Mijail M Bullaín Galardis

Raul C López Sánchez

Fatoumata Fall

Bettina Eichler- Löbermann

Ludovic Pruneau

Amadou M Bâ

Growth and physiological responses of ectomycorrhizal Coccoloba uvifera (L.) L. seedlings to salt stress

Keywords: Ectomycorrhiza, Seagrape, Photosynthesis, Chlorophyll, Water status

We studied the effect of the ectomycorrhizal (ECM) fungus Scleroderma bermudense on growth, photosynthesis and transpiration rates, chlorophyll fluorescence and content, K/Na and Ca/Na homeostasis, and water status of two provenances of Coccoloba uvifera L. (named also seagrape) seedlings exposed to four levels of salinity (0.0, 54.7, 164.1 and 273.5 mM NaCl equivalent to 0.02, 5, 15 and 25 dS m -1 , respectively). The results indicated that S. bermudense improved the salt tolerance in seagrape seedlings. There were no differences in terms of growth performance, nutritional and physiological functions between the two ECM seagrape provenances in response to salt stress. The reduction of the Na concentration and the increase of K and Ca favored a higher K/Na and Ca/Na ratio, respectively, in the tissues of the ECM seedlings. Furthermore, the beneficial effects of ECM symbiosis on the photosynthetic and transpiration rates, chlorophyll fluorescence and content, stomatal conductance and water status resulted in the improved growth performance of the seagrape provenances exposed to salt stress. From an ecological point of view, seagrape in symbiosis with S. bermudense may be used in the development of ornamental plantings and coastal windbreaks along beaches and roadsides in Cuba.

Introduction

Worldwide, 800 million hectares of soil are affected by salinity [START_REF] Munns | Mechanisms of salinity tolerance[END_REF]. This represents more than 6% of the terrestrial globe and approximately 20% of the total cultivated area. Land affected by salinity could increase to 50% of the total cultivated area by 2050 [START_REF] Courel | Suelos salinos y sódicos[END_REF]. Soil salinization is the excess accumulation of salts in the soil due to natural and anthropogenic causes. This major constraint to cultivation predominates in arid and semi-arid regions as well as in coastal regions, due to the inappropriate irrigation with a low-quality groundwater, the recovery of salts by capillarity and the sea level rise related to climate change [START_REF] Gamalero | The use of microorganisms to facilitate the growth of plants in saline soils[END_REF]Gopalakrisnan and Kumar, 2021). High concentrations of salts in soils cause an adverse impact on biodiversity, agricultural production and sustainable development [START_REF] Tester | Na⁺ tolerance and Na⁺ transport in higher plants[END_REF].

It is well known that salinity provokes osmotic and ionic stresses in plants [START_REF] Safdar | A review: Impact of salinity on plant growth[END_REF]. To cope with salt stress, halophytic plants develop mechanisms to reduce the toxicity of Na in the cytoplasm through restricting Na uptake, increasing the efflux of Na, and compartmentalizing Na in the vacuole [START_REF] Yang | Elucidating the molecular mechanisms mediating plant salt-stress responses[END_REF]. Plants also develop mechanisms to increase K in the cytoplasm to maintain a suitable K/Na and Ca/Na ratio and to prevent cellular damages and nutrient deficiency [START_REF] Munns | Mechanisms of salinity tolerance[END_REF]. The K ion is the major osmotically solute in the guard cells of open stomata and controls the influx of CO2 and the efflux of water vapor and O2, which can directly influence both photosynthetic and transpiration rates. The Ca ion acts as a second messenger in salt stress signaling [START_REF] Munns | Mechanisms of salinity tolerance[END_REF].

Several strategies have been established to overcome salt-stress problems such as a selection of salt-tolerant plants [START_REF] Larcher | Plants under stress[END_REF], a desalination of soil by leaching excessive salts [START_REF] Zeng | Soil salt leaching under different irrigation regimes: Hydrus-1D modelling analysis[END_REF] and a mycorrhizal inoculation [START_REF] Chen | Arbuscular mycorrhizal symbiosis alleviates salt stress in black locust through improved photosynthesis, water status, and K+/Na+ homeostasis[END_REF]. There is considerable evidence that arbuscular mycorrhizal (AM) fungi can improve plant growth and nutrition in soils subject to a range of saline stress [START_REF] Chen | Arbuscular mycorrhizal symbiosis alleviates salt stress in black locust through improved photosynthesis, water status, and K+/Na+ homeostasis[END_REF][START_REF] Chandrasekaran | Impact of arbuscular mycorrhizal fungi on photosynthesis, water status, and gas exchange of plants under salt stress-a meta-analysis[END_REF][START_REF] Shi-Chu | Improving plant growth and alleviating photosynthetic inhibition from salt stress using arbuscular mycorrhizal fungi in alfalfa seedlings[END_REF][START_REF] Heidarianpour | Positive effects of co-inoculation with Rhizophagus irregularis and Serendipita indica on tomato growth under saline conditions, and their individual colonization estimated by signature lipids[END_REF]. The AM fungi can also enhance photosynthetic and transpiration rates, chlorophyll fluorescence and content, stomatal conductance, antioxidant metabolism, K/Na and Ca/Na ratio, and water status of the plants under salt stress [START_REF] Lin | Effects of arbuscular mycorrhizal fungi on the growth, photosynthesis and photosynthetic pigments of Leymus chinensis seedlings under salt-alkali stress and nitrogen deposition[END_REF][START_REF] Chandrasekaran | Impact of arbuscular mycorrhizal fungi on photosynthesis, water status, and gas exchange of plants under salt stress-a meta-analysis[END_REF][START_REF] Li | Arbuscular mycorrhizal fungi alleviate drought stress in C3 (Leymus chinensis) and C4 (Hemarthria altissima) grasses via altering antioxidant enzyme activities and photosynthesis[END_REF]. In contrast, little is known about the role of the ectomycorrhizal (ECM) symbiosis in enhancing salt tolerance of trees [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF][START_REF] Bois | Growth, compatible solute and salt accumulation of five mycorrhizal fungal species grown over a range of NaCl concentrations[END_REF][START_REF] Shi | Ectomycorrhizal fungi reduce the light compensation point and promote carbon fixation of Pinus thunbergii seedlings to adapt to shade environments[END_REF]Guerrero-Gala ń et al., 2019;[START_REF] Zwiazek | Role of urban ectomycorrhizal fungi in improving the tolerance of lodgepole pine (Pinus contorta) seedlings to salt stress[END_REF].

ECM symbiosis develops a fungal mantle around root tip from which an extracellular hyphal network facilitates the absorption of water and nutrients by the fungus. It develops also a Hartig net between cortical cells where fungus and host plant interact: fungus provides water and nutrients to the host plant, which, in return, provides carbohydrates to the fungus [START_REF] Bâ | Ectomycorrhizal symbiosis of tropical African trees[END_REF]. It is also a key factor for improved tolerance of woody plants to salt stress through the exclusion and compartmentation of Na in an extracellular hyphal network, improving K for plant growth and the water status of plants through the activation of both fungal and plant aquaporins. This leads to a higher K/Na ratio in ECM plants than in non-ECM plants under salt stress [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF]Guerrero-Gala ń et al., 2019).

Coccoloba uvifera (L.) L. (Polygonaceae), also named seagrape, is a woody plant often subject to high levels of salinity along the Atlantic, Caribbean and Pacific coasts of the American tropics and subtropics (Parrota, 1994). It is an important ECM tree for edible fruits and mushrooms, ornamental plantings and coastal windbreaks along Caribbean beaches and roadsides [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF][START_REF] Séne | Ectomycorrhizal fungal communities of Coccoloba uvifera (L.) L. mature trees and seedlings in the neotropical coastal forests of Guadeloupe (Lesser Antilles)[END_REF][START_REF] Séne | A pantropically introduced tree is followed by specific ectomycorrhizal symbionts due to pseudo-vertical transmission[END_REF]. In a previous study, [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF] showed that Na and Cl uptake together with a concomitant increase of P and K absorption and a higher water status in ECM plants may be important salt-alleviating mechanisms for one provenance of seagrape seedlings inoculated with Scleroderma bermudense under salt stress. However, growth responses of tree seedlings to ECM inoculation can vary from one provenance to another [START_REF] Bâ | Growth responses of Afzelia africana Sm. seedlings to ectomycorrhizal inoculation in a nutrient-deficient soil[END_REF], and some key physiological mechanisms (i.e. stomatal conductance, transpiration and photosynthetic rates, gas exchange, chlorophyll fluorescence and content, water status) by which S. bermudense can improve the growth of seagrape seedlings under salt stress were not taken into account by this study. Here, we analyzed some physiological parameters such as photosynthesis and transpiration rates, chlorophyll fluorescence and content, K/Na and Ca/Na homeostasis, and water status of the plants to assess the effect of the established S. bermudense on plant growth of two provenances of seagrape under salt stress, in order to improve understanding of the mechanisms regarding the alleviation of salt toxicity in ECM seagrape. This ECM fungus was selected as inoculum because it dominated ECM fungal communities in the seagrape coastal forests [START_REF] Séne | Ectomycorrhizal fungal communities of Coccoloba uvifera (L.) L. mature trees and seedlings in the neotropical coastal forests of Guadeloupe (Lesser Antilles)[END_REF] and was the most tolerant ECM fungi to high NaCl concentrations [START_REF] Bâ | Alleviation of salt stress by Scleroderma bermudense in Coccoloba uvifera seedlings[END_REF].

