Abdul-Hamid Soubra

Didier Galvani

Pierre Regenass
email: pierre.regenass@ensais.u-strasbg.fr

THREE-DIMENSIONAL ACTIVE EARTH PRESSURES

Keywords:

A theoretical approach to evaluate the active earth pressures taking into account the three-dimensional effect is presented. The analysis considers the general case of a frictional and cohesive (c, φ) soil subjected to a vertical surcharge loading q acting on the ground surface. This approach is based on the kinematical method of the limit analysis theory. A translational soil-wall movement is considered in the present analysis. A three-dimensional kinematically admissible failure mechanism M1, composed of a single rigid block is proposed. This mechanism is an extension into three dimensions of the classical twodimensional Coulomb mechanism. The three-dimensional active earth pressure coefficients K aγ , K ac and K aq representing the effect of soil weight, cohesion and surcharge loading are obtained by numerical optimisation. The numerical results so obtained show the influence of the different geometrical and mechanical characteristics on the three-dimensional active earth pressures.

INTRODUCTION

The problem of the two-dimensional active earth pressures acting on rigid retaining structures has been widely studied by several investigators 1, 2, 3, 4 . The review of existing literature has shown that little attention is given to the 3-D aspects. In this paper, a theoretical approach to evaluate the three-dimensional active earth pressures is presented. This approach is based on the kinematical method of the limit analysis theory. A three-dimensional kinematically admissible failure mechanism M1 is proposed. The analysis considers the general case of a frictional and cohesive (c, φ) soil with an eventual surcharge loading q acting on the ground surface. The numerical results of the three-dimensional coefficients are presented and discussed.

THE UPPER AND LOWER-BOUND THEOREMS OF LIMIT ANALYSIS

As is well known, the limit theorems of the limit analysis theory enable us to determine upper and lower bound solutions for the stability problems of a rigid perfectly plastic material.

While the lower-bound method is complex due to the fact that it requires the construction of a complete stress field, the upper-bound method is simpler: Equating the rate of external work to the rate of internal energy dissipation for a kinematically admissible velocity field gives an unsafe solution of the collapse or limit load.

A kinematically admissible velocity field is one that satisfies the flow rule, the velocity boundary conditions and compatibility. During plastic flow, energy is dissipated by general plastic yielding of the soil mass, as well as by sliding along velocity discontinuities where jumps in the normal and tangential velocities may occur. Note that the velocity field at collapse is often modelled by a mechanism of rigid blocks that move with constant velocities. Since no general plastic deformation of the soil mass is permitted to occur, the energy is dissipated solely at the interfaces between adjacent blocks which constitute velocity discontinuities.

UPPER-BOUND APPROACH

It is well known that the three-dimensional nature of the active earth pressure problem has the favourable effect of decreasing the active earth pressures exerted on the wall. In this paper, the decrease of the active earth pressure coefficients due to the decrease of the wall breadth is investigated using the kinematical approach of the limit analysis theory.

Assumptions

The following assumptions have been made in the analysis: 1. The wall of dimensions bxh (b = breadth; h = height) is vertical and the backfill is horizontal; 2. A translational soil-wall movement is assumed; 3. The soil is homogeneous and isotropic. It is assumed to be an associated flow rule Coulomb material obeying Hill's maximal work principle;

4. The angle of friction δ at the soil-structure interface is assumed to be constant. This hypothesis is in conformity with the translational kinematics assumed in this paper; 5. A tangential adhesive force P ad is assumed to act at the soil-structure interface. The intensity of this force is bh c φ δ tan tan ;

The velocity at the soil-structure interface is assumed to be tangential to the wall (see Chen 5 ). Collins 6 and Mroz and Drescher 7 assumed that the interfacial velocity is inclined at δ to the wall in order to respect the normality condition. Both hypotheses lead to the same result of the limit load (see Drescher and Detournay 8 , Michalowski 9 and Soubra 10 ).

Failure mechanism

M1 is an extension into three dimensions of the classical two-dimensional Coulomb mechanism. This mechanism is translational and is composed of a single rigid block (Figure 1). It is limited by the lower plane AA'D'D and the two lateral planes ABD and A'B'D'. Two different cases may occur depending on the height to breadth ratio h/b. Figure 1 shows the failure mechanism for small h/b values (Case I) where the lateral planes do not intersect and figure [START_REF] Caquot | Traité de mécanique des sols[END_REF] shows the second case where the lateral planes intersect along the vertical straight line TT'. For the case I, the soil mass AA'BB'DD' moves with velocity V 1 inclined at an angle of α-φ to the horizontal direction (Fig. 3), the wall moves with velocity V 0 and V 0,1 represents the relative velocity at the soil-structure interface. All these velocities are parallel to the vertical symmetrical plane xOy. For the case II, the kinematics is similar to that of Case I. Using the velocity hodograph shown in Fig. 3, we have:

( ) φ α - = cos V V 0 1 (1) ( ) 0 1 , 0 V . tan V φ α - = (2)
Note that the velocity V 1 should also make an angle φ with the lateral plane ABD (respectively A'B'D') in order to respect the normality condition. This imposes that the angle between the vector V 1 and its orthogonal projection on the lateral plane ABD (respectively A'B'D') must be equal to φ. This condition yields the orientation of the lateral planes ABD and A'B'D' for a given inclination α of the lower plane AA'DD'. It can be shown that the dihedral angle ξ [cf. Fig. 1] between the lateral plane ABD (respectively A'B'D') and the vertical plane xOy can be expressed as

( ) φ α φ φ ξ - - = 2 2
sin cos sin tan [START_REF] Sokolovski | Static of granular media[END_REF] This mechanism is defined by a single angular parameter α, the dihedral angle between the lower plane AA'D'D and the horizontal plane.

