2.1 ERA-Interim

We use ERA-Interim re-analysis data (Dee et al., 2011) covering the period 1979–2018. ERA-Interim is originally produced with a T255 spectral horizontal resolution and 60 hybrid σ–p levels in the vertical. We interpolated the data horizontally to a 1^∘ by 1^∘ grid and vertically to pressure and isentropic levels. The ERA-Interim data are provided at 6-hourly time intervals; in this study however, we aggregated all data to a daily temporal resolution. Besides the T2m fields, we also use potential vorticity (PV), total precipitation, 250 hPa meridional winds and sea ice concentration. Furthermore, we remove a (40-year) linear trend from all JJA T2m data at each grid point. Our analyses hereafter are based on the detrended data, except for Figs. 2, 8 and 9, which are more easily understood based on the non-detrended data (Figs. 2 and 8) or where the absolute T2m values are important (Fig. 9).

Figure 2Steps in computing ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values at the grid point closest to Zurich, Switzerland (47^∘ N, 9^∘ E). Values for the 1994 summer are highlighted in red. Panel (a) shows ERA-Interim T2m at 47^∘ N, 9^∘ E, for all 40 ERA-Interim summers. The sorted T2m values ($T_{d,k}^{\mathit{\text{ERAI}}}$) are shown in panel (b) and the ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values in panel (c). Note that for illustrating purposes Fig. 2 presents non-detrended T2m data.

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2.2 CESM

Besides ERA-Interim, the Community Earth System Model version 1 (CESM; Hurrell et al., 2013) is used to perform present-day climate simulations using restart files from the CESM large ensemble project (CESM-LE.; Kay et al., 2015). We use atmospheric fields at daily temporal resolution, with a horizontal resolution of approximately 1^∘ and 30 vertical levels. The original CESM-LE data contain a 35-member ensemble of simulations started on 1 January 1920 and integrated forward in time until 2100. These 35 “macro-ensemble” members were rerun for the period from 1 January 1990 to 31 December 1999 in order to obtain temporally high-resolution three-dimensional model output. To further increase the number of simulated JJA seasons, a “micro-ensemble” with additional 35 members was initialized from member one of the macro-ensemble, on 1 January 1980, by adding an O(10⁻¹³) perturbation to the initial atmospheric temperature field of each micro-ensemble member. These additional micro-ensemble runs are then integrated forward in time until 31 December 1999. Fischer et al. (2013) have shown that at the latest after a decade, the micro-ensemble members exhibit a similar spread in atmospheric variables compared to members of the macro-ensemble. Thus, for the period 1990–1999, the micro-ensemble members can be regarded as additional independent members, yielding a total of 70 ensemble members covering the 10-year period from 1990 to 1999, i.e. 700 years of present-day climate. As for ERA-Interim data, a linear trend is removed from all JJA T2m data at each grid point and in each ensemble member. Note, however, that due to the ensemble set-up, this trend is calculated over only 10 years.

2.3 Decomposing a seasonal T2m anomaly to quantify the season's substructure

To examine the substructure of a particular July–August (JJA) season k, we decompose its seasonal T2m anomaly (SA_k) into contributions from the ranked D daily T2m values of season k, where D is the number of days in season k (e.g. for JJA, D=92). We thus aim to quantify how much each rank day (i.e. coldest day, second-coldest day, etc.) of season k contributes to the seasonal anomaly SA_k. This decomposition of SA_k is illustrated for the example grid point 47^∘ N, 9^∘ E (near Zurich, Switzerland), in Fig. 2 and introduced more formally below. It is applied to both data sets separately in exactly the same fashion, and, therefore, a superscript M∈{ERAI,CESM} will only be used where it is necessary to explicitly distinguish between the two data sets. All the important statistical quantities used in this study are summarised in Table 1. Furthermore, bear in mind that all these quantities are calculated at each grid point individually.

Table 1Definitions and descriptions of important quantities used in this study.

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We start by ranking all daily mean T2m values within their respective season k (Fig. 2a, b) and computing seasonal means (SM_k), i.e.

where T_d,k is the daily mean T2m value with rank d in season k (i.e. the temporal ordering of the days is lost; see Fig. 2b). At each grid point we thus compute K^ERAI=40 seasonal mean values for ERA-Interim and K^CESM=700 values for CESM.

