Alexis Bougouin
email: alexis.bougouin@univ-lehavre.fr

Olivier Roche

Raphaël Paris

Herbert E Huppert

Experimental insights on the propagation of fine-grained geophysical flows entering water

Keywords: laboratory experiments, granular media, fluidization process, geophysical flows, gravity currents, tsunamis

Gas-fluidized granular flows down an inclined plane entering water are investigated experimentally and compared to those propagating in air

• The presence of the water body promotes the generation of a granular jet above the free surface and a particle-driven gravity current underwater

• The granular jet and the current are well described by ballistic motion theory and the physics of gravity currents, respectively

Introduction

Geophysical granular flows driven by gravity occur frequently on the Earth's surface, as a result of climatic, tectonic, or volcanic events [START_REF] Delannay | Granular and particle-laden flows: from laboratory experiments to field observations[END_REF]. Granular flows contribute significantly to the global sediment cycle and the shape of landscapes.

When they occur near the sea, a lake or a river, these solid-fluid mixtures may enter water, generate tsunamis, and propagate underwater [START_REF] Løvholt | On the characteristics of landslide tsunamis[END_REF]. The subaqueous flows are known to be more highly mobile compared to subaerial flows and they may cause severe damage to submarine facilities [START_REF] De Blasio | Understanding the high mobility of subaqueous debris flows[END_REF]. To predict their potential impact offshore, it is essential to understand the flow dynamics of subaerially-initiated granular flows after impacting water.

Many field observations reported the situation of submarine landslides evolving to turbidity currents (e.g., [START_REF] Heezen | Turbidity currents and submarine slumps, and the 1929 Grand Banks earthquake[END_REF][START_REF] Heezen | Orleansville earthquake and turbidity currents[END_REF][START_REF] Mulder | Numerical modelling of a mid-sized gravity flow: the 1979 Nice turbidity current (dynamics, processes, sediment budget and seafloor impact)[END_REF][START_REF] Talling | Onset of submarine debris flow deposition far from original giant landslide[END_REF][START_REF] Hsu | Turbidity currents, submarine landslides and the 2006 Pingtung earthquake off SW Taiwan[END_REF]. There are different ways to define these currents, but they broadly consist of very dilute particle-laden flows in which grains are mainly suspended by the turbulent water [START_REF] Kneller | The structure and fluid mechanics of turbidity currents: a review of some recent studies and their geological implications[END_REF]. In some cases, the dilute clouds can overlay dense near-bed layers, as recently shown by field measurements [START_REF] Paull | Powerful turbidity currents driven by dense basal layers[END_REF][START_REF] Simmons | Novel acoustic method provides first detailed measurements of sediment concentration structure within submarine turbidity currents[END_REF][START_REF] Wang | Direct evidence of a high-concentration basal layer in a submarine turbidity current[END_REF]. It is more difficult to find such reports about the transformation of a subaerial granular mass after entering water, because interest has mainly focused on the wave generation (e.g., [START_REF] Weiss | Hybrid modeling of the megatsunami runup in Lituya Bay after half a century[END_REF]. However, it is expected that granular flows composed of fine materials (e.g., fine-grained landslides, ash-rich pyroclastic flows) are particularly predisposed to generate subaqueous turbidity currents when they enter water because particles have low settling velocity and they can be easily suspended by the turbulent water. To the best of our knowledge, such an event has never been observed and only field works on volcaniclastic deposits have discussed turbidity currents formed by pyroclastic flows entering water. During the 12-13 July 2003 paroxysmal dome collapse of Soufrière Hills volcano, more than 200×10 6 m 3 of hot pyroclastic material entered the ocean. Phreatic explosions were observed and a cloud surge, decoupled from the main flow, travelled 2 to 3 km over the water surface [START_REF] Edmonds | Inland-directed base surge generated by the explosive interaction of pyroclastic flows and sea water at Soufrière Hills volcano[END_REF]. This phreatic activity promoted the ingestion of sea water by the pyroclastic flow that evolved as a cool and dense water-saturated granular flow underwater [START_REF] Trofimovs | Anatomy of a submarine pyroclastic flow and associated turbidity current: July 2003 dome collapse, Soufrière Hills volcano, Montserrat, West Indies[END_REF]. Additionally, a grain-size sorting process occurred within the subaqueous flow, which led to a fine-particle overload in the upper part and the formation of distal turbidity currents that propagated for more than 30 km on gentle slopes of the sea floor [START_REF] Trofimovs | Submarine pyroclastic deposits formed at the Soufrière Hills volcano, Montserrat (19952003): What happens when pyroclastic flows enter the ocean[END_REF][START_REF] Trofimovs | Anatomy of a submarine pyroclastic flow and associated turbidity current: July 2003 dome collapse, Soufrière Hills volcano, Montserrat, West Indies[END_REF]. Similar transformation of a subaerially-initiated pyroclastic flow into a subaqueous turbidity current was proposed by [START_REF] Whitham | The behaviour of subaerially produced pyroclastic flows in a subaqueous environment: evidence from the Roseau eruption, Dominica, West Indies[END_REF] for explaining Roseau Ash submarine deposits (Dominica Island, Lesser Antilles). The volcaniclastic turbidite deposit was found up to 250 km from the source, at 3000 m water depth. [START_REF] Cas | Subaqueous pyroclastic flows and ignimbrites: an assessment[END_REF] reported the possible scenarios when hot pyroclastic flows interact with the sea, and most of them highlight the formation of turbidity currents.

Most experimental studies on granular masses entering water considered dry, heavy, and coarse grains leading to dense granular flows depositing rapidly underwater (e.g., [START_REF] Fritz | Landslide generated impulse waves. 2. Hydrodynamic impact craters[END_REF][START_REF] Viroulet | Tsunami generated by a granular collapse down a rough inclined plane[END_REF][START_REF] Miller | Tsunamis generated by long and thin granular landslides in a large flume[END_REF]. The generation of particledriven gravity currents by initially-subaerial fine-grained flows has therefore been poorly investigated. [START_REF] Mcleod | Behaviour of particle-laden flows into the ocean: experimental simulation and geological implications[END_REF] first considered different grain-fluid mixtures, initially maintained above the free-surface and released over an inclined plane, which entered fresh water. They concluded that both negatively and positively buoyant gravity currents can be obtained separately or together, by varying the grain and fluid densities of the mixture. These different flow dynamics have also been reported for homogeneous gravity currents released into a two-fluid layer [START_REF] Monaghan | Gravity currents descending a ramp in a stratified tank[END_REF][START_REF] Legros | On the emplacement of ignimbrite in shallowmarine environments[END_REF][START_REF] Monaghan | Gravity current interaction with interfaces[END_REF][START_REF] Wells | The long-term circulation driven by density currents in a two-layer stratified basin[END_REF][START_REF] Cortés | Experimental observations of the splitting of a gravity current at a density step in a stratified water body[END_REF]. [START_REF] Freundt | Entrance of hot pyroclastic flows into the sea: Experimental observations[END_REF] focused on the thermal effects of volcanic ash heated up to 400 • C, entering water at ambient temperature. Steam explosions and major ash-cloud surges propagating above the water surface were obtained with increasing temperature. By contrast, the dynamics of particledriven gravity currents was mainly controlled by the mass flux of the granular flow and the water depth, while it was not significantly affected by the temperature. In fact, heat transfers mainly modified the early stages of the gravity currents, which propagated dominantly along the water surface at high temperature. Despite these studies, there is still a lack on the description of the granular dynamics after such flows enter water, which is the focus of the present study.