Work equation

As shown in Figure [START_REF] Chen | Limit analysis solutions of earth pressure problems[END_REF], the external forces contributing to the rate of external work consist of the active earth force P a , the weight of the soil mass in motion W and the surcharge q acting on the ground surface. Energy is dissipated at the soil-wall interface and at the lower and lateral planes between the material at rest and the material in motion. By equating the total rate of external work to the total rate of energy dissipation along the different velocity discontinuities, one obtains

b h q K b h c K b h K P aq ac a a ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ - ⋅ ⋅ ⋅ = 2 2 γ γ (4) O x y V 0 V 0,1 V 1 φ α V 0 V 1 φ α - φ α π + - 2 V 0,1 q P a W δ P ad
where K aγ , K ac and K aq are the active earth pressure coefficients due to soil weight, cohesion and surcharge loading respectively. These coefficients are function of φ, δ and h/b.

NUMERICAL RESULTS

The critical active earth pressure coefficients are obtained by numerical optimisation. The numerical results have shown that the K ac and K aq coefficients are related by the following relationship (see theorem of corresponding states of Caquot and Kérisel 2 ):

φ δ tan cos 1 aq ac K K - = (5)
Thus, in the following sections, only K aγ and K aq coefficients will be presented; K ac may be computed using equation [START_REF] Chen | Limit analysis and soil plasticity[END_REF]. Figure [START_REF] Chen | Limit analysis and soil plasticity[END_REF] shows the variation of both K aγ and K aq with h/b when φ=30° and δ/φ=1. It can be easily shown that as the length h/b increases, the active earth pressure coefficients decrease. It should be emphasised that when two-dimensional problems (i.e. small values of h/b) are used, the active earth pressure coefficients given by the present analysis are identical to those of two-dimensional analysis given by Chen 5 . This figure also shows the limit values of h/b which separate cases I and II. Figure [START_REF] Collins | The upper-bound theorem for rigid/plastic solids to include Coulomb friction[END_REF] shows the cross-sections through xOy and the traces in plan view of M1 mechanism for φ=20°, 30° and 40°, δ/φ=0 and for two values of h/b (h/b=1 and 2.5). The value h/b=1 corresponds to Case I where the lateral planes do not intersect and h/b=2.5 corresponds to Case II where the trace of the failure mechanism in plan view is a triangle.

Table 1 and table 2 [cf. Appendix]

show some values of K aγ and K aq coefficients for various governing parameters φ, δ and h/b for practical use in geotechnical engineering. As expected, the active earth pressure coefficients decrease with increasing φ, δ and h/b.

CONCLUSIONS

A one block translational kinematically admissible failure mechanism has been considered for the calculation of the three-dimensional active earth pressures acting on rigid retaining walls of limited breadth. The method used is the kinematical approach of the limit analysis theory. It is shown that the three dimensional active earth pressure coefficients decrease with increasing φ, δ and h/b. Two design tables relating the active earth pressure coefficients to various governing parameters φ, δ and h/b are given for practical use in geotechnical engineering.

APPENDIX

δ/φ δ/φ δ/φ δ/φ K aγ γ γ γ φ φ φ φ 0 1/3 1/2 2/

Figure 1 :

 1 Figure 1: Failure mechanism M1 for small h/b values (Case I)

Figure 2 :

 2 Figure 2: Failure mechanism M1 for large h/b values (Case II)

Figure 3 :

 3 Figure 3: Velocity field and velocity hodographs (Case I)

Figure 4 :

 4 Figure 4: Free body diagram (Case I)

Figure 5 :

 5 Figure 5: K aγ γ γ γ and K aq versus h/b for φ φ φ φ=30° and δ/φ δ/φ δ/φ δ/φ=1

Figure 6 :

 6 Figure 6: Cross-sections and traces in plan view of M1 mechanism for φ φ φ φ=20°, 30° and 40° when δ/φ δ/φ δ/φ δ/φ=0 and h/b=1 and 2.5.

Table 2 : K aq values for various governing parameters

 2

	3	1

Abdul-Hamid Soubra * , Didier Galvani * , and Pierre Regenass *


Sophie Defontaine

Isabel Jalón-Rojas

Physical processes matters! Recommendations for sampling microplastics in estuarine waters based on hydrodynamics

Keywords: Microplastics, estuary, sampling, monitoring, physical processes, hydrodynamics

Physical processes matters! Recommendations for sampling microplastics in estuarine waters based on hydrodynamics Sophie Defontaine (1) and Isabel

Introduction

Estuaries form the connection between marine and fluvial waters. They are highly productive ecosystems that provide important environmental, social and economic services [START_REF] Barbier | The value of estuarine and coastal ecosystem services[END_REF]. Estuaries are, however, highly susceptible to both natural and human disturbances. Like other aquatic systems around the world, estuaries are vulnerable to plastic pollution [START_REF] Browne | Spatial patterns of plastic debris along estuarine shorelines[END_REF]. In particular, the abundance and risks associated with microplastic particles and fibres (MPs), defined as plastics lower than 5 mm [START_REF] Frias | Microplastics: finding a consensus on the definition[END_REF], are of high concern.

Their similar dimension to sediments and planktonic organisms make them easily ingestible by the aquatic biota. The ingestion of MPs can be responsible for gut abrasion and blockage, as well as intoxication by sorbed contaminants or toxic additives used in the compounding of plastics [START_REF] Andrady | Microplastics in the marine environment[END_REF][START_REF] Kazour | Juvenile fish caging as a tool for assessing microplastics contamination in estuarine fish nursery grounds[END_REF]. Living organisms may also develop on the MP surface in the form of biofilms [START_REF] Amaral-Zettler | Ecology of the plastisphere[END_REF]. MPs may thus act as dispersal vectors of pathogens in moving estuarine waters [START_REF] Forero-López | Plastisphere on microplastics: In situ assays in an estuarine environment[END_REF].

