Jean Vérité
email: jean.verite@univ-lemans.fr

Édouard Ravier

Olivier Bourgeois

Paul Bessin

Stéphane Pochat

New metrics reveal the evolutionary continuum behind the morphological diversity of subglacial bedforms

Keywords: Glacial geomorphology, ribbed bedforms, drumlins, MSGLs, continuum, sinuous forms

Understanding the formation of subglacial bedforms is primordial to constrain ice-meltwater-bed interactions and the dynamics of past and present ice sheets. However, the difficulty to observe active subglacial bedforms below present-day ice sheets implies that their formation and evolution are essentially deduced from the inversion of morphological and sedimentological data from palaeosubglacial bedforms. In this study, the morphological characteristics of subglacial bedforms are explored with a new approach based on the combination of three dimensionless morphometric indices i.e., circularity index, sinuosity index and elongation component ratio. We measure the spatial distribution of these indices on an unpublished database composed of ~13,500 digitized bedforms taken from two selected portions of the Irish Ice Sheet and Laurentide Ice Sheet beds considering that (1) all subglacial bedforms in each region were formed under a single ice flow configuration and (2) the bedforms may have developed either orthogonal, parallel, or oblique to ice displacement directions. Our results reveal a morphometric and spatial continuum along which ribbed bedforms of various lengths and shapesranging from circular to elongated flow-transverse formsare incipient bedforms that can evolve into transitional sinuous ribbed bedforms, then into predominantly flowparallel sinuous bedforms that progressively realign to form streamlined bedforms. Assuming the glaciological contexts of selected study areas, this continuum may provide a new geomorphic criterion to constrain spatial variations in subglacial conditions (e.g. ice flow velocity, bedrock characteristics, meltwater pressure).

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Introduction

Glacial geomorphologists have historically classified subglacial bedforms in different classes, based on their differences in shapes, dimensions and orientations in relation to a presupposed direction of ice flow (i.e. ice displacement direction). Linear ridges supposedly parallel to the ice flow direction have been called streamlined bedforms and include drumlins [START_REF] Hill | The distribution of drumlins in County Down, Ireland[END_REF][START_REF] Menzies | A review of the literature on the formation and location of drumlins[END_REF], flutes [START_REF] Shaw | The Athabasca fluting field, Alberta, Canada: implications for the formation of large-scale fluting (erosional lineations)[END_REF][START_REF] Eyles | Rock drumlins and megaflutes of the Niagara Escarpment, Ontario, Canada: a hard bed landform assemblage cut by the Saginaw-Huron Ice Stream[END_REF], and mega-scale glacial lineations (MSGLs;[START_REF] Clark | Mega-scale glacial lineations and cross-cutting ice-flow landforms[END_REF][START_REF] Canals | Deep sea-floor evidence of past ice streams off the Antarctic Peninsula[END_REF] by order of increasing elongation ratios. Periodic ridges supposedly orthogonal to the ice flow direction have been defined as ribbed bedforms [START_REF] Stokes | Ribbed bedforms on palaeo-ice stream beds resemble regular patterns of basal shear stress ('traction ribs') inferred from modern ice streams[END_REF][START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF], and include bedforms that have been variously called ribbed moraines [START_REF] Cowan | Ribbed moraine: till-fabric analysis and origin[END_REF][START_REF] Hättestrand | Ribbed moraines in Sweden-distribution pattern and palaeoglaciological implications[END_REF],

Rogen moraines [START_REF] Lundqvist | Problems of the so-called Rogen moraine[END_REF][START_REF] Shaw | Genesis of the Sveg tills and Rogen moraines of central Sweden: a model of basal melt out[END_REF] and mega-ribs [START_REF] Greenwood | Glacial landforms of extreme size in the Keewatin sector of the Laurentide Ice Sheet[END_REF] depending on their sizes and locations of first description. Sub-circular mounds without a determinable orientation have been referred to as circular bedforms [START_REF] Knight | Landform modification by palaeo-ice streams in east-central Ireland[END_REF][START_REF] Greenwood | Subglacial bedforms of the Irish ice sheet[END_REF], mammillary hills [START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF], ovoid forms [START_REF] Smith | Problems of bias in mapping linear landforms from satellite imagery[END_REF] or hummocky ribbed bedforms [START_REF] Hättestrand | Ribbed moraines in Sweden-distribution pattern and palaeoglaciological implications[END_REF][START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF]Möller & Dowling, 2015). While parallelism between streamlined bedforms and ice flow directions has been confirmed below modern ice sheets and in front of retreating ice lobes (e.g. [START_REF] King | Seismic and radar observations of subglacial bed forms beneath the onset zone of Rutford Ice Stream, Antarctica[END_REF][START_REF] Johnson | Active drumlin field revealed at the margin of Múlajökull, Iceland: a surge-type glacier[END_REF], the relationship between orientation of other bedform classes and ice flow directions has not been proven.

Although these different morphological classes have sometimes been attributed to different subglacial formation processes, the alternative hypothesis that they may be part of a same morphometric, and perhaps genetic, continuum has emerged from the observation of spatial and morphometric transitions between ribbed and streamlined bedforms (Fig. 1) [START_REF] Carl | Ribbed moraine-drumlin transition belt[END_REF][START_REF] Aylsworth | Bedforms of the Keewatin ice sheet, Canada[END_REF][START_REF] Dyke | The Laurentide Ice Sheet and an introduction to the Quaternary geology of the Canadian Shield. Quaternary Geology of Canada and Greenland[END_REF][START_REF] Knight | Landform modification by palaeo-ice streams in east-central Ireland[END_REF][START_REF] Fannon | Numerical simulations of drumlin formation[END_REF], between circular and ribbed bedforms [START_REF] Hättestrand | Ribbed moraines in Sweden-distribution pattern and palaeoglaciological implications[END_REF][START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF][START_REF] Greenwood | Subglacial bedforms of the Irish ice sheet[END_REF][START_REF] Vérité | Formation of murtoos by repeated flooding of ribbed bedforms along subglacial meltwater corridors[END_REF], between different types of streamlined bedforms (e.g. [START_REF] Rose | Drumlins as part of a glacier bedform continuum[END_REF][START_REF] Clark | Size and shape characteristics of drumlins, derived from a large sample, and associated scaling laws[END_REF]Stokes et al., 2013;[START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF][START_REF] Sookhan | Drumlins and megascale glacial lineations as a continuum of subglacial shear marks: A LiDAR based morphometric study of streamlined surfaces on the bed of a Canadian paleo-ice stream[END_REF], and between circular, ribbed and streamlined bedforms [START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF][START_REF] Markgren | Problems of moraine morphology: Rogen moraine and Blattnick moraine[END_REF][START_REF] Ely | Do subglacial bedforms comprise a size and shape continuum?[END_REF].

Several additional classes have also been proposed to describe more specific or more complex bedforms, such as smooth-sided/drumlinized ribbed moraines (Fig. 1b) [START_REF] Carl | Ribbed moraine-drumlin transition belt[END_REF][START_REF] Lundqvist | Moraine morphology: terminological remarks and regional aspects[END_REF] A c c e p t e d P r e -P r o o f 3

Dunlop & Clark, 2006), Blattnick moraines (Fig. 1c) [START_REF] Markgren | Problems of moraine morphology: Rogen moraine and Blattnick moraine[END_REF], transverse/asymmetrical drumlins (Fig. 1d) [START_REF] Shaw | Drumlin formation related to inverted melt-water erosional marks[END_REF][START_REF] Boulton | A theory of drumlin formation by subglacial sediment deformation[END_REF][START_REF] Knight | Morphological and morphometric analyses of drumlin bedforms in the Omagh Basin, north central Ireland[END_REF], crescentic/parabolic ridges and drumlins (Fig. 1e) [START_REF] Boulton | A theory of drumlin formation by subglacial sediment deformation[END_REF][START_REF] Lundqvist | Rogen (ribbed) moraine-identification and possible origin[END_REF][START_REF] Smalley | The shape of drumlins, their distribution in drumlin fields, and the nature of the sub-ice shaping forces[END_REF][START_REF] Maclachlan | Quantitative geomorphological analysis of drumlins in the Peterborough drumlin field, Ontario, Canada[END_REF][START_REF] Eyles | Erosional origin of drumlins and megaridges[END_REF], barchan-shaped (Fig. 1f), blocky angular (Fig. 1g) and jagged-type ribbed moraines (Fig. 1h) [START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF], extensional ribbed moraines (Fig. 1i) [START_REF] Wagner | Ribbed moraines and subglacial geomorphological signatures of interior-sector palaeo-ice sheet dynamics[END_REF], oblique ribbed bedforms (Fig. 1j) [START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF] and channeled drumlins (Fig. 1k) [START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF] among others. Since most of these complex bedforms are sinuous, i.e. have a non-rectilinear crestline, and/or have apparently no clear orientation relative to the local ice flow direction, they do not fit with usual bedform classes and have therefore been excluded from most of the existing models of subglacial bedform continuum governed by ice flow direction and velocity [START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF][START_REF] Stokes | Formation of mega-scale glacial lineations on the Dubawnt Lake Ice Stream bed: 1. size, shape and spacing from a large remote sensing dataset[END_REF].

