figure 3(j)). This could arise from how Arctic warming influences cold outbreaks across Europe. In all models, the differences in shifting velocity are mainly driven by the gradient patterns. Furthermore, while the overall shifting velocity may differ from model to model, the dependencies with the quantiles (i.e. the slope of the curves in figure 3) are mostly similar. While multi-model ensemble simulations would increase the accuracy and the delineation of uncertainty ranges [55], the similarity of the results across models indicates that relying on a single model is sufficient for the purpose of this study.

Figure 5 displays two-dimensional histograms where the color scale indicates the amount of grid cells that display a pair of given velocities for $\overbrace{T2m}$ and for the 2nd or 98th percentiles. As the shifting velocity $\vec{v}$ is a vector, Panels a–d focus on the velocity magnitude, while panels e–h display the shifting directions. The diagonal corresponds to a perfect equality between the two shifting velocities: the clustering of data along this line illustrates the similar shifting velocities of the median and the extreme percentiles, as discussed above (figure 1). This is particularly the case over land for both both magnitudes (figures 5(c) and (d)) and directions (figure 5(g) and (h)).

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Over marine regions, shifting velocities are slightly more dispersed. The percentiles 2 (figure 5(a)) and 98 (figure 5(b)) respectively display marginally slower and faster shifts than the median, consistent with the results showed in figure 1. For cold extremes (Panel a), the stripe deviating down from the diagonal corresponds to coastal regions and to the Atlantic Ocean West of Iceland. Regarding hot extremes (Panel b), the stripe above the diagonal corresponds to the Atlantic Ocean. In spite of these small fluctuations, the overall shifting directions of the extreme percentiles and the median over marine regions are rather similar, although slightly more spread than those over land.

The same analysis has been carried out by considering the percentiles 1 (resp. 99), 5 (95), and 10 (90) instead of 2 (98) as the cold (hot) extremes. Similar results are found, although the features described above are slightly stronger for more extreme percentiles. We also performed the same analysis with the annual mean as the central value, and ±0.5σ, ±σ, ±1.5σ, or ±2σ of the T2m, as the temperature extremes (figures S3–S5 for ±2σ). The results are very similar with each other and with those obtained considering quantiles and the median. This can be explained by considering that the yearly median and the mean of the daily T2m are extremely correlated, both in space ($R\geqslant 0.99$) and in time ($R\geqslant 0.999$). Their difference is limited to ±0.5 °C over almost 70% of the grid cells.

Therefore, the behavior of the shifting velocity of the temperature extremes defined with respect to the T2m PDF is to a large extent resilient to the particular definition of the extremes. This immunity covers the choice of the central trend (median or mean), the tails of the distribution (percentiles or standard deviations), as well as the cutoff chosen in the temperature distribution to define the extremes. This corroborates the robustness of our findings.

Isochrones defining the number of tropical nights per year generally shift much faster over marine regions, as well as over Russia and Northern Africa (figure 6(a)). The behavior is similar to that of $\overbrace{T2m}$ (figure 1(b)), in spite of a slightly faster shifting velocity. As a result, the ratio between the corresponding velocities is quite homogeneous, with an average value of 1.08 over all regions where it is defined (figure 6(d)). The slightly faster shifting velocity of the tropical nights as compared with $\overbrace{T2m}$ is also evidenced by a deviation of data above the diagonal in the two-dimensional histograms of figure 7(a). The only exceptions are the Atlantic Ocean off the Spanish coast and the Black Sea, where the tropical night isochrones locally shift up to 2–6 times faster than the median. In spite of these deviations on the velocity norm, the number of tropical nights shifts in the same direction as $\overbrace{T2m}$ (figure 7(d)). The dispersion of shifting directions mostly corresponds to the rounding errors related to low shifting velocities.

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Where they occur, the number of frost days also shift at velocity magnitudes and directions comparable to that of $\overbrace{T2m}$ (figures 6(b), 7(b) and (e)) over most of the domain. However, they shift slower than the median in both the North Sea and in the Atlantic Ocean south of Iceland (figure 6(e)).

Finally, the WDCI shifts typically 4 times slower (1 km yr⁻¹) over the whole domain than $\overbrace{T2m}$ as well as than the other extremes investigated in this study (figures 6(c), (f) and 7(c)). Slightly faster shifts are however observed over the Atlantic Ocean (1.7 km yr⁻¹ on average) and to a lesser extent over Russia (1.2 km yr⁻¹ on average).