Most works on the propagation of particle-driven gravity currents considered simple configurations using laboratory experiments, numerical simulations and theoretical descriptions. One of the most famous configurations corresponds to the propagation of a fixed volume of a heavy fluid released into a lighter fluid on a horizontal plane, the socalled lock-exchange problem (e.g., [START_REF] Huppert | The slumping of gravity currents[END_REF][START_REF] Huppert | The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface[END_REF][START_REF] Rottman | Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel[END_REF]. In this case, the motion of homogeneous gravity currents is driven by the density difference between the current and the surrounding fluid. Depending on the dominant forces, the flow dynamics of the current in the two-dimensional configuration is usually described in three distinct stages, well predicted by shallow layer/lubrication models: (i) a slumping regime dominated by inertia during which the flow-front velocity remains constant; (ii) an inertial self-similar regime characterized by a buoyancy/inertia balance for which the front position scales with time as x f ∝ t 2/3 ; and (iii) a viscous regime controlled by a balance between buoyancy and viscous dissipation in which the front evolves as x f ∝ t 1/5 [START_REF] Huppert | The slumping of gravity currents[END_REF][START_REF] Huppert | The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface[END_REF][START_REF] Rottman | Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel[END_REF]. This configuration has been extended to gravity currents propagating down an inclined plane, for which the gravity current is described by an acceleration phase followed by a deceleration phase (e.g., [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF][START_REF] Monaghan | Gravity currents descending a ramp in a stratified tank[END_REF][START_REF] Dai | Gravity currents down a slope in deceleration phase[END_REF][START_REF] Dai | Experiments on gravity currents propagating on different bottom slopes[END_REF][START_REF] Dai | Non-Boussinesq gravity currents propagating on different bottom slopes[END_REF]. When the current decelerates, the front position scales as x f ∝ t 2/3 or x f ∝ t 1/2 according to the preponderance of inertia or viscous effects, respectively [START_REF] Dai | Non-Boussinesq gravity currents propagating on different bottom slopes[END_REF]. The motion of these gravity currents is also well predicted by the thermal theory based on the mass and momentum conservation by considering the entrainment of the ambient fluid [START_REF] Beghin | Gravitational convection from instantaneous sources on inclined boundaries[END_REF]. Finally, for gravity currents down an inclined plane produced by a continuous inflow, the currents evolve with a constant front velocity depending mainly on the buoyancy flux, while the effect of the slope angle is unsignificant (e.g., [START_REF] Britter | The motion of the front of a gravity current travelling down an incline[END_REF][START_REF] Baines | Mixing in flows down gentle slopes into stratified environments[END_REF][START_REF] Baines | Mixing regimes for the flow of dense fluid down slopes into stratified environments[END_REF]. We recall that other simple configurations have also been considered, in which the above-mentioned stages were observed despite a modification of the scaling laws (e.g., [START_REF] Huppert | The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface[END_REF][START_REF] Zgheib | Dynamics of non-circular finite-release gravity currents[END_REF][START_REF] Zhu | Front dynamics of elliptical gravity currents on a uniform slope[END_REF]. The particle-driven gravity currents are fundamentally gravity currents whose density is greater than that of the surrounding fluid because of their suspended grains [START_REF] Meiburg | Turbidity currents and their deposits[END_REF][START_REF] Wells | Turbulence processes within turbidity currents[END_REF]. However, the flow dynamics of these currents are more complex than those of homogeneous gravity currents, i.e. the intrusion of a pure fluid into another fluid, because the particle concentration evolves with time and position due to the erosion/deposition of grains (e.g., [START_REF] Bonnecaze | Particle-driven gravity currents[END_REF][START_REF] Bonnecaze | Axisymmetric particle-driven gravity currents[END_REF]. In the present work, laboratory experiments on gas-fluidized granular flows entering a body of water are performed to understand better the dynamics of geophysical flows after they reach the sea, a lake or a river. More specifically, it focuses on granular flows composed of fine materials, which are predisposed to generate particle-driven gravity cur-rents underwater, as illustrated by the oblique view of Figure 1. The questions raised in the present contribution are the following. What is the effect of a water body on the propagation of subaerially-initiated fine-grained flows along an inclined plane? Does the water body play a dissipative or a driven role on the granular material? Can the general evolution of the granular material be described using simple theoretical models? In order to provide some answers, the paper is organized as follows. In Sec. 2, the experimental setup and materials used are presented. Sec. 3 highlights the influence of the presence of a water body on the granular dynamics through qualitative observations and a comparison between gas-fluidized granular flows propagating only in air and entering water. Finally, the evolution of the granular material, i.e. the initial granular jet above the water surface and the particle-driven gravity current underwater, is described using theoretical models (Secs. 4 and 5). Laboratory experiments are conducted in a horizontal transparent channel of rectangular cross-section with dimensions of 7 × 0.8 × 0.2 m 3 in the longitudinal, vertical and spanwise directions, respectively. A sketch of the experimental setup is given in Fig- ure 2. One side of the channel is equipped with a reservoir delimited by a sluice gate and a mobile vertical plate to vary the initial length L i of the granular column. In this study, the length and the height of the initial granular column are varied in the range L i = [9 : 34] ± 0.5 cm and H i = [10 : 42] ± 0.5 cm, respectively. The reservoir is connected to an inclined plane composed of a 1 m-subaerial ramp and an immersed ramp. Both the bottom of the reservoir and of the 1 m-subaerial ramp are equipped with a 12 µmporous plate to fluidize the granular flow by a vertical air flux before entering water (see black arrows, in Figure 2). Here, the granular material is fluidized with an air velocity of U f ∼ 4.7 ± 0.1 mm.s -1 , larger than the minimum fluidization velocity U mf ∼ 3.8 ± 0.1 mm.s -1 , which ensures that the granular flow is fully fluidized when it impacts water. The minimum fluidization velocity U mf is estimated prior the experiments as the minimum velocity for which the gas pore pressure counterbalances the pressure of a static granular column. The length of the immersed ramp varies as H o / sin θ, where the slope angle is set to θ = 5 • , 10 • , 15 • , 20 • , and 30 • and the water depth is varied up to H o = 38.5±0.1 cm. In the following, the zones containing the immersed ramp and the horizontal bottom are referred to as the near-and far-field regions, respectively. On the opposite side of the channel, a wave breaker is used to limit the reflected waves, which could affect the granular flow dynamics underwater. Indeed, the wave velocity is still much larger than that of the particle-driven gravity current.

Experimental setup

Apparatus

As discussed by [START_REF] Bougouin | Impact of fluidized granular flows into water: Implications for tsunamis generated by pyroclastic flows[END_REF], the initial conditions of the experiment control the flow parameters, namely the volume, the velocity, and the height of the granular flow. The volume per unit width is estimated as υ = (H i -h r )L i , where h r is the residual height of grains in the reservoir at the end of the experiment. The height profile of the granular flow is non-uniform along the inclined plane, and the front-height h f at the impact is therefore chosen as the flow height at 10 cm behind the front (more information is provided in the [START_REF] Bougouin | Impact of fluidized granular flows into water: Implications for tsunamis generated by pyroclastic flows[END_REF]'s Supporting Information). The constant front velocity u f of fluidized granular flows is estimated at the impact, from the slope of the linear trend between the front position and time. Finally, as both h f and u f vary together with H i , the flow rate per unit width q = h f u f is also defined. In this study, the control parameters are varied in the range υ = [1.5 : 13.2] ± 0.1 dm 2 (with 1 dm 2 = 10 -2 m 2 ), h f = [0.9 : 2.8] ± 0.4 cm, u f = [2 : 3.7] ± 0.1 m.s -1 , and q = [1.8 :

8.5] ± 1 dm 2 .s -1 , respectively.
At the initial time t = 0, the sluice gate is rapidly and entirely removed with a velocity of about 2 m.s -1 by releasing a suspended weight, which triggers the collapse of the granular column on the inclined plane. The flow dynamics of the granular material is recorded using a classical shadowgraph method with two Photron Fastcam 1024× 1024 pixel cameras and LED panels on the back side of the channel. This optical method allows vizualization of the interface between fluids due to different light absorptions. Here, the granular material, the water layer, and the ambient air are represented in black, gray and white, respectively (see Figure 3, for instance). The first camera records a large view of the experiment, while the second camera focuses on the flow dynamics of currents in the far-field region. The resolution obtained is 4 mm/pixel and 0.8 mm/pixel, respectively, and the acquisition rate is 250 Hz. Finally, the extraction process is performed using a Matlab routine based on a threshold method.

Materials

The granular material used corresponds to quasi-monodisperse spherical glass beads manufactured by Wheelabrator, with a diameter of d = 65 ± 10 µm and a density of ρ p = 2550±50 kg.m -3 , respectively. Before each experiment, the grains are weighted, poured into the reservoir, fluidized and then the height of the initial column is measured.

The particle concentration of the initial granular column is φ i = V grains /V column ∼ 0.56 ± 0.01, where V grains and V column denote the volume of grains and the total volume of the initial column, respectively. This corresponds to an initial bulk density of ρ =

φ i ρ p +(1-φ i )ρ a ∼ 1400 kg.m -3
, where ρ a ∼ 1.2 kg.m -3 is the density of ambient air.

Both the particle concentration and the density are expected to remain constant in the reservoir and during the flow propagation on the 1 m-subaerial ramp because of the sustained gas flow from the bottom. Some experiments of dense liquid flows entering water are also performed to compare the propagation of homogeneous gravity currents with that of particle-driven gravity currents. Sodium Chloride (NaCl) is therefore added to dyed water with a concentration of 325g/kg of water, leading to a dense saline liquid of ρ ∼ 1200 kg.m -3 measured by a DMA 35 Anton Paar electronic densimeter. In Sec. 5, we will show that the density of homogeneous gravity currents is roughly similar to that of particle-driven gravity currents, and their dynamics can therefore be compared together depending on the flow conditions at the impact. It should be remembered that no gravity currents would be generated by the entrance of a fluid into another fluid of the same density [START_REF] Bullard | Experimental analysis of tsunamis generated by the impact of landslides with high mobility[END_REF][START_REF] Clous | Génération de vagues par des coulées liquides : application aux écoulements pyroclastiques. 24ème Congrès français de mécanique[END_REF], while reversing buoyancy of currents would be observed with a lighter fluid into a heavy fluid [START_REF] Monaghan | Gravity currents descending a ramp in a stratified tank[END_REF][START_REF] Monaghan | Gravity current interaction with interfaces[END_REF]. For dense liquid flows, the initial parameters are H i = [16 : 40] cm, L i = 34 cm, H o = 26.4 cm and θ = 15 • , while the flow parameters at the impact are found in the range υ = [5.2 : 12.5] dm 2 , u f = [2.4 : 3.2] m.s -1 , h f = [0.9 : 2.5] cm, and q = [2.2 : 8.0] dm 2 .s -1 .

