Emna Berrich
email: emna.berrich@univ-nantes.fr

Jack Legrand
email: jack.legrand@univ-nantes.fr

EXPERIMENTAL INVESTIGATIONS OF COUETTE-TAYLOR-POISEUILLE FLOWS USING THE ELECTRO-DIFFUSIONAL TECHNIQUE

Couette-Taylor-Poiseuille flow CTPF consists on the superposition of Couette-Taylor flow to an axial flow. The CTPF flow hydrodynamics studies remain rather qualitative or numerical or are restricted to relatively low Taylor and/or axial Reynolds numbers. For more comprehensive and control of CTPF, especially for relatively high Taylor numbers and high axial Reynolds numbers, we investigated experimentally CTF with and without an axial flow, using the electro-diffusion ED method. This technique requires the use of Electro-Diffusion ED probe which allows the determination of the local mass transfer rate from the Limiting Diffusion current measurement delivered by the ED probe while it is polarized by a polarization voltage. From the local mass transfer (the Sherwood number), we determined the wall shear rate using different approaches. The results illustrate that low axial flow can generate a stabilizing effect on the CT flow. The time-evolutions of the local mass transfer and the wall shear rate are periodic. These evolutions characterize the waviness or the stretching of the vortices. However, Taylor Wavy Vortex Flow TWVF is destabilized under the effect of relatively important axial flow. The time-evolutions of wall shear rate are no longer periodic. Indeed, Taylor vortices are overlapped or completely destructed.

NOMENCLATURE

A C D Surface of the probe (m 2 ) Concentration (mol/m 3 ) Coefficient of diffusion (m 2 .s -1 ) Amplitude of the oscillation Constant used in the inverse method Directional angle relative to a pointer, fixed on the sandwich probe mantle (°) Angular velocity of the rotating plate (rad.s -1 ) Kinetic viscosity (m 2 .s -1 ) Dimensionless Temporal average Experimental Numerical Gradient Peclet number Reynolds number Sherwood number

INTRODUCTION

The Couette-Taylor Poiseuille CTP flow consists on the superposition of an axial flow to Couette-Taylor flow. An axial flow can have a stabilizing effect for Taylor Vortex Flow TVF [START_REF] Cornish | Flow of water through fine clearances with relative motion of the boundaries[END_REF]. It can delay the appearance of the first instability to TVF mode. However, an axial flow can also have a stabilizing effect on TVF flow for specific range of Re (0 < Reax < 15) but not another range (15 < Re < 25) [2]. In this paper, we present an experimental investigation using the Electro-Diffusion method to study CTP flows taking into account the history effect, for relatively large Taylor numbers and axial Reynolds numbers (Ta < 181.81) and (Reax < 52.75).

EXPERIMENTAL APPARATUS AND ELECTRO-DIFFUSION METHOD

The experimental installation of Couette-Taylor-Poiseuille is presented on Figure 1. It contains two coaxial cylinders, a rotating inner one and a fixed outer one. The radius of respectively the inner cylinder and the outer one are R1 = 85.5 mm±0.2 and R2 = 100 mm±0.2. The height is equal to H = 450 mm±0.5. The radius ratio is thus η = 0.855. The aspect ratio is Г = 31. The system is constituted also by a storage tank, a pump; a by-pass; a flowrate control valve, a flowmeter; a CT feeder tank; a drain valve; and an electrical motor which assure the inner cylinder motion.

Figure 1. Experimental installation

A ferri-ferrocyanure of potassium is used as electrochemical solution. Its concentration is equal to 25 mol/m 3 . An access of the sulfate of potassium K2SO4 (130 mol/m 3 ) is used as supporting electrolyte. A dispersion of 2.5% of Kalliroscope particles is added to visualize the flow. The 39 electrochemical sensors (38 Single probes and triple probe) must be flush with the internal wall of the outer cylinder end to ensure the polarographic study of vortex-wall interaction. Bonding and securing operations simple probes is relatively difficult due to the cylindrical geometry of the experimental device. Two neighboring sensors are spaced apart by an angle of 7 degrees on the circumference of the inner wall of the outer cylinder of the SCT (Figure2). The two horizontal rows are spaced by a distance of 10 mm. 14 single probes are arranged on a vertical row. A single pair of probes disposed vertically are spaced 10 mm. Two neighboring pairs of single probes are arranged vertically spaced 54 mm.

RESULTS AND DISCUSSION

The Taylor number characterizes the instabilities apparition. It can be written as:

int int int Re i cent R Ta R            (1)
Were the azimuthal Reynolds number is defined by:

Re a R      (2)
1. Effect of axial flow on mass transfer evolution

Indirect protocol

We defined an experimental protocol to study the protocol history effect on the hydrodynamic CTPF flows behaviour: we firstly impose an ascendant axial flow to the fluid at rest on the CTS gap. When the regime is established, we superposed the azimuthal flow. We named this protocol "the indirect protocol". The axial Reynolds number is defined as: .2 Re

ax ax d   ( 5
)
Where uax is the axial velocity imposed to the Taylor Vortex flow and υ is the kinematic viscosity. We firstly present a case of low axial flow without any azimuthal motion. The mass transfer time evolution is presented in Figure 3 for Reax = 2.64. As we can see, the Sherwood numbers values are positive for all the axial and horizontal coordinates (probes positions).

The time evolution of mass transfer for low (Reax = 1 and Ta = 45.45), (Reax = 2.64 and Ta = 90.90) and relatively high Reax (Reax = 20.76 and Ta = 181.81) are respectively presented in Figure 4 to 6. Ta = 45.45 corresponds to the apparition of the first instability.

The visualization using Kalliroscope [START_REF] Berrich | On the stability of Taylor -Couette Flow with Axial Flow[END_REF] illustrates that at low axial flow (Reax = 0.94), the Taylor vortices start to move in the axial flow direction. At Reax = 1.45, helical modes with an increasing number of helices which have a "helicity" opposite to the imposed axial flow are developed. The helical vortices celerity varies between 1 to 1.3 times the axial flow rates. This confirms the results of [START_REF] Johnson | Hydrodynamic stability of flow between rotating porous cylinders with radial and axial flow[END_REF].

When the axial Reynolds number increases, the mass transfer becomes having sinusoidal evolution. This variation characterize vortices apparition. It increases when the Taylor number increases. For a same proportional forces effect (axial/azimuthal), the Sherwood number becomes more important when the axial Reynolds number increases.

Direct protocol

We initially impose an azimuthal motion to the fluid at rest. When the regime is established, we superposed the axial flow. We named this protocol "the direct protocol". The time evolution of the Sherwood number for relatively low axial Reynolds numbers at the first instability (Ta = 45.45 and Reax = 0.94), (Ta = 181.81 and Reax = 1.10), and high Reynolds number (Ta = 181.81 and Reax = 52.75).As we can see, the vortices manifests their presence by the mass transfer variation.

6 7 8 F G R

Low axial flow effect on CF instability

The wall shear rate evolution is illustrated for low Reynolds numbers with an azimuthal flow (Reax = 1 and Ta = 45.45), respectively in Figure 13-15. In spite of a well developed axial flow, the vortices appear.

High axial flow effect on WTVF

The wall shear rate evolution is illustrated for relatively high Reynolds numbers with an azimuthal flow: Reax = 20.76 and Ta = 181.81 respectively in Figure 19-21. The vortices appear even if an axial flow corresponding to Reax = 20.76 is well developed. The shear rate depends on the special disposition of the probes.

Indirect protocol

Low axial flow effect on CT instability

The wall shear rate evolution is presented for Ta = 45.45 and Reax = 0.94 in Figure 22-24. The wall shear rate values are lower than the indirect protocol case.

High axial flow effect on WVF

The wall shear rate evolution is presented for Ta = 181.81 and Reax = 52.75 in Figure 28-30. Comparing to low Reynolds number case, the wall shear rate decreases when we increase the axial Reynolds number. There is vortices detachment.

CONCLUSION

Mass transfer and wall shear rate time evolution for different horizontal and vertical CTPF positions, according to direct and indirect protocol are presented. The history effect are illustrated for relatively large range of Taylor and axial Reynolds numbers. Axial flow have a stabilizing effect on vortices or a destabilizing effect according to the torsional flow (Ta) and axial flow (Re) ratio. In such destabilizing cases, the vortices are either folded or stretched or overlapped or completely destructed. In other cases, it persists.

  of oscillation (Hz) Gap thickness between the front and the back segments (m) Length of the gap between the rotating disk and the fixed one (m) Current (A) width of the probe (m) Perimeter of the probe (m) Radial distance (mm) Wall shear stress (s -1 ) Time (s) Velocity (m.s -1 ) Coordinates (m)

Figure 2 .

 2 Figure 2. Electrochemical probes DISPOSITION FROM 1 to 14 and from A to X

Figure 6 .Figure 5 .Figure 4 .Figure 3 .

 6543 Figure 6. Time evolution of the Sherwood number for Reax = 20.76 and Ta = 181.81

1 .Figure 9 .Figure 8 .Figure 7 .

 1987 Figure 9. Time evolution of the Sherwood number for Ta = 181.81 and Reax = 52.75

Figure 18 .Figure 19 .Figure 16 .

 181916 Figure 18. Time evolution of wall shear rate (Deslouis et al. method) for Reax = 2.64 and Ta = 90.90

Figure 27 .Figure 26 .Figure 28 .Figure 25 .

 27262825 Figure 27. Time evolution of wall shear rate (Deslouis et al. method) for Ta = 181.81 and Reax = 1.10

. Low axial flow effect on WTVF

  The wall shear rate evolution is presented for low Reynolds numbers with an azimuthal flow: Reax = 2.64 and Ta = 90.90, respectively in Figure16-18. The vortices appear when we inilially impose an axial Reynolds corresponding to Reax = 2.64. Leveque solution shear rate remains always positive. It is attenuated comparing to Sobolik et al., and Deslouis solutions.

	t(s) Figure 11. Time evolution of wall shear rate (Sobolik et al. method) for Reax = 2.64 1 1.5 2 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 7 8 F G R S 2.1.30.5 S LEVEQUE (s-1) 1.6 6	0.5 Figure 13. Time evolution of wall shear rate (Leveque method) for 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t(s) S LEVEQUE (s-1) 6 7 8 F G R S G 40 60 6 7 F 8 Reax = 1 and Ta = 45.45 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -80 -60 -40 -20 40 60 80 t(s) S 6 7 8 F G R 0 20 DESLOUIS (s-1) S
												R
										6	SOBOLIK (s-1)	0 20	S
		60								7	S	-20
										8
		40								F G		-40
	SOBOLIK (s-1)	0 20								R S		-60	0.2	0.3	0.4	0.5 t(s)	0.6	0.7	0.8
	for Reax = 2.64 Figure 12. Time evolution of wall shear rate (Delouis et al. method) -20 S	Figure 14. Time evolution of wall shear rate (Sobolik et al. method) for Reax = 1 and Ta = 45.45
		-40
												6
		-60	0.4	0.45	0.5	0.55 t(s)	0.6	0.65	0.7	0.75		60	7 8 F
	Figure 17. Time evolution of wall shear rate (Sobolik et al. method) for Reax = 2.64 and Ta = 90.90	DESLOUIS (s-1)	0 20 40	G R S
											S
												-20
												-40
												-60
												0.2	0.4	0.6	0.8	1	1.2
												t(s)
											Figure 15. Time evolution of wall shear rate (Delouis et al. method) for Reax = 1 and Ta = 45.45

. Low axial flow effect on WVF The

  wall shear rate evolution is presented for Ta = 181.81 and Reax = 1.10 in Figure25-27. The wall shear rate increases when we increase the Taylor number for a same axial Reynolds number. It has a quasi-sinusoidal evolution characterizing that the vortices persists.