Materials and methods

Plant material and germination

Mature fruits of C. uvifera were collected from Las Coloradas beach (provenance LC) (19º55'31,9" N, 77º41'14,1" W) and Punta Tomate beach (provenance PT) (21º. 16'16.6'' N, 76º31'18.5'' W) in Cuba. The fruits were washed with tap water to eliminate surrounding pulp. The seeds were air dried at room temperature for 72 hours on cardboard trays. To break dormancy, the seeds were scarified in 95% sulfuric acid (H2SO4) for two hours and every 20 minutes the container was shaken to achieve the uniform action of the acid on the surface of the seeds [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF]. Then, they were rinsed with abundant distilled water, kept in water for 24 hours and transferred to the sterile substrate incubating them for 5 to 8 days at room temperature before transplanting. The substrate is a sandy soil collected along the Yao River (20º15'32.3" N,76º45'19.3" W) located in the town known as La China in Cuba. The sandy soil was sterilized at 121 °C and 1.2 kg/cm 2 pressure in an autoclave. The nutrient contents of the heat-sterilized sandy soil were as follows (ppm): 65 K, 25.4 Na, 80.19 Ca, 29.57 Mg, 3.1 Olsen-P, pH (H 2 O) 7.5 and electrical conductivity (EC) 0.84 dS m -1 . The sandy soil was analyzed in the soil laboratory of the Ministry of Agriculture of Granma province (Cuba).

Fungal material, inoculation and experimental design

Mature sporocarps of Scleroderma bermudense, a gasteromycetous fungus, were collected in sandy soil from under a stand of C. uvifera along the Las Coloradas beach. An herbarium reference voucher UG-04 was given to the sporocarps. Spores were aseptically extracted directly from the gleba using a spatula and kept in sealed boxes at 4°C until use. The spore inoculum consisted of 0.12 g fresh weight of spores per plant.

Inoculation consisted of introducing 0.12 g of spores in a drain hole bored 2 cm above the bottom of the surface of sterile sandy soil. Then, pre-germinated seeds with approximately 2 cm long tap roots were planted in the same hole, one per polyethylene bag (long 20 cm, diameter 12 cm), with 1 kg of previously moistened and sterilized sandy soil.

The experiment was set up as completely 2x2x4 factorial design consisting of two provenances (LC and PT) of C. uvifera, two ECM inoculation treatments (inoculated and non-inoculated) and four salinity levels (0.0, 54.7, 164.1 and 273.5 mM NaCl equivalent to 0.02, 5, 15 and 25 dS m -1 , respectively) representative to the salt concentrations of sandy soils along the beach. In all, 16 treatments were compared with ten replicates per treatment.

During the first two months, seedlings were well irrigated with tap water without NaCl to achieve adequate mycorrhization. Then, over the course of 1 month, seedlings were subjected to salinization at the four salt levels by adding NaCl until reaching the levels of predetermined salinity for each treatment. For this purpose and to avoid osmotic shock, a volume of 25 ml from 2 dS m -1 of NaCl solution was gradually added to the soil every 3 days to increase the initial EC from 0.02 (0 mM NaCl) to 5, 15 and 25 dS m -1 during 4 weeks. The sandy soil was leached with tap water every week to reduce salt accumulation. A volume of 25 ml of fresh salt solution was added immediately after each leaching to keep a constant NaCl concentration in the soil. Salinity was controlled weekly by the method of estimating salinity [START_REF] Torres | Medición de pH y Conductividad Eléctrica en Sustratos[END_REF], using a portable conductivity meter (HANNA. HI 9033, Rumania). [START_REF] Chandrasekaran | Impact of arbuscular mycorrhizal fungi on photosynthesis, water status, and gas exchange of plants under salt stress-a meta-analysis[END_REF] define soil salinity in three categories according to USDA Natural Resources Conservation Service: low soil salinity has an EC≤4 dS m -1 , moderate soil salinity ranged from 4 to 8 dS m -1 , and higher than 8 dS m -1 was high salinity.

The experiment was carried out over 6 months (from September 2018 to February 2019) in nursery conditions at temperatures of 25ºC-30ºC with a period of exposure to sunlight of approximately 12 hours in the open air (civil time). Seedlings were harvested after they had been grown under salt stress conditions for 3 months.

Photosynthesis and gas exchange parameters

The photosynthetic rate (A), transpiration rate (E), stomatal conductance (gs) and substomatal CO2 (Ci) were non-destructively measured using a portable open flow gas exchange system (ADC BioScientific. LCpro-SD, United Kingdom) from 9:00 to 11:00 a.m. (civil time). The fourth fully expanded leaf of each plant (ten plants per treatment, n = 10) was used for the measurements. Photosynthesis and gas exchange parameters were measured at a temperature of 35.5 ºC, a relative humidity of 79%, a CO2 concentration of 414.54 ppm and an active photosynthetic irradiation of 1,950 µmol m -2 s -1 .

Chlorophyll fluorescence and content

The minimal fluorescence (Fo), maximum fluorescence (Fm) and variable fluorescence (Fv= Fm-Fo) were measured with a portable chlorophyll fluorimeter (Hansatech Instruments. RS232, United Kingdom) according to the manufacturer's instructions. For this, the fourth fully expanded leaf of each plant (ten plants per treatment, n = 10) was used for the measurements. Before measuring, the leaves were dark-adapted for 30 min using the clips provided with the kit. After darkening the leaves, the Fo was recorded and a saturating pulse of radiation (3500 µmol m -2 s -1 with the help of three light-emitting diodes of 650 nm) was given for (from 10 µs to 1 s) to determine Fm. The data obtained from Fo, Fm and Fv permeated to determine two biophysical parameters that describe the photochemistry of PII: the maximum quantum yield of photosystem II (Fv/Fm) and the performance index on absorption basis (PIABS) (Table 5).

The chlorophyll content was determined on the fourth fully expanded leaf of each plant (ten plants per treatment, n = 10) using SPAD-502 chlorophyll meter (Konica-Minolta. Japan) from 9:00 to 11:00 a.m. (civil time) according to the manufacturer's instructions. The reading is made in an arbitrary unit that is proportional to the leaf chlorophyll concentration [START_REF] Jiang | A correlation analysis on chlorophyll content and SPAD value in tomato leaves[END_REF].

Water status and plant biomass

The foliar (Ψwf) and the xylem (Ψwx) water potential were measured at midday (civil time) with a Scholander pressure chamber (PMS. 615, USA) [START_REF] Scholander | Sap pressure in vascular plants: negative hydrostatic pressure can be measured in plants[END_REF] at the end of the harvest. One and a half hours before measuring the xylem potential, the leaves were covered with hermetic ziplock bags made of high-density polyethylene with a reflective cover. The leaf area was determined with the area meter (ADC BioScientific. AM.350, United Kingdom).

At the end of the harvest, the leaves, stems and roots of seedlings (ten seedlings per treatment, n=10) were separated to measure the length of the stem, collar diameter and dry weight (5 days at 80°C). The relative water content (RWC) was determined after measuring the fresh mass (FM), dry mass (DM) and turgid mass (TM) in leaf discs (diameter 15 mm) from the fourth fully expanded leaf of each plant (ten plants per treatment) was used for the measurement. The turgid mass was the mass of the leaves after their saturation in water at 4 °C in the dark. The relative water content was calculated according to [START_REF] Morgan | Osmoregulation and water stress in higher plants[END_REF] and using the following formula:

RWC = [(FM -DM)/(TM -DM)] x100.
The relative mycorrhizal dependency (RMD) was calculated at a given NaCl level [START_REF] Plenchette | Growth response of several plants species to mycorrhiza in a soil of moderate P fertility. I. Mycorrhizal dependency under field conditions[END_REF] using the following formula: RMD = [Biomass of ECM plants -Biomass of non-ECM plants] ×100/Biomass of ECM plants.

Ectomycorrhizal colonization

To determine the percentage of mycorrhization, a sample of ten lateral roots (per seedling and per treatment) was washed gently and dispersed in a petri dish containing water, and the number of root tips of ECM roots and non-colonized roots was counted under a stereomicroscope at ×100 magnification. ECM colonization was evaluated [(number of ECM roots/total number of roots) ×100] and confirmed by microscope (×400) examination of root tips to determine the presence of a fungal mantle and Hartig net, according to [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF].

Concentrations of Na, K and Ca

The roots, stems and leaves (ten plants per treatment, n = 10) were dried at 80 ºC until constant weight. They were crushed in a blade mill to particles smaller than 0.5 mm and 0.2 g were digested in a mixture of HClO4/HNO3 (v/v 1:5) diluted with 100 mL of distilled water.

To determine the concentration of Na, K and Ca, each sample was analyzed in an atomic absorption spectrophotometer (PinAAcle. 900T. USA) in the Department of Plant Sciences, Faculty of Agriculture, University of Rostock (Germany).

Statistical analysis

Prior to the ANOVA analysis, Kolmogorov-Smirnov and Shapiro Wilk's test, and Levene and Bartlett´s test were performed for normality and homogeneity of variances, respectively.