Scaling issues of geophysical granular flows

In this section, we propose to discuss the relevance and limitations of the experiments, which aim to help the understanding of fine-grained geophysical flows propagating down a mountainside, entering the sea, a lake or a river, and then propagating underwater far away.

The novelty of this study lies in the consideration of gas-fluidized granular flows, which ensures a more suitable modelling of geophysical flows taking their high mobility into account. Indeed, natural granular flows are known to exhibit energy dissipation lower than at the laboratory scale, for which several physical mechanisms have been proposed, i.e. basal lubrication (e.g., [START_REF] Bowden | The mechanism of sliding on ice and snow[END_REF][START_REF] Shreve | Leakage and fluidization in air-layer lubricated avalanches[END_REF][START_REF] Goren | Long runout landslides: the role of frictional heating and hydraulic diffusivity[END_REF], acoustic or mechanical fluidization (e.g., [START_REF] Melosh | Acoustic fluidization: A new geologic process?[END_REF][START_REF] Davies | Spreading of rock avalanche debris by mechanical fluidization[END_REF][START_REF] Collins | Acoustic fluidization and the extraordinary mobility of sturzstroms[END_REF], high gas pore pressure (e.g., [START_REF] Major | Debris-flow deposition: Effects of pore-fluid pressure and friction concentrated at flow margins[END_REF][START_REF] Sparks | Grain size variations in ignimbrites and implications for the transport of pyroclastic flows[END_REF], dynamic fragmentation (e.g., T. R. [START_REF] Davies | The role of rock fragmentation in the motion of large landslides[END_REF], and material entrainment (e.g., [START_REF] Hungr | Entrainment of debris in rock avalanches: an analysis of a long run-out mechanism[END_REF][START_REF] Sovilla | Field experiments and numerical modeling of mass entrainment in snow avalanches[END_REF], among others. At the laboratory scale, a granular flow fluidized at the source propagates faster and farther than a non-fluidized granular flow with the same initial conditions [START_REF] Roche | Experimental observations of water-like behavior of initially fluidized, dam break granular flows and their relevance for the propagation of ash-rich pyroclastic flows[END_REF]. In case of fluidization along the flow path and for gentle slopes, the fluidization process also prevents the deposition of grains along the bottom plane, and the entire volume of the granular flow thus enters water (Bougouin et al., 2019). This behavior is attributed to the high gas pore pressure generated by the fluidization process, which reduces the internal friction within granular flows and therefore promotes the high mobility. The dynamics of experimental and natural granular flows can be compared through the Froude number F r = u f /(gh f ) 1/2 , which is found in the range F r = [6 : 9] corresponding to the upper range of natural flows [START_REF] Delannay | Granular and particle-laden flows: from laboratory experiments to field observations[END_REF]. Moreover, the bulk flow to water density ratio is about ρ/ρ f ∼ 1.4 which is consistent with that of most natural flows [START_REF] Delannay | Granular and particle-laden flows: from laboratory experiments to field observations[END_REF], even if ash-rich pyroclastic flows and dry snow avalanches may be less dense than water. This experimental configuration seems therefore to be a relevant modelling for investigating most geophysical granular flows entering water, despite some limitations.

In the present configuration, the fluidization process is facilitated using fine, spherical, and monodisperse beads, which is a crude assumption about natural flows usually composed of different grain sizes, densities, and shapes. In this way, some physical mechanisms such as the grain-size sorting process in or out of the water, the reversing buoyancy and the coarse grain spreading underwater cannot be reproduced here. The grain size to flow length ratio of natural flows cannot also be maintained at the laboratory scale, because the grain size used in the experiments is usually similar to those found in the field. Moreover, natural pyroclastic flows can have high temperatures up to 500-600 • when they enter the sea. [START_REF] Freundt | Entrance of hot pyroclastic flows into the sea: Experimental observations[END_REF] concluded that the main consequence of high temperature was the generation of phreatic explosions and major ash-cloud surges propagating over the water surface, while particle-driven gravity currents were slightly affected close to the shoreline. However, an open question still remains about the formation of welded deposits, which could not be addressed here [START_REF] Sparks | The entrance of pyroclastic flows into the sea. II. Theoretical considerations on subaqueous emplacement and welding[END_REF][START_REF] Cas | Subaqueous pyroclastic flows and ignimbrites: an assessment[END_REF][START_REF] Kokelaar | Subaqueous explosive eruption and welding of pyroclastic deposits[END_REF]. In summary, the present setup ensures a suitable modelling of dense, cold or moderate-heated, fine particle-rich geophysical flows entering water. At the initial time t = 0, the initial column of grains is released, collapses and forms a gas-fluidized granular flow on the subaerial inclined plane until reaching water. The impact between the granular flow and the water body generates (i) an initial granular jet corresponding to grains ejected above the water surface, (ii) a leading and largest wave, and (iii) a turbulent mixing zone from which a particle-driven gravity current is spontaneously formed (Figure 3). These observations are direct consequences of the presence of a water body, which have already been observed for fine particle-fluid mixtures entering water [START_REF] Mcleod | Behaviour of particle-laden flows into the ocean: experimental simulation and geological implications[END_REF][START_REF] Freundt | Entrance of hot pyroclastic flows into the sea: Experimental observations[END_REF][START_REF] Allen | Characteristics of submarine pumice-rich density current deposits sourced from turbulent mixing of subaerial pyroclastic flows at the shoreline: field and experimental assessment[END_REF][START_REF] Bougouin | Impact of fluidized granular flows into water: Implications for tsunamis generated by pyroclastic flows[END_REF].

By contrast, both the granular jet and the gravity current have never been reported in the literature for coarse granular materials [i.e., d > O(10 -3 ) m] under similar flow conditions (e.g., Fritz et al., 2003a;[START_REF] Heller | Scale effects in subaerial landslide generated impulse waves[END_REF][START_REF] Viroulet | Tsunami generated by a granular collapse down a rough inclined plane[END_REF][START_REF] Zitti | Impulse waves generated by snow avalanches: Momentum and energy transfer to a water body[END_REF][START_REF] Miller | Tsunamis generated by long and thin granular landslides in a large flume[END_REF]. It suggests therefore that both the granular jet and the particle-driven gravity current are mainly controlled by the grain-fluid interaction. In fact, the flow dynamics of particle-driven gravity currents along the bottom plane looks more like that of homogeneous gravity currents penetrating a two-fluid layer [START_REF] Monaghan | Gravity currents descending a ramp in a stratified tank[END_REF][START_REF] Legros | On the emplacement of ignimbrite in shallowmarine environments[END_REF][START_REF] Monaghan | Gravity current interaction with interfaces[END_REF][START_REF] Wells | The long-term circulation driven by density currents in a two-layer stratified basin[END_REF][START_REF] Cortés | Experimental observations of the splitting of a gravity current at a density step in a stratified water body[END_REF].

In this configuration, a splitting of the homogeneous gravity current can even be obtained, which could be related to the formation of the granular jet above the water surface and of the particle-driven gravity current underwater. However, some caution has to be exercised here because the splitting of homogeneous gravity currents was attributed to lighter fluid incorporated into the current by turbulent entrainment [START_REF] Wells | The long-term circulation driven by density currents in a two-layer stratified basin[END_REF][START_REF] Cortés | Experimental observations of the splitting of a gravity current at a density step in a stratified water body[END_REF], which is not expected to occur for gas-fluidized granular flows propagating in air.

Comparison with gas-fluidized granular flows in air

In this section, we highlight the effect of a water body on the granular dynamics and the associated deposits by comparison of gas-fluidized granular flows propagating only in air and entering water, under the same initial conditions. (b) Front-velocity u f /(gHi cos θ) 1/2 as a function of the front position x f /x A , where x A is the streamwise coordinate of the shoreline. The gray area corresponds to u f /(gHi cos θ) 1/2 = [1.4 : 2]; the solid line is a slope -3/2; and the dotted-dashed lines delimit the subaerial, near-and farfield regions.

