F Hendrickx
email: frederic.hendrickx@edf.fr

E Sauquet
email: eric.sauquet@irstea.fr

Impact of warming climate on water management for the Ariège river basin (France)

Keywords: water management, climate change, France, hydropower, low flow, Garonne Mots clefs gestion de l'eau, changement climatique, France, hydro-électricité, étiage, Garonne

The French national project IMAGINE2030 aims to assess future water availability in the Garonne River basin (southwest France) by taking account of changes in both climate and water management in the 2030s.Within this project, two mountainous drainage basins located in the Pyrenees were examined to assess the specific impact of climate change on reservoir management. The Salat River basin at Roquefort, is considered as a proxy (representative of a natural basin), whereas the Ariège River at Foix is influenced by hydropower production in winter and by water releases to sustain low flows in summer. The Cequeau rainfall-runoff model, combined with a simplified model of reservoir management operations, was calibrated on present-day conditions and forced with climate projections derived from the IPCC AR4 report. The results show that a warming climate over the basins induces a decrease in mean annual runoff, a shift to earlier snow melting in mountainous areas and more severe low-flow conditions. The simulations show a decrease in electricity generation. Under two water management scenarios (one "business-as-usual" and the other incorporating an increased downstream water demand in compliance with requirements for increased minimum flow), simulations for the Ariège River basin suggest an earlier filling of the reservoir is necessary in winter to anticipate the increased release from reservoirs in summer to support minimum flow farther downstream..

Located in southwest France, the Garonne River basin (approx. area: 65 000 km²) is one of the five major French river basins. This basin is surrounded by mountains (the Pyrenees to the south and the Massif Central to the east) and by a wide alluvial plain on the northern side. River flow regimes observed within the basin [START_REF] Sauquet | Estimating mean monthly runoff at ungauged locations: an application to France[END_REF] are influenced by different climates ranging from mountainous conditions in the upper part of the basin to continental rainy conditions in the lowlands. In addition river flows are greatly altered by water management (abstraction, reservoirs, derivations, etc.).

The severe low flows recently observed during 2003, 2005 and 2006 have indicated the vulnerability of the systems for allocating water between the main users of this resource, whose interests may conflict. Competition within and between sectors is increasing, and water resources are already under heavy pressure. With economic development, the need and demand for water will probably undergo changes. In parallel, the European Water Framework Directive and the French policy on water management seek to ensure compliance with good ecological status before 2015, restricting water availability for water users.

Several studies have already investigated the impact of climate change on water resources in southwest France. On the basis of seven climate projections from seven different global climate models (GCMs) running with CO2 concentrations reaching near twice present-day concentrations by 2050, [START_REF] Caballero | Hydrological sensitivity of the Adour-Garonne River basin to climate change[END_REF] suggest a decline in low flow by 13%±11% for the Garonne River at Lamagistère from the present to the mid-21st century. Discharges in spring and autumn are also reduced, whereas discharges in winter are projected to increase for most of the climate projections and the 16 river basins under study. Impacts on low flows are moderate, since the effect of a warmer climate and drier summers penalising water resources is partially compensated by wetter winters. The Safran-Isba-Modcou (SIM) hydrometeorological modelling system was applied to simulate future discharge response to a perturbed climate. The SIM model was recently re-used by [START_REF] Boé | Projected changes in components of the hydrological cycle in French river basins during the 21st century[END_REF] to examine the impact of future climate on the hydrological cycle of the main French river basins, and to assess uncertainties linked to climate models and downscaling techniques. The forcing data were composed of a set of 17 downscaled climate scenarios. The Garonne River basin was found to be the most severely impacted basin, with declining precipitation and increasing temperature for all the seasons resulting in a significant reduction of low flows by the 2050s (e.g. the relative change in mean summer discharge for the Garonne River basin at Lamagistère is projected to lie between -22% and -42%). [START_REF] Tisseuil | Statistical downscaling of river flows[END_REF] developed a new downscaling approach to derive river flow characteristics directly from GCM outputs, applying it to southwestern France. The results show a pronounced decrease in mean summer discharge of basins with rainfall-fed river flow regimes, including the Garonne River basin at Tonneins (51 500 km²) close to its outlet.. All these recent studies suggest a considerable modification of water availability for the future, with a clear tendency towards more severe low flows. However, none of these studies has explicitly taken into account reservoir operations within the basin, or has questioned the sustainability of present-day water management. The present study is intended to fill these gaps, and also forms part of a French national project IMAGINE2030: "ClImate and water MAnaGement: uncertainties on water resources for the Garonne rivEr basin in 2030". This project [START_REF] Sauquet | secheresses, canicules rares et leurs impacts sur les usages de l'eau[END_REF] involves: the National Research Institute of Science and Technology for Environment and Agriculture (Irstea, formerly Cemagref); Electricité de France, one of the major water stakeholders as electricity producer; and the Adour Garonne Water Agency. Through an integrated study, IMAGINE2030 aims to assess both current and future risks of water shortage in the 2030s, while taking into account changes in climate and human activities. The study area encompasses nine sub-basins of the Garonne River basin upstream from Lamagistère (32 500 km²) (Fig. 1).

This paper focuses on the most innovative aspect of the project. We develop here the first model for simulating hydropower reservoir management in France, which allows us to assess the impact of climate change on river flow regime, applying it to a case study in the Pyrenees: the Ariège River at Foix (1340 km²). This sub-basin is located in the mountainous part of the Garonne River basin. The river flow regime is influenced by a series of high elevation reservoirs primarily dedicated to producing hydro-electric power. Due to their high elevation, this series of reservoirs mostly relies on snowmelt. Moreover, there are water transfers from one year to another between the Ebro River basin, located on the southern side of the Pyrenees in Spain, and the headwaters of the Ariège River basin in France. For comparison, we also show the results obtained for a neighbouring basin, i.e. the Salat River basin at Roquefort, with a similar catchment drainage area (1570 km²) but not significantly affected by human activities.

In the present study, we first describe the data used. The rainfall-runoff model and the simplified model of hydropower reservoir management are then outlined, along with the results of their application to the case study discussed in the two first sections. The following section is dedicated to the methodology of constructing transient daily climate projections. We then show the projected changes, including the effects due to both reservoir operations and climate change. We also discuss the main sources of uncertainty affecting projections, before drawing some general conclusions.

DATA

Historical climatological data are extracted from the 8-km-resolution atmospheric reanalysis over France performed with the Safran gauge-based system (Quintana-Seguí et al. 2008, Vidal et al. 2010). The variables of interest for this study are daily mean temperature and daily mean total precipitation.

Observed daily discharge time series for the Salat River at Roquefort and for the Ariège River at Foix are available in the French database HYDRO (http://www.hydro.eaufrance.fr/) for the periods 1970-2006 and 1970-2005, respectively. Inflows for the Ariège River at Foix were naturalized over the 1990-2004 sub-period using historical records of reservoir level variations, as well as released discharges for hydropower production and overflows for all dams located in the headwaters. The curve resulting from this correction is shown in comparison with the mean annual hydrographs for the Ariège River (middle of the right panel in Fig. 1): during spring and early summer, reservoirs store inflows naturally produced by snowmelt; in winter, stored water is released to meet unscheduled high peak demands for energy and, in summer, reservoirs give water back to meet multipurpose needs such as agriculture, recreation, drinking water or ecological services. Between 25% and 33% of the mean annual flow observed at Foix is actually involved in water management operations in the main head reservoirs. The natural river flow regime of the two basins is predominantly affected by snowmelt processes. Both basins have a mean annual runoff of about 40 m 3 s -1 .

For this study, a total of 34 climate projections from the Intergovernmental Panel on Climate Change Fourth Assessment Report (AR4) (IPCC 2007) were used:

13 GCMs under the A1B SRES emissions scenarios, 10 GCMs under the A2 SRES emissions scenarios and 11 GCMs under the B1 SRES emissions scenarios. The climate simulations were obtained from the IPCC Data Distribution Center (http://www.ipcc-data.org).