All data were subjected to two-way (inoculation and salinity) or three-way (provenance, inoculation and salinity) analysis of variance, and mean values were compared using Tukey´s test with the InfoStat software version 2008 [START_REF] Di Rienzo | InfoStat (version[END_REF]. The Pearson's correlation coefficients between dependent variables were determined by the same software.

Results

Root colonization and plant growth

There was not a significant effect of provenances on ECM colonization (P= 0.886) (S1 Table ).

Therefore, we analyzed the two factors (inoculation and salinity) and their interactions (Figure 1). The two factors (inoculation and salinity) and their interactions had a significant effect (P ≤ 0.05) on mycorrhization (Figure 1). The ectomycorrhizas of the S. bermudense were characterized by a white and smooth mantle and abundant mycelial strands. The ECM colonization by the S. bermudense varied from 45.8% to 81.0% and from 48.4% to 80.2% depending on the provenances of the seagrape. However, the extent of ECM colonization was significantly affected by salt stress for each provenance, respectively (Figure 1).

The three factors (provenance, inoculation and salinity) and their interactions had significant effects (P ≤ 0.05) depending on studied growth parameters (Table 1). There was no significant effect of the provenance on these parameters. However, the two factors (inoculation and salinity) and their interactions were significant for all parameters (Table 1). The effects of salinity on seagrape seedlings growth after inoculation with S.

bermudense did not vary with the seagrape provenances (Table 1). The ECM colonization by S. bermudense enhanced the seedlings growth regardless of the salt level. Stem and root length, stem diameter, number of leaves and leaf area were much higher for ECM than for non-ECM plants regardless of salt level (Table 1). Furthermore, the total biomass of ECM plants increased by 68% compared with the non-inoculated plants whatever the provenance of the seagrape and salt level. Although growth parameters declined in both ECM and non-ECM plants as salinity increased, all of them remained superior to that of non-ECM plants at each salt level. The ECM dependency on seagrape seedlings with S. bermudense showed an increasing trend with increasing NaCl levels (Table 1). However, the ECM dependence did not differ significantly among salt treatments. This result implies that although salt stress affected ECM colonization, S. bermudense enhanced the salinity tolerance of seedlings regardless the seagrape provenances.

Photosynthesis and gas exchange

The three factors (provenance, inoculation and salinity) and their interactions had significant effects (P ≤ 0.05) depending on parameters studied (Table 2). There was no significant effect of the provenance on these parameters except for gs (P=0.001). However, the two factors (inoculation and salinity) and their interactions were significant for all parameters (Table 2). In the absence of salt, A, gs, E and Ci parameters were higher for ECM than non-ECM plants, suggesting an enhancing of photosynthesis and transpiration in seagrape seedlings inoculated with S. bermudense regardless of the provenances (Table 2). In the presence of salt, there was a significant decrease of A, gs, E and Ci in the ECM and in the non-ECM seagrape provenances. However, at each level of salinity, A, gs, E and Ci parameters were higher for ECM than non-ECM plants.

Chlorophyll fluorescence and content

The three factors (provenance, inoculation and salinity) and their interactions had significant effects (P ≤ 0.05) depending on parameters studied (Table 3). There was no significant effect of the provenance on these parameters except for SPAD values (P<0.001). However, the two factors (inoculation and salinity) and their interactions were significant for all parameters (Table 3). In the absence of salt, basal fluorescence (Fo) values were significantly lower for ECM than non-ECM plants, whereas values of maximun fluorescence (Fm) were higher for ECM than non-ECM plants (Table 3). A Similar trend occurred for Fo and Fm values in ECM and non-ECM plants in the presence of salt (Table 3). In the absence of salt, Fv/Fm, PIABS and SPAD values were higher for ECM than non-ECM plants (Table 3).

Increased salinity reduced the Fv/Fm and PIABS in non-ECM plants more than those in ECM plants. However, in ECM plants, increased salinity did not cause a significant decrease in Fv/Fm and PIABS parameters. Values of Fv/Fm and PIABS were higher in ECM than in non-ECM plants under saline conditions.

Plant water status

The three factors (provenance, inoculation and salinity) and only the interactions of the two factors (inoculation and salinity) had significant effects on all parameters (Table 4).

The RWC values in the leaves were higher in ECM than non-ECM plants at all salinity levels regardless of seagrape provenances (Table 4). The RWC declined with exposure to salinity (Table 4). In non-ECM plants, the reduction in the RWC was 24% in provenance LC and 27% in provenance PT, whereas in ECM plants, the reduction was 20% in provenance LC and 22% in provenance PT.

In the absence of salt, leaf and xylem water potentials were higher for ECM than non-ECM plants regardless the provenances, suggesting an improvement of water status of seagrape seedlings inoculated with S. bermudense (Table 4). Salt stress however, reduced leaf and xylem water potentials in both ECM and non-ECM plants (Table 4).

3.5. Concentrations of Na, K, Ca, K/Na and Ca/Na ratio in leaves, stems and roots There was not a significant effect of provenances on Na, K, Ca, K/Na and Ca/Na ratio in leaves, stems and roots (S1 table ). Therefore, we analyzed the two factors (inoculation and salinity) and their interactions. They had a significant effect (P ≤ 0.05) on Na in leaves, stems and roots (Figure 2). In the absence of salt, there was less Na in leaves, stems and roots of ECM than non-ECM plants (Figure 2). Under saline conditions, non-ECM plants accumulated more Na than ECM plants at a given NaCl level regardless of seagrape provenance. The seagrape provenances did not differ regarding Na content of ECM plants (Figure 2). Na content was apparently higher in the leaves than in the stems and roots for all salt treatments.

The two factors (inoculation and salinity) and their interactions had significant effects (P ≤ 0.05) on K (Figures 3) and K/Na (Figure 4) in leaves, stems and roots. The K content in leaves, stems and roots was higher in ECM than in non-ECM plants regardless of the salinity level and the seagrape provenance (Figure 3). As a consequence of this, the K/Na ratio increased in the leaves, stems and roots of ECM plants compared to non-ECM plants (Figure 4).

The two factors (inoculation and salinity) and their interactions had a significant effect (P ≤ 0.05) on Ca (Figures 5) and Ca/Na (Figures 6) in leaves, stems and roots (Figures 5 and6).

There was more Ca in the leaves, stems and roots in ECM than non-ECM plants regardless of salinity (Figure 5). Therefore, the ratio Ca/Na ratio in leaves, stems and roots was higher in ECM than non-ECM plants (Figure 6).

Relations between datasets

Most of the morphological and physiological variables were significant positive or negative correlated between them (S2 table). For example, A was positively correlated to Fv/Fm (r= 0.89, P ≤ 0.01), Fv/Fo (r= 0.85, P ≤ 0.01), PI (r= 0.83, P ≤ 0.01) and total biomass (r= 0.35, P ≤ 0.05) (S2 table). The parameter E was also positively correlated to RWC (r= 0.59, P ≤ 0.01), and negatively correlated to the foliar Ψwf (r= -0.94, P ≤ 0.01) and Ψwx (r= -0.94, P ≤ 0.01). The C. uvifera seedlings showed a negative correlation between Na leaves and K contents in leaves (r= -0.81, P ≤ 0.01), shoots (r= -0.47, P ≤ 0.01) and roots (r= -0.76, P ≤ 0.01). There was also a negative correlation between Na accumulation in leaves and K/Na ratios in leaves (r= -0.92, P ≤ 0.01), shoots (r= -0.49, P ≤ 0.01) and roots (r= -0.70, P ≤ 0.01).

Discussion

In the present study, salinity caused a reduction in ECM colonization, which is in line with a previous study on seagrape [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF]. The reduction of the ECM colonization by the S. bermudense with increasing NaCl levels, tended to reduce total biomass of the ECM seedlings compared with the non-ECM plants, regardless of seagrape provenances. As a consequence of this, ECM dependency apparently increased with increasing NaCl levels. This suggests that ECM plants mitigated salt stress in both seagrape provenances and indicates a high symbiosis efficiency of S. bermudense once it was established. Indeed, it is well known that the ECM symbiosis plays a major role in helping tree seedlings to survive under salt conditions [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF][START_REF] Bois | Growth, compatible solute and salt accumulation of five mycorrhizal fungal species grown over a range of NaCl concentrations[END_REF][START_REF] Zwiazek | Role of urban ectomycorrhizal fungi in improving the tolerance of lodgepole pine (Pinus contorta) seedlings to salt stress[END_REF]Guerrero-Gala ń et al., 2019;[START_REF] Thiem | Halotolerant fungi stimulate growth and mitigate salt stress in Alnus glutinosa Gaertn[END_REF]. A similar effectiveness of the ECM symbiosis was also reported in other tree species, including Pinus spp. [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF][START_REF] Bois | Growth, compatible solute and salt accumulation of five mycorrhizal fungal species grown over a range of NaCl concentrations[END_REF][START_REF] Zwiazek | Role of urban ectomycorrhizal fungi in improving the tolerance of lodgepole pine (Pinus contorta) seedlings to salt stress[END_REF]Guerrero-Gala ń et al., 2019). For example, [START_REF] Zwiazek | Role of urban ectomycorrhizal fungi in improving the tolerance of lodgepole pine (Pinus contorta) seedlings to salt stress[END_REF] found that Pinus concorta was dependent on ECM fungi that could be helpful in alleviating effects of NaCl in urban soils. [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF] had also shown an increase of ECM dependency with an increasing salt level within in one seagrape provenance from Guadeloupe.

