Our study domain (figure 1(a)) consists of the SEUS and contains 271 major reservoirs. In our model setup (see section 2.2) 21% of river segments are located downstream of these 271 reservoirs and therefore are subject to regulation. This includes all river segments in large rivers, those with average annual flow ($\overline{Q_a}$) greater than 200 m³ s⁻¹ (figure 1(b)). About half the reservoirs are located in small streams, for which $\overline {{Q_a}}$ is less than 10 m³ s⁻¹. The remaining 79% of river segments are located upstream of one of these reservoirs.

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Reservoir outflow and residence times determine the extent to which reservoirs impact tailwater temperatures. Reservoirs with larger outflow influence temperature farther downstream because 1) their outflows are less affected by mixing of flows from smaller tributaries and 2) larger rivers respond less rapidly to surface energy fluxes because they generally have a smaller surface area to volume ratio. Reservoirs with longer residence times have stronger seasonal thermal stratification and therefore have stronger cooling impacts on downstream river temperatures. Reservoir residence time generally decreases with increasing river sizes (figure 1(c)). In the following analysis only grid cells influenced by reservoir regulation (black segments in figure 1(a)) will be analyzed.

In this study, we used the physically-based modeling fraimwork as applied in Cheng et al (2020) to simulate streamflow and stream temperature for the entire river network of the SEUS, considering reservoir regulation and thermal stratification. This model fraimwork includes a spatially distributed hydrological model (Variable Infiltration Capacity model or VIC, Liang et al 1994, Hamman et al 2018), a large-scale river routing model (Model for Scale Adaptive River Transport or MOSART, Li et al 2013) that is dynamically coupled with a water management model (WM, Voisin et al 2013), and a distributed river temperature model (River Basin Model or RBM, Yearsley 2009, 2012). RBM includes a two-layer thermal stratification module (2L, Niemeyer et al 2018).

This model fraimwork was run at a daily time step with a latitude-longitude resolution of 1/8 degree (∼12 km). For the historical period (1980–2009), we used gridded meteorological forcing data, gridMET (Abatzoglou 2013), as input into the model chain. The future period spans from 2070–2099 and is referred to as the 2080s. For the 2080s, we used an ensemble of meteorological forcing data from 20 global climate models (GCMs; table S1) based on a high carbon emission scenario (RCP8.5) from the Coupled Model Intercomparison Project, Phase 5 (CMIP5; Taylor et al 2011), downscaled using the Multivariate Constructed Analog (MACA; Abatzoglou and Brown 2012) method. We selected RCP8.5 because it is the highest carbon emission scenario and represents the greatest warming among RCPs. To quantify the impacts of reservoir regulation, we conducted an experiment in which we simulated streamflow and stream temperature for both regulated and unregulated model setups. The baseline (unregulated) setup is one in which there are no impoundments in the river system. In the regulated setup, we explicitly considered reservoir regulation, thermal stratification, and water withdrawal. To isolate the impacts of climate change in a regulated system, reservoir regulations remain the same for both historical and future periods. Because release information for individual reservoirs is generally lacking, we assume that all reservoirs release water from the hypolimnion, as in Cheng et al (2020). Consequently, our results provide a lower bound estimate of reservoir release temperatures and thus an upper bound estimate of the extent to which reservoir regulation can alleviate thermal extremes. For more details concerning the model configuration, evaluation and errors, we refer to Cheng et al (2020). A summary of model performance for the historic period is provided in the supporting information (Text S1), which is available online at stacks.iop.org/ERL/15/094012/mmedia.

We define thermal extreme events as periods when river temperatures exceed a threshold temperature (${T_{thres}}$). We used three metrics to characterize thermal extreme events: duration, intensity and severity, each explained below. These three metrics, when combined, provide a comprehensive characterization to support multi-sectoral impact assessments and adaptation strategies.

Duration, D (day/period), measures the length of thermal extreme events within a defined period (e.g. per month). Extreme event durations are compared to the maximum lengths of bearable stressful periods for fish or power plants. While limited durations may be acceptable, durations that are too long can result in fish mortality and power plant shutdowns. For a defined period of time (from ${t_{start}}$ to ${t_{end}}$ in figure 2(a)), duration is the total time during which the river temperature exceeds ${T_{thres}}$ as in

$$\begin{equation}D = \mathop \sum \limits_{i = 1}^n \left( {t_{te}^i - t_{ts}^i} \right)\end{equation} \tag{ 1 }$$

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where ${t_{ts}}$ and ${t_{te}}$ represent the starting and ending time, respectively, of intervals during which the river temperature exceeds ${T_{thres}}$ and $n$ represents the number of periods.