The climatological seasonal mean (C) is also calculated from the ranked daily mean T2m values (T_d,k) as

Hereby, $\frac{\mathrm{1}}{K}\sum _{k=\mathrm{1}}^K{T}_{d,k}$ is the average T2m value of all K days with rank d in their respective season, e.g. for d=1 the average coldest day of the season and for d=92 the average hottest day of the season. Hence, C is computed as the mean over the average T2m values for each rank. These rank day T2m means (bold grey contour in Fig. 2b) are hereafter referred to as

Using the RDM_d, the seasonal T2m anomaly of any season k (SA_k) can be decomposed into contributions from each of the D rank days:

where in the last equality the rank day anomaly of the day with rank d in season k is introduced as ${\mathit{\text{RDA}}}_{d,k}=T_{d,k}-{\mathit{\text{RDM}}}_d$. In other words, the seasonal mean anomaly SA_k is expressed as the average rank day anomaly (see also Fig. 2c).

This decomposition of SA_k thus allows assessing the exact contribution from each (ranked) day of season k to SA_k. For example, if for a particular season k SA_k=1 K and ${\mathit{\text{RDA}}}_{\mathrm{92},k}=\mathrm{3}$ K (i.e, the hottest day of season k is 3 K warmer than the respective rank day mean) this day contributed $\mathrm{3}/\mathrm{92}=\mathrm{0.0326}$ K, or 3.26 %, to the seasonal anomaly SA_k. In the following we split the 92 d of each JJA season k into three parts according to their rank and focus on the relative contributions to SA_k from the coldest, middle and hottest third of the 92 d of season k by calculating

The notation [x] hereby stands for x rounded to the nearest integer. For computing contributions to SA_k from the middle and hottest thirds of the summer days (SF_middle,k and SF_hot,k), the sum in Eq. (5) runs from $\left[\frac{D}{\mathrm{3}}\right]+\mathrm{1}$ to $\left[D\frac{\mathrm{2}}{\mathrm{3}}\right]$ for SF_middle,k and from $\left[D\frac{\mathrm{2}}{\mathrm{3}}\right]+\mathrm{1}$ to D for SF_hot,k. By construction, the sum of the three fractions amounts to 1.

2.4 Identification and substructure of extreme summers

Extremely hot summers at each grid point in the Northern Hemisphere are identified in the ERA-Interim (CESM) data set as the five (35) hottest JJA seasons, yielding two sets of extreme summers, ${\mathbb{X}}^M=\mathit{\{}{k}_{\mathrm{1}},\mathrm{\dots },k_{N^M}\mathit{\}}$, M∈{ERAI,CESM}, with N^ERAI=5 and N^CESM=35 members, respectively. Hence, ERA-Interim extreme summers correspond to the 12.5 % hottest summers (5 out of 40), while the CESM extreme summers correspond to the 5 % hottest summers (35 out of 700).

An analogous procedure to that described in Sect. 2.3 is employed to quantify the contributions from each of the three thirds of the extreme-summer days to the average T2m anomaly of the N considered extreme summers. The mean of these extreme summers (XM) is calculated as $\mathit{\text{XM}}=\frac{\mathrm{1}}{N}{\sum }_{k\in \mathbb{X}}{\mathit{\text{SM}}}_k$ and is used to compute the mean anomaly of these extreme summers, $\mathit{\text{XA}}=\mathit{\text{XM}}-C$. The relative contributions from the three thirds of the summer days to the extreme-summer anomaly XA are calculated as, for example,

The quantities XF_cold, XF_middle and XF_hot again add up to 1 and quantify the relative contributions from the three thirds to the average T2m anomaly of all extreme summers at a particular grid point. Note that the quantities XF_cold, XF_middle and XF_hot characterise the mean extreme-summer substructure at a particular grid point, while SF_cold,k, SF_middle,k and SF_hot,k characterise the substructure of a single season k.

3.1 Extreme-summer T2m anomalies

Figures 3a and 4a depict the average T2m anomalies during extreme summers in the two data sets (XA^ERAI and XA^CESM, respectively). In both data sets, XA exhibits considerable spatial variability. The ERA-Interim extreme summers have temperature anomalies of up to 3 K over western Russia, while over some tropical ocean areas XA^ERAI is less than 0.5 K (Fig. 3a). The XA^CESM field exhibits a generally similar spatial pattern to XA^ERAI, with larger values over land than over the oceans (Fig. 4a). However, XA^CESM generally exceeds XA^ERAI, as the summers 𝕏^CESM are statistically more extreme than the summers 𝕏^ERAI. In the following, we decompose the extreme-summer T2m anomalies (XA) shown in Figs. 3a and 4a using the methodology described in Sect. 2.3 and 2.4, first at few selected grid points and then for all Northern Hemisphere grid points.