Complex bedforms have frequently been interpreted as resulting from the remolding of bedforms or the superimposition/crosscutting of several generations of bedforms such as drumlins and ribbed bedforms [START_REF] Rose | Drumlin measurements: a test of the reliability of data derived from 1:25,000 scale topographic maps[END_REF][START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF][START_REF] Knight | Landform modification by palaeo-ice streams in east-central Ireland[END_REF][START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF][START_REF] Mitchell | Drumlin map of the Western Pennines and southern Vale of Eden, Northern England, UK[END_REF][START_REF] De Angelis | Palaeo-ice streams in the Foxe/Baffin sector of the Laurentide Ice Sheet[END_REF]Greenwood & Clark, 2009), in response to changes in ice flow direction [START_REF] Clark | Mega-scale glacial lineations and cross-cutting ice-flow landforms[END_REF]Stokes et al., 2006a;[START_REF] Brown | The glacial geomorphology of the north-west sector of the Laurentide Ice Sheet[END_REF] or/and velocity [START_REF] Stokes | Geomorphological map of ribbed moraines on the Dubawnt Lake Palaeo-Ice stream bed: a signature of ice stream shut-down[END_REF][START_REF] Stokes | Superimposition of ribbed moraines on a palaeo-ice-stream bed: implications for ice stream dynamics and shutdown[END_REF].

A demonstration that all subglacial bedform classes, including usual and complex ones, may or may not be part of a same spatial, morphometric and perhaps evolutionary continuum is missing.

Previous attempts were based on two main preliminary assumptions: (1) elongated bedforms may be classified a-priori as either parallel or orthogonal to ice flow direction; (2) complex (oblique and/or sinuous) bedforms cannot form under constant ice flow directions. Without invoking changes in ice flow directions, physical modeling (Fig. 1j) [START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF], glacial geomorphology (Fig. 1k) [START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF] and numerical modelling [START_REF] Ely | Numerical modelling of subglacial ribs, drumlins, herringbones, and mega-scale glacial lineations reveals their developmental trajectories and transitions[END_REF] recently demonstrated that complex bedforms, referred to as oblique ribbed bedforms, channeled drumlins and diagonal ribs, can develop under the same ice flow direction as transverse ribbed bedforms and streamlined bedforms respectively. Therefore, we perform here a systematic morphometric analysis of ~13,500 subglacial bedforms mapped in selected portions of the Irish and Laurentide Ice Sheet beds, only assuming that smoothsided/drumlinized ribbed moraines [START_REF] Carl | Ribbed moraine-drumlin transition belt[END_REF], (c) Blattnick moraines [START_REF] Markgren | Problems of moraine morphology: Rogen moraine and Blattnick moraine[END_REF], (d) transverse/asymmetrical drumlins [START_REF] Shaw | Drumlin formation related to inverted melt-water erosional marks[END_REF], (e) parabolic drumlins [START_REF] Boulton | A theory of drumlin formation by subglacial sediment deformation[END_REF] for mapping ; Eyles et al., 2016 for denomination), (f) barchan-shaped, (g) blocky angular and (h) jagged-type ribbed moraines [START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF], (i) extensional ribbed moraines [START_REF] Wagner | Ribbed moraines and subglacial geomorphological signatures of interior-sector palaeo-ice sheet dynamics[END_REF], (j) oblique ribbed bedforms [START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF] and (k) channeled drumlins [START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF].

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Data and method

Study areas

The two study areas are selected for (i) their great diversity of subglacial bedforms and (ii) the occurrence of complex forms at the spatial transitions between fields of ribbed and streamlined bedforms (Fig. 2). The first study area (11,300-km²) The second study area (2,100-km²) is located west of the Hudson Bay in northern Canada, between Firedrake Lake and Boyd Lake (Fig. 2). The area was covered by the Keewatin Ice Dome of the Laurentide Ice Sheet (LIS) during the last Wisconsinian glaciation and the surficial geology is characterized by a 10 m thick (at most) layer of subglacial till deposited over a Precambrian bedrock [START_REF] Shilts | Keewatin Ice Sheet-Re-evaluation of the traditional concept of the Laurentide Ice Sheet[END_REF][START_REF] Dyke | The Laurentide Ice Sheet and an introduction to the Quaternary geology of the Canadian Shield. Quaternary Geology of Canada and Greenland[END_REF]. Ribbed and streamlined bedform tracts laterally alternate, radiating outwards from the Keewatin ice divide [START_REF] Shilts | Canadian shield[END_REF][START_REF] Aylsworth | Bedforms of the Keewatin ice sheet, Canada[END_REF] and Nalepa, 2010). From these DEMs, we derive and superimpose unidirectional hillshade maps and residual relief maps [START_REF] Hillier | Residual relief separation: digital elevation model enhancement for geomorphological mapping[END_REF] with a Geographic Information System (GIS) software (Fig. 3a). The 1-m contour lines of residual relief data are manually delineated to produce bedform outlines. Lines of maximum elevation, manually delineated within each outlined bedform, represent the bedform crest line (Fig. 3a). The manual digitization is performed by a single mapper in order to avoid bias related to mapping experience, with a graphic tablet and using a freehand drawing tool. The digitization is performed at a constant 1:10 000 scale to counteract differences in DEM resolutions for the two selected study areas. A last step of manual correction carried out from hillshaded DEMs by removing outlier features and adding bedforms smaller than the radius of residual relief, provide the final modifications to the dataset.

Data sources and processing

Morphometric analysis

Current morphometric techniques on subglacial bedforms

Subglacial bedform shapes are classically characterized by their dimensions, orientations relative to the supposed ice flow direction, and degrees of elongation. Hollingworth (1931) defined the 'length/width' ratio on drumlins, which has been later referred to as the 'elongation ratio' (e.g. [START_REF] Doornkamp | Numerical analysis in geomorphology: an introduction[END_REF][START_REF] Mills | Morphometry of drumlins in the northeastern and north-central USA[END_REF][START_REF] Rose | Drumlins as part of a glacier bedform continuum[END_REF]. Later, the elongation ratio was abundantly used to distinguish different types of streamlined bedforms such as flutes, drumlins, mega-flutes, mega-drumlins, lineations and MSGLs [START_REF] Boyce | Drumlins carved by deforming till streams below the Laurentide ice sheet[END_REF][START_REF] Knight | Morphological and morphometric analyses of drumlin bedforms in the Omagh Basin, north central Ireland[END_REF]O Cofaigh et al., 2002;[START_REF] Stokes | Are long subglacial bedforms indicative of fast ice flow?[END_REF][START_REF] Briner | Supporting evidence from the New York drumlin field that elongate subglacial bedforms indicate fast ice flow[END_REF][START_REF] Clark | Size and shape characteristics of drumlins, derived from a large sample, and associated scaling laws[END_REF][START_REF] Dowling | Morphometry and core type of streamlined bedforms in southern Sweden from high resolution LiDAR[END_REF]. The 'length/width' ratio has also been used to characterize the shape and dimensions of different subclasses of ribbed bedforms, such as ribbed/Rogen moraines [START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF]

Doornkamp and

Definition of morphometric indexes

Using the same method as Vérité et al. ( 2022), we measure the perimeters and areas of bedform outlines and the curvilinear and straight lengths of bedform crest lines (Fig. 3b). In order to characterize the complex shapes of bedform outlines and crestlines, we use these dimensioned shape indices to calculate their circularity (ICIR) (Table 1.A) and sinuosity (ISIN) (Table 1.B) respectively.

The maximal height (hMAX), defined as the amplitude, is also measured for each bedform.