	Figure 20. Time evolution of wall shear rate (Sobolik et al. method) for Reax = 20.76 and Ta = 181.81 1 1.5 2 0 0.5 1 1.5 2 2.5 S LEVEQUE (s-1) 7 8 F G R S 2.2.20.5 3 6	0.8 Figure 22. Time evolution of wall shear rate (Leveque method) for 1 1.2 1.4 1.6 1.8 2 2.2 2.4 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t(s) S LEVEQUE (s-1) 6 7 8 F G R S 60 80 6 7 8 Ta = 45.45 and Reax = 0.94 0.8 1 1.2 1.4 1.6 1.8 2 -200 -150 -100 50 100 t(s) S 6 7 8 F G R -50 0 DESLOUIS (s-1) S
						t(s)					40	G F
		100							6 7	SOBOLIK (s-1) S	-20 0 20	R S
									8		-40
									F
		50							G R		-60
	SOBOLIK (s-1)	0							S		-80	0.4	0.6	0.8 t(s)	1	1.2
	S	Figure 21. Time evolution of wall shear rate (Deslouis et al. method) for Reax = 20.76 and Ta = 181.81 -50	for Ta = 45.45 and Reax = 0.94 Figure 23. Time evolution of wall shear rate (Sobolik et al. method)
		-100
											80
											6
		-150	0.4	0.5	0.6	0.7	0.8	0.9	1		60	8 7
						t(s)					40	G F
										DESLOUIS (s-1)	-20 0 20	R S
										S
											-40
											-60
											-80
											0.1	0.2	0.3	0.4	0.5	0.6	0.7
											t(s)
											Figure 24. Time evolution of wall shear rate (Deslouis et al. method) for Ta = 45.45 and Reax = 0.94

ACKNOWLEDGMENTS

This work was supported by the laboratories GEPEA UMR CNRS 6144 (University of Nantes), LAMIH (UMR CNRS 8201 (University of Valenciennes) and the "Agence Nationale de la Recherche (ANR), France", by the grant n° ANR-08-BLAN-0184-02. These supports are gratefully acknowledged.


Rogelio Chovet
email: rogelio.chovet@univ-valenciennes.fr

Fethi Aloui
email: fethi.aloui@univ-valenciennes.fr

Laurent Keirsbulck
email: laurent.keirsbulck@univ-valenciennes.fr

GAS-LIQUID FOAM THROUGH STRAIGHT DUCTS AND SINGULARITIES: CFD SIMULATIONS AND EXPERIMENTS

Some industrial processes are associated with the flow of aqueous foams inside horizontal channels. Examples are found in the oil, food and cosmetic industries. This type of flow presents an important pressure loss, originated from the shear stress exerted by the channel walls. Foam flow is one of the most complex fluids. In a macroscopic point of view, the physical-chemical interaction between the bubbles can be related to some non-Newtonian models (Bingham law, power law, etc.) or an apparent viscosity. These last can represent the internal deformations of fluid elements when shear stress is applied. An experimental facility able to create this type of flow is not so easy to design. Many parameters must be taken into consideration. So, Computational Fluid Dynamics (CFD) constitutes an ideal technique for analyzing this kind of problem. The aim of this study is to validate the use of Computational Fluid Dynamics in order to correctly predict the pressure losses and the velocity fields of a foam flowing through a straight channel and singularities (fence and half-sudden expansion). Simulations for a realistic scenario: two-phase flow, change in the surface tension, bubble size, were undertaken. Obtained results showed that simulations are not able to accurately reproduce for such a complex fluid, the important aspects of this study, such as the pressure losses and the velocity fields. Therefore, an approximation to a Bingham fluid was made. For a foam flow quality of 70% and a velocity of 2 cm/s, the numerical results are justified by experimental evidence. Experiments have been done and predictions for the flow * Address all correspondence to this author.

behavior are extrapolated. Results show that the software is able to recreate the behavior of foam flow through a straight channel and singularities. However, this approach is extremely sensitive to the choice of several parameters, like the apparent viscosity, the yield stress, the viscosity consistence, etc.

NOMENCLATURE

Roman letters e x Unit vector of the Ox axis P Static pressure (Pa) P 0 Static pressure at the exit of the duct (Pa) u Velocity component in the x direction ( cm / s ) u Velocity vector ( cm / s ) v Velocity component in the y direction ( cm / s ) x Horizontal tranverse coordinate (mm) y Vertical transverse coordinate (mm) Greek letters γ Yield ( 1 / s ) k Viscosity Consistency (Pa.s) ρ Density ( kg / m 3 ) τ Stress (Pa) τ c Yield Stress (Pa)

INTRODUCTION

The word foam originates from the medieval german veim. It is a substance that is formed by trapping pockets of gas in a liquir or a solid. In most of them, the gas volume is larger than the liquid one. The liquid is located in thin films separating the regions of gas. They are in general opaque, remarkably stable, and even elastic, as is evident for the shaving foam. The mechanical properties of foams (low density, high specific surface, disjoining pressure, foam-wall viscous friction) are important in several industrial applications. In oil recovery, for example, foam is used efficiently to drag petroleum out of porous media while augmenting the well pressure. The improve of these industrial processes will only be achieve with a better understanding of the foam rheology and accurate aproximations of foam flow behavior.

Rheology is the study of the deformation and flow of complex fluids that can exhibit both liquid and solid behavior. Although foams are mostly compose of gas, its rheological properties can be assimilated to those of a complex fluid [START_REF] Cantat | Foams: Structures and Dynamics[END_REF]. Aqueous foams typically exhibit a combination of the following three types of behavior:

Elastic: Foam stores up mechanical energy, later on it releases it to return to its initial state. Plastic: Beyond a certain stress point the deformation is irreversible and the foam will acquire and retain a new shape. Viscous: The foam flows like a viscous liquid, dissipating an amount of energy that depends on the deformation rate. This rheological behavior is a consequence of the high interfacial area [START_REF] Cohen-Addad | Viscoelastic response of a corsening foam[END_REF]. Due to the many different length-and timescales, foam rheology is particularly difficult to model. For the macroscopic scale (the foam sample is considered as a continuous medium), constitutive laws describe the relation between stress, deformation and flow. If we gradually increase the strain applied to the foam, the elastic stress becomes a non-linear function of the strain and in the end the material yields. in order to model the steady flow of materials that behave both as a solid or a liquid (depending on the applied stress) over a large range of strain rates, we must account for viscous stresses. The Bingham model is often used, for this kind of fluids the rheological behavior is expressed in Eqn. [START_REF] Cantat | Foams: Structures and Dynamics[END_REF].

γ = 0 if τ < τ y τ = τ c + kγ if τ > τ c (1)
In order to study the foam flow behavior several approaches have been used. Theoretical studies have been limited because of the complexity of all the parameters involved in foams physycs and mathematics (plateau border stability, disjoining pressure, coalescence, drainage, ripening, etc.). Despite the complexity, important results are available [START_REF] Weaire | The rheology of foam[END_REF]. The experimental approach has also numerous disadvantages, from the measurement methods to the foam flow stability regulation [START_REF] Aloui | Wet foam flow under a fence located in the middle of a horizontal duct of square section[END_REF][START_REF] Aloui | Experimental investigation of a wet foam flow through a horizontal sudden expansion[END_REF][START_REF] Chovet | Wet foam flow in horizontal square duct and through singularities[END_REF]. For all these reasons numerical approach seems more appropiate. There have been some foam simulations. However, these ones were focused in different subjects: foam generation or deterioration, physical properties, complex rheological behavior and for heat exchanging intentions [START_REF] Herzhaft | Measurement and modeling of the flow behavior of aqueous foams using a recirculation pipe rheometer[END_REF][START_REF] Feitosa | Gas and liquid transport in steady-state aqueous foam[END_REF][START_REF] Ireland | Coalescence in steady-state rising foam[END_REF][START_REF] Gabrielaitiene | Modelling of aqueous foam flow for developing energy-efficient heat exchanger applications[END_REF]. Also, none of them have made an intensive study over an horizontal foam flow and compared both, experimental and numerical, results.

For this analysis, Computational Fluid Dynamics, CFD, is an appropiate method for obtaining and approximation of the foam behavior when flowing through a straight channel and singularities (fence and hal-sudden expansion) . CFD allows the consideration of different parameters, for non-Newtonian flows, such as velocity, pressure losses, shear stress, parameters that directly affect the flow regime, fluid mobility, and pressure profile along the channel. In particular, conclusion will be based in the evolution of the velocity and pressure profiles along the channel and in presence of singularities, then they will be compared against the experimental results obtained for the same conditions.

GEOMETRY AND NUMERICAL PARAMETERS

The dimensions of the numerical geometry were related to the experimental ones [START_REF] Aloui | Wet foam flow under a fence located in the middle of a horizontal duct of square section[END_REF][START_REF] Aloui | Experimental investigation of a wet foam flow through a horizontal sudden expansion[END_REF]. The length of the channel variates as a function of the singularity used. For the square configuration it is 3.13 m, for the half-sudden is 3.87 m and for the fence 3.41 m. The inlet of the channel has a square section of 21 x 21 mm 2 . The measurement section is located 1.3 m downstream of the conducts input. At this level the two singularities were placed. The fence (Figure 1) presents a height of 10 mm and an escape angle of 45 • . As for the half-sudden expansion (Figure 2), its upstream has the same square section as the fence (21 x 21 mm 2 ). However, with a section ratio of 0.5 the downstream area is 21 x 42 mm 2 .