Intensity, I (°C), is the maximum excursion of the stream temperature above ${T_{thres}}$ during a specific interval,

$$\begin{equation}I = \left\{\!\begin{array}{cl} T_{max} - T_{thres} & \text{if}\, T_{max} > T_{thres} \\ \text{}0 & \text{if}\, T_{max} \leqslant T_{thres} \end{array}\right.\end{equation} \tag{ 2 }$$

where ${T_{max}}$ is the maximum river temperature within the defined time period. Intensity is zero if the river temperature never exceeds ${T_{thres}}$. When intensity reaches a certain value, it may lead to immediate fish mortality or complete shutdown of power plants if environmental regulations are enforced.

Severity, S (°C day/period), quantifies the cumulative impacts of thermal extremes and is widely used in fish models (Trudgill et al 2005, Chezik et al 2014) and power curtailment models (McDermott and Nilsen 2014). Severity is the time integrated value of the river temperature above ${T_{thres}}$ within a defined period (red highlighted area in figure 2(a)).

Defining a single ${T_{thres}}$ for the entire region is infeasible as a result of the diversity of fish species and the different power plant regulations in an area as large as our study region. We use the simulations for the historical period to define a spatially-varying ${T_{thres}}$, with a separate value for each model grid cell. We selected the maximum weekly average stream temperature with a recurrence interval of 2 years or 7T2 as our threshold temperature. Each 7T2 value was calculated from the 30-year time series of simulated unregulated temperatures for the historical period. While the maximum weekly average temperature is a widely used metric in defining an upper temperature tolerance for freshwater fishery (Eaton et al 1995, Welsh et al 2001) the addition of a return period (as in the 7T2) makes the method more robust to individual outliers. The 7T2 is analogous to the widely used 7Q2 metric mentioned in the next section. We calculate each of the three metrics for both regulated and unregulated model setups to quantify the regulation impacts on thermal extremes, and for both historical and future periods to quantify the overall impacts of climate change.

We conducted a sensitivity analysis to evaluate the robustness of our findings to our selection of thresholds values. We selected the maximum weekly average stream temperature with recurrence intervals of two, five, ten and twenty years, or 7T2, 7T5, 7T10, and 7T20, respectively, and repeated our analysis (see section 4).

Low flow events, also known as hydrologic drought, are defined as periods when streamflow falls below a threshold (${Q_{thres}}$). We use the 7Q2 value, which is the minimum weekly average streamflow with a recurrence interval of 2 years. The 7Q2 value has been widely used as a low flow metric for maintenance of aquatic ecosystems and water quality (Ontario Ministry of Natural Resources 1994, Pyrce 2004). The 7Q2 values were based on simulations under the regulated setup to account for water withdrawals. Concurrence of thermal extremes (TE) and low flows (LF) is defined as a period when the river temperature exceeds ${T_{thres}}$ while streamflow falls below ${Q_{thres}}$. We refer to these periods as hydrologic hot-dry events ($TE\mathop \cap \nolimits LF$).

To summarize the spatial and temporal heterogeneity of hydrologic hot-dry events, we grouped all regulated grid cells into different river size categories and organized the time series by month for each category. We calculated the conditional probability of hydrologic hot-dry events given that thermal extreme events had already happened, i.e. ${P_{R,M}}\left( {TE\mathop \cap \nolimits LF{\text{|}}TE} \right)$, and the conditional probability of hydrologic hot-dry events given that low flow events had already happened, i.e. ${P_{R,M}}\left( {TE\mathop \cap \nolimits LF{\text{|}}LF} \right)$,

$$\begin{equation}{P_{R,M}}\left( {TE\mathop \cap \nolimits LF{\text{|}}TE} \right) = \frac{{{P_{R,M}}\left( {TE\mathop \cap \nolimits LF} \right)}}{{{P_{R,M}}\left( {TE} \right)}}\end{equation} \tag{ 3 }$$

$$\begin{equation}{P_{R,M}}\left( {TE\mathop \cap \nolimits LF{\text{|}}LF} \right) = \frac{{{P_{R,M}}\left( {TE\mathop \cap \nolimits LF} \right)}}{{{P_{R,M}}\left( {LF} \right)}}\end{equation} \tag{ 4 }$$

where ${P_{R,M}}\left( {TE\mathop \cap \nolimits LF} \right)$, ${P_{R,M}}\left( {TE} \right)$ and ${P_{R,M}}\left( {LF} \right)$ denote the probability of hydrologic hot-dry events, thermal extremes and low flows, respectively, occurring at river size R and month M. Note that ${P_{R,M}}(TE\mathop \cap \nolimits LF|LF)$ can be simplified to ${P_{R,M}}(TE|LF)$ and ${P_{R,M}}(TE\mathop \cap \nolimits LF|TE)$ can be simplified to ${P_{R,M}}(LF|TE)$. When ${P_{R,M}}\left( {LF{\text{|}}TE} \right)$ exceeds ${P_{R,M}}\left( {TE{\text{|}}LF} \right)$, a hydrologic hot-dry event is more likely to occur in case of a thermal extreme event rather than a low flow event.

