Raphaël Moncelon
email: raphael.moncelon1@univ-lr.fr

Marie Gouazé

Philippe Pineau

Eric Beneteau

Martine Bréret

Olivier Philippine

François-Xavier Robin

Christine Dupuy

Edouard Metzger

Coupling between sediment biogeochemistry and phytoplankton development in a temperate freshwater marsh (Charente-Maritime, France): Evidence of temporal pattern

Keywords: Freshwater marsh, phytoplankton, early diagenesis, nutrient fluxes, nitrogen, iron, phosphorus Mn Fe P Sd

In freshwater systems, sediment can be an important source for the internal loading of phosphate (PO 4 ). The limiting character of this element in such system leads to consider this phenomenon in terms of eutrophication risks and water quality stakes. A four-months followup (January, March, April and May 2019) was carried out in a strong PO 4 limited secondary channel from an artificial irrigation system of Charente Maritime (France) to link the mobilization of remineralization products in the upper 6 cm layer of sediment (conventional core slicing/centrifugation and DET probes) and the phytoplankton biomass dynamics in the water column. Results showed congruent patterns between the temporal succession of the organic matter mineralization processes in the sediment and the primary biomass dynamics in the water column. In January and March (considered in winter), PO 4 proved to be retained by adsorption onto iron oxides in anoxic sediment since pore water nitrate inhibited for about a month the respiration of metal oxides in the first cm of sediment, thus limiting PO 4 availability and the phytoplankton growth. In April and May (early spring), after exhaustion of pore water nitrate, the dissolutive reduction of iron oxides released PO 4 into pore water generated a significant diffusive outgoing flux from the sediment to the water column with a maximum in April (-1.10E-04±2.81E-05 nmol cm -2 s -1 ). This release coincided with the nanophytoplankton bloom (5.50 µg Chla L -1 ) and a potential increase of PO 4 concentration in the water column. This work provides some insight on the importance of benthic-pelagic coupling in anthropogenic systems. This conceptual model has to be deployed on other sites of interest where internal loading of P takes precedence over external inputs and nitrate mitigation drives its benthic recycling and ultimately its bioavailability. This is to be essential to characterize the aquatic environment quality in order to limit eutrophication risks.

Introduction

The importance of wetland ecosystems has been widely recognized for services such as carbon catchment control [START_REF] Chmura | Global carbon sequestration in tidal, saline wetland soils[END_REF][START_REF] Sousa | Productivity and nutrient cycling in salt marshes: Contribution to ecosystem health[END_REF] or water purification led by primary producers [START_REF] Smith | Eutrophication of freshwater and coastal marine ecosystems a global problem[END_REF]. Phytoplankton structure and dynamics has been well studied in both marine environments [START_REF] Legendre | Plankton and nutrient dynamics in marine waters[END_REF][START_REF] Hlaili | Planktonic food webs revisited: Reanalysis of results from the linear inverse approach[END_REF] and freshwater ecosystems [START_REF] Tortajada | Variability of fresh-and salt-water marshes characteristics on the west coast of France: A spatio-temporal assessment[END_REF][START_REF] Masclaux | Planktonic food web structure and dynamic in freshwater marshes after a lock closing in early spring[END_REF]David et al., 2020a[START_REF] David | Ecological succession and resilience of plankton recovering from an acute disturbance in freshwater marshes[END_REF]. Its development can depend on bottom-up factors such as water temperature, light penetration or nutrient availability [START_REF] Reid | Phytoplankton of the North Sea and its dynamics: A review[END_REF] and/or top-down factors such as grazing or parasitism [START_REF] Reid | Phytoplankton of the North Sea and its dynamics: A review[END_REF][START_REF] Lynn | The journal of eukaryotic microbiology[END_REF], or macrophyte competition [START_REF] Barrow | Macrophytes moderate the taxonomic and functional composition of phytoplankton assemblages during a nutrient loading experiment[END_REF]. The variations of the physical-chemical parameters in the water column and their consequences on phytoplankton dynamics are well documented in marine coastal zones [START_REF] Justić | Changes in nutrient structure of riverdominated coastal waters: Stoichiometric nutrient balance and its consequences[END_REF][START_REF] Dupuy | Community structure and grazing of the nano-microzooplankton on the continental shelf of the Bay of Biscay[END_REF] and continental freshwater environments [START_REF] Soballe | A large-scale comparison of factors influencing phytoplankton abundance in rivers, lakes, and impoundments[END_REF][START_REF] Søndergaard | Nitrogen or phosphorus limitation in lakes and its impact on phytoplankton biomass and submerged macrophyte cover[END_REF]. Temporarily, two blooms are generally mentioned in saline environments, in early spring and late summer [START_REF] Morais | Phytoplankton dynamics in a coastal saline lake (SE-Portugal)[END_REF][START_REF] Toompuu | Seasonal variation of average phytoplankton concentration in the Kattegat -A periodical point model[END_REF]. However, little is known about marsh ecosystems. Research focusing on the variation of water column settings on these dynamics shows a close relation to physical and chemical parameters of the environment both in freshwater or brackish environments. While the phytoplankton dynamics seems to depend less on the season in brackish environments but more on the renewal of water [START_REF] Tortajada | Variability of fresh-and salt-water marshes characteristics on the west coast of France: A spatio-temporal assessment[END_REF], significant temporal succession has been highlighted in freshwater marshes [START_REF] Masclaux | Planktonic food web structure and dynamic in freshwater marshes after a lock closing in early spring[END_REF].

By their position in the watershed, marshes receive huge amounts of nutrients coming from the continent by soil leaching, such as nitrogen (N) and phosphorus (P), essential for primary producers' growth [START_REF] Redfield | The biological control of chemical factors in the environment[END_REF][START_REF] Søndergaard | Nitrogen or phosphorus limitation in lakes and its impact on phytoplankton biomass and submerged macrophyte cover[END_REF]. Among these elements, P is widely recognized as the principal factor of phytoplankton growth, especially in freshwater systems [START_REF] Schindler | Eutrophication and recovery in experimental lakes: Implications for lake management[END_REF][START_REF] Søndergaard | Hypolimnetic Nitrate Treatment to Reduce Internal Phosphorus Loading in a Stratified Lake[END_REF][START_REF] Kraal | Sedimentary iron-phosphorus cycling under contrasting redox conditions in a eutrophic estuary[END_REF]. Human activities can alter the growth-limiting nutrients, especially through wastewater or various surfaces runoff [START_REF] Withers | Delivery and cycling of phosphorus in rivers: A review[END_REF] and can cause eutrophication [START_REF] Schindler | Eutrophication and recovery in experimental lakes: Implications for lake management[END_REF][START_REF] Smith | Eutrophication of freshwater and coastal marine ecosystems a global problem[END_REF]. The reduction of external growth-limiting nutrients as P from shift in management framework would lead to an increase in the importance of sediment as an internal source [START_REF] Perkins | The potential for phosphorus release across the sediment-water interface in an eutrophic reservoir dosed with ferric sulphate[END_REF]. In some lowland river systems, P released from organic matter (OM) remineralization in the sediment may even be the main source of P in the water column [START_REF] Van Dael | Sediment respiration contributes to phosphate release in lowland surface waters[END_REF] and thus for the phytoplankton productivity. This loading from sediment could also determinate the P limitation in freshwater ecosystems such as lakes [START_REF] Wu | Internal cycling, not external loading, decides the nutrient limitation in eutrophic lake: A dynamic model with temporal Bayesian hierarchical inference[END_REF].

Many techniques exist for calculating diffusive fluxes at the sediment-water interface (SWI), such as the in situ use of benthic chambers, or ex situ measures through the use of Fick's law [START_REF] Grenz | Comparaison de deux méthodes de mesure de flux de nutriments à l'interface eau-sédiment : méthode des peepers et méthode des chambres benthiques[END_REF][START_REF] Viollier | Benthic biogeochemistry: State of the art technologies and guidelines for the future of in situ survey[END_REF]. The latter allow an evaluation of redox processes in the sediment through the modelling of sediment profiles [START_REF] Grüneberg | Effects of nitrogen and phosphorus load reduction on benthic phosphorus release in a riverine lake[END_REF].

More recent techniques such as the use of 2D gel probes provide results (flux and profile modeling) with better vertical and horizontal resolution [START_REF] Jézéquel | Two-dimensional determination of dissolved iron and sulfur species in marine sediment pore-waters by thin-film based imaging. Thau lagoon (France)[END_REF][START_REF] Santner | Two decades of chemical imaging of solutes in sediments and soils -a review[END_REF][START_REF] Thibault De Chanvalon | Two dimensional mapping of iron release in marine sediments at submillimetre scale[END_REF]. They make it possible to document the intense gradients that can be expected during P remobilization.

Benthic phosphorus dynamics is well documented, in both marine and freshwater systems [START_REF] Boström | Exchange of phosphorus across the sediment-water interface[END_REF][START_REF] Caraco | A comparison of phosphorus immobilization in sediments of freshwater and coastal marine systems[END_REF][START_REF] Anschutz | Burial efficiency of phosphorus and the geochemistry of iron in continental margin sedime[END_REF][START_REF] Ekholm | Does control of soil erosion inhibit aquatic eutrophication[END_REF], and depends on the evolution of the oxide stocks in the sediment. The iron cycle is of particular interest in understanding the phosphate (PO 4 ) remobilization, having the capacity of forming relatively stable Fe(OOH)-P complexes preventing its dissolution [START_REF] Herzsprung | Flood effects on phosphorus immobilisation in a river water filled pit lake-Case study Lake Goitsche (Germany)[END_REF]. The dynamics of elements interacting with the iron cycle in sediment and influencing its mobilization, such as oxygen, nitrates, manganese, and sulphur, is therefore a major issue in understanding the mechanisms governing the release of PO 4 into the water column [START_REF] Borggaard | Influence of organic matter on phosphate adsorption by aluminium and iron oxides in sandy soils[END_REF][START_REF] Caraco | A comparison of phosphorus immobilization in sediments of freshwater and coastal marine systems[END_REF][START_REF] Herzsprung | Flood effects on phosphorus immobilisation in a river water filled pit lake-Case study Lake Goitsche (Germany)[END_REF][START_REF] Xiao | Experimental study on adsorption of phosphorus on sediment components with different particle sizes[END_REF].

