A Gay

C Padron

S Meyer

D Beaufort

E Oliot

S Lallemand

B Marcaillou

M Philippon

J-J Cornée

F Audemard

J-F Lebrun

F Klingelhoefer

B Mercier De Lepinay

P Münch

C Garrocq

M Boucart

M Laigle

L Schenini

Giant seabed polygons and underlying polygonal faults in the Caribbean Sea as markers of the sedimentary cover extension in the Grenada Basin

> Seabed giant polygons were identified in the Grenada Basin, covering the widest area (55000 km²) ever found on Earth > The local ellipse of strains provided major orientations for extension needed for polygons to initiate > Polygonal faults orientations are indicative of the modern strain state in the upper sedimentary column, defining two major tectonic domains

Introduction

Polygonal fault systems (PFS) have been recognised in many basins worldwide [START_REF] Klitgord | Jurassic seismic stratigraphy and basement structure of the western Atlantic magnetic quiet zone[END_REF][START_REF] Clausen | Small scale faulting as an indicator of deformation mechanism in the Tertiary sediments of the northern Danish Central Trough[END_REF][START_REF] Cartwright | Episodic basin-wide hydrofracturing of overpressured Early Cenozoic mudrock sequences in the North Sea Basin[END_REF][START_REF] Oldham | Lake Hope 3D: a case study[END_REF][START_REF] Lonergan | The geometry of polygonal fault in Tertiary mudrocks of the North Sea[END_REF][START_REF] Clausen | Development of intraformational (Oligocene-Miocene) faults in the northern North Sea: influence of remote stresses and doming of Fennoscandia[END_REF][START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF][START_REF] Hansen | Development of a major polygonal fault system in Upper Cretaceous chalk and Cenozoic mudrocks of the Sable Subbasin, Canadian Atlantic margin[END_REF][START_REF] Gay | Cessation/reactivation of polygonal faulting and effects on fluid flow in the Vøring Basin, Norwegian Margin[END_REF]Gay et al, 2009;[START_REF] He | Polygonal faults in the Sanzhao sag of the Songliao basin: their significance in hydrocarbon accumulation[END_REF][START_REF] Sun | Polygonal faults and their implications for hydrocarbon reservoirs in the southern Qiongdongnan Basin, South China Sea[END_REF][START_REF] Cartwright | Diagenetically induced shear failure of fine-grained sediments and the development of polygonal fault systems[END_REF][START_REF] Laurent | High-resolution architecture of a polygonal fault interval inferred from geomodel applied to 3D seismic data from the Gjallar Ridge, Vøring Basin, Offshore Norway[END_REF][START_REF] Ghalayini | Using polygonal layer-bound faults as tools to delimit clastic reservoirs in the Levant Basin offshore Lebanon[END_REF]. They are a special type of non-tectonic normal faults forming polygons in plane view and prisms in 3D, similar to other environments, such as thermal contraction of cooling lavas, ice-wedge polygons [START_REF] Lachenbruch | Mechanics of thermal contraction cracks and ice-wedge polygons[END_REF] or desiccation cracks [START_REF] Weinberger | Initiation and growth of cracks during desiccation of stratified muddy sediments[END_REF].

Four main hypotheses are proposed to explain the PFS formation: i) syneresis related to colloidal properties of fine-grained sediments [START_REF] Dewhurst | The development of polygonal fault systems by syneresis of colloidal sediments[END_REF], ii) density inversions and associated hydrofracturing [START_REF] Watterson | Geometry and origin of a polygonal fault system[END_REF], iii) smectite-rich clays causing residual friction at low burial depth [START_REF] Goulty | Mechanics of layer-bound polygonal faulting in fine-grained sediments[END_REF][START_REF] Goulty | Geomechanics of polygonal fault systems: a review[END_REF] and iv) grain dissolution in uncemented media inducing a decrease in horizontal stress that leads to shear failure and shear strain localization [START_REF] Shin | Contraction-driven shear failure in compacting uncemented sediments[END_REF]2010). Other mechanisms different from lithology variation have been referenced in the literature as responsible for polygonal fault initiation, propagation or reactivation, such as sediment loading [START_REF] Gay | Cessation/reactivation of polygonal faulting and effects on fluid flow in the Vøring Basin, Norwegian Margin[END_REF]. In any case, this kind of shrinkage is due to diagenetic reactions in the host-rock unit during early burial causing volumetric contraction of fine-grained clay-rich sediments accommodated by small scale normal faults [START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF][START_REF] Neagu | Measurement of diagenetic compaction strain quantitative analysis of fault plane dip[END_REF][START_REF] Ireland | Structure of a silica diagenetic transformation zone: the Gjallar Ridge, offshore Norway[END_REF][START_REF] Davies | Differential compaction due to irregular topology of a diagenetic reaction boundary: a new mechanism for the formation of polygonal faults[END_REF][START_REF] Wrona | Kinematics of polygonal fault systems: Observations from the Northern North Sea[END_REF].

PFS are usually interpreted as layer-confined because they occur within sub-horizontal intervals, a few hundreds of m thick, associated with lithological variations of the host sediments. They do not abruptly end at a specific stratigraphical horizon and they can locally connect to major faults structuring the basin [START_REF] Gay | Cessation/reactivation of polygonal faulting and effects on fluid flow in the Vøring Basin, Norwegian Margin[END_REF]. Some of these faults almost reach the modern seabed, thus representing the top of a modern or recently active polygonal fault system [START_REF] Hansen | 3D seismic expression of fluid migration and mud remobilization on the Gjallar Ridge, offshore mid-Norway[END_REF][START_REF] Ireland | Structure of a silica diagenetic transformation zone: the Gjallar Ridge, offshore Norway[END_REF]. Even if a polygonal fault interval (PFI) is generally buried from a few tens of meters to hundreds of meters, only a few examples have been reported outcropping at the present-day seabed such as in the Hatton Basin [START_REF] Berndt | Kilometre-scale polygonal seabed depressions in the Hatton Basin, NE Atlantic Ocean: Constraints on the origin of polygonal faulting[END_REF].

In the investigated area, extending from the Saba Bank southwest of Virgin Islands in the north Grenada Basin (NGB) to the south Grenada Basin (SGB) west off Grenada island, the Grenada Basin is in the backarc basin of the subducting North American Plate beneath the Caribbean Plate (Fig. 1). Giant polygons, 1 to 5 km wide, have been identified outcropping at the seabed during the GARANTI Cruise in May-June 2017. They cover the widest area of outcropping polygonal faults ever found on Earth, with a surface of ~55000 km². They are directly linked to a unique underlying 1000 m thick PFI that thins towards the modern volcanic arc where it abruptly ends and towards the North where sedimentary sequences are getting thinner. However, the PFI also extends towards the Venezuela Basin covering a total area of ~75000 km² in the study area, meaning that its total extent is probably much more.

Recent studies have attempted to link PFS with structural context using topology of the faults (Morley & Binazirnejad, 2020 and references therein) and/or rose diagram orientations (Jimahantakul et al., 2020 and references therein). Even if the fault pattern can be influenced by the slope of underlying units, causing different amounts of horizontal stress anisotropy within the faulted units [START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF][START_REF] Li | Seismic analysis of polygonal fault systems in the Great South Basin, New Zealand[END_REF], the fault orientations are generally used as a direct marker of extension related to local paleo-stress fields during PFS formation [START_REF] Laurent | High-resolution architecture of a polygonal fault interval inferred from geomodel applied to 3D seismic data from the Gjallar Ridge, Vøring Basin, Offshore Norway[END_REF][START_REF] Ho | Formation of linear planform chimneys controlled by preferential hydrocarbon leakage and anisotropic stresses in faulted fine-grained sediments, offshore Angola[END_REF]. The large extent of acquired data from north to south in the Grenada Basin gives the opportunity to investigate whether these polygonal faults are related to the current state of stress in the sedimentary cover by using a novel approach involving the centre-to-centre method. This method applied to adjacent polygons provided unexpected results with the evolving orientation of the sedimentary cover extension in the entire Grenada Basin.

Data and methods

Deep-penetration multichannel seismic reflection (MCS) data were acquired onboard R/V L'Atalante during the GARANTI cruise in 2017 [START_REF] Lebrun | GARANTI cruise, L'Atalante R/V[END_REF]) (Fig. 1). MCS data were collected using a 6473 in 3 airgun array of 16 seismic sources emitting signals with a 9-40 Hz frequency range, and a 4.5 km long, 720-channel solid streamer. The data were quality-controlled and binned in common midpoint (CMP) gathers every 12.5 meters using the SolidQC software of IFREMER and they were processed using the Geovation Software. EM122 multibeam bathymetric data and high-resolution CHIRP profiles were also recorded along all seismic profiles. Additional profiles BOL30, 124VB-11 and 124VB-12 were used to connect our interpretation with previous studies in the SGB (Fig. 1).

Clay mineralogy of the core samples GR-07 and GR-08 collected during the GARANTI cruise (Fig. 1) was determined from X-ray diffractograms of oriented powder mounts of the bulk material and the less than 2 μm granulometric fraction which is usually considered as representative of the clay fraction of sedimentary rocks.

Oriented preparations are a prerequisite for a detailed characterization of the d00 reflections of phyllosilicates.

The disaggregation of the bulk sediments and the dispersion of clay particles was made by ultrasonic treatment in distilled water without any preliminary grinding to strongly limit the contamination with fine grained fragments of detrital minerals. The less than 2 μm fraction was then extracted from the previous suspension by centrifugation. X-ray diffractograms were acquired on a Bruker D8 Advance diffractometer (40 kV and 40 mA) coupled with a copper anticathode (Cu Kα 1+2 radiation) in the 2-30 o 2Θ angular range with 0.02°2Θ steps and a counting time of 2 seconds per step. Relative humidity was not controlled during data acquisition.

SEM observation of clay particles in sediments was performed on freshly broken fragments of core previously coated with a carbon film using a JEOL JSM IT500 scanning electron microscope equipped with Secondary Electron (SE), Backscatter Electron (BSE) detectors and coupled with a Bruker linxeye Energy Dispersive X-ray Spectrometer (EDX). The analytical conditions for quantitative EDX analysis of clay minerals were as follows: acceleration voltage 15 kV, current beam 1 nA, counting time 60 s, working distance 11 mm, and analytical area of ~ 2 μm. The standards used for the EDX quantitative analysis consisted of albite (Na, Al, Si), almandine (Mg, Fe), diopside (Ca), orthoclase (K) and spessartite (Mn). Matrix corrections were performed using an integrated program called PhiRhoz correction. The reproducibility of the standard analyses was 1.5% for all chemical elements, except Na, for which the reproducibility was 3%.