We complement this morphometric protocol to describe the diversity of elongation and orientation of subglacial bedforms relative to the ice flow direction (Fig. S1). We assume that all bedforms were formed under constant ice flow fields, considering no cross-cutting relationships of streamlined bedforms are observed (discussed in Section 4.2). Using a method inspired from those developed by

Ng & Hughes (2019), we use streamlined bedforms as proxies for reconstructing ice flow fields and local ice flow directions. To do so, we derive the average orientation of the long-axis of streamlined bedforms, sampled in a hexagonal grid, to build the resulting vectors of ice flow direction in each hexagonal grid and to attribute a local ice flow direction to each bedform located in the hexagonal grid. In areas where streamlined bedforms are absent, we extrapolate ice flow directions from adjacent areas, with a method based on the principle of nearest neighborhood (Figs. 45). Since the orientation

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of the long-axis of streamlined bedforms varies spatially, the nearest neighborhood in a hexagonal grid is more convenient than in a rectangular grid [START_REF] Birch | Rectangular and hexagonal grids used for observation, experiment and simulation in ecology[END_REF]. The hexagonal grid area is chosen to ensure the presence of a minimum of 10 subglacial bedforms in each cell.

To allow the possibility that bedforms can be oblique to ice flow directions and to quantify this obliquity, we develop a new morphometric index called "elongation component ratio", computed from transverse (i.e. across-flow) and longitudinal (i.e. along-flow) components for each bedform (Fig 3b).

Using minimum bounding rectangles, we automatically measure the lengths of transverse and longitudinal components for the long axis (i.e. TC1 and LC1) and the short axis (i.e. TC2 and LC2) of each bedform (Fig. 3b). From the ratio between the maximal transverse (TCMAX) and maximal longitudinal components (LCMAX), we compute the "elongation component ratio" (IELO) (Table 1.C), which integrates both orientation and elongation of bedforms (Fig. S2).

𝐼 𝐶𝐼𝑅 = 4 × 𝐴 × 𝜋 𝑃²
Compares the perimeter (P) of a bedform outline of a given area (A) with the perimeter of a circle of an identical area [START_REF] Burgess | Morphometric comparisons between Rogen terrain and hummocky terrain[END_REF].

ICIR : [

A morphometric continuum of subglacial bedforms

The results of our morphometric measurements in northeastern Ireland (n = 10,100; Fig. 4) and in northern Canada (n = 3,400; Fig. 5) have been compiled in a single dataset. First, there is no cluster or gap but a continuum in the distribution of bedform morphometric values (Fig. 6; Figs. S2,S3). All analyzed bedforms stand along a continuum between four end members, which are schematically portrayed in red in Fig. 6 and correspond to minimal and maximal values of the dimensionless morphometric indices presented in Table 1. The 1 st end-member corresponds to linear bedforms transverse to the ice flow direction with a ISIN close to 0 and a IELO approaching -∞. The 2 nd end member corresponds to circular bedforms showing a ICIR = 1, a ISIN = 0 and a IELO = 0. The 3 rd end member corresponds to linear bedforms parallel to the ice flow direction with a ISIN close to 0 and a IELO approaching +∞. Finally, the 4 th end member corresponds to bedforms displaying a highly sinuous shape (ISIN close to 1) without any dominant elongation component (IELO = 0).

Between the 1 st and 2 nd end members lies a category corresponding to bedforms transverse to the Between the 2 nd and 3 rd end members lies a category corresponding to bedforms parallel to the ice flow direction (0 < IELO < +∞) with an ovoid shape and a linear crest line (ISIN ~ 0). Typically, these bedforms have a homogenous TCMAX ranging from 200 to 400 m, and a LCMAX ranging from 500 to 2000 m as they approach the 3 rd end member. Their amplitudes hMAX ranging from 4 to 14 m, decrease towards the 3 rd end member. These bedforms are typically referred to as streamlined bedforms and Other bedforms, representing 44% of the morphometric cluster, are neither parallel nor transverse to the ice flow direction and have more sinuous crest lines than linear ribbed and streamlined bedforms (Fig. 6). Between the 1 st and 4 th end members, bedforms are sinuous (ISIN > 0) and predominantly flowtransverse (IELO < 0). These bedforms are morphologically related to linear ribbed bedforms: they share a mainly transverse orientation relative to the ice flow direction (i.e. TCMAX > LCMAX) (Fig. 6)

and have a similar TCMAX ranging between 700 and 1400 m. However, they differ from linear ribbed bedforms by their higher LCMAX ranging between 500 and 850 m, and their higher crest line sinuosity, typically ranging between 0.1 and 0.35. On average, their ISIN is increasingly high as their IELO increases and tends to 0, and their ICIR decreases as their ISIN increases (Fig. 6). These bedforms are typically 9 to 20 m in amplitude. Examples of sinuous and predominantly flow-transverse bedforms have usually been reported as ribbed moraines although their sinuous appearance and their shape complexity have led some authors to create sub-categories of ribbed bedforms with new descriptive terms, such as Blattnick moraines [START_REF] Markgren | Problems of moraine morphology: Rogen moraine and Blattnick moraine[END_REF], drumlinized/smooth-sided ribbed moraines [START_REF] Carl | Ribbed moraine-drumlin transition belt[END_REF][START_REF] Lundqvist | Moraine morphology: terminological remarks and regional aspects[END_REF][START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF], blocky angular or jagged-type ribbed moraines [START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF]) (Fig. 1). Considering their morphological similarities with A c c e p t e d P r e -P r o o f 13 linear ribbed bedforms but also their higher degree of complexity, we refer to these sinuous and predominantly flow-transverse bedforms as sinuous ribbed bedforms.

Between the 3 rd and 4 th end members, when the LCMAX exceeds the TCMAX, bedforms are sinuous (ISIN > 0) and predominantly parallel to the ice flow direction (IELO ≥ 0). On average, their ISIN decreases as their IELO increases and their ICIR decreases as their ISIN increases (Fig. 6). Their TCMAX typically ranges between 350 and 750 m and is half of the linear and sinuous ribbed bedforms ones, while their LCMAX typically ranging between 550 and 1200 m overlaps those of streamlined bedforms.

These bedforms display lower amplitude than sinuous ribbed bedforms (hMAX = 7-17 m). Although sinuous and predominantly flow-parallel bedforms are more similar to streamlined bedforms than ribbed bedforms, their TCMAX are on average two times longer and their crest line sinuosity (ISIN = 0.05-0.25 m) is much higher than those of streamlined bedforms. 14 Some are highly sinuous with a U-shape, resembling bedforms referred to as crescentic/parabolic drumlins [START_REF] Boulton | A theory of drumlin formation by subglacial sediment deformation[END_REF][START_REF] Lundqvist | Rogen (ribbed) moraine-identification and possible origin[END_REF][START_REF] Smalley | The shape of drumlins, their distribution in drumlin fields, and the nature of the sub-ice shaping forces[END_REF][START_REF] Maclachlan | Quantitative geomorphological analysis of drumlins in the Peterborough drumlin field, Ontario, Canada[END_REF][START_REF] Eyles | Erosional origin of drumlins and megaridges[END_REF], barchan-shaped ribbed moraines [START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF] or channeled drumlins [START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF] in the literature; while others are slightly sinuous, oblique to the flow direction and similar to bedforms described in the literature as oblique ribbed bedforms [START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF], extensional ribbed moraines [START_REF] Wagner | Ribbed moraines and subglacial geomorphological signatures of interior-sector palaeo-ice sheet dynamics[END_REF], transverse/asymmetrical drumlins [START_REF] Shaw | Drumlin formation related to inverted melt-water erosional marks[END_REF][START_REF] Boulton | A theory of drumlin formation by subglacial sediment deformation[END_REF][START_REF] Knight | Morphological and morphometric analyses of drumlin bedforms in the Omagh Basin, north central Ireland[END_REF] or drumlinized moraines [START_REF] Carl | Ribbed moraine-drumlin transition belt[END_REF][START_REF] Brown | The glacial geomorphology of the north-west sector of the Laurentide Ice Sheet[END_REF] (Fig. 1).

Spatial distribution of subglacial bedform morphometrics

To investigate whether the morphometric continuum described in Section 3.1 is supported by a spatial continuum, we describe now the distribution of bedform shapes in the two study areas. To do so, we explore the evolution of bedform morphological indices at the transition between flowperpendicular and flow-parallel bedforms, both at regional (Figs. 78) and local scales (Figs. 9-10; Fig. S4).