THE CEQUEAU RAINFALL-RUNOFF MODEL

The distributed rainfall-runoff model known as Cequeau [START_REF] Charbonneau | The Cequeau model: description and examples of its use in problems related to water resources management[END_REF]) is adopted here for predicting natural daily discharge. This conceptual model has already been successfully applied to simulate observed river flows under various climatic conditions (e.g. Ayadi and Bergaoui 1998[START_REF] Hendrickx | Impact of climate change on the hydrology of the Rhône catchment (in French)[END_REF][START_REF] Manoha | Climate change impact on the activities of Électricité de France (in French)[END_REF].

The Cequeau model comprises a runoff production function that generates runoff from precipitation (rain or snow) at each cell based on a series of interconnected conceptual reservoirs. The transfer function is represented by a reservoir cascade that transforms direct and subsurface runoff (provided by the production function) into surface runoff in the drainage network.

The Thornthwaite formulation [START_REF] Thornthwaite | An approach toward a rational classification of climate[END_REF] is used to estimate daily potential evapotranspiration. Snow accumulation is simulated within each grid cell at daily time steps using precipitation and air temperature data. The snowmelt equation is based on the temperature index degree-day method developed by the US Army Corps of Engineers (USACE 1956). The Cequeau model estimates the components of the water balance, including snowpack melt and accumulation, at each time step for each cell. The ability to simulate snow-related processes-which is fundamental to correctly reproduce runoff in mountainous areas-is one of the reasons for selecting the Cequeau model for this impact study.

Inputs correspond to the daily temperature and rainfall data provided by the Safran re-analysis at each point of the 8-km grid. The Nash-Sutcliffe efficiency criterion, NSE [START_REF] Nash | River flow forecasting through conceptual models. 1. A discussion of principles[END_REF], calculated on the natural and simulated daily discharge series, is used as a goodness-of-fit measure. A split-sample procedure [START_REF] Klemeš | Operational testing of hydrological simulation models[END_REF]) is carried out to calibrate Cequeau against naturalized or natural flow by evaluating parameters in turn for two sub-periods (with a preceding 1-year model spin-up period). Results for the Roquefort River basin show small differences in performance over the two sub-periods; e.g. the NSE criterion is 0.86 over the period 1971-1988 in calibration and 0.85 for the period 1989-2006 in validation, despite contrasted climate conditions for calibration and validation (change in mean annual temperature of 1°C and in total annual precipitation of 13%). Conversely, performance in validation for the Ariège River basin is very sensitive to the choice of calibration period. This could be due to the short period of available inflows and to the uncertainties related to the naturalization procedure. Considering the entire period of record, the NSE criterion is 0.79 and 0.86 for the Ariège and Salat River basins, respectively. Figure 2 illustrates the ability of Cequeau to reproduce three pth percentiles Q d p derived from the empirical distribution of daily discharges for each Julian day, associated with exceedence probabilities of p = 5%, 50% and 95%, respectively. The marked variability of observed Q d 95 values for the Ariège River basin compared to the Salat River basin is clearly an artefact of the naturalization procedure. Overall, the main aspects of seasonality within the year are satisfactorily captured. The Cequeau rainfall-runoff model demonstrates its ability to reproduce the basic characteristics of river flow regimes and, implicitly, the dominant hydrological processes-related to snowmelt-in the two Pyrenean case studies.

SIMPLIFIED MODEL OF HYDROPOWER RESERVOIR MANAGEMENT

Numerous works on the impact of climate change on water resources can be found in the literature. However, few studies have examined potential changes in hydropower production [START_REF] Minville | Behaviour and performance of a water resource system in Québec (Canada) under adapted operating policies in a climate change context[END_REF]. [START_REF] Schaefli | Climate change and hydropower production in the Swiss Alps: quantification of potential impacts and related modelling uncertainties[END_REF] and [START_REF] Finger | Projections of future water resources and their uncertainty in a glaciarized catchment in the Swiss Alps and the subsequent effects on hydropower production during the 21st century[END_REF] simulated dam operations and power generation of reservoirs located in the Swiss Alps. There are no well-defined management rules, and energy demand is not explicitly included in these applications. Both studies conclude that climate change will lead to decrease of glacier extent, as well as of hydropower production. Other published works used simplified representations of current dam operations. The ColSim reservoir operation model developed by [START_REF] Hamlet | Effects of climate change on hydrology and water resources in the Columbia River Basin[END_REF] was later used by [START_REF] Payne | Mitigating the effects of climate change on the water resources of the Columbia River basin[END_REF], [START_REF] Markoff | Impact of climate change on Pacific Northwest hydropower[END_REF] and [START_REF] Hamlet | Effects of projected climate change on energy supply and demand in the Pacific Northwest and Washington State[END_REF] to simulate hydropower production at different time slices in the Pacific Northwest and Washington State, USA. [START_REF] Christensen | Effects of climate change on the hydrology and water resources of the Colorado River Basin[END_REF] and, more recently, [START_REF] Christensen | A multimodel ensemble approach to assessment of climate change impacts on the hydrology and water resources of the Colorado River Basin[END_REF] ran a simplified version of the Colorado River Simulation System (USDOI 1985), forced by monthly inflows simulated by the Variable Infiltration Capacity (VIC) macroscale hydrological model. The CVmod model was developed by Van Rheenen et al. (2004) to simulate water storage in the main reservoirs located within the Sacramento-San Joaquin River basin (California) at a monthly time step. [START_REF] Fortin | Impact des changements climatiques sur les plans de gestion des lacs Saint-François et Aylmer au sud du Québec (in French)[END_REF] also developed a model that represents operating rules for two multipurpose reservoirs (hydropower generation, water supply, flood control in spring, ecological targets in spring, recreational uses in summer and autumn) in Québec, Canada. [START_REF] Payne | Mitigating the effects of climate change on the water resources of the Columbia River basin[END_REF], [START_REF] Christensen | Effects of climate change on the hydrology and water resources of the Colorado River Basin[END_REF]), Van Rheenen et al. (2004) and [START_REF] Fortin | Impact des changements climatiques sur les plans de gestion des lacs Saint-François et Aylmer au sud du Québec (in French)[END_REF] have tested different technical measures, including timing shifts in reservoir refill and changes in authorized maximum water levels, for mitigating the negative impacts of climate change. [START_REF] Markoff | Impact of climate change on Pacific Northwest hydropower[END_REF] used results from ColSimand also from another model, Genesys (NPCC 2003), on a reduced set of hydrological projections to derive simple empirical formulas linking changes in energy production and changes in seasonal meteorological forcing; they found the hydropower systems to be more sensitive to change in precipitation than to temperature modification. [START_REF] Hamlet | Effects of projected climate change on energy supply and demand in the Pacific Northwest and Washington State[END_REF] demonstrated that changes in energy production are positively correlated with changes in river flow regime, i.e. seasonal trends in discharge and in energy production share the same sign. Under the assumption of increase of future energy demand in parallel with population growth and the development of air-conditioning, greater difficulties are expected in adapting both water and energy management in summer in the next decades. More sophisticated models based on maximization of benefits have been developed by [START_REF] Turgeon | Solving a stochastic reservoir management problem with multilag autocorrelated inflows[END_REF] and applied later by [START_REF] Minville | Adaptation to climate change in the management of a Canadian waterresources system exploited for hydropower[END_REF][START_REF] Minville | Behaviour and performance of a water resource system in Québec (Canada) under adapted operating policies in a climate change context[END_REF] on the Peribonka River (Québec), by [START_REF] Madani | Modeling California's highelevation hydropower systems in energy units[END_REF] and by Vicuña et al. (2008), both with applications in California (Madani andLund 2010, Vicuña et al. 2011). These two study areas in Canada and the USA may experience a shift in timing of natural high flows and a reduction in snow contribution due to temperature increase. [START_REF] Minville | Adaptation to climate change in the management of a Canadian waterresources system exploited for hydropower[END_REF][START_REF] Minville | Behaviour and performance of a water resource system in Québec (Canada) under adapted operating policies in a climate change context[END_REF] and [START_REF] Madani | Estimated impacts of climate warming on California's high-elevation hydropower[END_REF] pointed out a possible rise in unproductive spills that may result in losses in reservoir efficiency. Three scenarios for hydrology corresponding to three different climate change scenarios were run through the monthly-based optimization model EBHOM [START_REF] Madani | Modeling California's highelevation hydropower systems in energy units[END_REF]. The results show that, under the dry and warm climate, profits are expected to reduce with respect to the baseline simulation for the two high-elevation systems studied in California. The study by Vicuña et al. (2011) suggests a decrease in revenue as a consequence of the reduction in annual inflows.

