Ana Casado
email: ana.casado@uns.edu.ar.

Jean-Luc Peiry

Alicia M Campo

Geomorphic and vegetation changes in a meandering dryland river regulated by a large dam, Sauce Grande River, Argentina

Keywords: flow regulation, geomorphic changes, vegetation changes, dryland rivers, Sauce Grande River, Paso de las Piedras Dam

This paper investigates post-dam geomorphic and vegetation changes in the Sauce Grande River, a meandering dryland river impounded by a large water-conservation dam. As the dam impounds a river section with scarce influence of tributaries, sources for fresh water and sediment downstream are limited. Changes were inspected based on (i) analysis of historical photographs/imagery spanning pre-(1961) and post-dam (1981, 2004) channel conditions for two river segments located above and below the dam, and (ii) field survey of present channel conditions for a set of eight reference reaches along the river segments. Whilst the unregulated river exhibited active lateral migration with consequent adjustments of the channel shape and size, the river section below the dam was characterized by (i) marked planform stability (93 to 97%), and by (ii) vegetation encroachment leading to alternating yet localized contraction of the channel width (up to 30%). The present river displays a moribund, stable channel where (i) redistribution of sediment along the river course no longer occurs and (ii) channel forms constitute a remnant of a fluvial environment created before closing the dam, under conditions of higher energy. In addition to providing new information on the complex geomorphic response of dryland rivers to impoundment, this paper represents the very first geomorphic assessment of the regulated Sauce Grande and therefore provides an important platform to underpin further research assessing the geomorphic state of this highly regulated dryland river.

Introduction

Drylands -which include dry-subhumid, semiarid, arid, and hyperarid regions -cover 40% of the Earth's surface and contain almost 40% of the global population [START_REF] Unem | Global Drylands: a UN system-wide response[END_REF].

Population growth and changing living standards force increased water allocation for urban, agricultural and industrial use [START_REF] Young | Flow variability in large unregulated dryland rivers[END_REF][START_REF] Schmandt | How sustainable are engineered rivers in arid lands?[END_REF]. As a result, water resources in drylands tend to be heavily exploited through dams, weirs, canals, and other structures [START_REF] Davies | Dryland rivers: their ecology, management and conservation[END_REF]. Although most dams in drylands are multipurpose dams, with hydropower and flood control as common primary functions, many of them operate as water-conservation structures to support irrigation and/or drinking water supply.

Water-conservation dams maintain the reservoir as full as possible and impound the entire runoff volume in periods of reservoir filling [START_REF] Petts | Impounded Rivers[END_REF]. Thus, the disparity between natural and regulated flow regimes may be particularly striking in drylands [START_REF] Walker | A perspective on dryland river ecosystems[END_REF].

One particular concern is that dryland fluvial processes (and therefore the fluvial response to impoundment) may be very different from those generally accepted in more humid regions (Tooth, 2000b[START_REF] Tooth | Dryland fluvial environments: assessing distinctiveness and diversity from a global perspective[END_REF][START_REF] Nanson | A global perspective on dryland rivers: perceptions, misconceptions and disctinctions[END_REF]. Dryland rivers are characterized by extreme variability in flow and sediment transport [START_REF] Davies | Dryland rivers: their ecology, management and conservation[END_REF][START_REF] Bunn | Flow variability in dryland rivers: boom, bust and the bits between[END_REF][START_REF] Young | Flow variability in large unregulated dryland rivers[END_REF]. Long periods of little or no flow are interspersed with floods of high, sometimes extreme magnitude [START_REF] Tooth | Equilibrium and nonequilibrium conditions in dryland rivers[END_REF], short duration [START_REF] Graf | Fluvial processes in dryland rivers[END_REF], and low predictability [START_REF] Poff | Implications of streamflow variability and predictability for lotic community structure: a regional analysis for streamflow patterns[END_REF]. Floods control erosion, transport, and deposition processes [START_REF] Bull | Dryland river characteristics and concepts[END_REF] and therefore constitute the major determinant of dryland channel shape and size (Tooth, 2000b). As morphogenetic floods exhibit highly skewed frequency (Tooth, 2000b), complete adjustment of dryland channel form to process is sometimes inhibited [START_REF] Bull | Dryland river characteristics and concepts[END_REF]. Feedback mechanisms between channel form and process are rarely found in drylands (Tooth, 2000a), and therefore researchers have a tendency to presume that dryland rivers are in an unstable, nonequilibrium state [START_REF] Tooth | Equilibrium and nonequilibrium conditions in dryland rivers[END_REF].

The unstable character of natural dryland rivers challenges assessing the impacts of flow regulation on dryland channel forms. Notable advances on the complex response of dryland rivers to impoundment have been made in the United States and Australia, as well as in central Asia, South Africa, and South America in minor extent. For example, [START_REF] Friedman | Downstream effects of dams on channel geometry and bottomland vegetation: regional patterns in the Great Plains[END_REF] found that regulated braided rivers in the American Great Plains tend to narrow owing to vegetation encroachment (such as has occurred in the Orange River, South Africa; [START_REF] Blanchon | La stabilité des formes fluviales de l'Orange, entre variabilité naturelle et impacts des grands barrages (secteur Boegoeberg-Augrabies, Afrique du Sud)[END_REF], whereas meandering rivers tend to reduce their migration rates (e.g., the upper Missouri; [START_REF] Shields | Reservoir effects on downstream river channel migration[END_REF]. In the American Southwest (e.g., Green River; [START_REF] Merritt | Riparian vegetation and channel change in response to river regulation: a comparative study of regulated and unregulated streams in the Green River Basin, USA[END_REF][START_REF] Grams | Streamflow regulation and multi-level flood plain formation: channel narrowing on the aggrading Green River in the eastern Uinta Mountains, Colorado and Utah[END_REF], the most common response to impoundment involved reduced channel capacity by aggradation and vegetation invasion of lateral deposits, such as has occurred in dryland rivers of Australia (e.g., Cudgegong River; [START_REF] Benn | Complex channel response to flow regulation: Cudgegong River below Windamere Dam, Australia[END_REF], the Mediterranean Basin (e.g., Medjerda River in Tunisia; [START_REF] Zahar | Impacts of large dams on downstream flow conditions of rivers: Aggradation and reduction of the Medjerda channel capacity downstream of the Sidi Salem dam (Tunisia)[END_REF], and South America (e.g., Chubut River in Argentina; [START_REF] Kaless | Cambios hidrológicos y morfológicos en el río Chubut aguas debajo de la presa Florentino Ameghino[END_REF]. The studies cited above, among many others, have certainly contributed to the understanding of form to process relationships in regulated dryland rivers. Yet most of the previous research has centred on locations where tributaries contribute water and sediment below the dam, and very few studies have investigated geomorphic adjustments in flood-driven, regulated dryland rivers where tributary influence below the dam is scarce. This paper examines geomorphic and vegetation adjustments in a meandering dryland river where (i) flow regulation is extreme and (ii) sources for flow and sediment below the dam are limited to erratic reservoir discharges and reworked local alluvium because the river flows without influence of tributaries. It aims to address two fundamental questions: (i) how do flood-driven, dryland channels adjust their morphology in highly regulated, low stream power settings with scarce influence of tributaries? and (ii) how does riparian vegetation interact with (and react to) altered river hydrology and morphology in dry, highly regulated fluvial settings? The study centres on the Sauce Grande River below the Paso de las Piedras Dam (central-eastern Argentina), where prior geomorphic assessment is limited to reconstructions of Quaternary fluvial processes at the broad scale of the river basin. Change in channel geometry (planform and cross section) and riparian vegetation structure are quantified simultaneously above and below the dam using a sequence of rectified aerial photographs and high-resolution imagery spanning pre-and post-dam channel conditions. Changes as a function of distance downstream from the dam are also examined.

Materials and methods

Study site

The Sauce Grande River collects its waters on the eastern slope of the Sierra de la Ventana Range and flows down into the Atlantic Ocean draining a basin area of ~4600 km 2 (Fig. 1).

The climate in the majority of the river basin is dry-subhumid. Mean annual rainfall decreases from 800 mm in the uplands to 640 mm in the lowlands, and mean annual potential evapotranspiration is 1030 mm [START_REF] Paoloni | Caracterización hidrológica de la cuenca del río Sauce Grande en primera fase[END_REF]. Interannual rainfall variability is marked and linked primarily to alternating phases of ENSO [START_REF] Scian | Episodios ENSO y su relación con las anomalías de precipitación en la pradera pampeana[END_REF], inducing episodes of drier-and wetter-than-normal climate [START_REF] Campo | Análisis de las variaciones anuales de precipitación en el Suroeste bonaerense, Argentina[END_REF][START_REF] Bohn | Análisis de los periodos secos y húmedos en el sudoeste de la provincia de Buenos Aires (Argentina)[END_REF]. Recurrence of drought and increasing water demand owing to population growth very seriously impact on local water resources [START_REF] Andrés | Informe técnico sobre aspectos relevantes de la problemática del agua en la región de Bahía Blanca[END_REF]. Fig. 1. Map of the Sauce Grande River basin showing main regional features (left) and the river segments selected for analysis (right). The location of reference river reaches (R) along each river segment is also illustrated. US, MS and LS in the left map designate the upper, middle, and lower river sections, respectively.