Ireland

In the northern part of the Irish study area, flow-perpendicular (IELO < 0) and linear (ISIN < 0.1) bedforms, i.e. linear ribbed bedforms, gather in a 70-km long and 30-km wide field lying along the ice divide (Figs. 7a-b). To the south, flow-parallel (IELO > 0) and linear (ISIN < 0.1) bedforms, i.e.

streamlined bedforms, gather in a 60-km long and 40-km wide field (Figs. 7a-b). The progressive increase in IELO from north to south highlights that streamlined bedforms are increasingly elongated, ranging from typical drumlin to MSGL dimensions (Fig. 7a). Circular bedforms, i.e. bedforms with high ICIR, tend to concentrate within the regional field of linear ribbed bedforms but are absent within the regional field of streamlined bedforms (Figs. 7c,10a). The transition between ribbed and streamlined bedforms is gradual and characterized by an overall increase in sinuosity index (ISIN) of bedform crest lines (Fig. 7b). Within this transitional field of bedforms, another transition from north to south is observed between predominantly flow-perpendicular (IELO < 0) and sinuous bedforms (ISIN > 0.1), i.e. sinuous ribbed bedforms, and predominantly flow-parallel (IELO > 0) and sinuous bedforms (ISIN > 0.1) (Figs. 7a-b). Sinuous bedforms with an oblique orientation relative to the local ice flow direction frequently occur along such transitional zones, forming an "opening gate" pattern (Fig. 10b). Down-ice of these oblique and sinuous bedforms, streamlined bedforms are commonly aligned following the same oblique orientation forming an "en-echelon" pattern (Fig. 10b).

Within a portion located in the regional-scale field of streamlined bedforms in the southern part of the Irish study area (Fig. 9b), some linear and sinuous ribbed bedforms gather in sub-circular spots,

Canada

This site is characterized by alternating tracts of streamlined and ribbed bedforms (Figs. 5,8).

These tracts are about 4-km wide and several tens of kilometers long; they are elongated parallel to the local ice flow direction and meltwater drainage features (eskers and meltwater channels). Along across-flow profiles to these tracts (Fig. 9c), tracts of ribbed bedforms characterized by low values of IELO and linear to low sinuous crest lines correlate with the position of drainage features. Streamlined bedforms characterized by a high IELO and linear crest lines are located in between drainage features.

Laterally, sinuous ribbed bedforms and predominantly flow-parallel sinuous bedforms occur at the transition between linear ribbed bedform and streamlined bedform tracts. Some of these sinuous bedforms exhibit an obliquity to the local ice flow direction on both tract sides (Fig. 10c). Along across-flow profiles (Fig. 9c), streamlined bedforms with maximal IELO in the core of the corridor are bounded by streamlined bedforms with lower IELO, some of which exhibit an "en-echelon" pattern (Fig. 10c). The spatial gradation of IELO highlights there is no break in the distribution of streamlined bedforms morphometrics when increasing the IELO, which is materialized by an increasing LCMAX with

A c c e p t e d P r e -P r o o f

19 a constant TCMAX of 300 m (Fig. 6a). This gradation demonstrates the morphometric and spatial continuity between drumlins and MSGLs.

Discussion

The morphometric approach developed in this study, based on dimensionless indices allowing the possibility that bedforms can be sinuous and oblique to the ice flow directions, reveals that all the analyzed bedforms can be integrated in a morphometric continuum. Scatterplots (Fig. 6), maps (Figs.

7-8) and spatial transects of morphometric indices (Fig. 9) demonstrate a continuum of bedforms between: (i) circular bedforms, (ii) linear ribbed bedforms, (iii) streamlined bedforms and (iv) a variety of morphologically intermediate sinuous bedforms which were hitherto mostly interpreted as superimposed or multiphased bedforms.

A unifying sequence of bedform evolution

Based on the morphometric (Fig. 6) and spatial (Figs. 7-10) continuum of subglacial bedform shapes, we explore how the variations of the dimensionless morphometric indices could help deciphering the subglacial bedform evolution in a time and space grid. Keeping in mind the initial assumption that the formation of all bedforms is not necessarily synchronous but associated with constant ice flow trajectories, we suggest that circular to linear ribbed bedforms are the first bedforms to form, then become more sinuous and more elongated in the ice flow direction as they evolve. As the sinuosity index of their crest line (ISIN) increases, we suggest that they then evolve into predominantly flow-parallel bedforms, until progressively paralleling the local ice flow direction to form streamlined bedforms with increasing elongation component ratio (IELO) (Figs. 1112).

Three major assumptions are required to reconstruct the chronological frame of this bedform evolution: (1) ribbed bedforms, ranging from circular to linear forms, are incipient bedforms and their IELO is independent of their degree of bedform evolution; (2) sinuous bedforms are transitional bedforms between ribbed and streamlined bedforms along a sequence of increasing degree of bedform evolution;

(3) streamlined bedforms arise from the fragmentation of transitional sinuous bedforms and their IELO increases as their degree of bedform evolution increases. We translate these assumptions into )) × (𝐼 𝐸𝐿𝑂 + 1) (Eq. 1)

Other propositions of equation quantifying the degree of bedform evolution and more details are presented in Fig. S5. These three assumptions and the bedform evolution sequence are discussed hereafter.

Ribbed bedforms with various elongation come first

Circular bedforms are circular forms of ribbed bedforms

Along the ordinate axis of the scatter plot in where streamlined bedforms are abundant and do not particularly gather at the transition between ribbed and streamlined bedforms (Fig. 7). For our two study areas, those spatial distributions do not corroborate interpretations that circular bedforms are transitional forms between ribbed and streamlined bedforms [START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF][START_REF] Ely | Do subglacial bedforms comprise a size and shape continuum?[END_REF]. Considering the spatial relationships between subglacial bedforms, we suggest to insert circular bedforms in a morphometric and spatial continuum of ribbed bedforms ranging from circular to linear forms. In agreement with previous observations reported by [START_REF] Hättestrand | Ribbed moraines in Sweden-distribution pattern and palaeoglaciological implications[END_REF], [START_REF] Greenwood | Subglacial bedforms of the Irish ice sheet[END_REF], [START_REF] Möller | The importance of thermal boundary transitions on glacial geomorphology; mapping of ribbed/hummocky moraine and streamlined terrain from LiDAR, over Småland, South Sweden[END_REF] and

Linear ribbed bedforms are incipient bedforms

Numerical models simulating the formation of subglacial bedforms demonstrate that ribbed bedforms with linear crest lines are the first bedforms to form, and constitute incipient bedforms in evolutionary sequences [START_REF] Chapwanya | Numerical computations of a theoretical model of ribbed moraine formation[END_REF][START_REF] Barchyn | Subglacial bed form morphology controlled by ice speed and sediment thickness[END_REF][START_REF] Fannon | Numerical simulations of drumlin formation[END_REF].

Numerical models also demonstrated that circular bedforms can emerge from the random dislocation of ribbed bedforms without any increases of their width and sinuosity during the early stages of evolutionary sequences, constituting incipient bedforms just as ribbed bedforms (Fannon, 2020 p.139-140;[START_REF] Ely | Numerical modelling of subglacial ribs, drumlins, herringbones, and mega-scale glacial lineations reveals their developmental trajectories and transitions[END_REF]. Similarly, physical simulations demonstrated the synchronous formation of ribbed bedforms with various values of their maximal transverse components and various shapes, ranging from circular to highly-elongated forms [START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF][START_REF] Vérité | Formation of murtoos by repeated flooding of ribbed bedforms along subglacial meltwater corridors[END_REF]. Considering (i) these modeling observations and (ii) the high overlap between the spatial distribution of circular and linear ribbed bedforms with variable elongation (Fig. 7), we propose that incipient ribbed bedforms arise and elongated forms as resulting from a single process, we propose that ribbed bedforms with linear crest lines correspond to the first stage of subglacial bedform evolution (Fig. 12a), whatever their size and shape [START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF]Greenwood andKleman, 2010, Ely et al., 2016).

Ribbed bedforms evolve into sinuous transitional bedforms

According to the initial size and shape of the ribbed bedform and to the later stages of bedform transformation, an infinite number of bedform evolutions can be followed (Fig. 11), leading to variations in the metrics of bedform sequences (Fig. 12 ; three of them are represented). Sequences n° 2 and 3 aim to respectively represent the evolution of a low-elongated and a highly-elongated ribbed bedform, and are built using typical bedform assemblages observed along spatial sequences. Based on evolution sequences n°2 and 3, we suggest that elongated ribbed bedforms, characterized by a linear crest line when they initially form, become increasingly sinuous (with increasing IELO) as their degree of evolution increases and as they are initially highly-elongated (Fig. 12). Numerical simulations of subglacial bedforms demonstrated that, whatever the glaciological and hydrological conditions simulated, ribbed bedforms become increasingly stretched towards the ice flow direction, sinuous and oblique as they evolve before dislocating into streamlined bedforms (Supplementary materials of

Fannon et al. 2017; Fannon, 2020 p.139; Ely et al., 2022).