The coordinate systems for both, the fence and the halfsudden expansion were located at the height of the singularity. Therefore all measurement done downstream of it will have a positive value in the x direction and all done upstream a negative one.

In order to decrease the CPU time calculations and require- ments, all three cases were assumed to have a 2D geometry. Therefore, the depth of the geometry was assumed as infinite (x coordinate).

For the grid construction, mesh seeds are located in the axial and vertical direction of the models, then hexahedral elements are generated. Hexahedral elements allow better results than tetrahedral ones in terms of computing time and accuracy; therefore; the former types of elements are selected for this study.

For each geometry a small refinement was done for the interest regions, at the fence and at the expansion. A grid was considered adequate when the average velocity keeps a relative difference lesser than 0.1% when compared with the results obtained by using a refined grid with the double number of elements. For this analysis, refinement is done by altering the size of the element through the whole geometry. The mentioned average velocity is evaluated at the inlet of duct and at 1.5 m from it.

A foam inside a straight duct can present three different regimes, depending on its velocity [START_REF] Blondin | Etude experimentale d'une mousse humide en coulement en conduite horizontale de section rectangulaire[END_REF]:

One-dimensional flow: For this regime the flow behaves as a whole, it moves as a block or a piston. The velocity vectors have only one uniform axial component in the flow direction. Three-dimensional flow: it is obtained when the established flow velocity vector has an axial component that depends on

u = u. e x = cte. e x ( 2
u = u(y, z). e x ( 4
)
To establish the practical relevance of these considerations, the numerical simulations were done undertaking a regime close to the one (one-dimensional) used for the experimental measurements [START_REF] Aloui | Wet foam flow under a fence located in the middle of a horizontal duct of square section[END_REF][START_REF] Aloui | Experimental investigation of a wet foam flow through a horizontal sudden expansion[END_REF].

Simulations with the Eulerian-Eulerian multiphase models were made. They took into consideration all the important parameters that a foam flow could have: a continuous phase and a disperse phase, diminution of the surface tension for the liquid phase, bubble size, drainage, wall lubrication force, drag force, etc. Despite all the data available inside the experimental results, the software was not able to accurately reproduce the viscoelasto-plastic behavior of such a complex fluid. However, in a macroscopic point of view, the physical-chemical interactions between the bubbles can be related to some non-Newtonians models, Bingham, Power Law, etc.

Therefore, an isothermal and laminar Bingham fluid will try to represent the conduct of an aqueous foam with a void fraction of 70% (70/100 air volume) and a bubble diameter of 0,5 mm. Table [START_REF] Cantat | Foams: Structures and Dynamics[END_REF] shows the rheological parameters that describe each Bingham fluid that better aproaches each experimental case (half-sudden expansion and fence).

RESULTS AND DISCUSSIONS

The results obtained from the simulations are presented in terms of streamlines, velocity profiles and pressure gradients for the two singularities studied (half-sudden expansion and fence). These ones will be compared against the experimental results of Aloui et al. [START_REF] Aloui | Wet foam flow under a fence located in the middle of a horizontal duct of square section[END_REF][START_REF] Aloui | Experimental investigation of a wet foam flow through a horizontal sudden expansion[END_REF].

Streamlines

The streamlines for the fence and the half sudden expansion were calculated from the mean veolicty fields from the CFD simulations (Figure 3). They show the creation of death zones for both geometries. The presence of the singularities causes the For this cases foam flow can be divided in two regions: the principal foam flow and the dead zones were the bubbles are almost motionless.

Pressure distribution

The pressure distribution is an essential element for modelling any kind of flow. Therefore, it is important to find out if our numerical simulations are able to recreate this behavior. Figs. 4 and5 show the regular pressure evolution along the duct containig the half-sudden expansion and the fence for both experimental and numerical data. They have the same comportment, far from the singularities the static pressure varies linearly at the upstream and downstream of the duct.

Local averaged velocity profiles

In order to realize the nature of the foam flow, upstream and downstream of the singularities, we have compared the axial velocity component profiles. Figs. 6 and7 show the profiles for the half-sudden expansion and the fence respectively. We notice that the axial velocity presents the one-dimensional regime explained before. For the numerical results, the flow tends to shear more near the wall and the non-slipping condition is observed. However, all profiles approach considerable close.

Figs. 8 and9 represent the comparison of some profiles of the axial velocity component along the lateral side of the duct, downstream of the expansion and both downstream and upstream of the fence. For the half-sudden expansion we note that the flow slows down as we move away from the singularity until it reaches a balance value (1.1 cm/s) and it remains constant until the exit of the duct. As for the fence, we remark an acceleration as we move closer to the singularity reaching a value of 4.3 cm/s. Then the flow returns to the same velocity that it had before passing the obstacle (2.15 cm/s). The presence of the death zones, for both cases is clearly noticed near the singularities where the axial velocity tends to 0. The absence of a negative axial velocity The numerical results present a better approach for the half-sudden expansion case that for the fence one.

For the same plane the vertical velocity component profiles were obtained (Figs. 10 and11). In the half-sudden expansion we notice that there are some vertical velocities that are more important in the immediate vicinity of the singularity. They are related to the upward movement of the flow trying to occupy the rest of the duct. These velocities also show that after the section expansion a fully developed flow is reached when moving away from the singularity. For the fence, as in a nozzle, the reduction of the section creates a descending vertical movement towards the passage section below the singularity. The foam leaves the obstacle to occupy the rest of the duct while decelerating gradually until reaching a constant velocity and returning to the onedimensional flow. The numercial results present the same behavior as the experimental ones. However, some particular points present high deviations, x = 20.7 mm for the half-sudden expansion and x =-5 mm for the fence. Fig. 12 shows the velocity components evolutions along the x direction near the top side of the half-sudden expansion. In the axial component graph we can clearly observe that the death zone, are with a zero velocity, for the numerical results has a size two times bigger than the experimental one. This means that the foam flow adapts faster than the Bingham fluid to the section change and it quickly starts to fills the void created by the singularity. However, they both seem to reach a fully developed flow near the same velocity (0.98 cm/s) and both curves behave equally. For the vertical component the same phenomenon is witnessed. The death zone prolongs farther away for the CFD simulations and then flow acceleration occurs filling the empty space at the top of the expansion. A maximum velocity is achieved; this one is bigger for the experimental results. The highest deviation between the experimental and the numerical results is 24%. As for the fence Fig. 13 shows the longitudinal profiles in the vicinity of the singularity. The first one represents the axial velocity. We can observe that both, foam flow and Bingham fluid, slow down gradually upstream of the obstacle. Later on they stop completely at the bottom of the corner formed by the fence and the duct (-30 mm < x < 0 mm). In the downstream of the fence, the flows start to accelerate continuously, coming out of the second death zone, to reach a constant velocity, with a value close The death zones at the both side of the fence can also be noted, more clearly for the numerical results. The deviation for the vertical profiles is much higher than for the axial ones, reaching a maximum difference of almost the double between the experimental and numerical data. The graphs appearence show once more that foam flow has a capacity to adapt to section change faster than the Bingham fluid. Because these experimental results were only available for the half-sudden expansion, the variation of the axial and vertical velocity components along the expansion direction (x direction) are represented in Figs. 14 and15 for different positions on the y direction. From the axial velocity profiles, we observe that the velocity all along the height of the channel descends until reaching the same value creating the piston flow. This behavior is expected for this kind of flow. This deceleration is due to the increase of the flow area section, after this change the flow tries to reach an equilibrium point and a minimal uniform speed far from the singularity. The same situation is equally described by the vertical profiles. We see the flow vertical acceleration filling the void after the expansion and then a transition period until the piston flow is achieved and the vertical velocity is null. Both numerical and experimental plots present the sama appearence, and the maximum deviation between them is 16%.

CONCLUSIONS

In this paper, CFD simulations of Bingham fluids through a half-sudden expansion and a fence were compared against experimental results of a foam flow in the same conditions. The parameters choose to undertake this comparison were the flow streamlines, the pressure distribution along the whole channel length and the velocity profiles near the two singularities. The transport characterization of foam flow raises at least two prob- To be able to simplify the rheological problem by relating such a complex fluid to a simpler one is an important step for simulations of foam flow and behavior comprehension. Both the CFD and the experimental results graphics present the same appearance when compared one against the other. The fluids, foam flow and Bingham, behave equally when faced against a half-sudden expansion and a fence. One of the most important aspect of any rheological model is the interaction between the wall and the fluid, wall shear stress, velocity gradient and pressure losses. The pressure distribution for the both geometries is exactly the same, with a maximum deviation of 5%. As for the velocities profiles, the slipping condition that bubbles have over the channel walls plays a major role in the difference between the numerical and the ex- In order to do this, 3D simulation should be undertaken for this same conditions and the use of more elaborated models is recommended, such as the Power Law rheological model and even a visco-elasto-plastic model.

FIGURE 1 :

 1 FIGURE 1: LATERAL VIEW OF THE FENCE [mm]

FIGURE 2 :

 2 FIGURE 2: LATERAL VIEW OF THE HALF-SUDDEN EX-PANSION [mm]

)

  Two-dimensional flow: It is obtained when the established flow has an axial velocity component that only depends of the y coordinate u = u(y). e x[START_REF] Weaire | The rheology of foam[END_REF]

FIGURE 3 :FIGURE 4 :

 34 FIGURE 3: STREAMLINES OF THE MEAN VELOCITY FIELDS: a) HALF-SUDDEN EXPANSION b) FENCE

FIGURE 5 :FIGURE 6 :

 56 FIGURE 5: EVOLUTION OF THE STATIC PRESSURE FOR THE FENCE

FIGURE 7 :FIGURE 8 :

 78 FIGURE 7: PROFILE OF THE AXIAL VELOCITY COMPO-NENT FOR THE HALF SUDDEN EXPANSION: (a) AT THE UPSTREAM AND (b) DOWNSTREAM

FIGURE 9 :

 9 FIGURE 9: AXIAL VELOCITY PROFILE COMPONENTS AT DIFFERENT DISTANCES FOR THE FENCE

FIGURE 10 :

 10 FIGURE 10: VERTICAL VELOCITY PROFILE COMPO-NENTS AT DIFFERENT DISTANCES FOR THE HALF-SUDDEN EXPANSION

FIGURE 11 :FIGURE 12 :FIGURE 13 :

 111213 FIGURE 11: VERTICAL VELOCITY PROFILE COMPO-NENTS AT DIFFERENT DISTANCES FOR THE FENCE

FIGURE 14 :

 14 FIGURE 14: AXIAL VELOCITY IN THE VICINITY OF THE HALF-SUDDEN EXPANSION AT DIFFERENT HEIGHTS

FIGURE 15 :

 15 FIGURE 15: VERTICAL VELOCITY IN THE VICINITY OF THE HALF-SUDDEN EXPANSION AT DIFFERENT HEIGHTS

TABLE 1 :

 1 RHEOLOGICAL PROPERTIES OF FLUIDS IN THIS STUDYGeometryρ( kg / m 3 ) τ c (Pa) k (Pa.s)

	Half-Sudden Expansionl	300.22	6.22	0.001
	Fence	300.22	4.6	0.001
	the z and y coordinates.