Hydrological context

The natural river flow regime is perennial flashy (rainfed) and event-driven. Mean daily flow is 3.4 m 3 s -1 , but floods may reach more than 1000 m 3 s -1 in a few hours [START_REF] Schefer | Los recursos hídricos y el abastecimiento de agua[END_REF]. The Paso de las Piedras Dam (initiated in 1972 andcompleted in 1978) impounds the middle river section for water supply to the cities of Bahía Blanca and Punta Alta (Fig. 1). Its reservoir has a surface area of 36 km 2 and a maximum storage capacity of 328 m 3 10 -6 for 25 m depth [START_REF] Schefer | Los recursos hídricos y el abastecimiento de agua[END_REF]. Annual yield to meet water demand is 65 m 3 10 -6 ; this is about 60% of the mean annual inflow volume (107.2 m 3 10 -6 ). The remaining volume is conserved within the reservoir to assure water supply in periods of drought; flow release downstream occurs only in periods of full storage. Water is conducted either through the bottom gate (controlled flow releases) or through the overflow spill weir (uncontrolled release).

Figure 2 illustrates monthly runoff volumes at La Toma (Fig. 1) and monthly volumes of reservoir discharge as a surrogate of flow magnitude upstream and downstream from the dam.

Gauged and estimated reservoir discharges indicate that the reservoir has absorbed the full range of inflow volumes over 19 years of record (48%). In years recording reservoir discharges, annual runoff volumes were reduced by 54%, annual maxima were reduced by 42%, and annual minima were reduced by 86%. The reservoir history registers only two episodes of overflow spills (1984 and 2002), during which the flood wave was attenuated by 31 and 34%, respectively. In the absence of flow release, downstream flow averages 0.26 m 3 s - 1 and originates from reservoir seepage below the dam structure. Except for some ephemeral tributaries, the Sauce Grande River receives no permanent flow inputs until its confluence with de las Mostazas Creek (lower river section; Fig. 1) about 110 river km downstream from the dam closure. [START_REF] Makhlouf | A two-parameter monthly water balance model for French watersheds[END_REF]. Monthly volumes of reservoir discharge over the period 1973-1988 were calculated by solving the reservoir water balance equation (missing data). Daily reservoir volumes were illustrated to provide an idea of the hydrological magnitude of the dam relative to that of the river. Circles indicate episodes of overflow spills.

Geomorphic context

In its upper-middle course, the Sauce Grande River exhibits clear meandering patterns, low longitudinal gradient (0.002, in average), and pool-riffle bed morphology. It flows on the bottom of a broad and deeply incised valley [START_REF] Rabassa | Variación regional y significado geomorfológico de la densidad de drenaje en la cuenca del río Sauce Grande, provincia de Buenos Aires[END_REF] excavated on Pliocene-Pleistocene loess deposits [START_REF] Quattrocchio | Cuaternario del Sur de la provincia de Buenos Aires: Estratigrafía e inferencias paleoambientales[END_REF]. Three terrace remnants along the valley (T0 to T2; Fig. 3) give evidence of at least three superimposed sequences of valley incision and compound fill, including gravitational, aeolian, and ephemeral fluvial deposits [START_REF] Zavala | Estratigrafía y evolución geológica del río Sauce Grande[END_REF][START_REF] Quattrocchio | Landscape evolution and climate changes in the Late Pleistocene-Holocene, southern Pampa (Argentina): Evidence from palynology, mammals and sedimentology[END_REF]. Fig. 3. Schematic cross sections of the Sauce Grande River showing typical geomorphic and geologic characteristics within the upper and middle river sections. Geologic features were modified from [START_REF] Borromei | Geología y palinología de los depósitos cuaternarios en el valle del río Sauce Grande[END_REF], Zavala and[START_REF] Zavala | Estratigrafía y evolución geológica del río Sauce Grande[END_REF][START_REF] Quattrocchio | Landscape evolution and climate changes in the Late Pleistocene-Holocene, southern Pampa (Argentina): Evidence from palynology, mammals and sedimentology[END_REF]. Not to scale. Downstream variations in channel forms and size are apparent (Fig. 3) and conceivably linked to the attenuation of floods and flow transmission losses characteristic of dryland environments (Tooth, 2000a). In the upper section, the river channel is wide, shallow, and develops on the bottom of the incised valley. Migration of the meanders and point bar building outward led to formation of small terrace steps that have changed the original configuration of the valley bottom. In the middle river section, the valley broadens markedly, the terraces lose altitude, and the degree of confinement decreases. The river flows through a narrow, deep, and tortuous channel that constitutes a remnant of an ancient braided morphology [START_REF] Borromei | Geología y palinología de los depósitos cuaternarios en el valle del río Sauce Grande[END_REF]. Former braids were covered by aeolian deposits, and the present floodplain constitutes an abandoned portion of the original braided channel.

Scales of analysis

This paper quantifies post-dam geomorphic and vegetation changes across multiple scales (Fig. 4). Changes with time were inspected using a chronological sequence of historical aerial photographs (1961,1981) and high-resolution imagery (Ikonos, 2004) defining two comparative periods. The first period spans a pre-disturbance phase, characterized by unregulated flow conditions, and a phase of early disturbance linked to dam completion in 1978 and initial filling of the reservoir (Fig. 2). The second period spans the post-disturbance phase where the dam controls the magnitude, frequency, duration, and variability of downstream flows. Changes from unregulated to regulated river conditions for a given time step were inspected by contrasting two river segments located above and below the dam (~26-and 41-km length, respectively). In addition, field-based descriptions of present channel conditions (2011-2012) were performed for eight reference reaches of 1-km length located above (3 reaches) and below (5 reaches) the dam (Fig. 1). Fig. 4. Spatial and temporal scales used in geomorphic and vegetation analysis. Quantification of change within the regulated river segment (RSD) is performed with time (RSD → RSD") and with respect to the unregulated river segment upstream (RSU). Upstream and downstream river segments were delineated by the extent of the river corridor in each time step.

GIS-and field-based analyses

We have mapped fluvial forms and land cover types within each river segment and for each time step used in analysis using ArcGIS (ESRI©). Mapped fluvial forms included the river channel and major floodplain features such as abandoned meanders, oxbow lakes, and terraces (Fig. 4). In this paper, the term 'river corridor' encompasses the river channel (delimited by the top portions of cutbanks and point bars) and the active floodplain, i.e., the alluvial surface next to the channel, separated from the channel by banks, and built on materials transported and deposited by the present regime of the river (Graf, 1988, p. 214).

Mapped land cover types include: (i) surfaces with none or little vegetation (bare surfaces);

(ii) surfaces covered by water; (iii) agricultural lands, including crops and pastures; (iv) grasslands, dominated by graminoid species of Stipa and important associations of pampas grasses (Cortaderia seollana); and (v) woodlands, clearly dominated by native species of willow (Salix humbodltiana). In vegetation analysis, the term 'riparian vegetation' refers explicitly to grasses and woods developing within the river corridor. Field-based analysis of present channel conditions in the reference reaches used the stream reconnaissance scheme devised by [START_REF] Thorne | Guidelines for the use of stream reconnaissance record sheets in the field[END_REF] and included: (i) survey of the channel geometry (slope, planform, and cross section); (ii) description of channel forms and materials, and (iii) interpretation of vertical and lateral relations of channel to floodplain.

Quantification of geomorphic and vegetation changes

Geomorphic changes

Geomorphic analysis considered two aspects of the channel geometry: the channel planform and the channel width as indicator of channel capacity. Quantification of planform change used measures of channel activity [START_REF] Wellmeyer | Quantifying downstream impacts of impoundment on flow regime and channel planform, lower Trinity River, Texas[END_REF]. Overlay of channel bank lines between consecutive time steps [T1 : T2] provided three distinct types of channel area: (i) the area occupied during both time periods (planform stability), (ii) the area occupied only in the second time step (channel creation), and (iii) the area occupied only in the first time step (channel abandonment). Percent of channel stability, creation, and abandonment were calculated relative to total channel area in T1. Rates of channel creation (or abandonment) were calculated as channel area gained (or lost) in T1 : T2 divided by the time interval length [T2 -T1]. To determine rates of stream lateral migration in terms of length per unit time [START_REF] Shields | Reservoir effects on downstream river channel migration[END_REF], we measured the area enclosed by successive stream centrelines (active migration area, Am). Rates of stream lateral migration were then calculated as the ratio of Am to the stream centreline length in T1, divided by the time interval length [T2 -T1].

Quantification of width change used series of transects splitting the channel every 200 m of channel length. This provided two types of measure: absolute channel width, given by the length of each transect, and relative channel width, calculated as the area enclosed between consecutive transects divided by channel length. Differences in width [w] between consecutive time steps [wT2 -wT1] provided a net measure of channel narrowing (negative differences) or widening (positive differences) along each river segment.

Vegetation changes

Changes in the composition of the fluvial corridor were inspected by overlaying successive land cover maps. This provided a measure of the proportion (%) of riparian landscape gained or lost by each land cover type. The direction and the strength of land cover changes with time were assessed based on cross-tabulation of results and interpretation of transition diagrams. Changes in riparian vegetation were measured in terms of (i) surface area gained or lost by each vegetation type (changes in vegetation structure) and (ii) transitions to (or from) other landscape units (vegetation dynamics). This permitted identification of patterns of vegetation establishment (nature and extent) along the river course and with time.