From north to south, five subzones (A to E) have been selected based on available data (seismic profiles, multibeam bathymetry and CHIRP profiles) in order to perform a statistical analysis on the length and orientation of polygon edges. The centre-to-centre distances between every pair of polygons were measured and plotted on a polar diagram using a sheet of tracing film repeatedly placed with its centre over an object, similar to the method of strain analysis [START_REF] Ramsay | Folding and Fracturing of Rocks[END_REF]. The relative positions of all surrounding objects are plotted until a central vacancy field emerges, defining well-defined fabric ellipses. The method requires the assumption of a statistically isotropic initial distribution of objects in which objects are unlikely to be closer than a minimum distance apart. It means that fabric ellipses are due to anisotropic conditions during polygon initiation, with a minor axis parallel the direction of the minimum horizontal stress.

Geological settings

The present-day crescent-shaped Grenada Basin (according to [START_REF] Mann | Caribbean sedimentary basins: Classification and tectonic setting from Jurassic to present[END_REF][START_REF] Picard | Contrasting sedimentary processes along a convergent margin; the Lesser Antilles arc system[END_REF] results from the latter 56-Ma geodynamical evolution of the Caribbean plate movement forming the Greater Arc of the Caribbean (GAC) subduction zone [START_REF] Ladd | Seismic stratigraphy of the Bahamas[END_REF][START_REF] Iturralde-Vinent | Paleogeography of the Caribbean region: implications for Cenozoic biogeography[END_REF][START_REF] Audemard | Key issues on the post-Mesozoic southern Caribbean Plate boundary[END_REF][START_REF] Pindell | The Greater Antillean Arc: Early Cretaceous origin and proposed relationship to Central American subduction mélanges: Implications for models of Caribbean evolution[END_REF][START_REF] Boschman | Kinematic reconstruction of the Caribbean region since the Early Jurassic[END_REF][START_REF] Münch | Pliocene to Pleistocene vertical movements in the forearc of the Lesser Antilles subduction: Insights from chronostratigraphy of shallow-water carbonate platforms (Guadeloupe archipelago)[END_REF][START_REF] Legendre | Trench bending initiation: Upper plate strain pattern and volcanism. Insights from the Lesser Antilles arc, St. Barthelemy Island, French West Indies[END_REF].

It is bounded to the west by the Aves Ridge, to the east by the Lesser Antilles Arc (LAA), to the south by the shallow Venezuelan continental shelf and to the north by the Saba Bank (Fig. 1). Currently, the American plates are subducting below the Caribbean plate at a mean rate of 2 cm/y [START_REF] Demets | GPS geodetic constraints on Caribbean-North America plate motion[END_REF]. The LAA subduction zone marks the eastern boundary of the Caribbean Plate, whereas the Aves Ridge corresponds to the southern part of the remnant GAC that was exposed between approximately 88 and 59 Ma (Fox et 1971; [START_REF] Bouysse | Aves swell and northern Lesser Antilles rodge: Rock-dredging results from ARCANTE 3 cruise[END_REF][START_REF] Bouysse | Subduction of Atlantic aseismic ridges and Late Cenozoic evolution of the Lesser Antilles island arc[END_REF][START_REF] Neill | Origin of the Aves Ridge and Dutch-Venezuelan Antilles: Interaction of the Cretaceous 'Great Arc' and Caribbean-Colombian Oceanic Plateau[END_REF]. The Grenada Basin has long been considered a typical backarc basin between the Aves Ridge and the LAA. However, its morphology varies from a rough ~1000-2000 m deep bathymetry and a Moho ~25 km deep in the North and a flat ~3100 m bathymetry and a ~10-15 km deep Moho in the South [START_REF] Gómez-García | 3D Modeling of Vertical Gravity Gradients and the Delimitation of Tectonic Boundaries: The Caribbean Oceanic Domain as a Case Study: Geochemistry[END_REF][START_REF] Padron | Deep structure of the Grenada Basin from wide-angle seismic, bathymetric and gravity data[END_REF]. This is a marginal basin partly underlain by oceanic crust [START_REF] Christeson | Crustal structure of the Caribbeannortheastern South America arc-continent collision zone[END_REF][START_REF] Allen | The role of arc migration in the development of the Lesser Antilles: A new tectonic model for the Cenozoic evolution of the eastern Caribbean[END_REF][START_REF] Padron | Deep structure of the Grenada Basin from wide-angle seismic, bathymetric and gravity data[END_REF], adjacent to an oceanic island arc, that receives volcaniclastic debris from the bordering volcanic arc and, to a lesser degree, the remnant arc [START_REF] Carey | A model of volcanogenic sedimentation in marginal basins[END_REF][START_REF] Parra | Two potential sources for Holocene clay sedimentation in the Caribbean Basin: The Lesser Antilles Arc and the South American continent[END_REF][START_REF] Murray | Diagenesis in tephra-rich sediments from the Lesser Antilles Volcanic Arc: pore fluid constraints[END_REF].

However, previous sedimentological and geochemical studies have indicated that Grenada Basin sediments are originated from two principal sources: volcanogenic material from the LAA, and terrigenous material eroded from the South American continent [START_REF] Pautrizel | Modalités de la sédimentation au Quaternaire récent dans la Mer des Petites Antilles (Fosse de Grenade--Ride des Oiseaux--Bassin du Venezuela)[END_REF][START_REF] Kinder | Some aspects of the Caribbean circulation[END_REF][START_REF] Bowles | Orinoco and Amazon river sediment input to the eastern Caribbean Sea[END_REF][START_REF] Parra | Two potential sources for Holocene clay sedimentation in the Caribbean Basin: The Lesser Antilles Arc and the South American continent[END_REF].

Seabed giant polygons and related polygonal faults

Seabed polygons are separated from the neighbouring polygons by 700-1500 m wide and 10-60 m deep furrows (compared to the regional seafloor) with a flat bottom (Fig. 2). The polygons are ranging from 2000 to 3000 m on average. However, in the SGB the seabed polygons are much wider by a factor of 50-to-100%

where they can reach 5000 m in length, suggesting that the size of polygons depends on the location within the basin. On profile AB (GA-35 close-up view in sub-zone B), seafloor furrows are generally seen as small depressions affecting underlying horizons down to 200-300 ms TWT below seafloor (Fig. 2). The vertically stacked furrows are consistently located above small-scale faults affecting an interval ranging from 700 to 1200 ms TWT. The faults are characterized by various reflectors offsets, typically a few tens of ms TWT and they reach the seabed where they bound furrows. On high-resolution Chirp profiles, and despite a 20X vertical exaggeration, the polygons have very steep flanks (~3°), defining depressions (or furrows) that are 800-1500 m wide and 40 m deep compared to the 0,3° smooth regional slope (Fig. 3). Very high amplitude reflections occur right beneath the seabed creating large hyperbolae. A seismic profile displayed at the identical horizontal scale shows that each depression is bounded by faults reaching the seabed (Fig. 3). The polygonal faults are characterized on seismic profiles by an intense dimming of reflections on both edges of the fault planes suggesting that fluids are currently migrating upward. In more details all faults are normal and they are organised in set of 2-5 parallel faults (Fig. 4). Within the set of faults, the depressed reflections are vertically stacked and the offset along the fault plane is decreasing towards the seabed. At depth, the throw along a fault plane is progressively decreasing at the base of the polygonal fault interval where the lowest horizons do not appear affected (Fig. 4). The maximum throw along the fault planes has been identified at the transition between Early and Middle Pliocene (Fig. 4). over the Grenada Basin. Units were correlated and described in [START_REF] Padron | Deep structure of the Grenada Basin from wide-angle seismic, bathymetric and gravity data[END_REF].

Nature of sediments

The present-day Antillean sediments are mainly of mafic-affinity, rich in pyroxenes and amphiboles but poor in quartz and biotite (in agreement with the andesitic composition of the volcanogenic material related to the LAA activity), whereas the South American supply is essentially of felsic-affinity, composed of quartz and phyllosilicate minerals, such as biotite, muscovite, and chlorite [START_REF] Pautrizel | Modalités de la sédimentation au Quaternaire récent dans la Mer des Petites Antilles (Fosse de Grenade--Ride des Oiseaux--Bassin du Venezuela)[END_REF]. Clay mineralogy of two grab cores (GR-07 and GR-08) collected during the GARANTI cruise display very similar patterns. The bulk analysis shows a high clay content, mostly composed of smectite, illite/muscovite and kaolinite with subordinate chlorite and pyrophyllite, mixed with non-clay minerals such as pyroxenes, plagioclases, quartz and calcite (Fig. 5). The disappearance of chlorite and pyrophyllite X-ray reflections as well as the strong decrease in the peak intensity of illite/muscovite present in the XRD pattern of the fraction <2 µm indicate that these minerals are coarse grained and probably related to clastic metamorphic-derived source (Fig. 5). The XRD patterns of the fraction <2 µm are similar for both samples, with a very high smectite content and lower amounts of kaolinite and illite/muscovite. Based on cation exchange capacity measurement for the bulk material, the smectite content has been estimated to about 50% in both samples.

Figure 5: XRD analysis of grab cores GR-07 and GR-08. Top: bulk analysis displaying very similar patterns with a high clay content (smectite: sm, illite: Il, chlorite: ch and kaolinite: ka) and some other minerals (pyroxenes: py, plagioclases: pl, quartz: qz and calcite: ca). Bottom: analysis of the fraction <2 µm for both samples GR-07 (blue curves) and GR-08 (green curve) displaying very high smectite content, high kaolinite peaks and low illite peaks.

The SEM photographs shows that the sediments are dominated by smectite and kaolinite (Fig. 6). The calcite content is provided by the pelagic sedimentation of foraminifers although other minerals, such as feldspars (plagioclases), pyroxenes and quartz represent a minor contribution. The collected seabed sediments are primarily composed (>50%) of smectite at the grab cores location.

Chemical point analyses of the clay particles observed with SEM indicate that smectite is dioctahedral and is rich in Al and Fe with a layer charge mostly satisfied by K in interlayer position (Fig. 6). Such a chemical composition is similar to that of the potassic ferriferous beidellites characterized in the volcanogenic clay material in recent marine sediments [START_REF] Desprairies | Relation entre la composition des smectites d'altération sousmarine et leurs cortèges de terres rares[END_REF][START_REF] Parra | Origin and evolution of smectites in recent marine sediments of the NE Atlantique[END_REF]. In these sediments, smectite is generally derived from the diagenetic transformation of volcanogenic material (including volcanic ashes) into clay minerals. Conversely, kaolinite is usually derived from the weathering or the hydrothermal alteration of Al-bearing silicates in low pH environments and it is considered as a clastic material in marine sediments.