The parameters A, gs, E and Ci were significant positive correlated with the improved total biomass in the leaves, stems and roots of ECM plants exposed to saline conditions. Indeed, the presence of ECM fungus, seagrape provenances had higher stomatal conductance, transpiration and photosynthetic rates, and sub-stomatal CO 2 concentration in the mesophyll than those of non-ECM plants under salt stress. This result suggests that ECM fungus can elevate the photosynthetic ability of seagrape seedlings thanks to an improving of the stomatal conductance and gas exchange capacity under salt stress. Our study is in consistence with work of [START_REF] Shi-Chu | Improving plant growth and alleviating photosynthetic inhibition from salt stress using arbuscular mycorrhizal fungi in alfalfa seedlings[END_REF] which showed that increasing salt concentration led to a significant decrease of A via a decrease in the gs, which was less important in AM than non-AM alfalfa. The decrease of gs due to the stomatal closure is often related to the water status of plants [START_REF] Mohamed | Stomata and xylem vessels traits improved by melatonin application contribute to enhancing salt tolerance and fatty acid composition of Brassica napus L. plants[END_REF]. Under water deficit conditions due to salt stress, the plants reduce transpiration rate more in non-ECM than ECM plants by closing stomata to reduce water loss. Furthermore, in this study, a higher gs in the ECM vs non-ECM plants under salt stress was positively correlated to the increased of CO 2 diffusion through the stomata and water absorption. As a result, the photosynthesis and water status may be better in ECM plants. Measurements of RWC, Ψwf and Ψwx are considered informative on water status and transpiration rate in plants under salt stress [START_REF] Larcher | Plants under stress[END_REF][START_REF] Chen | Arbuscular mycorrhizal symbiosis alleviates salt stress in black locust through improved photosynthesis, water status, and K+/Na+ homeostasis[END_REF]. It is also well known that the accumulation of salts in the root zone causes an increase in the osmotic potential and, consequently, the water potential decreases in the rhizosphere [START_REF] Augé | Hydraulic conductance and water potential gradients in squash leaves showing mycorrhiza-induced increases in stomatal conductance[END_REF]. Here, despite their higher evaporative leaf surface, the ECM plants had higher RWC, Ψwf and Ψwx than non-ECM plants regardless of salinity and seagrape provenances, suggesting that the extensive hyphal extension developed by S. bermudense allowed higher water absorption and hydraulic conductivity in roots of seagrape even when water potential is low [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF][START_REF] Augé | Hydraulic conductance and water potential gradients in squash leaves showing mycorrhiza-induced increases in stomatal conductance[END_REF][START_REF] Lehto | Ectomycorrhizas and water relations of trees: a review[END_REF].

Our study also indicated that the response of ECM plants to the ratio of Fv/Fm, PIABS and chlorophyll content was greater than those in non-ECM plants, suggesting that the ECM symbiosis could enhance the chlorophyll concentration and fluorescence of seagrape leaves, which is consistent with the results of other studies (Sheng et al., 2018). Indeed, plants trap photons in chlorophylls of PSII from sunlight and use them to split H2O into O2, H + and eduring light reaction phase of photosynthesis. Moreover, salt stress decreases the concentration of chlorophyll and inhibits the electron transport chain to produce ATP and NaDPH [START_REF] Shi-Chu | Improving plant growth and alleviating photosynthetic inhibition from salt stress using arbuscular mycorrhizal fungi in alfalfa seedlings[END_REF]. Similarly, to our study, previous works showed that the AM fungus-plant symbiosis with a greater chlorophyll than that in non-AM, presented also higher rates of photosynthesis under salt stress [START_REF] Elhindi | The impact of arbuscular mycorrhizal fungi in mitigating salt-induced adverse effects in sweet basil (Ocimum basilicum L.)[END_REF].

Here, results show that ECM plants kept a higher K/Na ratio in leaves, shoots and roots and confirmed that K competed for the absorption site of Na on the cell membrane [START_REF] Garcia | The role of mycorrhizal associations in plant potassium nutrition[END_REF]. Therefore, ECM symbiosis can facilitate K whereas preventing Na absorption and translocation in shoot and leaves of seagrape seedlings to maintain a high cytosolic K/Na ratio which is a key feature of plant salt tolerance. Our results were in consistence with works showing that the ECM symbiosis enhances the growth of host plants by promoting uptake water and nutrients by the host under salt stress [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF][START_REF] Bois | Growth, compatible solute and salt accumulation of five mycorrhizal fungal species grown over a range of NaCl concentrations[END_REF]Guerrero-Gala ń et al., 2019). In this respect, concentrations of Na clearly decreased and K content increased in ECM-plants compared to non-ECM plants, concomitantly with an increasing of the ratio K/Na ratio to maintain higher cell turgor [START_REF] Bandou | The ectomycorrhizal fungus Scleroderma bermudense alleviates salt stress in seagrape (Coccoloba uvifera L.) seedlings[END_REF][START_REF] Chen | Arbuscular mycorrhizal symbiosis alleviates salt stress in black locust through improved photosynthesis, water status, and K+/Na+ homeostasis[END_REF]. Moreover, K ion is involved in regulating stomatal opening and osmotic potential in the vacuoles. The results here showed that competition of K due to colonization by S. bermudense may induce the decrease of Na, thus enhancing salt tolerance of ECM plants. Alongside with Ca, K is also an essential nutrient for plant growth [START_REF] Evelin | Mitigation of salinity stress in plants by arbuscular mycorrhizal symbiosis: current understanding and new challenges[END_REF]. Indeed, in the present study, the ECM plants had higher concentrations of K and Ca than non-ECM plants particularly under salt stress conditions. The Ca ion can also act as a second messenger in salt stress signaling [START_REF] Munns | Mechanisms of salinity tolerance[END_REF][START_REF] Evelin | Mitigation of salinity stress in plants by arbuscular mycorrhizal symbiosis: current understanding and new challenges[END_REF]. However, the absorption of Ca could be limited through a competition with an elevated Na concentration in the rhizosphere under salt stress [START_REF] Evelin | Mitigation of salinity stress in plants by arbuscular mycorrhizal symbiosis: current understanding and new challenges[END_REF]. Less Ca absorption and translocation than Na within the plant leads to a decrease in the Ca/Na ratio in salt stressed plants. [START_REF] Evelin | Contribution of Glomus intraradices inoculation to nutrient acquisition and mitigation of ionic imbalance in NaCl-stressed Trigonella foenumgraecum[END_REF][START_REF] Evelin | Mitigation of salinity stress in plants by arbuscular mycorrhizal symbiosis: current understanding and new challenges[END_REF] suggested that the Ca/Na ratio could also be increased by AM colonization and also suggested that Ca/Na ratio could be an indicator of salt tolerance in plants. This statement is consistent with the present study, which showed a higher Ca/Na ratio in ECM than non-ECM plants. However, the mechanism involved is not well known and needs further works [START_REF] Evelin | Mitigation of salinity stress in plants by arbuscular mycorrhizal symbiosis: current understanding and new challenges[END_REF].

In conclusion, the results indicate that the ECM fungus S. bermudense improved the salt tolerance in seagrape seedlings. There was no difference in terms of growth performance or nutritional and physiological functions between the two ECM seagrape provenances in response to salt stress. The reduction of the Na concentration and the increase of K and Ca favored a higher K/Na and Ca/Na ratio, respectively, in the tissues of the ECM seedlings.

Additionally, the beneficial effects of ECM symbiosis on the photosynthetic and transpiration rates, chlorophyll fluorescence and content, stomatal conductance and water status resulted in the improved growth performance of the seagrape provenances exposed to salt stress. From an applied point of view, transplanting of the ECM seagrape to such degraded sites not only may benefit the individual plant but, more importantly, may result in the development of ornamental plantings and coastal windbreaks along beaches and roadsides in Cuba. Different letters indicate significant differences for P ≤ 0.05. Values are the means±SD (n=10).

Figure 1 .

 1 Figure 1. Effect of ectomycorrhizal inoculation and salinity on the mycorrhization of seedlings from each provenance (Las Coloradas and Punta Tomate) of C. uvifera. Bars topped with different letters are significant different according to the Tukey HSD test at P ≤ 0.05. Vertical bars indicate standard deviations of mean values (n = 10).

Figure 2 .

 2 Figure 2. Effect of ectomycorrhizal inoculation and salinity on Na concentration in leaves, stems and roots of seedlings from each provenance (Las Coloradas and Punta Tomate) of C. uvifera. Bars topped with different letters are significant different according to the Tukey HSD test at P ≤ 0.05. Vertical bars indicate standard deviations of mean values (n = 10).

Figure 3 .

 3 Figure 3. Effect of ectomycorrhizal inoculation and salinity on K concentration in leaves, stems and roots of seedlings from each provenance (Las Coloradas and Punta Tomate) of C. uvifera. Bars topped with different letters are significant different according to the Tukey HSD test at P ≤ 0.05. Vertical bars indicate standard deviations of mean values (n = 10).

Figure 4 .