Vertical oxidant successions in the first stages of biotic OM mineralization in sediment, called early diagenesis, are well known in marine environments [START_REF] Froelich | Early oxidation of organic matter in pelagic sediments of the eastern equatorial Atlantic: suboxic diagenesis[END_REF][START_REF] Aller | Conceptual models of early diagenetic processes: The muddy seafloor as an unsteady, batch reactor[END_REF].

However, temporal succession of these processes is poorly documented (Thibault de Chanvalon et al., 2016). Moreover, retrospective reports on the response of pelagic nutrient and biota are quite frequent [START_REF] Jeppesen | Lake responses to reduced nutrient loading -An analysis of contemporary long-term data from 35 case studies[END_REF][START_REF] Welch | Internal phosphorus loading in shallow lakes: Importance and control[END_REF], but rarely combined with the in situ sediment analysis [START_REF] Grüneberg | Effects of nitrogen and phosphorus load reduction on benthic phosphorus release in a riverine lake[END_REF][START_REF] Liu | Phosphorus released from sediment of Dianchi Lake and its effect on growth of Microcystis aeruginosa[END_REF].

The high productivity of the Marans' site already observed despite the strong P limitation [START_REF] Masclaux | Planktonic food web structure and dynamic in freshwater marshes after a lock closing in early spring[END_REF] testifies to the importance of internal P recycling in this system. The objective of this study is to propose for the first time an interdisciplinary seasonal benthospelagos in situ monitoring in a strong P-limited coastal freshwater marsh. It is of interest to target such P limited site as environmental policies focuses on limiting P release into natural aquatic systems which consequence is the increase of water bodies characterized by high N/P ratios. The main hypothesis of this work lies on the importance of temporal variation of benthic mineralization processes for the phytoplankton development through generated nutrient fluxes at the SWI. Benthos-pelagos coupling was assessed through a combined ecological and biogeochemical approach, with temporal phytoplankton biomass measurements and nutrient and metal oxides profiling. Diffusive fluxes were obtained from porewater chemical gradients from core slicing (0.5 cm resolution) and 2D-DET gel deployment (submillimeter resolution).

Material and methods

Study site

The Charente Maritime marshes (French Atlantic west coast) present a high diversity of freshwater marsh types, differing by their locality, their role and the way they are managed. A eutrophic freshwater marsh, based on Tortajada classification (2011), used for anthropogenic activities (e.g. field irrigation) was chosen to carry out this work. Among the artificial hydrographic networks, the study site took place in a secondary channel of the Sèvre Niortaise River marshes (46.282°, 0.969°) located closed to the commune of Marans (Fig. S1 as supplementary material). The station is surrounded by crops and cattle lands, and the management (i.e. replenishment, drainage and dredging) is provided by a trade union association.

Sampling strategy and parameters studied

The sediment and the water column compartments were sampled simultaneously and monthly from winter to early spring 2019. The sampling campaigns were realized in January (the 30 th ), March (the 5 th ), April (the 2 nd ) and May (the 1 st ).

Water column

 Abiotic parameters

At each campaign, in situ temperature, salinity and oxygen concentration in the water column were measured with multiparametric sensors (VWR). In addition, these three parameters were monitored continuously (one measurement per 15 minutes) between January (the 30 th ) and May (the 1 st ) with HOBO U-26 sensor.

 Phytoplankton compartment

A volume of 15 L of surface water was collected for the phytoplankton sampling in the middle of the channel. The Chlorophyll a biomass (Chla) was evaluated for three-size classes of phytoplankton: micro-: >20 µm, nano-: 3 to 20 µm and picophytoplankton: <3 µm. Filters, made in triplicate (20 µm, 3 µm and 0.7 µm), were used to separate the 3 different size fractions by fractional filtration. The biomass of Chla of each size class was determined by fluorimetry according to the Yentsch and Menzel method (1963).

Sediment compartment

 Core sampling and processing Four cylindrical cores were collected using PVC tubes (8.2 cm diameter, 20 cm long) with taking care to keep a water thickness of at least 5 cm above the sediment to preserve the sediment-water interface and for bottom water chemistry. Three cores were sliced every 5 mm to a depth of 6 centimetres (i.e. 12 layers) under nitrogen atmosphere. Porewater was extracted from each sediment layer by centrifugation (20 minutes at 2 700g and at in situ temperature) then filtered through 0.2 µm filters. Overlying and pore waters were conserved at -20 °C for nutrients analyses, an aliquot was conserved at 5°C for silica analysis in both bottom and pore waters. Aliquots were immediately used for alkalinity measurement and HNO 3 acidification (0.01 mol L -1 ; Suprapur Merck) for metal analysis. Nutrients i.e. nitrates (NO 3 -), nitrites (NO 2 -), soluble reactive phosphorus (SRP), ammonium (NH 3 ) and silica (Si) concentrations were determined by a colorimetric autoanalyser Scalar Seal (whit a detection limit of 0.02 µmol L -1 ). The acidified aliquot was analyzed by ICP-AES (Thermo Scientific iCAP 6300 Radial) to measure dissolved iron (Fe d ), manganese (Mn d ) and sulphur (S d ).

Alkalinity was determined according to the colorimetric [START_REF] Sarazin | A rapid and accurate spectroscopic method for alkalinity measurements in sea water samples[END_REF] method.

Absorbance measurements were carried out with a spectrophotomer (SPECTROstar Nano, BMG LABTECH).

The remaining sediment (from centrifugation) was then frozen at -20°C. Samples were freeze-dried and manually grounded for solid phase analyses. Once the samples have been reweighed, the porosity ( ) was determined for use in fluxes modeling (see next section).

An aliquot (~100 mg) was used for the ascorbate extraction of reactive manganese (Mn 3+ and Mn 4+ ), iron (amorphous Fe 3+ ) and P according to the method used in Kostka and Luther (1994) and [START_REF] Anschutz | Burial efficiency of phosphorus and the geochemistry of iron in continental margin sedime[END_REF].

 High resolution methods

-Microprofiling

A core was placed in an aquarium thermoregulated at the in situ temperature and air was injected at the overlying water to maintain O 2 saturation and keep a thin benthic boundary layer. Oxygen was analysed with Clark-type electrodes (50 µm tip diameter, [START_REF] Revsbech | An oxygen microsensor with a guard cathode[END_REF] connected to a Unisense multimeter and a motorized vertical micromanipulator. A two-point calibration was made from saturated overlying water and anoxic sediment.

-2D-DET probes

In April, two supplementary cores were collected for a 2D-DET probes deployment. This method uses a Diffusive Equilibrium in Thin films (DET) probe [START_REF] Davison | Distribution of dissolved iron in sediment pore waters at submillimetre resolution[END_REF], coupled with a colorimetric technique to visualize in two-dimensions the repartition of dissolved nitrite iron and P concentrations along the core [START_REF] Jézéquel | Two-dimensional determination of dissolved iron and sulfur species in marine sediment pore-waters by thin-film based imaging. Thau lagoon (France)[END_REF][START_REF] Pagès | Representative measurement of two-dimensional reactive phosphate distributions and codistributed iron(II) and sulfide in seagrass sediment porewaters[END_REF], Metzger et al., 2016). Firstly, a DET gel probe which samples the dissolved chemical species from the pore water at high resolution fixed to a Plexiglas plate was introduced into the sediment core. This probe was made of a 185 mm×125 mm×0.8 mm polyacrylamide thin film (Krom et al., 1994) laid on an acrylic support and a PVDF porous (0.2 µm) membrane that maintained and protected the gel in order to minimize lateral heterogeneity. For a single support, 2 gels were placed back to back to make a double-faced probe. The first probe had a gel dedicated to nitrite and another to iron, the second had a probe dedicated to P, the other to iron. After 5hours equilibrium time, gel probes were quickly processed to minimize lateral diffusion within the sampling gel and subsequent loss of 2D resolution. It consisted in superimposing each gel into a specific reagent gel previously equilibrated with NO 2 , PO 4 and Fe colorimetric reagents (Griess, molybdate, ferrozine reagents, see references cited above). Twenty minutes after contact, a picture was taken with a standard scanner. Images were analyzed by densitometry of different RGB spectral bandwidths using ImageJ software. The iron was treated in the blue channel. This channel was chosen despite its lower sensibility compared to the green one in order to avoid saturation (range between 0 and 500 µmol L -1 for the blue channel, between 0 and 200 µmol L -1 for the green one). Phosphates and nitrites were analyzed in the red and green channel respectively.

 Diffusive fluxes modeling NO 3 -, NO 2 -, ƩNH 3 , SRP, Si, Fe d and oxygen (O 2 ) diffusive fluxes at the SWI were determined using the Profile software [START_REF] Berg | Interpretation of measured concentration profiles in sediment pore water[END_REF]. Based on a transport-reaction model, this software allowed to describe the sedimentary column as a succession of layers with a specific reaction rate for a dissolved chemical species and the fluxes above and below that sediment column (modeling vertical domain). This work focused on the vertical diffusion and the transport was assumed to be dominated by molecular diffusion (without bioturbation), as described in [START_REF] Lewandowski | Two-dimensional small-scale variability of pore water phosphate in freshwater lakes: Results from a novel dialysis sampler[END_REF], since the sediment is impermeable (silt-clay). The flux (F) was calculated at the SWI thanks the Fick Law as:

(1) with the sediment porosity, Ds the diffusion coefficient, C the element concentration and z the depth of calculation. In Profile, Ds was calculated using [START_REF] Rasmussen | Microelectrode studies of seasonal oxygen uptake in a coastal sediment: role of molecular diffusion[END_REF], as below:

(2) Ds = D/(1+3(1-))
Diffusion coefficient D was determined with previous studies varying with temperature.