Seabed polygon organization

The polygons in the sub-zones A and B are very similar in size and shape (Fig. 7). However, they are 10-15 m deep in sub-zone A and 40 m deep in sub-zone B compared to the regional seafloor. They are characterized by 3 mains directions oriented N170-N10°, N30-50° and N90-110°, with lengths of 1260/1274, 1190/1208 and 1345/1374 ±15 m respectively. The basic shape is a polygon with a major axis of ~2400/2500 m in length in sub-zones A and B. In the NGB the polygons are very similar in shape and orientation. The ellipses obtained from the centreto-centre method show a minor axis oriented N40° indicating the main direction of anisotropic contraction of polygons. In the SGB the polygons are more elongated and their orientation progressively rotates southward.

Seafloor furrows are deeper in the north than in the south by a factor of two. The minor axes are oriented N120°, N144° and N163° respectively, indicating that they are progressively changing towards the SGB.

Consequently, two main areas are defined in the Grenada Basin in the north and the south by both the general shape of polygons and the direction of the minor axis obtained from the centre-to-centre method. Polygons are very similar in size and shape in sub-zones A and B in the NGB, although they are getting longer from sub-zone C to E towards the SGB. This is confirmed by best fit ellipses obtained from the centre-tocentre method, giving a minor axis oriented N40° in sub-zones A and B whereas it is turning from N120° in sub-zone C to N144° in sub-zone D and N163° in sub-zone E.

Discussion

Origin of sediments and sequences

The volcanic arc is volumetrically the most important source for sediments, with abundant volcaniclastics being produced by explosive subaerial and/or subaqueous eruptions and secondary erosion of the arc complex [START_REF] Carey | A model of volcanogenic sedimentation in marginal basins[END_REF]. Except for direct ash-fall deposits, the recent sediments in the flanks of the Grenada Basin are composed of volcanogenic debris emplaced by turbidity currents and debris-flows coming from both the LAA [START_REF] Carey | Deep-sea evidence for distribution of tephra from the mixed magma eruption of the Soufriere of St. Vincent, 1902: ash turbidites and air fall[END_REF]1980;[START_REF] Sigurdsson | Volcanogenic sedimentation in the Lesser Antilles arc[END_REF][START_REF] Deplus | Submarine evidence for large-scale debris avalanches in the Lesser Antilles Arc[END_REF][START_REF] Brunet | Composition, geometry, and emplacement dynamics of a large volcanic island landslide offshore Martinique: from volcano flank-collapse to seafloor sediment failure?[END_REF][START_REF] Friant | Submarine landslides around volcanic islands: a review of what can be learned from the Lesser Antilles Arc[END_REF][START_REF] Seibert | Seafloor morphology and sediment transfer in the mixed carbonate siliciclastic environment of the Lesser Antilles forearc along Barbuda to St. Lucia[END_REF] and from the Aves Ridge [START_REF] Bader | Initial Reports of the Deep Sea Drilling Project, Site 30[END_REF][START_REF] Holcombe | Caribbean marine geology; ridges and basin of the plate interior[END_REF]. The hemipelagic sedimentation mixing dispersed ash and clastic clays coming from South America is the dominant process in the deeper part of the Grenada Basin (Sen [START_REF] Gupta | Diagenesis in tephra-rich sediments from the Lesser Antilles volcanic arc; pore fluid constraints[END_REF], entering the Grenada basin by northward flowing Caribbean ocean currents [START_REF] Corredor | Cyclonic eddy entrains Orinoco river plume in Eastern Caribbean[END_REF]. This mixed sedimentation is active since middle Miocene [START_REF] Aitken | Evolution of the Grenada and Tobago basins and implications for arc migration[END_REF] defining unit U3 (Fig. 8) identified on seismic sections as chaotic and semi-continuous tabular high amplitude reflectors [START_REF] Padron | Deep structure of the Grenada Basin from wide-angle seismic, bathymetric and gravity data[END_REF]. The LAA is composed of mostly andesitic igneous rocks [START_REF] Macdonald | The lesser Antilles chain: a study in arc magmatism[END_REF] whose weathering products carry a distinctive signature. Smectites and kaolinite are the dominant clay minerals produced by the weathering of Lesser Antillean igneous rocks, with over twice as much smectite produced as kaolinite. Another typical weathering characteristic of an igneous island arc terrane is the absence of illite or illite-smectite mixed layers. Hydrothermal illite and illite-smectite mixed-layers may form in the geothermal fields related to the volcanic activity, as documented on the islands of Guadeloupe to Grenada [START_REF] Parra | Two potential sources for Holocene clay sedimentation in the Caribbean Basin: The Lesser Antilles Arc and the South American continent[END_REF]Mas et al, 2003;Mas et al. 2006;[START_REF] Murray | Diagenesis in tephra-rich sediments from the Lesser Antilles Volcanic Arc: pore fluid constraints[END_REF]. However, the spatial extension of the geothermal areas is too limited to exert a significant contribution to the clay material accumulated in marine sediments. Hence, high smectite and low illite contents are typical of sediments with a volcanogenic (Lesser Antillean) source [START_REF] Pautrizel | Modalités de la sédimentation au Quaternaire récent dans la Mer des Petites Antilles (Fosse de Grenade--Ride des Oiseaux--Bassin du Venezuela)[END_REF][START_REF] Parra | Two potential sources for Holocene clay sedimentation in the Caribbean Basin: The Lesser Antilles Arc and the South American continent[END_REF]. Crustal terranes of the South American continent made of granites and gneisses act as the source area for non-volcanic abiogenic sediments deposited in the Caribbean. Tropical weathering of these continental rocks produces four dominant clay minerals: illite, kaolinite, smectites, and chlorite. This fine-grained terrigenous material is carried in suspension to the southeast Caribbean by major rivers, particularly the Amazon and the Orinoco [START_REF] Bowles | Orinoco and Amazon river sediment input to the eastern Caribbean Sea[END_REF]. Thus, a clay mineral assemblage characterized by high abundances of illite, kaolinite, chlorite and minor pyrophyllite is typical of a South American terrigenous source, as is the presence of detrital quartz [START_REF] Pautrizel | Modalités de la sédimentation au Quaternaire récent dans la Mer des Petites Antilles (Fosse de Grenade--Ride des Oiseaux--Bassin du Venezuela)[END_REF]. Some deep channels across the Aves Ridge permitted the transport of the Venezuela Basin bottom water (commonly labeled Caribbean Bottom Water or CBW) into the Grenada Basin [START_REF] Kinder | Some aspects of the Caribbean circulation[END_REF].

Unit U2 (Early to Middle Miocene) was fed by the Orinoco whereas its delta front was oriented North-South towards the Grenada basin. During Tertiary times the Orinoco drainage basin and its associated delta has migrated eastward over more than 500 km because of the eastward deformation in north Venezuela due to the Caribbean plate/South America relative movements [START_REF] Beck | Eocene paleogeography of the southeastern Caribbean: relations between sedimentation on the Atlantic abyssal plain at site 672 and evolution of the South America margin[END_REF][START_REF] Pindell | Foundations of Gulf of Mexico and Carribean evolution : eight controversies resolved[END_REF][START_REF] Audemard | Key issues on the post-Mesozoic southern Caribbean Plate boundary[END_REF]. Presently, the turbidite system issued from the present-day Orinoco delta develops at the eastern edge of the East Caribbean active margin partly above the large southern part of the Barbados accretionary prism and downslope at the front of this prism, within the Demerara abyssal plain [START_REF] Deville | Tectonics and sedimentation interactions in the east Caribbean subduction zone; an overview from the Orinoco Delta and the Barbados accretionary prism[END_REF]. Unit U2 is marked by a shallowing of the facies corresponding to a progressive infilling of the Grenada Basin [START_REF] Padron | Deep structure of the Grenada Basin from wide-angle seismic, bathymetric and gravity data[END_REF]. Sediments in Unit U1 are deep-water, pelagic and volcanogenic shale and siltstone, with some biogenic limestone in the deeper parts [START_REF] Ysaccis | Tertiary evolution of the northeastern Venezuela offshore[END_REF], whereas seismic reflections in Unit U0 are very chaotic but they are well organized in some places, possibly related to cretaceous sediments. In the NGB, the polygonal faults reach the seafloor, except in the vicinity of deep turbiditic channels where they do not develop at all (Fig. 8). The PFI is getting progressively thicker southward as its base is getting deeper. However, the PFI abruptly ends at the toe of the Saba Bank (not shown in this study but visible on profile GA-15C in [START_REF] Cornée | Lost islands in the northern Lesser Antilles: a possible milestone in the Cenozoic dispersal of terrestrial organisms between South-America and the Greater Antilles[END_REF].

In the western part of the SGB, the PFS are affecting unit U3 (Fig. 8) and giant polygons are cropping out at the seabed. The base of PFI is not concordant with the base of unit U3 and, as shown on Fig. 4, it is undulating over the Grenada Basin within the top of unit U2, depending on the depth at which the faults have propagated downward. In the centre of the SGB the PFS do not reach the seafloor nor unit U2 where the PFI appears buried (500 ms TWT below seafloor) and thinner (700 ms TWT) defining a lenticular shape. The PFS abruptly end eastward at a distance of ~40-70 km from the LAA. On the Aves ridge, the PFI appears thinner by a factor of 2, which is correlated with thinner sedimentary sequences.

The total area covered by both outcropping and buried polygonal faults is about 75000 km² (55000 + 20000 km² respectively), defining a crescent shape more or less concordant with the shape of the Grenada Basin (Fig

1).