These findings and conclusions demonstrate the diversity of the impact of climate change on hydropower using projections for climate, water use and electricity demand (e.g. attributable to air conditioning use, residential heating, etc.). The approach proposed within the IMAGINE2030 project is similar to that of the models applied in these previous works. A water demand prioritization model coupled with the Cequeau rainfall-runoff model was developed to account for the links between water demand, local hydropower production and the French electricity grid.

One of the particularities of the French electricity production system is that most of the energy demand (approx. 80%) is supplied by nuclear power plants. The baseline capacity for nuclear electricity production has to be supplemented by hydropower to ensure matching between supply and demand in real time. This flexible energy source is mainly used to manage peak demands for heating in winter during cold weather conditions and, occasionally, to a lesser extent, at other times. Electricity demand is highly dependent on air temperature, with a different sensitivity between winter and summer: energy demand is increased by +1700 MW per -1°C in winter and by +400 MW per +1°C during hot summers over France. In addition to producing electricity, reservoirs located in the Ariège drainage basin tend to moderate the extremes of low flow (water is released to increase downstream discharges).

The relationships between reservoir operations in the Pyrenees within the context of the French electricity grid are too complicated to model in detail, as we do not have the necessary information about the individual plants of the hydropower system. Instead, we adopt a simplified modelling approach, which takes into account only the main drivers of water storage and release at the basin scale. The model was developed bearing in mind that input data should be easily available to facilitate studies on the impact of climate change. The main characteristics of the model are: (a) to manage reservoir operations, which consists essentially of addressing the issue of whether water should be released in the short term, or stored for use at a later stage; (b) to ignore electricity prices and only be driven by meteorological and hydrological conditions; (c) to be based on dynamic programming very similar to the operational tools used by Electricité De France for water management, an approach that involves optimizing the quantity of released water, R, that maximizes a benefit function, B, using the following recursive equation established for a given time step, t:

( ) ( ) ( ) t t t t t t R t t I R S B R P S B + - + ⋅ = +1 max ( 1
)
where S is the stored volume, P the daily benefit derived from a release, R the daily released volume, and I the daily inflow volume. The benefit function B expresses the economic value of storage S at a time step t, given the initial and final target states of the system. Equation ( 1) thus allows us to determine the optimal trajectory of the stored volume over the period considered. For this application, • In operational mode, reservoir operations are adjusted at least three times a day based on updated information, including weather forecasts and the availability of energy facilities. The short time step of the decision-making process cannot be achieved easily. A daily time step was finally used for calculation. This may result in a slightly less-than-optimal simulation of water allocation. • The marginal cost of electricity is estimated using a simple empirical formula.

Different tests led to the following expression for the benefit of daily release, P:

( ) France 15 ; 0 max t t T P - = (2)
where T France (°C) is the air temperature in France. • The Cequeau rainfall-runoff model provides information on the input of water, I, into a reservoir at a daily time step. • A regulatory constraint stipulating a minimum daily discharge of 8 m 3 s -1 at Foix is added to satisfy downstream multi-purpose uses in summer, such as irrigation systems, navigation, drinking water supply, ecological services [START_REF] Cavitte | Dry period water management plans on Adour-Garonne basin, an example of partnership between basin authorities and the agriculture world[END_REF]. Other constraints related to the French water laws of 1992 and 2006 exist, but they are less restrictive for water management than the obligation to guarantee at least 8 m 3 s -1 at Foix. • All the reservoirs of the Ariège River basin are aggregated into a single reservoir.

• The model is run deterministically with a known future, i.e. assuming perfect foresight of both future natural inflows, I and air temperature in France, T France .

The results obtained should be interpreted as the optimal path for integrated water management, especially in the context of a perturbed climate. Figure 3(a) compares daily observed and simulated discharges, based on the same statistics as displayed in Fig. 2. The results demonstrate the relatively good performance of this simplified model and its ability to represent the seasonal influence of reservoir management on the natural hydrological river flow regime. The NSE criterion computed on observed and simulated daily regulated discharges reaches 0.71 for the period 1972-2005. This performance is consistent with the degree of refinement in the representation of reservoir management. In addition to daily discharge statistics, Fig. 3(b) also shows the stored water volume. While no observed data are available for comparison, the obtained trajectories are nevertheless consistent with the total reservoir storage capacity of 200 hm 3 and with the known reservoir operations; storage is simulated between day 106 (15 April) and day 301 (26 October), including the major filling period during snowmelt in spring, whereas release is observed at other times.

RESAMPLING TECHNIQUE TO DERIVE PERTURBED METEOROLOGICAL SCENARIOS

Only monthly climatic time series (Vidal and Sauquet 2010) were available at the beginning of the IMAGINE2030 project. This data set consists of 34 monthly gridded local projections obtained by the Bias-corrected Local Mapping (BLM) procedure (Vidal and Wade 2008) performed on 34 IPCC AR4 runs. The application to the Garonne River basin makes use of observed data and the grid of the Safran meteorological archive. Results are given as monthly projections of 8-km gridded precipitation and air temperature over the study area for each GCM. Figure 4 shows the transient moving-average temperature and precipitation throughout the 1980-2030 period. Table 1 summarizes the results for specific periods, including the historic baseline taken throughout as the climate averaged over the period 1970-1989. A comparison with statistics derived from the Safran re-analysis demonstrates that current conditions are reasonably well reproduced by bias-corrected GCMs. On a seasonal basis, there is good agreement between projections in temperature, with increases ranging on average from +1°C to +2°C for the 2021-2040 period. The data presented in Fig. 4 and Table 1 also suggest that potential changes in future precipitation are more uncertain. Dispersion in the projections globally increases with greater distance into the future. Dispersion between projected precipitations is high, especially in winter. The magnitudes of the changes in precipitation are weak, except in summer, for which the most notable changes are projected, i.e. a maximum decrease of 33% in precipitation together with an increase in temperature by up to +4°C over the 34 GCMs. Using the average of the 34 AR4 climate projections, the Garonne River basin will experience a warmer and drier climate for the 2021-2040 period: 75% of the models predict a decline in annual precipitation, and all the models point to an upward trend in mean annual temperature.

Daily data were required as model inputs. A simple way to overcome this problem is to apply the delta-change method, certainly the simplest and most frequently-used technique, despite its well-known limitations (e.g. Diaz-Nieto and Wilby 2005). Our wish was to develop projections based on the most reliable information provided by the GCMs (projections of air temperature are more consistent than projections of precipitation), while ensuring realistic values in meteorological forcings. The approach applied uses non-parametric resampling partly conditioned by the available monthly GCM outputs.