O 2 fluxes modeling were made using data from microprofiling, D was taken from the Unisense gas table. The NO 3 -, NO 2 -, ƩNH 3 and SRP coefficient diffusion were taken from [START_REF] Li | Diffusion of ions in sea water and in deep-sea sediments[END_REF], and the Si one from [START_REF] Rebreanu | The diffusion coefficient of dissolved silica revisited[END_REF]. Modeling was based on data from the core slicing and 2D-DET profiles. In case of too few points with quantified concentrations a linear model was used to calculate the flux "manually" [START_REF] Boudreau | Diagenetic models and their implementation[END_REF] using the formula (1) and Ds as :

(3) Ds = D/(1-ln( 2 ))
In this work, the negative flux values corresponded to fluxes leaving the sediment, the positive values to fluxes entering the sediment. O 2 , NO 3 -, ƩNH 3 and Si fluxes were calculated through profiles obtained with the core slicing method. NO 2 -, SRP and Fe fluxes were calculated through both core slicing method profiles and 2D-DET probes. The fluxes calculated from the 2D-DET probes represented the width-integrated average fluxes of the gel at three interfaces positioning: 0 (SWI), SWI+1 mm and SWI-1mm. The flux was modeled from the mean concentrations per gel line. The overlying water was not used for the calculation of the gradients at the interface, because of the very strong chemical gradients near the diffusive boundary layer (DBL) and not reflecting its chemical properties.

Statistical tests

All statistical tests were made with the software R studio (1.2.1335). Analyses of Variance (ANOVA) and Tuckey post hoc tests (threshold α = 0.05) were made to determinate significant differences between replicates. To respect the validity of this statistical test, data could be log transformed. If residuals did not followed normality, Kruskal Wallis and Wilcoxon post-hoc tests were made.

Results

Physical-chemical parameters of the water column

Temperature increased regularly between the end of January and May, rising from 8°C to 15°C on average in the channel (Fig. 1). Discrete sampling points indicated (measured with WTW sensor) 7.5, 10, 14 and 14°C for January, March, April and May respectively.

Conductivity was relatively stable around 850 µS cm -1 , which represents a salinity around 0.5. Dissolved oxygen increased between late January and late February (approximately 5 to 17 mg L -1 ). From April to May, it decreased on average from about 17 to 10 mg L -1 . Discrete sampling indicated subsequently 9, 11, 14.5 and 11.3 mg L -1 for January, March, April and May respectively.

Nitrogen was mainly represented by NO 3 -with the highest concentration (668 µmol L -1 ) observed in March (Tuckey test, F(3) = 7.58, p < 0.05) (Table . 1). NO 2 -concentration was the highest in April (20.43 µmol L -1 ) (Tuckey test, F(3) = 14.06, p < 0.05). ƩNH 3 concentration was higher in April (79 µmol L -1 ) than those of January and March (Kruskal-Wallis test,  2 (3) = 8.66, p < 0.05). SRP increased over time, from 0.00 to 0.37 µmol L -1 in March and May respectively. However, no significant differences were noted despite the increase (Kruskal-Wallis test,  2 (3) = 5.64, p < 0.05). Finally, Si concentration in April (154 µmol L -1 ) was higher than for March (Kruskal-Wallis test,  2 (3) = 3.17, p = 0.05). According to Redfield ratio calculations (Fig. S2 as supplementary material), the study site was only phosphatelimited for planktonic primary producers, all along the survey.

Phytoplankton dynamics

From January to March, the phytoplankton biomass was between 1 and 3.4 µg Chla L -1 depending on the size class, and dominated by the microphytoplankton (Fig. 2). However, no significant difference was observed between the three size classes neither in January (ANOVA, F(2) = 1.21, p > 0.05) nor in March (ANOVA, F(2) = 3.55, p > 0.05). In April, a significant nanophytoplankton development was observed with 5.5 µg Chla L -1 (Tuckey post Hoc test, p < 0.001). A similar pattern was observed in May, with a dominant nanophytoplankton biomass, four times higher than that of the other two size classes (4 and 1 µg Chla L -1 respectively) (Tuckey post Hoc test, p < 0.001).

Within each size class, the April nanophytoplankton and picophytoplankton biomasses (5 and 3 µg Chla L -1 respectively) were higher than in January, March and May (Tuckey post Hoc test, p < 0.03 and p < 0.001 respectively).

Sediment geochemistry

In late January, NO 3 -mean concentration was around 200 µmol L -1 with high spatial variability between 0 and 2 cm depth and decreased sharply to reach about 5 µmol L -1 at 3.5 cm depth (Fig. 3). Same pattern was observed for NO 2 -with a maximum mean concentration in the top layer (43 µmol L -1 ) which decreased rapidly to reach the detection limit at 3.5 cm depth. Early in March, only the first sediment layer showed detectable nitrate and nitrite with 37 and 6 µmol L -1 respectively. Then, no sample showed detectable concentrations.

Mn asc showed a maximum concentration in late January in the first layer of sediment (3.2 mmol kg -1 ) and remained constant downwards at 1.5 mmol kg -1 . For March, April and May, values ranged between 1.0 to 1.5 mmol kg -1 . Mn d presented a linear increase along the core in January, April and May between 1 to 60 µmol kg -1 . A peak was observed in March between 0 and 3 cm, with a maximum of 35 µmol kg -1 .

Fe asc was higher in January and May around 25 mmol kg -1 and lower in March and April around 20 mmol kg -1 , until 4 cm depth. In late January, Fe d was near the detection limit from 0 to 5 cm, and in March a first peak was observed at 1.5 cm (47 µmol kg -1 ) at the same depth as the manganese one. In April, a strong mobilization was detected between the surface and 4 cm deep with a peak of 152 µmol kg -1 . It then increased to 238 µmol kg -1 in the deepest layers. In May, at the same depth dissolved iron peaked at 70 µmol kg -1 , to then reach 232 µmol kg -1 at 6 cm depth. In April and May, Fe d increased from 4 and 3 cm depth respectively to reach more than 200 µmol L -1 at the core bottom.

P asc ranged between 4 and 5 mmol kg -1 in January, March and May, without any real trend with depth. It was higher in April to reach between 5 and 6 mmol kg -1 . SRP was under detection limit in January. A first peak of 3 µmol L -1 appeared at 1.5 cm depth in March (concomitant with manganese and the first iron peak) to increase in April, with a strong mobilization in the first 2 cm with a maximum of 32 µmol L -1 . In May, a weaker and deeper peak was observed (3 cm depth, 22 µmol L -1 ). These two peaks seemed to coincide with the globally increase of dissolved iron mobilization in the first 4 cm of sediment, especially in April.

Sulphate profiles showed generally increasing concentrations with depth, with the highest concentrations observed in January (2 to 8 mmol kg -1 at surface and depth respectively). In March, April and May, concentrations ranged from about 0.5 mmol kg -1 at the surface to 3 mmol kg -1 at depth. Concentrations in April were the lowest and showed the smallest gradient (between 0.5 and 1.5 mmol kg -1 ) along the sediment.

For both January and March, ƩNH 3 concentrations ranged from nearly 50 µmol L -1 to 130 µmol L -1 between top layer and 6 cm deep respectively, with a slight increase in the last centimeter. ƩNH 3 concentrations in April and May were about 4 times higher than those obtained in winter in the surface layer (193 and 175 µmol L -1 respectively), and presented a maximum peak in May at 3 cm depth to reach 439 µmol L -1 .

Si had the same vertical trend for every month with a slightly increasing concentration along both the core column and time. Maxima observed were at around 5 cm depth, with 411, 496, 583 and 682 µmol L -1 for January, March, April and May respectively.

The alkalinity increased over months and was positively correlated with ƩNH 3 concentration in porewater (p < 0.001, r 2 = 0.76). In January, it ranged between 6.5 and 7.7 mmol kg -1 at the surface and core bottom layer respectively, with a slight increase in the last centimeter. It then increased from Marsh (between 7.0 and 8.9 mmol kg -1 ) to April (between 10.5 and 13.5mmol kg -1 ).

The 2D representation of dissolved iron, nitrite and SRP in the sediment pore water was made in April (Fig. 4). Both iron gels (probe 1 and 2) showed an increasing concentration with depth (Fig. 4), with mobilization starting at nearly 0.5 cm below the SWI and increased sharply from 15 to 170 µmol L -1 at 1.5 cm depth for probe 1 and from 30 to 170 µmol L -1 at 2.5 cm depth for probe 2. The concentrations then increased slightly to reach a maximum of 1, nitrite was only present in a thin layer between the SWI and 1 cm depth (maximum concentrations of about 65 µmol L -1 ). The average SRP (probe 2) presented a strong mobilization between 1 and 6 cm with a maximum concentration about 90 µmol L -1 . The lateral heterogeneity was more pronounced on probe 1(e.g. Fe d -depleted and NO 2 -rich zones between 5-9 cm and 0-2 cm depth respectively, on a 2 cm width).

The Fe asc /P asc ratio is an interesting indicator to understand the effects of the adsorption site saturation on the remobilization of P in the porewater [START_REF] Anschutz | Burial efficiency of phosphorus and the geochemistry of iron in continental margin sedime[END_REF], Thibault de Chanvalon et al 2016). Between 1 and 5 cm deep, in January and May Fe asc /P asc ratio was quite similar around 6. (Fig. 5). In March and April, this ratio was lower around 5 and 4 respectively.

Benthic fluxes and high spatial resolution consideration

Incoming oxygen fluxes were observed (around 10 -2 nmol cm -2 s -1 ), with maximum fluxes in May significantly higher than those of March (Kruskal-Wallis test,  2 (3) = 12.19, p = 0.03). The nitrogen nutrient fluxes were most heterogeneous, recording ingoing and outgoing fluxes depending on the element studied. Ingoing NO 3 -and NO 2 -fluxes were observed for each date with a significantly higher flux in January (4.53E -03 nmol cm -2 s -1 ;

Tuckey test, F(3)= 6.48, p<0.05 and 5.06E -04 nmol cm -2 s -1 ; Tuckey test, F(3) = 5.87, p < 0.05 for NO 3 -and NO 2 -respectively). Lastly, ƩNH 3 fluxes tended to be outgoing for each date. No significant differences were noted between dates (Kruskal-Wallis test,  2 (3) = 4.59, p = 0.20).

A slight SRP influx in January and efflux between March and May was observed. The outgoing flux in April (-1.10E -04 nmol cm -2 s -1 ) was significantly higher and 10 times greater compared to ones in March (Tuckey test, F(3) = 12.51, p < 0.005). The Si fluxes showed an outgoing flux for each date. No significant differences were noted (Tuckey test, F(3) = 2.06, p = 0.18). Finally, the iron flowed out regardless of the date. The February flux (-2.10E -06 nmol cm -2 s -1 ) was significantly lower than those of March, April and May (Tuckey test, F(3) = 10.38, p <0.05).