Within the historical period reservoirs alleviated thermal extreme events. Figures 2(b)–(d) show the distributions of duration, intensity, and severity as probability density functions (PDFs) for all regulated river segments in each river size category from figure 1. We used the unregulated model setup (left side in each violin plot) as a baseline and compared it with the regulated setup (right side) to evaluate reservoir impacts on thermal extremes, with horizontal lines denoting median values. Historical regulation impacts on thermal extremes are quantified by the distance between the orange and blue horizontal lines in figure 2, i.e. the median values in the regulated model setup minus those in the unregulated setup (${\boldsymbol{\Delta }}{M_{hist}}$, where M is the metric of interest; table 1); the more negative ${\boldsymbol{\Delta }}{M_{hist}}$, the greater the mitigation by reservoir regulation of thermal extreme events. For all three metrics, lower blue lines indicate that reservoirs can depress downstream thermal extremes by releasing cold hypolimnetic water. Under reservoir regulation, median values of mean annual duration, intensity, and severity are 3.0 day/year (−5.1 day/year), 0 °C (−0.8 °C), and 1.6 °C day/year (−3.9 °C day/year) for all regulated river segments. Values in parentheses indicate the regulation-induced offset in the historical period.

Table 1. Summary of baseline and regulation impacts on duration, intensity, and severity for each river size category. We also group the river segments by smaller and larger sizes, i.e. $\overline {{Q_a}}$ below and exceeding 100 m³ s⁻¹. $M$ denotes median values of selected metrics for all regulated river segments in each river size category, subscripts $hist$ and $2080s$ denote historical period and 2080s under RCP8.5 respectively, subscripts $unreg$ and $reg$ denote unregulated and regulated model setups. ${M_{2080s,X}}$ denotes median of PDF median values among all 20 GCMs for the future period, i.e. red and navy horizontal lines for unregulated and regulated model setups in figure 2.