 4 Figure 4. Effect of ectomycorrhizal inoculation and salinity on K/Na concentration in leaves, stems and roots of seedlings from each provenance (Las Coloradas and Punta Tomate) of C. uvifera. Bars topped with different letters are significant different according to the Tukey HSD test at P ≤ 0.05. Vertical bars indicate standard deviations of mean values (n = 10).

Figure 5 .

 5 Figure 5. Effect of ectomycorrhizal inoculation and salinity on Ca concentration in leaves, stems and roots of seedlings from each provenance (Las Coloradas and Punta Tomate) of C. uvifera. Bars topped with different letters are significant different according to the Tukey HSD test at P ≤ 0.05. Vertical bars indicate standard deviations of mean values (n = 10).

Figure 6 .

 6 Figure 6. Effect of ectomycorrhizal inoculation and salinity on Ca/Na concentration in leaves, stems and roots of seedlings from each provenance (Las Coloradas and Punta Tomate) of C. uvifera. Bars topped with different letters are significant different according to the Tukey HSD test at P ≤ 0.05. Vertical bars indicate standard deviations of mean values (n = 10). 592

Table 1 .

 1 Effect of provenance, inoculation and salinity on morphological variables in C. uvifera seedlings.

	Provenance of C. uvifera	Inoculation treatment	Salinity (dS m -1 )	Stem length (cm)	Stem diameter (mm)	Number of leaves	Root length (cm)	Total biomass (g)	Leaf area (cm 2 )	ECM dependency (%)
	Las Coloradas	Non-mycorrhizal	0.02 5	16.10±1.30fg 13.68±0.84cdef 3.00±0.56bc 4.20±0.45cd 27.90±2.75abc 3.30±0.24cd 5.60±0.55efg 26.80±1.79ab	2.76±0.22cd 100.36±31.48bc -2.41±0.45bc 72.38±21.45c -
			15	11.04±1.01abc	2.31±0.21a	3.40±0.55bc 30.40±2.79bcd	1.26±0.30a	41.77±6.05d	-
			25	9.04±0.43a	2.17±0.23a	2.40±0.55ab 27.04±1.39ab	0.97±0.27a	41.97±8.09d	-
		Mycorrhizal 0.02	19.42±2.10h	4.94±0.21g	7.40±0.55i	41.00±3.59e	4.04±0.60e	237.71±43.05a	30.62±8.95abc
			5	18.20±2.02gh	4.11±0.14ef	6.80±0.45hi	40.00±2.45e	3.06±0.37d	142.75±22.95ab 19.29±21.89a
			15	14.20±1.45def	3.64±0.22de 5.80±0.45fgh 37.20±4.48de	2.27±0.21b	97.21±3.24abc	43.86±14.84bc
			25	12.58±1.11bcde 3.03±0.13c	5.20±0.45def 37.30±3.53de	1.94±0.37b	81.26±17.31bc	47.56±20.39bc
	Punta de Tomate	Non mycorrhizal	0.02 5	15.08±1.59ef 12.96±0.53bcde 3.33±0.18cd 5.00±0.00def 31.00±2.15bcd 3.92±0.14cd 5.40±0.55ef 30.50±1.41bcd	2.78±0.16cd 99.92±32.03bc 2.41±0.31bc 71.71±21.36c	--
			15	10.92±0.04ab	2.43±0.19ab 3.20±0.45abc 28.10±3.65abc	1.25±0.22a	40.90±5.08d	-
			25	8.98±0.36a	1.92±0.19a	2.20±0.45a	20.82±3.19a	0.97±0.14a	42.03±7.93d	-
		Mycorrhizal 0.02	18.92±1.43h	4.60±0.26fg	7.00±0.00i	35.60±4.52cde	3.98±0.32e	239.02±42.39a	29.88±7.17abc
			5	17.92±1.40gh	4.03±0.50ef	6.60±0.55ghi 36.90±4.42de	3.01±0.28d	141.36±20.18ab 19.82±10.62ab
			15	13.90±0.96def	2.98±0.14cd 5.60±0.55efg 34.50±6.43bcde 2.37±0.23bc 96.84±2.92abc	46.99±10.91bc
			25	11.92±0.47bcd	2.93±0.18bc 4.60±0.55de 33.84±2.45bcde 1.93±0.26b	80.52±17.25bc	48.77±8.95c
	Provenance			P= 0.091	P= 0.059	P= 0.162	P= 0.013	P= 0.995	P= 0.914	P= 0.739
	Inoculation			P< 0.001	P < 0.001	P < 0.001	P < 0.001	P < 0.001	P < 0.001	P < 0.001
	Salinity			P < 0.001	P < 0.001	P < 0.001	P= 0.002	P < 0.001	P < 0.001	P < 0.001
	Provenance x Inoculation		P= 0.933	P= 0.009	P= 0.064	P= 0.048	P= 0.954	P= 0.980	P= 0.739
	Provenance x Salinity		P= 0.901	P= 0.995	P= 0.103	P= 0.164	P= 0.958	P= 0.999	P= 0.977
	Inoculation x Salinity		P= 0.123	P= 0.003	P= 0.023	P= 0.182	P < 0.001	P < 0.001	P < 0.001
	Provenance x Inoculation x Salinity	P= 0.852	P= 0.335	P= 0.263	P= 0.047	P= 0.635	P= 0.702	P= 0.572

Table 2 .

 2 Effect of provenance, inoculation and salinity on stomatal conductance (gs), photosynthetic (A) and transpiration (E) rates and sub-stomatal CO2 (Ci) in leaves of C. uvifera seedlings.

	Provenance	Inoculation	Salinity	gs	E	A	Ci
	of C. uvifera	treatment	(dS m -1 )	(mol m -2 s -1 )	(mol m -2 s -1 )	(µmol m -2 s -1 )	(vpm)
	Las	Non-	0.02	0.07±0.02cd	0.96±0.22cd 1.26±0.09d	374.80±1.23e
	Coloradas	mycorrhizal	5	0.05±0.01bc	0.54±0.14ab 0.86±0.07c	345.80±1.03c
			15	0.04±0.01ab	0.49±0.02ab 0.50±0.03b	325.10±0.88b
			25	0.04±0.0048ab 0.43±0.04a 0.22±0.02a	274.50±1.08a
		Mycorrhizal 0.02	0.16±0.02f	2.61±0.25e 1.96±0.02e	432.10±1.85g
			5	0.13±0.0048ef 1.93±0.06de 1.78±0.13e	431.00±1.25g
			15	0.07±0.01cde 1.07±0.09cd 0.89±0.05c	394.80±1.23f
			25	0.06±0.0032c 0.71±0.05bc 0.57±0.01b	361.90±1.60d
	Punta de Tomate	Non-mycorrhizal	0.02 5	0.06±0.02c 0.04±0.01ab	0.95±0.26cd 1.25±0.11d 0.52±0.16ab 0.83±0.08c	376.10±1.20e 346.40±0.84c
			15	0.03±0.01a	0.47±0.07a 0.51±0.05b	323.80±0.79b
			25	0.03±0.01a	0.41±0.06a 0.19±0.05a	273.90±0.88a
		Mycorrhizal 0.02	0.15±0.02f	2.60±0.27e 1.91±0.02e	431.70±1.57g
			5	0.12±0.01def 1.91±0.08de 1.79±0.15e	430.80±1.03g
			15	0.06±0.01bc	1.05±0.09cd 0.87±0.09c	396.10±0.74f
			25	0.06±0.01cd	0.69±0.08bc 0.54±0.05b	363.10±0.88d
	Provenance			P= 0.001	P= 0.476	P= 0.335	P= 0.231
	Inoculation			P < 0.001	P < 0.001	P < 0.001	P < 0.001
	Salinity			P < 0.001	P < 0.001	P < 0.001	P < 0.001
	Provenance x Inoculation		P= 0.076	P= 0.930	P= 0.777	P= 0.231
	Provenance x Salinity		P= 0.702	P= 0.998	P= 0.911	P= 0.876
	Inoculation x Salinity		P < 0.001	P < 0.001	P < 0.001	P < 0.001
	Provenance x Inoculation x Salinity	P= 0.257	P= 0.510	P= 0.462	P= 0.032
	Different letters indicate significant differences for P ≤ 0.05. Values are the means±SD
	(n=10).

Table 3 .

 3 Effect of provenance, inoculation and salinity on minimal fluorescence (Fo), maximum fluorescence (Fm), maximum quantum yield of photosystem II (Fv/Fm), performance index on absorption basis (PIabs) and chlorophyll quantification (SPAD) in leaves of C. uvifera seedlings.