Those numerical simulations, supporting our model of bedform evolution, agree with recent physical modeling works describing increasingly sinuous and oblique ribbed bedform through experimental runs [START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF][START_REF] Vérité | Formation of murtoos by repeated flooding of ribbed bedforms along subglacial meltwater corridors[END_REF]. These works support the hypothesispostulated from scatterplots (Fig. 6) and morphometrical transects through bedform fields (Fig. 9)that sinuous ribbed bedforms and predominantly flow-parallel sinuous bedforms could represent transitional bedforms resulting from the evolution of linear ribbed bedforms (Figs. 12b-c). This interpretation also correlates with the description of complex bedforms in intermediate positions along spatial sequences unifying ribbed and streamlined bedforms along palaeo-ice sheet beds [START_REF] Carl | Ribbed moraine-drumlin transition belt[END_REF][START_REF] Markgren | Problems of moraine morphology: Rogen moraine and Blattnick moraine[END_REF][START_REF] Aylsworth | Bedforms of the Keewatin ice sheet, Canada[END_REF][START_REF] Knight | Landform modification by palaeo-ice streams in east-central Ireland[END_REF]. Linear ribbed bedforms progressively evolve through complex transitional bedforms with variable sinuosities and orientations, towards increasingly elongated streamlined bedforms; while circular ribbed bedforms progressively evolve into a unique or several streamlined bedforms without complex transitional bedforms. The degree of bedform evolution, defined from the combination of their IELO and ISIN, is a quantitative measure of this progressive evolution. Three examples of bedform evolution sequences are proposed depending on the initial IELO of the ribbed bedform (evolution sequence n°3 is presented in Fig. 11b). The color of each bedform corresponds to the color associated with its degree of evolution indicated in Fig. 11b.

Streamlined bedforms arise from sinuous transitional bedform fragmentation and are increasingly elongated as they evolve

In Ireland and Canada, sinuous oblique bedforms and "en-echelon pattern" of drumlinsexamples of which have been independently mentioned in the literature [START_REF] Shaw | Drumlin formation related to inverted melt-water erosional marks[END_REF] 12c-d).

The fragmentation first leads to the formation of distinct and individualized low-elongated and flow-parallel bedforms we consider as being proto-drumlins [START_REF] Clark | Size and shape characteristics of drumlins, derived from a large sample, and associated scaling laws[END_REF]. Figure 6 shows there is no break in the distribution of streamlined bedforms morphometrics when increasing the IELO, demonstrating continuity between drumlins and MSGLs. In Ireland and Canada, this continuity is also spatial as increasing IELO gradientswhich is materialized by an increasing LCMAX with a constant TCMAX (~300 m)occur parallel (i.e. in the down-ice direction; Fig. 9a) or transverse to the local ice flow direction (Fig. 9c). In addition to several studies highlighting the spatial [START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF][START_REF] Boyce | Drumlins carved by deforming till streams below the Laurentide ice sheet[END_REF][START_REF] Hart | Identifying fast ice flow from landform assemblages in the geological record: a discussion[END_REF][START_REF] Stokes | Are long subglacial bedforms indicative of fast ice flow?[END_REF][START_REF] Briner | Supporting evidence from the New York drumlin field that elongate subglacial bedforms indicate fast ice flow[END_REF][START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF] and morphometric [START_REF] Rose | Drumlins as part of a glacier bedform continuum[END_REF][START_REF] Clark | Size and shape characteristics of drumlins, derived from a large sample, and associated scaling laws[END_REF][START_REF] Stokes | Formation of mega-scale glacial lineations on the Dubawnt Lake Ice Stream bed: 1. size, shape and spacing from a large remote sensing dataset[END_REF][START_REF] Ely | Do subglacial bedforms comprise a size and shape continuum?[END_REF] continuity between drumlins and MSGLs, we quantitatively demonstrate that the intensity of streamlined bedform elongation parallel to the ice flow direction illustrates their degree of evolution (Figs. 1112).

Other potential evolutionary sequences

Complex sinuous bedforms, corresponding to transitional bedforms, do not systematically occur within subglacial bedform assemblages along palaeo-ice sheet beds, suggesting that other evolutionary sequences without sinuous bedforms should also exist.

Considering evolutionary sequence n°1 (Fig. 12), initial ribbed bedforms with a circular shape will evolve into single or multiple increasingly elongated streamlined bedforms, without sinuous

Sinuous bedforms vs superimposed bedforms along palaeo-ice sheet beds: a time-transgressive issue

This model of bedform continuum and the interpretation of sinuous/complex bedforms as transitional bedforms between ribbed and streamlined bedforms rely on the main and strong assumption of a steady ice flow configuration. The formation of most complex bedforms could alternatively be explained by multiple and asynchronous ice flow trajectories.

In Ireland and Canada (Figs. 4-5), the "steady" assumption is supported by the lack of crosscutting flow-sets of streamlined bedforms within areas of several thousands of square kilometers around the study sites. Complex bedforms have alternatively been interpreted as superimposed drumlins on a pre-existing parent ice-remolded bedform (e.g. Clark, 1993), thus suggesting crosscutting ice flow directions. We rather suggest that complex bedforms display shapes and orientations consistent with their transitional position along spatial evolutionary sequences deciphered from morphometrical analysis, especially when surrounding bedform fields do not record converging streamlined bedforms. The observation of "opening gate" patterns (see Fig. 9) associated with some oblique bedforms displaying a symmetric pattern with respect to the ice flow direction could be consistent with at least three cross-cutting ice flow direction suggested by "superimposed drumlins". The steady flow trajectory and the bedform continuum model we determine using our database is not ubiquitous and it should be noted that complex subglacial forms might also derive from the remolding of pre-existing bedforms and/or the superimposition of bedform generations driven by changes in ice flow direction (e.g. [START_REF] Rose | Superimposed drumlins[END_REF][START_REF] Rose | Drumlins as part of a glacier bedform continuum[END_REF][START_REF] Clark | Mega-scale glacial lineations and cross-cutting ice-flow landforms[END_REF][START_REF] Knight | Morphological and morphometric analyses of drumlin bedforms in the Omagh Basin, north central Ireland[END_REF]Stokes et al., 2006a). Within study areas where morphological evidence would clearly indicate time-transgressive subglacial records with changes in ice flow directions, the evolutionary models we determined from our database would indeed not be relevant to reconstruct past glacial dynamics. However, the areas presented in this article demonstrate that the integration of these complex bedforms into a single continuum model could help in revising the number of ice-sheet flow trajectories currently envisaged in Ireland (Greenwood and Clark, 2009a;[START_REF] Clark | Growth and retreat of the last British-Irish Ice Sheet, 31 000 to 15 000 years ago: the BRITICE-CHRONO reconstruction[END_REF], in Canada [START_REF] Winsborrow | Ice streams of the Laurentide ice sheet[END_REF][START_REF] De Angelis | Palaeo-ice streams in the Foxe/Baffin sector of the Laurentide Ice Sheet[END_REF] and probably in many other areas.

Glaciodynamics significance of the bedform evolution degree

The sequences of bedform evolution have sometimes been interpreted as spatial variations in the duration of ice streaming process [START_REF] Boyce | Drumlins carved by deforming till streams below the Laurentide ice sheet[END_REF][START_REF] Jamieson | Subglacial processes on an Antarctic ice stream bed. 2: Can modelled ice dynamics explain the morphology of mega-scale glacial lineations?[END_REF][START_REF] Zoet | Factors that contribute to the elongation of drumlins beneath the Green Bay Lobe, Laurentide Ice Sheet[END_REF], bed lithology [START_REF] Aylsworth | Bedforms of the Keewatin ice sheet, Canada[END_REF][START_REF] Wellner | Distribution of glacial geomorphic features on the Antarctic continental shelf and correlation with substrate: implications for ice behavior[END_REF]O Cofaigh et al., 2002;[START_REF] Greenwood | The sensitivity of subglacial bedform size and distribution to substrate lithological control[END_REF][START_REF] Mckenzie | Streamlined subglacial bedform sensitivity to bed characteristics across the deglaciated Northern Hemisphere[END_REF], bed topography [START_REF] Bouchard | Subglacial landforms and deposits in central and northern Quebec, Canada, with emphasis on Rogen moraines[END_REF][START_REF] Franke | Bed topography and subglacial landforms in the onset region of the Northeast Greenland Ice Stream[END_REF], bed thickness [START_REF] Kerr | Origin of drumlins on the floor of Lake Ontario and in upper New York State[END_REF][START_REF] Barchyn | Subglacial bed form morphology controlled by ice speed and sediment thickness[END_REF] or meltwater pressure [START_REF] Shaw | The meltwater hypothesis for subglacial bedforms[END_REF][START_REF] Lewington | New Insights into Subglacial Meltwater Drainage Pathways from the ArcticDEM[END_REF]. In most cases, they have been interpreted as spatial variations in ice flow velocity from the A c c e p t e d P r e -P r o o f 27 onset area to the ice stream core [START_REF] Dyke | The Laurentide Ice Sheet and an introduction to the Quaternary geology of the Canadian Shield. Quaternary Geology of Canada and Greenland[END_REF][START_REF] King | Seismic and radar observations of subglacial bed forms beneath the onset zone of Rutford Ice Stream, Antarctica[END_REF] and even along or across the ice stream itself [START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF][START_REF] Hart | Identifying fast ice flow from landform assemblages in the geological record: a discussion[END_REF][START_REF] Briner | Supporting evidence from the New York drumlin field that elongate subglacial bedforms indicate fast ice flow[END_REF][START_REF] Stokes | Formation of mega-scale glacial lineations on the Dubawnt Lake Ice Stream bed: 1. size, shape and spacing from a large remote sensing dataset[END_REF][START_REF] Hillier | Exploring explanations of subglacial bedform sizes using statistical models[END_REF][START_REF] Jamieson | Subglacial processes on an Antarctic ice stream bed. 2: Can modelled ice dynamics explain the morphology of mega-scale glacial lineations?[END_REF][START_REF] Hermanowski | An erosional origin for drumlins of NW Poland[END_REF][START_REF] Van Landeghem | Bed erosion during fast ice streaming regulated the retreat dynamics of the Irish Sea Ice Stream[END_REF][START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF][START_REF] Sookhan | Drumlins and megascale glacial lineations as a continuum of subglacial shear marks: A LiDAR based morphometric study of streamlined surfaces on the bed of a Canadian paleo-ice stream[END_REF].