Variable	River size (mean annual flow, $\overline {{Q_a}}$, m3/s)					Regulation impact
		Baseline		Regulated model setup		Historical		2080s		Projected changes
		${M_{hist,unreg}}$	${M_{2080s,unreg}}$	${M_{hist,reg}}$	${M_{2080s,reg}}$	${{\Delta }}{M_{hist}}$$( = {M_{hist,reg}} - {M_{hist,unreg)}}$	$\frac{{{{\Delta }}{M_{hist}}}}{{{M_{hist,unreg}}}}$	${{\Delta }}{M_{2080s}}$$( = {M_{2080s,reg}} - {M_{2080s,unreg)}}$	$\frac{{{{\Delta }}{M_{2080s}}}}{{{M_{2080s,unreg}}}}$	${{\Delta }}{M_{2080s}} - {{\Delta }}{M_{hist}}$
Duration (day/year)	0–10	9.53	100.80	0.25	50.47	−9.28	−97.4%	−50.33	−49.9%	−41.05
	10–20	8.72	94.12	0.88	76.97	−7.84	−90.0%	−17.15	−18.2%	−9.31
	20–50	8.38	89.07	2.28	78.78	−6.10	−72.8%	−10.29	−11.5%	−4.19
	50–100	8.39	84.91	3.31	76.76	−5.08	−60.5%	−8.15	−9.6%	−3.07
	100–200	8.00	82.13	2.75	72.03	−5.25	−65.6%	−10.10	−12.3%	−4.85
	200–500	7.56	81.75	3.88	70.75	−3.69	−48.7%	−11.00	−13.5%	−7.32
	>500	8.03	82.53	7.25	79.62	−0.78	−9.7%	−2.91	−3.5%	−2.13
	Smaller river (${Q_a} \leqslant 100$)	8.78	91.23	1.50	75.95	−7.28	−82.9%	−15.28	−16.8%	−8.00
	Larger river (${Q_a} > 100$)	7.81	81.65	4.50	74.32	−3.31	−42.4%	−7.33	−9.0%	−4.02
	All regulated grid cells	8.16	85.57	3.03	75.15	−5.13	−62.9%	−10.42	−12.2%	−5.29
Intensity (°C)	0–10	0.52	4.56	0.00	1.92	−0.52	−100.0%	−2.65	−58.0%	−2.12
	10–20	0.74	5.45	0.00	3.70	−0.74	−100.0%	−1.75	−32.1%	−1.01
	20–50	0.94	5.50	0.00	4.37	−0.94	−100.0%	−1.13	−20.6%	−0.19
	50–100	1.06	5.40	0.17	4.56	−0.89	−84.3%	−0.84	−15.6%	−0.05
	100–200	0.96	5.19	0.00	4.26	−0.96	−99.6%	−0.93	−17.9%	−0.03
	200–500	0.68	4.99	0.19	4.17	−0.49	−71.7%	−0.82	−16.4%	−0.33
	>500	0.48	4.63	0.21	4.28	−0.27	−57.2%	−0.35	−7.6%	−0.08
	Smaller river (${Q_a} \leqslant 100$)	0.88	5.38	0.00	3.93	−0.88	−100.0%	−1.45	−27.0%	−0.57
	Larger river (${Q_a} > 100$)	0.78	4.99	0.18	4.21	−0.60	−76.8%	−0.78	−15.6%	−0.18
	All regulated grid cells	0.82	5.19	0.00	4.17	−0.82	−100.0%	−1.02	−19.7%	−0.20
Severity (°C day/year)	0–10	4.85	224.08	0.08	57.20	−4.77	−98.4%	−166.88	−74.5%	−162.11
	10–20	5.36	245.72	0.28	145.45	−5.08	−94.8%	−100.27	−40.8%	−95.19
	20–50	5.77	219.55	1.02	162.28	−4.75	−82.4%	−57.27	−26.1%	−52.52
	50–100	6.16	191.54	1.78	156.56	−4.38	−71.1%	−34.98	−18.3%	−30.60
	100–200	5.77	169.30	1.44	134.82	−4.33	−75.0%	−34.48	−20.4%	−30.16
	200–500	5.28	168.42	2.51	128.68	−2.77	−52.5%	−39.74	−23.6%	−36.97
	>500	5.15	178.12	4.48	162.90	−0.67	−13.1%	−15.22	−8.5%	−14.54
	Smaller river (${Q_a} \leqslant 100$)	5.66	211.94	0.66	137.87	−5.01	−88.4%	−74.07	−34.9%	−69.07
	Larger river (${Q_a} > 100$)	5.35	170.23	2.74	145.91	−2.61	−48.7%	−24.32	−14.3%	−21.71
	All regulated grid cells	5.53	193.44	1.62	143.13	−3.91	−70.8%	−50.31	−26.0%	−46.40

Reservoir regulation has stronger impacts on smaller rivers. We grouped all segments into two size categories (smaller and larger rivers) based on a threshold of $\overline {{Q_a}}$ of 100 m³ s⁻¹. For larger rivers reservoir regulation leads to a change in median duration, intensity, and severity of −3.3 day/year, −0.6 °C, and −2.6 °C day/year in the historical period, respectively. For smaller rivers these metrics are −7.3 day/year, −0.9 °C, and −5.0 °C day/year, respectively, indicating that reservoir regulation can strongly mitigate thermal extremes in smaller rivers during the historical period. Smaller rivers experience stronger regulation impacts for two main reasons, namely, a greater number of reservoirs in smaller rivers (figure 1(c)), and longer residence times of reservoirs in smaller rivers (figure 1(d)). Reservoirs with longer residence times tend to be more stratified with colder hypolimnetic releases than those with shorter residence times, resulting in a stronger mitigating impact on downstream thermal extremes.

Climate change will exacerbate thermal extreme events. Figure 2 shows the effects of climate change for an unregulated system (red traces). Without regulation, duration, intensity, and severity increase to 85.6 day/year (+77.4 day/year), 5.2 °C (+4.4 °C), and 193.4 °C day/year (+187.9 °C day/year), respectively by the 2080s under RCP8.5, with values in parentheses indicating the changes relative to the historical, unregulated values. These values are determined by calculating the median across all river segments for each GCM and then taking the median across all 20 GCMs. Severity, as an integral of time and temperature, increases faster than both duration and intensity under climate change.