The fluxes calculations via the modeling of 2D gel concentrations showed the impact of the core scale heterogeneity on the fluxes and the importance of interface positioning (Table 3). At the 0 interface, NO 2 -and SRP presented an incoming flux of the order of 10 -5 and 10 -6 µmol cm -2 s -1 respectively. A Fe d efflux was observed around -2E -04 nmol cm -2 s -1 .

The change in the interface positioning generated variability in flux directions. For NO 2 -, a +/-1mm positioning generated a mean outgoing (10 -5 µmol cm -2 s -1 ) and incoming (10 -4 µmol cm -2 s -1 ) flux respectively. For SRP, these two positions produced an outgoing flux of the same magnitude order (10 -5 µmol cm -2 s -1 ). Only the averaged iron fluxes were always outgoing whatever the interface considered, which showed a better homogeneity in the upper part of the gel.

Discussion

Phytoplankton dynamics and nutrients availability

The phytoplanktonic dynamics observed at Marans can be qualified as relatively classic for a freshwater system [START_REF] Tortajada | Variability of fresh-and salt-water marshes characteristics on the west coast of France: A spatio-temporal assessment[END_REF]. A poor development in winter regardless of the size classes was explained by the unfavorable physical conditions (low temperature and probably luminosity) despite huge nitrates values in March (Table 1) probably coming from near agricultural soil leaching [START_REF] Tortajada | Variability of fresh-and salt-water marshes characteristics on the west coast of France: A spatio-temporal assessment[END_REF]. The nanophytoplankton bloom observed in April could be attributed to much more favorable physical conditions (increase in temperature and probably luminosity), but also in nutrient charging of the system, and especially in dissolved phosphorus (see below for link benthos/pelagos coupling) as strong limiting factor [START_REF] Kraal | Sedimentary iron-phosphorus cycling under contrasting redox conditions in a eutrophic estuary[END_REF][START_REF] Horppila | Sediment nutrients, ecological status and restoration of lakes[END_REF]. These very weak concentrations of SRP, not exceeding 0.4 µmol L -1 , were of the same magnitude order as those measured in the work of David et al (2020a) in a comparable environment. Such weak concentration could suggest a constant upstream to downstream flow of water preventing its accumulation or a rapid withdrawal by autotrophic organisms [START_REF] Masclaux | Planktonic food web structure and dynamic in freshwater marshes after a lock closing in early spring[END_REF].

The P limitation at Marans could induce competition between different phytoplanktonic communities. The high surface area/volume ratio and the active metabolism of nanophytoplankton and picophytoplankton make them more efficient to uptake nutrients than larger cells (microphytoplankton) (Legendre et Rasoulzadegan 1995;[START_REF] Masclaux | Planktonic food web structure and dynamic in freshwater marshes after a lock closing in early spring[END_REF] which could explain the dominance of nanophytoplankton at early spring in this study.

However, these dynamics over time was not entirely consistent to those observed in comparable systems. If a slight spring development of nano-microphytoplankton was observed in David et al. (2020a), the real difference appeared to occur in March, when a strong microphytoplankton development has been observed in other works, in addition to the pico-nanophytoplankton [START_REF] Masclaux | Planktonic food web structure and dynamic in freshwater marshes after a lock closing in early spring[END_REF]. The absence of such a development in this work would suggest that a possible bloom took place between sampling dates, in line with the potential fast phytoplankton response to environmental changes [START_REF] Thyssen | Short-term variation of phytoplankton assemblages in Mediterranean coastal waters recorded with an automated submerged flow cytometer[END_REF].

Indeed, the decrease of nitrate in the water column from March to May (668 to 371 µmol L -1 ) as well as the increase of the dissolved O 2 concentration (Fig 1) could indicate a nutrient uptake by primary producers as observed in April that would have started before [START_REF] Masclaux | Planktonic food web structure and dynamic in freshwater marshes after a lock closing in early spring[END_REF]. Moreover, P limitation in Marans did not allow a strong development with regard to the Redfield ratio [START_REF] Justić | Changes in nutrient structure of riverdominated coastal waters: Stoichiometric nutrient balance and its consequences[END_REF], but shift in such limitation could be possible, attributed to the endogenous recycling of P [START_REF] Wu | Internal cycling, not external loading, decides the nutrient limitation in eutrophic lake: A dynamic model with temporal Bayesian hierarchical inference[END_REF] through diagenetic processes.

Temporal succession of dominant diagenetic processes

Initial condition : winter situation

The high NO 3 -concentration in the water column and the high NO 3 --NO 2 -concentration in January between the interface and 3 cm depth suggested water column as the main nitrate source for the sediment, rather than nitrification in the sediment of low-nitrate environments [START_REF] Middelburg | Denitrification in marine sediments: A model study[END_REF]. The strong NO 3 --NO 2 -gradient in these sediment layers (Fig. 3) led to consider denitrification as the main process for nitrogen removal since the only slight reduction to ammonia (DNRA) occur in the sediment [START_REF] Binnerup | Denitrification, dissimilatory reduction of nitrate to ammonium, and nitrification in a bioturbated estuarine sediment as measured with 15 N and microsensor techniques[END_REF][START_REF] Rysgaard | Nitrification and denitrification in lake and estuarine sediments measured by the 15N dilution technique and isotope pairing[END_REF] Here, denitrification appeared to indirectly inhibit the SRP remobilization, by inhibiting the reduction of iron oxides [START_REF] Wauer | P-immobilisation and phosphatase activities in lake sediment following treatment with nitrate and iron[END_REF][START_REF] Petzoldt | Nitrogen emissions into freshwater ecosystems: Is there a need for nitrate elimination in all wastewater treatment plants?[END_REF][START_REF] Schauser | Effects of nitrate on phosphorus release: Comparison of two Berlin lakes[END_REF][START_REF] Grüneberg | Effects of nitrogen and phosphorus load reduction on benthic phosphorus release in a riverine lake[END_REF] whereas Mn and Fe cycles quickly dominated early diagenetic processes in coastal marine sediment [START_REF] Pastor | Transient early diagenetic processes in Rhône prodelta sediments revealed in contrasting flood events[END_REF]. Indeed, the only trace-state presence of SRP congruent with the non-detection of Fe d and the relatively large stock of iron oxide along the profile (Fig. 3) suggested a low iron oxide reactivity. The high NO 3 concentration in the surficial sediment in winter, plus the probable thin oxic layer, could promote iron oxidation and explain the maximum Fe asc values and the Mn asc peak at the sediment surface (Fig. 3) [START_REF] Hansen | Effects of aluminum, iron, oxygen and nitrate additions on phosphorus release from the sediment of a Danish softwater lake[END_REF][START_REF] Burgin | Beyond carbon and nitrogen: How the microbial energy economy couples elemental cycles in diverse ecosystems[END_REF]. These high oxides concentrations allow a great amount of adsorption sites for PO 4 , which would be immediately uptaken during OM mineralization. The deeper Fe asc values could be explained by a ferrous fraction Fe 2+ associated with iron monosulphide (Kostka and Luther, 1994;[START_REF] Anschutz | La distribution du phosphore inorganique dans les sédiments modernes du golfe de Gascogne[END_REF]. The constant increasing gradient of Mn d could be generated by dissolutive manganese reduction during OM mineralization following the appearance of favourable redox conditions.

Another possible source of dissolved manganese could be the reduction of manganese oxides by reduced iron diffusion from a deeper source. Moreover, the concomitant low P asc (Fig. 3)

and low SRP values in January could suggest that mineralization was still low at this time of the year. Moreover, in considering ƩNH 3 and alkalinity in porewater as a tracer of anaerobic mineralization processes, their slight increase in the last cm seemed to indicate the occurrence of such processes under the sampling area (Fig. 3).

Spring time evolution

In March, the slight presence of nitrate and nitrite in sediment surface indicated that the denitrification became shallower. Overlying waters were more concentrated (Table 1)

probably due to inputs from watershed, suggesting that denitrification was relatively fast, limiting diffusion of nitrate and nitrite downwards. Moreover, the non-accumulation of ƩNH 3 between January and March (Fig. 3) would be consistent with annamox reaction for the nitrogen removal, possible as long as NO 3 -was not totally depleted [START_REF] Canfield | Pathways of organic carbon oxidation in three continental margin sediments[END_REF].

However weak gradients of ƩNH 3 in the surficial sediment made this hypothesis unlikely. The peak of Mn d also suggested a consumption of manganese oxides for mineralization of OM in the first 3 cm (Fig. 3) as evidenced by the decrease in Mn asc at the surface between January and March, which could lead to the first remobilization of phosphorus explained by the slight peak of SRP at 1.5 cm depth. The presence of a first Fe dissolved peak concurrent to the dissolved Mn one also suggested the beginning of the consumption of Fe 3+ hydroxides as a terminal electron acceptor for the degradation of OM. This seemed to be consistent with the decrease of Fe asc in the sediment at that time. This decrease could also be attributed to the non-efficient Fe 2+ reoxidation due to the absence of NO 3 -and the thin oxic layer. The quantities of oxides were certainly still too high to permit a significant diffusive P flux into water column (Table 2) as shown by the high ratio Fe asc /P asc (Fig. 5) and the rather weak mineralization.

The total depletion of nitrate and nitrite in the sediment from April (Fig. 3) and the increase in temperature (Fig. 1) were favourable factors for triggering the chemical and microbial reduction of iron hydroxides [START_REF] Jensen | Phosphorus cycling in a coastal marine sediment, Aarhus Bay, Denmark[END_REF] as OM mineralization pathway.

The sharp increase in Fe d in the porewaters within the first 4 cm in April and the second peak in the first 3 cm in May was an evidence of the iron oxides consumption (Fig. 3). As a result, the phosphate sorbed onto iron hydroxides was released into interstitial waters and seemed to be the main route for SRP mobilization, rather than the dissimilatory sulphate reduction observed in estuarine systems [START_REF] Pan | Cyclical patterns and (im)mobilization mechanisms of phosphorus in sediments from a small creek estuary: Evidence from in situ monthly sampling and indoor experiments[END_REF]. The phosphate mobilization attributed to the 4 first cm in spring was the highest and most superficial in April, according to P asc diminution in the first cm (Fig. 3). A significant SRP efflux was observed at that time (Table 2). 2D gels realization in April allowed to validate and strengthen the link between iron and P cycle observed via core slicing profile within the upper part of the sediment in Marans (Fig. 4) [START_REF] Pan | Effects of tide and season changes on the iron-sulfur-phosphorus biogeochemistry in sediment porewater of a mangrove coast[END_REF]. Moreover, the overall trend of Fe d to increase in depth, as for Mn d , suggested more favourable redox conditions for the OM mineralization via the iron reduction route.