In Ireland, streamlined bedform tractsseveral 10s of kilometers long and widehave been interpreted as palaeo-corridors of fast ice flow [START_REF] Synge | The Quaternary period in Ireland-an assessment[END_REF][START_REF] Stokes | Geomorphological criteria for identifying Pleistocene ice streams[END_REF]Greenwood and Clark, 2009a), which probably correspond to palaeo-ice streams draining the Lowland Ice Dome between 20 and 17 ka BP [START_REF] Clark | Growth and retreat of the last British-Irish Ice Sheet, 31 000 to 15 000 years ago: the BRITICE-CHRONO reconstruction[END_REF]. Morphometric and spatial analyses conducted in Ireland suggest that the bedform distribution reveals longitudinal and lateral gradients of bedform evolution relative to the ice flow direction (Figs. 78), certainly recording ice flow velocity pattern along and across an ice stream. Indeed, the transition between slow ice near the ice dome (i.e.

onset area) and at ice stream margins (i.e. shear margins) to fast ice in the ice stream core is respectively recorded through transitions between ribbed, sinuous and streamlined bedforms (Fig. 9b).

Our model, encompassing the degree of bedform evolution in a spatio-temporal continuum, is convincing as it can be explained by transitions in ice flow velocity classically monitored in modern ice streams (e.g. Sergienko & Hindmarsh, 2013; [START_REF] Zheng | The possible transition from glacial surge to ice stream on Vavilov Ice Cap[END_REF].

Small fields of linear to sinuous ribbed bedforms contained within streamlined bedform tracts in Ireland (Fig. 10) indicate abrupt and very localized changes in the degree of bedform evolution. We interpret this local anomaly in bedform evolution as the morphological expression of slow ice patches within fast-flowing ice corridors, as has commonly been reported for ice stream sticky spots [START_REF] Alley | In search of ice-stream sticky spots[END_REF]. We notice that palaeo-sticky spots have mainly been recognized through the superposition of ribbed bedforms on top of streamlined bedforms [START_REF] Stokes | Geomorphological map of ribbed moraines on the Dubawnt Lake Palaeo-Ice stream bed: a signature of ice stream shut-down[END_REF][START_REF] Stokes | Ice stream sticky spots: a review of their identification and influence beneath contemporary and palaeo-ice streams[END_REF][START_REF] Stokes | Ribbed bedforms on palaeo-ice stream beds resemble regular patterns of basal shear stress ('traction ribs') inferred from modern ice streams[END_REF]. There are no cross-cutting relationships or superposition in the sticky spots revealed in Ireland, instead we suggest that streamlined bedforms and ribbed bedforms are synchronous and that patches of bedform of lower evolution reveal sustained sticky conditions related to nearby bedrock topographic highs.

In Canada, previous studies suggest that the position of the study area below the Keewatin Ice Dome intersects meltwater corridors that developed below the retreating Laurentide Ice Sheet [START_REF] Storrar | A map of large Canadian eskers from Landsat satellite imagery[END_REF][START_REF] Lewington | A model for interaction between conduits and surrounding hydraulically connected distributed drainage based on geomorphological evidence from Keewatin, Canada[END_REF]. Moreover, Lewington (2020) suggested that corridors of ribbed

Conclusion

Combined mapping and morphometric analysis of 13,500 subglacial bedforms enabled the morphological evolution of subglacial bedforms to be explored. The results of this study are consistent with previous works on the subglacial bedform continuum [START_REF] Lundqvist | Studies of drumlin tracks in Central Sweden[END_REF][START_REF] Aario | Classification and terminology of morainic landforms in Finland[END_REF][START_REF] Rose | Drumlins as part of a glacier bedform continuum[END_REF][START_REF] Ely | Do subglacial bedforms comprise a size and shape continuum?[END_REF], 2022[START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF], but go further into the integration of the whole morphological diversity of subglacial bedforms and in the description of transitional steps characterizing the bedform re-alignment between flow-transverse ribbed bedforms and flow-parallel streamlined bedforms. The spatial relationships between bedform evolutionary sequences reported in Ireland and Canada and their assumed glaciological contexts (e.g.

ice flow velocity, meltwater pressure, bed topography and lithology) suggest the degree of bedform evolution could be used to (1) explore factors controlling subglacial bedform formation and evolution and (2) reconstruct the past glacial dynamics with greater detail. This work opens up new perspectives to bridge the gap between subglacial bedform morphology, subglacial processes and ice dynamics.

Further work is needed to explore the genetic significance of this morphometric continuum that could contribute to increase the resolution of reconstructions of past glacial dynamics and ice-meltwater-bed interactions.

  parallel to the local ice flow direction. We quantify the complexity and variability of subglacial bedform shapes in these regions with three dimensionless morphometric indices and a spatial analysis of subglacial bedform distribution. We propose that all subglacial bedforms stand along an overall morphometric and spatial continuum that corresponds to an evolutionary continuum.

Figure 1 .

 1 Figure 1. (a) Schematic representation, as currently envisioned, of the subglacial bedform continuum between ribbed, circular and streamlined bedforms (drumlins and MSGLs) (after Aario, 1977 and Stokes et al., 2013).(b) to (j) Examples of complex subglacial bedforms commonly associated with fields of ribbed and streamlined bedforms but excluded from the existing models of subglacial bedform continuum. They are referred to as (b) smoothsided/drumlinized ribbed moraines[START_REF] Carl | Ribbed moraine-drumlin transition belt[END_REF], (c) Blattnick moraines[START_REF] Markgren | Problems of moraine morphology: Rogen moraine and Blattnick moraine[END_REF],(d) transverse/asymmetrical drumlins[START_REF] Shaw | Drumlin formation related to inverted melt-water erosional marks[END_REF], (e) parabolic drumlins[START_REF] Boulton | A theory of drumlin formation by subglacial sediment deformation[END_REF] for mapping ; Eyles et al., 2016 for denomination), (f) barchan-shaped, (g) blocky angular and (h) jagged-type ribbed moraines[START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF], (i) extensional ribbed moraines[START_REF] Wagner | Ribbed moraines and subglacial geomorphological signatures of interior-sector palaeo-ice sheet dynamics[END_REF], (j) oblique ribbed bedforms[START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF] and (k) channeled drumlins[START_REF] Sookhan | LiDAR-based quantitative assessment of drumlin to mega-scale glacial lineation continuums and flow of the paleo Seneca-Cayuga paleo-ice stream[END_REF].

  is located in northeastern Ireland, where a layer of subglacial till, commonly more than 15 m thick and deposited below the Irish Ice Sheet (IIS) during the last Devensian glaciation, covers the Paleozoic sedimentary bedrock (McConnell & Gatley, 2006).Large-scale streamlined bedform tracts that have been interpreted as corridors of fast ice flow[START_REF] Stokes | Geomorphological criteria for identifying Pleistocene ice streams[END_REF] Greenwood & Clark, 2009a) and possibly ice streams[START_REF] Synge | The Quaternary period in Ireland-an assessment[END_REF] are reported south of the northern ice divide of the Lowland Ice Dome between 20 and 17 ka BP(Clark et al., 2022) and downstream from ribbed and circular bedform fields[START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF][START_REF] Greenwood | Subglacial bedforms of the Irish ice sheet[END_REF]. Based on frequent apparent superimposition and crosscutting relationships between bedforms over the study area, the geomorphological record has been interpreted as a superimposition of several configuration of ice-sheet flow trajectories(Greenwood & Clark, 2009a,b).