Error analysis

Errors in quantification of geomorphic and vegetation change originate from (i) distortions in the geometry of aerial photographs used as a data source and (ii) digitizing procedures.

Scanned aerial photographs were first corrected for scale and terrain distortions using ground control point orthorectification in ErMapper. Orthorectification used high-resolution imagery ( 2004) and a 10-m contour-derived DEM [START_REF] Casado | Evaluación de la calidad de modelos digitales de elevación derivados de curvas de nivel para aplicaciones hidrológicas[END_REF]. Orthorectified photographs were then transformed in ArcGIS using the adjust transformation algorithm as it optimizes for global and local accuracy. The RMSE from transformation of single photographs averaged 2.2 m (SD = 1.8 m) for 1961 photographs and 1.2 m (SD = 1.0 m) for 1981 photographs; mean RMSE for all photographs was 1.7 m (SD = 1.5 m). Errors associated with digitizing procedures were estimated using the method of [START_REF] Downward | A methodology for quantifying river channel planform change using GIS. Variability in Stream Erosion and Sediment Transport[END_REF]. Positional errors were calculated from redigitizing the same feature 50 times using 2004 Ikonos imagery as source.

For a digitizing scale of 1:2500, errors averaged 0.7 m (SD = 0.4 m). Mean errors from transformation of the photographs and digitizing procedures can be accumulated to give a total mean error of 2.4 m. The total error margin for an exceedance probability of 10% is 4.8 m.

Results

Geomorphic change

Planform change

The unregulated river exhibited marked channel activity (Table 1; Table 1 Rates of channel and stream activity within the unregulated and regulated river segments by comparative period Channel migration (m²10 -2 ) Unregulated river segment Regulated river segment 1961-1981 1981-2004 1961-1981 1981- Conversely, the river channel below the dam exhibited marked planform stability during both comparative periods (93 to 97%). Channel activity during the period 1961-1981 was closely related to waterworks conducted in the vicinity of the dam (e.g., deviation of the river course from its original position to the dam outlet; Fig. 5, Ex. 5). Localized translation of meander bends occurred in the middle-upper river segment, and two abandoned bends were observed at river km 2 and 13 below the dam. Whereas the cutoff morphology in the first case suggests a human-related origin, the latter was conceivably cutoff by high flows occurring during the pre-disturbance phase (Fig. 5, Ex. 6). Meander cutoff and abandonment of meander bends were dominant processes during the post-disturbance phase ). Yet four out of five cutoffs displayed a human-related origin (e.g., two major cutoffs facilitate canalization of dam spills; Fig. 5, Ex. 7). Percentage of channel abandonment was five times that of channel creation (7.4 and 1.4%, respectively), and the channel sinuosity was reduced by 2.7%.

Width change

Upstream from the dam, coupled channel widening and narrowing occurred during both comparative periods with a tendency to narrowing during the second comparative period (Fig. 6). Width changes occurred most likely in river sections exhibiting active lateral migration.

Thus, many channel sections narrowed owing to lateral accretion and outward migration of meander bends (up to 50% during 1961-1981, and up to 60% during 1981-2004), and river sections dominated by cut bank erosion increased in width by up to 62% (1961-1981) and 55% (1981-2004). Below the dam, the channel revealed a tendency to narrowing during both comparative periods (Fig. 6). The period 1961-1981 was characterized by marked contraction of the channel width (12 to 33%) owing to deviation and artificial straightening of the river course in the vicinity of the impoundment. During the period 1981-2004, the channel width contracted in 31 sites (15% of cases). The percentage of channel narrowing decreased in the downstream direction from 44 to 4% and responded to two distinct, localized processes.

Immediately below the dam and over ~4 river km, the channel narrowed by between 9 and 44% owing to artificial meander cutoff, building of artificial levees, and other dam-related waterworks. Downstream from river km 4, the position of bank lines remained relatively unchanged. However, contraction of the channel width (6 to 30%) was inferred by vegetation encroachment over former bare channel surfaces.

Vegetation changes

An overall tendency to vegetation growth was observed simultaneously upstream and downstream from the dam (Table 2). Yet there were marked upstream-downstream differences concerning (i) the spatial extent of vegetation growth, (ii) the nature of vegetation transitions, and (iii) the patterns of vegetation establishment.

First, the expansion of woodlands within the river corridor downstream from the dam was four times smaller than that observed within the unregulated river upstream (91 and 397% between 1961 and 2004, respectively), and the surface area occupied by woodlands in 2004 was 53% of that observed upstream (Table 2). Yet the proportion of woodlands as percentage of total river corridor surface area in 2004 was similar for both river environments (23 and 28%, respectively). Second, the river corridors upstream and downstream from the dam revealed contrasting trends in vegetation dynamics (Fig. 7). Other than the expansion of agricultural lands over grasslands (583% between 1961 and 2004; Table 2), the most important transitions downstream occurred from water surfaces to woodlands (38%). This suggests a tendency for woody vegetation establishment within the river channel, near to the water surface. A moderated tendency for woody vegetation growth over former bare banks (18%) and grasslands (16%) was also identified. Yet the expansion of woodlands over these landscape units was notably less significant than that observed within the river corridor upstream from the dam (Fig. 7). Upstream, woodlands expanded within the river channel and the floodplain, covering 33% of former grasslands and 57% of former bare banks.

Present channel conditions

Present channel forms clearly differentiate river reaches located above and below the dam.

The river channel above the dam (Fig. 9) is broad and shallow ( 27

Discussion

Geomorphic response of the Sauce Grande River to flow regulation

The construction and operation of the Paso de las Piedras Dam led to substantial changes in the hydrology of the Sauce Grande River downstream. The frequency and magnitude of floods were dramatically reduced, and the permanence of low flows increased significantly.

Although the sediment trapping efficiency of the reservoir may be assumed as near 100% [START_REF] Petts | Impounded Rivers[END_REF][START_REF] Williams | Downstream effects of dams on alluvial rivers[END_REF], this study postulates that under conditions of extreme reduction in flow discharge, the impacts of the dam on the sediment transport capacity of the stream were more significant than those on the sediment load.

If reduction in flow discharge is greater than that in sediment load, then one would expect a dominance of aggradation processes over erosion and scour [START_REF] Petts | Complex response of river channel morphology subsequent to reservoir construction[END_REF][START_REF] Brandt | Classification of geomorphological effects downstream of dams[END_REF], as well as a tendency to channel narrowing and vegetation encroachment over bank deposits increasing the channel roughness [START_REF] Petts | Dams and geomorphology: Research progress and future directions[END_REF]. The geomorphic response of the Sauce Grande River, however, was characterized by (i) marked channel stability, (ii) vegetation encroachment reflecting (and influencing) channel stability, and (iii) localized reduction of the pre-dam channel capacity owing to vegetation growth. The following sections expand on these findings and compare the geomorphic response of the Sauce Grande River to other impounded rivers in dryland regions (Table 3).

Channel stability

One of the most common consequences of impoundment of meandering rivers in drylands is the reduction of lateral migration rates (e.g., the upper Missouri and many other meandering rivers within the American Great Plains; Table 3). Within the regulated Sauce Grande, however, lateral channel migration was not reduced but inhibited. Any evidence of lateral channel activity constrained to the phase of pre-and early disturbance and major planform changes observed during the post-disturbance phase -such as meander cutoff and abandonment of meander bends -revealed a human-related origin in 80% of cases. Planform stability has been reported for very few meandering rivers (e.g., the lower Murray; Table 3) and appears to be a more common consequence within regulated braided rivers (e.g., Orange

and Durance rivers; Table 3).

Moreover, studies on impounded dryland rivers subject to reduced and overloaded flows have reported a substantial reduction of the channel capacity owing to lateral and vertical accretion (e.g., Rio Grande, Green, and Medjerda rivers; Table 3). In opposition, the regulated Sauce Two interrelated processes explain marked planform and cross section stability in the river section below the dam. First, the morphology of the unregulated Sauce Grande is driven by floods of high relative magnitude and low frequency common to most dryland rivers [START_REF] Graf | Fluvial processes in dryland rivers[END_REF]Tooth, 2000b;[START_REF] Nanson | A global perspective on dryland rivers: perceptions, misconceptions and disctinctions[END_REF][START_REF] Tooth | Dryland fluvial environments: assessing distinctiveness and diversity from a global perspective[END_REF]. Since the Paso de las Piedras Dam retains nearly the full range of floods, the processes promoting migration of meanders bends and adjustments of the channel cross section downstream were heavily truncated. As a result, the broad morphology of the regulated Sauce Grande remained unchanged since dam closure and is likely to exhibit stable conditions until a geomorphically significant flow event occurs.

Second, reduction of the magnitude and frequency of morphogenetic flows should promote channel aggradation. However, and as reported by [START_REF] Petts | Complex response of river channel morphology subsequent to reservoir construction[END_REF], channel aggradation below dams requires introduction of sediment from tributaries and/or redistribution of sediment within the channel. Within the regulated Sauce Grande both processes are unlikely. The river below the dam flows for the majority of its course without influence of tributaries and, in the absence of reservoir discharges, sources for water and sediment are limited to (i) reservoir seepage, (ii) groundwater inflows, and (iii) overland flow along the river course. Combined with the endogenous production of organic matter, these processes could explain bed deposition of very fine materials. On the other hand, redistribution of sediment within the regulated river channel is unlikely because the regulated flow is incompetent to rework the pre-dam channel forms.