	Provenance of C. uvifera	Inoculation treatment	Salinity (dS m -1 )	Fo (bits)	Fm (bits)	Fv/Fm (bits)	Fv/Fo (bits)	PI abs	Chlorophyll quantification (SPAD)
	Las Coloradas	Non-mycorrhizal	0.02 5	310.40±14.47c 374.20±32.94d	1496.00±72.95c 1442.80±10.69bc 0.74±0.03bc 2.88±0.35bc 1.29±0.11b 0.79±0.02cde 3.84±0.43de 2.05±0.0034c 54.26±0.77e 51.78±0.84d
			15	378.60±34.22d	1341.60±38.00b	0.72±0.03b	2.57±0.41ab 1.00±0.08b	46.20±0.66b
			25	383.20±25.06d	1018.40±64.90a	0.62±0.05a	1.68±0.34a	0.28±0.02a	40.02±0.29a
		Mycorrhizal 0.02	270.20±7.85ab	1809.60±124.06e 0.85±0.01fg 5.71±0.67hi	5.12±0.80f	60.76±0.69g
			5	284.80±19.61abc 1649.00±10.79d	0.83±0.01efg 4.81±0.44fgh 3.94±0.20e	56.86±0.66f
			15	288.40±8.56abc 1509.20±7.50c	0.81±0.01ef	4.24±0.18efg 3.41±0.20e	54.00±0.50e
			25	301.60±12.74bc 1412.40±46.96bc 0.79±0.02cde 3.69±0.34cde 2.81±0.15d	48.64±0.69c
	Punta de Tomate	Non-mycorrhizal	0.02 5	306.20±15.02bc 1491.20±73.21c 370.80±31.52d 1438.60±11.37bc 0.74±0.02bcd 2.90±0.35bcd 1.30±0.10b 0.80±0.02de 3.89±0.45ef 2.10±0.05c	53.80±0.77e 50.89±0.96d
			15	376.00±33.30d	1336.60±39.78b	0.72±0.03b	2.59±0.42ab 1.01±0.08b	45.40±1.01b
			25	380.80±24.49d	1014.60±65.23a	0.62±0.05a	1.68±0.34a	0.25±0.02a	39.26±0.69a
		Mycorrhizal 0.02	262.00±7.38a	1806.80±123.13e 0.86±0.02g	5.91±0.68i	5.08±0.78f	59.83±0.95g
			5	281.80±18.43abc 1644.20±10.62d	0.83±0.01efg 4.85±0.42gh 3.90±0.23e	55.92±0.71f
			15	286.60±9.61abc 1501.80±7.56c	0.81±0.01ef	4.25±0.20efg 3.44±0.34e	53.31±0.51e
			25	298.80±12.74abc 1403.00±35.60bc 0.79±0.01cde 3.70±0.31cde 2.78±0.20d	47.78±0.54c
	Provenance			P= 0.451	P= 0.687	P= 0.781	P= 0.631	P= 0.925	P < 0.001
	Inoculation			P < 0.001	P < 0.001	P < 0.001	P < 0.001	P < 0.001	P < 0.001
	Salinity			P < 0.001	P < 0.001	P < 0.001	P < 0.001	P < 0.001	P < 0.001
	Provenance x Inoculation		P= 0.932	P= 0.950	P= 0.980	P= 0.817	P= 0.831	P= 0.576
	Provenance x Salinity		P= 0.990	P= 0.999	P= 0.998	P= 0.964	P= 0.996	P= 0.914
	Inoculation x Salinity		P= 0.001	P < 0.001	P < 0.001	P= 0.522	P= 0.017	P < 0.001
	Provenance x Inoculation x Salinity	P= 0.741	P= 0.407	P= 0.543	P= 0.867	P= 0.625	P= 0.358

Acknowledgements

We thank two reviewers and editor for their helpful improvement in the text. We thank also Mr Montoya RY and Mr Salcedo YM for their helping in laboratory and to collect fruits of C. uvifera and sporocarps of S. bermudense in the field. This work was funded by PHC CARLOS J. FINLAY 2017-2018.


Sandra Geara
email: sandra.geara@emse.fr

Sylvain Martin

S Adami

W Petry

J Allenou

B Stepnik

Olivier Bonnefoy

A new SPH density formulation for 3D free-surface flows

Keywords: SPH, Free Surface, Density correction, Curvature, Surface Tension

Smoothed Particle Hydrodynamics (SPH) is a mesh-free Lagrangian numerical method that was first introduced independently in 1977 by Lucy [START_REF] Lucy | A numerical approach to the testing of the fission hypothesis[END_REF] and Gingold and Monaghan [START_REF] Gingold | Smoothed particle hydrodynamics: theory and application to non-spherical stars[END_REF] to solve astrophysical problems. Since 1977, the SPH method has been developed and improved significantly to model a wide range of problems, especially in fluid dynamics. This method models a continuous fluid by discretizing it with a series of fluid particles that move through space and time. The continuity of the fluid and its properties are recovered by the spatial convolution of the physical properties of the particles by a smoothing kernel function.

For free surface flows, the standard SPH approximations suffer from the lack of full support. This mainly affects the density estimation and, consequently, potential pressure oscillations. Many techniques were proposed in the literature to overcome this problem.

The first attempt to apply SPH to free surface flow was introduced by Monaghan [START_REF] Monaghan | Simulating Free Surface Flows with SPH[END_REF] for the simulation of wave run-up and breaking in shallow water. Monaghan used the continuity equation to calculate the change rate of density as a function of the velocity gradient. He also introduced the XSPH velocity variant algorithm to improve the particle distribution. In XSPH, the velocity of each particle is modified to take into account the average velocity of all nearby particles. A second common approach to smooth out the pressure oscillations is to apply a density filter. Colagrossi and Landrini [START_REF] Colagrossi | Numerical simulation of interfacial flows by smoothed particle hydrodynamics[END_REF] applied a periodic re-initialization of the density field based on the Moving Least Squares Approach (MLS) proposed by Belytschko et al. [START_REF] Belytschko | On the completeness of meshfree particle methods[END_REF] and Dilts [START_REF] Dilts | Moving least-squares particle hydrodynamics II: conservation and boundaries[END_REF]. This proposed kernel correction ensures a consistent interpolation of the density field. Bonet and Lok [START_REF] Bonet | Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations[END_REF] considered this kernel correction unsuitable for explicit schemes. Instead, they proposed a simpler correction of the density field by considering a constant correction instead of a linear one or what is known as the Shepard filter. These density corrections methods should be applied every n time steps, where n is typically chosen between 20 and 50. Bonet and Lok [START_REF] Bonet | Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations[END_REF] also introduced another correction technique based on the modification of the kernel gradient by a correction matrix. Various schemes for applying the kernel gradiant correction can be found in the literature [START_REF] Shao | An improved SPH method for modeling liquid sloshing dynamics[END_REF][START_REF] Zhang | Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two-and three-29 dimensional spaces[END_REF]. A combination of these two techniques, i.e. the constant kernel correction and the kernel gradient correction, is also possible. Furthermore, Molteni and Colagrossi [START_REF] Molteni | A simple procedure to improve the pressure evaluation in hydrodynamic context using the SPH[END_REF] and Ferrari et al. [START_REF] Ferrari | A new 3D parallel SPH scheme for free surface flows[END_REF] introduced an artificial density diffusion term to smooth out the numerical noise of the pressure field.

The intensity of this numerical correction is to be defined according to the problem at hand. More recently, Seo et al. [START_REF] Seo | The particle-attached element interpolation for density correction in smoothed particle hydrodynamics[END_REF] proposed to correct the density of each particle based on shape functions adopted from the interpolation scheme of FEM. Several review papers discussing free-surface flows in SPH have been published, amongst other [START_REF] Gomez-Gesteira | State-of-the-art of classical SPH for free-surface flows[END_REF][START_REF] Violeau | Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future[END_REF].

All these procedures give good results, some of them are more suitable than the others depending on the application. However, these corrections can be insufficient when dealing with surface tension force. In this case, the force should be applied on surface particles at each time step, thus it is important to estimate the density of these particles accurately.

For applications where the surface tension force is not the dominant force, Calderon et al. [START_REF] Calderon-Sanchez | A geometric formulation of the Shepard renormalization factor[END_REF] developed very recently a geometrical formulation that improves the Shepard correction coefficient. In their work, the dimensionality of the problem is reduced by defining the kernel in such a way that the volume integral is cast into a surface integral. Despite the promising results, this method still contains complex and computationally expensive numerical evaluation [START_REF] Kostorz | A semi-analytical boundary integral method for radial functions with application to Smoothed Particle Hydrodynamics[END_REF].

In this work, we propose a new density evaluation method. The idea behind this method is to calculate analytically a coefficient that represents the weight of the missing particles from the support domain and use it to correct the density summation for particles near the free surface. This new correction factor depends on the distance to the surface as well as on the curvature of the free surface. A variant of this method has originally been proposed by Herant [START_REF] Herant | Dirty tricks for sph[END_REF] and also used by Vanhala and Cameron [START_REF] Vanhala | Numerical Simulations of Triggered Star Formation. I. Collapse of Dense Molecular Cloud Cores[END_REF] in order to implement boundary conditions in cases where boundaries are only used for confinement. To our best knowledge, this method has never been extensively described in the literature and never been used to simulate free surface flows with surface tension.

SPH governing equations and numerical model

In the Lagrangian description, the Navier-Stokes equations read as

dρ dt = -ρ∇ • v (1)
and

d v dt = g + 1 ρ [-∇P + F ν + F s ] (2)
where P and g are the material pressure and body force, respectively. F ν denotes the viscous force and F s is the surface tension force. In weakly compressible SPH, the pressure is related to the density by means of an Equation-Of-State (EOS)

P = ρ 0 c 2 s γ ρ ρ 0 γ -1 (3)
The exponent γ is usually taken equal to 7 for water. The artificial sound of speed c s is estimated based on a scale analysis of the Navier-Stokes equation presented by Morris et al. [START_REF] Morris | Modeling Low Reynolds Number Incompressible Flows Using SPH[END_REF].