The OM mineralization could also be identified via the sharp increase in alkalinity [START_REF] Sholkovitz | Interstitial water chemistry of the Santa Barbara Basin sediments[END_REF] and mineralization products in sediment porewaters (ƩNH 3 , Si) until May (Fig. 3). The maximum concentrations of P asc in April (Fig. 3) could be explained by SRP sorption on the hydroxides during their diffusion throughout the sediment. The minimum Fe asc /P asc ratio (Fig. 5), following the consumption of metal oxides, could indicate a kind of saturation of the Fe adsorption sites with respect to P [START_REF] Anschutz | Burial efficiency of phosphorus and the geochemistry of iron in continental margin sedime[END_REF]. The excess in mobilisation of SRP in the upper part of the sediment could explain the diffusive flux into the water column in April and May (Table 2).

However, an Fe asc /P asc ratio < 2 showed the non-sufficiency of FeOOH to prevent the diffusion of P into the water column in lake systems [START_REF] Gächter | Why the phosphorus retention of lakes does not necessarily depend on the oxygen supply to their sediment surface[END_REF]. The constant Fe asc /P asc > 3 in this work could suggest the existence of another phase that precipitates with Fe 3+ such as FeS x , which could be consistent with the potential sulphate reduction supposed by the decrease of soluble sulphur compounds over time (Fig 3). The increasing of Fe d in depth in April and May (Figs 3,4) could be coming from the Fe 2+ resolubilisation following abiotic reduction between S 2-and a Fe 3+ source or FeS x dissolution [START_REF] Johnston | Sulfur, iron and carbon cycling following hydrological restoration of acidic freshwater wetlands[END_REF] below the iron hydroxide respiration zone.

The increase of Fe asc concentrations in May in the first 3 cm may also suggest a highly crystallised ferric form that would not react with P and thus explain the non-correlation between Fe asc and P asc in this sediment layer. On this assumption, the probable non-saturation of the surface sites of iron oxides by phosphate ions, (high Fe asc /P asc ratio) could allow Fe 3+ to recrystallize [START_REF] Anschutz | Burial efficiency of phosphorus and the geochemistry of iron in continental margin sedime[END_REF] and explained the slight SRP efflux observed in May (Table 2). Moreover, the more important residence time of dissolved P compared to Fe d in sediment could explain this Fe asc -P asc phase shifting in May [START_REF] Millero | Oxidation kinetics of Fe (II) in sea water[END_REF][START_REF] Neupane | Kinetics of competitive adsorption/desorption of arsenate and phosphate at the ferrihydrite-water interface[END_REF][START_REF] Giles | The mobility of phosphorus, iron, and manganese through the sediment-water continuum of a shallow eutrophic freshwater lake under stratified and mixed water-column conditions[END_REF][START_REF] Smolders | Internal Loading and Redox Cycling of Sediment Iron Explain Reactive Phosphorus Concentrations in Lowland Rivers[END_REF][START_REF] Ding | Internal phosphorus loading from sediments causes seasonal nitrogen limitation for harmful algal blooms[END_REF] and favor P mobilization (Thibault de Chanvalon et al., 2016).

The chemical species distribution presented vertical and horizontal heterogeneity at centimeter scale and could make difficult the interpretation of their layering in the sediment column. This heterogeneity could be directly observed through 2D-DET probes, especially concerning iron and nitrite gels (Fig. 4). These distributions can be impaired by macroorganisms burrows or root systems in the sediment which can strongly influence its geochemistry by promoting O 2 and nitrate inputs [START_REF] Christensen | Comparison of iron, manganese, and phosphorus retention in freshwater littoral sediment with growth of Littorella uniflora and benthic microalgae[END_REF][START_REF] Frederiksen | Oxygen dynamics in the rhizosphere of Zostera marina: A two-dimensional planar optode study[END_REF][START_REF] Cesbron | Simultaneous 2D imaging of dissolved iron and reactive phosphorus in sediment porewaters by thin-film and hyperspectral methods[END_REF][START_REF] Aller | Worm tubes as conduits for the electrogenic microbial grid in marine sediments[END_REF] or impacting the vertical and horizontal distribution of solids [START_REF] Thibault De Chanvalon | Manganese, iron and phosphorus cycling in an estuarine mudflat, Loire, France[END_REF]. Such heterogeneity is also evidenced at the station scale by the variability between core triplicates.

Sediment and biochemical parameters retroaction of the water column

The flux directions from profiles out of the core slicing method were relatively consistent with the surface gradients observed via mean nutrient profiles in the sediment, indicating a good fit between actual and modelled data. The high-resolution method tends to decrease fluxes by reducing the chemical gradients at the interface (Table 3). The heterogeneity in chemical species distribution (evidenced on the nitrite gel) (Fig. 4), makes the placement of the interface critical and explains the flux variations on a small scale (Table 3).

The NO 3 -increase in the water column in March in spite of its ingoing flux toward sediment seemed to be largely explained by allochthonous inputs. The influx in April and May could be in part explained by large nitrogen demand for denitrification in the upper layers of the sediment in winter. With regard to the degradation products of OM, the permanent ƩNH 3 outflows coincided with the progressive loading of the water column until

April. The reverse of this trend in May despite the strong remobilization of ammonium at depth, and the not significant increase in SRP in the water column despite the SRP efflux in March and April could reflect a vegetation pumping of ammonium and P (explaining the downward shift of the P peak during this period, Fig. 3) or even primary producer bloom (Sand-Jensen and Borum, 1990;[START_REF] Rozan | Iron-sulfur-phosphorus cycling in the sediments of a shallow coastal bay: Implications for sediment nutrient release and benthic macroalgal blooms[END_REF]. Moreover, forms of water stratification which can exist in pounds or even marshes [START_REF] Condie | Estimating stratification in shallow water bodies from mean meteorological conditions[END_REF][START_REF] Chimney | Patterns of vertical stratification in a subtropical constructed wetland in south Florida (USA)[END_REF] could counteract the link between fluxes at the interface and water column response, especially through the phosphate availability [START_REF] Crockford | Storm-triggered, increased supply of sediment-derived phosphorus to the epilimnion in a small freshwater lake[END_REF]. However, the shallow depth of the marshes, the site's windward catch and the variations in water level caused by management of marsh water may favour the mixing and thus allow efficient nutrient accessibility throughout the water column.

However, this work strongly suggested the link between the outflow of SRP and the triggering of the nanophytoplankton development in April by reducing P limitation of this system, in addition to the physical parameters being favorable to such development. It highlighted the link between phytoplankton dynamics and mineralization processes by showing congruence between temporal patterns of benthic SRP regeneration and small phytoplanktonic cells (i.e. nano/picophytoplankton) development. Indeed, the evolution of N/P ratios strongly suggests a prevalence of internal recycling of P rather than external input for the development of phytoplankton. Although temporal patterns between benthic regeneration and planktonic production have potentially been shown in coastal environments (Kemp and Boyton, 1984;[START_REF] Cowan | Sediment-water oxygen and nutrient exchanges along the longitudinal axis of Chesapeake Bay: seasonal patterns, controlling factors and ecological significance[END_REF], the high nitrate content at Marans influenced the dynamics of this system, both at the sediment compartment and the water column scale, which could decrease benthos-pelagos interactions [START_REF] Kelly | Benthic-pelagic coupling and nutrient cycling across an experimental eutrophication gradient[END_REF]. Indeed, denitrification processes inhibited for about a month the respiration of metal oxides and thus the potential release of SRP into the water column. This phenomenon suggested a completely different behavior of the sediment compartment in lower nitrate systems, and therefore a different response of the chemical and biological parameters of the water column. However, the conceptual model presented in this work could be deployed in other sites of interest in order to propose new tools for observatories to characterize the aquatic environment quality and predict future trends.

Conclusions and perspectives



The strong phosphate limitation and huge nitrogen content in water column suggested a high nitrate control on the temporal patterns of diagenetic processes and phytoplankton development.



Denitrification processes inhibited phosphates mobilisation in winter by preventing iron hydroxide reduction and seemed to be the main route for nitrogen removal in the sediment.

 Depletion in nitrates in the sediment from early spring triggered iron hydroxide respiration and phosphate mobilisation on the upper part of the sediment. A significant phosphate efflux was observed in April, congruent with nanophytoplankton development.



The multiscale heterogeneity on chemical species distribution highlighted the importance of biotic parameters on the distribution and mobilisation of such species in the sediment.



The complexity in evaluating nutrient flux feedback on the water column leads to the necessity of a better understanding of the nutrient catchment control by autotrophs. Font: Times New Roman, 12.



No color required.

Table 3

Integrated average NO 2 -(probe 1), SRP and Fe d (probe 2) fluxes at the 2D-gel interface as a function of the interface positioning (-1; 0; +1 mm). Font: Times New Roman, 12.

Color required online only. Font: Times New Roman, 12.

Color required online only. Font: Times New Roman, 12.

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Color required online only.

Si/P Si/N Si/N

FontFig. 1 .

 1 Fig. 1. Water column temperature (Temp), conductivity (Cond) and dissolved oxygen (DO) of Marans monitored continuously from January to May. Black arrows corresponded to the field campaigns. The red lines represented the LOWESS smoother of the data (Clevelend, 1981).

Fig. 2 .

 2 Fig. 2. Barplot showing the average chlorophyll a (Chla) biomasses (mean ± sd) by size class (micro: microphytoplankton (> 20 µm), nano: nanophytoplankton (3-20 µm), pico: picophytoplankton (< 3 µm)) at Marans in January, March, April and May 2019. The letters highlighted the significant differences between the dates at fixed size class. The stars highlighted the significant differences between the size classes at fixed date (ANOVA and Tuckey post Hoc tests, α<0.05, p<0.001 : ***).