Vegetation dynamics

In opposition to what was expected, results revealed a synchronized tendency to afforestation of the river corridor upstream and downstream from the dam. Vegetation growth involved establishment of woody species such as willows (Salix humboldtiana, native species) and poplars (Populus sp., exotic species) and conceivably responded to a cycle of wetter climate that affects central-eastern Argentina since the 1970s [START_REF] Penalba | Interdecadal and interannual variations of annual and extreme precipitation over central-northeastern Argentina[END_REF]. As argued by [START_REF] Braatne | Analyzing the impacts of dams on riparian ecosystems: a review of research strategies and their relevance to the Snake River through Hells Canyon[END_REF], cycles of dry and wet climate drive episodes of decline and recovery in riparian communities, and the authors outlined that these natural cycles provide a variable baseline upon which impacts of damming and flow-regulation are superimposed (p. 278).

Moreover, results revealed that the extent of the riparian zone downstream from the dam has been reduced for much of the river course (and keeps reducing) because of the progression of agricultural lands over the floodplain. The interacting influence of land use changes appends a second factor superimposing the effects of dam operation on riparian vegetation.

Irrespective of climate and land use changes influencing vegetation dynamics at the scale of the river basin, vegetation structure and successions along the river corridor are primarily driven by a combination of hydrologic and geomorphic factors [START_REF] Hupp | Riparian vegetation and fluvial geomorphic processes[END_REF][START_REF] Hughes | Floodplain biogeomorphology[END_REF][START_REF] Corenblit | Reciprocal interactions and adjustments between fluvial landforms and vegetation dynamics in river corridors: A review of complementary approaches[END_REF]. Accordingly, altered hydrology and morphology within the regulated Sauce Grande explain upstream-downstream differences in the spatial extent of vegetation growth, in the nature of vegetation transitions, and in patterns of vegetation establishment with distance downstream. . Hydrologic factors. Flow regulation caused two major changes in the river hydrology below the dam that have had direct influence on patterns of vegetation establishment. First, and as reported for many regulated rivers in dryland regions (Table 3), the maintenance of base flow levels has encouraged vegetation to establish on the channel bed and banks. Second, the reduction of the magnitude and frequency of floods resulted in a disruption of lateral processes of river-floodplain connectivity (productivity and disturbance) maintaining riparian succession and diversity [START_REF] Ward | Ecological connectivity in alluvial river ecosystems and its disruption by flow regulation[END_REF][START_REF] Ward | Riverine landscapes: biodiversity patterns, disturbance regimes, and aquatic conservation[END_REF]. Recruitment of riparian pioneers no longer occurs, and woody vegetation establishes within the main channel, near to the water surface. In contrast to other impounded rivers in drylands, where altered flood regimes caused substantial changes in the composition of riparian communities (e.g., invasion of exotic, drought-tolerant species in the Green and Bill Williams rivers; Table 3), the quasisuppression of floods in this agricultural stream has encouraged farmers to extend their practices to the river channel. At present, the floodplain is no longer 'active' and the riparian zone is much reduced because of the advancement of agricultural lands.

Geomorphic factors. Downstream differences in patterns of vegetation establishment clearly reflect (and influence) channel planform stability. Channel migration promotes establishment of riparian pioneers by providing new bare and moist surfaces suitable for vegetation establishment and growth [START_REF] Hupp | Riparian vegetation and fluvial geomorphic processes[END_REF][START_REF] Friedman | Downstream effects of dams on channel geometry and bottomland vegetation: regional patterns in the Great Plains[END_REF][START_REF] Tockner | River flood plains are model ecosystems to test general hydrogeomorphic and ecological concepts[END_REF]. Within the river upstream, this aspect was evidenced by landscape transitions from bare bank surfaces to woodlands (57%). Downstream from the dam, lateral channel stability prevented formation of new channel surfaces suitable for vegetation establishment, and therefore vegetation development was constrained to the pre-dam channel shape. Immediately below the dam, the valley-like channel forms permitted proliferation of willows over riparian surfaces that are no longer subject to flood disturbance. Farther downstream, the channel is narrower and deeper, which makes the area available for vegetation establishment relatively small. Hence, willows develop discontinuously near the water surface where the bank slope decreases and the soil moisture is high. This aspect explains the high percentage of transitions from water surfaces to woodlands (38%) within the river corridor downstream.

Reduced channel capacity

Most studies on regulated dryland rivers have documented reduced channel capacity owing to channel aggradation, with vegetation invasion and stabilisation of lateral deposits (e.g., Mangrove Creek and Cudgegon River in Australia, Chubut River in Argentina, Mejerda River in Tunisia, and many others; Table 3). Within the regulated Sauce Grande, where redistribution of sediment no longer occurs and so stable conditions prevail, reduced channel capacity was primarily related to alternating yet localized vegetation encroachment on the channel banks. Vegetation encroachment was apparent over the first 10 river km below the dam, there where the pre-dam channel shape provided suitable surfaces for vegetation establishment. Downstream thereafter, the channel is too narrow to allow vegetation growth, and discontinuous patterns of vegetation establishment prevailed. In channel sections with little vegetation, field survey revealed formation of small terrace edges on channel banks (e.g., R6, Fig. 10). Although these terraces are too small to be captured by photointerpretation analysis, they suggest that the banks that have been mapped as a basis for measuring width were conceivably pre-dam banks that have been abandoned as a result of narrowing. If these banks were abandoned, then they would not display any changes in width, but the river would have formed a narrower channel within the abandoned banks. This aspect is evidenced by the fact that the river is actively wandering within the channel (Table 1) and hence suggests that it has created a new, narrower channel within the old one.

Present geomorphic state

The present river constitutes a low-energy stream characterized by marked channel stability.

The floodplain is reduced or absent for much of the river course, and vegetation encroachment on channel banks leads to localized contraction of the pre-dam channel capacity. As outlined above, marked channel stability responds to extreme flow regulation in a river section where the influence of tributaries is scarce. Yet the dam history registers erratic, albeit significant episodes of reservoir discharge and two episodes of overflow spill that resulted in downstream flooding. The question at this point is why these episodes of relative high flow did not initiate channel adjustments? We have two possible explanations for apparent channel stability during periods of flow release from the dam. First, the river channel in the vicinity of the dam has been strongly altered by humans to facilitate evacuation of reservoir discharges (e.g., artificial channel deepening, meander cutoff, levee building). Humans have done what the river would have done by itself; the difference in this instance is that the sediment recovered was not redistributed downstream but rather deposited into the levees constructed along the river course to prevent overbank flowing. Second, vegetation growth alongside the stream has conceivably increased the channel roughness and the resistance to erosion of bank materials. In dryland rivers, where complexity and irregularity in morphology, hydrology, and riparian vegetation are the norm [START_REF] Sandercock | Vegetation in dryland river channels and its interaction with fluvial processes[END_REF], biostabilisation of channel banks and maturation of individuals may contribute to increased thresholds for geomorphic activity [START_REF] Corenblit | Biogeomorphologic succession dynamics in a Mediterranean river system[END_REF].

In opposition to these assumptions, however, artificial levee building has been reported to increase rates of bank erosion because the unconsolidated sediment was easily eroded once bank-full flows occurred [START_REF] Leeks | Channel change, fluvial geomorphology and river engineering: the case of the Aton Trannon, mid-Wales[END_REF]. Furthermore, the comprehensive review of Tooth (2000b) on dryland rivers outlined that even channels heavily vegetated with perennial or ephemeral trees, shrubs and grasses are sometimes subject to flood-related changes (p. 83). These aspects raise two important questions. First, even if vegetation did provide an element of resistance to bank erosion, was the maturation of individuals rapid enough to prevent the levees from being eroded? Second, even if channel banks were biostabilised, was vegetation establishment significant enough to increase thresholds of geomorphic activity? As reported by [START_REF] Petts | Long-term consequences of upstream impoundment[END_REF], the potential for change in regulated rivers depends not only on the channel morphology and materials, but also on the frequency and intensity of competent discharges below the dam. In drylands, this aspect is particularly relevant because a 'competent discharge' may be an infrequent, large flood unrepresentative of a more common range of floods (Tooth, 2000b). Accordingly, marked channel stability within this highly regulated dryland river may be interpreted within a model of channel adjustment relative to the frequency and intensity of competent floods with time and distance from dam closure (Fig. 11). The unregulated Sauce Grande is in a transient, unstable state (sensu [START_REF] Tooth | Equilibrium and nonequilibrium conditions in dryland rivers[END_REF] where complete adjustment of form to process is inhibited until a geomorphically significant flood event occurs. After impoundment by a large surface-area, water-storage reservoir such as Paso de las Piedras, competent floods downstream are unlikely. The channel morphology will not change, and downstream flows will simply accommodate within the existing channel form. As the regulated Sauce Grande receives no major tributaries, the probability of competent floods decreases with distance downstream and so stability will prevail throughout all reaches (Fig. 11). Reservoir discharges during dry spells are unlikely, and if a reservoir discharge is to occur, its magnitude will be most likely below the threshold for geomorphic activity. Thus, the time-lag before channel changes initiate may be considerably longer than in more humid rivers having the propensity to accommodate change, and longer than in dryland rivers regulated by dams having less effects on the frequency and intensity of competent floods downstream. In the absence of competent floods with time, the channel morphology will not change and the river will exhibit stable, moribund conditions. [START_REF] Thorne | Geomorphological river channel reconnaissance for river analysis, engineering and management[END_REF] defined moribund channels as those that are not strictly alluvial because the observed channel form is the result of processes that operated in the past under conditions of higher energy and/or more abundant sediment supply, so that the geometry and features of the channel are relics of a fluvial environment that no longer exists (p. 471). Accordingly, channel forms will not reflect the regulated flow regime but the effects of the last major flood before closing the dam, and the only remarkable change throughout the reaches will be determined by vegetation invasion and stabilisation of channel banks in sections where the pre-dam channel shape allows vegetation establishment (Fig. 11). Assuming that dam operation procedures will not change in the future, moribund stability will remain indefinitely and if recovery is to occur at all, further human intervention through river restoration is essential (Thorne et al., 1996, p. 471).