Estuaries are critical areas for plastic pollution due to their interface nature between the ocean and land. They are convergence areas of marine-based plastic pollution [START_REF] Kuczenski | Plastic gear loss estimates from remote observation of industrial fishing activity[END_REF], land-based plastic pollution from rivers [START_REF] Lebreton | River plastic emissions to the world's oceans[END_REF], and also intrinsic estuarine-based plastic pollution from industries, cities, fishing, and port activities [START_REF] Napper | Potential microplastic release from the maritime industry: Abrasion of rope[END_REF]. In a similar way that estuaries are sinks for sediments, they can represent a temporary or permanent sink for MPs [START_REF] Fok | Hong kong at the pearl river estuary: A hotspot of microplastic pollution[END_REF][START_REF] Nel | Citizen science reveals microplastic hotspots within tidal estuaries and the remote scilly islands, united kingdom[END_REF][START_REF] Simon-Sánchez | River deltas as hotspots of microplastic accumulation: The case study of the ebro river (nw mediterranean)[END_REF]. Estuarine hydrodynamic processes can form convergence zones of MPs or Estuarine MP Maxima (EMPM) within the estuary [START_REF] Díez-Minguito | Observations and idealized modelling of microplastic transport in estuaries: the exemplary case of an upwelling system (ría de vigo, nw spain)[END_REF][START_REF] Bermúdez | Unravelling spatio-temporal patterns of suspended microplastic concentration in the natura 2000 guadalquivir estuary (sw spain): Observations and model simulations[END_REF]. Even after being trapped during long periods of time, MPs can still be flushed from estuaries by extreme events at time scales from annual to pluriannual [START_REF] Tramoy | Transfer dynamic of macroplastics in estuaries-New insights from the Seine estuary: Part 1. Long term dynamic based on date-prints on stranded debris[END_REF]. The high ecological values of estuaries together with their double role as an ocean source and a sink have motivated numerous research questions and attracted the recent attention of the interdisciplinary plastic research community [START_REF] Gallagher | Microplastics in the solent estuarine complex, uk: an initial assessment[END_REF][START_REF] Gray | Microplastic in two south carolina estuaries: Occurrence, distribution, and composition[END_REF][START_REF] Hitchcock | Microplastic pollution in estuaries across a gradient of human impact[END_REF][START_REF] Sadri | On the quantity and composition of floating plastic debris entering and leaving the tamar estuary, southwest england[END_REF][START_REF] Zhao | Suspended microplastics in the surface water of the yangtze estuary system, china: first observations on occurrence, distribution[END_REF].

In situ observations of microplastic concentrations in the abiotic compartment is a key approach to address emergent research questions on plastic pollution at the land-sea continuum, e.g.: (1) the sources and generation of MPs; (2) the pathways by which MPs reach estuarine, coastal and ocean waters; (3) the abundance, distribution and fate of MPs through the continuum ecosystems; (4) the transport mechanisms driving spatio-temporal variations and promoting the flushing of MPs to the ocean; (5) MPs ageing, weathering and biofouling; and (6) ecosystems exposition and risks, among others. Numerous protocols have been developed to study MP contamination in ocean surface waters or beaches [START_REF] Besley | A standardized method for sampling and extraction methods for quantifying microplastics in beach sand[END_REF][START_REF] Hanke | Guidance on monitoring of marine litter in european seas: a guidance document within the common implementation strategy for the marine strategy framework directive[END_REF][START_REF] Masura | Laboratory methods for the analysis of microplastics in the marine environment: Recommendations for quantifying synthetic particles in waters and sediments[END_REF][START_REF] Miller | Recovering microplastics from marine samples: a review of current practices[END_REF][START_REF] Miller | Recommended best practices for collecting, analyzing, and reporting microplastics in environmental media: Lessons learned from comprehensive monitoring of san francisco bay[END_REF]. However, these protocols or strategies may not be suitable to evaluate MP contamination in estuaries and transitional waters. Most of these protocols are based on the assumption that MPs are floating on the surface or have been deposited on the bed depending on their density. The complex three-dimensional hydrodynamics in estuaries can make this assumption, and therefore these protocols, inappropriate. Table 1 compares sampling strategies previously implemented in the water matrix of different estuaries around the world, highlighting that most of them focused on surface water.

The specific hydrodynamics and properties of estuarine waters play a crucial role in the distri-bution, transport and trapping of MPs [START_REF] Malli | Transport mechanisms and fate of microplastics in estuarine compartments: A review[END_REF]Jalón-Rojas et al.). Estuarine waters are characterised by a changing density due to varying salinity and temperature that will largely impact the buoyancy of MPs [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF]. Estuarine waters are also rich in sediment and living organisms (blooms) that may favour MP flocculation. [START_REF] Andersen | Flocculation of PVC microplastic and fine-grained cohesive sediment at environmentally realistic concentrations[END_REF] suggested that the estuarine residence times of some MPs are long enough for sediment adhesion and possibly biofouling to occur. Flocculation and biofouling can modify the dynamical behaviour of particles [START_REF] Andersen | Flocculation of PVC microplastic and fine-grained cohesive sediment at environmentally realistic concentrations[END_REF][START_REF] Jalón-Rojas | Effects of biofilms and particle physical properties on the rising and settling velocities of microplastic fibers and sheets[END_REF] and even induce the sinking of initially buoyant MPs [START_REF] Kaiser | Effects of biofouling on the sinking behavior of microplastics[END_REF][START_REF] Laursen | Rapid flocculation and settling of positively buoyant microplastic and fine-grained sediment in natural seawater[END_REF]. The variety of forcing influencing estuarine hydrodynamics at different time scales, such as wind, tide, river discharge, and waves, also largely affects MP transport. However, only a few studies in the literature specify the tidal phase, current velocities, or water properties (e.g. salinity, turbidity) at the time of sampling (e.g. [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF]; [START_REF] Gasperi | Projet Plastic-Seine: Flux et impacts des microplastiques dans l'estuaire de la Seine[END_REF], see Table 1). Compared to river or ocean transport, the three-dimensional estuarine circulation greatly increases the complexity of the dynamical behaviour of MPs and vertical transport plays a key role. Due to this complexity and the youth of this research field, the contamination, distribution and fluxes of MPs in estuaries are largely unknown.

In this manuscript, we sustain that to keep gaining knowledge on MP pollution in estuaries, monitoring protocols should be linked to estuarine hydrodynamics and their typical spatio-temporal scales. We also identify the lack of consistent and standardised methods and protocols for these systems (Table 1) as a limitation to inter-compare different systems and keep gaining understanding. The present work provides technical recommendations and suggestions for monitoring MPs in estuarine waters, focusing on sampling strategies; i.e. digestion, density separation, extraction, counting and chemical characterisation techniques being out of the scope of this note. Here, we present some of the most used techniques with their advantages and disadvantages depending on the type of estuary to be studied. In light of this, this note aims to answer the following questions : (i) when to monitor MPs in estuaries, (ii) where to monitor MPs in estuaries, (iii) and how to monitor MPs in estuaries. Good practices when sampling MPs are also discussed.