Fig,Font

  Fig. 3. Dissolved NO 3 -, NO 2 -, Mn d -Mn asc , Fe d -Fe asc , SRP (P d )-P asc , SO 4 (S d ), ƩNH 3 , Si d , and alkalinity (Alk) profiles (mean ± sd) along the sediment column from 0 to 6 cm depth at Marans in January, March, April and May 2019. No data for alkalinity in May. The representation of standard deviation (n=3) indicates the spatial variability in the sediment. The arrows represented the concentration of the element under consideration in the water column. Font: Times New Roman, 12. Color required online only.

Fig. 4 .Font

 4 Fig. 4. 2D pore water distribution of iron (Fe d ) (probe 1 and 2), nitrite (NO 2 ) (probe 1) and soluble reactive phosphorus (SRP) (probe 2) concentrations of cores at Marans in April, and their associated 1D profile (black bold line represented the mean concentration, red lines, standard deviation). The horizontal black line represented the SWI. The color scale on the right of each image represented the concentrations in µmol L -1 . The probes came from two different cores. Font: Times New Roman, 12. Color required online only.

Fig. 5 .

 5 Fig. 5. [Fe] asc /[P asc ] ratio along the sediment column from 0 to 6 cm at Marans in January, March, April and May 2019.

Fig. S1 .

 S1 Fig. S1. Study region and sampling site (red point). The satellite image highlights the presence of crops around the study site (Google Earth). Blue lines represent the principal hydrographic network.

Fig. S2 .

 S2 Fig. S2. Potential nutrient limitation following the Redfield ratios for Marans in January, March, April and May 2019. The black lines represent the limitation thresholds of ratio pairs. The stations in the colored zones are N-limited (blue), P-limited (green) and Si-limited (pink).

Table 2

 2 Monthly in situ monitoring of the coupling between the sedimentary compartment and the water column in a temperate freshwater marsh. O 2 , average nutrient and iron (Fe 2+ ) sediment-water interface fluxes (mean ± sd) at Marans in January, March, April and May 2019. For O 2 , values corresponded to averages of 10 fluxes estimated on one core per date. For nutrients and Fe 2+ , values correspond to averages of 3 fluxes estimated on each individual core. BQL corresponded to values below quantification limit.

	An annual monitoring in such anthropogenic environment, duplicated on less nitrate-
	rich systems would provide a broader view of the importance and prediction of
	sediment-water column coupling in Charente Maritime freshwater marshes and
	temperate marshes in general.

Acknowledgments

This study was supported by the « Ministère de l'Enseignement Supérieur et de la Recherche », the water agencies Loire-Bretagne and Adour-Garonne, conseil général de Charente-Maritime, European Union. This research was supported by a PhD grant from CDA Agglomération of La Rochelle. We acknowledge the AutoAnalyzer plat-form for analysis of nutrients at LIENSs laboratory and ICPMS plat-form for metal analysis at Nantes. Thanks, are extended to the UNIMA (Union des Marais de Charente-Maritime), FMA and stakeholders for their expertise on the Charente-Maritime marshes.


Meriem Tamart
email: meriem.tamart@u-pem.fr

Francois Sarrazin

Elodie Richalot

Julien Matthieu Davy
email: matthieu.davy@univ-rennes1.fr

Julien De Rosny
email: julien.derosny@espci.fr

Diffuse Field Cross-Correlation in a Reverberation Chamber

I. INTRODUCTION

With technological advancement and the use of numerous integrated antennas, research on novel antenna characterization techniques has become a real need. We operate here the technique of diffuse-field cross-correlation to passively characterize the coupling between two antennas. This method was developed in different areas of theoretical and experimental physics and it has been applied to all types of waves (seismic [START_REF] Campillo | Long-Range Correlations in the Diffuse Seismic Coda[END_REF], acoustic [START_REF] Roux | Extracting coherent wave fronts from acoustic ambient noise in the ocean[END_REF], elastic [START_REF] Chehami | E xtraction des fonctions de Green par corrélation de bruit : application pour la détection et l'imagerie de défaut dans des plaques minces réverbérante[END_REF] and electromagnetic [START_REF] Davy | Green's FunctionRetrievalwith-AbsorbingProbesinReverberating Cavities[END_REF]). The use of this technique requires the reception of a diffuse field, meaning that the field received at location is homogeneous and isotropic. This diffuse field can be obtained either by a particular distribution of sources, or by diffusers introduced into the medium (multiple diffusion), or then quite simply by the reverberant nature of the medium.

The work presented here takes advantage of this technique to passively characterize the coupling between two antennas by noise correlation in a Reverberation Chamber (RC). We exploit the properties of the RC to generate a diffuse field using a single source and a mechanical mode stirrer. Averaging over different realizations, the reconstructed signal is expected to converge towards the impulse response between two receiving antennas.

II. RESULTS

A. Experimental Setup

The experiment was made in the RC of the ESYCOM laboratory (2.751m x 2.951m x 2.354m, LUF around 400 MHz) as depicted in

B. Data Processing

Once the measurement data have been recovered, they are processed in the frequency domain because the calculation of the frequency correlation function is faster. First, we eliminate the amplitude information from the Z ij parameters by working only with the information contained in the phase Φ ij of the signals. This allows for equivalent contributions from all frequencies in the working band. The effectiveness of phase information has already been demonstrated in [START_REF] Derode | Ultrasonic pulse compression with one-bit time reversal through multiple scattering[END_REF].

Let Z in (1) be the new parameters taking into account only the phase information, the indices a and b for the receiving antennas, and c for the source antenna. The C ab crosscorrelation is then calculated as in [START_REF] Roux | Extracting coherent wave fronts from acoustic ambient noise in the ocean[END_REF]. The cross-correlation function is then converted in the time domain using an inverse Fourier transformation and it is averaged over the different stirring positions and excitation antenna orientations.

Z ij (f ) = exp (j.Φ ij ) (1)
and :

C ab (f ) = Z ac Z bc * (2)
Figure 2 displays the reconstructed impulse response obtained by calculating the correlation function (red curve) and by direct measurement of the impulse responses between the antennas a and b (blue dash-dotted curve for the causal response and black dotted curve for the anti-causal one). As expected, the results obtained are well superimposed and the inter-correlation function converges towards the average impulse response between the antennas so that we can recover the ballistic wave and the first echoes in the RC.

C. Convergence Analysis

Now, we study the reconstruction quality. The similarity of the reconstructed impulse response C ab calculated for different realizations with the reference impulse response obtaind with 120 stirer positioins, C ab is evaluated via the Pearson correlation coefficient S (3) as a function of the number of available RC realizations. The results displayed in Fig. 3 shows that the correlation results obtained with 60 positions and 120 stirring positions are close, which means that in this case 60 RC realizations are sufficient to converge towards the impulse response between the two AUT. In this communication, we have used the diffuse field cross-correlation technique to passively estimate the impulse response between two antennas using a single source located in an RC. The experimental validation of this technique was exposed through measurements in the RC of the ESY-COM laboratory with two receiving horn antennas. We have presented the measurement results obtained to validate the principle of reconstruction of the impulse response by studying the influence of the number of realizations on the convergence of the field correlation towards the impulse response between the antennas. In a perspective, this work will be deepened to use the technique of cross-correlation in order to characterize the coupling between two antennas in reception which do not have the possibility of being used in transmission.

S = Re f C ab (f ) C * ab (f ) ( f |C ab (f )| 2 df )( f | Cab (f )| 2 df ) (3)

Fig. 1 .

 1 The two antennas under test (AUT) are two horn antennas (a and b) facing each other and separated with a distance of 45 cm. They are placed inside the working volume of the RC and they are used in receiving mode. The source antenna (c) is directed to a mechanical mode stirrer in order to generate a diffuse field. The measurements are carried out at 100001 frequency points uniformly spaced on the frequency band of [1-8] GHz for 120 regularly space stirrer positions over a whole rotation, and two different tilts of the excitation antenna are considered (namely of Xdeg and Ydeg in regard to the horizontal plane).

Fig. 1 .

 1 Fig. 1. Experimental setup in ESYCOM laboratory's RC.

Fig. 2 .

 2 Fig. 2. Calculated cross-correlation function (red) and measured impulse responses (blue and black).

Fig. 3 .

 3 Fig. 3. Similarity Coefficient as a function of the number of equally spaced stirrer positions over a whole rotation


Jean-Baptiste Keck

Georges-Henri Cottet

Eckart Meiburg
email: meiburg@engineering.ucsb.edu

Iraj Mortazavi

Christophe Picard

Double-diffusive sedimentation at high Schmidt numbers: Semi-Lagrangian simulations

When particle-laden freshwater is placed above clear saltwater, double-diffusive sedimentation can arise. Navier-Stokes direct numerical simulations by Burns and Meiburg showed that this process can be dominated by either Rayleigh-Taylor or double-diffusive fingering instabilities. Based on twodimensional simulations, those authors identify a single dimensionless parameter that can be employed to distinguish between these regimes. Here we develop a high-performance semi-Lagrangian computational approach that enables us to extend these high Schmidt number simulations to three dimensions, and to confirm the validity of their proposed scaling law for three-dimensional flows.

I. INTRODUCTION

When a less dense layer of warm and salty water is placed above a denser layer of cold and less salty water, the unstable stratification of the more slowly diffusing salt can trigger double-diffusive fingering instabilities [START_REF] Turner | Salt fingers across a density interface[END_REF][START_REF] Turner | Double diffusive phenomena[END_REF]. An interesting variation of this problem concerns the situation when the more slowly diffusing scalar has a settling velocity associated with it, such as when particle-laden freshwater overlies clear saltwater. These conditions may give rise to double-diffusive sedimentation [START_REF] Green | The importance of double diffusion to the settling of suspended material[END_REF][START_REF] Chen | Particle flux through sediment fingers[END_REF][START_REF] Hoyal | The influence of diffusive convection on sedimentation from buoyant plumes[END_REF][START_REF] Hoyal | Settling-driven convection: a mechanism of sedimentation from stratified fluids[END_REF][START_REF] Maxworthy | The dynamics of sedimenting surface gravity current[END_REF][START_REF] Parsons | Enhanced sediment scavenging due to double-diffusive convection[END_REF][START_REF] Parsons | Hyperpycnal plume formation from riverine outflows with small sediment concentrations[END_REF][START_REF] Manville | Vertical density currents: a review of their potential role in the deposition and interpretation of deep-sea ash layers[END_REF], which can strongly alter the effective settling rate of the sediment. This situation was addressed in the linear stability analyses by [START_REF] Burns | Sediment-laden fresh water above salt water: linear stability analysis[END_REF][START_REF] Yu | Convective instability in sedimentation: Linear stability analysis[END_REF] and in the subsequent fully nonlinear simulations by the same groups of authors [START_REF] Yu | Convective instability in sedimentation: 3-d numerical study[END_REF][START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF]. For constant gradient base states, [START_REF] Alsinan | A settling-driven instability in two-component, stably stratified fluids[END_REF][START_REF] Reali | Layer formation in sedimentary fingering convection[END_REF] furthermore show that the presence of the settling velocity can trigger an entirely new instability mode along with a secondary layering instability, an issue that was addressed in further detail by the recent investigation of [START_REF] Ouillon | Settling-driven large-scale instabilities in double-diffusive convection[END_REF].