Yet a geomorphically significant flood is likely to occur with time (Fig. 11). If so, degradation may occur immediately below the dam (reach 4) increasing channel depth and width.

Differential erosion by meander migration may produce localised changes of the channel width in reach 5. Given the low energy of the river section below the dam, the eroded material will be deposited in reach 6 and, within the distal reaches, the channel morphology will not change. Tooth (2000a) reported that in dry, lowland zones with absence of tributaries, channels were stable because of the effects of flow transmission losses and attenuation of flood hydrographs. Although the most obvious spatial comparison contrasts river reaches upstream and downstream from a dam [START_REF] Braatne | Analyzing the impacts of dams on riparian ecosystems: a review of research strategies and their relevance to the Snake River through Hells Canyon[END_REF], channel stability in this lowgradient lowland river section probably existed prior to dam closure and will remain unless a high-magnitude, low-frequency flood event occurs.

Conclusions

This paper provided a detailed geomorphic and vegetation assessment within an impounded meandering dryland river where (i) flow regulation is extreme and (ii) the influence of tributaries in the river section below the dam is scarce. In addition to providing new information on the complex geomorphic response of dryland rivers to impoundment, it provides the first geomorphic assessment of the regulated Sauce Grande River. Results from this investigation revealed that aside from human-related adjustments of the channel planform and cross section, the river response to upstream impoundment was characterized by marked channel stability. Because the reservoir impounds nearly the full range of flows and the totality of sediment delivered from the headwaters, redistribution of sediment in the river section below the dam no longer occurs and stable conditions prevail. The present river morphology constitutes a remnant of a fluvial environment created before closing the dam, and the only remarkable change observed since dam closure was reduced channel capacity owing to vegetation encroachment on the channel banks. Vegetation growth alongside this much-reduced stream has been encouraged by maintenance of base flow levels and reduced flood disturbance. Yet vegetation develops only in river sections where the pre-dam channel shape is large enough to allow establishment of pioneers. In other words, vegetation establishment is influenced by (and reflects) channel stability.

These findings serve as an important platform to enable further research assessing the present geomorphic state of the regulated Sauce Grande River, and they lead to a number of additional questions that require further attention. While the temporal dimension could be assessed robustly, the spatial dimension requires further research efforts to understand the nature of the relationships between form and process within the regulated river as well as the influence of transmission losses (evaporation and infiltration) and attenuation of the flood hydrograph on the downriver channel morphology. One major challenge for further research is the lack of historical flow data along the river course. This makes estimations of pre-and post-dam channel hydraulics very difficult to achieve. Thus, detailed field observations and sediment analyses will be required to identify temporal and spatial variations in the relationship between stream power and critical thresholds for geomorphic effectiveness of regulated flows below the dam. Similarly, this study outlined some interconnections between regulated flow, channel forms, and vegetation dynamics. However, detailed vegetation analyses are required to assess the complex mechanisms by which vegetal successions in this regulated river influence (and are influenced by) channel morphology.

Fig. 2 .

 2 Fig. 2. Monthly runoff volume upstream from the dam and monthly volumes of reservoir

  Fig. 5). Although new channel surfaces were created by meander migration (10.6%) and cutoff (2.1%), the period 1961-1981 was characterized by abandonment of channel surfaces owing to meander translation (15.1%; Fig. 5, Ex. 1) and abandonment of meander bends (3.1%; Fig. 5, Ex. 2). Channel activity during the period 1981-2004 was clearly dominated by meander extension and translation. High percentages of channel abandonment (19.8%) indicate a tendency to channel straightening. Yet increased bend amplitude in some river sections was notable (Fig. 5, Ex. 3), and three major cutoffs in the middle-lower river segment (Fig. 5, Ex. 4) give evidence of the geomorphic effectiveness of floods during the second comparative period.

Fig. 5 .

 5 Fig. 5. Channel and stream activity within the unregulated and regulated river segments by

Fig. 7 .

 7 Fig. 7. Trends in vegetation dynamics within the river corridors upstream and downstream

Fig. 8 .

 8 Fig. 8. Surface area occupied by woodlands by river reach (R) and time step. Pie charts

  < w:d < 37), and the floodplain is well developed for much of the river course. Channel width averages 53 m (±16 m), and floodplain width averages 135 m with high variation (±94 m). The channel morphology is clearly meandering for all reaches; variations in sinuosity (1.1 < 1.5), meander wavelength (150 m < 600 m), and meander width (20 m < 200 m) are probably linked to local variations in slope and degree of confinement within the incised valley. Bed material is coarse in riffles (cobbles and coarse gravel) and finer in pools (small gravels, sand and silt); banks are composed of noncohesive deposits in point bars (sand and gravel) and of layered, cohesive paleofluvial and aeolian deposits in cut banks (fine sand, silt and clay). Bank profiles reveal active cut bank erosion including bank undercut and toe scour (especially in R3; Fig. 9); channel sections exhibiting vertical banks display coupled toe accumulation by mass wasting. Lateral accretion in point bars is a more dominant process. Outward migration of the meanders led to formation of terrace steps contained within the valley, and hence topographically lower than the preincision surface.The river channel below the dam is narrow and deep (7 < w:d < 10), and the floodplain is much reduced (Fig.10). Mean channel width is 21 m (±6 m), and mean floodplain width is 53 m (±30 m). Except for R5 where meanders are very tortuous (S = 2.8), sinuosity increases in the downstream direction from 1.3 to 2.2. Bed material is composed of small gravel in riffles and of silt/clay in pools; banks exhibit cohesive layers of fine 'spring' and ephemeral fluvialaeolian deposits. Bank profiles display stable conditions for most river reaches, except for R4 where mass wasting occurs in vertical banks with little vegetation cover. Immediately below the dam, any evidence of channel activity is related to human intervention. For example, artificial bed deepening in R4 facilitates evacuation of dam outlets and spills, and artificial levees (built by dredging floodplain material) prevent bank overflowing. Outside the vicinity of the impoundment, bed deposition of very fine materials (silt/clay and organic matter) leads to localized reduction of the channel depth (e.g., R6; Fig.10).

Fig. 9 .

 9 Fig. 9. Geomorphic features, land cover types, and representative cross sections for two reference reaches located upstream from the Paso de las Piedras Dam. The locations of the reference reaches along the unregulated river segment upstream are shown in Fig. 1.

Fig. 10 .

 10 Fig. 10. Geomorphic features, land cover types, and representative cross sections for two reference reaches located downstream from the Paso de las Piedras Dam. The locations of the reference reaches along the regulated river segment downstream are shown in Fig.1.

  photointerpretation analysis (i.e., a true reduction of the channel width by redistribution of

Fig. 11 .

 11 Fig. 11. Adjustment of the channel morphology to extreme flow regulation below water-

Fig. 11 .

 11 Fig. 11. Adjustment of the channel morphology to extreme flow regulation below water-

Table 2

 2 Surface area by land cover type within the unregulated and regulated river segments per time step; rates of change with time were calculated based on absolute surface differences per time interval; highest percentages of change by cover type are indicated in bold

		Surface area (ha)	Difference (ha)	Rate of change (%)
	Cover type	1961	1981	2004	1961-81 1981-04 1961-81 1981-04
	Unregulated river segment upstream from the dam
	Agricultural lands	24.9	10.7	8.8	-14.2	-1.9	-57.0	-18.0
	Water surfaces	50.4	149.9	157.3	99.4	7.4	197.1	4.9
	Bare soils	66.1	33.3	10.9	-32.8	-22.4	-49.6	-67.3
	Grasslands	250.2	192.7	121.8	-57.5	-70.9	-23.0	-36.8
	Woodlands	23.4	28.5	116.4	5.1	87.9	21.8	308.0
	Regulated river segment downstream from the dam
	Agricultural lands	12.7	45.1	86.8	32.4	41.6	255.2	92.2
	Water surfaces	28.8	26.9	20.8	-1.9	-6.1	-6.6	-22.7
	Bare soils	1.0	2.1	0.2	1.1	-1.9	102.4	-92.8
	Grasslands	191.3	155.5	96.5	-35.8	-59.0	-18.7	-37.9
	Woodlands	32.4	36.6	62.0	4.2	25.4	12.9	69.3

Table 3

 3 Examples of geomorphic changes of regulated dryland rivers by world region

	River	Impoundment Purpose	Channel adjustments*	Citation
	North America
	Colorado	Glen Canyon	Multipurpose d +/-, bed armouring/aggradation, w -,	Howard and
	(Colorado)	(1963)		CC -, s -, marsh development	Dolan (1981);
					Stevens et al.
					(1995); Grams
					et al. (2007)

*[d] is channel depth, [w] is channel width, [CC] is channel capacity, [n] is channel roughness, [s] is channel slope, and [S] is stream sinuosity. The direction of change is indicated by [o] no significant change, [+] increase, and [-] decrease.