When to monitor MP in estuaries?

Two elements can affect the spatio-temporal variability of MPs in an estuary: flow patterns and MP sources. The environmental forcing (e.g., tides, river discharge, wind, waves) affecting the flow and the underlying physical processes driving the transport of suspended particulate matter (e.g., density stratification, exchange flow, tidal mixing, stokes drift, tidal pumping; Fig. 1) are well-known from decades of research in estuarine physics and sediment transport [START_REF] Winterwerp | Introduction to the physics of cohesive sediment dynamics in the marine environment[END_REF][START_REF] Scully | Sediment pumping by tidal asymmetry in a partially mixed estuary[END_REF][START_REF] Jay | Estuarine variability[END_REF][START_REF] Geyer | The estuarine circulation[END_REF][START_REF] Burchard | Sediment trapping in estuaries[END_REF]. MP inputs from various sources (fluvial, marine and local) introduce additional temporal variability in MP abundance and distribution. This variability may depend on the environmental forcing (e.g. river and surface runoff inputs), or on human activities schedule (e.g. boat-based sources, sewage output). Advanced knowledge of the local hydrodynamics and the potential sources is therefore required to determine when to collect MPs.

Various key time scales should be considered when planning samplings in estuaries, namely tidal, fortnightly and seasonal time scales. The particular importance of a given time scale is strongly related to the dominant forcing of the estuarine dynamics and is, therefore, site-specific [START_REF] Jay | Estuarine variability[END_REF][START_REF] Jalón-Rojas | Comparison of environmental forcings affecting suspended sediments variability in two macrotidal, highly-turbid estuaries[END_REF]. In estuaries dominated by tides, MP concentration can vary significantly between ebb and flood tides [START_REF] Oo | Horizontal variation of microplastics with tidal fluctuation in the chao phraya river estuary, thailand[END_REF]. This difference can be related to various processes such as tidal asymmetry (the flood or ebb dominance of currents). For example, in a flood-dominated estuary, sinking MPs could have a higher rate of resuspension and mixing during flood tides. Microplastic concentration may also vary over the tidal cycle in stratified and salt-wadge estuaries where tidal motion is weaker. This can be due to variations in currents and salinity driven by tidal variations in vertical mixing and the along-channel density gradient. [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF] show that the transport of MPs in suspension may be contained by the pycnocline due to the turbulence damping induced by density stratification, whereas periods of intense mixing led to homogeneous concentration along the water column. However, fewer studies sampled microplastics at different tidal phases (Tab. 1). Given the importance of the tidal time scale, we recommend sampling always at the same tidal phases to allow the inter-comparison of samples and, as far as possible, sampling at different key tidal phases (e.g. flood, ebb, high water, low water).

Estuarine physical processes may change from spring to neap tides or from equatorial to tropical tides [START_REF] Defontaine | Variability of subtidal flow in a narrow meandering stratified estuary[END_REF][START_REF] Valle-Levinson | Fortnightly switching of residual flow drivers in a tropical semiarid estuary[END_REF]. Microplastic concentration may therefore vary at the fortnightly time scale [START_REF] Stead | Identification of tidal trapping of microplastics in a temperate salt marsh system using sea surface microlayer sampling[END_REF], analogously to suspended sediment concentrations [START_REF] Jalón-Rojas | Turbidity in the fluvial Gironde Estuary (southwest France) based on 10-year continuous monitoring: sensitivity to hydrological conditions[END_REF]. In addition, estuaries can be subject to the seasonal variability of forcing (e.g. river flow, wind) that may impact the stratification, residual currents, and therefore MP transport trends. In particular, the estuarine residual circulation (which includes several processes such as density-driven circulation, wind-driven circulation, internal asymmetry, non-linear tidal motion and the Stokes drift) can be subject to seasonal changes related to the annual river flow cycle and the precipitation/evaporation balance [START_REF] Jay | Estuarine variability[END_REF]. To take into account all these variability time scales, it is highly recommended to monitor during full tidal cycles, under contrasting tidal ranges and river flow conditions.

Other forcing and processes such as wind gusts, sea breeze, harbour seiches and waves may also play critical roles in MP dispersion at different time scales. [START_REF] Browne | Spatial patterns of plastic debris along estuarine shorelines[END_REF] show that a prevailing wind direction can be responsible for MP accumulation at downwind sites of the Tamar estuary. Sánchez-Hernández et al. ( 2021) also showed that MP abundance was higher during periods of strong winds. In addition, wind and waves may also trigger vertical mixing of plastic debris [START_REF] Kukulka | The effect of wind mixing on the vertical distribution of buoyant plastic debris[END_REF].

In summary, contrasting conditions of prevailing forcing (e.g. low/high river flow, spring/neap tides, wind/unwind conditions, heavy swell/calm sea conditions) should be considered when planning monitoring campaigns. Table 2 summarizes key recommendations about when sampling microplastics depending on the main forcing and hydrodynamic processes of a given site. The postprocessing of MP samples is expensive in terms of time and human resources, which can inevitably affect the final choice of the monitoring periods. When all the representative time scales of variability are not considered, results should be analysed keeping in mind that MP concentrations are only representative of the selected conditions.

The discussion of MP concentrations at the different key time scales should also consider the variability of MP inputs from sources (Tab.2). Sources such as urban storm-water runoff or wastewater effluents may have a variable discharge depending on the weather forecast (rainfall) or the peak period of city occupation (e.g. tourism). Significant links between rainfall and MP concentration have been found in the literature as it drains the land-based MP pollution through rivers to estuaries [START_REF] Lima | Seasonal distribution and interactions between plankton and microplastics in a tropical estuary[END_REF][START_REF] Hitchcock | Storm events as key moments of microplastic contamination in aquatic ecosystems[END_REF]. The wet season could therefore represent a period of a strong abundance of MPs. However, the dry season may represent a period of denser population and thus an increase of MP inputs from surrounding cities and shores. Collection during the wet and dry seasons is therefore relevant to investigate potential seasonal differences in contamination.