Figure 3Extreme-summer T2m anomaly and extreme-summer substructure for selected grid points in ERA-Interim. Panel (a) depicts XA^ERAI; panels (b)–(e) show ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ for the five ERA-Interim extreme summers in colours and for the remaining summers in light grey. Crosses in panel (a) indicate the grid points for which the ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values are shown in panels (b)–(e).

Figure 4Extreme-summer T2m anomaly and extreme-summer substructure for selected grid points in CESM. Panel (a) displays XA^CESM, and panels (b)–(e) show in red the maximum and minimum (dotted) 90th and 10th percentile (dashed) and the median (solid red) ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{CESM}}}$ of the 35 CESM extreme summers. The 5th- to 95th-percentile range of the ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{CESM}}}$ of all JJA seasons is depicted in grey. Crosses in panel (a) indicate the grid points for which the rank day anomalies are shown in panels (b)–(e).

3.2 Extreme-summer substructures at selected grid points

The rank day anomalies (${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$) for the five ERA-Interim extreme summers at a grid point located in eastern India (21^∘ N, 81^∘ E; Fig. 3a, b) reveal a similar substructure in at least four of the extreme summers. The largest ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values (up to 5 K) occur in the hottest 30 d of each season, while for the 60 coldest summer days in each extreme summer, ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ does not exceed 1.5 K. The contributions of the coldest, middle and hottest third of all extreme-summer days to XA^ERAI at this grid point (i.e. ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$, ${\mathit{\text{XF}}}_{\mathrm{middle}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$) are 13 %, 20 % and 67 %, respectively. For the 2005 summer, the contributions were −1 %, 6 % and 95 %, and, hence, almost the entire seasonal T2m anomaly resulted from the hottest 30 d of the summer being hotter than normal.

A comparison between the ERA-Interim and CESM extreme-summer substructures at this grid point (Figs. 3b and 4b) reveals remarkable qualitative similarities between the extreme-summer substructure at 21^∘ N, 81^∘ E, in the two data sets. At this grid point, also the seasons 𝕏^CESM exhibit their largest ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{CESM}}}$ values for the 30 hottest summer days. Moreover, despite the different number of seasons in the two data sets, the ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$, ${\mathit{\text{XF}}}_{\mathrm{middle}}^{\mathit{\text{CESM}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ values of 11 %, 24 % and 65 %, respectively, are not far off the respective values for the seasons 𝕏^ERAI. Figures 3b and 4b further reveal that the largest ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{CESM}}}$ values reach much larger values (up to 8 K) than the ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values, which is an expected result, since the seasons 𝕏^CESM are statistically more extreme than the seasons 𝕏^ERAI.

Considering now the grid point 39^∘ N, 116^∘ W, in Nevada, US, we find a substantially different ERA-Interim extreme-summer substructure compared to eastern India (Fig. 3b, c), with largest extreme-summer ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values in the coldest third of the summer days and ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}=\mathrm{49}$ %, ${\mathit{\text{XF}}}_{\mathrm{middle}}^{\mathit{\text{ERAI}}}=\mathrm{31}$ % and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}=\mathrm{20}$ %. Also for this grid point, the mean substructure of CESM extreme summers is similar to that of ERA-Interim extreme summers, with ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}=\mathrm{42}$ %, ${\mathit{\text{XF}}}_{\mathrm{middle}}^{\mathit{\text{CESM}}}=\mathrm{33}$ % and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}=\mathrm{25}$ % (Fig. 4c). Thus, at this grid point, all thirds of the T2m distribution contribute to extreme summers, but the contribution from the coldest third is overproportionately large (i.e. considerably larger than 33 %). Hence, the re-analysis and the climate model data both suggest that the suppression of cool summer days (leading to coldest days of the summer that are milder than usual) is a key ingredient for extreme summers at 39^∘ N, 116^∘ W.

A further extreme-summer substructure is apparent at the grid point closest to Paris, France (49^∘ N, 2^∘ E; Figs. 3d, 4d). At this grid point, the ERA-Interim extreme summer of 2018 was characterised by ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values of 1.5–2 K for almost all ranks; i.e. this summer resulted from an almost uniform shift in the entire T2m distribution. Moreover, this grid point also illustrates that clearly distinct extreme-summer substructures can occur at the same grid point. While the extreme summer of 2003 exhibited particularly large anomalies in the coldest and the hottest third (${\mathit{\text{SF}}}_{\mathrm{cold},\mathrm{2003}}^{\mathit{\text{ERAI}}}=\mathrm{34}$ %, ${\mathit{\text{SF}}}_{\mathrm{middle},\mathrm{2003}}^{\mathit{\text{ERAI}}}=\mathrm{28}$ % and ${\mathit{\text{SF}}}_{\mathrm{hot},\mathrm{2003}}^{\mathit{\text{ERAI}}}=\mathrm{38}$ %), the contribution from the coldest third to the extreme summer 1995 was negative, and the middle and top third were responsible for the entire seasonal anomaly (${\mathit{\text{SF}}}_{\mathrm{cold},\mathrm{1995}}^{\mathit{\text{ERAI}}}=-\mathrm{15}$ %, ${\mathit{\text{SF}}}_{\mathrm{middle},\mathrm{1995}}^{\mathit{\text{ERAI}}}=\mathrm{49}$ % and ${\mathit{\text{SF}}}_{\mathrm{hot},\mathrm{1995}}^{\mathit{\text{ERAI}}}=\mathrm{66}$ %; Fig. 3d).