The SPH method is based on the discretization of the domain with a set of particles used to interpolate continuous field functions. The contribution or weight of each particle is determined by a kernel function. The value of any field function f at a position r can be estimated according to the following summation form

f ( r) ≈ N j m j ρ j f ( r j )W (| r -r j |, h), (4)
where m j and r j are the mass and position of particle j, respectively. W represents the weighting kernel function with the smoothing length h.

Here, we use the cubic spike kernel function with a compact support of 3h. We prefer this kernel for its stability against compression [START_REF] Sirotkin | A new particle method for simulating breakup of liquid jets[END_REF] and it has the advantage of being a one piece function which simplifies the analytical calculation of the correction factor C, defined later in Section 3.

W ij =    α(1 -r 3h ) 3 if r h ≤ 3 0 otherwise (5) with α =          2 3h dim=1 9 10πh 2 dim=2 15 27πh 3 dim=3 (6)
Various SPH formulations can be obtained depending on the assumptions and purpose of the simulation [START_REF] Monaghan | Smoothed particle hydrodynamics[END_REF]. In our study, we used the formulation proposed by Adami [START_REF] Adami | A new surface-tension formulation for multi-phase SPH using a reproducing divergence approximation[END_REF]. The momentum equation is written as

d v i dt = 1 m i j -(V 2 i + V 2 j ) pij ∇W ij + 2ν i ν j ν i + ν j v ij r ij ∂W ij ∂r ij + F (s) i m i (7)
where V and ν are the volume and dynamic viscosity of each particle, respectively. v ij = v i -v j is the relative velocity between particles i and j

and

r ij = | r i -r j | is the distance between the two particles. F (s) i
is the surface tension force of particle i, and pij is the density-weighted inter-particle

averaged pressure pij = ρ i p j + ρ j p i ρ i + ρ j (8)
In the present work, the Continuum Surface Force (CSF) approach is used to model the surface tension force. This approach was initially proposed by Brackbill [START_REF] Brackbill | A continuum method for modeling surface tension[END_REF] and then extended by Morris et al. [START_REF] Morris | Simulating surface tension with smoothed particle hydrodynamics[END_REF] to be applied in the framework of SPH. The surface tension force is expressed as a volumetric force applied only on particles close to the interface

F (s) = σk nδ s (9)
Here, δ s is the surface-delta function used to smooth the surface tension force over a band of particles near the free surface, n is the normal vector and k is the curvature. In general and for this work, it is taken equal to the norm of the color gradient factor.

Due to the lack of full support near the free-surface, additional correction should be applied to accurately estimate the normal direction and the curvature of the interface. In this work, the correction matrix for the kernel gradient [START_REF] Bonet | Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations[END_REF] is used only for the calculation of the local normal vector and the curvature is estimated following Sirotkin et al. [START_REF] Sirotkin | A new particle method for simulating breakup of liquid jets[END_REF]. The full details for the surface tension force can be seen in [START_REF] Sirotkin | A new particle method for simulating breakup of liquid jets[END_REF].

For single phase simulation, the color function of all liquid particles in the bulk is equal to c 0 j = 1. The smoothing of the color function gives

c i = j m j ρ j c 0 j W ij (10)
The value of the smoothed color function c i is theoretically equal to 1 for particles in the bulk with a full kernel support. Contrary, close to the free surface the number of neighbouring particles decreases and thus the value of c i will also decrease. In general, a threshold value of 0.9 is defined for detecting surface particles. However with this method voids inside the fluid phase can be detected as free-surface, to avoid this problem a more accurate surface tracking algorithms was used. The cover-vector technique was proposed by Barecasco et al. [START_REF] Barecasco | Simple free-surface detection in two and three-dimensional SPH solver[END_REF]. Each particle i is represented by a sphere and has a cover vector defined by : b

i = j r ij || r ij || .
For detecting surface particles, a cone of angle θ i (threshold angle) is considered around each b i . If at least one of the neighbouring particles j is inside the cone, then particle i is not considered as a surface particle, otherwise particle i belongs to the free surface. The value of θ i plays an important role in boundary particle detection, it is usually chosen equal to π 3 .

Once the surface particles are detected, the normal vector n i can be expressed as the gradient of the color function :

n i = ∇c i |∇c i | .
To overcome the problem of lacking full support for free-surface flows, the correction matrix to adjust the kernel gradient [START_REF] Bonet | Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations[END_REF], can be used.

For each particle i, the correction matrix is defined as

L i = j V j ∇W ij ⊗ r ij (11)
and the corrected kernel gradient is expressed as follows

∇ Wij = L -1 i ∇W ij (12)
In SPH, the local curvature for each particle is calculated as the divergence of the normalised normal vector as follows

k i = j ( n j -n i ). ∇ Wij (13)
Furstenau et al. [START_REF] Fürstenau | Free surface tension in incompressible Smoothed Particle Hydrodynamcis (ISPH)[END_REF] found that equation 13 overestimates the curvature by almost a factor of 2 in 3D simulations. To overcome this problem, they calculated the mean curvature based on a local coordinate system. First, the global curvature tensor is calculated by

Ξ ij = j min(R i , R j )( n j -n i ) ⊗ ∇ Wij (14)
Then, this global curvature tensor is rotated into the local coordinate system as follows : Ξ kl = T T ki Ξ ij T jl . The metric T is calculated as the scalar product of the basis vectors T ij = G i .L j , where G is the global basis vectors which form a unity matrix and L = ( n, t 1 , t 2 ), with t 1 and t 2 being two tangent vectors. After the transformation, the 3D matrix is reduced to a 2D matrix by deleting the row and column related to the normal vectors, in this case the first row and column. Then, the eigenvalues of the 2x2 matrix are calculated. They are considered as the principal curvature k 1 and k 2 . The mean curvature is finally obtained by k = 1 2 (k 1 + k 2 ).

The time integration scheme used in this work is the kick-drift-kick scheme also known as the velocity Verlet algorithm used by Monaghan [START_REF] Monaghan | Smoothed particle hydrodynamics[END_REF]. It starts with the prediction of the intermediate velocity

v (t+ 1 2 dt) = v (t) + 1 2 δt a (t) . (15)
Then, the position is updated by

r (t+dt) = r (t) + δt v (t+ 1 2 dt) . (16)
The new density and forces are calculated at this new position, the acceleration is deduced from Newton's second law of motion. Finally the velocity is updated by

v (t+dt) = v (t+ 1 2 dt) + 1 2 δt a (t+dt) . (17)
For stability reasons the time step δt should be limited. To satisfy all the conditions, the minimum of these four time steps is used [START_REF] Morris | Simulating surface tension with smoothed particle hydrodynamics[END_REF] δt ≤ 0.25

h c s + v ref (18)
-Surface tension condition

δt ≤ 0.25 ρh 3 2πσ 1/2 (19)
-Viscous diffusion condition

δt ≤ 0.125 ρh 2 ν (20)
-Body force condition

δt ≤ 0.25 h g 1/2 (21)

New density formulation

The novelty in this work is a new version of the density summation taking into account the proximity of a free surface by algebraic correction .

From the different density correction techniques cited above, the Shepard density filter is the most common one. In this case the density will be calculated using the density evolution formula

dρ i dt = j m j v ij • ∇W ij (22)
Then, the density is corrected every n time steps by the Shepard summation as follows

ρ i = m i j W ij j V j W ij (23)
Note, equation 23 can be written in the form of a corrective factor C i , which will depend in this work on the distance of each particle to the free surface and the local curvature.

ρ i = C i m i j W ij (24)
If we consider only the geometric aspect, then the correction coefficient C can be expressed as follows

C i ≈ V SD V SD -V empty (25)
where V SD and V empty are the volume of the support domain of particle i and the empty volume represented by the missing particles at the free surface, respectively. Figure 1 In fact, the impact of missing neighbours on the density calculation depends on their location. The impact of removing a distant neighbour is much smaller than the impact of a closer one. Therefore, the weight of each missing particle, represented by the kernel function, should be taken into account when expressing C i . V empty is modified to V empty W (r)dV and V SD is modified to V SD W (r)dV which is the integral of the kernel function over all the support domain. Equation 25 will be then modified to

C i = 1 1 - V empty W (r)dV V SD W (r)dV (26)
Because the kernel functions are normalized, the integral of W over all the support domain should theoretically be equal to 1. The integral over the empty volume can be expressed as

V empty W (r)dV = Rc d W (r)2π(r -I)dr (27)
where I represents the distance between the center and the intersection of the two spheres. By writing I as a function of 1/k, r and d, we obtain

I = - r 2 + ( 1 k -d) 2 -( 1 k ) 2 2( 1 k -d) (28)
Finally, by substituting equation 28 in equation 27, we obtain

V empty W (r)dV = Rc d W (r)2πr r + r 2 + ( 1 k -d) 2 -( 1 k ) 2 2( 1 k -d) dr (29)
For 3D simulations, the integral over the empty volume can be expressed as a function of δ = d h and = 1 |k|h . d represents the distance between particle i and the nearest surface particle and k is the curvature of this surface particle. The kernel function used in this case is the Spike with R c = 3h (Equation 5). Two cases must be considered.