One of the relevant macroscopic analyses to compare the dynamics of granular flows is the tracking of the front position x f and the front-velocity u f with time and position (Figure 4). The presence of a water body significantly affects the temporal evolution of the flow front beyond the initial height H i , the latter being already reported for water dam-break flows (e.g., [START_REF] Ritter | Die fortpflanzung der wasserwellen[END_REF][START_REF] Laubert | Experiments to dambreak wave: Horizontal channel[END_REF][START_REF] Ancey | An exact solution for ideal dam-break floods on steep slopes[END_REF] and subaerial granular flows (e.g., [START_REF] Roche | Experimental observations of water-like behavior of initially fluidized, dam break granular flows and their relevance for the propagation of ash-rich pyroclastic flows[END_REF][START_REF] Mangeney | Erosion and mobility in granular collapse over sloping beds[END_REF][START_REF] Farin | Fundamental changes of granular flow dynamics, deposition, and erosion processes at high slope angles: Insights from laboratory[END_REF][START_REF] Bougouin | Impact of fluidized granular flows into water: Implications for tsunamis generated by pyroclastic flows[END_REF]. At early times, all curves collapse together corresponding to the propagation of granular flows down the subaerial inclined plane. During this stage, the flow front slightly accelerates along the plane, as predicted by the theory for a frictionless fluid flow down an inclined plane [START_REF] Ancey | An exact solution for ideal dam-break floods on steep slopes[END_REF]. We recall that the acceleration phase of channeled granular flows is expected to end when the gravity is balanced by sidewall friction [START_REF] Brodu | New patterns of highspeed granular flows[END_REF]. For the sake of simplicity, the averaged front velocity can also be defined in the range [START_REF] Dressler | Comparison of theories and experiments for the hydraulic dam-break wave[END_REF][START_REF] Jánosi | Turbulent drag reduction in dam-break flows[END_REF][START_REF] Leal | Dam-break wave-front velocity[END_REF][START_REF] Roche | Experimental observations of water-like behavior of initially fluidized, dam break granular flows and their relevance for the propagation of ash-rich pyroclastic flows[END_REF][START_REF] Bonometti | Wall effects in non-Boussinesq density currents[END_REF][START_REF] Bougouin | Collapse of a neutrally buoyant suspension column: from Newtonian to apparent non-Newtonian flow regimes[END_REF]. During the second phase, gas-fluidized granular flows either enter water and then propagate in the form of subaqueous particle-driven gravity currents, or they continue to flow down the subaerial inclined plane without the sustained gas flow from the bottom. The main consequence of the water body is to slow down the spreading of the granular material. A gross estimate of the normalized front velocity with the normalized front position gives u f /(gH i cos θ) 1/2 ∝ (x f /x A ) -3/2 [solid line, in Figure 4(b)]. However, the flow dynamics of the currents will be more fully investigated in Sec. 5. Finally, the gas-fluidized granular flows in air stop rapidly on the horizontal bottom, while the gravity currents continue slowly to propagate underwater [Farfield region, in Figure 4 The morphology of the granular flow can also be investigated here, because it is particularly modified by the presence of the water body (Figure 5). More specifically, the major effect of water is to generate thick and dilute particle-driven gravity currents underwater compared to dense and thin gas-fluidized granular flows in air. Moreover, the inertial dimensionless form used here allows the collapse of the height profile of gasfluidized granular flows obtained for different H i , which does not hold true for subaqueous particle-driven gravity currents.

u f /(gH i cos θ) 1/2 = [1.4 : 2] [gray area, in
Finally, at sufficiently long times (i.e., from a few seconds to tens of seconds), the granular material stops forming a final deposit along the bottom surface (Figure 6). Again, the morphology of the granular deposit strongly differs for gas-fluidized granular flows propagating in air and those entering water. The presence of the water body leads to a thick and voluminous granular mass along the inclined plane probably due to a dense granular flow, while a thin layer of grains, initially suspended in the gravity current, is deposited on the horizontal bottom. By contrast, the subaerial granular flows form only a massive deposit at the end of the inclined plane. The inset of Figure 6 shows that the presence of water reduces significantly the runout of granular flows, as already reported for coarse grains falling into water [START_REF] Mazzanti | The dynamics of coastal landslides: Insights from laboratory experiments and theoretical analyses[END_REF][START_REF] Viroulet | Tsunami generated by a granular collapse down a rough inclined plane[END_REF][START_REF] Miller | Tsunamis generated by long and thin granular landslides in a large flume[END_REF]. The runout is defined here as h(x -x B = R f ) = H i /10 to be free of the thin and very elongated granular layer deposited by the current. Overall, the presence of the water body therefore plays a dissipative role by slowing and reducing the spreading of the granular material, even if a low amount of grains are still transported by the turbulent fluid far away.

Discussion of results

Our laboratory experiments on fine-grained granular flows entering a water body have revealed the following results. Overall, the presence of water promoted the formation of a granular jet, corresponding to grains ejected above the water surface and then falling onto it, and a particle-driven gravity current underwater. Moreover, hydrodynamics forces mainly played a dissipative role by slowing and reducing the spreading of the granular mass underwater, even if a low amount of grains were still transported by the turbulent fluid as a gravity current far away. These observations highlight the complexity of granular flows impacting water, with in particular, a splitting of granular masses.

These results are now discussed to help the interpretation of some submarine deposits in the natural environment.

In the field, both the morphology and the internal structure of deposited material represent a suitable marker of the dynamics of granular flows. In the case of fine-grained geophysical flows entering water, it is expected from our experiments to obtain distinct submarine deposits, recording successively gravity currents, dense basal flows, and granular jets. The 1995-2010 eruption of Soufrière Hills volcano represents an invaluable case study of pyroclastic flows entering the sea with volcanic activity having been intensively

documented (e.g., [START_REF] Young | Overview of the eruption of Soufriere Hills volcano, Montserrat, 18 July 1995 to[END_REF][START_REF] Trofimovs | Submarine pyroclastic deposits formed at the Soufrière Hills volcano, Montserrat (19952003): What happens when pyroclastic flows enter the ocean[END_REF][START_REF] Trofimovs | Anatomy of a submarine pyroclastic flow and associated turbidity current: July 2003 dome collapse, Soufrière Hills volcano, Montserrat, West Indies[END_REF][START_REF] Friant | Submarine deposition of volcaniclastic material from the 1995-2005 eruptions of Soufrière Hills volcano, Montserrat[END_REF], 2010;[START_REF] Trofimovs | Submarine pyroclastic deposits formed during the 20th May 2006 dome collapse of the Soufrière Hills Volcano[END_REF]Wadge et al., 2014, among others). A large part of volcaniclastic materials was transported into the sea, and associated submarine deposits have been particularly well described through bathymetric survey data and core samples. Overall, massive coarse-grained deposits were observed close to the slope break, which was interpreted as the sign of dense granular flows propagating slowly underwater (Le [START_REF] Friant | Submarine deposition of volcaniclastic material from the 1995-2005 eruptions of Soufrière Hills volcano, Montserrat[END_REF][START_REF] Friant | Eruption of Soufrière Hills (1995-2009) from an offshore perspective: Insights from repeated swath bathymetry surveys[END_REF][START_REF] Trofimovs | Submarine pyroclastic deposits formed during the 20th May 2006 dome collapse of the Soufrière Hills Volcano[END_REF]. A grain-size sorting process also promoted the formation of dilute turbidity currents composed of fine materials, which deposited as thin and elongated masses far away from the shoreline [START_REF] Trofimovs | Submarine pyroclastic deposits formed at the Soufrière Hills volcano, Montserrat (19952003): What happens when pyroclastic flows enter the ocean[END_REF][START_REF] Trofimovs | Anatomy of a submarine pyroclastic flow and associated turbidity current: July 2003 dome collapse, Soufrière Hills volcano, Montserrat, West Indies[END_REF][START_REF] Trofimovs | Submarine pyroclastic deposits formed during the 20th May 2006 dome collapse of the Soufrière Hills Volcano[END_REF]. Our experimental observations of submarine gravity currents and deposit morphologies are consistent with those reported from the field. Additionally, they highlight that grain-size sorting processes are not needed to form both proximal massive deposits and distal turbidites. Finally, deposits of granular jets have never been mentioned in the literature, suggested that they were probably hardly recognized.

Frictionless ballistic motion theory

For the purpose of describing the trajectory of the granular jet crest, let us first consider a single grain ejected with an initial velocity u i and an initial angle α (with respect to the horizontal plane), i.e. (X c , Y c ) = (0, 0) and t -t A = 0 with t A the time at the impact [see Figure 7(a)]. To first order, the drag force of the ambient air on the grain motion can be neglected. Hence, the single grain is only subject to its own weight and momentum conservation reduces to (d 2 X c /dt 2 , d 2 Y c /dt 2 ) = (0, -g) in the hori-zontal and vertical directions, respectively. Then one can simply obtain the temporal evolution of the horizontal and vertical positions of the grain as

X c = u i cos α(t -t A ), (1)
Y c = - g 2 (t -t A ) 2 + u i sin α(t -t A ). (2)
Additionally, the spatial evolution can also be expressed by combining equations

(1) and ( 2), which leads to

Y c = - g 2u i 2 cos 2 α X c 2 + tan αX c . (3)
It can also be useful to predict the position (X m c , Y m c ) and the time T m at which the maximum height is reached as discussed later in Sec. 4.3. This is obtained when dY c /dt = dY c /dX j = 0, at t = T m and X c = X m c . Using equations ( 1), ( 2) and (3), we obtain

X m c = cos α sin α g u i 2 , Y m c = sin 2 α 2g u i 2 , T m = sin α g u i . (4)
It should be noted that each quantity depends only on the initial conditions of the ballistic trajectory, i.e. the initial velocity u i and the initial angle α. Now, we assume that the grain motion of the jet crest can be related to that of a single grain with a ballistic trajectory, which can be anticipated to be a crude assumption in view of dense granular flows impacting the water body. In fact, this suggests that grain-grain interaction can be neglected, at least in the upper region of the granular jet. In the following, it will be shown that this model captures most of the physical mechanisms, giving a fairly good description of the spatio-temporal evolution of the granular jet.

