## 2 Materials and Methods
### Heat Waves
As an example, the Tmax_c value for 16 March is calculated considering all days from 1 March until 30 March for the 20 years. In this study, HW are identified for the period starting on 1 May and ending on 31 October (extended summer), since these are the warmest months. In addition, all HW are considered if they have days within this period even if they begin or end outside it. All HW are identified using Tmax_c calculated for HIST of the respective model, since the objective of this study is to evaluate HW changes for MED and LONG relative to HIST. Only then may this information be used in impact and adaptation studies. Furthermore, the intensity, duration, and time of the year of HW should also be considered because these features can be relevant to evaluate their impact [7]. Other derived quantities relevant to quantify HW heat stress are calculated. In this study, we consider the following measures related to HW:  
* NWAVE--Number of HW. * NDAYS--Number of HW days. * DUR--Duration (days) of a HW. * Tmax_ave (\({}^{\circ}\)C)--Tmax averaged over all days of a HW. * Tmin_ave (\({}^{\circ}\)C)--Tmin averaged over all days of a HW. * Tmax_max (\({}^{\circ}\)C)--Maximum Tmax of all days of a HW. * INT (intensity) (\({}^{\circ}\)C)--Tmax minus Tmax_c (i.e., excess of Tmax relative to Tmax_c) averaged over all days of a HW. Note that the same INT may represent distinct conditions. For example, INT equal to 3 \({}^{\circ}\)C may be present in a HW with Tmax_ave of 35 \({}^{\circ}\)C and Tmax_c of 32 \({}^{\circ}\)C or in a HW with Tmax_ave of 30 \({}^{\circ}\)C and Tmax_c of 27 \({}^{\circ}\)C.  
* \(\mathrm{Tmax\_sum}\)(\({}^{\circ}\)C-day)--Sum of \(\mathrm{Tmax}\) of all days of a HW. This variable quantifies, in absolute terms, the accumulated temperature during a HW that is relevant to heat stress and is similar to growing degree days used in phenology. For example, a \(\mathrm{Tmax\_sum}\) of 300 \({}^{\circ}\)C-day may represent a HW of 10 days with \(\mathrm{Tmax}\) of 30 \({}^{\circ}\)C in each day or a HW of 8 days with \(\mathrm{Tmax}\) of 37.5 \({}^{\circ}\)C in each day. * RF (Recovery Factor) (\({}^{\circ}\)C)--Average daily temperature range during a HW. RF represents the average relief people may experience considering night time and daytime temperatures. RF is relevant since it incorporates minimum temperature, which contributes to assess the dissipation of heat overnight after a HW day [55; 56]. Similar heat excess indices have been derived by [PERSON] and [PERSON] [57]. Again, RF may represent distinct conditions. For example, a RF of 10 \({}^{\circ}\)C may be associated to a HW with \(\mathrm{Tmax\_ave}\) of 35 \({}^{\circ}\)C and \(\mathrm{Tmin\_ave}\) of 25 \({}^{\circ}\)C or a HW with \(\mathrm{Tmax\_ave}\) of 30 \({}^{\circ}\)C and \(\mathrm{Tmin\_ave}\) of 20 \({}^{\circ}\)C.  
Some of these measures are relative (INT and RF), while others are absolute (i.e., NWAVES, NDAYS, \(\mathrm{Tmax\_ave}\), \(\mathrm{Tmin\_ave}\), \(\mathrm{Tmax\_max}\), and \(\mathrm{Tmax\_sum}\)). They are complementary since they characterize various aspects of HW relevant for heat stress. We use two measures to quantify the intensity of a HW. INT is a measure of HW intensity relative to the respective threshold (\(\mathrm{Tmax\_c}\)). \(\mathrm{Tmax\_sum}\) is an absolute measure of HW intensity. \(\mathrm{Tmax\_sum}\) differentiates between 2 HW with different thresholds, for example, one in May and the other in August. DUR times INT (INT is defined relative to \(\mathrm{Tmax\_c}\)) gives the excess accumulated temperature difference (\(\mathrm{Tmax\_-Tmax\_c}\)). However, this product (DUR*INT) does not distinguish 2 HWs, one in May (for example, with \(\mathrm{Tmax\_c}\) of 26 \({}^{\circ}\)C) and the other in August (for example with \(\mathrm{Tmax\_c}\) of 30 \({}^{\circ}\)C), whereas the \(\mathrm{Tmax\_sum}\) for both HW displays this difference.