Irene Calderón-Sanou

| Lucie Zinger

| Mickael Hedde

Camille Martinez-Almoyna

Amelie Saillard

Julien Renaud

| Ludovic Gielly

Norine Khedim

Clement Lionnet

| Marc Ohlmann

| Orchamp Consortium

Tamara Münkemüller

Wilfried Thuiller

Alfried Vogler

Energy and physiological tolerance explain multi-trophic soil diversity in temperate mountains

Aim: Although soil biodiversity is extremely rich and spatially variable, both in terms of species and trophic groups, we still know little about its main drivers. Here, we contrast four long-standing hypotheses to explain the spatial variation of soil multitrophic diversity: energy, physiological tolerance, habitat heterogeneity and resource heterogeneity.

Location: French Alps.

Methods:

We built on a large-scale observatory across the French Alps (Orchamp) made of seventeen elevational gradients (~90 plots) ranging from low to very high altitude (280-3,160 m), and encompassing large variations in climate, vegetation and pedological conditions. Biodiversity measurements of 36 soil trophic groups were obtained through environmental DNA metabarcoding. Using a machine learning approach, we assessed (1) the relative importance of predictors linked to different ecological hypotheses in explaining overall multi-trophic soil biodiversity and (2) the consistency of the response curves across trophic groups.

Results:

We showed that predictors associated with the four hypotheses had a statistically significant influence on soil multi-trophic diversity, with the strongest support for the energy and physiological tolerance hypotheses. Physiological tolerance explained spatial variation in soil diversity consistently across trophic groups, and was an especially strong predictor for bacteria, protists and microfauna. The effect of energy was more group-specific, with energy input through soil organic matter strongly affecting groups related to the detritus channel. Habitat and resource heterogeneity had overall weaker and more specific impacts on biodiversity with habitat heterogeneity affecting mostly autotrophs, and resource heterogeneity affecting bacterivores, phytophagous insects, enchytraeids and saprotrophic fungi.

Main Conclusions: Despite the variability of responses to the environmental drivers found across soil trophic groups, major commonalities on the ecological processes

| INTRODUC TI ON

With the ever-increasing availability of biodiversity information, a global synthesis on the major ecological determinants of broadscale biodiversity patterns is starting to emerge [START_REF] Belmaker | Relative roles of ecological and energetic constraints, diversification rates and region history on global species richness gradients[END_REF][START_REF] Braga | Spatial analyses of multi-trophic terrestrial vertebrate assemblages in Europe[END_REF][START_REF] Pontarp | The latitudinal diversity gradient: Novel understanding through mechanistic ecoevolutionary models[END_REF][START_REF] Thuiller | Productivity begets less phylogenetic diversity but higher uniqueness than expected[END_REF]. This general understanding is pivotal to forecast how biodiversity responds to natural and anthropogenic changes [START_REF] Mcgill | Fifteen forms of biodiversity trend in the Anthropocene[END_REF][START_REF] Urban | Soil nematode abundance and functional group composition at a global scale[END_REF]. Yet, most of the empirical support is grounded on specific aboveground macroorganisms, in particular vertebrates and plants. Comparatively, soil biodiversity has been largely less studied [START_REF] Guerra | Blind spots in global soil biodiversity and ecosystem function research[END_REF], although it represents one quarter of global diversity and is essential for decomposition, nutrient cycling or carbon sequestrations [START_REF] Delgado-Baquerizo | Multiple elements of soil biodiversity drive ecosystem functions across biomes[END_REF][START_REF] Wagg | Soil biodiversity and soil community composition determine ecosystem multifunctionality[END_REF].

Therefore, it remains unclear whether the ecological hypotheses that hold true for aboveground systems, such as the energy or the habitat heterogeneity hypotheses, also apply to the massive bulk of belowground biodiversity [START_REF] Bardgett | Patterns and determinants of soil biological diversity[END_REF][START_REF] Decaëns | Macroecological patterns in soil communities[END_REF].

Historically, the complexity of studying the soil compartment, for example, complex physical structure [START_REF] Young | Interactions and self-organization in the soil-microbe complex[END_REF], taxonomic impediment [START_REF] Decaëns | Macroecological patterns in soil communities[END_REF], scale of approach [START_REF] Bardgett | Patterns and determinants of soil biological diversity[END_REF][START_REF] Ettema | Spatial soil ecology[END_REF][START_REF] Thakur | Towards an integrative understanding of soil biodiversity[END_REF], has hindered the integration of soil biodiversity into a broader ecological hypothesis testing framework. Yet, our ability to study soil biodiversity at large spatial scales is constantly improving with joint taxonomic efforts, the development of new sampling technologies (e.g. eDNA metabarcoding) and the increase of collaborative databases and initiatives (e.g. Drilobase, Earth microbiome project, Global Soil Biodiversity Initiative). Global-scale analyses have thus recently emerged for several soil organism groups (e.g. [START_REF] Tedersoo | Global diversity and geography of soil fungi[END_REF] for fungi; [START_REF] Delgado-Baquerizo | A global atlas of the dominant bacteria found in soil[END_REF] for bacteria; Phillips et al., 2019 for earthworms;[START_REF] Phillips | Global distribution of earthworm diversity[END_REF] for nematodes; [START_REF] Oliverio | The global-scale distributions of soil protists and their contributions to belowground systems[END_REF] for protists), unveiling their environmental drivers. Yet, whether soil biodiversity at all its taxonomic and trophic levels responds to the same ecological drivers as aboveground diversity and follows similar trends remains to be tested. For such tests, the integration of spatial scales and the scale at which organisms are analysed together is pivotal [START_REF] Thakur | Towards an integrative understanding of soil biodiversity[END_REF][START_REF] White | Methods and approaches to advance soil macroecology[END_REF]. Indeed, the way environmental parameters drive local diversity can depend on the spatial extent (e.g. [START_REF] Steiner | Cyclic assembly trajectories and scale-dependent productivity-diversity relationships[END_REF], or the taxonomic or trophic groups being studied (e.g. [START_REF] Boyero | Global distribution of a key trophic guild contrasts with common latitudinal diversity patterns[END_REF][START_REF] Peters | Predictors of elevational biodiversity gradients change from single taxa to the multi-taxa community level[END_REF][START_REF] Tedersoo | Global diversity and geography of soil fungi[END_REF].

Among the hypotheses formulated to explain the spatial variation of biodiversity, theory and support from empirical studies on plants and other aboveground organisms have led to four major ecological hypotheses: the "energy hypothesis", the "physiological tolerance hypothesis", the "habitat heterogeneity hypothesis" and the "resource heterogeneity hypothesis" (Figure 1). Yet, these hypotheses have been seldom tested in a single framework for soil organisms [START_REF] Decaëns | Macroecological patterns in soil communities[END_REF][START_REF] Thakur | Towards an integrative understanding of soil biodiversity[END_REF], and even less at the scale of the whole soil biota. Observing diversity patterns of soil organisms in nature, that is, the relationship between various relevant predictors and soil diversity, is a first step to test whether these ecological hypotheses apply to the wide range of soil organisms [START_REF] Shade | Macroecology to unite all life, large and small[END_REF].

The "energy hypothesis" predicts a positive relationship between diversity and energy. An increasing amount of energy (i.e. thermic, solar or chemical) promotes diversity across trophic levels by increasing speciation rates and/or the number of species populations, and thereby reducing local extinction [START_REF] Evans | Species-energy relationships at the macroecological scale: A review of the mechanisms[END_REF][START_REF] Wright | Species-energy theory: An extension of speciesarea theory[END_REF]).

An extension of the hypothesis predicts a hump-shaped relationship with a decrease in diversity at high energy levels due to exclusive competition [START_REF] Mittelbach | What is the observed relationship between species richness and productivity[END_REF]. Plant productivity is traditionally used as a primary energy measure, because it accounts for water limitations in the transformation of solar energy into available resources, and because plants are the main basal resource (primary producers) for aboveground organisms [START_REF] Currie | Predictions and tests of climate-based hypotheses of broad-scale variation in taxonomic richness[END_REF][START_REF] Evans | Species-energy relationships at the macroecological scale: A review of the mechanisms[END_REF]. Yet, in the soil compartment, soil organic matter (SOM) is also a major source of energy fuelling the soil food web [START_REF] Moore | Detritus, trophic dynamics and biodiversity[END_REF]. The local amount and content of SOM is driven by multiple drivers such as plant community composition, climate or parent material [START_REF] Wiesmeier | Soil organic carbon storage as a key function of soils -A review of drivers and indicators at various scales[END_REF], and not only by plant productivity.

Considering both solar energy and SOM, hereafter referred as primary and secondary energy, respectively, is thus essential to test the energy-diversity relationship for the soil biota. Therefore, since most soil organisms are thought to be weakly limited by competition due to their limited mobility and the complexity of the soil matrix [START_REF] Ettema | Spatial soil ecology[END_REF][START_REF] Wardle | The influence of biotic interactions on soil biodiversity[END_REF], it could be expected that soil diversity increase monotonously with available energy.

The "physiological tolerance hypothesis" states that favourable environmental conditions support higher biodiversity because a wider range of strategies can persist under such conditions (i.e. tighter niche packing), while only a few well-adapted species can tolerate stressful conditions [START_REF] Currie | Predictions and tests of climate-based hypotheses of broad-scale variation in taxonomic richness[END_REF][START_REF] Spasojevic | Inferring community assembly mechanisms from functional diversity patterns: The importance of multiple assembly processes[END_REF].

structuring soil biodiversity emerged. We conclude that among the major ecological hypotheses traditionally applied to aboveground organisms, some are particularly relevant to predict the spatial variation in soil biodiversity across the major soil trophic groups.