Application to the experiments

The validity of the predictive model is now assessed from experimental data. 3). This supports the idea that the trajectory of the jet crest could be simply described from a frictionless ballistic trajectory of grains. Finally, the insets of Figure 8 show the spatio-temporal evolution of the jet crest generated by dense liquid flows impacting a water body (salt water into fresh water, here). In this case, the jet corresponds to a volume of dense liquid ejected above the water surface. The dynamics of liquid jets is equivalent to that of granular jets, as shown by the collapse of the experimental data and the solid lines. In the present configuration, the jet dynamics is therefore mainly dependent on the flow conditions (e.g., flow velocity, slope angle), regardless of the rheology of the flowing material (e.g., Newtonian fluid, granular material).

In order to describe the spatio-temporal evolution of the granular jet, the initial parameters (u i , α) of the model have to be quantified from the flow conditions. The temporal evolution of the horizontal and vertical positions of the granular jet crest, i.e. X c (t) and Y c (t), are fitted by equations ( 1) and (2), respectively, for which u i and α can be and α ∼ 48 • , respectively. To overcome the problem of time and space offsets, the fitted relations are not set to (Xc, Yc) = (0, 0), at t -t A = 0.

adjusted. To overcome the problem of time and space offsets, the fitted relations are not set to (X c , Y c ) = (0, 0), at t -t A = 0. A similar method is also realized with the spatial evolution of the granular jet crest, i.e. Y c (X c ), using equation (3). Figure 9 shows an example of the good agreement between the experimental data and the fitted theoretical predictions. Here, the initial parameters of the model are equal to (a) and (b) u i ∼ 3.0 m.s -1 and α ∼ 53 • and (c) u i ∼ 3.1 m.s -1 , and α ∼ 48 • , respectively. As two different values of u i and α are obtained by the extracted method, the values and errorbars used in the following correspond to the averaged value and the standard deviation, respectively. closed symbols). Here, the variation of u f is only attributed to the slope angle θ, as the initial height of the column is kept constant. This observation seems to be inconsistent with the conclusions drawn from Figure 10(a), for which u i and u f increase together.

In fact, this could suggest that the driving effect of u f on the initial velocity u i is here counterbalanced by a dissipative effect of θ. In the inset of Figure 10(b), it is shown that α increases with θ. This means that grains substain a change of their direction more abrupt at larger θ (i.e., from α + θ ∼ 40 • to α + θ ∼ 100 • with θ = [5 : 30] • ) which could strongly promote dissipation in the system.

Discussion of results

Experiments showed that the spatio-temporal evolution of the position of the granular jet crest above the water surface can be well described using the theoretical prediction of a frictionless ballistic motion. This simple model requires only two parameters, i.e. the initial velocity u i and the initial angle α of the ballistic trajectory, which are di-rectly related to the flow-front velocity u f and the slope angle θ of the inclined plane.

In the following, the predictive model is applied to a recent natural event to assess its accuracy for geophysical applications.

Stromboli volcano (Italy) has persistent explosive activity with several hundreds of moderate-intensity events per day, and some notable paroxysmal phases. The 03 July and 28 August 2019 paroxysmal explosions produced eruptive columns of more than 5 km height, whose collapse generated pyroclastic density currents travelling rapidly along the northern flank of the volcano and entering the Mediterranean Sea [START_REF] Giudicepietro | Geophysical precursors of the July-August 2019 paroxysmal eruptive phase and their implications for Stromboli volcano (Italy) monitoring[END_REF]. Recorded images of the impact of pyroclastic density currents into water showed the generation of volcaniclastic jets above the sea level, from which the maximum height In any case, this result suggests that the theoretical prediction of a frictionless balistic motion could be successfully used for granular jets generated by fine-grained geophysical flows entering water to provide an estimate of their impact velocity, which is unvaluable for the hazard assessment of tsunami generation.

Flow dynamics of the particle-driven gravity current

In this section, we focus on the propagation of particle-driven gravity currents along the inclined plane and the horizontal bottom underwater, referred to as the near-and far-field regions, respectively (Figure 11). The current is characterized by the height profile h(x, t), the front position x f , and the bulk volume concentration of particles φ as a function of the flow rate per unit width q and the volume per unit width υ of the granular flow, the slope angle θ and the water depth H o . It should be remembered that the gas-fluidized granular flow enters the water body at x f = x A and t = t A , and then the current reaches the slope break at x f = x B and t = t B .

Near-field region: gravity current over an inclined plane

The first stage of the flow dynamics of particle-driven gravity currents underwater corresponds to the flow propagation along the inclined plane in the near-field region.

It can already be anticipated that the water depth H o could be disregarded here, being sufficiently far from the inclined plane. This is supported by results (Figure S1, in the Supporting Information) and the present configuration is therefore reduced to three control parameters here, i.e. q, υ and θ.

Figure 12 shows the temporal evolution of the front position x f -x A varying (a)

the flow rate per unit width q, (b) the volume per unit width υ, and (c) the slope angle θ of the inclined plane. In the first two cases, other control parameters are strictly kept constant while, in the latter case, the flow-front velocity u f , and therefore the flow rate q, varies with the slope angle θ of the inclined plane. First, it can be clearly observed that both q and υ affect the propagation of the front position, while θ can be disregarded here. More specifically, q modifies the whole propagation of the current, with in particular, a front propagation slower at lower q than at larger q [from light to dark gray, in Figure 12(a)]. By contrast, the influence of υ is only clear for t 0.5 s, from which all data deviate from the master curve. In this case, the flow front of currents propagates slower at lower υ than at larger υ [from light to dark gray, in Figure 12(b)]. This tran- sient behavior can probably be attributed to the end of the granular inflow. Finally, the influence of q and υ is hardly observed from the height profile of currents (insets of Fig-

ure 12), which supports the idea that the front position evolution is more relevant to quantify the flow dynamics of currents in the near-field region.

These results show that, as long as the entire granular volume has not yet entered the fluid layer, the flow rate per unit width q mostly controls the front dynamics of the gravity current beneath the free-surface. This observation can be related to a homogeneous gravity current produced by a constant flux over an inclined plane, for which the front position scales as x f ∝ (g ′ Q) 1/3 t, with g ′ = 2g(ρ -ρ f )/(ρ + ρ f ) the reduced gravity, ρ and ρ f the densities of the current and of the surrounding fluid, respectively, and Q the flow rate per unit width [START_REF] Britter | The motion of the front of a gravity current travelling down an incline[END_REF]. It is difficult to consider this scaling in the present configuration, given that the flow rate per unit width Q beneath the free-surface and the bulk density ρ of currents are unknown. However, we make the somewhat crude assumptions that the flow rate per unit width q at the impact is related to the flow rate per unit width Q beneath the free-surface, and that the bulk volume concentration of particles φ, and therefore the reduced gravity g ′ , do not vary significantly in the range of parameters considered here. The latter issue will be more fully discussed in Sec. 5.2. Under these assumptions, the front position x f -x A is plotted as a function of (gq) 1/3 (t-t A ) in Figure 13, for the same set of experiments shown in Figure 12(a). Surprisingly, a very good collapse of the experimental data is obtained for different flow rates per unit width q, which supports the use of this scaling. More specifically, it is not necessary to consider the reduced gravity g ′ , which means that the particle concentration φ of currents is fairly constant here. The inset of Figure 13 shows that all experimental data are included in a low interval delimited by two scaling laws, i.e.

x f -x A = λ(gq) 1/3 (t -t A ) with λ = 1.3 and 2 (gray area). At large times, the front position deviates from the linear trend, which is probably caused by the end of the granular inflow. The dynamics could tend towards the power-law evolution x f ∝ t 2/3 of a finite volume released over an inclined plane [START_REF] Dai | Experiments on gravity currents propagating on different bottom slopes[END_REF][START_REF] Dai | Non-Boussinesq gravity currents propagating on different bottom slopes[END_REF], represented by the dashed line in the inset of Figure 13. However, the inclined plane is not long enough in this study to fully support this interpretation. the volume per unit width υ (q ∼ 3.3 dm 2 .s -1 , θ = 15 • ), and (c) the slope angle θ (q = [2.6 : 4.3] dm 2 .s -1 , υ ∼ 6.2 dm 2 ). Insets: Height profile of currents along the inclined plane, at x f -x A ∼ 1 m.