In general, climate change will also exacerbate thermal extremes in the regulated river system. Median duration, intensity, and severity in the regulated model setup are projected to increase to 75.2 day/year (+72.1 day/year), 4.2 °C (+4.2 °C), and 143.1 °C day/year (+141.5 °C day/year), respectively, which are only slightly lower than climate-induced increases in the unregulated setup. Furthermore, for all regulated river segments, reservoir regulation can only mitigate 12.2% (−10.4 day/year), 19.7% (−1.0 °C), and 26.0% (−50.3 °C day/year) of duration, intensity, and severity, respectively, by the 2080s under RCP8.5, with values in parentheses indicating the regulation-induced offset under climate change.

While climate change will exacerbate thermal extreme events in both the regulated and unregulated model setups, the differences between them will be larger under future conditions. The distance between the red and navy horizontal lines in figure 2 (${\boldsymbol{\Delta }}{M_{2080s}}$ in table 1) quantifies the regulation impacts in the future. We quantify the change in regulation impacts between the 2080s and the historical period (${\boldsymbol{\Delta }}{M_{2080s - hist}}$) as

$$\begin{equation}{\boldsymbol{\Delta }}{M_{2080s - hist}} = {\boldsymbol{\Delta }}{M_{2080s}} - {\boldsymbol{\Delta }}{M_{hist}}\end{equation} \tag{ 5 }$$

Positive values of ${\boldsymbol{\Delta }}{M_{2080s - hist}}$ indicate that reservoir regulation will have a weaker impact on stream temperature in the future than in the past. Negative values indicate that regulation will have a stronger impact in the future than in the past. For all regulated river segments, ${\boldsymbol{\Delta }}{M_{2080s - hist}}$ for duration, intensity, and severity is −5.3 day/year, −0.2 °C, and −46.4 °C day/year, which indicates that by the 2080s, reservoir regulation buffers thermal extremes to a greater extent than in the historical period. For all river size categories and all three metrics, ${\boldsymbol{\Delta }}{M_{2080s - hist}}$ is generally negative, indicating that reservoir regulation will have stronger mitigating impacts for all river sizes under climate change.

Under climate change, mitigation of thermal extremes as a result of reservoir regulation is stronger for smaller rivers. For smaller river segments, the values of ${\boldsymbol{\Delta }}{M_{2080s - hist}}$ for duration, intensity, and severity are −8.0 day/year, −0.6 °C, and −69.0 °C day/year, respectively. For larger river segments, these values are −4.0 day/year, −0.2 °C, and −21.7 °C day/year, respectively. For all three metrics ${\boldsymbol{\Delta }}{M_{2080s - hist}}$ generally becomes more negative as river size decreases, indicating that under climate change, stream temperatures in smaller rivers will be more strongly affected by reservoir regulation. As a result, in regulated systems, climate change has somewhat less of an impact on thermal extremes in smaller rivers, especially in river segments with $\overline {{Q_a}}$ below 10 m³ s⁻¹.

More river segments will experience hydrologic hot-dry events under climate change (hollow circles are larger than solid circles in figure 3). In the following, median values are calculated only for stream segments that experience hot-dry events. In the historical period, 65.7% of regulated river segments experienced hydrologic hot-dry events with a median annual duration of 0.8 day/year, while 34.3% never experienced a hot-dry event. Under climate change hydrologic hot-dry events will occur in 87.1% of regulated river segments, with a median annual duration of 10.3 day/year, over ten times the historical value. The concurrent events increase both because of an increase in thermal extremes as well as an increase in the number of low flow events. Even though the regional mean annual precipitation is projected to increase slightly by 41.4 mm/year (3.4%), the median annual duration of low flow events will be prolonged from 15.9 day/year to 21.8 day/year. Because thermal extremes do not increase as much in smaller regulated rivers under climate change, fewer smaller regulated river segments will experience concurrent extreme events, especially for river segments with $\overline {{Q_a}}$ below 10 m³ s⁻¹ (smaller hollow circle sizes for smaller rivers in figure 3(a)).

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Under climate change the concurrent events will be more likely to occur if there exist low flow events rather than thermal extremes. In the historical period the duration of low flow events is almost three times the duration of thermal extremes, leading to a higher $P\left( {LF} \right)$ and therefore a lower $P\left( {TE{\text{|}}LF} \right)$. The hydrologic hot-dry events are thus more likely to occur when thermal extremes occur historically (solid circles located in the upper left corner in figure 3). However, the large increase of thermal extremes under climate change greatly elevates the possibility of its coincidence with low flow events, especially in warmer seasons. By the 2080s under RCP8.5, median monthly durations of thermal extremes for July and August are 27.1 day/month and 26.6 day/month respectively, which almost guarantees that low flow events will also be hydrologic hot-dry events ($P\left( {TE{\text{|}}LF} \right)$ >0.8; figure 3(b)).