Polygonal faults orientations as a marker of basin extension

The main hypothesis for polygonal faults formation considers a finite bed length extension and the development of conjugate fault planes with extensional offsets (apparent normal faults) due to a bulk volume loss of pore fluids. This process leads to a thinned sequence, which is partly compacted [START_REF] Cartwright | Volumetric contraction during the compaction of mudrocks: a mechanism for the development of regional-scale polygonal fault systems[END_REF][START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF]. The consequences are that the complex polygonal fault systems in mudstone dominated sequences formed due to volumetric contraction and concomitant fluid expulsion [START_REF] Shin | Contraction-driven shear failure in compacting uncemented sediments[END_REF][START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF][START_REF] Gay | Cessation/reactivation of polygonal faulting and effects on fluid flow in the Vøring Basin, Norwegian Margin[END_REF] that leads to shrinkage in mud-dominated sequences [START_REF] Cartwright | Volumetric contraction during the compaction of mudrocks: a mechanism for the development of regional-scale polygonal fault systems[END_REF][START_REF] Dewhurst | The development of polygonal fault systems by syneresis of colloidal sediments[END_REF]. Normal faulting related to burial and sediment loading is not a common process that has been already proposed for rocks deformation, probably because soft sediments like clays have a specific rheological behaviour. The dominant orientation of minor axis of fabric ellipses obtained from the centre-to-centre method as well as their bending in the slope direction suggests the interaction of gravity-driven shearing of the compacting interval. For example, sedimentary structures, such as dewatering pipes and sand volcanoes [START_REF] Waldron | Determination of finite strain in bedding surfaces using sedimentary structures and trace fossils: a comparison of techniques[END_REF], may display diffuse outlines but nonetheless show anticlustered distributions. A radial stress tension related to the volumetric contraction of clay could allow to reduce the lithostatic mean stress, also increasing the differential stress, and then to reach shear failure along a Coulomb-type conventional envelope. However, this process assumes that 1) the faulted material has a common frictional envelope, 2) the lithostatic stress state prior to faulting due to compaction was sub-critical (i.e. close to shear failure), and/or 3) the stress tension related to the volumetric change is large enough. In such unlithified and uncompacted clay sediments, the coefficient of Earth pressure at rest (Ko) is generally large (0.45-0.8) implying a little differential stress due to the vertical loading [START_REF] Earl | Assessment of the behaviour of field soils during compression[END_REF][START_REF] James | Discussion on development of polygonal fault systems: a test of hypotheses[END_REF]. This suggests that a high horizontal tension due to internal contraction is needed to get the failure envelope [START_REF] Cartwright | Volumetric contraction during the compaction of mudrocks: a mechanism for the development of regional-scale polygonal fault systems[END_REF], coupled to an external regional extension [START_REF] Laurent | High-resolution architecture of a polygonal fault interval inferred from geomodel applied to 3D seismic data from the Gjallar Ridge, Vøring Basin, Offshore Norway[END_REF][START_REF] Bureau | Characterization of interactions between a pre-existing polygonal fault system and sandstone intrusions and the determination of paleo-stresses in the Faroe-Shetland basin[END_REF]. This model suggests that progressive sediment loading acts as a centrepiece in the initiation and propagation of polygonal faults [START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF][START_REF] Gay | Cessation/reactivation of polygonal faulting and effects on fluid flow in the Vøring Basin, Norwegian Margin[END_REF][START_REF] Reiche | High-resolution seismic stratigraphy, sedimentary processes and the origin of seabed cracks and pockmarks at Nyegga, mid-Norwegian margin[END_REF]. It also implies that, at great burial depths, the compaction through the process of volumetric contraction ends and the dissolution-precipitation mechanisms become dominant [START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF]. [START_REF] Clausen | Development of intraformational (Oligocene-Miocene) faults in the northern North Sea: influence of remote stresses and doming of Fennoscandia[END_REF] have suggested that PF developed within a broadly polygonal array due to gravitational sliding influenced by far-field tectonic stresses, even if [START_REF] Wrona | Kinematics of polygonal fault systems: Observations from the Northern North Sea[END_REF] have recently suggested that such processes played no or an imperceptible role in the growth of this specific North Sea system. However, the hypothesis that layer-parallel displacements of these faults are accommodated by regional extension of the host strata has been confirmed in various basins, such as in the Lower Congo Basin [START_REF] Gay | Polygonal faults-furrows system related to early stages of compaction -Upper Miocene to present sediments of the Lower Congo Basin[END_REF] or in the Angola basin [START_REF] Ho | Formation of linear planform chimneys controlled by preferential hydrocarbon leakage and anisotropic stresses in faulted fine-grained sediments, offshore Angola[END_REF], and more recently by field studies showing that a radial extension may account for PFS development [START_REF] Antonellini | Polygonal deformation bands[END_REF][START_REF] Petracchini | Syn-thrusting polygonal normal faults exposed in the hinge of the Cingoli anticline, northern Apennines, Italy[END_REF].

Syn-thrusting non-diagenetic polygonal normal faults were reported in the Cingoli anticline. The PF nucleated into multi-directional stretching processes that are consistent with the fault slip indicators [START_REF] Petracchini | Syn-thrusting polygonal normal faults exposed in the hinge of the Cingoli anticline, northern Apennines, Italy[END_REF]. This study also reported that faults are not systematically parallel to the general trending anticline axis suggesting an evolving pattern of PFS influenced by structural paleo-stresses conditions. So, the development of PFS at the seabed could thus be indicative of the present-day strain state within the sedimentary column [START_REF] Tuckwell | The control of stress history and flaw distribution on the evolution of polygonal fracture networks[END_REF][START_REF] Ho | Formation of linear planform chimneys controlled by preferential hydrocarbon leakage and anisotropic stresses in faulted fine-grained sediments, offshore Angola[END_REF][START_REF] Jitmahantakul | Seismic characteristics of polygonal faults systems in the Great South Nasin, New Zealand[END_REF].

In the NGB, the strain ellipses show a similar minor axis orientation of N40° in both areas A and B, meaning that all polygons are contracting following the same major direction. It is now commonly admitted that such direction of contraction is parallel to the minimum principal stress σ3 [START_REF] Petracchini | Syn-thrusting polygonal normal faults exposed in the hinge of the Cingoli anticline, northern Apennines, Italy[END_REF][START_REF] Ho | Formation of linear planform chimneys controlled by preferential hydrocarbon leakage and anisotropic stresses in faulted fine-grained sediments, offshore Angola[END_REF], which represent a direction of extension within the sedimentary column. So, we assume here that the PFI is currently in extension following one major direction of N40°. The forearc-backarc domain has undergone a regional subsidence and a NE-SW extension since Late Miocene, evidenced in the Northern Lesser Antilles forearc [START_REF] Boucard | Paleogene V-shaped basins and Neogene subsidence of the Northern Lesser Antilles Forearc[END_REF] and in the intra-arc Kalinago Basin [START_REF] Cornée | Lost islands in the northern Lesser Antilles: a possible milestone in the Cenozoic dispersal of terrestrial organisms between South-America and the Greater Antilles[END_REF]. The N40° extension evidenced from PF orientations thus may indicate that the upper sedimentary cover (unit U3 at least) is submitted to similar processes in the backarc. During this period, uplifts phases were moderate compared to this subsidence, which has submerged islands and previously uplifted areas [START_REF] Bouysse | Opening of the Grenada backarc basin and evolution of the Caribbean Plate during the Mesozoic and early Paleogene[END_REF][START_REF] Mann | Actively evolving microplate formation by oblique collision and sideways motion along strike-slip faults: an example from the northeastern Caribbean plate margin[END_REF][START_REF] Feuillet | Active faulting induced by slip partitioning in Montserrat and link with volcanic activity: New insights from the 2009 GWADASEIS marine cruise data[END_REF]Philippon et al., 2020a;[START_REF] Legendre | Trench bending initiation: Upper plate strain pattern and volcanism. Insights from the Lesser Antilles arc, St. Barthelemy Island, French West Indies[END_REF][START_REF] Cornée | Paleogene carbonate systems of Saint Barthélemy, Lesser Antilles: stratigraphy and general organization[END_REF][START_REF] Cornée | Lost islands in the northern Lesser Antilles: a possible milestone in the Cenozoic dispersal of terrestrial organisms between South-America and the Greater Antilles[END_REF].

The extensive deformation has triggered deeply-rooting NW-SE faults in the forearc [START_REF] Boucard | Paleogene V-shaped basins and Neogene subsidence of the Northern Lesser Antilles Forearc[END_REF] and in the Kalinago Basin [START_REF] Cornée | Lost islands in the northern Lesser Antilles: a possible milestone in the Cenozoic dispersal of terrestrial organisms between South-America and the Greater Antilles[END_REF]. This is possibly explained by thermal relaxation related to the cessation of arc activity in the northeastern LAA [START_REF] Macphee | Pre-Wisconsinan mammals from Jamaica and models of late Quaternary extinction in the Greater Antilles[END_REF][START_REF] Jany | Geological interpretation of combined seabeam, gloria and seismic data from Anegada passage (Virgin Islands, north Caribbean)[END_REF][START_REF] Philippon | Caribbean intra-plate deformation: Paleomagnetic evidence from St. Barthelemy island for Post-Oligocene rotation in the Lesser Antilles forearc[END_REF] and/or by distal tectonics influence of subduction dynamics [START_REF] Boucard | Paleogene V-shaped basins and Neogene subsidence of the Northern Lesser Antilles Forearc[END_REF]. The more recent and continuous subsidence since Late Miocene -Early Pliocene may explain shallower depths of both the present-day seafloor and the base of unit U3, with an average depth of 2 s TWT (Fig. 9). The isopach map of unit U3 shows a constant thickness of about 0.7-0.8 s TWT, even in the area of Saba Bank, indicating a homogeneous subsidence in the entire area (Fig. 9).

In the SGB, the ellipses show a change in direction of the minor axis, indicating varying orientations of the extension of the PFI pointing towards the centre of the SGB where the curved depocenter of unit U3 follows the curvature of the modern volcanic arc (Fig. 9). The major faults identified in the SGB [START_REF] Speed | Lesser Antilles arc and adjacent terranes[END_REF][START_REF] Pindell | Geological evolution of the Caribbean region: a plate tectonic perspective[END_REF][START_REF] Pindell | Tectonic evolution of the Gulf of Mexico, Caribbean and northern South America in the mantle reference frame: An update[END_REF][START_REF] Aitken | Evolution of the Grenada and Tobago basins and implications for arc migration[END_REF][START_REF] Padron | Deep structure of the Grenada Basin from wide-angle seismic, bathymetric and gravity data[END_REF] do not seem to control the orientation of PF, indicating that the upper part of the sedimentary column, particularly unit U3, is actually in extension following a general subsidence in the south. In this area, the subsidence has significantly increased from Late Miocene to present, possibly explained by a southeastward regional tilting and/or a greater sediment compaction in response to sediment loading in the basin [START_REF] Padron | Deep structure of the Grenada Basin from wide-angle seismic, bathymetric and gravity data[END_REF]. The base of unit U3 is characterized by a sharp change in average depth from 2 to 5 s TWT west off Dominica island and a slight deepening to the South where it reaches about 6 s TWT west off Grenada island (Fig. 9).

However, the isopach map of unit U3 shows a curved NE-SW depocenter with a maximum thickness of 2,3 km in the southwest indicating a higher subsidence rate in the south.