Figure 13. Near-field region: front position x f -x A of the particle-driven gravity currents as a function of (gq) 1/3 (t -t A ) [based on [START_REF] Britter | The motion of the front of a gravity current travelling down an incline[END_REF]], for the same set of experiments shown in Figure 12(a) (inset: for the whole set of experiments, in a log-log representation). Gray area:

x f -x A = λ(gq) 1/3 (t -t A ) delimited by λ = 1.3 and 2; (---) x f -x A ∝ (t -t A ) 2/3 .

Far-field region: gravity current over a horizontal bottom

The front of the particle-driven gravity current reaches the slope break, at x f -

x B = 0 and t -t B = 0. Then, it propagates on the horizontal bottom until it reaches the end of the channel, referred to as the far-field region. Figure 14 shows x f -x B as a function of t-t B for different flow rates per unit width q and volumes per unit width υ of the granular flow, while the water depth and the slope angle are set to H o ∼ 26.5 cm and θ = 15 • , respectively. Additionally, the front dynamics of homogeneous gravity currents generated by the impact of dense liquid flows into water is shown, for different q and υ (blue circles, in inset of Figure 14). Note that, at early times of the propagation, the partial mixing between the dense liquid and water prevents the tracking of the invisible front of the currents.

At early times, the front position x f -x B increases faster for larger υ (from light to dark gray symbols), due to the flow dynamics of the current in the near-field region.

Then, at sufficiently long times, all experimental data collapse on a master curve with a linear trend (dashed line), regardless of q and υ. In contrast, at sufficiently low υ 2.3 dm 2 (cross symbols, in inset of Figure 14), the front position deviates from the linear trend and tends towards a slope of 2/3 (solid line). Surprisingly, the constant frontvelocities of both particle-driven gravity currents (black symbols, in Figure 14) and homogeneous gravity currents (blue circles, in inset of Figure 14) are similar for different q and υ. More specifically, one obtains dx f /dt ∼ 0.35 m.s -1 (dashed lines), which suggests that both currents have similar densities, and this point will be further discussed in the following. with H o , and in particular, it is found that h ∼ 0.37H o (solid line). Note that h is slightly lower for particle-driven gravity currents (full circles) than for homogeneous gravity currents (cross), which could be attributed to the different mixing efficiencies at the interface. Finally, the dotted line represents the theoretical value h = 0.5H o , for homogeneous gravity currents in the full-depth lock-exchange [START_REF] Ungarish | A shallow-water model for high-Reynolds-number gravity currents for a wide range of density differences and fractional depths[END_REF][START_REF] Bonometti | Slumping of non-Boussinesq density currents of various initial fractional depths: a comparison between direct numerical simulations and a recent shallow-water model[END_REF]. Experimental data of both particle-driven and homogeneous gravity currents are lower than the theoretical prediction, which is partly consistent with other experimental studies [START_REF] Rottman | Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel[END_REF][START_REF] Lowe | The non-Boussinesq lockexchange problem. Part 1. Theory and experiments[END_REF]. In particular, [START_REF] Lowe | The non-Boussinesq lockexchange problem. Part 1. Theory and experiments[END_REF] showed that the decrease of the density ratio ρ f /ρ between the surrounding fluid and the current could decrease the parameter γ (see their Figure 12), but it cannot fully explain the values obtained here.

In view of these results, the motion of particle-driven gravity currents in the farfield region can be roughly related to that of homogeneous gravity currents in the fulldepth lock-exchange, far enough from the shoreline where the propagation in the nearfield region does not affect yet the front dynamics. In the present work, most of currents exhibit a constant height and velocity of the front similar to the slumping regime while, at low υ, the front position evolves as x f ∝ t 2/3 according to the inertial self-similar regime. In both regimes, the preponderance of inertia over viscous effects can be quan- with g ′′ = g(ρ -ρ f )/ρ f , can be used here [START_REF] Ungarish | A shallow-water model for high-Reynolds-number gravity currents for a wide range of density differences and fractional depths[END_REF]. It should be remembered that this relation is applicable as long as the settling of grains is sufficiently slow along the time and the particle concentration is not too high, i.e. φ 0.3 [START_REF] Hallworth | Abrupt transitions in highconcentration, particle-driven gravity currents[END_REF]. These assumptions seem to be reasonable here because the constant front velocity of currents is broadly maintained until x f -x B ∼ 2 m corresponding to the tracked propagation, and the bulk particle concentration of the currents is thereafter estimated of the order of O(0.1) [Figure 16(e)]. Now, the above-mentioned equation of the front velocity can be reversed to estimate the bulk concentration of particle-driven gravity currents because the density and the concentration are related by ρ = φρ p + (1φ)ρ f , where ρ p = 2550 kg.m -3 and ρ f = 1000 kg.m -3 are the densities of grains and water, respectively. Recall that the bulk particle concentration of currents corresponds to φ = V grains /V current , where V grains ans V current denote the volume of grains and the total volume of the current, respectively. In this way, one obtains

φ = 4ρ f (ρ p -ρ f )gH o dx f dt 2 .
(5)

No clear trend is obtained between the bulk volume concentration of particles φ and the flow rate per unit width q, the volume per unit width υ, and the water depth . More precisely, it is found that φ ∼ 0.13 ± 0.035 (dashed line and gray area), but some caution has to be exercised with the obtained value because some crude assumptions were made through the model. This result suggests, however, that an increase of the front velocity, the front height, or the volume of the granular flow do not increase the bulk particle concentration φ of the currents interpreted as a critical value. This critical value of φ corresponds to a current density of ρ ∼ 1.20 ± 0.09 g.cm -3 , similar to that of homogeneous gravity currents in our experiments. This could explain that the front-velocity of both particle-driven and homogeneous gravity currents is similar in the far-field region (see Figure 14). Moreover, the constant value of φ supports the idea that the relative gravity can be disregarded using Britter and Linden's scaling in the nearfield region (see Figure 13). Finally, [START_REF] Freundt | Entrance of hot pyroclastic flows into the sea: Experimental observations[END_REF] also estimated the density of particledriven gravity currents generated by experimental volcanic ash flows entering water based on the front dynamics of the currents. He reported a bulk density of currents evolving from ρ ∼ 1.15 -1.25 g.cm -3 proximally to < 1.01 g.cm -3 near the tank end, corresponding to bulk particle concentrations from 0.16 -0.27 proximally to 0.01 distally.

The strong decrease of the particle concentration along the current propagation is at-tributed to the high settling velocity of the ash material sieved at 4 mm, which is much smaller in our experiments with glass beads of 65 µm. In both cases, the bulk particle concentration of currents proximally is estimated to be of the same order of magnitude, i.e. φ ∼ 0.16 -0.27 for [START_REF] Freundt | Entrance of hot pyroclastic flows into the sea: Experimental observations[END_REF]'s experiments and φ ∼ 0.13 ± 0.035 for our experiments, while quantitative variations can be probably attributed to the different grain properties (e.g., size, density, shape).

Discussion of results

In the natural environment, there are few measurements of particle concentration in subaqueous particle-laden flows due to the difficulty of predicting them. For turbidity currents in which grains are mainly suspended by the turbulent fluid, available field measurements showed that the volume concentration was usually lower than 0.01 (Talling et al., 2013, and references herein). The indirect concentration measurements of submarine currents caused by the 1929 Grand Banks event, however, give higher values of ∼ 0.03-0.05 [START_REF] Stevenson | Reconstructing the sediment concentration of a giant submarine gravity flow[END_REF]. Moreover, recent works reported that the dilute clouds of such currents can sometimes overlay dense basal layers [START_REF] Paull | Powerful turbidity currents driven by dense basal layers[END_REF][START_REF] Simmons | Novel acoustic method provides first detailed measurements of sediment concentration structure within submarine turbidity currents[END_REF][START_REF] Wang | Direct evidence of a high-concentration basal layer in a submarine turbidity current[END_REF]. In fact, grains can theoretically be suspended by the fluid turbulence up to a volume concentration of about 0.1, above which grains settle due to the importance of grain-grain interaction [START_REF] Bagnold | Auto-suspension of transported sediment: turbidity currents[END_REF]. This threshold is in line with the critical concentration obtained in our experiments, while other experimental studies also observed a threshold of the particle concentration in fluid-particle systems.