Finally, the grid point 58^∘ N, 35^∘ E, in western Russia (Fig. 3e) illustrates that, occasionally, the temperature variability during individual seasons can be fundamentally different from all other seasons at a particular grid point. Such a “regime shift” could be observed during the extreme summer of 2010, which was characterised by ${\mathit{\text{RDA}}}_{d,\mathrm{2010}}^{\mathit{\text{ERAI}}}$ values in excess of 4 K for ranks ∼40–92 (${\mathit{\text{SF}}}_{\mathrm{cold},\mathrm{2010}}^{\mathit{\text{ERAI}}}=\mathrm{1}$ %, ${\mathit{\text{SF}}}_{\mathrm{middle},\mathrm{2010}}^{\mathit{\text{ERAI}}}=\mathrm{46}$ % and ${\mathit{\text{SF}}}_{\mathrm{hot},\mathrm{2010}}^{\mathit{\text{ERAI}}}=\mathrm{53}$ %). For these ranks, the ${\mathit{\text{RDA}}}_{d,\mathrm{2010}}^{\mathit{\text{ERAI}}}$ values were almost twice as large as for the second hottest summer in these ranks (1981). The truly exceptional nature of the 2010 summer at 58^∘ N, 35^∘ E (e.g. Barriopedro et al. 2011; Fig. 3e), becomes even more evident when comparing its ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ values with those of the CESM extreme summers at the same grid points (Fig. 4e). For some ranks, none of the 700 CESM JJA seasons reach ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{CESM}}}$ values of comparable magnitude to those observed during the 2010 summer at this grid point. Some implications of this finding will be discussed in Sect. 4.

In summary, the mean extreme-summer substructure at these four grid points is qualitatively remarkably similar for the 5 hottest ERA-Interim summers and the 35 hottest CESM summers. On the one hand, this similarity implies that the rank day anomaly patterns presented in Fig. 3b–e are not artefacts of the rather short ERA-Interim period but must instead result from physical processes that shape the local extreme-summer substructure. On the other hand, these similarities suggest that the CESM is able to correctly capture the processes that generate the distinct extreme-summer substructures at these example grid points. We next compare the mean ERA-Interim and mean CESM extreme-summer substructures at all grid points in the Northern Hemisphere by considering the spatial patterns of ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$, ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$, ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$.

3.3 Spatial variability in ERA-Interim and CESM extreme-summer substructure

If extreme summers resulted from a uniform shift in the entire T2m distribution, all three thirds of the T2m distribution would contribute equally (i.e. 33 %) to XA^ERAI. However, the ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ field (Fig. 5a) reveals a complex pattern of coherent regions with increased (>33 %) or decreased (<33 %) contributions from the hottest third of extreme-summer days to XA^ERAI. Land areas where particularly large ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ values are found include the central US; the UK; parts of northeastern Europe, India and southeastern Asia; and the southern Sahel region (Fig. 5a). In some of these areas, ${\mathit{\text{SF}}}_{\mathrm{hot},k}^{\mathit{\text{ERAI}}}$ exceeded ${\mathit{\text{SF}}}_{\mathrm{middle},k}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{SF}}}_{\mathrm{cold},k}^{\mathit{\text{ERAI}}}$ during at least four out of the five ERA-Interim extreme summers (stippling in Fig. 5a). In these regions, at least four out of the five extreme summers thus exhibited a similar substructure. However, it is important to bear in mind that in other regions the substructure of individual extreme seasons (i.e. SF_cold,k, SF_middle,k and SF_hot,k) may differ from the mean extreme season substructure characterised by XF_cold, XF_middle and XF_hot. Furthermore, also in parts of the northern North Pacific and northern North Atlantic, ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ is substantially increased and reaches up to 60 %. In many regions, however, ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ is less than 33 %, indicating that in these regions, extreme summers do not arise primarily from the hottest 30 d of the summer being hotter than they are climatologically.