In addition, wastewater effluent contamination may also vary between weekends and weekdays, and between flood and ebb tides (e.g. WWTP discharge during ebb tide to favour flushing). Samplings only on weekdays should be envisaged [START_REF] Miller | Recommended best practices for collecting, analyzing, and reporting microplastics in environmental media: Lessons learned from comprehensive monitoring of san francisco bay[END_REF].

Where to monitor MP in estuaries?

As in many environmental pollution studies, the number of sampling sites and their location is of the foremost importance to ensure a truly representative study by probabilistic sampling. However, as explained above for the sampling frequency, the particularly expensive cost and time-consuming laboratory analyses that follow MP collection may be a strong limitation. Compared to lakes, rivers or the ocean, MP distribution in estuaries can strongly vary spatially in the three dimensions over small spatial and temporal scales [START_REF] Malli | Transport mechanisms and fate of microplastics in estuarine compartments: A review[END_REF]Jalón-Rojas et al.). A compromise should be found between spatial statistical coverage and time/financial costs. The site selection should then be restricted to a few sites as representative of the whole estuary as possible. Table2 also summarizes key recommendations on sampling locations depending on the major forcing and hydrodynamic processes.

Intratidal and subtidal circulation may vary along the longitudinal and lateral axes as well as water properties affecting the dispersion of suspended particles such as MPs [START_REF] Lam | Spatial variation of floatable plastic debris and microplastics in the pearl river estuary, south china[END_REF].

Different locations along and across the estuarine channel should thus be considered (Tab.2). Even if the choice of the locations should be based on the specific characteristics of the study site, a general recommendation is to sample at different sites along the longitudinal density gradient, as physical processes driving the transport of MPs and MP buoyancy vary over this gradient [START_REF] Malli | Transport mechanisms and fate of microplastics in estuarine compartments: A review[END_REF]; Jalón-Rojas et al.) (Fig. 1). For instance, density-driven circulation is a physical process directly related to the density gradient. In estuaries characterized by decreasing salinity from the ocean toward the upper estuary, light MPs might be transported as wash load to the ocean by a surface seaward flow, while dense MPs might be transported as bed load by a landward flow near the bottom [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF]. Wind stress can also promote a two-layer circulation [START_REF] Li | Effects of winds on stratification and circulation in a partially mixed estuary[END_REF]. Recent studies based on numerical and idealized models have suggested that estuarine circulation may form hotspots of microplastics, also called Estuarine MicroPlastic Maxima (EMPM) [START_REF] Díez-Minguito | Observations and idealized modelling of microplastic transport in estuaries: the exemplary case of an upwelling system (ría de vigo, nw spain)[END_REF][START_REF] Bermúdez | Unravelling spatio-temporal patterns of suspended microplastic concentration in the natura 2000 guadalquivir estuary (sw spain): Observations and model simulations[END_REF]. In estuaries or estuarine regions dominated by tides, tidal pumping can also generate longitudinal circulation and MP trapping [START_REF] Stead | Identification of tidal trapping of microplastics in a temperate salt marsh system using sea surface microlayer sampling[END_REF]. In flood-dominated estuaries, the tidal wave may become increasingly asymmetric (higher flood than ebb currents) along the estuarine channel depending on the competition between friction and channel convergence. Non-buoyant MPs may be only resuspended and transported landward during floods if the bed shear stress during ebbs does not exceed the critical shear stress. This net landward transport of MPs may also form EMPM (Jalón-Rojas et al.). Readers can refer to [START_REF] Burchard | Sediment trapping in estuaries[END_REF] for a whole explanation of estuarine trapping mechanisms. Longitudinal gradients and hotspots of MPs can therefore be key estuarine features and should be considered when planning monitoring campaigns.

The specific location of EMPMs may be difficult to predict as it may depend on the physical properties of the particles. In other words, different EMPM may happen for different types of MPs (Jalón-Rojas et al.), but more research is needed to understand this phenomenon. In some estuaries, EMPM might correspond to estuarine turbidity maximum [START_REF] Díez-Minguito | Observations and idealized modelling of microplastic transport in estuaries: the exemplary case of an upwelling system (ría de vigo, nw spain)[END_REF].

These potential zones of accumulation should be considered in the conception of field campaigns and during the data analysis. In particular, when using filtering processes, the huge quantity of sediments or living organisms contained in estuarine waters may lead to the clogging of filters or nets [START_REF] Hanke | Guidance on monitoring of marine litter in european seas: a guidance document within the common implementation strategy for the marine strategy framework directive[END_REF]. If such areas should not be avoided during sampling, the choice of larger mesh sizes for filtration may be an option. It must be noted that cleaning nets or sieves during field campaigns is very challenging.

Estuary cross-sections are generally composed of a navigation channel and shoals, where the hydrodynamics may largely vary and induce differential advection of suspended matter [START_REF] Mcsweeney | Lateral variability of sediment transport in the d elaware e stuary[END_REF]. Banks with vegetation can be sinks of MPs [START_REF] Carmen | Microplastic retention by marine vegetated canopies: simulations with seagrass meadows in a hydraulic flume[END_REF][START_REF] Stead | Identification of tidal trapping of microplastics in a temperate salt marsh system using sea surface microlayer sampling[END_REF]. A particular phenomenon of great importance for the accumulation and transport of MPs is estuarine fronts [START_REF] Largier | Estuarine fronts: how important are they?[END_REF]. Fronts are related to strong convergence currents at surface waters that form visible lines of foam and debris, promoting lateral gradients of MP concentration [START_REF] Wang | Accumulation, transformation and transport of microplastics in estuarine fronts[END_REF]. To adequately account for the lateral variation of contamination, sampling locations should also be spread across the channel, at least two points to compare channel and shoals or the regions inside and outside the front (Tab.2). It should be noted that trawling nets can be difficult to operate in the front and can become rapidly clogged due to the high litter abundance [START_REF] Green | A comparison of sampling methods for seawater microplastics and a first report of the microplastic litter in coastal waters of ascension and falkland islands[END_REF]. On the other hand, locations with morphological specificity (e.g. sills) or with distinctive hydrodynamic features (e.g. bends) should be avoided unless studied on purpose, as they may introduce site-specific phenomena (Tab.2).