Factors controlling the initiation and propagation of polygonal faults

Lithological analyses conducted on wells penetrating polygonal fault intervals show that the amount of shrinkage in layers appears to increase as the grain size decreases and smectite content increases [START_REF] Dewhurst | The development of polygonal fault systems by syneresis of colloidal sediments[END_REF][START_REF] Gay | Cessation/reactivation of polygonal faulting and effects on fluid flow in the Vøring Basin, Norwegian Margin[END_REF]. This composition forms a fine texture of flocculated particles, 2 μm in size, submitted to shrinkage upon drying leaving voids that are about 5 μm in width [START_REF] Forsberg | Mineralogical and microstructural development of the sediments on the Mid-Norwegian margin[END_REF]. This process occurs at a microscale, and could explain 1) the spontaneous contraction of mud-dominated sediments, leading to the formation of normal faults at a wider scale and 2) the development of polygonal faults outcropping at the seabed, meaning that the volumetric contraction can form during early compaction, leading to a rapid bulk volume loss. The PF initiation and propagation in the Grenada Basin can be summarized in an evolutionary model taking into account their morphology both at the seabed and on seismic profiles (Fig. 10). Unit U2 was fed by clastic metamorphic material coming from the Orinoco delta while it was oriented NS, which is not compatible with spontaneous volumetric sediment contraction at seabed (stage 1). During Late Miocene the lateral shift of the Orinoco river on the eastern flank of the LAA has limited the coarser metamorphic material input into the Grenada basin (Diaz de [START_REF] Diaz De Gamero | The changing course of the Orinoco river during the neogene: a review[END_REF][START_REF] Escalona | Tectonics, basin subsidence mechanisms, and paleogeography of the Caribbean-South American plate boundary zone[END_REF]. The base of unit U3 was more and more fed by smectite-rich sediments derived from the volcanic arc and/or coming from the South American continent through the CBW current (stage 2) and the contraction started during the Middle Pliocene (stage 3). The process of contraction has gone on as long as sediments deposited at the seabed had the composition required for PF development [START_REF] Gay | Cessation/reactivation of polygonal faulting and effects on fluid flow in the Vøring Basin, Norwegian Margin[END_REF][START_REF] Jackson | Can polygonal faults help locate deep-water reservoirs[END_REF]. Increased dewatering during burial implied more displacement along faults [START_REF] Jitmahantakul | Seismic characteristics of polygonal faults systems in the Great South Nasin, New Zealand[END_REF], which would continue to grow as long as the dewatering cell contracted volumetrically (stage 4). The PF propagated both upward and downward although the progressive reduction in bed length through contraction was balanced by an incremental increase in the amount of displacement distributed along the faults (stage 5). Fault propagation is much easier towards the seabed than at depth due to the confinement pressure, and the throw maximum along major PF planes is always found in the lower tier of the affected interval. At depth, secondary faults are initiated due to sediment loading in order to better drain the centre of polygons (stage 5). They display lower amount of displacement than the major polygonal faults at the same stratigraphic level, confirming that the nucleation of PF occurred where/when the throw is maximum (i.e. Middle Pliocene) (stage 6a). The kinematic model of polygonal fault growth in which the propagation of faults is discontinuous during basin infilling leads to a 4D interpretation of the whole polygonal faut system. In the eastern part of the SGB, the polygonal fault interval appears buried and the PFI is thinning towards the east, cross-cutting stratigraphic horizons. The eastern part of the Grenada Basin can be considered as a fossil interval of PF (stage 6b), whereas polygonal faulting remains active to the west and although the initiation point is at the same stratigraphic level both in east and west. This could reflect 1) a progressive westward migration of the extension domain affecting the upper part of the sedimentary column or 2) the western area remains dominated by clay-and smectite-rich sedimentation, whereas the eastern area has received different lithologies. Flank collapse events have occurred all along the LAA, resulting in debris avalanches, some of them involving large volumes of material [START_REF] Deplus | Submarine evidence for large-scale debris avalanches in the Lesser Antilles Arc[END_REF][START_REF] Brunet | Composition, geometry, and emplacement dynamics of a large volcanic island landslide offshore Martinique: from volcano flank-collapse to seafloor sediment failure?[END_REF]. A striking characteristic of the deposits in the Grenada Basin is the presence of a thick chaotic unit in seismic data, about 250 ms TWT in thickness, west off emerged islands in the deep basin. Some of the debris avalanches have evolved into debris flows and turbidity currents, feeding the eastern deep basin with coarser material [START_REF] Deplus | Submarine evidence for large-scale debris avalanches in the Lesser Antilles Arc[END_REF]. At this point, the change in sediment input (and composition) seems the most probable cause for stopping PF propagation in the east Grenada Basin.

Model of polygonal faults in the asymmetric basin of Grenada

Major differences in depths, morphologies and PF between the NGB and the SGB are evidenced along a NS profile (GH) (Fig. 11 > The basin infilling is controlled by the available space for sedimentation. Due to repeated episodes of uplift and emersion in the north, the thickness of the sedimentary cover is relatively thin with a maximum of about 3 s TWT south of Saba Bank and west of Antigua. Unit U3 appears isopachous with a seabed slightly deepening to the south, indicating that the present-day subsidence is homogeneous in the area. The continuous subsidence in the south since Eocene has led to a very thick sedimentary cover of about 6 s TWT between the Aves Ridge and the Grenada Island. Unit U3 is thickening southward with a relative flat seabed, indicating an actual differential subsidence which is stronger in the south.

> Classical normal faults usually propagate parallel to the principal stress σ1, while hydraulic fractures open in parallel to the intermediate horizontal stress σ2 and against the minimum principal stress σ3 [START_REF] Cosgrove | The expression of hydraulic fracturing in rocks and sediments[END_REF]. However, PFS do not follow such behavior due to the anisotropy of unlithified cohesive fine-grained sediments and the anisotropic stress attributed to perturbations of the regional stress field can control the PF orientations [START_REF] Ho | Formation of linear planform chimneys controlled by preferential hydrocarbon leakage and anisotropic stresses in faulted fine-grained sediments, offshore Angola[END_REF]. Volumetric contraction is a spontaneous process forming polygons by joints growing from an initiation point in all directions at the same time, such as in cooling lava flows [START_REF] Aydin | Evolution of polygonal fracture patterns in lava flows[END_REF]. In sedimentary basins, σ1 is generally vertical and corresponds to the lithostatic vertical stress (σv) and we assume here that σ2 and σ3 are respectively perpendicular and parallel to the minor axis of fabric ellipses obtained from the centre-to-centre method. In the north, the PFI abruptly ends at the toe of Saba Bank, a lithified reefal and red algal platform [START_REF] Cornée | Lost islands in the northern Lesser Antilles: a possible milestone in the Cenozoic dispersal of terrestrial organisms between South-America and the Greater Antilles[END_REF] in which PF cannot develop. PF are controlled by a general N40° extension of the present-day backarc basin. In the south, PF are following the deepest point of subsidence actually towards the south. The active interval of PF has continued growing until the present day in the centre and in the west of the Grenada Basin whereas it has stopped or never developed in the east due to coarser and/or less smectite-rich input coming from the active LAA.

> Between the NGB and the SGB, the change in depths, morphologies and polygonal faults orientations is sharp, suggesting that both basins are presently submitted to very different tectonic regimes. The boundary between both PF domains is estimated to be west off Dominica island, which is consistent with a NGB underlain by about 90 Ma subducted lithosphere, whereas the subducted lithosphere is up to 120 Ma old beneath the SGB [START_REF] Müller | A Global Plate Model Including Lithospheric Deformation Along Major Rifts and Orogens Since the Triassic[END_REF]. This is confirmed by a basement that remains high in the north, reaching a maximum of about 4 s TWT in depth, although it is strongly deepening in the south, reaching a maximum of 10 s TWT. This limit between two underlying geodynamic domains may actually be recorded by shallow PFI in the backarc domain.

Conclusion

A regional acquisition of seismic data and multibeam imagery at a basin scale provided new insights on the recent Grenada Basin history and improved the understanding of post-depositional processes occurring in PFI.

We have identified giant seabed polygons covering the widest area ever found on Earth, with a surface of approximately 55000 km², and even up to 75000 km 2 taking into account those which do not reach the seafloor in the centre and the eastern Grenada Basin. Moreover, the western extent of the PFS area is limited by our investigation area but we suspect that most of the Aves Ridge and even its western flank is concerned.

The horizon depths, seabed and thickness morphologies are very different in the NGB and the SGB, which is consistent with their respective and very different geodynamic histories. The NGB underwent repeated episodes of uplift (including aerial exposures at some places) and subsidence although the SGB is on a continuous state of subsidence since Eocene. They are both subsiding since middle Pliocene but at different rates.

Surprisingly, different orientations of polygons have been evidenced in both domains. The polygons are very regular in the north, whereas they are more elongated in the south. The centre-to-centre method applied to seabed polygons provided major orientations of extension needed for polygons to initiate. In the NGB minor axes are oriented N40°, indicating a general NE-SW extension of the upper part of the sedimentary cover. This could indicate that the backarc basin is currently submitted to similar processes than the forearc. In the SGB, minor axes are progressively turning towards the south, pointing out the actual maximum subsidence point.

In both cases, these observations imply that the development of PFS at the seabed could thus be indicative of the present-day strain state within the upper sedimentary column. Further investigations, including extensive seismic acquisitions and sampling, must be considered in order to completely map the extent of both active and fossil/buried intervals in the entire Aves Ridge area as PFI could be a marker of both actual or passed strain states within affected intervals, which is of high interest for the characterization of tectonic contexts.

Aknowledments:

This work was supported by the INSU TelluS-SYSTER grant call 2017, the GAARAnti project (ANR-17-CE31-0009), the GARANTI Cruise (2017). We are indebted to Saba Bank Resources N.V. and managing director Clark Gomes-Casseres for the provision of seismic lines of the Saba Bank. We gratefully thank the captain and crew of R/V L'Atalante, as well as the technical staff of Genavir for having successfully completed the acquisition of seismic data and dredge samples during the GARANTI cruise (https://doi.org/10.17600/17001200). Multichannel seismic processing was performed with Geovation software of CGG and Seismic Unix. All geophysical data and GARANTI cruise data are available on demand at SISMER (www.ifremer.fr/sismer/).

Figure 1 :

 1 Figure 1: Regional map of the eastern Caribbean Sea displaying the extension of giant seabed polygons (green line) and underlying polygonal faults (blue line) developing in the north Grenada Basin (NGB) and the south Grenada Basin (SGB). The multibeam bathymetry data and corresponding MCS profiles (noted GA-) were acquired during the GARANTI cruise in 2017. Additional seismic profiles (BOL30, 124VB-11 and 124VB-12) were used to extend the interpretation in the south. The grab cores GR-07 and GR-08 were collected within the area of seabed polygons. The shaded relief map is extracted from GEBCO datasets.

  ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

Figure 2 :

 2 Figure 2: Top: Shaded relief map of seafloor along the seismic profile GA-35 showing 2 to 3 km wide polygons bounded by ~40 m deep furrows. Bottom: Seismic profile AB (GA-35 close-up view) displaying polygonal faults affecting a ~1000 ms TWT thick interval. The polygonal faults are organized in set of smallthrow normal faults distributed beneath seafloor furrows. The furrows are the result of stacked depressions down to ~300 ms TWT below seafloor.