The chosen algorithm is the modified k-nearest neighbour (KNN) approach [START_REF] Lall | A nearest neighbor bootstrap for resampling hydrologic time series[END_REF], which provides stochastically a consistent set of multivariable time series sampled from observed past situations. To generate the vector of climatic variables for day t + 1, we first make use of a meteorological historical archive to select a set of k days with characteristics analogous to those simulated for day t. The potential candidates for k analogous situations are all days included within the N time windows of days that are centred around the target ordinal date corresponding to t of all the N available years. Days are ranked according to the Mahalanobis distance measuring their similarity to day t. Finally, one of the k closest neighbours is randomly selected and the observed values for the day following that neighbour are considered as the values for day t + 1. [START_REF] Lall | A nearest neighbor bootstrap for resampling hydrologic time series[END_REF] suggest giving more weight to closer neighbours in the probability distribution. Each of the analogous days d(i), i = 1, ... , k ranked according to its degree of similarity with day t has the following probability of being randomly selected:

( ) ( )

∑ = = = k j j i i d t 1 1 1 Prob (3)
Here, we adopt the procedure suggested by Yates et al. (2003) and [START_REF] Sharif | Simulating climate change scenarios using an improved k-nearest neighbor model[END_REF] to develop an ensemble of precipitation and temperature time series conditioned according to trends identified in climate projections. The KNN algorithm is modified so that some years -corresponding to certain meteorological situations -are favoured rather than others. A list is first established of N years, which are sorted with respect to prescribed additional criteria (e.g. from the driest to the wettest period for all years). A N-dimension vector (x i , i = 1, ... , N) is randomly generated:

( ) [ ] 1 1 + - × = S i r N INT x
(4) where r ∼ U(0,1) is a random number and S is a positive shape parameter to be fixed. Lastly, the analogous day among the k-nearest neighbours with associated rank j in the sorted list of records is repeated as many times (nd j , j = 1, ... , N) as j appears in the randomly generated vector (x i , i = 1, ... , N). No bias is simulated when S = 1. A bias toward the selection of days with high rank is observed when S > 1. Conversely, a bias toward the selection of days with low rank is observed when S < 1. The potential candidates for the k analogous situations are thus all the days included within the N time windows repeated nd j times according to the rank j of the year of record.

Adaptations have been required to develop transient climate projections. We first consider that the effect of climate change has become apparent since a specific year (year start ) and, second, that these changes are modelled by the parameter S, which is assumed to be a function of time as follows: with r 1 ∼ U(0,1). The term (0.5 + r 1 ) is introduced to represent the spread in GCMprojected changes. Each KNN scenario differs from the others by the random value attributed to r 1 . The function f and the four parameters S 0 account for the long-term trend specific for each season. Equation ( 6) is the only novelty introduced in the KNN algorithm published by Yates et al. (2003) and [START_REF] Sharif | Simulating climate change scenarios using an improved k-nearest neighbor model[END_REF].

A set of twenty 70-year daily projections, conditioned by these 34 downscaled IPCC AR4 model simulations, has been developed by applying the modified KNN approach. For the application:

• The decision was taken to bias the KNN algorithm with respect to the most robust trend, i.e. to constrain the biasing by the evolution in average air temperature. The four parameters S 0 (DJF), S 0 (MAM), S 0 (JJA) and S 0 (SON) are calibrated in such a way that the seasonal trends computed on the ensemble medians of temperature fit with trends identified on the downscaled AR4 projections for the Garonne River basin at Lamagistère. By imposing this constraint over the whole basin, consistency is ensured between the climate projections obtained for the sub-basins studied in the project IMAGINE2030. • Daily precipitation and air temperature data were generated for each 8-km grid cell within the Garonne River basin upstream from Lamagistère. • Each of the N available years was divided into decades: (dec i,j , i = 1, . . . , 36, j = 1, ... , N). For fixed i; (dec i,j , j = 1, . . . , N) values were ranked from the coldest rank 1) to the warmest (rank N). Each of the k-nearest neighbours observed in decade dec i,j with associated rank m is repeated nd m times in the final list of candidate days. • year start is fixed at 1990. • f is given by a simple linear model:

) 1990 2040 ( ) 1990 year ( ) year ( - - = f (7)
The final results are displayed in Fig. 4 and Table 1. The parameters S 0 (DJF), S 0 (MAM), S 0 (JJA) and S 0 (SON) are fixed at 1, 1.25, 4.5 and 2, respectively. As expected, S 0 (JJA) is the highest value, since most changes in temperature are expected in summer and S 0 (DJF) is taken as equal to 1, because no strong change is projected for this season. Figure 5 displays the number of times (nd j , j = 1, ... , N) each day is repeated according to its rank j in the list of potential candidates for the k-nearest neighbours in 2040 for each seasonal shape parameter S 0 . The time series of daily temperature in France (T France ) associated with each KNN scenario is composed of observed values for the sampled days.

The results reported in Fig. 4 and Table 1 show an acceptable fit between changes in median temperature derived from the BLM climate projections and those obtained applying the KNN algorithm. The dispersion characterized by the interdecile range is slightly less well reproduced, with a tendency towards a systematic underestimation of the dispersion.

It is also noteworthy that the bias imposed on temperature leads to scenarios that are on the whole drier and less dispersed than those obtained by BLM. This tendency is more apparent for the end of the simulation period. Here, we can point out a major difference between the BLM and KNN procedures. Future precipitation and temperature are established independently when applying the BLM method, while the KNN algorithm preserves correlations that have been previously observed between these two variables. In the absence of physically-based constraints, the BLM procedure projects links between precipitation and temperature that differ from the actual ones. Temperature and precipitation used as input for the models and obtained by the KNN procedure are established with the strong assumption of stationarity in the physical processes. The consequence is a possible risk of masking changes in variability.

IMPACT ON WATER MANAGEMENT

In this study, simulations of the natural basin behaviour are performed using the Cequeau rainfall-runoff model of the two basins forced by transient climates provided by the KNN algorithm. The increase in mean air temperature leads to an impact on snow cover and snowmelt, i.e. a trend towards reduced winter snow storage and a shorter snowmelt season due to an earlier timing of snowmelt. Figure 6 displays the changes in hydrological regime during the period 1970-2040. Table 2 summarizes the simulations of mean seasonal flows QJFM, QAMJ, QJAS and QOND for time periods in the future. Seasons are defined to match the present-day periods of low flow and high flow. This table reports the average number of days per year N <VCN10(5) with daily discharge below the current annual minimum 10-day flow with a 5-year return period, VCN10(5). The value for VCN10( 5) is commonly adopted as the first warning level in French drought management plans. For both basins, VCN10(5) is computed from natural observed discharges and is fixed at 8 m 3 s -1 . The two percentiles, Q95 and Q50, which are derived from inter-annual flow duration curves are also included. "Nat" refers to natural or naturalized discharges, "Safran" to simulated discharges obtained with the Safran re-analysis and "KNN" to simulated discharges obtained with the KNN scenarios. To remove quantitative biases, changes are presented in terms of anomalies calculated for each KNN projection by difference between statistics established for the 1972-1989 period and the different periods of interest. The historic baseline differs slightly from the one chosen for the climate due to the availability of records and the 1-year spin-up period required for the models.

The main change for the two studied drainage basins is observed in spring: the peak due to snow melting is diminished and the decrease in spring total precipitation leads to a reduction in flows by about 25%. It can be noted that projected changes are greater in the Ariège River basin than in the Salat River basin, clearly due to a more pronounced contribution of snowmelt within the Ariège headwaters. These changes in spring discharges accelerate the basin depletion and, combined with both warmer and drier summers, they result in a 30% reduction in summer low flows during the period 2021-2040. There is a tendency for the change in the number of days with discharge below VCN10(5) to increase consistently with decrease in QJAS. As a consequence local authorities will face more frequent and severe droughts on both basins. New strategies for water management will have to be implemented with solutions that differ from one basin to the other as no regulation infrastructure exists within the Salat River basin. By contrast, no significant change is observed in winter. Under the KNN climate warming scenarios, there is a progressive decline in Q50 and Q95. The trends displayed for the two basins share the same pattern.

After calibration of the models, simulations of reservoir management are performed using KNN climate projections. The data in Fig. 6(c) and Table 3 show that reservoir operations have a marked effect on the seasonality of changes-similar to the effect observed on present-day natural discharges-resulting in a reduction of the variability from month-to-month of the future anomalies on the natural river flow regime (Fig. 6(b)). Figure 7 shows the changes in two percentiles derived from the annual flow duration curves representative of low to median flows. By the 2030s, changes in median regulated discharges Q y 50 are of the same order as those found for the natural river flow regime. However, low flows represented by Q y 95 are not projected to decrease to the same extent because of the constraint imposed on minimum flows. Under the selected warming scenarios, the change in natural Q y 95 indicates that there is an increased risk of daily discharges falling to below 8 m 3 s -1 by the 2030s (Fig. 7(a)). Nevertheless, the stored volume in the reservoirs is still sufficient to maintain Q y 95 above 8 m 3 s -1 (Fig. 7(b)).