Acknowledgements

The authors thank the ADA Authority, Marta Crisafulli, and the rest of the ADA staff for permitting access to the Paso de las Piedras Dam complex, for providing assistance with equipment installation and survey, and for providing flow data. The authors also thank Paul Ledger and Johannes Steiger for providing assistance and constructive comments to the manuscript. Funding was provided to Ana Casado by the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) post-graduate scholarship. This paper

would not have the present form without the valuable comments and suggestions of Paul Grams, Carlos Zavala, and four anonymous reviewers.

(Cuaternario), provincia de Buenos Aires, Argentina. Revista de la Asociación Geológica 853 Argentina,56,[25][26][27][28][29][30][31][32][33][34][35][36][37]


Cécile Daversin

Christophe Prud'homme

Simultaneous empirical interpolation and reduced basis method. Application to non-linear multi-physics problem

Simultaneous empirical interpolation and reduced basis method. Application to non-linear multi-physics problem

Introduction

The demand in terms of real time simulations and uncertainty quantification is fast growing area in engineering, together with the size and the complexity of the considered problems. Reduced order methods, and in particular the reduced basis methods, play a critical role at breaking complexity.

Especially designed for real-time and many-query simulations, the Reduced Basis (RB) method offers an efficient evaluation of quantities of interest and covers a large range of problems among which non-affinely parametrized Partial Differential Equations (PDE). Based on the so-called offline/online decomposition, this method distinguishes the parameter independent terms whose computation is costly due to their dependence on the finite element dimension. Allowing to compute the latter only once, such a decomposition is not necessarily available in particular for nonaffine/non-linear problems. The Empirical Interpolation Method (EIM) is classically used prior to the RB methodology to recover an affinely parametrized problem ensuring the availability of the offline/online decomposition. We proposed a simultaneous EIM-RB (SER) method [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF] to circumvent the possibly dissuasive additional cost required by the EIM buidling step. Building together the affine decomposition as well as the RB approximation space, SER indeed requires only but a few finite elements solves.

Following up [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF], this paper reports a finer analysis of SER and its variants along with its expanded use to non-linear multi-physics problem. In particular we introduce a multilevel SER, SER( ) which improves significantly SER. After a reminder of the SER method in the context of non-affinely parametrized PDEs, we first give an overview of the Cécile Daversin IRMA, 7 rue René Descartes, 67084 Strasbourg cedex, France, e-mail: daversin@math.unistra.fr Christophe Prud'homme IRMA, 7 rue René Descartes, 67084 Strasbourg cedex, France, e-mail: prudhomme@unistra.fr investigated variants. All of them are based on the development of an error indicator initially designed to guide the construction of the RB approximation space from a Greedy algorithm. The second part illustrates these variants with results obtained on a benchmark introduced in [START_REF] Grepl | Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations[END_REF] on which the SER preliminary results shown in [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF] were based. The last part focuses on the application of SER and SER( ) to a non-linear multi-physics problem from the HiFiMagnet project aiming to design an efficient model for high field magnets [START_REF] Daversin | Full Three-Dimensional Multiphysics Model of High-Field Polyhelices Magnets[END_REF].

A simultaneous EIM-RB method

Let u(µ) be the solution of a non-linear and non-affinely parametrized PDE, where µ denotes the p-vector of inputs defined in the parameter space D ⊂ R p . The non-affine parametrization comes from the dependance of the PDE on at least one non-affine function w(u(µ), x; µ). Considering X ⊂ H 1 (Ω ) an Hilbert space whose scalar product is denoted as (•, •) X , the variational formulation of the PDE consists in finding u(µ) ∈ X as a root of a functional r such that r(u(µ), v; µ; w(u(µ), x; µ)) = 0 ∀v ∈ X.

(1)

Preliminaries

We denote by X N ⊂ X the finite element approximation space of dimension N in which the approximation u N (µ) of u(µ) resides. The non linearity of the considered PDE is handled through iterative methods. The following description relies on a Newton algorithm for which we introduce j the Jacobian associated with the functional r of (1), and k u(µ) the solution at k-th Newton's iteration. The problem (1) then consists in finding δ k+1 u(µ)

≡ k+1 u(µ) -k u(µ) ∈ X N such that j(u, v; µ; k u(µ); w( k u(µ), x; µ))δ k+1 u(µ) = -r( k u(µ), v; µ; w( k u(µ), x; µ)) (2)

Empirical interpolation method

As previously mentionned, the reduced basis method is based on an offline/online strategy assuming the existence of an affine decomposition of (2). The dependance of j and r on w(u(µ), x; µ) stands in the way of the availability of such decomposition. In this context, the Empirical Interpolation method (EIM) is widely used to recover an affinely parametrized problem from (2) building an affine approximation w M (u(µ), x; µ) of w(u(µ), x; µ) reading as

w M (u(µ), x; µ) = M ∑ m=1 β M m (u(µ); µ)q m (x) (3)
whose approximation coefficients β M m results from the resolution of a M × M system ensuring the exactness of w M at a set of interpolation points {t i } M i=1 . To this end, we first introduce a subset Ξ of D from which a sample SM = { μ1 , . . . , μM } ∈ D M is built. The EIM approximation space WM = span{ ξ m ≡ w(u( μm ), x; μm ), 1 m M} in which the approximation w M (u(µ), x; µ) shall reside consists of the set of w evaluations on SM elements. As starting point of the EIM algorithm, the first sample point μ1 is picked in Ξ assuming ξ 1 = 0.

ξ 1 = w(u( μ1 ), x; μ1 ), t 1 = arg sup x∈Ω | ξ 1 (x)|, q 1 = ξ 1 (x) ξ 1 (t 1 ) (4)
The next sample points { μm } M m=2 are then selected through a Greedy algorithm as

μM = arg max µ∈Ξ inf z∈W M-1 ||w(.; .; µ) -z|| L ∞ (Ω ) (5)
leading to the EIM approximation space enrichment WM = WM-1 ⊕ span{ξ M }. The computation of the coefficients β M-1 m (u( μM ); μM ) allows to evaluate the residual r M (x) = w(u( μM ), x; μM )w M-1 (u( μM ), x; μM ) defining the interpolation point t M and the next basis function q M as

t M = arg sup x∈Ω |r M (x)|, q M (x) = r M (x) r M (t M ) (6)

Reduced basis method

Defined as the linear combination of the finite element solutions forming the RB approximation space W N = {ξ n } N i=1 , the reduced basis approximation u N (µ) of u(µ) reads as

u N (µ) = N ∑ i=1 u N,i (µ)ξ i (7)
Based on a sample

S N = {µ 1 , • • • , µ N } ⊂ D with N << N , W N is built from the set of finite element solutions {u N (µ i )} N i=1
which are orthonormalized, with respect to the scalar product of X, through a Gram-Schmidt algorithm. As u N (µ), the reduced basis approximation u N (µ) has to satisfy the equation [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF]. The computation the u N, j in the RB approximation space W N consists in solving the N × N reduced system (8) considering {ξ n } N i=1 as test functions

N ∑ j=1 j(ξ j , ξ l ; µ; k u N ; w( k u N , x; µ))δ k+1 u N, j -r(ξ l ; µ; k u N ; w( k u N , x; µ)) (8)
where δ k+1 u N, j ≡ ( k+1 u N, j -k u N, j ) and with 1 j, l N.

SER method

Offering both an efficient computation of W N and an efficient assembly of the reduced system (8) by means of the precomputation of the parameter independent terms of (2), the affine decomposition obtained through the EIM approximation of w(u(µ), x; µ) is a core enabler of the reduced basis method. The initialization step (4) requires one finite element solve which is unavoidable, subsequently the finite element solves are needed by the Greedy algorithm [START_REF] Daversin | A Reduced Basis Framework: Application to large scale non-linear multiphysics problems[END_REF] and the complexity scales with the size of the EIM trainset Ξ . The EIM offline step has a prohibitive cost.

Introduced in [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF], the simultaneous EIM-RB (SER) method reduces this cost using the readily available reduced approximation into the Greedy algorithm instead of finite element solves. Due to the lack of reduced approximation at the initialization step (4) and as mentioned previously, the finite element solve of the full non-linear problem [START_REF] Daversin | Full Three-Dimensional Multiphysics Model of High-Field Polyhelices Magnets[END_REF] for the first EIM approximation w 1 cannot be avoided. ( 2) is never solved afterwards. This leads to a first rough affine decomposition making the reduced basis methodology feasible, and it results in a first reduced approximation to be used in the EIM Greedy algorithm [START_REF] Daversin | A Reduced Basis Framework: Application to large scale non-linear multiphysics problems[END_REF]. From there, EIM and RB approximations spaces are enriched alternately using solely RB approximations making the number of finite element solves required drop to one for the EIM offline step.