Figure 3 :

 3 Figure 3: Very high-resolution seismic profile (CHIRP) at the identical horizontal scale than the corresponding seismic profile GA-35. The furrows are affecting the present-day seafloor and they are ~1500 m wide and ~40 m deep. The seabed furrows represent the top of an underlying polygonal fault system (PFS). The bottom of furrows is characterized by very high amplitude reflections compared to well-bedded horizons between furrows.

Figure 4 :

 4 Figure 4: Detailed view of seismic profile GA-35 displaying decreasing offsets along fault planes both upward and downward from. The maximum throw is located at the transition between Early and Middle Pliocene all

Figure 6 :

 6 Figure 6: Top and bottom: SEM photographs of samples GR-07 (blue) and GR-08 (green). Both samples are dominated by smectite and kaolinite but smectite represents more than 50% of the bulk volume. The calcite peaks are derived from pelagic foraminifers; Middle left: Representative chemical analyses of smectites and the corresponding calculated structural formulae. Total iron was arbitrarily considered as Fe 3+ ; Middle right: Al2O3-Fe2O3-MgO plots of the point chemical analyses of smectite particles from the two grab cores (GR-07 and GR-08) collected during the GARANTI cruise.

  The sub-zone C is characterized by polygon edges oriented N170-10°, N30-50° and N70-90° with lengths of 1070, 1274 and 1047 ±15 m respectively. This defines polygons with a major axis of ~2250 m in length on average and 40-60 m deep bounding seafloor furrows. The sub-zone D is characterized by polygon edges oriented N10-30°, N30-50° and N70-90° with lengths of 1245 m ±15 m for every direction. They define elongated polygons with a major axis of ~3100 m in length on average and 30 m deep bounding seafloor furrows. The sub-zone E displays polygon edges oriented N30-50°, N50-70° and N70-90° with lengths of 1653, 1935 and 1677 ±15 m respectively. They define very elongated polygons with a major axis of ~5000 m in length, clearly visible on the bathymetric map. The seafloor furrows are 20-30 m deep.

Figure 7 :

 7 Figure 7: Statistical analysis conducted on seafloor polygons bounded by furrows in sub-zones A to E.

  ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

Figure 8 :

 8 Figure 8: Top: Seismic profile GA-05 and its line-drawing displaying the PFI in the NGB. The polygonal faults do not develop where deep turbiditic channels are present. The PFI is getting thicker southward affecting almost all unit U2. Units U0 to U3 were correlated on GARANTI seismic data from previous studies and ODP/DSDP wells (Garrocq et al., 2021); Bottom: Seismic profile GA-29 and its line-drawing displaying the PFI in the SGB. In both the western part of the SGB and the Aves Ridge, the polygonal faults are reaching the seafloor. The base of the PFI varies from west to east generally affecting the top of U2. In the eastern part of the SGB the PFI is thinning eastward (both the top and the base of the PFI are crosscutting stratigraphic horizons) and abruptly ends 40-70 km off the volcanic arc.

  ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

  ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

Figure 9 :

 9 Figure 9: Left: isochron map of the base of U3 displaying a crescent shape following the LAA. There is a sharp change west off Dominica where U3 is strongly deepening of ~3 s TWT. From Dominica to the south of Grenada Island the base of U3 is slightly deepening to reach ~5,5 s TWT. U3 is very shallow on Saba Bank and on top of the Aves Ridge. Right: Isopach map of U3 showing a homogeneous thickness in the NGB. In the SGB, U3 progressively thickens southward, indicating a differential subsidence between north and south.

Figure 10 :

 10 Figure 10: Schematic sketch illustrating the initiation and propagation of PF starting at the seabed. The PFS started at the transition between Early and Middle Pliocene. They are then propagating both upward and downward during sedimentation and as long as the sediment composition remains clay-and smectite-rich and the extension is large enough to allow proper contraction of sediments.

  ): ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

Figure 11 :

 11 Figure 11: Model of PF occurrence in the Grenada Basin. Between north and south the change in depths,morphologies and polygonal faults orientations is sharp, suggesting that both basins are actually submitted to very different tectonic regimes. The best fit ellipses extracted from the centre-to-centre method allowed the identification of preferential contraction orientations in unit U3. It is assumed that the minor axis of the ellipse represents the main orientation of volumetric contraction of sediments and it can be related to the minimum principal stress σ3. σ1 and σ2 orientations are deduced from the orientation of σ3. The extension is N40° and remains constant in the NGB, although it is changing (N120° to N144° to N163°) in the SGB, following the

  ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

  al.,

	ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

ESSOAr | https://doi.org/10.1002/essoar.10506638.1 | CC_BY_NC_4.0 | First posted online: Thu, 1 Apr 2021 00:40:54 | This content has not been peer reviewed.

. Clay-mineral signatures of fossil and active hydrothermal circulations in the geothermal system of the Lamentin Plain, Martinique. Journal of Volcanology and Geothermal Research, 124, 195-218. Mas, A., Guisseau D., Patrier P., Beaufort D. Genter A., Sanjuan, B. et al. (2006). Clay minerals related to the hydrothermal activity of the Bouillante geothermal field (Guadeloupe). Journal of Volcanology and Geothermal Research, 158, 380-400. Morley, C.K. & Binazirnejad, H. (2020). Investigating polygonal fault topological variability: Structural causes vs image resolution. Journal of Structural Geology, 130, 103930.


Bérengère Podvin

Stéphanie Pellerin

Yann Fraigneau

Guillaume Bonnavion

Olivier Cadot

Low-order modelling of the wake dynamics of an Ahmed body

We investigate the large-scale signature of the random switches between two mirrored turbulent wake states of flat-backed bodies. A direct numerical simulation of the flow around an Ahmed body at a Reynolds number of 10000 is considered. Using Proper Orthogonal Decomposition, we identify the most energetic modes of the velocity field and build a low-dimensional model based on the first six fluctuating velocity modes capturing the characteristics of the flow dynamics during and in-between switches. In the absence of noise, the model produces random switches with characteristic time scales in agreement with the simulation and experiments. This chaotic model suggests that random switches are triggered by the increase of the vortex shedding activity. However, the addition of noise results in a better agreement in the temporal spectra of the coefficients between the model and the simulation.

Introduction

The flat-backed Ahmed geometry [START_REF] Ahmed | Some salient features of the time-averaged ground vehicle wake[END_REF]) is a useful model to study the complex afterbody aerodynamics and the related base suction of trucks, SUVs and other blunt base vehicles responsible of high drag [START_REF] Choi | Aerodynamics of heavy vehicles[END_REF]. The massive turbulent separation of the flat-backed Ahmed body is dominated by different motions associated with specific time scales [START_REF] Grandemange | Turbulent wake past a three-dimensional blunt body. part 1. global modes and bi-stability[END_REF][START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back ahmed body: flow bi-stability and spectral analysis[END_REF][START_REF] Haffner | Mechanics of bluff body drag reduction during transient near-wake reversals[END_REF]) -which will be expressed in the remainder of the paper in convective time units based on the fluid velocity and body height. First of all, the recirculation zone behind the body is generally instantaneously asymmetric and characterized by switches between the asymmetric quasi-stationary states. The duration of the switches is O(50) convective time units and the mean time between switches is O(1000) convective units. A second type of motion corresponds to vortex shedding in the wake, which can be split into lateral and vertical components and is characterized by nondimensional periods of O(5) convective units. A third type of motion can be associated with the pumping motion of the recirculation zone over time scales of O(10 -20) units. These results have been confirmed using different techniques to analyse the flow such as direct Fourier analysis on the velocity field or using proper orthogonal decomposition (POD) by identifying the combinations of the POD modes associated with these global dynamics. [START_REF] Pavia | Salient three-dimensional features of the turbulent wake of a simplified square-back vehicle[END_REF] made use of the filtering properties of the POD analysis to characterize experimentally the three-dimensional topology of the recirculation and succeed to observe its evolution during the switches. These results were confirmed from the numerical simulation of [START_REF] Fan | Experimental and numerical analysis of the bi-stable turbulent wake of a rectangular flat-backed bluff body[END_REF]. These three-dimensional characterizations completed the previous experimental two-dimensional wake measurements of [START_REF] Pavia | Evolution of the bi-stable wake of a square-back automotive shape[END_REF]; [START_REF] Perry | Influence of short rear end tapers on the wake of a simplified square-back vehicle: wake topology and rear drag[END_REF]. [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF] recently took advantage of the three-dimensional POD to correlate the different known global dynamics to the base suction (or base drag). They found that the larger the quasi-steady deviation of the wake, the larger the base drag, while large vortex shedding intensity was correlated to lower quasi-steady deviation and then to lower drag. Although their simulation reported the permament quasi-steady wake deviation (the reflectional symmetry-breaking state of [START_REF] Grandemange | Turbulent wake past a three-dimensional blunt body. part 1. global modes and bi-stability[END_REF]), it was not long enough to observe wake switching towards the opposite deviation. Recently, the experiment of [START_REF] Haffner | Mechanics of bluff body drag reduction during transient near-wake reversals[END_REF] has carefully examined wake switching in relation to base drag and concluded to a base drag reduction of 8% during the switch, a similar figure found by [START_REF] Grandemange | Turbulent wake past a three-dimensional blunt body. Part 2. Experimental sensitivity analysis[END_REF] using a passive control cylinder to symmetrize the wake or previous measurements of base drag due to [START_REF] Evrard | Fluid force and symmetry breaking modes of a 3d bluff body with a base cavity[END_REF] during a switch.

The first numerical wake switching was observed by Dalla [START_REF] Dalla Longa | Simulations of the bi-modal wake past three-dimensional blunt bluff bodies[END_REF] and found to be associated with a vortex loop shedding, also reported by [START_REF] Fan | Experimental and numerical analysis of the bi-stable turbulent wake of a rectangular flat-backed bluff body[END_REF]. In an attempt to understand the triggering mechanism for the switch, [START_REF] Hesse | Simulation of wake bimodality behind squareback bluffbodies using les[END_REF] uses different turbulent modelling and pointed out the role of the separation at the front. Whatever the mechanism is, the very long waiting time between the switches and the associated randomness are consistent with rare events of Poisson statistics [START_REF] Grandemange | Turbulent wake past a three-dimensional blunt body. part 1. global modes and bi-stability[END_REF][START_REF] Volpe | Experimental characterization of the unsteady natural wake of the full-scale square back ahmed body: flow bi-stability and spectral analysis[END_REF][START_REF] Cadot | On threedimensional bluff body wake symmetry breaking with free-stream turbulence and residual asymmetry[END_REF] and can be successfully modelled using a Langevin equation as first introduced by [START_REF] Rigas | Low-dimensional dynamics of a turbulent axisymmetric wake[END_REF][START_REF] Rigas | Diffusive dynamics and stochastic models of turbulent axisymmetric wakes[END_REF]. The modelling has been adapted to the Ahmed body [START_REF] Brackston | Stochastic modelling and feedback control of bistability in a turbulent bluff bodywake[END_REF][START_REF] Evrard | Fluid force and symmetry breaking modes of a 3d bluff body with a base cavity[END_REF][START_REF] Barros | Forcing symmetry exchanges and flow reversals in turbulent wakes[END_REF] to simulate the bistable dynamics. For these so-called stochastic models, an additional noise is applied on a nonlinear dynamical system and the probability to change equilibrium states is proportional to exp(-∆V σ ). The noise intensity σ and the potential barrier between the two equilibrium state ∆V , are deduced from the experimental signals using different identification methods [START_REF] Rigas | Low-dimensional dynamics of a turbulent axisymmetric wake[END_REF][START_REF] Boujo | Robust identification of harmonic oscillator parameters using the adjoint fokker-planck equation[END_REF]. Recently, an alternative low-dimensional modelling based on POD was obtained by [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF] by projecting the Navier Stokes equations on the most energetic POD modes. However, their POD basis did not contain switching events, as no switching occurred during the simulation. Adding noise to the reduced order model was necessary to produce the wake switching similarly to the stochastic models.