All the physical processes mentioned above are characterized by strong three-dimensional components. Vertical transport became particularly important in shallow waters. For instance, vertical mixing is a key process in transferring suspended particulate matter from the bottom to the surface and for keeping small particles in suspension [START_REF] Shamskhany | Entrainment and vertical mixing of aquatic microplastics in turbulent flow: The coupled role of particle size and density[END_REF]. Surface sampling is certainly not representative of MP contamination, underestimating MP fluxes by avoiding suspended MPs inside the water column [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF][START_REF] Gasperi | Projet Plastic-Seine: Flux et impacts des microplastiques dans l'estuaire de la Seine[END_REF]. For example, [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF] have shown that vertical density stratification affected MP abundance and size distribution through the water column of the Adour estuary: MP concentration and size distribution was different (similar) in surface and bottom waters during stratification (wellmixed) periods. We recommend sampling three depths in the water column to compare surface, mid-column and bottom contamination, especially in highly-stratified estuaries, or at least two depths when time and human resources are limited (Tab.2).

Local sources (e.g., sewage networks, industrial outflows, rainwater networks or marinas) should be also considered when planning sampling locations. Recent studies have shown that MP pollution is largely correlated to the proximity to urban areas [START_REF] Lebreton | River plastic emissions to the world's oceans[END_REF][START_REF] Rodrigues | Microplastic contamination in an urban estuary: abundance and distribution of microplastics and fish larvae in the Douro estuary[END_REF][START_REF] Yonkos | Microplastics in four estuarine rivers in the Chesapeake Bay, USA[END_REF]. Unless otherwise desired, the sampling locations should be placed away from such local sources to be as representative of the whole estuary pollution as possible (Tab.2).

However, the analysis of pollution pathways is particularly important for management and policy regulations, as well as for developing numerical studies. In this sense, it is particularly interesting to sample MP source outflows to estimate discharged loads to the estuary [START_REF] Bailey | Quantification and composition of microplastics in the raritan hudson estuary: comparison to pathways of entry and implications for fate[END_REF].

How to monitor MP in estuaries?

The choice of the sampling method is not trivial when sampling MPs as it generally defines the lower MP size within the sample. If a consensus has been reached regarding the larger size limit of MPs to be 5 mm, it is not the case with the lower size limit. Some studies have defined a lower size limit of 1 µm, and plastic particles being smaller as nanoplastics [START_REF] Thompson | Plastics, the environment and human health: current consensus and future trends[END_REF][START_REF] Frias | Microplastics: finding a consensus on the definition[END_REF]. However, the smaller size limit is defined operationally by the size of the finer mesh, sieve or filter pore used during sampling. Table 1 compares sampling methods employed in previous studies. These methods are diverse and imply different lowest-size limits. For example trawling nets classically only capture MPs greater than 300 µm (lowest mesh size), while the lowest size of MP collected with a pumping system or bottle sampling depends on the sieve size or filter pore, which can go down to 1 µm. Sampling techniques may be responsible for some sorting in MP collection. In particular, fibres are complex to collect due to the thin and elongated shape that allows them to pass through nets and sieves. Therefore, the different available sampling methods should be carefully evaluated during the study design. The two crucial characteristics to consider are the volume of analysed water and the finest mesh size, sieve or filter pore used. The possibility of sampling at different depths to capture the intrinsic vertical variability of estuaries, and during short periods to capture tidal variability, can also be important criteria to select a sampling method in estuaries. In this section, the advantages and disadvantages of the most commonly used techniques are presented.

Trawling nets. As classical protocols for sampling MPs were based on the hypothesis that most MPs are floating at surface waters, the most commonly used techniques are trawling nets such as plankton, neuston or manta nets [START_REF] Gallagher | Microplastics in the solent estuarine complex, uk: an initial assessment[END_REF][START_REF] Mceachern | Microplastics in tampa bay, florida: abundance and variability in estuarine waters and sediments[END_REF][START_REF] Simon-Sánchez | River deltas as hotspots of microplastic accumulation: The case study of the ebro river (nw mediterranean)[END_REF][START_REF] Yonkos | Microplastics in four estuarine rivers in the Chesapeake Bay, USA[END_REF]. Such nets are generally equipped with a rectangular opening and two floats to sample large volume in surface waters. On the other hand, bongo nets may be equipped with round openings and depressor weights to allow collection inside the water column. However, the sampling depth is hard to target precisely with a depressor weight. The volume of filtered water is estimated thanks to a flowmeter that is fixed on the opening frame [START_REF] Hanke | Guidance on monitoring of marine litter in european seas: a guidance document within the common implementation strategy for the marine strategy framework directive[END_REF][START_REF] Prata | Methods for sampling and detection of microplastics in water and sediment: a critical review[END_REF]. Large volumes of water can be easily sampled ensuring solid statistical data and reducing the impact of background contamination. Paired bongo nets present the advantage to allow sampling duplicates. Trawling net seems to be the most common technique used in estuaries (Tab. 1), which permits comparison between sites.

The mesh size of the net is one of the most restrictive elements of trawling nets. A standard mesh of 300 µm is generally mounted on such systems that impede finer MPs to be collected, leading to an underestimation of the contamination [START_REF] Green | A comparison of sampling methods for seawater microplastics and a first report of the microplastic litter in coastal waters of ascension and falkland islands[END_REF][START_REF] Lindeque | Are we underestimating microplastic abundance in the marine environment? a comparison of microplastic capture with nets of different mesh-size[END_REF]. However, some suppliers offer finer mesh sizes that may be considered. In the Seine River, a comparison between a plankton net equipped with an 80 µm mesh and a manta net equipped with a classical 330 µm mesh revealed that concentration may be largely underestimated (several orders of magnitude) with the larger mesh size (330 µm), even though a greater diversity of shape and types may be captured [START_REF] Dris | Microplastic contamination in an urban area: a case study in greater paris[END_REF]. Nevertheless, trawling nets with fine meshes are more susceptible to clogging, leading to a reduced volume of filtered water and to the collection of MPs finer than the mesh size [START_REF] Dris | Synthetic and non-synthetic anthropogenic fibers in a river under the impact of paris megacity: Sampling methodological aspects and flux estimations[END_REF]. To avoid such drawbacks, it is recommended to deploy fine mesh trawling in relatively clear waters, which is not usual in (turbid) estuaries.