The effect of climate change is perceptible in the trajectory of reservoir storage (Fig. 8(a)), which is modified by both the reduced discharge in spring and the increased requirement to maintain a minimal discharge in the river during the summer. The change in the reservoir trajectory is a consequence of the change in the main drivers of storage. On average, mean temperature over France T France is below 15°C for 263 days in the 1970-1989 period, while T France is below 15°C for 245 days in the 2021-2040 period. The reduction in the number of days when there is a benefit in carrying out releases for energy production is balanced by the increase in number of days when releases are required to guarantee minimal discharge (on average, discharge is below 8 m 3 s -1 for 9 days in the 1971-1989 period, cf. 35 days in the 2021-2040 period). Although the total number of days when stored water is in demand is comparable at the beginning and at the end of the simulation period, their modified distribution within the year is likely to impact reservoir operations. Storage rises relatively early in the year, and is reduced slightly in the late spring. In winter, it will become less easy to release water for energy demand. This will create a loss of flexibility. Under warming climate conditions, hydropower generated in the Pyrenees will not be able to contribute to the French electric grid as in the past: the annual production of electricity will be reduced by 15-20%, and reservoir flexibility during the winter demand peak periods will also be reduced.

Finally, we examine the sensitivity of the reservoir trajectory to changes in the minimal daily flow, which integrates higher downstream needs under a warmer and drier climate (Fig. 8(b)). This minimal daily flow imposed at Foix is fixed at 12 m 3 s -1 for the whole 1970-2040 period. This increased flow is consistent with the upward trend of 20% in water needs for irrigation projected for the 2030s [START_REF] Sauquet | secheresses, canicules rares et leurs impacts sur les usages de l'eau[END_REF]. Overall: • During summer, more demands are made of reservoirs to release water for downstream needs, since natural inflow is frequently below the value of 12 m 3 s -1 (Fig. 6(c)) observed earlier in the year. • Drawdown starts earlier compared to results obtained using the current constraint of 8 m 3 s -1 . • Increased daily minimum flow imposes higher storage within the year (cf. water level in the reservoirs at the end of winter). The minimum storage reached in March is less pronounced than with business-as-usual scenarios, which meet subsequent water demands (i.e. the next summer) as a first priority. These results suggest a less contrasted trajectory, demonstrating how more severe constraints restrict the flexibility of reservoir operations, and thus point to a new compromise between energy production and other water uses.

UNCERTAINTY ANALYSIS

Investigating uncertainty in climate change impact on river flows has been an increasing concern in the last few years. Many published works compare the different sources of uncertainty, including future emissions of greenhouse gases, GCM structures, downscaling procedures, structure of rainfall-runoff models, etc. A multimodel approach is often adopted to quantify individual sources of uncertainty and to then examine how they propagate to runoff estimates. Rainfall-runoff models are often considered to contribute less to the overall uncertainty on river flow (e.g. [START_REF] Kay | Comparison of uncertainty sources for climate change impacts: flood frequency in England[END_REF][START_REF] Prudhomme | Assessing uncertainties in climate change impact analyses on the river flow regimes in the UK. Part 2: Baseline climate[END_REF][START_REF] Najafi | Assessing the uncertainties of hydrologic model selection in climate change impact studies[END_REF][START_REF] Teng | Estimating the relative uncertainties sourced from GCMs and hydrological models in modeling climate change impact on runoff[END_REF], and the use of numerous climate models is strongly advised. Recent studies have shown that this hierarchy is partly dependent on the hydrological variable of interest (e.g. [START_REF] Ludwig | The role of hydrological model complexity and uncertainty in climate change impact assessment[END_REF][START_REF] Chen | Overall uncertainty study of the hydrological impacts of climate change for a Canadian watershed[END_REF][START_REF] Sauquet | Climate and hydrological uncertainties in projections of flood and low-flows in France[END_REF]. Uncertainty in GCM configuration and uncertainty stemming from hydrological model structure were considered the IMAGINE2030 project by using multiple climate scenarios and two rainfallrunoff models, respectively. The GR4J lumped rainfall-runoff model [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]) and the Cequeau model were applied in the same conditions in calibration and simulation. A less satisfactory fit between observed and simulated discharges was obtained with the GR4J model. This is reflected in the lower NSE of 0.80 obtained for the Salat River basin (in comparison to 0.86 obtained with the Cequeau model). The results also show that under modified climate conditions the GR4J model projects more substantial decrease in summer river discharge for the two Pyrenean river basins than the Cequeau model. The GR4J model produces the largest reductions in Q50. The magnitude of change is directly dependent on the choice of the GCM and of the hydrological model. Despite the differences in the results some confidence can be placed in the fact that increased temperature will lead to reductions in the proportion of snow that contributes to runoff and to reductions in low flows in the Pyrenees.

In this study, one method was considered to downscale and bias-correct the outputs of GCMs. Uncertainty related to this step was not examined. Thus, it is not possible to draw conclusions on the relative importance of the choice of the downscaling technique. [START_REF] Ducharne | Evolution potentielle du régime des crues de la Seine sous changement climatique[END_REF] have examined climate change impacts on river discharges in northern France, extensively exploring the sources of models uncertainty. Their results show that uncertainty related to hydrological modelling and uncertainty related to downscaling could be comparable. However, it is arguable whether this conclusion applies for the Pyrenean area because of the differences in hydro-climatic conditions.

Other potential sources of uncertainty have been identified within the project IMAGINE2030. Meteorological and hydrological data used to calibrate models under current condition are not free from uncertainty. The Safran meteorological archive was based upon historical data using the available information, which is known to be sparse at high elevation. Values are highly uncertain in the mountainous areas where estimates depend on the interpolation procedure. A new re-analysis covering the main French mountainous areas is now available at 1-km resolution [START_REF] Gottardi | Statistical reanalysis of precipitation fields based on ground network data and weather patterns: Application over French mountains[END_REF] and its use may contribute to quantifying the uncertainty in the data used for calibration purposes. Errors affecting observed discharge data series are linked to errors in water stage measurements and uncertainties in rating curves [START_REF] Lang | Extrapolation of rating curves by hydraulic modelling, with application to flood frequency analysis[END_REF]). In natural rivers, possible errors are typically around ±15%. Uncertainties in naturalized inflows have not been evaluated precisely, but it is evident that naturalized discharges are more uncertain than measured discharges recorded in rivers. We may suspect that the procedure leads to overestimated high flows with consequences in terms of models performance assessment. If the highest flows (above the percentile Q02) are excluded from the time series, the NSE criterion for the Cequeau model is 0.89 in calibration and 0.86 in validation, respectively. Model parameters are constant over time, assuming implicitly the stationarity of the hydrological processes operating in rainfall-runoff transformation. This hypothesis may not hold because land and vegetation cover will adapt in response to altered climatic conditions.

Uncertainty in parameter estimation was not investigated here. Recent works show that calibration under current climatic conditions may introduce significant biases in climate change impact studies [START_REF] Coron | Crash testing hydrological models in contrasted climate conditions: an experiment on 216 Australian catchments[END_REF][START_REF] Seiller | Multimodel evaluation of twenty lumped hydrological models under contrasted climate conditions[END_REF]. In our application to the Ariège River basin, we suspect a lack of robustness because naturalized inflows are only available for a period that is too short and too dependent of climatic conditions for adequate model calibration.