The origin of the noise responsible for these switchings still remains an open question. It has been addressed experimentally in [START_REF] Cadot | On threedimensional bluff body wake symmetry breaking with free-stream turbulence and residual asymmetry[END_REF] who attributed the main role to the fluctuation in the wake and not to the small-scale turbulent intensity of the free stream flow. Together with the numerical simulations observation [START_REF] Dalla Longa | Simulations of the bi-modal wake past three-dimensional blunt bluff bodies[END_REF][START_REF] Fan | Experimental and numerical analysis of the bi-stable turbulent wake of a rectangular flat-backed bluff body[END_REF], there are strong evidences that events such as vortex shedding may trigger the switch and one may wonder if a chaotic process involving the largest scales of the flow could generate the observed bistable dynamics. The flow behavior would be described by a nonlinear system as for stochastic models but it would not require additional noise to produce the bistable dynamics. This idea was explored by [START_REF] Varon | Chaotic dynamics of large-scale structures in a turbulent wake[END_REF] who showed that the low-frequency dynamics of the switch appeared to exhibit weakly chaotic behavior.

The aim of the present paper is to pursue the modelling based on the POD of [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF] but on a longer simulation that contains wake switchings. It will be shown that the POD basis is completed by a new POD mode associated with a switching event. The modelling leads to spontaneous and random switching through a chaotic dynamics implying directly the periodic vortex shedding. The article is organised as follows: section §2 contains a description of the numerical simulation along with a review of POD. The velocity POD modes are presented in section §3, then the model is elaborated and discussed in section §4 and the paper is concluded in section §5.

Methodology

The characteristics of the simulation are the same as in [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF]. The dimensions of the squareback Ahmed body are L = 1.124m, H = 0.297m, W = 0.350m. The ground height is taken equal to 0.334H in the simulation. The code SUNFLUIDh, which is a finite volume solver, is used to solve the incompressible Navier-Stokes equations. The Reynolds number based on the fluid viscosity ν, incoming velocity U and Ahmed body height H is 10 4 . The temporal discretization is based on a secondorder backward Euler scheme, with implicit treatment of the diffusion terms and explicit representation of the convective terms (Adams-Bashforth scheme). We use (512 × 256 × 256) grid points in, respectively, the longitudinal direction x, the spanwise direction y, and the vertical direction z.

The mesh was refined near the body surface and the wall in order to capture the dissipative scales (following [START_REF] Howard | Large eddy simulation of an Ahmed reference model[END_REF]; [START_REF] Fares | Unsteady flow simulation of the ahmed reference body using a lattice boltzmann approach[END_REF]; [START_REF] Minguez | High-order large-eddy simulation of flow over the "Ahmed body" car model[END_REF]). In the boundary layers, the cell sizes in the wall-normal and transverse directions ranged from η bl to 8η bl , where the viscous boundary layer thickness η bl ≈ 6.71Re -0.9 is the smallest wall scale in the simulation. The cell size in the downstream direction ranged from 5η bl and 15η bl . In the wake region, within a downstream distance of 2H, cell sizes varied from 2η w to 16η w in the transverse directions and from 10η w to 20η w along the downstream direction, where η w ≈ 1.2Re -0.75 is the Kolmogorov length scale. The simulation time considered in the paper was about 500 convective time units based on the velocity U and height H. All reference units will be based on these two quantities. The time step was set to ∆t = 5 10 -4 and the CFL number never exceeded 0.4. During the period considered two switches in the deviation could be observed. Figure 1 represents the mean velocity field in a mid-height (a) and a mid-span plane (b). In all figures the origin of the axes is taken at the top and foremost position of the Ahmed body in the vertical symmetry plane. The longitudinal axis is oriented downstream, the vertical axis pointing in the upward direction and the third component is chosen accordingly to obtain a right-handed coordinate system. Two switches were observed during the total length of the simulation. We note that in experiments [START_REF] Evrard | Fluid force and symmetry breaking modes of a 3d bluff body with a base cavity[END_REF] or in simulations [START_REF] Hesse | Simulation of wake bimodality behind squareback bluffbodies using les[END_REF] it was not unusual to observe inter-switch times of only a few hundred convective time units, which is consistent with the fact that the advent of switches is governed by a Poisson distribution of mean 1000 time units [START_REF] Grandemange | Turbulent wake past a three-dimensional blunt body. part 1. global modes and bi-stability[END_REF].

The main tool of analysis used in this paper is Proper Orthogonal Decomposition (POD) [START_REF] Lumley | The structure of inhomogeneous turbulent flows[END_REF]), which we apply to the velocity field u(x, t) defined on a spatial domain D. The field can be expressed as a superposition of spatial modes

u(x, t) = k 0 a v k (t)φ k (x) (2.1)
where the spatial modes φ k are orthogonal (and can be made orthonormal), i.e D φ k (x).φ m (x)dx = δ km , while the amplitudes a k are uncorrelated. The modes can be ordered by decreasing energy λ 1 λ 2 . . . λ k =< ã2 k >, where < . > represents a time average. Since the full velocity field is considered and the database is symmetrized, the mode 0 corresponds to the symmetric part of the time-averaged field [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF]), which we checked was essentially equal to the time-averaged field (since the coefficient a 0 is constant within less than 0.3%). The amplitudes a v k can be obtained from the knowledge of the spatial modes by projection of the vector field u onto the spatial modes

a v k (t) = Ω u(x, t).φ k (x)dx. (2.2)
One can obtain the modes from a set of N snapshots by solving Cij .A jk = λ k A ik , where A jk = a k (t j ) and C is the temporal autocorrelation matrix: Cij = 1 N D u(x, t i ).u(x, t j )dx (method of snapshots). When appropriate we will consider normalized amplitudes defined as

a k = a v k / √ λ k .
The decomposition was applied to the velocity field, using the symmetrization procedure described in [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF] in order to guarantee all POD modes to have the same reflectional symmetry y → -y as the geometry. Consequently the POD modal basis is made to respect the symmetry of the forcing geometry. Results shown in this paper were based on a set of 1200 snapshots, 600 of which are extracted from the simulation, and 600 are obtained through the reflection symmetry. The original 600 snapshots span 300 convective time units, 200 of which correspond to the duration of the first switch. Comparison with another set of snapshots based on 2 × 200 samples acquired spanning the duration of the second switch did not show any significant differences in the shape of the modes.

POD analysis of the velocity field

The POD modes of the full velocity field are shown in figure 2(a,b) as the component φ kx of the modes k. They can be compared with those obtained using a database that does not contain any switching, described in [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF]. The main difference in the decompositions is the emergence of a new mode (mode 2 in figure 2a,b), that will be called the switch mode (please note that modes are indexed from 0 in the present paper and 1 in [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF]). The new second mode creates a strong dissymmetry between the upper and the lower part of the near wake. The mode is symmetric and is essentially streamwise invariant in the far wake. The temporal spectra of the amplitudes are also represented for each mode in figure 2(c). Modes 1 and 2 are characterized by low frequencies. We can notice a slight local increase in the spectrum around the nondimensional frequency (or Strouhal number) of 0.2 -we will come back to this point in the next section.

The next four modes (modes 3 to 6) correspond to Von Kármán shedding modes in the horizontal and in the vertical direction. Modes 3 and 4 are characterized by a nondimensional frequency of 0.19 and modes 5 and 6 are characterized by a frequency of 0.23. These four modes also display a slight peak at 0.8, which can be seen to match the frequency of the Kelvin-Helmholtz vortices observed along the body sides and originating from the separations at the nose in the modes 3,4,5,6.

It is possible to describe the large-scale dynamics with a relatively low number of modes as evidenced by figure 3 which shows three instantaneous fields at the beginning (a), in the middle (b) and at the end (c) of the first switch. Each time stamp of the switch is defined as a v 1 maximum, zero and minimum. The times are materialized with red vertical lines in the shaded area of figure 4. The full fields (left side of figure 3) can be compared with their reconstructions (right side of figure 3) based on the mean (figure 1) and the 6 most energetic POD modes (figure 2a,b). The time evolution of the first two fluctuating amplitudes a v 1 and a v 2 are represented in figure 4. We can see that the coefficient of the switch mode a v 2 is maximum during the switch. In the next section, we examine if the large-scale signature of the switch can be reproduced with a low-dimensional model.

Low-dimensional model

Derivation

We give a brief review of the derivation process (see also [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF]; [START_REF] Soucasse | Reduced-order modelling of radiative transfer effects on rayleigh-bénard convection in a cubic cell[END_REF]). The Navier-Stokes equations are projected onto the basis of the spatial velocity modes φ k for a given truncation k ∈ [1, . . . , N T ] and a model, or system of N T ODEs is obtained. The model is of the form The linear term L km is due to the viscous dissipation L km = ν∆φ m .φ k dx and is found to be essentially diagonal. The quadratic terms Q kmp can be written in symmetric form as

ȧv k = L km a v m + Q kmp a v m a v p + T k . (4.1)
Q kmp = 1 2 (2 -δ mp ) (φ p .∇φ m ) + φ m .∇φ p ).φ k dx. (4.2)
They represent the nonlinear interaction of the resolved modes, including the mean field corresponding to mode 0. T k is a closure term representing the contribution of the unresolved stresses (associated with the modes excluded from the truncation) to the evolution of the amplitude a k . As has been shown in [START_REF] Podvin | Precursor for wind reversal in a square rayleigh-bénard cell[END_REF], the closure term can be modelled as

T k = -α k N T p=1 (λ p + |a v p | 2 )a v k + k
, where λ p is the energy of mode p, and k is a noise-like perturbation. The value of α k is determined by requiring that for some equilibrium state a v,eq p inferred from the DNS, the total contribution

A k = L kk a v k + Q kk0 a v k + T k cancels when a v k = a v,eq
k for k = 1, . . . , 6, and k = 0, which yields:

α k = L kk + Q kk0 N T p=1 (λ p + |a v,eq p | 2 ) (4.3) which yields A k = α k a v k N T p=1 (|a v,eq p | 2 -|a v p | 2 ). (4.4)
Since equation (4.4) is derived from relatively crude modelling assumptions, some adjustment in the definition of A k may be needed in the model.