Figure 5Spatial variability in the extreme-summer substructure in ERA-Interim and CESM. Panels (a) and (b) depict ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$, respectively, while ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$ are shown in panels (c) and (d). Stippled areas in all panels indicate grid points at which the same third of the distribution contributes the largest fraction of all thirds to at least 80 % of the extreme summers (i.e. similar substructure in at least 80 % of the extreme summers). Black crosses as in Fig. 3a.

In fact, in many regions it is the contribution to XA^ERAI from the coldest third of the summer (${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$) that is substantially increased (Fig. 5c), for example in the southwestern US, the northern Sahel region, Pakistan and parts of Greenland. Moreover, increased ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ values are also found in the southern North Pacific and the southern North Atlantic as well as over the Arctic Ocean (Fig. 5c). Overall, Fig. 5c clearly demonstrates that the coldest third of all summer days contributes a substantial fraction to XA^ERAI in most regions (more than 25 % over 83 % of the Northern Hemisphere land area in ERAI). In fact, in 46 % of the Northern Hemisphere land area, ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ exceeds ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$; i.e. the coldest third of extreme summers contributes more to XA^ERAI than the hottest third. Consequently, in these regions the mechanisms that suppress unusually cool summer days must be considered when assessing the physical causes of extremely hot summers.

Comparing these results, which are derived from ERAI with results based on CESM, i.e. ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ (Fig. 5a, b) as well as ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$ (Fig. 5c, d), unravels strikingly similar patterns in many regions. For example, both data sets agree (even quantitatively) that extreme summers in India and southeastern Asia come about primarily by the hottest summer days being hotter than they are climatologically, while the coldest third of extreme-summer days only contributes a marginal fraction to the respective XA. Also in the western and central US, XF_cold and XF_hot agree very well between the two data sets, with the cool summer days contributing an overproportionately large fraction to XA in the western US and the hot summer days in the central US. Further areas of remarkable agreement between ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$ (Fig. 5c, d) are the high Arctic and the northern Sahel region. Moreover, in 49 % of the Northern Hemisphere land area, ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$ exceeds ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$, which compares well with the 46 % of the land area in which ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ exceeds ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$. Figure 5 thus clearly reveals that the CESM reproduces many features of the observed extreme-summer substructure and its variability in space to a remarkable degree.

However, there are also some areas of notable differences between ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ as well as ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$. For example over Greenland, Saudi Arabia and the northern North Atlantic, there are substantial differences between ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$ (Fig. 5c, d). Moreover, over the northern North Pacific as well as the high Arctic, the ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ patterns agree only qualitatively but not quantitatively (Fig. 5a, b). It is important to note, though, that some differences in the XF_cold and XF_hot fields for the two data sets are expected due to the different sample sizes, even if the model was perfect. In the remainder of this paper we aim to explain statistical and physical reasons behind selected aspects of the spatial variability in XF_cold and XF_hot.

3.4 A statistical explanation for the observed extreme-summer substructures

Figures 3b and c and 4b and c clearly illustrate that, at the selected grid points in India (21^∘ N, 81^∘ E) and in the US (39^∘ N, 116^∘ W), some rank days are climatologically much more variable than others. Importantly, this is the case not just for extreme summers but is rather a climatological characteristic of the local temperature variability. For example, at 21^∘ N, 81^∘ E, the hottest 30 d of the summer are much more variable than the colder days. The 5th- to 95th-percentile range of the ${\mathit{\text{RDA}}}_{\mathrm{80},k}^{\mathit{\text{CESM}}}$ values is roughly 4 times larger than that of the ${\mathit{\text{RDA}}}_{\mathrm{10},k}^{\mathit{\text{CESM}}}$ values (Fig. 4b). At 39^∘ N, 116^∘ W, the largest rank day variability is found for lower ranks, and the 5th- to 95th-percentile range of the ${\mathit{\text{RDA}}}_{\mathrm{80},k}^{\mathit{\text{CESM}}}$ values is roughly 2 times smaller than the same percentile range of the ${\mathit{\text{RDA}}}_{\mathrm{10},k}^{\mathit{\text{CESM}}}$ values (Fig. 4c). Similar ratios are found when comparing the spread of ${\mathit{\text{RDA}}}_{\mathrm{80},k}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{RDA}}}_{\mathrm{10},k}^{\mathit{\text{ERAI}}}$ for these two grid points (Fig. 3b, c). Moreover, at both grid points extreme summers occur when the most variable rank days are particularly hot (Figs. 3b and c and 4b and c). Hence, from a statistical point of view, the extreme-summer substructure at these two particular grid points appears to be largely determined by the local “rank day variability pattern” – that is, the contributions to XA from the distinct rank days during extreme summers depend on how variable the respective values T_d,k are climatologically.