The reservoir model is the result of large simplifications: (a) we simplify the hydropower facilities into one single reservoir; and (b) the technical limitations, e.g. the downstream maximum turbine flows, have not been considered. There is also uncertainty associated with equation ( 2) that models the marginal cost of electricity. This function results from the current equilibrium in the energy mix and from electricity consumption needs throughout the year, and has been partly validated against historical data. Under future climate, equation ( 2) may not hold if electricity consumption changes, or if more renewable energy is produced. The results are obtained using dynamic programming in a known future leading to optimistic predictions in terms of water allocation. Furthermore, in the light of possible future water scarcity, rules defining water allocation will certainly adapt, leading to new constraints for reservoir management in response to more frequent droughts..

CONCLUSION

In this study, we analysed the sensitivity of natural low flows and reservoir management to climate change. The Cequeau rainfall-runoff model was calibrated under present-day conditions to provide natural daily river flows under a perturbed climate consistent with IPCC AR4 monthly precipitation and temperature scenarios developed in the framework of a multi-model downscaling approach. A set of twenty 70-year daily meteorological scenarios was established representing a non-stationary, but progressively warming climate for the Garonne River basin. These projections were produced using a modified KNN algorithm calibrated against monthly air temperature trends projected by GCMs. A model was specifically developed to simulate reservoir operations. This model runs on a daily time step and solves the [START_REF] Bellman | Dynamic programming[END_REF] equation to optimize the choice between "storage today and future release for hydropower" and "release now". Under present conditions, the simplified model of hydropower reservoir management applied to the Ariège River basin reproduces fairly well the "business-as-usual" pattern of water storage and release. Under perturbed climate conditions, the results show a decline of around 15% in annual water resources for the two Pyrenean basins. The contribution of snow storage and snowmelt to runoff are both reduced, and the increase in temperature leads to more severe summer low flows. The results including reservoir management operations show a substantial effect of climate change including: (a) a decrease in energy production (due to the decrease in annual inflows), (b) an earlier filling of the reservoir and (c) an increase in water release in summer to meet summer needs. For the energy producer, full compliance with the required minimum discharge downstream from the reservoir leads to less flexibility for hydropower management during winter peak demand, whatever the future water demand scenario considered in this study case. This feasibility study offers an opportunity to develop and calibrate new tools for assessing the impact of climate change in France on energy production and water management. The results provide a first insight into the future hydro-electricity production for the 2030s, while including simplifying assumptions. For a more complete simulation over the Garonne River basin, we need to make additional assumptions on energy consumption, such as the development of cooling in summer, which may lead to a double peak in electricity demand. The evolution in population and the changes in agricultural practices that are likely to impact on water demand should also be examined. In the context of this feasibility study, we only explore "business-as-usual" water management. Another limitation is that reservoir operations are simulated assuming known future conditions, which, in practice, is not possible in real reservoir management. Therefore, the results from the simplified model of hydropower reservoir management may be somewhat optimistic.

Uncertainties have been partly addressed in this study. It is evident that additional work is needed to provide a more complete view of the sources of uncertainties for the future of the Garonne River basin. However, the results of the IMAGINE2030 project have convinced the French Adour-Garonne Water Agency to support other projects on climate change and to draw-up adaptation strategies for water management within the Garonne River basin using other tools ("Garonne2050" http://www.garonne2050.fr and "Adapt'eau" http://www.adapteau.fr). Developments of the models applied in IMAGINE2030 are in progress thanks to other case studies. Ongoing research in the framework of the French national project R2D2-2050 (Risk, water Resources and sustainable Development within the Durance River basin in 2050; https://r2d2-2050.cemagref.fr/) will provide an opportunity to adapt and improve this simplified model under a different climate context, to develop future scenarios for water-related sectors (energy, economic activities, agriculture) at the regional scale, to study possible options for adaptation of reservoir operations to water availability and water needs in the 2050s, and to examine in more depth the uncertainties related to climate change for large French river basins with both complex and highly-altered river flow regimes.

FIGURES

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

  Van Rheenen, N.T., et al., 2004. Potential implications of PCM climate change scenarios for California hydrology and water resources. Climatic Change, 62, 257-281. Vicuña, S., Dracup, J.A., and Dale, L., 2011. Climate change impacts on two high-elevation hydropower systems in California. Climatic Change, 109 (Supplement 1), S151-S169, doi:10.1007/s10584-011-0301-8. Vicuña, S., et al., 2008. Climate change impacts on high elevation hydropower generation in California's Sierra Nevada: a case study in the upper American river. Climatic Change, 28 (Supplement 1), S123-S137, doi:10.1007/s10584-007-9365-x Vidal, J-P. and Sauquet, E., 2010. Multimodel ensemble climate projections for the Garonne River basin, France. Proceedings of the 11th international meeting on statistical climatology, Edinburgh, 12 July 2010, 167-168. Vidal, J.-P. and Wade, S.D., 2008. A framework for developing high-resolution multi-model climate projections: 21st century scenarios for the UK. International Journal of Climatology, 28 (7), 843-858, doi:10.1002/joc.1593. Vidal, J.-P., et al., 2010. A 50-year high-resolution atmospheric reanalysis over France with the Safran system. International Journal of Climatology, 30 (11), 1627-1644, doi:10.1002/joc.2003 Yates, D.S., et al., 2003. A technique for generating regional climate scenarios using a nearestneighbor algorithm. Water Resources Research, 39 (7), 1199, doi:10.1029/2002WR001769.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. Location of the two studied sub-basins (outlined in red) within the Garonne River basin upstream of Lamagistère (bottom left panel). Long-term average monthly discharge is indicated at three gauging stations under observed conditions (red) and either naturalized or natural conditions (blue) in the three panels on right.

Fig. 5 .

 5 Fig. 5. Bias in KNN procedure expressed in terms of (nd j , j = 1, ... , N).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Year-to-year variation in Qy95 (dark grey/blue) and Qy50 (light grey/red) for the Ariège River basin at Foix: (a) natural conditions; and (b) including reservoir operation management). Dots ( ) indicate observations. The fine lines show results for a given KNN projection, the thick lines show the means of the sets of scenarios.

Table 1 .

 1 TABLES Mean values of seasonal precipitation and temperature, with range in brackets (min; max), derived from sets of climate projections for the Garonne River basin at Lamagistère. DJF: Dec.-Feb., MAM: Mar.-May, JJA: Jun.-Aug., SON: Sep.-Nov.

			Precipitation (mm)			Temperature (°C)
	Period	DJF	MAM	JJA	SON	DJF	MAM	JJA	SON

Table 2 .

 2 Mean changes in characteristics of river flow regime, with range (min; max) in brackets for the sets of climate projections. JFM: Jan.-Mar., AMJ: Apr.-Jun., JAS: Jul.-Sep., OND: Oct.-Dec..

				The Salat River basin at Roquefort					The Ariège River basin at Foix
	Period	N <VCN10(5)	Q95	Q50	QJFM	QAMJ	QJAS	QOND	N <VCN10(5)	Q95	Q50	QJFM	QAMJ	QJAS	QOND

Acknowledgements

The authors would like to thank Richard Turcotte and an anonymous reviewer for their helpful comments that improved the manuscript. Project IMAGINE2030 (contract 07-0000796) was partly supported by funding from the "Risque Décision Territoire" programme of the French Ministry for Ecology. The authors wish to thank all participants in the project, in particular Jean-Philippe Vidal (Irstea, UR HHLY), who performed. the downscaling of AR4 climate projections, Anne-Laure Jeanpierre (Irstea, UR HHLY), who implemented hydrological modelling tools under the R system (http://cran.r-project.org/), and René Samie and Anne Dupeyrat (EDF R&D, LNHE), who carried out naturalization of the observed flows. The authors also acknowledge Météo-France for providing the Safran data set for research purposes and the French Adour-Garonne Water Agency for technical support. Dr M.S.N. Carpenter post-edited the English style.


Olivier Sorlin
email: sorlin@ganil.fr

Nuclear Structure Studies of Neutron-Rich Nuclei: Astrophysical Implications

Keywords: spin-orbit force, drip-line, r-process, transfer and knock-out reactions

It is proposed here to investigate three major properties of the nuclear force that influence the amplitude of shell gaps, the nuclear binding energies as well as the nuclear β-decay properties far from stability, that are all key ingredients for modeling the r-process nucleosynthesis. These properties are derived from experiments performed in different facilities worldwide, using several various state-ofthe-art experimental techniques including transfer and knockout reactions. Expected consequences on the r process nucleosynthesis as well as on the stability of super heavy elements are discussed.