Seven-mode truncation

In what follows we consider a seven-mode truncation for the full field (N T = 6). Since the amplitude of the mean field is constant, i.e. a 0 = 1, the model is six-dimensional. The model includes the deviation mode, the switch mode and the Von Kármán shedding modes. We note that the model is different from [START_REF] Podvin | Proper orthogonal decomposition analysis and modelling of the wake deviation behind a squareback ahmed body[END_REF]'s which did not include the switch mode a 2 . If only the deviation mode is retained in the model, the system is formally analogous to [START_REF] Rigas | Diffusive dynamics and stochastic models of turbulent axisymmetric wakes[END_REF]'s model. The quadratic interactions coefficients were thresholded: values smaller than 5×10 -3 were not included in the model. It remains to determine the small-scale modelling parameters α k and k . The noise level rms σ k =< 2 k > 1/2 was chosen to be equal to a constant value σ for all modes and was estimated to be on the order of σ ∼ α k N T p 1 λ p ∼ 0.1. In order to determine α k , we use equation ( 4.3). The equilibrium state was defined as a eq 1 = ±1, a eq 2 = 0 (no switch mode), |a eq 2i-1 | 2 + |a eq 2i | 2 = 2 for i = 2, 3 (constant vortex shedding intensity). Given the eigenvalues λ 2 = 0.057, λ 3 = 0.015 and λ i ∼ 0.01 for i = 3, . . . , 6, equation (4.3) yields values of α k in the range [0.85, 1.25]. Since the small scales modelling is relatively crude, we used a single value α k = α = 1 for all modes. The system can then be written in normalized form as: ȧ0 = 0 (4.5) ȧ1 = A 1 + 0.011a 1 a 2 + 0.019a 0 a 3 + 0.023a 0 a 4 + 0.045a 3 a 5 + 0.005a 1 a 6 + 1 (4.6) ȧ2 = A 2 + 0.12a 0 a 5 + 0.11a 0 a 6 + 0.007a 2 a 3 -0.018a 2 1 + 0.018a 2 0 + 2 (4.7) ȧ3 = A 3 + 1.08a 0 a 4 -0.02a 0 a 1 + 0.037a 1 a 6 -0.025a 2 a 4 + 3 (4.8) ȧ4 = A 4 -1.08a 0 a 3 -0.16a 0 a 1 -0.04a 1 a 5 + 0.025a 2 a 3 + 4 (4.9)

ȧ5 = A 5 + 1.18a 0 a 6 -0.13a 0 a 2 + 0.035a 2 a 3 -0.013a 1 a 3 + 0.029a 1 a 4 -0.013a 2 2 + 5 (4.10) ȧ6 = A 6 -1.18a 0 a 5 -0.26a 0 a 2 -0.026a 2 a 5 -0.02a 1 a 3 + 6 , (4.11)
where the numerical quadratic coefficients Qkmp are equal to andA k is based on the following nondimensional adaptation of (4.4) to account for the energy transfer to the unresolved scales:

√ λmλp λ k Q kmp ,
A k = α k a k N T p=1 (β k |a v,eq p | 2 -|a v p | 2 ) = αa k 6 p=1 λ p (β k |a eq p | 2 -|a p | 2 ), (4.12)
where β k = 1 for k 2 and β k = 0.8 for k 3. The mode a 0 corresponding to the mean velocity is equal to 1. We point out that the evolution of a mode of a given symmetry is governed by interactions of modes obeying the same symmetry. The deviation mode is therefore controlled by its interaction with the switch mode, but also by cross-interactions between vortex shedding modes of opposite symmetries.

Results

The model was integrated over 20000 convective time units with and without noise. We first consider the integration without noise (σ = 0). The time series are represented in figure 5(a) and can be compared with the normalized amplitudes obtained from the simulation in figure 4. We can see that the model displays chaotic dynamics characterized by switches. The inter-switch time is random and on the order of a few thousand units, in agreement with [START_REF] Grandemange | Turbulent wake past a three-dimensional blunt body. part 1. global modes and bi-stability[END_REF]. The duration of the switch itself is O(100) convective units, in good agreement with the simulation. The intensity of the vortex shedding increases drastically during the switches (figure 5a, bottom rows). This suggests that the shedding modes symmetrize the wake. At the onset of the switch, a 2 increases then decreases sharply. It becomes strongly positive and reaches its maximum when a 1 goes through zero, in agreement with the DNS (figure 4). The model therefore does not only display spontaneous switches, but also captures the specific dynamics of the switch mode a 2 . We then consider the integration of the model with noise. Figure 5(b) shows the evolution of the amplitudes for a model with a noise rms of σ = 0.07. As expected, the inter-switch time interval decreased with an increasing noise level but remained on the order of 10 3 units for σ ∈ [0.05, 0.1]. As can be seen in figure 5(b), the noise may trigger a switch without any shedding intensity increase as shown for the first switch example (see caption). However, the second switch example is clearly associated with an a 5 increase corresponding to the shedding in the vertical direction.

The addition of noise leads to time evolutions which are closer to the DNS amplitudes, as can also be seen in figure 6, which compares the phase portraits in the (a 1 , a 2 ) space for the model without noise, the model with noise and the DNS. We note that when noise is added to the model, a 2 can take negative values in-between switches, which is more consistent with DNS observations (there is still a discrepancy, which is due to the choice of the particular equilibrium a eq 2 = 0). This is confirmed by comparison of the spectra shown figure 7. The characteristic frequencies associated with vortex shedding are clearly identified in the model without noise, while an excellent agreement in both the dominant frequencies and in the general shape and levels of the spectra is observed between the model with noise and the DNS. The model therefore captures the main time scales of the dynamics, including those of the switch.

Further insight into the physics of the model can be obtained by artificially manipulating the energy levels of the modes. The results of these numerical experiments are presented in figure 8. Excluding the switch mode from the model (see figure 8a) i.e. setting a 2 = 0 suppresses switches in the absence of noise -the system remains near one of the equilibrium states -and strongly limits them in the presence of noise which was obtained with the same noise perturbation as in figure 5(b). This is confirmed by linear stability of the quasi-steady state, according to which the equilibrium is stable if the switch mode is omitted from the model. The influence of the vortex shedding intensity can be examined by modifying the eigenvalues λ i → sλ i for i = 3, . . . , 6. This leads to modifications in the quadratic terms Qkmp as well as in A k , with a new equilibrium state |a eq 2i-1 | 2 + |a eq 2i | 2 = 2s for i = 2, 3 and a new value of α in order to satisfy equations (4.3,4.4). Integration of the model without noise is presented in figure 8(b) for s = 1.5 and s = 0.5. Spontaneous switching is more frequent when the energy is increased by 50% (s = 1.5), as can be seen from comparison with figure 5, and is suppressed when the vortex shedding energy is reduced to 50% (s = 0.5) of its expected value. This suggests that both the switch and vortex shedding modes are key players in the switching process. We speculate that the vortex shedding modes could be considered as a periodic forcing that introduces chaotic behavior at longer time scales in the 2-D system constituted by the deviation and the switch modes, in a manner that could be reminiscent of forced nonlinear oscillators such as the Duffing or the Van der Pol equation [START_REF] Guckenheimer | Nonlinear oscillations, dynamical systems and bifurcations of vector fields[END_REF].

Conclusion

We have investigated switches of the wake deviation in the flow around an Ahmed body using POD analysis of the velocity field. Along with the deviation and vortex shedding modes that were previously detected in the simulation, a new mode, called the switch mode, was identified. The effect of the mode during the switch is to speed up fluid on the upper part of the wake, and to slow it down on its lower part. A sixdimensional model consisting of the deviation, switch and four modes associated with vortex shedding was educed from the Navier-Stokes equations. In the absence of noise, the model displays chaotic dynamics, with switches occurring intermittently on a time scale of O(10 3 ) convective units and lasting for about 100 units. The model captures the strong variations of the switch mode, which reaches its maximum when the deviation amplitude is zero, and predicts a sharp increase of the vortex shedding intensity at the onset of the switch. The agreement with the simulation is further strengthened when noise is added to the model, as it results in a very good prediction of the temporal spectra of the large scales. Numerical experiments highlight the importance of both switch and vortex shedding modes in the switching process.

Figure 1 .

 1 Figure 1. Mean velocity (a) mid-height horizontal plane; (b) mid-span vertical plane -The black lines correspond to the isovalue 0.

Figure 2 .

 2 Figure 2. Cross-sections of POD velocity mode streamwise component:(a) on mid-height plane; (b) on mid-span plan for the symmetric modes k = 2, 3, 4 and at the position y = 0.4H for the antisymmetric modes k = 1, 5, 6. The black line corresponds to the contour Ux = 0; (c) Amplitude temporal spectra -the red dashed lines correspond to the frequencies f = 0.2 and f = 0.8.

Figure 3 .

 3 Figure 3. Velocity in a horizontal mid-plane at (a) the beginning, (b) during when a v 1 = 0, and (c) at the end of the switch; left) instantaneous field; right) reconstructed field using the mean and the 6 first fluctuating modes.

Figure 4 .

 4 Figure 4. Amplitudes of the POD velocity modes (a) a v 1 ; (b) a v 2 . The vertical lines in the shaded area corresponding to the switch represent the times of figure 3.

Figure 5 .

 5 Figure 5. Amplitudes of POD modes 1, 2, 3 and 5 integrated in the model (a) without and (b) with noise. Vertical dashed lines locate time at which a1 = 0 for two switching examples.

Figure 6 .

 6 Figure 6. Phase portraits in the (a1, a2) space: (a) in model without noise; (b) in model with noise level σ = 0.07; (c) in DNS.

Figure 7 .

 7 Figure 7. Comparison of the spectra of the velocity mode amplitudes a k in the DNS and in the model without and with a noise level σ.

Figure 8 .

 8 Figure8. Amplitudes of POD modes 1 and 3 in modified models: (a) models with no switch mode a2 = 0 with (solid line) and without noise (dotted line),(b) models with no noise and modified vortex shedding energy sλi, i = 3, . . . 6 (see text) for s = 1.5 (solid line) and s = 0.5 (dotted line).