Burns and Meiburg [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF] use three-dimensional Navier-Stokes simulations at a diffusivity ratio of 25 to demonstrate the emergence of an interfacial 'nose region' containing both salt and sediment, and whose density can exceed that of the clear salt water below, cf. fig. 1.

Hence, there is the potential for Rayleigh-Taylor instabilities to evolve at the lower boundary of the nose region. Depending on the ratio of the nose height H and the effective salinity interface thickness l s , the authors argue that the flow is dominated by double-diffusive fingering when H/l s < O (0.1), and by Rayleigh-Taylor instabilities when H/l s > O (0.1). (dash-dotted line), salinity (dashed line) and total density (solid line). The particle settling velocity can result in the formation of a nose region with a maximum total density. Unfortunately, numerical resolution requirements limited these three-dimensional simulations to unphysically low values of the Schmidt number Sc = ν 0 /κ s , which reflects the ratio of the kinematic viscosity ν 0 to the diffusivity of salt κ s . The three-dimensional simulations of [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF] employed values of Sc = 0.7 and 7, respectively, as opposed to a realistic value of 700 for salt in water. To access a larger but still somewhat low Schmidt number of 70, the authors resorted to two-dimensional simulations. Results from these ensemble-averaged simulations suggest that the ratio H/l s is a linear function of a single dimensionless grouping that can be interpreted as the ratio of the rates at which sediment flows into and out of the nose region. However, because the proposed scaling law was obtained on the basis of two-dimensional simulations, its validity for three-dimensional flows at realistic Schmidt numbers remains an open question.

To address this issue, the present investigation develops a novel computational strategy based on a semi-Lagrangian approach that allows for the efficient use of hardwarde accelerators, which enables us to extend the fully three-dimensional simulations to more realistic Schmidt number values. After a brief review of the physical problem, we will introduce the semi-Lagrangian computational approach based on the vorticity-velocity formulation of the flow equations. The implementation of this approach on hardware accelerators such as GPUs (Graphics Processing Units) will be discussed, along with performance data that demonstrate its ability to simulate realistic double-diffusive flows at very moderate cost.

Subsequently, we will focus on how the physics of the three-dimensional double-diffusive sedimentation process changes as we increase the Schmidt number.

The results demonstrate that the scaling law proposed by [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF] based on two-dimensional simulations also hold for three-dimensional flows, so that a single dimensionless parameter can predict whether a given flow will be dominated by double-diffusive fingering or by Rayleigh-Taylor instabilities.

II. PHYSICAL PROBLEM

The problem set-up is identical to that of Burns and Meiburg [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF], in that we consider a less dense layer of particle-laden fresh water above a clear, denser later of salt water, as shown in fig. 1. Here x and y refer to the horizontal directions, whereas z indicates the vertical coordinate. Initially, the interface between the two layers is located at z = 0.

We assume that the overall density ρ(x, t) depends linearly on salinity S and sediment concentration C, with associated density expansion coefficients α and γ, respectively. The sediment consists of small monodisperse particles with negligible inertia, which are convected by the superposition of the fluid velocity u(x, t) and the Stokes settling velocity V st . The current investigation aims to explore situations where the effective sediment diffusivity κ c is substantially smaller than that of the salinity κ s . We invoke the Boussinesq approximation, and consequently obtain the set of governing equations as

ρ = ρ 0 + ρ = ρ 0 (1 + αS + γC) , ∇ • u = 0 , ρ 0 ∂u ∂t + (u • ∇) u = ν 0 ∆u -∇P + ρg , ∂S ∂t + (u • ∇) S = κ s ∆S , ∂C ∂t + (u • ∇) C -V st ∂C ∂z = κ c ∆C . ( 1a
) (1b) (1c) (1d) (1e)
Here, ρ 0 is a constant that indicates the density of clear freshwater and t represents time.

We follow the approach of Burns and Meiburg [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF] to render the above equations nondimensional. By taking the curl of the momentum equation, we thus obtain the set of governing dimensionless equations in the form

ρ = R s S + C , ω = ∇ × u , ∇ • u = 0 , ∂ω ∂t + (u • ∇) ω = ∆ω + (ω • ∇)u -∇ × ρ êz , ∂S ∂t + (u • ∇) S = 1 Sc ∆S , ∂C ∂t + (u • ∇) C -V p ∂C ∂z = 1 τSc ∆C , (2a) (2b) (2c) (2d) (2e) (2f)
where ω(x, t) denotes the vorticity.

The governing dimensionless parameters are given as

Dimensionless parameters Terminology

R s = αSmax /γCmax Stability ratio τ = κs /κc Diffusivity ratio Sc = ν 0 /κs Schmidt number V p = Vst /(ν 0 g ) 1 3
Dimensionless settling velocity We impose vanishing wall-normal derivatives of the salinity at those walls, along with a noflux condition for the sediment concentration at the top wall and a vanishing wall-normal derivative at the bottom wall. We remark that the simulations always terminate before any sediment plumes encounter the bottom wall.

The flow is at rest initially, with salinity and sediment concentration fields in the form of smooth error function profiles as shown on horizontally averaged densities in fig. 1.

Initial fields are given as

u 0 (x, y, z) = 0 , ω 0 (x, y, z) = 0 , C 0 (x, y, z) = 1 2 1 + erf z -δ(x, y) l 0 , S 0 (x, y, z) = 1 -C 0 (x, y, z) . (3a) (3b) (3c) (3d)
We choose the initial profile thickness as l 0 = 1.5, which represents less than one per cent of the total domain height. A small horizontal random perturbation δ(x, y) of amplitude ±l 0 /20 is imposed on the interface location, in order to trigger the growth of unstable modes.

Only stability ratios R s > 1 are considered, so that the system is initially gravitationally stable.

III. SEMI-LAGRANGIAN COMPUTATIONAL APPROACH

The computational approach is based on the velocity-vorticity form of the Navier-Stokes equations in the Boussinesq approximation, and a discretization of the advection-diffusion equations for the vorticity, salinity and sediment concentration fields by a semi-Lagrangian particle method, as will be described in the following.

Particle methods for advective systems discretize the advected quantities via Lagrangian computational elements, referred to as 'particles' (not to be confused with the sediment particles) that move with their respective local velocities [START_REF] Raviart | An analysis of particle methods[END_REF]. The use of particle methods for simulating variable-density flows dates back to at least [START_REF] Anderson | A vortex method for flows with slight density varaition[END_REF], who considered the case of a vortex sheet coinciding with a density jump. To handle general flows while ensuring accuracy at all times, the Lagrangian particles can be remeshed on a regular grid via interpolation kernels with strong conservation properties [START_REF] Cottet | Vortex methods: theory and practice[END_REF]. By remeshing at each time step, we obtain a forward, conservative, semi-Lagrangian method [START_REF] Cottet | TVD remeshing schemes for particle methods[END_REF][START_REF] Cottet | High order semi-lagrangian particle methods for transport equations: numerical analysis and implementation issues[END_REF]. These methods were applied to various flows including porous media [START_REF] Mimeau | Direct numerical simulations of three-dimensional flows past obstacles with a Vortex penalization method[END_REF][START_REF] Mimeau | Passive control of the flow around a hemisphere using porous media[END_REF]. In the present investigation we employ particles to solve equations (2d), (2e) and (2f), where the quantities carried by the particles are ω, S and C. If V = (ω, S, C), one advective step can be written as

V n+1 i = j V n j Λ x i -x n+1 j ∆x .
Here the superscripts n and n + 1 refer to two successive time steps, x j , V j respectively denote the grid points along with the values of V at those points, x n+1 j is the location at time n + 1 of the particle initialized at x j at time n, and Λ denotes the interpolation kernel used to remesh the particles. The regular Cartesian grid with mesh size ∆x on which the particles are remeshed is also used to compute at the end of each time step the velocity from the vorticity, and to calculate vortex stretching, diffusion and the right hand side of (2d).

Following [START_REF] Cottet | TVD remeshing schemes for particle methods[END_REF], we use a direction-by-direction splitting for the advection of particles and the following one-dimensional formula for the remeshing kernel Lagrangian particle advection is performed by an explicit fourth order Runge-Kutta method combined with trilinear velocity interpolation, while remeshing is carried out with the formula given above. This accounts for the advective part of (2d), (2e) and (2f). The stretching term in the right hand side of equation (2d) is discretized by fourth order explicit central finite differences. The diffusion terms present on the right hand sides of equations (2d), (2e) and (2f) are solved through an unconditionally stable Fourier-based spectral method adapted to each individual boundary conditions. A similar spectral solver is used to recover vorticity from velocity at the end of each iteration.

Λ(x) =                          1 - 5
For performance reasons, the directional splitting approach used in the semi-Lagrangian particle method to deal with the advection part of the system is further generalized to all finite-difference based operators for vortex stretching and the buoyancy term. These operators can be readily split directionally, as they do not introduce spatial cross-derivatives.

Between the treatments of the individual directions, the data are rearranged in memory so that the next direction to be computed becomes contiguous in memory. Data permutation improves the achieved memory bandwidth of the numerical routines and is particularly beneficial when implementing numerical methods with low arithmetic intensity [START_REF] Carvalho | The gap between processor and memory speeds[END_REF][START_REF] Ofenbeck | Applying the roofline model[END_REF]. Narrow dotted lines represent (a) the location z = -V p t moving with the Stokes settling velocity; and (b) the least-squares fit of the interface thickness to a κ t 1/2 diffusive behavior.