K E Y W O R D S

environmental DNA metabarcoding, French Alps, macroecology, random forest, soil biodiversity, trophic groups Temperature is one of the most acknowledged factors constraining the "thermal niche" of organisms. Yet, compared to aboveground temperature, soil temperatures are buffered making it more difficult to isolate its effect on soil biodiversity. For example, in mountain environments, soil temperature is strongly regulated by snow cover and duration [START_REF] Carlson | Modelling snow cover duration improves predictions of functional and taxonomic diversity for alpine plant communities[END_REF]. In the absence of snow, soil frost might impact the structure and activity of soil communities [START_REF] Schostag | Bacterial and protozoan dynamics upon thawing and freezing of an active layer permafrost soil[END_REF][START_REF] Sulkava | Effects of hard frost and freeze-thaw cycles on decomposer communities and N mineralisation in boreal forest soil[END_REF]. In addition, soil organisms often rely on other abiotic conditions such as water availability, heavy metal content and pH that can generate stressful conditions at extreme values, for example, drought, toxicity, acidity [START_REF] Gans | Computational improvements reveal great bacterial diversity and high metal toxicity in soil[END_REF][START_REF] Xu | Seasonal exposure to drought and air warming affects soil collembola and mites[END_REF]. Indeed, soil pH is recognized as a major driver of soil microorganisms diversity [START_REF] Fierer | The diversity and biogeography of soil bacterial communities[END_REF].

While the stressful environmental factors may differ, the general response form to stress should be the same for above and belowground diversity.

The "habitat heterogeneity hypothesis" postulates that increasing habitat heterogeneity provides larger niche space or dimensionality that can be finely partitioned and sustain more coexisting species [START_REF] Stein | Environmental heterogeneity as a universal driver of species richness across taxa, biomes and spatial scales[END_REF][START_REF] Tews | Animal species diversity driven by habitat heterogeneity/diversity: The importance of keystone structures[END_REF]. Traditionally, the "habitat heterogeneity hypothesis" is tested at the landscape scale where biodiversity increases with habitat or vegetation diversity [START_REF] Stein | Environmental heterogeneity as a universal driver of species richness across taxa, biomes and spatial scales[END_REF]. However, soils can harbour a high degree of heterogeneity at much smaller grains than those considered aboveground [START_REF] Young | Interactions and self-organization in the soil-microbe complex[END_REF], and this partly explains their remarkably high biodiversity [START_REF] Ettema | Spatial soil ecology[END_REF][START_REF] Nielsen | The enigma of soil animal species diversity revisited: The role of small-scale heterogeneity[END_REF]. On a microscale, habitat heterogeneity can be structural, that is, associated with the size distribution of the pores, which is controlled by soil texture and compaction (i.e. bulk density). Pore size distribution varies within and between soil types, and can influence habitat conditions by modulating nutrient availability, gas diffusion and soil water holding capacity [START_REF] Ranjard | Quantitative and qualitative microscale distribution of bacteria in soil[END_REF][START_REF] Six | A history of research on the link between (micro)aggregates, soil biota, and soil organic matter dynamics[END_REF], parameters that may affect the diversity of soil organisms [START_REF] Nielsen | The enigma of soil animal species diversity revisited: The role of small-scale heterogeneity[END_REF][START_REF] Xia | Soil microbial diversity and composition: Links to soil texture and associated properties[END_REF] and their interactions [START_REF] Erktan | The physical structure of soil: Determinant and consequence of trophic interactions[END_REF]. The effects of soil texture and compaction on the diversity might vary between soil organisms with different sizes or life-history strategies [START_REF] Seaton | Soil textural heterogeneity impacts bacterial but not fungal diversity[END_REF] or whether there are ecosystem engineers able to modify the soil structural properties [START_REF] Decaëns | Macroecological patterns in soil communities[END_REF][START_REF] Six | A history of research on the link between (micro)aggregates, soil biota, and soil organic matter dynamics[END_REF].

The "resource heterogeneity hypothesis" follows the same rationale as the habitat hypothesis. An increase in resource heterogeneity can lead to an increase in diversity [START_REF] Steiner | The effects of prey heterogeneity and consumer identity on the limitation of trophic-level biomass[END_REF][START_REF] Heidrich | Heterogeneity-diversity relationships differ between and within trophic levels in temperate forests[END_REF][START_REF] Dal Bello | Resource-diversity relationships in bacterial communities reflect the network structure of microbial metabolism[END_REF]. We acknowledge that resource heterogeneity can be intrinsically linked to the habitat F I G U R E 1 Overview of the four big ecological hypotheses and theoretical predictions tested in this study within the soil biodiversity context. Each hypothesis is introduced in a coloured box, the predictors used to represent each hypothesis are given at the end of the boxes in a frame heterogeneity, which makes it difficult to separate them. As for aboveground, soil basal resources can take different forms, but their heterogeneity can be well approximated by plant functional diversity since it explains variation in SOM composition, type of potential mycorrhiza, root exudates and direct resources for phytophages [START_REF] Anderson | Inter-and intra-habitat relationships between woodland cryptostigmata species diversity and the diversity of soil and litter microhabitats[END_REF][START_REF] Eviner | Functional matrix: A conceptual framework for predicting multiple plant effects on ecosystem processes[END_REF][START_REF] Hooper | Interactions between aboveground and belowground biodiversity in terrestrial ecosystems: Patterns, mechanisms, and feedbacks[END_REF]. For higher trophic level groups (secondary and tertiary consumers), the diversity in potential prey might be taken as a proxy for resource heterogeneity.

Here, we tested the above outlined macroecological biodiversity hypotheses and estimated their relative importance in explaining soil biodiversity patterns across most soil trophic groups.

We built on a large-scale observatory network across the French Alps (Orchamp) that provides soil biodiversity measurements from environmental DNA metabarcoding across seventeen elevational gradients ranging from low to very high altitude (280-3160 m), and harbouring very contrasting climatic, vegetation and pedological conditions (Figure S1). Mountainous systems are well suited to test empirically large-scale drivers of biodiversity as they include wide ranges of environmental conditions and high biotic turnover over a reduced spatial scale [START_REF] Mccain | Elevational gradients in species richness[END_REF]. Instead of focusing on specific taxonomic orders, we followed a multi-trophic approach to test the above hypotheses on most trophic groups representative of soil ecosystems. After selecting the predictors related to the ecological hypotheses, we used a machine learning approach to account for complex interactions between predictors and soil biodiversity and corrected for remaining spatial dependencies that may originate from processes that have not been considered, such as missing abiotic factors or dispersal limitations. More specifically, we used biodiversity patterns to assess (1) the relative importance of predictors linked to different ecological hypotheses in explaining overall multi-trophic soil biodiversity and (2) the consistency of the response curves across trophic groups.

| MATERIAL AND ME THODS

| Study site and sampling design

The data come from the French Alps long-term observatory, Orchamp (www.orcha mp.osug.fr, Appendix S1), made of multiple elevational gradients distributed across the whole French Alps (ca. 40,500 km 2 ) and representative of the environmental conditions of the region. Each elevational gradient has a homogenous exposure and slope, and consists of four to nine 30 × 30 m plots separated by 200 m of altitude, on average. In this study, we used data gathered from 2016 to 2018, corresponding to 17 gradients (Figure S1), 90 plots and 540 soil samples. Plant species abundances were quantified at the vegetation peak (mostly in July or August) along a linear transect crossing each plot using the pin-point method [START_REF] Jonasson | Evaluation of the point intercept method for the estimation of plant biomass[END_REF]. A second 4-m-wide transect was dedicated to soil sampling at the end of the summer season. Soil was sampled from 3 subplots (2 × 2 m) selected across the transect where we collected around ten soil cores of 5 cm in diameter that were separated into two soil layers, that is, surface (ca. 1-8 cm depth) and subsurface (ca. 8-16 cm depth), which could be differentiated in most cases by a change in the colour. The ten soil cores were pooled together and homogenized by separating the two layers to make a biological sample per soil layer per subplot, to obtain a total of six samples per plot.

| Soil sample processing

Each soil sample was separated into two components. The main part was sieved at 2mm and used to measure soil physicochemical properties (soil pH, SOM content and soil C/N) as described in [START_REF] Martinez-Almoyna | Climate, soil resources and microbial activity shape the distributions of mountain plants based on their functional traits[END_REF]. The other part was used for environmental DNA, where DNA was extracted from a 15 g aliquot and processed in the field using the procedure described in [START_REF] Taberlet | Soil sampling and isolation of extracellular DNA from large amount of starting material suitable for metabarcoding studies[END_REF], [START_REF] Taberlet | Environmental DNA: For biodiversity research and monitoring[END_REF]. We used six DNA markers to have a complete overview of the soil biota, including two universal markers (euka02 for eukaryote, bact01 for bacteria) and fourth clade-specific markers (fung02 for fungi, inse01 for insect, olig01 for oligochaete and coll02 for collembola). Information on the markers and molecular analyses including PCR, library preparation and sequencing steps are detailed in Appendix S2. A standardized bioinformatic pipeline was then applied [START_REF] Calderón-Sanou | From environmental DNA sequences to ecological conclusions: How strong is the influence of methodological choices?[END_REF], using the OBITools software [START_REF] Boyer | obitools: A unix -inspired software package for DNA metabarcoding[END_REF] and the R package "metabaR" [START_REF] Zinger | metabaR: An r package for the evaluation and improvement of DNA metabarcoding data quality[END_REF], to remove contaminants and errors and to get the taxonomic composition in terms of Molecular Operational Taxonomic Unit (MOTU) of each sample (Appendix S2).

| Diversity of trophic groups

The obtained MOTUs were classified into 36 trophic groups. We chose to distinguish not only trophic levels but also phylogenetic distant groups of the same trophic level, as they may have different preys/predators or exhibit different resource acquisition strategies (e.g. bacterivorous nematodes vs. protists, or predatory mites versus insects, [START_REF] Potapov | Trophic consistency of supraspecific taxa in below-ground invertebrate communities: Comparison across lineages and taxonomic ranks[END_REF], following [START_REF] Calderón-Sanou | Cascading effects of moth outbreaks on subarctic soil food webs[END_REF].