Figure 2 .

 2 Figure 2. Location of study areas relative to the maximum extension of the northern hemisphere ice sheets during the Last Glacial Maximum[START_REF] Batchelor | The configuration of Northern Hemisphere ice sheets through the Quaternary[END_REF].

  and paralleling meltwater corridors described in the Keewatin area by Storrar et al. (2013) and Lewington et al., (2019, 2020).

Figure 3 .

 3 Figure 3. (a) Hillshaded residual relief derived from a digital elevation model and from which bedform outlines and crest lines are manually digitized. (b) Measured quantities for the calculation of dimensionless indices: elongation component ratio (IELO), circularity index (ICIR) and sinuosity index (ISIN). We conduct subglacial bedform mapping from open-data DEMs of the LIS bed in Canada (10-m

  Measures the sinuosity of the crest line (i.e. ratio between the curvilinear length of the crest line and its straight length; Schumm, 1963), normalized to the sinuosity of an equilateral triangle 1(Vérité et al.

  ice flow direction (0 > IELO > -∞), with a linear crest line (ISIN ~ 0) and a slightly undulating outline: these bedforms are referred as linear ribbed bedforms since they share all characteristics defined by Dunlop andClark (2006). Linear ribbed bedforms have a TCMAX ranging from 600 to 1250 m, a LCMAX ranging from 300 to 550 m and an amplitude hMAX ranging from 9 to 19 m. By approaching the 2 nd end member, linear ribbed bedforms become less elongated and more circular. These observations are consistent with the hypothesis of a morphometric continuity among linear ribbed bedforms, as suggested by Hättestrand (1997),[START_REF] Dunlop | The morphological characteristics of ribbed moraine[END_REF], Möller and Dowling (2015) and Vérité et al. (2021, 2022).

  A c c e p t e d P r e -P r o o f 11 form a continuum between bedforms classically subdivided in drumlins and MSGLs depending if they are respectively low-or highly-elongated, as suggested by Stokes et al. (2013) and Ely et al. (2016).

Figure 4 .

 4 Figure 4. (a) Hillshaded residual relief and (b) interpretative map with digitized bedforms of a portion of the Irish Ice Sheet bed in northeastern Ireland. Reconstruction of local ice flow directions (solid red lines) from sampled glacial lineations in a grid of 25-km² hexagons based on the assumption that the formation of streamlined bedforms is associated to a unique configuration of ice flow trajectories. Where streamlined bedforms are absent (white hexagons), the average flow direction is extrapolated (dotted red lines) from surrounding glacial lineations (grey hexagons). The ice divide (black stippled line) revealed by diverging flowsets is consistent with ice sheet reconstructions from Greenwood and Clark (2009a,b).

Figure 5 .

 5 Figure 5. (a) Hillshaded residual relief and (b) interpretative map with digitized bedforms of a portion of the Laurentide Ice Sheet bed in northern Canada, between Firedrake Lake and Boyd Lake. Reconstruction of local ice flow directions in a grid of 9-km² hexagons with the same method as described in Fig. 4.

Figure 6 .

 6 Figure 6. Plots of elongation component ratio (IELO) and sinuosity index (ISIN). (a) Irish (dark grey dots) and Canadian bedforms (light grey dots). Selected bedform outlines picked up from the database are shown at each node of the grid to illustrate how ISIN and IELO reflect their shapes. (b) Dot color refers to the circularity index (ICIR) value of bedforms, reflecting its relation with both IELO and ISIN. Both graphs highlight a morphometric continuum of bedforms in which four end members, corresponding to the minimal and maximal values of the dimensionless morphometric indices (IELO, ISIN and ICIR; see Table1), are highlighted by red circles. Note that subglacial bedforms standing along the ordinate axis (ISIN ~ 0) correspond to bedforms typically integrated in current models of bedform continuum, unrevealing a wider a morphometric continuum of bedforms.

Figure 7 .

 7 Figure 7. Hexagonal grid maps of average morphometric indices (IELO, ISIN and ICIR) for bedforms observed along the IIS bed in northeastern Ireland. Sinuous bedforms (ISIN > 0.1) without predominant elongation component (IELO ~ 0) gather in an intermediate position between linear ribbed bedforms (ISIN < 0.1; IELO < 0) and streamlined bedforms (ISIN < 0.1; IELO > 0). Circular bedforms, characterized by high circularity values, tend to gather in area where IELO is negative, in association with bedforms.

Figure 8 .

 8 Figure 8. Hexagonal grid maps of averaged morphometric indices (IELO, ISIN and ICIR) for bedforms observed along the LIS bed in Canada. Flow-parallel corridors of streamlined bedforms (IELO > 0, ISIN < 0.1 and ICIR ~ 0.4) alternate with corridors of ribbed bedforms (IELO < 0, ISIN < 0.1 and ICIR > 0.65) and sinuous bedforms (IELO ~ 0 and ISIN > 0.1).Within a portion located in the western part of this transitional zone (Fig.9a), an along-flow

approximately 3 -

 3 km in diameter. Both along-flow and across-flow profiles of morphometric indices show striking symmetrical trends in bedform gradation: streamlined bedforms, predominantly flow-and linear / sinuous ribbed bedforms respectively occur from the borders to the core of the circular spot.

Figure 9 .

 9 Figure 9. Interpretative maps with digitized bedforms of the Irish study area and associated profiles of elongation component ratio IELO and sinuosity index ISIN along transitional sections between flow-perpendicular and flow parallel bedforms (cf. locations on Figs. 7-8). Profiles display raw values of morphometric variables for each bedforms (vertical bars) and moving average curves (5-values). (a) Sections located along the regionalscale transition between fields of linear ribbed bedforms (in the central / northern part of the study area) and streamlined bedforms (in the southwestern part of the study area). (b) Sections located within the regional-scale field of streamlined bedforms, in the southwestern part of the study area, and displaying local transition with spots of ribbed bedforms. (c) Across-flow sections intersect ribbed bedform tracts laterally alternating with streamlined bedform tracts. All profiles illustrate the progressive gradation, whether on regional-or local-scale, between linear flow-perpendicular bedforms (IELO ≤ 0; ISIN ~ 0) and linear flow-parallel bedforms (IELO > 0; ISIN ~ 0) transitioning through sinuous bedforms with no dominant elongation component (IELO ~ 0; ISIN > 0). Additional maps and morphometric profiles of similar transitional sections are provided in Fig. S4.

  Eq. 1) we applied on our bedform database (Fig.11), representing our mathematical interpretation of the degree of bedform evolution:

Fig. 6 ,

 6 circular bedforms are intermediate bedforms among the range of bedforms displaying linear crest lines (i.e. ISIN < 0.10). This morphometric continuity is reflected (i) by their IELO, which is intermediate between those of linear ribbed bedforms and streamlined bedforms, and (ii) their average diameter, which is similar to the average TCMAX of streamlined bedforms and to the average LCMAX of linear ribbed bedforms. The morphometric continuity between linear ribbed, circular and streamlined bedforms corroborates the size and shape bedform continuum proposed by Ely et al. (2016). However, although circular bedforms are scattered within an area overlapping those of linear ribbed bedforms, circular bedforms are absent in the area

  [START_REF] Vérité | Formation of ribbed bedforms below shear margins and lobes of palaeo-ice streams[END_REF][START_REF] Vérité | Formation of murtoos by repeated flooding of ribbed bedforms along subglacial meltwater corridors[END_REF], we therefore consider circular forms as circular ribbed bedforms.

Figure 11 .

 11 Figure 11. (a) Plots of elongation component ratio (IELO) and sinuosity index (ISIN) of bedforms mapped in this study with different bedform populations. (b)A degree of subglacial bedform evolution (from dark blue for the lower degree of evolution to red for the higher one) is calculated for each bedform using Eq. 1 and is based on the evolution of elongation component ratio (IELO) and sinuosity index (ISIN) along spatial sequences. Dotted black lines are graduations of the colored scale materializing the degree of bedform evolution. An example of morphological evolution sequence illustrates how a linear ribbed bedform progressively evolves and fragments through time into several sinuous and streamlined bedforms with increasing degrees of bedform evolution. The color of each bedform corresponds to the color associated with its degree of evolution; grey dots indicate the morphometric values of each bedform along the evolution sequence.

  with a random maximal transverse component. Corroborating works from Hättestrand (1997) and Möller and Dowling (2015) interpreting a continuum of ribbed bedforms from circular to transverse A c c e p t e d P r e -P r o o f 22

Figure 12 .