For the purpose of better understanding subaerial dilute pyroclastic density currents, [START_REF] Weit | Experimental measurement of the solid particle concentration in geophysical turbulent gas-particle mixtures[END_REF][START_REF] Weit | Maximum solid phase concentration in geophysical turbulent gasparticle flows: Insights from laboratory experiments[END_REF] performed experiments to study the spatial distribution of solid grains into a gas-particle turbulent system. In a steady vertical flow in which grains were poured gradually, the bulk particle concentration of the dilute gas-particle suspen-sion increased with the added granular mass before it reached a threshold. Then, the excess of grains could not be suspended by the turbulent fluid and clusters were formed and fell forming a basal concentrated granular bed. The coexistence of dilute and dense regions controlled by clustering instabilities has also been observed for gas-particle gravitydriven flows, which suggested that the upper dilute part was in its saturated-state [START_REF] Breard | Coupling of turbulent and non-turbulent flow regimes within pyroclastic density currents[END_REF]. Although clustering is poorly understood, it is usually attributed to dissipation of the granular temperature by collisional contacts or to hydrodynamic instabilities caused by the relative motion between the grains and the surrounding fluid (more details in Fullmer & Hrenya, 2017, and references herein). In both above-mentioned experiments, the critical volume concentration of the dilute region was a few percent, much lower than in our experiments. Yet, subaerial pyroclastic density currents and subaqueous turbidity currents share common characteristics, such as the turbulent-multiphase nature and the potential formation of dense granular regions [START_REF] Doronzo | A fluid dynamic model of volcaniclastic turbidity currents based on the similarity with the lower part of dilute pyroclastic density currents: evaluation of the ash dispersal from ash turbidites[END_REF].

For grains suspended in water by a vertical oscillating grid, the particle concentration also saturated to a critical value depending on the oscillation frequency, but the clustering instabilities disappeared [START_REF] Bennett | Turbulence modulation by suspended sediment in a zero mean-shear geophysical flow[END_REF][START_REF] Bennett | Turbulence suppression by suspended sediment within a geophysical flow[END_REF].

In view of these findings, the particle-driven gravity currents in our experiments could be related to physical mechanisms similar to those mentioned above. The impact between the granular flow and water generated a turbulent mixing zone, in which grains were suspended by the turbulent fluid. The bulk volume concentration of particles exceeded a critical value, leading to a saturated upper region in which the excess of grains probably fell, forming a basal concentrated region. For all experiments, a thick and voluminous granular deposit was obtained along the inclined plane, which could be interpreted as the final state of the dense granular flow (see Figure 6). Unfortunately, the flow dynamics of the dense basal region cannot be observed directly because of the visual limitations of the optical shadowgraph method. Under the influence of gravity, the upper dilute suspension at critical particle concentration collapsed on the inclined plane forming the particle-driven gravity current. Compared to gas-particle turbulent systems with critical volume concentrations less than ∼ 0.03-0.04, the high value of the critical concentration observed here could be attributed to different grain-fluid interaction. It is probable that the clustering instabilities control the critical concentration of gas-particle flows, in response to different settling rates between individual particles and clusters. This physical mechanism is mainly controlled by the grain inertia through the Stokes number, which is usually defined as the ratio between the particle inertial response time and the turbulent timescale [START_REF] Warhaft | Laboratory studies of droplets in turbulence: towards understanding the formation of clouds[END_REF]. For liquid-particle systems, however, further work will be necessary to highlight the physical mechanisms controlling the critical concentration of particles.

Conclusion

Well-controlled laboratory experiments focusing on the entrance of gas-fluidized granular flows into water have been performed, and backed up by theoretical models, to understand better the motion of geophysical granular flows going down a mountainside, then entering the sea, a lake or a river, and finally progagating underwater for what can be a considerable distance. The consideration of fluidized granular flows ensured a more suitable modelling of highly-mobile fine-grained geophysical flows, which are predisposed to reach the coast and generate turbidity currents. First, the presence of a water body promoted the generation of a granular jet over the free-surface, a leading and largest wave, and a turbulent mixing zone from which a particle-driven gravity current was spontaneously formed. Overall, hydrodynamic forces played a dissipative role by slowing and reducing the spreading of the granular mass along the bottom plane. However, a low amount of grains were still transported by the turbulent fluid like a gravity current propagating far away, showing both the dissipative and driving role of the fluid. Then, the granular jet and the particle-driven gravity current were analyzed by tracking the spatio-temporal evolution of the jet crest and the current front, respectively. The trajectory of the granular jet was well described by the theoretical prediction of frictionless ballistic motion, for which the initial parameters of the model depended on the flow-front velocity and the slope angle of the inclined plane. The flow dynamics of particle-driven gravity currents was described in two distinguished regions beneath the water surface, namely along the inclined plane and the horizontal plane, referred to as the near-and far-field regions, respectively. In the near-field region, the propagation of the current mainly depended on the flow rate per unit width q and the temporal evolution of the front position scaled with time as x f ∼ (gq) 1/3 t, with g the gravitational acceleration, in agreement with earlier findings. Then, the granular inflow was stopped and the finite volume affected the motion of the current from the above-mentioned scaling. In the far-field region, the evolution of the front position was similar to that of homogeneous gravity currents in the full-depth lock-exchange configuration. The particle-driven gravity current first evolved with a constant-front velocity in the so-called slumping regime, then the front decelerated and its position scaled as x f ∼ t 2/3 in the inertial self-similar regime. During the slumping regime, the constant flow-front velocity was only controlled by the height of the current depending on the water depth. More specifically, the bulk particle concentration of the current was roughly constant, i.e. φ ∼ 0.13 ± 0.035 (in volume), in the range of parameters considered. This result is interpreted as a critical concentration reached in turbulent fluid-particle flows, above which the excess of particles cannot be maintained by the turbulent fluid.

This study provides the conclusions that highly mobile fine-grained geophysical flows entering water are predisposed to generate granular jets above the free surface and dilute particle-driven gravity currents underwater. Both can be fairly well predicted using simple theoretical models capturing most of the physical mechanisms, which are of interest for geophysical purposes. Lastly, the bulk particle concentration of gravity currents generated by rapid granular flows entering water reaches a threshold, which needs further work to investigate the influence of grains properties (e.g., density, diameter) and to support this outcome by direct measurements.

Figure 1 .

 1 Figure1. Oblique view of a particle-driven gravity current, generated by a gas-fluidized granular flow down an inclined plane entering water. The dashed line indicates the bottom of the inclined plane above the level of the undisturbed water surface. On the left, the subaerial granular flow is relatively thin, dense and rapid before evolving to a thick and dilute particle-driven gravity current underwater. An impulse wave is also generated on the water surface, which has already been studied in[START_REF] Bougouin | Impact of fluidized granular flows into water: Implications for tsunamis generated by pyroclastic flows[END_REF].

Figure 2 .

 2 Figure 2. Sketch of the experimental setup with the initial parameters (the height Hi and the length Li defining the initial dimensions of the granular column, the slope angle θ of the inclined plane, and the water depth Ho). Black arrows represent the air flux through the 12 µm-porous bottom plane for the fluidization process. The streamwise coordinates x A and x B indicate the transition from the 1 m-subaerial ramp to the immersed ramp, and the end of the inclined plane that connects to the horizontal bottom, respectively. The zones containing the immersed inclined plane and the horizontal bottom are referred to as the near-and far-field regions, respectively.

3

  Figure 3. Snapshots of the entrance of a gas-fluidized granular flow into water, for Hi = 22 cm, Li = 34 cm, Ho = 38.5 cm, and θ = 15 • , at four different times from the opening of the sluice gate. A movie of the same experiment and a compilation of experimental movies from different camera viewpoints are also available in the Supporting Information.

Figure 4 .

 4 Figure 4. (a) Front position x f /Hi as a function of normalized time (g cos θ/Hi) 1/2 t for gasfluidized granular flows propagating in air (opened symbols) and entering water (closed symbols), with Hi = 10 cm (diamonds), Hi = 22 cm (squares) and Hi = 30 cm (circles). The other initial parameters are set to Li = 34 cm, θ = 15 • , and Ho = 26.6 cm (in the presence of water).

Figure 4

 4 Figure 4(b)], in agreement with values found in the case of dam-break flows on a horizontal plane[START_REF] Dressler | Comparison of theories and experiments for the hydraulic dam-break wave[END_REF][START_REF] Jánosi | Turbulent drag reduction in dam-break flows[END_REF][START_REF] Leal | Dam-break wave-front velocity[END_REF][START_REF] Roche | Experimental observations of water-like behavior of initially fluidized, dam break granular flows and their relevance for the propagation of ash-rich pyroclastic flows[END_REF] Bonometti

  (b)].

Figure 5 .

 5 Figure 5. Temporal evolution of the flow height h/Hi at the slope break x -x B = 0, for gasfluidized granular flows propagating in air (-) and entering water (---), with Hi = 10, 22, 30 and 42 cm (from light to dark gray). The other initial parameters are set to Li = 34 cm, θ = 15 • , and Ho = 26.6 cm (in the presence of water).

Figure 6 .

 6 Figure 6. Morphology of granular deposits along the bottom surface, for gas-fluidized granular flows propagating in air (-, opened circles) and entering water (---, closed circles), with Hi = 22, 30 and 42 cm (from light to dark gray). The other initial parameters are set to Li = 34 cm, θ = 15 • , and Ho = 26.6 cm (in the presence of water). Inset: Runout R f /Li, defined as h(x -x B = R f ) = Hi/10, as a function of the aspect ratio a = Hi/Li of the initial column.