The databases used for the trophic and functional assignments were FungalTraits [START_REF] Põlme | FungalTraits: A user-friendly traits database of fungi and fungus-like stramenopiles[END_REF], for fungal MOTUs assigned at the genus level and FUNGuild [START_REF] Nguyen | FUNGuild: An open annotation tool for parsing fungal community datasets by ecological guild[END_REF] for the rest of fungi, FAPROTAX [START_REF] Louca | Decoupling function and taxonomy in the global ocean microbiome[END_REF] S1. The MOTU diversity of each trophic group was estimated per sample using the exponential of the Shannon entropy (i.e. Shannon diversity), which represents "the effective number of MOTUs" as it penalizes rare sequences that could be artefacts in eDNA data. Shannon diversity leads to more robust ecological results and to diversity estimates that are more similar to those assessed from conventional sampling approaches [START_REF] Calderón-Sanou | From environmental DNA sequences to ecological conclusions: How strong is the influence of methodological choices?[END_REF].

| Environmental predictors

We used two environmental predictors to represent each ecological hypothesis (Figure 1), with the condition of having a final set of weakly correlated predictors (see Figure S2 for a visualization of the correlation between all initially considered parameters).

| Energy hypothesis

It was separated into primary (solar energy) and secondary energy (SOM), and two predictors were selected for each category. Solar radiation and the Normalized Difference Vegetation Index (NDVI) were used to represent the primary energy predictors. Solar radiation directly measures the amount of solar energy arriving into the Earth's surface, while NDVI estimates the amount of solar energy that is transformed by photoautotrophic organisms into available resources accounting for water limitations [START_REF] Evans | Species-energy relationships at the macroecological scale: A review of the mechanisms[END_REF]. We did not add mean annual temperature as sometimes done to represent energy [START_REF] Clarke | Climate, energy and diversity[END_REF] since it was strongly correlated to NDVI (Figure S2). Solar radiation was calculated per plot as the sum of the daily surface incident direct and diffuse shortwave radiation accumulated over 10 years, from 2008 to 2018. NDVI was estimated from the surface spectral reflectance at a resolution of 250 m from MODIS (Moderate Resolution Imaging Spectroradiometer), available online: https://lpdaac.usgs.gov/produ cts/mod09 q1v00 6/. Raw NDVI times series were pre-processed following [START_REF] Choler | Growth response of temperate mountain grasslands to inter-annual variations in snow cover duration[END_REF], and we kept the mean yearly sum of NDVI greater than 0.2 over 2009-2019, as the final predictor for the analyses measured at the plot level. To represent secondary energy, we used the SOM content and the C/N ratio, measured from the soil samples. The former indicates the total amount of organic matter available in the soil, while the latter is a proxy for nutrient availability or SOM decomposability [START_REF] Cleveland | C:N: P stoichiometry in soil: Is there a "Redfield ratio" for the microbial biomass?[END_REF], meaning that soils with low C/N rates have potentially more readily available energy than soils with high C/N ratio, if we account for nutrient stoichiometric constraints.

| Physiological tolerance hypothesis

We used soil pH and the freezing degree days (FDD) to represent potential sources of abiotic physiological stress for soil organisms. The pH has been described as an important limiting physiological factor of soil communities [START_REF] Fierer | The diversity and biogeography of soil bacterial communities[END_REF][START_REF] Räty | Earthworms and pH affect communities of nematodes and enchytraeids in forest soil[END_REF].

The FDD summarizes the duration and intensity of ground freezing events and it has been addressed as a good candidate to model the thermal niches [START_REF] Choler | Winter soil temperature dependence of alpine plant distribution: Implications for anticipating vegetation changes under a warming climate[END_REF]. FDD was calculated per plot as the annual sum of average daily degrees below zero, modelled within the first soil horizon (1 cm depth) and averaged over 2008-2018.

| Habitat heterogeneity hypothesis

Clay percentage in soil and bulk density were selected to represent the microscale habitat heterogeneity. Clay percentage characterizes the soil texture and thus reflects the granulometry distribution, the aeration, ability of soil to retain water and more globally the physical properties of the soil [START_REF] Hao | Soil density and porosity[END_REF][START_REF] Seaton | Soil textural heterogeneity impacts bacterial but not fungal diversity[END_REF]. Soil texture might affect diversity differently across trophic groups with different sizes or life-history strategies [START_REF] Seaton | Soil textural heterogeneity impacts bacterial but not fungal diversity[END_REF][START_REF] Vreeken-Buijs | Relationships of soil microarthropod biomass with organic matter and pore size distribution in soils under different land use[END_REF]. For example, the diversity of mesofauna could be expected to increase in coarse-textured soils (i.e. with low clay percentage), where the higher availability of larger pores provides more different habitats to be potentially colonized by these organisms [START_REF] Vreeken-Buijs | Relationships of soil microarthropod biomass with organic matter and pore size distribution in soils under different land use[END_REF]. Bulk density reflects soil compaction and porosity as it accounts for the amount of soil per volume unit when removing water and air spaces [START_REF] Hao | Soil density and porosity[END_REF]. Compact soils, with higher values of bulk density, have relatively lower total pore space and organic matter content, thus providing a lower heterogeneity of habitats. Both variables were measured from a soil pit carried out next to the plot). Three soil replicates were collected with a volumetric cylinder (100 cm3) from the superficial horizon.

They were dried at 105°C for 24 h and sieved to 2 mm. The mass of dry soil (mS) contained in the cylinder as well as the mass of coarse elements greater than 2 mm (mEG) were measured. The formula applied for the calculation of bulk density is as follows (Equation 1), with V cyl for the volume of the cylinder. The bulk density of the three replicates were averaged.

| Resource heterogeneity hypothesis

For decomposers, detritivores and plant symbionts, we used two metrics of plant functional diversity as predictors, that is, the functional richness and the functional divergence [START_REF] Villéger | New multidimensional functional diversity indices for a multifaceted framework in functional ecology[END_REF], calculated for each plot using the R package "FD" [START_REF] Laliberté | A distance-based framework for measuring functional diversity from multiple traits[END_REF]. Functional richness represents the total trait space filled by all the plant species present in the community (here the plot). Functional divergence describes how specie's abundances are distributed within the functional trait volume. To estimate these two metrics, we used our own trait measurement values for species (median values across individuals) present in our botanical surveys.

We included the following traits: specific leaf area (SLA), leaf carbon and nitrogen ratio, root depth (extracted from [START_REF] Landolt | Flora indicativa: Okologische Zeigerwerte und biologische Kennzeichen zur Flora der Schweiz und der Alpen[END_REF], vegetative plant height and woodyness index. For the rest of the soil groups (except autotrophs), we selected two predictors measuring prey diversity (exponential of the Shannon entropy) of the focal trophic group. For omnivores (i.e. tardigrades, rotifers and protists), we used the MOTU's diversity of bacteria and the MOTU's diversity of fungi. For bacterivores, we used the MOTU's diversity and phylum's diversity of bacteria. For fungivores and zooparasites, we used the MOTU's diversity and class diversity of fungi and metazoans, Solar radiation and FDD were calculated from the SAFRAN-SURFEX/ISBA-Crocus-MEPRA reanalysis [START_REF] Durand | Reanalysis of 44 Yr of climate in the french alps (1958-2002): Methodology, model validation, climatology, and trends for air temperature and precipitation[END_REF][START_REF] Vannier | Calcul d'une évapotranspiration de référence spatialisée pour la modélisation hydrologique à partir des données de la réanalyse SAFRAN de Météo-France[END_REF], a model which addresses meteorological and snow conditions in mountainous regions based on large-scale topographical features.

| Spatial structure

Given the hierarchical sampling design of the data (two soil layers within plots within gradients), we accounted for the overall spatial structure of the samples to avoid having spatial autocorrelation issues [START_REF] Dray | Community ecology in the age of multivariate multiscale spatial analysis[END_REF]. We defined a set of spatial predictors representing the residual spatial structure (i.e. the left-out spatial structure not explained by the environmental predictors) to include in the models. This approach aims to reduce the spatial autocorrelation that could remain in the residuals and to identify potential spatial structures with a strong influence on soil diversity. We did so using Moran's eigenvector maps (MEM), a method based on computing the principal coordinates of a matrix of geographic neighbours [START_REF] Dray | Spatial modelling: A comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM)[END_REF]. The obtained eigenvectors are orthogonal and have a straightforward interpretation as each of them represents a spatial pattern at a given scale that can be ranked from broad spatial structures to fine spatial structures. We identified 18 MEM-variables describing significant spatial autocorrelation (only positive eigenvalues, [START_REF] Dray | Spatial modelling: A comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM)[END_REF] based on the Euclidean geographic distances between each subplot using the function dbmem from the R package "adespatial" [START_REF] Dray | adespatial: Multivariate Multiscale Spatial Analysis[END_REF]. MEM 1 to 8 described broad scale spatial structures, while MEM 9 to 18 represented intermediate to fine spatial structures (Figure S3). To remove the imprint of the environment on these MEMs, we modelled with a random forest each of the 18 MEMs as a function of our environmental predictors and extracted the residuals of these relationships. These residuals thus represented the spatial structure not explained by our environmental predictors (e.g. missing predictors, dispersal limitations). This approach differs from partialling out the spatial component of diversity and compare the pure effect of environment, the pure effect of space and the shared explained variance [START_REF] Borcard | Partialling out the spatial component of ecological variation[END_REF]. Here, we argue that space is likely affecting environment and that environment is then affecting biodiversity. The shared explained variance of space and environment is thus relevant for our hypotheses. We treat the pure effect of space as a statistical nuisance as we cannot link it to ecological processes, given that we jointly analyse taxa with very different dispersal abilities. We made sure that it was properly accounted for to avoid residual spatial autocorrelation [START_REF] Dray | Community ecology in the age of multivariate multiscale spatial analysis[END_REF].

| Random forest

To model the diversity of each trophic group as a function of the predictors representing our four hypotheses and the residual spatial structure, we used random forest models [START_REF] Breiman | Random forests[END_REF], which are particularly suited when nonlinear relationships and complex interactions among predictors are expected. Random forest analyses were run with the R package "party" [START_REF] Hothorn | Unbiased recursive partitioning: A conditional inference framework[END_REF] with the cforest_unbiased function, which avoids bias introduced by heterogeneity in scale and number of categories among predictors [START_REF] Strobl | Bias in random forest variable importance measures: Illustrations, sources and a solution[END_REF]. The number of trees was set to 1,000 and the number of variables randomly sampled as candidates at each split (mtry) was tuned using the function train of the R package "caret" [START_REF] Kuhn | caret: Classification and Regression Training[END_REF] Table S2). Variable importance was estimated as the mean decrease in accuracy using the function varimp. The method allows assessing relative variable importance, by identifying the covariates which, when removed, ensure a significant drop of prediction power [START_REF] Strobl | Bias in random forest variable importance measures: Illustrations, sources and a solution[END_REF]. It thus avoids any over-fitting and allows sound inference. Overall explained variance (r-square) was calculated by extracting the coefficient of determination between predictions and observations. The shape of the relationship between the diversity and the predictors was assessed with partial dependent plots obtained from the R package "iml" [START_REF] Molnar | iml: An R package for interpretable machine learning[END_REF], which estimate the marginal effect of a given predictor while accounting for the average effect of the other predictors in the model. We considered that a relationship was relevant, when the predictor had a predictive importance higher than 25%. The predictive importance was assessed by permuting each predictor one by one and then evaluating how the prediction was affected.