The parameter selection process is then based on the last reduced basis approximation u M-1

μM = arg max µ∈Ξ inf z∈W M-1 ||w(u M-1 (µ); .; µ) -z|| L ∞ (Ω ) (9)
used as well to build the current EIM basis function

ξ M = w (u M-1 ( μM ); x; μM ) . (10)
The residual r M (x) = w(u M-1 ( μM ), x; μM )w M-1 (u M-1 ( μM ), x; μM ) is also computed from u M-1 giving t M and q M from (6).

Error estimation

The Greedy algorithm used in the EIM offline step -either based on finite element solve [START_REF] Daversin | A Reduced Basis Framework: Application to large scale non-linear multiphysics problems[END_REF] or on reduced basis approximation (14) -rests on the maximum value of a functional related with the approximation error. In the absence of such an error indicator, the RB sample S N on which the RB approximation space is based is built from a random selection process. This is the case for the preliminary results displayed in [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF]. In order to improve the reliability of the reduced basis approximation, we build W N from a Greedy algorithm based on an error indicator. In the context of SER, this should improve the quality of the reduced basis approximation used in the EIM algorithm and then the affine decomposition especially during the first steps of SER.

The definition of an error bound for non-linear but affinely parametrized problems is given in [START_REF] Veroy | A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations[END_REF], based on the norm of the Riesz representation Y r of the residual r such that

(Y r , v) X = r(v) (11)
The definition of such an error bound is not readily feasible for non-linear non affinely parametrized problems. Let us now introduce r a f f N,M as the evaluation of (1) from the reduced basis approximation u N and the EIM approximation w M which served to build it.

r a f f N,M (µ) = r(u N (µ), v; µ; w M (u(µ), x; µ)) (12)
Inspired from [START_REF] Veroy | A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations[END_REF] but not providing an error bound, the norm of the Riesz representation Y r

a f f N,M (11) of r a f f N,M (12)
provides an error indicator. We then use this indicator to drive construction of S N in the Greedy algorithm

µ i = arg max µ∈D Y r a f f N,M (•; µ) X (13)

Some SER variants

Besides its use in the S N building process (13) consisting in a first SER variant, the previously introduced error indicator serves as a quantifier of the reduced basis approximation accuracy through the SER offline procedure. Various alternatives based on this error indicator have been investigated, whose most relevant ones are detailed in the following. They are still illustrated from the results obtained on the 2D benchmark introduced in [START_REF] Grepl | Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations[END_REF].

r-adaptation

We remind that the SER methodology consists of the simultaneous enrichment of EIM and RB approximation spaces whose basis functions are alternately built one by one. A first alternative consists in changing the frequency of the affine decomposition updates, corresponding to perform the alternate build per groups of size r. In this context, r = 1 corresponds to the initial SER method while r = M stands for the standard RB methodology. Intermediary stages with 1 < r < M were investigated in [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF] with r constant for the whole offline step. The error indicators come to a criterion providing guidance to perform a smart adaptation of r during the SER process introduced as r-adaptation method. As mentionned, the Greedy algorithm used both in EIM (14) and RB (13) offline stage selects the maximizer of a functional given as an indicator of the approximation error. The r-adaptation method rests on the increment of the corresponding maximum value between two updates. As a quantifier of the gain in term of accuracy, the resulting criterion aims to continue the enrichment until a relevant decrease of the approximation error. This adaptation process distinguishes the update frequency of EIM and RB approximation spaces which are considered independently each with their own criterion.

Hybrid Greedy algorithm

As mentionned, the accuracy of the reduced basis approximation plays a key role in the SM building step especially for the first EIM basis functions. Indeed, a reduced basis approximation of poor quality could damage the EIM approximation and consequently the quality of the affine decomposition. In this context, we propose to assess the quality of the reduced basis approximation from the error indicator used in (13) for each parameter of the trainset. In this way, the reduced basis approximation is employed solely on parameters for which it is sufficiently relevant. A parametric finite element solve based on the current affine decomposition is used for the remaining parameters to benefit from the precomputations while considering the whole trainset all through the SER offline step. This leads to an hybrid method combining the use of finite element (14) and RB approximations (5) within the EIM Greedy algorithm.

Multilevel SER( )

The last variant we propose rests on the application of the SER methodology several times during the offline step. The first level exactly corresponds to the previously introduced SER method whose Greedy algorithm employed for EIM is described in (14). Once the EIM and RB approximation spaces completed, the multilevel SER method consists of restarting the algorithm while benefiting from the reduced basis approximation coming from the previous level. Considering u N the reduced basis approximation obtained at the -th level, the EIM Greedy algorithm becomes

μM = arg max µ∈Ξ inf z∈W M-1 ||w(u -1 N (µ); .; µ) -z|| L ∞ (Ω ) (14)
Coming from the whole W N approximation space, the reduced basis approximations used in the EIM Greedy algorithm from the second level offer EIM approximations of better quality. Based on this consideration, the resulting reduced basis approximations are expected to be more accurate as well.

Numerical experiments

Based on the 2D non-linear and non affinely parametrized benchmark of [START_REF] Grepl | Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations[END_REF], the following numerical experiments illustrate the proposed SER variants. We consider the 2D domain Ω =]0, 1[ 2 and the parameter space D = [0.01, 10] 2 , this problem consists in finding u such that

-∆ u + µ 1 e µ 2 u -1 µ 2 = 100 sin(2πx) sin(2πy) in Ω with µ = (µ 1 , µ 2 ) ∈ D (15)
The non-affinely parametrization of the problem (15) resides in the function g(u, x; µ) = µ 1 e µ 2 u -1

µ 2 . Based on a training set Ξ ⊂ D of size 225, its EIM approximation g M = M ∑ i=1 β M
m (u, µ)q m (x) allows to recover the required affine decomposition. Let's consider the absolute error on the solution u and on the output s defined as the average of u over

Ω ε u M,N = u N -u N L 2 ε s M,N =| s N -s N | (16)
where • N is the finite element solution/output of the initial problem (1) and • N the reduced basis solution/output.

The following tables display the maximum of the absolute errors (16) obtained on a set a 1000 realizations selected randomly in D. The table 1b reproduces the results in [START_REF] Grepl | Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations[END_REF] using the standard method with an RB approximation space built from (13), confirming the relevance of the proposed error indicator. Regarding the SER method, the table 2a displays the results coming from the use of a random selection process within the construction of the RB approximation space the table 2b illustrates the use of a Greedy algorithm in this context. Although slightly higher than the errors obtained in table 1b, these errors come close the ones obtained in table 1a for a reduced computational cost. Nevertheless, the impact of the previously introduced error indicator on this application turns out to be limited. The tables 3a and 3b combine the previous Greedy algorithm used to build the RB approximation space with the first two SER variants. The table 3a focus on the r-adaptation method while the table 3b investigates the previously introduced ranking of parameters depending on the reduced basis approximation reliability within the EIM Greedy algorithm. Compared to previous SER results displayed in table 2, neither of these variants results in a significative improvement in the error. A similar behaviour is observed combining these two variants. The tables 4a to 4d display the results obtained with the multilevel SER( )) method. As expected, the results coming from the first level (table 4a) are similar to ones of table 2a. The observed disparity comes from the random selection process performed at RB offline stage which could result in a slightly different RB approximation space.

N M max(ε u M,N ) max(ε s M,N ) 5
N M max(ε u M,N ) max(ε s M,N ) 5
N M max(ε u M,N ) max(ε s M,N ) 5
From the second level illustrated by table 4b, we observe a significant decrease of the error which comes up to the standard case introduced in table 1a. However, the error is no longer visibly evolving at the next levels as shown in tables 4c and 4d. As to the results obtained with SER multilevel method combined with the use of the previous error indicator through a Greedy algorithm within the RB offline stage, they are illustrated in tables 5a to 5d. Their analysis results in the same conclusion as for table 4. The previous analysis is supported by the convergence study of the considered EIM approximation illustrated in figure 1. The graph of figure 1 plots the maximal value of the functional used in the EIM Greedy algorithm depending on the number of basis functions which compose the EIM approximation space. Compared with the EIM convergence obtained using the standard RB methodology, we can indeed notice a significant error decrease from the second level. The next levels remains comparable to the latter, already very close to the EIM approximation performed from the standard method. The HiFiMagnet project [START_REF] Daversin | Full Three-Dimensional Multiphysics Model of High-Field Polyhelices Magnets[END_REF] aims at developing an efficient multi-physics model for high field magnets. We investigate the pertinence of the SER method on a 3D non-linear electro-thermal model. Considering the multilevel variant which has just been introduced, the initial SER method is denoted as SER(1) in the following.