Introduction

A suitable description of the r-process nucleosynthesis requires in particular nuclear structure information from about thousands of nuclei close-to or at the drip line, that are, despite the tremendous progresses achieved in producing rare ions in radioactive ion beam facilities, hardly or not accessible experimentally. This results in large uncertainties in predicting r-process abundances, that add to those related to the identification of the r-process sites and to their hydrodynamical modeling. Two complementary approaches exist to reach this nuclear structure endeavor. The first consists in measuring properties relevant to the r-process, such as atomic masses, lifetimes, neutron-delayed emission probabilities, and level schemes, for nuclei potentially paving the r-process path or as close as possible to it. The recent measurement of more than 100 lifetimes along the rprocess path [1] has lead to a significant improvement on the previous deficits in fitting the r-process curve close to the r-process peak A 130. These experimental results can then be compared to various theoretical predictions, and from this comparison, those that closely reproduce the experimental data can be selected as being probably the most relevant to be used further away from stability. The second consists in finding suitable nuclei, not necessarily involved in the r-process, in which the relevant properties of the nuclear force can be tested and constrained, in order to be applied to yet unexplored regions of the chart of nuclides. In this review, it is proposed in Sect.2 to propose a way to constraint the density and isospin dependence of the spin-orbit force toward the neutron drip line and in the region of super-heavy elements using the bubble nucleus 34 Si, in Sect.3 to reach a better shell evolution predictability in the region of the r-process path based on the hierarchy of nuclear forces derived from the study of nuclei with A ≤ 80, in Sect.4 to evaluate the change in effective proton-neutron interaction at the drip line (as compared to stable nuclei), through the study of 26 F.

Density and isospin dependence of the nuclear spin-orbit force

The spin orbit (SO) force was first introduced in 1949 by M. Goeppert-Mayer and O. Haxel et al. [2] as a result of a strong • s coupling, that is attractive for nucleons having their orbital angular momentum aligned with respect to their spin ( j ↑ = + s) and repulsive in case of anti-alignment ( j ↓ =s). Large gaps are created between the j ↑ and j ↓ orbits at nucleon numbers 6, 14, 28, 50, 82 and 126 for =1-6, which could not be explained otherwise. The shell gaps 50, 82 and 126 are strongly connected to the s and r-process abundance peaks and their reduction far from the valley of stability would influence the position, height and shape of the r-process peaks (see e.g. [3]). Relativistic Mean Field (RMF) models later introduced a more fundamental description of the nuclear spin-orbit interaction [4] derived from Dirac equation, whose non-relativistic derivation writes:

V s τ (r) = -(W 1 ∂ r ρ τ (r) + W 2 ∂ r ρ τ τ (r)) • s (1)
In this expression, the SO interaction contains three important components: the • s term already introduced earlier to account for shell gaps, the derivative of the nuclear density ρ(r) for like or unlike (τ τ ) particles, and the isospin dependence through the W 1 /W 2 ratio that depends on the σ, ω, ρ meson coupling constants. The density and isospin dependences of the SO interaction are subject to large uncertainties between relativistic and non-relativistic models with W 1 /W 2 ratio ranging from 1 to 2 (see e.g. the discussions in [5][6][7][8]). In particular, there is moderate isospin dependence of the SO interaction in the RMF approach (W 1 /W 2 1), meaning for instance that a proton depletion will affect in the same manner the proton and neutron SO force. On the other hand, non-relativistic Hartree-Fock approaches have a large isospin dependence (W 1 /W 2 =2) that would for instance reduce by a factor of two the effect of a proton depletion on the neutron SO force.

These components of the SO force could not be constrained experimentally so far, as most of the nuclei studied in the valley of stability similarly exhibit constant proton and neutron density profiles in their interior, and a sharp drop of the same slope at their surface (see Fig. 1). Significant changes in density profiles are however expected at the drip-line and in the region of super-heavy nuclei. Dripline nuclei are expected to display increased surface diffuseness (see Fig. 1) making the derivative of the nuclear density along the surface weaker and thus the SO splitting smaller. This increased diffuseness would cause a reduction of the neutron SO shell-gaps far from the valley of stability, where the r-process occurs, the intensity of which is strongly model-dependent [5]. Some of the super-heavy elements (SHE) exhibit a central proton depletion because of the large Coulomb repulsion between protons. In addition, according to RMF calculations, nuclei in the vicinity of 292 120 172 are expected to exhibit large proton and neutron central density depletions [8,9], despite being reduced when correlations are taken into account [10]. Central depletions adds a component of the SO interaction in the center of the nucleus of opposite sign to the one at the surface that globally reduced the amplitude of the SO splitting. The amplitude of this reduction in this region of SHE, and herewith the existence of more or less pronounced stabilizing gaps originating from this force, strongly depends on the unconstrained W 1 /W 2 ratio. Moreover, the mass distribution arising from the fission of heavy or super-heavy elements, essential for modeling fission cycling in neutron-star mergers, also likely depend on the density and isospin dependence of the SO force and its stabilizing effect at the time nuclei undergo scission.

Though not directly involved in the r-process nucleosynthesis, an ideal candidate to study these poorly constrained properties of the SO force is the 34 Si nucleus. Being radioactive, 34 Si was produced in a fragmentation reaction at the NSCL/MSU facility and selected in flight using the A1900 spectrometer. Proton knockout reaction was used in inverse kinematics to determine, from the measured (-1p) cross section, that the proton 2s occupancy in 34 Si is only 10% of that of the stable 36 S nucleus [START_REF] Mutschler | accepted in Nature Physics[END_REF]. Being peaked at the center of the nucleus, this lack of 2s protons induces a significant central depletion, as shown in the insert of Fig. 1. On the other hand, neutron density distributions 34 Si, obtained from one-proton knockout reaction, display a central depletion, is visualized here using Relativistic Hartree-Fock Bogoliubov (RHFB) calculations with PKO2 energy density functional [START_REF] Ebran | [END_REF], which predict very similar proton and neutron occupancies to those deduced experimentally.

of the two nuclei are similar, without central depletion. It follows that, with a proton but no neutron central depletion, 34 Si is especially suited to study density and isospin dependence of the SO force.

The central proton density depletion in 34 Si drives an additional (interior) component of the SO force, with a sign opposite to that at the surface. Therefore, low-nucleons, that can probe the interior of the proton bubble, should encounter a weaker overall SO interaction (that scales with the W 1 /W 2 ratio), and display a significantly reduced SO splitting. The 34 Si (d, p) 35 Si reaction was used in inverse kinematics at the GANIL facility to show that the neutron 2p 3/2 -2p 1/2 splitting in 35 Si is reduced by a factor of about two [13], as compared to the neighboring N = 21 isotones of 41 Ca and 37 S. Such a sudden change, connected to that in central density, demonstrates for the first time the density-dependent of the SO force. This result will be used in the future to contraint the strength of SO force in a unique manner, and to explore consequences in the r-process nucleosynthesis and in the modeling of SHE.

Shell evolution and the hierarchy of nuclear forces

The nuclear force can be casted into central, spin-orbit and tensor parts. When moving along the chart of nuclides, protons and neutrons occupy orbits with different combinations of quantum numbers (n, , j), with n the number of nodes of the radial wave function, the orbital momentum and j the total spin value. It follows that in a given chain of isotones, various components of the proton-neutron nuclear force are explored, inducing either no change in shell structure, or on the contrary a significant reduction of shell gaps and even a shell re-ordering.