A single random forest model was run for each trophic group with the same set of predictors, that is, solar radiation, NDVI, SOM, C/N ratio, percentage of clay, bulk density, two variables corresponding to resource heterogeneity (variable across trophic groups, and excluded for autotrophs) and the 18 residual spatial structure predictors. All analyses were run in the R statistical environment (R Core Team, 2020). The predictors underlying the tested ecological hypotheses explained a significant part of the spatial variation of diversity of most trophic groups. The overall explained variance varied from 29%

| RE SULTS

for detritivorous insects to 79% for arbuscular mycorrhizal fungi (Figure 2a, Table S2). The residual spatial structure explained much less variance than the environmental predictors, confirming the relevance of the latter to predict soil biodiversity. Only the diversity of predatory and phytophagous insects, and photoautotrophic protists was better explained by pure broad residual spatial structures than by the environment (MEM7, Figure S3).

We found that predictors associated with the energy and the physiological tolerance hypotheses were generally the most important, even so the relative importance of the predictors did vary between soil trophic groups in different trophic positions or from different body size categories (Figure 2b, Figure 3). The energy hypothesis was particularly important for consumers, that is, tertiary and secondary consumers and plant symbionts, and less important for autotrophs (Figure 3a). In particular, the secondary energy predictors related to SOM explained a large part of the diversity of most fungivores and detritivorous insects well-linked to the detritus channel. When looking at the tendencies per category of body size, the energy hypothesis was more important for metazoans of all sizes and fungi diversity, while the physiological tolerance hypothesis explained most variation for bacteria, protists and microfauna (Figure 3b). The habitat heterogeneity hypothesis had a higher importance for autotrophs compared to the other groups. The resource heterogeneity hypothesis was especially important for bacterivores (both protists and nematodes), phytophagous insects, enchytraeids and soil saprotrophic fungi.

In general, we found that the partial response curves of diversity to predictors were consistent across most soil trophic groups (Figure 4) and in agreement with predictions (Figure 1), with some few exceptions. The diversity of most trophic groups including zooparasites protists and fungi, metazoans consumers and ectomycorrhizal fungi, strongly increased with NDVI, but decreased for photolithoautotroph bacteria, phytophagous protists and earthworms (Figure 4a). The steepest changes in soil diversity across the NDVI gradient occurred in the transition from forest (high NDVI) to alpine grasslands (low NDVI). Groups for which diversity strongly increased with solar radiation included zooparasite bacteria, phytophagous protists and earthworms. All trophic groups primarily feeding on detritus positively increased in diversity with SOM (Figure 4b).

The diversity of several groups was also influenced by the C/N ratio: diversity decreased for herbivorous and bacterivorous nematodes, and root endophyte and arbuscular mycorrhizal fungi, but increased for ectomycorrhizal fungi and fungivorous nematodes (Table S3). With the exception of rotifers and tardigrades, all trophic groups responding to pH increased in diversity in more alkaline soils (Figure 4c). This positive relationship had a sigmoid form for all groups, but both the inflection points and associated slopes strongly varied across trophic groups. Saprotrophic, root endophytes and phytoparasitic fungi, and also photolithoautotrophic bacteria were positively affected by the soil clay content, and chemolithoautotrophic bacteria were positively affected by soil bulk density (Table S3). All phytophagous insects, saprotrophic fungi and bacterivore groups responded positively to resource heterogeneity, that is, plant functional richness and bacteria diversity respectively (Figure 4d).

Enchytraeids responded positively to plant functional divergence (aka. resource heterogeneity).

| DISCUSS ION

Testing ecological hypotheses has largely contributed to our understanding on how biodiversity is structured on Earth. However, generality can only be claimed if a significant part of biodiversity is covered. Here, we add an important missing piece to the general picture by testing several major ecological hypotheses simultaneously on the majority of trophic groups inhabiting the soil and along sharp environmental gradients which allow some generalization to be made. Our results confirm that the main environmental drivers of soil biodiversity are variable across soil trophic groups and depend on their resource or physiological requirements. Yet, we also find major commonalities in the ecological processes structuring soil biodiversity. Overall, the energy and physiological tolerance hypotheses had the strongest support from soil multi-trophic biodiversity.

Our results are in agreement with previous studies finding that an increase in primary energy increases the diversity of soil organisms such as protists [START_REF] Oliverio | The global-scale distributions of soil protists and their contributions to belowground systems[END_REF], metazoans [START_REF] Peters | Predictors of elevational biodiversity gradients change from single taxa to the multi-taxa community level[END_REF], soil predators [START_REF] Binkenstein | Multi-trophic guilds respond differently to changing elevation in a subtropical forest[END_REF] and fungi (Tedersoo et al., 2014, Figure 2b). Our results also reveal that secondary energy, related to soil organic matter, has a positive effect on soil biodiversity, especially for fungivorous and detritivorous animals, in agreement with earlier work [START_REF] Binkenstein | Multi-trophic guilds respond differently to changing elevation in a subtropical forest[END_REF][START_REF] Canedoli | Evaluation of ecosystem services in a protected mountain area: Soil organic carbon stock and biodiversity in alpine forests and grasslands[END_REF][START_REF] Caruso | Oribatid mites show how climate and latitudinal gradients in organic matter can drive large-scale biodiversity patterns of soil communities[END_REF]. We found that the relative importance between primary and secondary energy varies across trophic groups, with no clear trends across trophic levels, suggesting that both energy channels are at play across the soil food web. However, some groups responded to specific energy predictors in a way that differs from the predictions of the "energy hypothesis" (Figure 1).

For example, the diversity of earthworms, phytophagous fungi and photolithoautotroph bacteria decreased with increasing NDVI.

Part of these divergent trends between diversity and NDVI might be explained by the transition from forest to grassland in the NDVI gradient in our study system, for example, alpine grassland soils are more suitable for autotrophic bacteria adapted to high elevation stressful conditions [START_REF] Guo | Diversity and distribution of autotrophic microbial community along environmental gradients in grassland soils on the Tibetan Plateau[END_REF]. Otherwise, a negative interaction between ectomycorrhizal fungi and phytophagous fungi could explain the decrease in diversity of the latter (Figure 4a). Indeed, ectomycorrhizal fungi can provide protection against pathogens to their plant hosts, thus reducing the incidence of phytophagous fungi and their diversity [START_REF] Antunes | Chapter 9 -Mycorrhizas as nutrient and energy pumps of soil food webs: Multitrophic interactions and feedbacks[END_REF][START_REF] Wang | Distinct biogeography of different fungal guilds and their associations with plant species richness in forest ecosystems[END_REF].

Other divergent, but not unexpected, trends were found along the TA B L E 1 Information on the environmental DNA data characterizing each trophic group, including the DNA marker used to sample each group and the final number of reads, families (orders for protists), MOTUs and Shannon diversity obtained in total across the French alps and per sample (mean C/N ratio gradient, that is, ectomycorrhizal fungi and fungivorous nematodes were more diverse in soils with more recalcitrant organic matter (i.e. higher C/N ratio). This result reflects the differences in the energetic requirements or life-history traits of the different groups that may complexify generalizations of energy-related mechanisms. Contrary to other decomposers, ectomycorrhizal fungi can degrade recalcitrant organic complexes by using energy from their hosts [START_REF] Lindahl | Ectomycorrhizal fungi -potential organic matter decomposers, yet not saprotrophs[END_REF]. An increase in ectomycorrhizal fungi diversity could presumably cascade on fungivore nematodes diversity. Furthermore, while we show that energy has mainly a positive influence on soil biodiversity, the underlying mechanisms remain to be tested. For example, the more individual hypothesis states that greater energy availability allows a community to contain a larger number of individuals, and hence of a larger number of species with viable population size [START_REF] Wright | Species-energy theory: An extension of speciesarea theory[END_REF]. Quantifying species abundance or biomass would be needed to test this hypothesis, but this information is unfortunately not yet available with eDNA metabarcoding data [START_REF] Taberlet | Environmental DNA: For biodiversity research and monitoring[END_REF], and would be extremely challenging to obtain for the wide range of organisms studied here.

± SD)
Physiological tolerances, mainly to soil pH, were also a strong predictor of the diversity of soil organisms, especially for organisms living in the aqueous phase of the soils. Indeed, in the study system, the diversity of groups of bacteria, protists and microfauna was more constrained by pH-induced stress rather than limited by energy or habitat and resource heterogeneity (Figure 3b), in accordance with previous studies highlighting the importance of pH for soil microbes [START_REF] Fierer | The diversity and biogeography of soil bacterial communities[END_REF][START_REF] Karimi | Biogeography of soil bacteria and archaea across France[END_REF] and invertebrates [START_REF] Bastida | Climatic vulnerabilities and ecological preferences of soil invertebrates across biomes[END_REF][START_REF] Räty | Earthworms and pH affect communities of nematodes and enchytraeids in forest soil[END_REF]. The sigmoid trend observed between diversity and pH might correspond to the first half of the humpback curve expected from the theory (Figure 1). Indeed, our sampling had relatively few sites with alkaline soils, and did not include soils with pH >8, levels from which other studies have observed a decrease of diversity (e.g. [START_REF] Fierer | The diversity and biogeography of soil bacterial communities[END_REF].

Our results revealed consistent decreases of diversity in more acidic soils, but also different tolerance thresholds across soil trophic groups. The strong effect of soil pH might also be the sum of multiple linked factors not considered in this study including bedrock type and plant communities [START_REF] Roy | Microbes on the cliff: Alpine cushion plants structure bacterial and fungal communities[END_REF]. Contrarily, FDD had a minor effect on soil biodiversity. Limited effect of freezing events on soil biodiversity has previously been reported, and may result from the frost resistance [START_REF] Männistö | Bacterial and fungal communities in boreal forest soil are insensitive to changes in snow cover conditions[END_REF][START_REF] Stres | Frequent freeze-thaw cycles yield diminished yet resistant and responsive microbial communities in two temperate soils: A laboratory experiment[END_REF] or the rapid recovery of soil communities [START_REF] Sulkava | Effects of hard frost and freeze-thaw cycles on decomposer communities and N mineralisation in boreal forest soil[END_REF].

Theoretically, this low importance could be due to a scale mismatch between the measured soil communities (subplots are 4m 2 large) and the climatic data resolution (~300m). However, between the available in situ temperature HOBOs and the climatic data used here showed very consistent patterns, rendering the scale mismatch hypothesis unprobeable. Otherwise, a change in composition or activity, without changes in local diversity, might also have occurred and remains to be tested [START_REF] Schostag | Bacterial and protozoan dynamics upon thawing and freezing of an active layer permafrost soil[END_REF][START_REF] Stres | Frequent freeze-thaw cycles yield diminished yet resistant and responsive microbial communities in two temperate soils: A laboratory experiment[END_REF].