We next assess whether the local rank day variability pattern also explains the extreme-summer substructure at other Northern Hemisphere grid points. To do so, we consider the variance (V) of the RDA_d,k values of all ranks and all JJA seasons at a particular grid point:

Here we used the fact that the mean of the RDA_d,k values is by construction equal to zero, and thus their variance reduces to the average of the squared RDA_d,k values of all d and all k. The contribution from the coldest third of summer days to V is then

and the contributions from the middle and hottest third of the summer days to V are computed analogously.

Figure 6The variance of ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{ERAI}}}$ and its contributions from the coldest and hottest third of summer days. Panel (a) depicts V^ERAI, and panels (b) and (c) show ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{VF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$, respectively. Green contours in panels (b) and (c) depict C^ERAI gradient magnitudes of 6 and 12 K 10⁻⁶ m⁻¹. The C^ERAI gradient magnitudes have been computed as first-order central differences and are only plotted over oceans. Black crosses as in Fig. 3a.

Figure 7The variance of ${\mathit{\text{RDA}}}_{d,k}^{\mathit{\text{CESM}}}$ and its contributions from the coldest and hottest third of summer days. Panel (a) depicts V^CESM, and panels (b) and (c) show ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ and ${\mathit{\text{VF}}}_{\mathrm{cold}}^{\mathit{\text{CESM}}}$, respectively. Black crosses as in Fig. 3a.

The fields of V^ERAI and V^CESM (Figs. 6a, 7a) resemble the XA^ERAI and XA^CESM fields (Figs. 3a, 4a), as large rank day anomalies are a prerequisite for large seasonal T2m anomalies. Furthermore, comparing ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ (Figs. 5a and 6b) clearly reveals that wherever the contribution from the hottest third of the summer days to XA^ERAI is increased (${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}>\mathrm{33}$ %), the rank day variability in the hottest third (quantified by ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$) contributes overproportionately to V^ERAI. Figures 5c and 6c illustrate that the same relationship also holds for ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{VF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$: regions where milder-than-normal cool summer days contribute overproportionately to XA^ERAI (i.e. ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}>\mathrm{33}$ %) exhibit increased ${\mathit{\text{VF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ values. Figures 5b and d and 7b and c confirm this finding also for the CESM data. We thus conclude that in both data sets, the extreme-summer substructure is largely determined by the local rank day variability pattern.

Furthermore, comparing the patterns of ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ (Figs. 6b, 7b) reveals agreement in the same regions where also the patterns of ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ (Fig. 5a, b) agree, and, conversely, disagreement between ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ also results in disagreement between ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$. For example, the ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ fields (and the ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ fields) are almost identical in India and southeastern Asia, the northern Sahel, the western US or eastern Europe (cf. Fig. 6b with Fig. 7b and Fig. 5a with b). Over Saudi Arabia or the northern North Atlantic, however, the patterns of ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{VF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$ (and of ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{ERAI}}}$ and ${\mathit{\text{XF}}}_{\mathrm{hot}}^{\mathit{\text{CESM}}}$) do not agree particularly well. In summary, while the CESM correctly reproduces the local rank day variability pattern in most regions, differences in the local rank day variability patterns between the two data sets also lead to differences in the extreme-summer substructures.

It is interesting to compare the VF_cold and VF_hot patterns presented in Figs. 6 and 7 with the skewness of the local daily temperature distributions, which has been studied extensively in the past (Donat and Alexander, 2012; Garfinkel and Harnik, 2017; Linz et al., 2018; Loikith et al., 2018; Loikith and Neelin, 2015; Ruff and Neelin, 2012). The upper tail of a positively skewed JJA T2m distribution, for example, is longer than the lower tail, which is the case if the hottest summer days are more variable than the coldest summer days (cf. Figs. 5b and c with Fig. S1 in the Supplement). Hence, explanations of distinct skewness in daily T2m distributions also help to understand differences in the rank day variability patterns and, subsequently, extreme-summer substructures. Garfinkel and Harnik (2017) showed that the winter low-level temperature distributions are positively skewed on the cold side of the Northern Hemisphere storm tracks, primarily because there the magnitude of warm-air advection exceeds that of cold-air advection. And, vice versa, the winter low-level temperature distributions are negatively skewed on the warm side of the Northern Hemisphere storm tracks, where the magnitude of cold-air advection exceeds that of warm-air advection. Consistent with their results, Figs. 6 and 7 depict more variable hot summer days to the north and more variable cold summer days to the south of the Northern Hemisphere storm tracks, where the horizontal gradients of T2m are particularly large (see in particular green contours in Fig. 6b, c).