  The entrance of a gas-fluidized granular flow into a water body generates first a granular jet above the water surface, before falling onto it. Here, we propose to describe the spatio-temporal evolution of the position of the jet crest (X c , Y c ) considering a ballistic trajectory. A sketch of this issue is shown in Figure7(a), with the flow parameters before entering water (upper) and the granular jet after the impact (bottom). It can already be anticipated that both the initial length L i , which only controls the volume of the granular flow, and the water depth H o could be disregarded here, as the granular jet is rapidly generated near the shoreline. The control parameters for this problem are therefore the slope angle θ of the inclined plane and the front velocity u f , the latter being controlled by the initial height H i of the column and the slope angle θ. shows snapshots of the typical evolution of a granular jet over the free-surface at different times from the impact. Note that the granular jet could also contain some water, but the proportion is expected to be low compared to those of grains and the ambient air.

Figure 7 .

 7 Figure 7. (a) Sketch of a gas-fluidized granular flow before (upper) and after (bottom) the impact with water. The trajectory of the granular jet is characterized by the crest position (Xc, Yc) reaching a maximum height at the position (X m c , Y m c ) and the time T m , and described by ballistic motion theory considering an initial velocity ui and an initial angle α. (b) Snapshots of the typical evolution of a granular jet for u f = 3.0 m.s -1 and θ = 15 • , at different times from the impact.

  Figure 8 shows a fairly good collapse of the spatio-temporal evolution of the granular jet crest, using (X m c , Y m c ) and T m as length and time scales. The duration of the jet is (tt A )/T m ∼ 1.5 here, from which the jet falls down the generated leading wave. Moreover, X c /X m c and Y c /Y m c have a quasi-linear and a 2-polynomial trend with the time (t-t A )/T m [solid lines, in Figs. 8(a)-(b)], which is consistent with the form of the theoretical predictions given by equations (1) and (2), respectively. Similar conclusions can also be drawn from the spatial evolution of the granular jet [Figure 8(c)] in comparison with equation (

Figure 8 .

 8 Figure 8. Evolution of (a) the horizontal position Xc/X m c and (b)-(c) the vertical position Yc/Y m c of the jet crest generated by gas-fluidized granular flows (insets: dense liquid flows) impacting a water body, as a function of (a)-(b) the time (t -t A )/T m and (c) the horizontal position Xc/X m c , where (X m c , Y m c ) and T m are the position and the time at which the maximum height of the jet crest is reached. The solid lines correspond to (a) Xc/X m c = (t -t A )/T m , (b) Yc/Y m c = -[(t -t A )/T m ] 2 + 2(t -t A )/T m , and (c) Yc/Y m c = -(Xc/X m c ) 2 + 2Xc/X m c , respectively.

Figure 9 .

 9 Figure 9. Spatio-temporal evolution of the granular jet crest, with u f = 3.0 m.s -1 and θ = 15 • . Squares and solid lines correspond to the experimental data and the theoretical predictions [equations (1), (2) and (3)] with (a)-(b) ui ∼ 3.0 m.s -1 and α ∼ 53 • , and (c) ui ∼ 3.1 m.s -1

Figure 10 .

 10 Figure 10. (a) Initial velocity ui (inset: initial angle α) as a function of the front-velocity u f of gas-fluidized granular flows (circles) and dense liquid flows (squares) at the impact with water, with θ = 15 • . The solid line is ui = 2u f -3, and the dashed line and gray area correspond to α = 50 ± 4 • . (b) Initial velocity ui (opened circles) and front-velocity u f (closed circles)as a function of the slope angle θ of the inclined plane, with Hi ∼ 22 cm. The solid lines are ui = 2.55 m.s -1 and u f = 0.06θ + 1.9, respectively. Inset: α as a function of θ, with α = 1.5θ + 23 (dashed line).

  of a granular jet crest could be approximately estimated as Y m c ∼ 200 m. In the present case, equation (4) can therefore be reversed to provide an estimate of the impact velocity of the pyroclastic flow, knowing that the values of both u f ∼ 70 m.s -1 and α ∼ 50 • are also extracted from videos of LBZ webcam of Laboratorio Geofisica Sperimentale (lgs.geo.unifi.it). The predictive model gives an impact velocity of u i ∼ 68 m.s -1 , with Y m c = 200 m and α = 50 • , which is in excellent agreement with the direct measurement of the flow velocity u f ∼ 70 m.s -1 . It should be mentioned, however, that recorded images suggest a lower streamwise position X m c of the maximum crest height compared to the vertical position Y m c , unlike the prediction of the model, but the estimate of X m c is poorly constrained.

Figure 11 .

 11 Figure 11. Sketch of the propagation of a particle-driven gravity current along a bottom plane with its macroscopic quantities [height profile h(x, t), front position x f and bulk volume concentration of particles φ], and the control parameters (flow rate per unit width q and volume per unit width υ of the granular flow, slope angle θ, and water depth Ho) of the problem. The streamwise coordinates x A and x B indicate the transition from the 1 m-subaerial ramp to the immersed ramp, and the end of the inclined plane that connects to the horizontal bottom, respectively. The zones containing the immersed inclined plane and the horizontal bottom are referred to as the near-and far-field regions, respectively.

Figure 12 .

 12 Figure12. Near-field region: temporal evolution of the front position x f -x A of the particledriven gravity currents by varying (a) the flow rate per unit width q (υ ∼ 3.1 dm 2 , θ = 15 • ),(b)

Figure 15 (

 15 Figure15(a) shows the temporal evolution of the height h of particle-driven gravity currents at x ∼ 3.5 m. The height h increases rapidly at the front and then remains roughly constant over time (dashed lines), despite Kelvin-Helmholtz instabilities at the interface. Moreover, the constant height of the current behind the front, noted h , is only dependent on the water depth H o , while the flow rate per unit width q, the volume per unit width υ, and the slope angle θ are varied. In Figure15(b), h increases linearly

Figure 14 .

 14 Figure14. Far-field region: temporal evolution of the front position x f -x B of the particledriven gravity currents (gray symbols) and homogeneous gravity currents (blue circles) as a function of time since arrival at the slope break, for different flow rates q and volumes per unit width υ. The water depth and the slope angle are set to Ho ∼ 26.5 cm and θ = 15 • , respectively. The dashed line is x f -x B = 0.35(t -t B ) and the solid line represents a slope of 2/3.

Figure 15 .

 15 Figure15. Far-field region: (a) Temporal evolution of the height h of particle-driven gravity currents at x ∼ 3.5 m, for different flow rates per unit width q and volumes per unit width υ of the granular flow, water depths Ho and slope angles θ. Here, the time t = 0 corresponds to the arrival time of the current front at the given position. The dashed lines indicate the constant height h of currents behind the front. (b) h as a function of Ho, for particle-driven gravity currents (full circles) and homogeneous gravity currents (cross). The lines are h = γHo, with the best fit γ = 0.37 (solid line) and the theoretical value γ = 0.5 (dotted line) for homogeneous gravity currents in the full-depth lock-exchange, respectively.

  tified by the Reynolds number Re = (ρ h /µ)(dx f /dt), where ρ ∼ ρ f and µ ∼ µ f are the density and the viscosity of currents estimated as those of water given that the bulk particle concentration remains sufficiently low [see Figure16(e)], and h and dx f /dt are the height and the front velocity of currents in the far-field region. The Reynolds number is in the range Re = [10 4 : 6 × 10 4 ], much larger than one. Now, let us consider that the theoretical relation of the constant flow-front velocity, i.e. dx f /dt = 0.5(g ′′ H o ) 1/2

H

  o , which suggests its independency on these different parameters [Figure 16(a)-(c)]. By contrast, the slope angle θ of the inclined plane seems to affect the concentration of currents, in the far-field region [Figure 16(d)]. More specifically, a steeper slope angle θ of the inclined plane promotes a larger bulk particle concentration of currents. Now, regardless of the role of θ, the bulk concentration of particle-driven gravity currents is roughly constant in the set of experiments, with an order of magnitude of φ ∼ O(0.1) [Figure 16(e)]

Figure 16 .

 16 Figure 16. Bulk particle concentration φ (in volume) of particle-driven gravity currents estimated by equation (5) in the far-field region, as a function of (a) the flow rate per unit width q, (b) the volume per unit width υ, (c) the water depth Ho, (d) the slope angle θ of the inclined plane, and (e) for all experiments of the present study. The dashed line and gray area correspond to the mean value and the standard deviation, i.e. φ ∼ 0.13 ± 0.035.

Acknowledgments

This work was funded by the ANR RAVEX (ANR-16-CE03-0002) project. The authors acknowledge the technical support of the Laboratoire Magmas et Volcans (Marc Nivoix, Eric Brut, Jean-Louis Fruquière, Cyrille Guillot) for their contribution in designing and constructing the experimental setup. We thank the anonymous reviewers for their constructive comments that helped to improve the initial manuscript. This is ClerVolc contribution n • 470. Data presented in this paper are partly in open access at https://doi.org/10.6084/m9.figshare.12137373 manuscript submitted to JGR: Oceans