IV. IMPLEMENTATION ON HARDWARE ACCELERATORS, PERFORMANCE

DATA AND VALIDATION GPU implementations of vortex particle methods for 2D flow simulations were first proposed in [START_REF] Rossinelli | Vortex methods for incompressible flow simulations on the gpu[END_REF][START_REF] Rossinelli | GPU accelerated simulations of bluff body flows using vortex particle methods[END_REF][START_REF] Rossinelli | MRAG-I2D: Multi-resolution adapted grids for remeshed vortex methods on multicore architectures[END_REF]. The present numerical method was implemented within a Python-based library dedicated to 3D flow simulations with semi-Lagrangian particle methods, on various types of accelerators such as GPUs [START_REF] Etancelin | Multi-CPU and multi-GPU hybrid computations of multi-scale scalar transport[END_REF]. Within this library, an iteration of the solver is described as a sequence of discretized mathematical operators that are solved successively.

Given a single compute node, each operator can run either on the CPU or on an available coprocessor, so that multiple operators can run concurrently in a hybrid CPU-GPU fashion. This sequence of operators is obtained from the topological sort of a direct acyclic graph of operators described by the user [START_REF] Keck | Numerical modelling and High Performance Computing for sediment flows[END_REF]. Here we use a single GPU configuration so that each iteration of the solver is driven by the CPU, while the computations take place on the GPU by following the OpenCL standard [START_REF] Tompson | An introduction to the opencl programming model[END_REF]. All variables such as velocity u, vorticity ω, salinity S and sediment concentration C are discretized on Cartesian grids.

Our library deploys an automatic performance-tuning strategy for most operators to ensure performance portability [START_REF] Zhang | Improving performance portability in opencl programs[END_REF]. Spectral operators are implemented by using the clFFT library, which makes use of its own auto-tuning techniques. Nonperiodic boundary conditions are handled via sine and cosine transforms, implemented on top of clFFT by using efficient O (N ) input pre-processing and output post-processing capabilities of the library. On a GPU with 32GB of dedicated memory, the above implementation is able to solve single-precision, two-dimensional problems with discretizations up to 32768x16384 (536M grid elements), and three-dimensional configurations up to 1537x512x512 (402M grid elements). Those three-dimensional simulations take 10.9s per time step on a single Nvidia Tesla V100 GPU. By comparison, a quad socket Intel Xeon E7-8860 v4 CPU-only platform needs 24 minutes per time step for the same OpenCL implementation coupled to the Intel MKL-FFT library for the spectral transforms and memory mapping. Threedimensional problems addressed below typically require about 1500 time steps, which take around 5 hours in our single GPU configuration.

Semi-Lagrangian particle methods have been validated in [START_REF] Lagaert | Hybrid spectral-particle method for the turbulent transport of a passive scalar[END_REF][START_REF] Etancelin | Model coupling and hybrid computing for multi-scale CFD[END_REF] for passive scalar transport at high Sc-values, and in [START_REF] Keck | Numerical modelling and High Performance Computing for sediment flows[END_REF] for the classical periodic, incompressible Taylor-Green vortex benchmark at Re = 1, 600. To demonstrate that the solver captures the sedimentation dynamics of interest here, fig. 2 gives a comparison of the flow statistics with results obtained by [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF]. In this figure horizontally averaged flow profiles and interface locations and thickness are computed following [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF]. Horizontally averaged profiles are computed by integrating each scalar field of interest •(x, t) in the horizontal plane

• (z, t) = xmax x min ymax y min •(x, y, z, t)dxdy .
We can then track the scalar interfaces for the salinity and sediment concentration fields as follows. Since the error function solves the laminar diffusion equation with constant diffusivity, we compute the least-squares fit of the above horizontally averaged profiles to an error function. For a given time t,

1. We fit C (z, t) against C f it (z, t) = 0.5 1 + erf z -z c (t) l c (t) 2. We fit S (z, t) against S f it (z, t) = 0.5 1 -erf z -z s (t) l s (t)
This gives z c , l c , z s and l s as functions of time, with the interface position z • defined as the location where • f it = 0.5, and the interface thickness given by l • . In our implementation this fit is performed every few time steps and takes as input the last known interface location z * • and thickness l * • , with the initial values set to z c = z s = 0 and l c = l s = l 0 . Figure 2 indicates that the temporal evolution of these four quantities agrees well with the results of [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF]. Further validation results are discussed in [START_REF] Keck | Numerical modelling and High Performance Computing for sediment flows[END_REF].

V. RESULTS

For purely double-diffusive fingering instabilities we would expect to observe a nearly symmetric flow pattern, with slender, low-salinity, clear fluid fingers rising upwards, and corresponding narrow, salty, particle-laden fingers descending from the interfacial region.

On the other hand, Rayleigh-Taylor instabilities centered at the density overhang associated with the lower nose boundary should result in strong, downward moving plumes, while any upward propagating plumes should be capped by the stable density stratification at the upper nose boundary. Figure 3 A detailed analysis of these three-dimensional simulation data demonstrates that the horizontally averaged salinity and sediment concentration profiles can be approximated quite well by error function profiles. We take the inflection point of the error function profile as the effective interface location, and the width of the error function profile as the effective interface thickness. In this way, we can evaluate effective interface locations and the thicknesses for salinity and sediment concentration as functions of time. Similar to the two-dimensional simulation profiles in fig. 2, the profiles from the three-dimensional simulations indicate that the center of the salinity interface rises upwards, while that of the sediment concentration descends, so that an effective nose region forms that contains both salt and sediment. The error function fits allow us to evaluate both the nose height H = z s -z c and the thickness l s of the salinity interface as functions of time. While both H and l s individually grow with time, after an initial transient phase the ratio H/l s tends towards a constant value that depends on the governing dimensionless parameters of the flow. Fig. 4 compares results for this ratio H/l s of the nose height to the salinity interface thickness, as a function of the dimensionless grouping V p Sc 1/2 /R s . Data from the present, three-dimensional simulations (plus-signs) are seen to closely match the two-dimensional results of [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF]. This confirms that the scaling law proposed by those authors based on two-dimensional simulations remain valid for fully three-dimensional flows.

VI. CONCLUSION

We have developed a semi-Lagrangian computational approach suitable for being efficiently implemented on hardware accelerators. In this way, we were able to conduct fully resolved three-dimensional simulations of double-diffusive sedimentation at high Schmidt numbers. Consistent with the earlier two-dimensional results of [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF], the present threedimensional simulations demonstrate the emergence of an interfacial nose region that contains both salt and sediment. While the height H of the nose region and the thickness l s of the salinity interface individually grow with time, the ratio H/l s approaches a constant value that depends only on the governing dimensionless parameters. Depending on whether this ratio is smaller or larger than O(0.1), the sedimentation process is dominated either by double-diffusive fingering or by Rayleigh-Taylor instabilities. The scaling law proposed by [START_REF] Burns | Sediment-laden fresh water above salt water: nonlinear simulations[END_REF] for H/l s as function of V p Sc 1/2 /R s on the basis of two-dimensional simulations, is seen to remain valid for three-dimensional flows.

As a next step, we plan to implement the semi-Lagrangian computational approach on hardware platforms with multiple GPUs that have recently become widely available. This will allow us to investigate considerably larger systems involving stratified turbulence and double-diffusive phenomena at increasingly realistic Schmidt numbers, in the context of applications in oceanography, limnology and astrophysics.

FIG. 1 :

 1 FIG. 1: Particle-laden freshwater above clear saltwater: (a) Problem setup, governing equations and boundary conditions; (b) Sketch of density profiles showing the particle

  three-dimensional computational domain of size Ω = [0, 128] × [0, 128] × [-128, 64] in the x-, y-and z-directions. Periodic boundary conditions are applied in the horizontal directions, along with slip walls at the top and bottom boundaries, Γ t and Γ b .

FIG. 2 :

 2 FIG. 2: Comparison of horizontally averaged interface location z • (left) and thickness l • (right), between the present study (solid lines) and the investigation of [14] (dashed lines).

  shows the temporal evolution of the C = 0.5 contour of the sediment concentration field, for V p = 0.04, R s = 2, and τ = 25. Results are presented for Schmidt numbers Sc = 3.5, 7, 14 and 28 (from left to right), and for times t = 100, 125 and 150 (from top to bottom). For the lowest Sc-value (left column) these figures indeed show slender, up-and downward moving fingers of roughly equal strengths, indicative of a double-diffusive fingering. At the largest value of Sc (right column), on the other hand, we observe plumes moving only in the downward direction. The concentration contour exhibits a nearly flat top, where upward plume motion is suppressed by the stable density stratification associated with the upper boundary of the nose region. This confirms the scenario that we had outlined above, in that the flow transitions from being primarily double-diffusive at low Sc-values to being dominated by Rayleigh-Taylor instabilities at large values of Sc.

FIG. 3 :

 3 FIG. 3: C(x, y, z) = 0.5 contours of the sediment concentration, for Schmidt numbers 3.5, 7, 14, 28 (left to right) and times 100, 125, 150 (top to bottom). V p = 0.04, R s = 2, τ = 25.

FIG. 4 :

 4 FIG. 4: Ratio H/l s of nose height to salinity interface thickness as function of V p Sc 1/2 /R s . The two-dimensional results of [14] (•, , and symbols) closely match the present, three-dimensional results ( symbols) for Sc ∈ {3.5, 7, 14, 28}, V p = 0.04, R s = 2, τ = 25.

ACKNOWLEDGMENTS

The authors would like to thank Peter Burns for providing useful additional numerical implementation details. EM gratefully acknowledges support through NSF grants CBET-1803380 and OCE-1924655, as well as by the Army Research Office through grant W911NF-18-1-0379. JBK and GHC gratefully acknowledge support by Programme Investissement d'Avenir under grant ANR-17-CE23-0024-01. Some of the computations have been performed using the GRICAD infrastructure (https://gricad.univ-grenoble-alpes. fr), which is partially supported by grant ANR-10-EQPX-29-01.

JBK performed the numerical computations and data postprocessing. Technical details of the numerical solver were handled by JBK and CP. EM provided most of the physical input and discussed the results. GHC and IM delivered their expertise in semi-Lagrangian methods. GHC, IM and CP supervised the findings of this work. All authors provided critical feedback and contributed to the final manuscript.