Electro-thermal model

The considered electro-thermal model consists of the coupling of the electrical potential V in the magnet with the resulting temperature T . Thus, the temperature T is the solution of the non-linear coupled and non affinely parametrized thermo-electric problem

-∇ • (σ (T )∇V ) = 0 -∇ • (k(T )∇T ) = σ (T )∇V • ∇V (17)
The non-linearity of ( 17) is coming from the dependance of the electrical (resp. thermal) conductivity σ (T ) (resp. k(T )) on temperature as well as the joule effect terms

σ (T ) = σ 0 1 + α(T -T 0 ) and k(T ) = σ (T )LT (18)
The temperature coefficient α and the Lorentz number L are proper to the material, and σ 0 represents the electric conductivity at reference temperature T = T 0 . Related to the current density j in the magnet, the current flow is modeled from a difference of electrical potential V D between the current input and the current output imposed as Dirichlet boundary conditions. Other boundaries are considered as electrically insulated through a homogeneous Neumann condition.

V = 0 on input,V = V D on output -σ (T )∇V • n = 0 on other boundaries (19)
The temperature increase due to the Joule effect is controled with a water cooling of the magnet corresponding to a forced convection condition on the concerned regions, based on the water temperature T w and on heat transfer coefficient h.

-k(T

)∇T • n = h(T -T w ) on cooled surfaces -k(T )∇T • n = 0 on other boundaries (20)
The input parameter µ = (σ 0 , α, L, j, h, Tw) ∈ R 6 combines material properties and operating conditions parameters, while the considered output is the mean temperature over the magnet acting as a critical parameter in terms of magnet design. The definition of σ (T ) (18) from the input parameters thus makes the model (17) non-affinely parametrized as well as non-linear. The SER methodology readily applies in this context.

Application to Bitter magnet

The first application focuses on the geometry illustrated in figure 2 whose mesh is composed of 15388 nodes. This geometry stands for a sector of a Bitter magnet commonly used in the context of high field magnet facilities. The ranges considered for the input parameter given in table 6 are chosen from physical considerations coming both from litterature and experimental measures. Performed in parallel on 8 processors, the results given hereafter are based on a set of 1000 realizations for which parameters are randomly chosen in the ranges of table 6. The non-linearity is handled by a fixed point iterative method, but this time with a Picard algorithm instead of a Newton method. Nevertheless, all previous considerations apply in the same way. As for the benchmark, we display the maximum of absolute errors (16) on solution and on output to be compared with reference results obtained with the standard method. The considered EIM trainset is of size 100.

The first tables 7a and 7b compare the SER1 method with the standard one, both based on a randomly selection process to build the RB approximation space. Turning to the previously introduced error estimator, the table 8 displays the errors obtained from its use into the parameter selection process. Besides its low impact on the 2D benchmark illustrated by the tables 2a and 2b, the Greedy algorithm used to build the RB approximation space has a significant influence on this application. Indeed, it results in errors whose order of magnitude comes close to ones obtained with standard RB methodology. The convergence study of the SER(1) method and its variants on this kind of application allows to go further in its analysis. To this purpose, the figure 3a focuses on the convergence study related with the EIM approximation considered for the electrical conductivity σ (T ). Besides the expected error decrease, this plot highlights the enhancement coming from the use of the error indicator in term of convergence. The figure 3b studies the relative L 2 error of the RB approximation depending on the number of basis functions. The resulting behaviour was expected as well. Regarding its impact on the previous 2D benchmark, we investigate as well the use of multilevel SER( ) method with l 1 on the electro-thermal problem. In this context, the table 9 compares the results obtained at first and second levels. In spite of the random selection process, the table 9a is in good agreement with the previous table 7b. The table 9b confirms the relevance of the multilevel variant since SER(2) already gives results which come close to those obtained with standard RB methodology. Table 9: SER multilevel -Random

N M max(ε u M,N ) max(ε s M,N ) 5
N M max(ε u M,N ) max(ε s M,N ) 5 5 2.
As for the figure 1 in the case of the 2D benchmark, the convergence study of the considered EIM approximations allows to go further in the analysis. To this end, the figure 4a (resp. 4b) compares the convergence resulting obtained from various level for the EIM approximation of σ (T ) (resp. k(T )). This study tends to confirm the preliminar results obtained on the benchmark which show pertinence of this SER variant.

Application to Polyhelix magnet

As an alternative of the previously mentionned Bitter magnets, the polyHelix are designed to produce high magnetic fields. Detailled in [START_REF] Daversin | Full Three-Dimensional Multiphysics Model of High-Field Polyhelices Magnets[END_REF], this technology rests on complex geometries leading to large problems in a numerical point of view. The saving in computational time offered by the SER method is thus all the more pertinent in this context. We propose to investigate its use on such a problem.

Based on a mesh composed of 2.2 millions of tetrahedra for approximatively 500000 nodes (figure 5), the next simulations are performed on 12 processors. The computer used to perform this experiment is composed of 2 multithreaded 6 cores CPUs and 141 GB of shared memory. The considered problem is similar to the one introduced in (17) for the Bitter magnet. The underlying non-linearity is handled by a Picard method with a given tolerance of 10 -6 . Except the parameter related with the current flow, the input data are similar to the previous experiment. The current density previously considered is this time replaced by the difference of potential V D which directly gives the Dirichlet boundary condition (19).

Input Range The next study is based on EIM and RB approximation spaces of size 5. It aims to compare the computational time necessary to perform the EIM Greedy algorithm with the standard RB methodology and with the introduced SER method. To this end, we focus on the EIM approximation related with the electrical conductivity σ (T ) for which the considered trainset is composed of 100 parameters.

σ 0 [50 × 10 6 , 50.2 × 10 6 ] S.m -1 α [3.3 × 10 -3 , 3.5 × 10 -3 ] K -1 L [2.5 × 10 -8 , 2.9 × 10 -8 ] V D [55, 65] V h [70000, 90000] W.m -2 .K -1 T w [293, 313] K
Consisting of the same finite element solve in both cases, the computational time related with the initialization of the EIM building step is approximatively 1760 seconds. The SER method uses the reduced basis approximation which results from the first EIM basis function while the standard methodology continue to use finite elements approximations. Regarding the first EIM Greedy algorithm for which the available reduced basis approximation rests on a single basis function, the mean time required to solve the reduced problem is 2.1 seconds compared with 2087 seconds for the corresponding finite element one. Regarding the whole set of resolutions performed within the EIM Greedy algorithm, this amounts to a factor close to 500 in terms of computational time.

Through each stage of the SER method, the EIM basis functions are built from a set of reduced basis approximations based on an enriched RB approximation space. In this context, the table 11 displays the mean time necessary for a single resolution and the resulting gain factor related with the whole EIM Greedy algorithm.

N Mean time per online realization (s) Gain factor observed for EIM Greedy algorithm Table 11: Performances of the SER method on a large scale multi-physics problem Still considering the mean temperature over the domain, the use of the standard methodology with N = M = 5 results in an maximal output error of 4.5 × 10 -2 . Only the SER method combined with the error indicator ensures the convergence of the online stage which itself results in a maximal error of 2.1 × 10 -1 compared with the finite element approximation. As to the initial SER method using a random selection process regarding the building of the RB approximation space, it has convergence issues for some input parameters. It illustrates the relevance of a smart selection process, -the error indicator.

Conclusion

Intended as a follow-up of [START_REF] Daversin | Simultaneous empirical interpolation and reduced basis method for non-linear problems[END_REF] introducing the SER method, this paper investigates some of its variants and in particular SER( ) as well as the benefits of using an error indicator to drive the SER process. We have proposed a methodology that can build simultaneously the affine decomposition of the original problem and the associated reduced basis model. Both (EIM and RB) are feeding each other. We recover convincing error convergence on EIM and RB on a standard benchmark problem as well as on 3D industrial nonlinear multi-physics application. There are still various aspects of the methodology that can be investigated, we believe that SER opens various opportunities, and there is of course the theoretical question of the a priori convergence of the SER method -the methodology takes advantage of the underlying low-dimensional structures of the nonlinear PDE systems simultaneously from the EIM and RB sides.

Fig. 1 :

 1 Fig. 1: SER multilevel -EIM convergence

  Input Range σ 0 [40 × 10 6 , 60 × 10 6 ] S.m -1 α [3.3 × 10 -3 , 3.5 × 10 -3 ] K -1 L [2.5 × 10 -8 , 2.9 × 10 -8 ] j [30 × 10 6 , 60 × 10 6 ] A.m-2 h [50000, 65000] W.m -2 .K -1 T w [293, 313] K Table 6: Input parameters ranges Fig. 2: Temperature

Fig. 3 :

 3 Fig. 3: SER method -EIM and RB convergence

Fig. 4 :

 4 Fig. 4: SER multilevel -Random -EIM convergence

Fig. 5 :

 5 Temperature

Table 2 :

 2 Use of Greedy algorithm for RB within SER

Table 3 :

 3 SER variants based on error indicator

Table 4 :

 4 SER multilevel -Random

Table 5 :

 5 SER multilevel -RB Greedy

Table 8 :

 8 SER(1) -RB Greedy

	5 7.66e+0	2.71e-2
	10 10 3.37e-1	7.82e-4
	15 15 4.85e-2	2.64e-4
	20 20 2.93e-2	3.40e-4
	25 25 5.23e-3	4.59e-5

Table 10 :

 10 Input parameters ranges

Acknowledgements

The authors would like to thank A.T. Patera (MIT) for the fruitful discussions we had as well as Vincent Chabannes (U. Strasbourg) and Christophe Trophime (CNRS). The authors acknowledge also the financial support of the ANR Chorus and the Labex IRMIA.