Trawling nets are towed at the rear of the vessel and have the benefit to be easy to deploy in coastal areas. However, they may be not appropriate for narrow channels or very busy shipping lanes. Such techniques may even be forbidden in some parts of the estuary to not disturb harbour activities and navigation. It is highly recommended to contact local authorities before deploying such equipment in estuaries. The European Commission recommends deploying trawling net out of the wake zone due to turbulence leading to unrepresentative sampling [START_REF] Hanke | Guidance on monitoring of marine litter in european seas: a guidance document within the common implementation strategy for the marine strategy framework directive[END_REF]. Michida et al. ( 2019) study recommends conducting trawling surveys in conditions where wave heights are under 0.5 meters as trawling nets can be relatively difficult to manipulate in rougher conditions.

Pumping system. Pumping systems generally consist in a high-capacity pump that pours waters through a set of sieves [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF][START_REF] Xu | Microplastic risk assessment in surface waters: A case study in the Changjiang Estuary, China[END_REF][START_REF] Zhao | Microplastic in three urban estuaries, China[END_REF]. In some cases, water samples are collected in jars to be filtered at the laboratory. This method presents different advantages. First, it allows us to precisely choose the finest MP size to be collected (e.g. 70 µm, 50 µm or 5 µm) as a function of the finest mesh size or filter pore. Second, this system can easily collect MPs at different depths in the water column. A pressure sensor and a depressor weight need to be fixed to the pump inlet to ensure a vertical fall and precisely estimate the depth of measurement. Third, this method is relatively easy to set up on the field and duplicates can readily be collected with a second pumping system working in parallel.

Nevertheless, the use of such technologies may introduce some contamination by the plastic components of the pumping system and can trigger the fragmentation of MPs due to shear stress during pumping [START_REF] Enfrin | Release of hazardous nanoplastic contaminants due to microplastics fragmentation under shear stress forces[END_REF][START_REF] Skalska | Riverine microplastics: Behaviour, spatio-temporal variability, and recommendations for standardised sampling and monitoring[END_REF]. Another major drawback is that, depending on the pump capacity, the operation may take hours to collect some cubic meters of water [START_REF] Defontaine | Microplastics in a salt-wedge estuary: Vertical structure and tidal dynamics[END_REF]. During such a long operation, the estuarine hydrodynamics may have changed and may not be representative of specific conditions. A compromise has to be found between a large volume of water to gain statistically reliable data and a short operating time representative of one hydrodynamic state.

Bottle sampling. Grab sampling is a commonly used technique to estimate suspended sediment concentration. It holds some advantages. For instance, it is simple to operate and allows us to collect all types of MPs, including those difficult to capture with other techniques such as fibres. Several studies have shown that fibres are more efficiently collected with bottle sampling than with nets [START_REF] Rebelein | Microplastic fibers-underestimated threat to aquatic organisms?[END_REF]. The subsequent filtration at the laboratory with fine pore filters can capture small MPs (down to 1 µm). Contamination is also lower compared to nylon net and pumping system [START_REF] Prata | Methods for sampling and detection of microplastics in water and sediment: a critical review[END_REF]. Nevertheless, laboratory analyses are longer than with other methods as no pre-sieving is realised in the field. The major drawback of this method is the very small volume of water that is analysed, which leads to the need for replicates and thus additional laboratory costs. [START_REF] Dubaish | Suspended microplastics and black carbon particles in the jade system, southern north sea[END_REF] have shown that replicates from bottle samples displayed larger heterogeneity, especially for films and fragments. Although fragments and films may be underestimated by grab sampling, the concentrations of fibres may yield between 3 and 4 orders of magnitude greater than those estimated by common zooplankton net methods [START_REF] Green | A comparison of sampling methods for seawater microplastics and a first report of the microplastic litter in coastal waters of ascension and falkland islands[END_REF]. Therefore, bulk sampling may be envisaged to study fibres, but it is less recommended when all types of MPs are investigated.

Automatic samplers, continuous plankton recorder and continuous-flow centrifuges.

Recently, new technologies have been developed to investigate microplastic pollution in a more automatic way. Automatic rosette water samplers or even ROV have been used for MP sampling in ocean waters (La [START_REF] Daana | Microplastics in sub-surface waters of the arctic central basin[END_REF][START_REF] Dai | Occurrence of microplastics in the water column and sediment in an inland sea affected by intensive anthropogenic activities[END_REF][START_REF] Choy | The vertical distribution and biological transport of marine microplastics across the epipelagic and mesopelagic water column[END_REF]. However, such massive equipment required a lift arm and is not adapted for shallow estuarine waters. Continuous flow centrifuges are able to separate particles denser than ambient waters from less dense particles and, when coupled with a filtration system, may be used for sampling MPs [START_REF] Hildebrandt | Evaluation of continuous flow centrifugation as an alternative technique to sample microplastic from water bodies[END_REF].

However, it lasts several hours to process 100 L of water, so it may not be adapted to estuarine dynamics time scales. In-situ filtration devices consisting of a high-capacity pump associated with a filtration device (e.g. in-line steel filters, mesh bag) have shown promising results in sampling MPs [START_REF] Li | Profiling the vertical transport of microplastics in the West Pacific Ocean and the East Indian Ocean with a novel in situ filtration technique[END_REF][START_REF] Liu | A novel method enabling the accurate quantification of microplastics in the water column of deep ocean[END_REF][START_REF] Karlsson | Comparison between manta trawl and in situ pump filtration methods, and guidance for visual identification of microplastics in surface waters[END_REF][START_REF] Harrold | A peristaltic pump and filter-based method for aqueous microplastic sampling and analysis[END_REF]. They can be equipped with a flowmeter and pressure sensor. However, such systems do not sample the surface microlayer, where light MPs may accumulate, to keep the pump inlet underwater during sampling [START_REF] Karlsson | Comparison between manta trawl and in situ pump filtration methods, and guidance for visual identification of microplastics in surface waters[END_REF]. To our knowledge, there are no field studies in the literature using such new technologies in estuarine waters. However, it could be interesting to compare them with the most commonly used methods. properly rinsed with pure water and covered with aluminium foil to avoid airborne contamination prior to any contact with the sample [START_REF] Green | A comparison of sampling methods for seawater microplastics and a first report of the microplastic litter in coastal waters of ascension and falkland islands[END_REF]. Synthetic clothing should be avoided when researchers recover and manipulate samples; cotton clothes should be favoured [START_REF] Hanke | Guidance on monitoring of marine litter in european seas: a guidance document within the common implementation strategy for the marine strategy framework directive[END_REF].