 12 Figure12. Model of morphological evolution of subglacial bedforms, derived from the morphometric and spatial continuum observed in our study. Subglacial bedforms initiate as circular to linear ribbed bedforms. Linear ribbed bedforms progressively evolve through complex transitional bedforms with variable sinuosities and orientations, towards increasingly elongated streamlined bedforms; while circular ribbed bedforms progressively evolve into a unique or several streamlined bedforms without complex transitional bedforms. The degree of bedform evolution, defined from the combination of their IELO and ISIN, is a quantitative measure of this progressive evolution. Three examples of bedform evolution sequences are proposed depending on the initial IELO of the ribbed bedform (evolution sequence n°3 is presented in Fig.11b). The color of each bedform corresponds to the color associated with its degree of evolution indicated in Fig.11b.

;

  Stokes et al., 2016; Clark et al., 2018)are observed along transition zones between ribbed and streamlined bedforms (Figs. 10b-c). Similar examples of sinuous oblique bedforms fragmenting into streamlined bedforms with en-echelon patterns have been numerically simulated by Fannon et al. (2017), FannonA c c e p t e d P r e -P r o o f 24 (2020; p.139) and Ely et al. (2022), suggesting that streamlined bedforms derive from the fragmentation of sinuous, sometimes oblique bedforms (Fig. 12; evolution sequence n°2 and 3). Our results strongly support these observations and simulations. On average, the maximal transverse components (TCMAX) of transitional bedforms and streamlined bedforms are respectively half and a quarter the TCMAX measured for ribbed bedforms. The evolution of the maximal longitudinal component (LCMAX) is anti-correlated with both TCMAX and amplitudes (hMAX) illustrating the progressive elongation parallel to the ice flow direction and thinning of bedforms. The process of bedform evolution leading to inversion of the dominant component (i.e. TCMAX towards LCMAX) is concomitant with an initial increase in bedform sinuosity. The fragmentation of sinuous bedforms in response to sustained stretching subdivides the bedforms into multiple smaller and less sinuous entities (Figs.

  This evolutionary sequence agrees with the continuum of streamlined bedforms proposed by Sookhan et al. (2021), which demonstrate that sub-circular and large drumlins may subdivide into smaller and more elongated ones without transitional sinuous forms. They indeed demonstrate that the evolution / subdivision of a large drumlin results in contiguous drumlins with individual linear crest lines. Ely et al. (2018) also suggested that drumlins emerge as circular forms before growing, migrating, elongating and potentially coarsening through drumlin amalgamation. Except for coarsening processes that are not included in this study, this is consistent with the evolution sequence n°1 proposed in Figure 12. The existence of such evolutionary sequences, not requiring the initial presence of ribbed bedforms, suggests that streamlined bedforms can form self-organized periodic patterns (i.e. regular spacing and alignment) despite the absence of preliminary sinuous bedform fragmentation processes, as suggested by Spagnolo et al. (2017) and Ely et al. (2018).

  ice flow trajectories along both selected sites are more likely not reliable for two main reasons. First, such hypothetical changes in ice flow directions are not consistent with the most recent reconstruction of ice flow geometry over the Irish Ice Sheet when ice was covering the study area, even during migration of ice divides and reorganization of the ice flow geometry during the early stages of deglaciation[START_REF] Clark | Growth and retreat of the last British-Irish Ice Sheet, 31 000 to 15 000 years ago: the BRITICE-CHRONO reconstruction[END_REF]. Second, the preservation potential of multiple generations of bedforms composed of unconsolidated sediments is questionable considering that changes in flow trajectories are generally associated with changes in ice flow dynamics and reconfiguration of the subglacial hydrological system, which could favor erosion, reshaping and even erasing of preexisting bedforms generations rather than their preservation (e.g. Clark, 1999;[START_REF] Benediktsson | Cross-cutting palaeo-ice streams in NE-Iceland reveal shifting Iceland Ice Sheet dynamics[END_REF].

  Keewatinwhich were identified alternating with corridors of drumlins by[START_REF] Shilts | Canadian shield[END_REF] Aylsworth &[START_REF] Aylsworth | Bedforms of the Keewatin ice sheet, Canada[END_REF] correspond to potential geomorphic markers of meltwater corridors. This suggests that the lateral variations in bedform morphologies observed across welldefined meltwater drainage features indicate the influence of meltwater pressure variations on the degree of bedform evolution (Fig.11).Assuming constant ice-flow trajectories and the model of bedform continuum is correct, the spatial diversity of bedform shapes, dimensions and orientation observed in Ireland and Canada indicates various degree of bedform evolution relative to spatial variations in subglacial conditions (e.g. ice flow velocity, bedrock characteristics, meltwater pressure).

First, a new

  morphometric approachbased on the computation of the circularity index (ICIR), sinuosity index (ISIN) and elongation component ratio (IELO) of subglacial bedformsreveals a morphometric and spatial continuum between ribbed bedforms (i.e. circular to elongated, flowtransverse and linear forms), streamlined bedforms (i.e. drumlins and MSGLs) and sinuous bedforms with various shapes and orientations (i.e. sinuous ribbed bedforms and predominantly flow-parallel sinuous bedforms). Complex, oblique and sinuous bedforms, which have been excluded from most existing models of subglacial continuum and commonly interpreted as resulting from the remolding/superimposition of more simple and younger bedforms, are part of this continuum and can form under the same ice flow trajectory as ribbed and streamlined bedforms.Second, we interpreted this morphological and spatial continuum as the expression of evolutionary sequences between a diversity of ribbed bedforms, ranging from circular to linear transverse, and streamlined bedforms. Intermediate sinuous bedforms, recording the progressive realignment and elongation in the ice flow direction, can correspond to transitional bedforms that result from the stretching of linear ribbed bedforms and fragment into several streamlined bedforms.Evolutionary sequences between circular ribbed bedforms and streamlined bedforms are also thought A c c e p t e d P r e -P r o o f 29 to exist without an intermediate stage of sinuous bedforms. The degree of bedform evolution along these evolutionary sequences can be quantified by a morphometric variable based on the elongation component ratio and the sinuosity index.

, mega-ribs (Greenwood & Kleman, 2010) and traction ribs (Stokes et al., 2016), for

	either orthogonal or parallel to ice displacement directions, the possibility that they may be oblique to
	ice displacement was ignored.
	Using the equations provided
	P r e -P r o o f
	c c e p t e d
	A
	which, unlike drumlins, the
	length was measured perpendicular to the ice flow direction. In an attempt to integrate ribbed, circular
	and streamlined bedforms in a unique morphometric dataset, Ely et al. (2016) redefined the elongation
	ratio as the down-ice length of bedforms (i.e. the length of streamlined bedforms and the width of
	ribbed bedforms) divided by their across-ice length (i.e. conversely). Assuming that bedforms are
	8

by Folk (1968) and Moellering & Rayner (1979) who

	respectively
	studied the shapes of sedimentary grains and catchment areas, Burgess et al. (2003) computed three
	dimensionless shape indices (compactness, circularity, and shape factor) on subglacial bedforms from
	their perimeters and areas to study the complex and contorted shapes of hummocky and Rogen
	moraines in Canada. Recently, Vérité et al. (2022) calculated the sinuosity index of bedform crest
	lines to demonstrate a morphometric continuum between ribbed bedforms and murtoos, assuming that
	murtoos are the most sinuous subglacial bedforms. Compared with length, width and elongation ratio,
	dimensionless shape indices used

by Burgess et al. (2003) and Vérité et al. (2022)

	(i.e. compactness,
	circularity, shape factor, sinuosity) describe the complexity of subglacial bedform outlines and crest
	lines but they do not allow to characterize their orientation with respect to ice flow.

Table 1 .

 1 Description of dimensionless morphometric indices.

	Shape indices	Formula	Signification	Range of values & typical values
	(A)
	Circularity
	index (ICIR)

These morphometric indices are defined so that they can be applied to all subglacial bedforms currently described, including hummocks and murtoos that are not mentioned in this study. The sinuosity ratio is here normalized to the sinuosity ratio of an equilateral triangle, corresponding to the ratio between the length of two sides and the length of one side. The shape of equilateral triangle typically corresponds to the shape of 'murtoos', considered as the most sinuous bedforms currently described in the glacial literature.

Acknowledgment

This study is part of the Ice-Collapse project (The dynamics of ice sheet collapse in deglaciation periods) funded by the French Agence Nationale de la Recherche through grant ANR-18-CE01-0009.

Data availability

All datasets used in this paper are available from the corresponding author on request.

Declaration of competing interests

The authors declare that they have no conflict of interest.