For positive curvatures,

V empty W (r)dV = (δ -3) 5 (-4δ 2 + 3δ + 28δ + 42 + 27) 20412(δ -) , (30)
and for negative curvatures

V empty W (r)dV = (δ -3) 5 (-4δ 2 + 3δ -28δ -42 + 27) 20412(δ + ) . ( 31
)
The same scheme can be used in 2D simulations by integrating over a surface instead of a volume. However, the analytical solution for surface integrals is surprisingly not as simple as for 3D cases, thus numerical in-tegration is necessary in these cases. The same problem was reported by Hérant [START_REF] Herant | Dirty tricks for sph[END_REF] although he used another kernel function for the integration. We limit our study here to 3D simulations.

It should be noted that when δ tends to or is bigger than , equation 30 can diverge. Because the second derivative of the color function is very sensitive to the particle distribution, the curvature k is limited to DIM -1

3h

. In 3D, k max = 2 3h . Plus, the maximum value of δ is defined by the radius of the support domain, which is in this case equal to 3. This means that :

0 < δ < 3 2 3 < < ∞
To avoid this problem, the correction coefficient C i is only calculated for values of bigger than the maximum value of δ. In this case, for values of below 3, the standard Shepard summation (Equation 23) is used for the density calculation. The implementation of this correction technique is detailed in algorithm 1.

Numerical validation

Cubic droplet

As first test case, we analyse is the transformation of a cubic droplet into a spherical droplet. The main purpose of this example is to test the new correction coefficient for estimating the density at the free surface. For this example, an initial cube of l x = l y = l z = 0.6 with a total number of 3375 particles is considered. Dimensionless properties were assigned to the fluid phase, see table 1. The spherical droplet formed after t=2 is presented in figure 2 for three different methods of density evaluation. The Young-Laplace law gives the theoretical pressure inside the droplet at equilibrium With the new method, the homogeneity of the density field is strongly improved. However, we can still see the effect of the initial square shape on the pressure profile, where we see a pressure variation in the regions around the corners which is mainly a geometric effect due to the initial particle stacking. By comparing the results of the three methods, it can be found that the new method outperforms the existing approaches for particle distribution at the interface. Figure 4 shows the variation of the average pressure inside the droplet as a function of time.

P = 2σ R droplet = 2σ L 4π 3 1/3 . (32)
The problem with applying the Shepard filter every n time steps is that the density of the surface particles can change instantaneously, which results in strong pressure fluctuations. As a consequence, the drop is unstable and keeps expanding. Accordingly, the pressure inside the droplet keeps increasing, see the corresponding curve (Evolution + Shepard) in figure 4. It should be noted that this new calculation method does not add much computational cost for the simulations because the curvature is already calculated for the surface tension force. It was found that the computational cost is increased by around 5% compared to the density evolution and by 2% compared to the density evolution with the Shepard summation. Table 2 shows the percentiles of the number of particles as a function of pressure, for a better comparison between the density calculation methods.

It can be seen that with the new method, the pressure variation is smaller, with almost all the particles having a pressure variation within 30% of the theoretical Laplace pressure.

Rayleigh Plateau instability

We consider the Rayleigh-Plateau instability for validation of our method in dynamic situations. The test case simulation was first presented by Dai and Schmidt [START_REF] Dai | Adaptive tetrahedral meshing in free-surface flow[END_REF] using a new moving mesh algorithm and then reproduced by Olejnik and Szewc [START_REF] Olejnik | Smoothed particle hydrodynamics modelling of the Rayleigh-Plateau instability[END_REF] using SPH with two phases. For this simulation, we consider a fully periodic cubic (L × L × L) domain containing in its center a liquid column of length L and a radius r = L/10. An initial perturbation is imposed by a sinusoidal velocity field applied to the liquid column in its longitudinal direction

u x = u 0 sin( 2πx L ). (33)
Because the particles are placed on a Cartesian grid and do not form a perfect cylinder, we let the system first relax by imposing a high viscosity. Then, we switch to the real properties of water before applying the velocity perturbation. The dimensionless numbers describing this case are the Reynolds number (Re=18) and the Weber number (We=1.4) calculated based on u 0 . The dimensionless time is defined by

T = t ρr D 0 σ . ( 34
)
The simulations have been performed with a total number of fluid particles equal to 18000, which corresponds to h = L/82. Figure 5 shows several particle snapshots at different times for the classical and the new approach.

Clearly, the new method shows more stable results compared to the classic density evolution with the density filter. It should be noted that the two pic-tures at the last row correspond to the particles positions after the break-up.

With this new density formulation, only 0.005% of the total particle number disintegrate (as compared to 1.2% for the standard method of the density evolution with the Shepard filter). Because the break-up does not occur at the same moment for both cases, the time T is not the same. The break-up time is defined as the time when the ligament diameter between droplets is composed of less than two particles.

We can clearly see from figure 5 that the break-up with the new method is much smoother. For the first case the unstable flying particles are seen at the moment of the break-up. In fact, the density evolution with no additional correction can accumulate errors with time. However, the additional main issue of this method, for this specific test case with fragmentation, is that the density could change drastically when a bulk particle becomes a surface particle, thus creating instabilities.

The correction coefficient for the Shepard technique was compared with the new correction coefficient. To compare both methods, the SPH simulations have been performed for different resolutions. Table 3 presents the number of flying particles for each case. The result shows that the new method presents stable results even for simulations with a lower resolution. For a quantitative comparison, the relative disturbance size ( rmax(t)-r 0 r 0 ) as a function of time is presented in figure 7 for the highest resolution. The results show that the growth dynamics for the two methods agree well with the reference data presented by Dai and Schmidt, even though the break-up time is not the same. Furthermore, the droplet diameter can be estimated from the volume of the liquid within the disturbance wavelength λ [START_REF]Handbook of Atomization and Sprays[END_REF].

d 3 droplet = 6λR 2 jet ( 35
)
For both cases, the droplet diameter after the break-up agrees well with the theoretical one, with an error around 5%. This new method proved its efficiency for simulating free surface flows undergoing strong topological variations under the effect of surface tension force. We found the method to work best when the interface is well resolved with smoothed shapes. In fact, the density calculation presented above depends significantly on the curvature and the distance of each particle from the free surface. Thus, these two parameters must be calculated with great precision in order to have the desired output result.

Conclusion

This paper presents a new method for estimating the density for free surface simulations dominated by surface-tension force in SPH. The main idea is to correct the density by calculating the missing volume of the support domain of each particle taking into account the local curvature and the distance from the free surface.

This technique was successfully applied to 3D test cases with free-surface flows and surface-tension effects: Square Droplet and Rayleigh-Plateau instability. The results for the square droplet show that the current method can accurately predict the density of free surface particles. Numerical results of the Rayleigh-Plateau instability show that the stability of the break-up of a liquid column is improved with this correction technique. This method can be easily extended to simulate other situations with jet break-up and liquid atomization.

The new correction factors is strongly dependent to an accurate curvature prediction, which still gives room for future improvement especially in terms of robustness.

ρ 0

 0 Speed of sound (m/s) F ν Viscous force (N/m 3 ) F s Surface tension force (N/m 3 ) g External body acceleration (m/s 2 ) Reference density (kg/m 3 ) σ Surface tension coefficient (N/m) 1. Introduction

  represents the support domain area and empty volume of a particle at a distance d from the free surface. The support domain is represented by the circle of radius R c and the empty volume is represented by the hatched moon. The curvature of the free surface is equal to k.

Figure 1 :

 1 Figure 1: Representation of the empty volume (hatched moon) for a particle at a distance d from the free surface.

Figure 2 :

 2 Figure 2: Particle position at t=2: a) Density evolution (equation 22) b) Density evolution with Shepard filter (n=30) and c) New correction based on curvature.

Figure 3 :

 3 Figure 3: Pressure inside the droplet as a function of the radius

Figure 4 :

 4 Figure 4: Average pressure inside the droplet as a function of time

Figure 6

 6 presents these two correction coefficients for surface particles as a function of particle position along the column axis. The main difference is that this coefficient is more dispersed in the case of the Shepard correction compared to the new technique presented in this article.

Figure 5 :

 5 Figure 5: Particle position at different time: a) Density evolution and Shepard filter (n=30) and b) New correction based on curvature.

Figure 6 :

 6 Figure 6: Correction coefficients as a function of the position of the particles along the jet axis

Figure 7 :

 7 Figure 7: Disturbance growth process in time: Comparing the growth rate as a function of time with the reference data from Dai and Schmidt

Table 1 :

 1 Properties of the fluid phase

ρ ν σ 1 0.2 1

Table 2 :

 2 Fraction in (%) of particles having a pressure within a certain range around P Laplace

	Method	Pressure	P ±1% P ±10% P ±30% P ±50%
	Density evolution	1.066	25	60	100
	New method (L/h = 15) 20	54	91	100

Table 3 :

 3 Number of "flying" particles for each method as a function of the system resolution

	Method	Resolution (L/h)	55	66	82
	Density evolution + Shepard summation 30 (0.6 %) 96 (1 %)	217 (1.2%)
	New method		4 (0.08 %) 1 (0.01 %) 1 (0.005 %)

The present method is found to increase the stability for surface tension driven simulations