While this argument explains differences in the rank day variability and the extreme-summer substructures in regions of strong surface temperature gradients, Figs. 5–7 also reveal numerous rather small-scale features that do not necessarily occur in regions of strong surface temperature gradients. We therefore next analyse the extreme-summer substructure and its causes in three example regions in more detail. Due to the similarity between the ERA-Interim and CESM extreme-summer substructures, we restrict this analysis to ERA-Interim data (except where mentioned otherwise).

3.5 (Examples of) physical causes of extreme-summer substructures

A particularly striking feature of Fig. 5 is the large contribution from the hottest third of the summer days to XA^ERAI in India, illustrated exemplarily for the grid point at 21^∘ N, 81^∘ E, in Fig. 3b. The general temperature evolution in JJA (i.e. considering all JJA seasons) at this grid point follows a particular sub-seasonal pattern (Fig. 8a). In early June, ERA-Interim T2m values are highly variable and range from 27 to almost 40^∘ C, with a mean of 35^∘ C on 1 June. Throughout June and the first half of July the climatological T2m drops to approximately 26^∘ C and remains at this level until the end of August. Moreover, during that period, the variability in T2m is much smaller than in early June. The extreme summers exhibit comparatively high temperatures primarily in June, while in July and August their T2m evolution does not differ substantially from other JJA seasons (Fig. 8a). The drop of T2m in June is associated with the onset of the Indian summer monsoon (Fig. 8b; e.g. Slingo, 1999). During most JJA seasons, precipitation starts to fall already during the first half of June. However, the extreme summers each featured very little precipitation for at least the first 20 d of June, which suggests that extreme summers at this grid point occur when there is an unusually late onset of the Indian summer monsoon at this particular location. Moreover, the rank day variability pattern at 21^∘ N, 81^∘ E, is easily understood from Fig. 8: the hottest days of the season mostly occur in June and are associated with dry conditions. The onset date of the monsoon determines how many dry (and thus very hot) days occur in a JJA season; i.e. an early onset of the Indian monsoon suppresses a large number of very hot days and a late onset increases this number, which leads to the large temperature variability seen in the warmest 30 d of the JJA season.

Figure 8The JJA temperature and precipitation evolution at 21^∘ N, 81^∘ E. Panels (a) and (b) depict non-detrended ERA-Interim T2m and accumulated precipitation at 21^∘ N, 81^∘ E, for all JJA seasons, respectively. The extreme summers are highlighted in colours. The dashed black line in panel (a) depicts the climatological calendar day mean T2m at 21^∘ N, 81^∘ E.

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Figure 9Arctic sea ice and local summer temperature variability. (a) ${\mathit{\text{XF}}}_{\mathrm{cold}}^{\mathit{\text{ERAI}}}$ (shading; only 70^∘ N–90^∘ N is shown) and mean 1979–2018 JJA ERA-Interim sea ice concentration (green contours indicate sea ice concentrations of 0.3, 0.5 and 0.7). (b) Empirical probability density function of non-detrended ERA-Interim T2m at 79^∘ N, 42^∘ E (red); 81^∘ N, 42^∘ E (grey); and 83^∘ N, 42^∘ E (blue). Crosses in panel (a) locate these three grid points.

A further noteworthy feature in Fig. 5 is the sharp boundary in the extreme-summer substructure around 75–80^∘ N, for example in the North Atlantic sector. North of this boundary, the coldest third of all extreme-summer days contributes up to 60 % to the extreme-summer anomaly (Fig. 5c, d). South of it, the contribution from the coldest third of extreme-summer days is much smaller. (Quantitatively, there is some disagreement between the CESM and ERAI extreme-summer substructures, but both data sets agree about the general pattern.) This sharp boundary in the extreme-summer substructure is co-located with the climatological sea ice edge in JJA (Fig. 9a). Examining the JJA T2m distributions at three grid points across this boundary (83^∘ N, 42^∘ W; 81^∘ N, 42^∘ W; and 79^∘ N, 42^∘ W) reveals that for T2m below −1^∘ C, their probability density functions (pdf's) of the daily T2m values are almost identical, which is not surprising due to their close spatial proximity. However, large differences in the three pdf's are found for T2m at about 0^∘ C and above. At 83^∘ N, i.e. north of the climatological sea ice edge (Fig. 9a), the pdf exhibits a very short upper tail with very little probability density exceeding +2^∘ C (i.e. the pdf is strongly negatively skewed), while at 79^∘ N (i.e. south of the climatological sea ice edge) the upper tail is much more variable. The geographical co-location of this extreme-summer substructure boundary and of the climatological sea ice edge is striking and suggests that the contrasting substructures arise because the sea ice buffers “warm” temperatures at 0^∘ C – that is, air with T2m >0^∘ C is cooled down to close to 0^∘ C by the induced sea ice melting. The same effect has also been shown to shorten the upper tail of the surface temperature pdf over snow-covered areas (Loikith et al., 2018).