When using trawling nets, they should be carefully rinsed between tows to reduce background contamination. Nets should be rinsed from the outside of the net and never through the net opening [START_REF] Hanke | Guidance on monitoring of marine litter in european seas: a guidance document within the common implementation strategy for the marine strategy framework directive[END_REF]. In order to investigate the potential background contamination, blank analyses should be carried out. It is recommended that laboratory analyses are carried out by a unique analyst to limit the analyst bias in the results [START_REF] Green | A comparison of sampling methods for seawater microplastics and a first report of the microplastic litter in coastal waters of ascension and falkland islands[END_REF]. During recovering and manipulation of samples on the field, the materials should be placed upwind to avoid additional contamination by the analyst. Trawling nets should be positioned by the side of the ship instead of at the rear when possible to avoid MP contamination from the ship and vertical mixing by wake and bow waves.

Replicates are standard practice to measure variability in sample collection and analysis. When small volumes of water are considered during sampling (e.g. grab sampling), replicate samples may be necessary to avoid heterogeneity in samples. [START_REF] Miller | Recommended best practices for collecting, analyzing, and reporting microplastics in environmental media: Lessons learned from comprehensive monitoring of san francisco bay[END_REF] recommend duplicating every ten samples or more if the study focuses on microfibers.

Another good but uncommon practice is to collect a wide range of physical parameters simultaneously to MP samples. MP abundance and distribution alone are very difficult to interpret and compare which may lead to biased conclusions. We highly recommend collecting additional data such as water levels, current intensity, water properties (e.g. salinity, turbidity, organic content) and weather forecast (e.g. rainfall, wind, waves).

Research priorities

Measuring the abundance and distribution of MPs in estuaries and identifying their sources is the primary step in gaining understanding of their dynamics and fate, evaluating environmental risks, and establishing future mitigation measures and management strategies. Nevertheless, as discussed in this work, sampling MPs in estuarine waters is not trivial. No established protocols have been developed so far to harmonise sampling methods in the water compartment. Given that different sampling methods and strategies can lead to concentration differences by orders of magnitude [START_REF] Green | A comparison of sampling methods for seawater microplastics and a first report of the microplastic litter in coastal waters of ascension and falkland islands[END_REF], there is a critical need to establish standardized sampling protocols that ensure consistency and the inter-comparison of systems. However, it is also important for the protocols to be flexible, as estuaries are complex environments with a wide range of physical, chemical, and biological characteristics. Protocols should therefore take into account all the particularities of a specific estuarine environment (complex 3D hydrodynamics, ETM, variety of MP sources ..).

Nevertheless, some elements may be harmonized such as the lower size limit of the collected MPs (e.g. mesh size, filter pores) and the minimum volume of sampled water. Although an optimal technique is highly dependent of the study site and the specific objectives, manual or automated high-capacity pumping system associated with any kind of filtering system presents a large number of advantages: easily available for deployment at different depths in complex areas (e.g. navigation, shoals with vegetation), collection of large volumes of water (statistically reliable data) in relative short period of time (representative of one hydrodynamic state). The monitoring protocol should cover the main time scales of variability (tidal cycles, spring/neap tidal cycles, contrasting conditions of river flow or wind) and the estuarine regions characterised by different hydrodynamic regimes.

While these time scales and representative sampling locations strongly depend on the dominant forcing and may vary among systems, the hydro-meteorological conditions under which MPs have been collected should be clearly stated in the papers (tidal phase, current velocity, wind, waves, salinity etc...). Even if this study focuses on microplastic pollution, similar considerations should be taken into account for sampling other kinds of pollutants in estuarine environments.

Unlike classical sediment concentration analyses, MP concentration analyses have substantial financial and time costs that may not be neglected in the field campaign design. A relevant perspective is to test new methods of automatic sampling of MPs, reducing financial and time costs during laboratory analyses and field sampling.

Table 1 :

 1 Comparison of sampling strategies in different estuaries around the world

Figure 1 :

 1 Figure 1: Key processes in regards to microplastic sampling strategies in estuaries

Table 2 :

 2 Recommendations about when and where to sample MP depending on the main characteristics and hydrodynamic processes of an estuarine system

	Estuarine characteristics	Hydrodynamic processes	When	Where
	Tides	tidal pumping, tidal straining, tidal mixing, Stokes drift	ebb/flood/slacks (semi-diurne, diurne), neap/spring	along the tidal intrusion, surface/mid depth/bottom
	Stratification	two-layer flow, gravitational circulation, stratification-induced turbulence damping	ebb/flood, spring/neap, wet/dry season	along the salinity intrusion, surface/mid depth/bottom
	River	tide-river interaction, convergence area	wet/dry season	fluvial area, convergence areas, surface/mid depth/bottom
	Fronts	shear fronts, tidal intrusion fronts	ebb and/or flood	inside/outside front, surface
	Wind	wind mixing, wind driven-circulation	wind/unwind conditions	banks in the downwind direction, surface/subsurface
	Waves	wave mixing, resuspension	swell/calm conditions	mouth of the estuary, banks at the entrance, surface/subsurface/bottom
	Morphology	lateral circulation, differential advection	ebb/flood	thalweg/shoals, away from sills, bends ...
	MP sources	-	ebb/flood, weekdays/weekends, tourist season wet/dry season	MP sources (WWTPs, sewage outflows, mouth/head of the estuary, industrial outflows ...), away from the sources

Acknowledgments

This work was supported by the French National Agency for Research in the context of PLAS-TINEST project (grant N°ANR-22-CE01-0011-01).