Shell disappearance or weakening could be viewed at first glance in the evolution of the first excited states (2 + ) of even-even nuclei as a function of neutron numbers N, as shown in the bottom part of Fig. 2. It took about 30 years to establish these curves, from experiments carried out worldwide 1 energies is found instead. This apparent breaking of magicity far from stability represents one of the major change of paradigm in nuclear physics that should also apply in regions where the r-process nucleosynthesis takes place. at radioactive ion beam facilities (see e.g. Refs [14][15][16]) at N=40 . It can be seen that, in the valley of stability, all isotopic chains (black curves) display a sudden increase in 2 + energy at the shell closures N=8, 20 and 40. The effect is remarkably preserved at N=20 from Z=20 down to Z=14. This observation led to the earlier paradigm that magic nuclei were immutable. On the other hand, far from stability, as seen with the blue and red curves, there is no more sign of increase in 2 + energy, as if all magic numbers no longer exist far from stability. It has been proposed in Ref. [17] that these shell gaps disappear in the same manner in all regions, owing in particular to a common hierarchy of forces that is present far from stability but no longer in the valley of stability. In particular, shell effects increase in an isotonic chain when the proton orbits having their angular momentum aligned with their spin value j ↑ are filled, e.g. f 7/2 . They are preserved afterwards. Important is to notice that the same forces are expected to be at play in particular in all regions circled in yellow in the chart of nuclei of Fig. 2. Of particular importance in astrophysics is the region below 132 Sn, where the A 130 r-process abundance peak is built. Following this hierarchy, a weakening of the N=82 is also expected there. However, as containing a large number of nucleons, the net effect of removing few of the 10 protons occupying the g 9/2 orbit from 132 Sn will not be as strong as the removal of for instance two protons from 14 C to 12 Be, where an abrupt change of shell structure is revealed from the sudden drop in 2 + energy at N=8 (see Fig. 2). The reduction of shell gaps and the reordering of orbits should arise from this hierarchy, impacting for instance the location of the (n,γ)-(γ,n) equilibrium in the r-process, as well as β-decay lifetimes and neutron-delayed emission probabilities P n . Many models and parameterizations are used nowadays to calculate r-process abundances. Some of them may have the suitable ingredients included that lead to this hierarchy, others not, leading to significant differences in predicting r-process abundances. It is important, when using a model, to confront it to the present observation of shell reduction and re-ordering in regions of medium-mass nuclei, where experimental data such as 2 + energies, reduced transitions probabilities B(E2), atomic masses and single-particle energies exist now.

Proton-neutron forces at the drip-line

When stellar processes involve nuclei at or close to the drip lines, as it does in neutron-star mergers or in the neutron stars' crust, proton-neutron interactions occur between deeply bound protons and unbound neutrons. The wave functions of the latter explore the continuum and have a poorer radial overlap with the protons confined in the nucleus interior. This probably leads to a globally weaker proton-neutron force than for stable nuclei, in which proton and neutron binding energies are similar. This effect is not yet treated in models and no, or very few, experimental case existed so far to quantify its amplitude as a function of the nucleon binding energy, angular momentum and coupling to other states in the continuum. The 26 F nucleus is one of the few ideal probes to quantify the effect of the proximity of continuum on the effective nuclear force. Indeed, as shown in Fig. 3a), it can be viewed as a deeply bound d 5/2 proton to which an unbound neutron d 3/2 of 770 keV [18] is added on top of the doubly closedshell core of 24 O, to form a multiplet of states J = 1 + -4 + . The binding energies of these four states were required to determine this mean proton-neutron interaction strength, derived from their J weighted-averaged energy values. In Fig. 3b), these binding energies are normalized to provide interaction values, Int(J), which are zero in the case that the protons and neutrons do not interact outside the core, and are negative otherwise. Shell model calculations using USDA interaction are shown in the red color curve in Fig. 3b). They display an upward parabola as a function of J, whose amplitude and mean interaction amount to about 1.6 MeV and -1.5 MeV, respectively. The energies of the four states of the multiplet were determined using four different experiments. The ground state binding energy was deduced from its atomic mass value [19]. The excitation energy of the J=2 + , 0.657 (7) MeV, was obtained using the in-beam γ-ray spectroscopy technique [20], while a J=4 + 2 ms-lifetime isomeric state was discovered at 0.643 MeV from the observation of its delayed M3 transition to the 1 + ground state [21] (Fig. 3c). The unbound J=3 + state was populated selectively at the GSI facility by the one-proton knockout reaction from a beam of 27 Ne interacting in a CH 2 target placed at the center of the Cristal Ball gamma array [22]. It decayed in-flight by the emission of a neutron ( 26 F (3 + ) → 25 F+n ), that was detected at forward angles in the segmented neutron array LAND. The momentum vector of the residue 25 F, determined at the dispersive focal plane of the Aladin spectrometer, was implemented in the invariant mass equation to determine its energy of 323(33) keV above the neutron emission threshold S n of 1.04 [START_REF] Ebran | [END_REF] MeV (see Fig. 3d). The experimental results reported in the blue curve of Fig. 3b) suggest a weaker mean interaction energy, as well as a more compressed parabola (but still not totally flat), as compared to the USDA shell model calculation [23] that does not take the proximity of the continuum into account. This comparison points to an effective reduction of the proton-neutron interaction at the drip line by about 30%. As strongly influencing the shell evolutions and lifetimes values at the drip line, this effect, not yet taken into account in theoretical models, must be more systematically studied and implemented in models to reach satisfactory predictions on the location of the drip lines and the structure of nuclei there. It is noted here that the atomic masses of the nuclei involved to conclude on a reduced strength of the proton neutron force, 24 O, 25 F and 26 F, must be confirmed.

Fig. 1 .

 1 Fig.1. Chart of nuclides showing the density distributions determined for stable, expected towards superheavy elements (SHE) and at the neutron-drip-line. The newly determined proton density distribution of34 Si, obtained from one-proton knockout reaction, display a central depletion, is visualized here using Relativistic Hartree-Fock Bogoliubov (RHFB) calculations with PKO2 energy density functional[START_REF] Ebran | [END_REF], which predict very similar proton and neutron occupancies to those deduced experimentally.

Fig. 2 .

 2 Fig. 2. Top: The regions of the chart of nuclides circled in red correspond to protons having their orbital momentum and spin aligned ( j ↑ ). The filling of these orbitals (e.g. d 5/2 , f 7/2 or g 9/2 ) coincide with major changes in the structure of nuclides (in particular their 2 + 1 energies) shown in the bottom part of the figure. Bottom: The rise of 2 + 1 energies at the neutron shell closures N= 8, 20 and 40 (black lines) no longer exist far from the valley of stability (red and blue lines), where a steep decrease of 2 +1 energies is found instead. This apparent breaking of magicity far from stability represents one of the major change of paradigm in nuclear physics that should also apply in regions where the r-process nucleosynthesis takes place.

Fig. 3 .

 3 Fig.3. a) Schematic picture of the structure of 26 F nucleus, that can be viewed as a core of 24 O on top of which a deeply bound proton d 5/2 and an unbound neutron d 3/2 interact. Their coupling give rise to an upward parabola of interaction energies Int(J) as a function of J ranging from 1 + to 4 + , whose calculated (red) and experimental (blue) values, are reported in b). Deviation in mean interaction energy, normalized to Int(J) =0 in case of no interaction, and in the amplitude of the parabola between experiment and theory point to a weaker effective proton neutron interaction at the drip line. c) Experimental γ spectrum showing on the top row, in the shaded area, the line corresponding to the decay of 4 + isomer at 643 keV. d) Neutron spectrum S n showing a resonance at 323(33) keV corresponding to the decay of the 3 + unbound state.

Acknowledgments It is a pleasure to address special thanks to the physicists deeply involved in the data analysis of the experiments presented here, such as G. Burgunder, A. Lepailleur, A. Mutchler, V. Vandebrouck, A. Lemasson, K. Wimmer, T. Aumann and C. Caesar. I also warmly thank the whole LISE team at GANIL, the S800/Gretina collaboration at NSCL/MSU, as well as the ALADIN/LAND collaboration at GSI for their enthusiasm, efficiency and constant support during all the stages of the experiments.