As a third example, we return to the grid point in Nevada, US (at 39^∘ N, 116^∘ W), where the rank day variability is largest for the cold summer days and extreme summers occur when the coldest 30 d exhibit mostly large positive rank day anomalies (Figs. 3c and 4c). Thus, at this grid point, milder-than-normal coldest days of the summer (or, equivalently, suppressed cool summer days) are a key ingredient for extreme summers. We therefore briefly explore why, at this grid point, the coldest summer days during extreme summers are warmer than normal.

Figure 10(a) T2m difference between the 100 climatologically coldest JJA days and the 100 coldest extreme-summer days (shading). Contours depict the composite PV field at 335 K (contours of 2, 3.5 and 5 PVU) for the 100 climatologically coldest JJA days (blue) and for the 100 coldest extreme-summer days (red). The yellow cross indicates 39^∘ N, 116^∘ W. Panels (b) and (c) depict composite Hovmöller diagrams of the anomalous 250 hPa meridional wind, averaged between 35 and 65^∘ N, and temporally centred on the 100 climatologically coldest JJA days (b) and on the 100 coldest extreme-summer days (c). Meridional wind anomalies are calculated relative to the 1979–2018 mean JJA meridional wind. The vertical line in panel (b) and (c) indicates 116^∘ W.

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We first investigate what makes the climatologically coldest summer days at 39^∘ N, 116^∘ W, particularly cold and then contrast them with the coldest summer days during extreme summers at 39^∘ N, 116^∘ W. A composite analysis of the upper-level flow during the 100 climatologically coldest ERA-Interim days of all 1979–2018 summers unravels a characteristic upper-level flow pattern: a highly amplified Rossby wave pattern over the eastern North Pacific and North America, with a breaking synoptic-scale trough covering 39^∘ N, 116^∘ W (Fig. 10a). The breaking Rossby wave causing the trough is part of a synoptic-scale and transient wave packet (Fig. 10b) which has just the right phasing such that the trough axis crosses 39^∘ N, 116^∘ W, when the amplitude of the trough is largest (Fig. 10b). This type of relatively small-scale trough, shown here with contours of potential vorticity on an isentrope in the upper troposphere (Fig. 10a), is relatively slow-moving (Fig. 10b), such that the induced northwesterly low-level flow along its western flank can lead to strong and persistent cold-air advection to the western US. Additionally, the low-level flow induced by the trough impinges on the topography at the US West Coast. Consequently, low-level air masses that are advected into the western US are most likely forced to ascend, which leads to adiabatic cooling of these already cool air masses and finally results in the climatologically coldest summer days at 39^∘ N, 116^∘ W.

The composites for the 100 coldest days during extreme summers, in contrast, do not reveal such a wave pattern (Fig. 10a and c). This indicates that the flow pattern characteristic of the climatologically coldest days at this grid point, i.e. the Rossby wave breaking and trough formation with the phasing discussed above, simply did not occur very often during extreme summers. Furthermore, a synoptic analysis of these 100 coldest extreme-summer days (not shown) reveals that the associated upper-level flow configurations are rather variable, some featuring troughs while others even exhibited low-amplitude ridges, resulting in the rather zonal composite upper-level flow apparent in Fig. 10a and c.

The reason why such highly amplified troughs with the right phasing did not occur during extreme summers at 39^∘ N, 116^∘ W, is currently unclear and at the same time challenging to assess. Possibly, the exact longitude where the synoptic-scale waves have been triggered (Röthlisberger et al., 2018) as well as the strength and longitudinal extent of the North Pacific jet, which modulates the waves' downstream propagation and breaking behaviour (e.g. Drouard et al. 2015), might have played a role. However, both the jet strength and the characteristics of the transient waves propagating along the jet are strongly modulated by lower-frequency processes such as the Madden–Julian Oscillation (Moore et al., 2010) and the El Niño–Southern Oscillation (Drouard et al., 2015; Shapiro et al., 2001). This example thus illustrates that a seamless approach, combining processes on different timescales, is most likely required to fully reveal the physical causes of extreme summers.