The "habitat heterogeneity" and the "resource heterogeneity" hypotheses weakly explained the spatial variation in diversity of soil trophic groups compared to "energy" and "physiological tolerance",

TA B L E 1 (Continued)

with notable exceptions. Saprotrophic, root endophytes and phytoparasitic fungi, as well as autotrophic bacteria were highly affected by habitat heterogeneity. We found that these groups tended to be more diverse in fine-textured soils (higher clay percentage), which usually exhibit greater water retention capacity but also more recalcitrant and stable organic matter [START_REF] Ranjard | Quantitative and qualitative microscale distribution of bacteria in soil[END_REF][START_REF] Six | A history of research on the link between (micro)aggregates, soil biota, and soil organic matter dynamics[END_REF]. Previous studies have shown that soil texture can influence bacterial and fungal diversity, with subgroups of taxa responding differently to the proportion of soil particles (i.e. clay, sand, silt) [START_REF] Karimi | Biogeography of soil bacteria and archaea across France[END_REF][START_REF] Seaton | Soil textural heterogeneity impacts bacterial but not fungal diversity[END_REF][START_REF] Xia | Soil microbial diversity and composition: Links to soil texture and associated properties[END_REF]. Our results showed that such differences are also visible when considering different trophic groups of fungi and bacteria.

The importance of "habitat heterogeneity" could be expected to vary across soil trophic groups, as the spatial scale at which heterogeneity is perceived by organisms of different sizes or different lifestyles can be highly variable [START_REF] Heidrich | Heterogeneity-diversity relationships differ between and within trophic levels in temperate forests[END_REF]. Here again, perhaps the scale at which we measured heterogeneity was not relevant for some specific groups. When looking at the effect of resource heterogeneity, prey's diversity was remarkably important for bacterivores. Strong associations between bacterivore protists (or nematodes) and bacteria, but our results and previous studies point to noticeable differences in the factors shaping the diversity of these groups [START_REF] Oliverio | The global-scale distributions of soil protists and their contributions to belowground systems[END_REF][START_REF] Xiong | A global overview of the trophic structure within microbiomes across ecosystems[END_REF]. Moreover, the strong response of saprotrophic fungi to plant functional diversity could be explained by a trophic specialization, in accordance with a recent study showing a high degree of specialization to specific soil and litter compounds for some saprotrophic fungi (Algora [START_REF] Gallardo | Litterinhabiting fungi show high level of specialization towards biopolymers composing plant and fungal biomass[END_REF]. The significant association does not necessarily imply the realization of a trophic interaction, but it is a first step in assessing whether such interactions exist, leave signals in diversity distribution and can give us insights into the degree of food speciation in the focus trophic group.

To conclude, our near-complete coverage of soil biodiversity across trophic groups and across large and steep environmental gradients provides consistent and novel insights on the macroecological rules shaping the distribution of belowground biodiversity. Building on the efficiency of environmental DNA analyses combined with the wealth of existing knowledge on soil organisms, we showed that energy and physiological tolerance are the most plausible hypotheses to explain the spatial distribution of soil diversity at a regional scale. Interestingly, we found strong commonalities between trophic groups in their response to environmental drivers that should be later compared to aboveground organisms living in the same locations (e.g. ground-dwelling arthropods, pollinators or herbivores). Should belowground and aboveground compartments respond differently to environmental drivers, it will complexify their management under humaninduced pressures. Finally, identifying how these patterns in local diversity translate into compositional changes and interaction network structuration in space will be of crucial importance to understand soil biodiversity assembly and how it might be affected by ongoing environmental changes. Only groups for which the predictors had a predictive importance higher than 25% were represented. The predictive importance of the predictor was assessed by permuting each predictor one by one and then evaluating how the prediction was affected. Taxonomic groups are abbreviated as Bacteria (B.), Collembola (C.), Earthworms (Earth), Fungi (F.), Insects (I.), Mites (M.), Nematodes (N.) Protozoa (P.), Rotifera (Rotif) and Tardigrada (Tardi). The rest of the trophic component is abbreviated as arbuscular mycorrhizal (arb), bacterivore (bac), detritivore (det), ectomycorrhizal (ect), epigeic (epi), euedaphic-hemiedaphic (e-h), fungivore (fun), herbivore (her), heterotroph (het), omnivore (omn), photolitoautotroph (pho), phytoparasite or phytophageous (phy), predator (pre), protistivore (pro), saprotroph (sap) and zooparasite (zoo)

  and Wardeh et al. (2015) database of host-pathogen interactions for bacteria, NEMAguild and Nemaplex (http://nemap lex.ucdav is.edu/) for nematodes. The main references used included Adl et al. (2019) for protists, Rainford and Mayhew (2015) for insects and Potapov et al. (2016) for Collembola. The most abundant taxonomic clades composing each trophic group are shown in Table

  used the diversity at these two taxonomic levels, because MOTUs diversity might be redundant depending on the level of generalism of the focal trophic group, that is, a predator might be indifferent to two closely related species. For predators, we used the MOTU's diversity and class diversity of a subgroup of metazoans (or protists for protistivores), in which we excluded the focal group and we only considered the category of size that could potentially be a prey for the focal group (e.g. only micro-metazoans for predatory nematodes).

F

  Relative importance of competing hypotheses in explaining the alpha diversity of soil trophic groups. (a) Total r-squared of the random forest model for each trophic group. Colours represent the relative importance of the environmental versus the spatial predictors. Environmental predictors correspond to all the biotic and abiotic variables used to test the ecological hypotheses, and spatial predictors correspond to the residuals of the spatial structure when removing the effect of the environment. (b) Relative importance of the environmental predictors used to test the ecological hypotheses (colour key). The relative variable importance is the mean decrease in squared error, rescaled to sum the total r2 (a) or 1 (b). Letters correspond to broad taxonomic groups: Bacteria (B.), Protozoa (P.), Metazoa (C.: Collembola, I.: Insects, M.: Mites, N: Nematodes) and Fungi (F.). Symbols indicate the size category for fauna groups and bacteria diversity have been recently reported (Oliverio et al., 2020; Xiong et al., 2021), and could indicate a degree of trophic specialization in bacterivorous protists. Co-variation in diversity might also indicate shared habitat preferences between protists

F

  Boxplots of the relative importance of ecological hypotheses by trophic position and body size category. Relative importance of the four ecological hypotheses tested in this study across groups categorized by trophic position (a) or body size category (b). The values of relative importance correspond to the mean decrease in squared error from the random forest per trophic group, rescaled to sum the total r-square F I G U R E 4 Predicted diversity of soil trophic groups as a function of the environmental predictors representing the ecological hypotheses. Partial dependence plots showing the marginal effect of the predictors representing the ecological hypotheses on the diversity of soil trophic groups based on the random forest model results. The predictors represented are (a) NDVI, with the colours corresponding to the transition from alpine grassland (yellow) to forest (green), (b) Soil organic matter, (c) pH and (d) plant functional richness and Bacteria MOTUs diversity. Diversity was standardized by the maximum diversity observed per group to have a comparable scale across groups.

Trophic position Trophic group DNA marker Total reads (per sample) Total families/orders (per sample) Total MOTUs (per sample) Total Shannon diversity (per sample)

	108.4 (23.1 ± 9.8)	43.7 (10.3 ± 5.6)	55.5 (8.7 ± 4.1)	2,352.1 (611.9 ± 205.1)	200.2 (9.8 ± 5.6)	185.7 (10.4 ± 6.2)	435 (14 ± 8)	207 (6.5 ± 4.3)	70.4 (2.5 ± 2.6)	60.8 (3 ± 2.3)	57.6 (1.6 ± 0.8)	254.1 (13.4 ± 12)	336.6 (6.1 ± 5.4)	39.8 (3.2 ± 1.8)	140.3 (35.5 ± 16.6)	168.4 (9.5 ± 5.4)	55 (1.4 ± 0.7)	41.3 (3.6 ± 2)	37.1 (3.6 ± 2.2)	17.8 (5.1 ± 2.1)	370.5 (62.9 ± 23.6)	66.7 (2.7 ± 1.4)	53.1 (2.8 ± 1.9)	22.8 (5.4 ± 2.4)	21.7 (2 ± 1.1)	27.1 (4.3 ± 2.6)	5.8 (2 ± 1)	5.8 (2.1 ± 1)	97.7 (26.9 ± 11)
	3,148 (80.1 ± 47.3)	715 (22.3 ± 14.8)	374 (13.2 ± 7.3)	94,308 (2,980.1 ± 1,148.9)	1,305 (37.4 ± 17.7)	969 (25.7 ± 16.8)	3,587 (75.5 ± 37.1)	1,163 (20.1 ± 11.9)	1,151 (10.8 ± 8.6)	2,386 (24.3 ± 19.5)	240 (3.1 ± 2.3)	2,426 (36.6 ± 36.8)	2,795 (27.5 ± 28.5)	174 (5.6 ± 3.4)	5,539 (159.3 ± 89.3)	806 (21 ± 12.6)	172 (2.2 ± 1.7)	135 (5 ± 3.1)	167 (5.1 ± 3.6)	176 (10.2 ± 4.3)	2,764 (108 ± 53.3)	704 (7.9 ± 7.4)	636 (8.7 ± 11)	170 (10.7 ± 5)	73 (2.4 ± 1.4)	237 (6.6 ± 4.6)	39 (2.4 ± 1.5)	52 (3.1 ± 1.9)	1,677 (66.2 ± 25)
	20 (9.5 ± 1.9)	20 (3.9 ± 1.8)	12 (2.4 ± 1)	432 (161 ± 23.9)	62 (12.2 ± 4.6)	58 (9.6 ± 4.3)	44 (14 ± 4.2)	102 (11.4 ± 5.2)	6 (1.6 ± 0.6)	1 (1 ± 0)	29 (1.9 ± 1)	10 (2.9 ± 1.7)	38 (7 ± 4.8)	6 (2.9 ± 1.2)	38 (20 ± 3.5)	73 (11.3 ± 5.4)	10 (1.5 ± 0.8)	10 (2.6 ± 1.3)	18 (3 ± 1.6)	15 (6.5 ± 2.1)	22 (10.6 ± 2.1)	5 (2.7 ± 1.1)	3 (1.8 ± 0.5)	22 (4.1 ± 2)	7 (1.4 ± 0.7)	2 (1.6 ± 0.5)	N/A	N/A	7 (4 ± 0.5)
	269,418 (517.1 ± 606.8)	51,400 (98.8 ± 123.8)	34,083 (65 ± 91.3)	10,082,382 (19,352 ± 12,905.5)	1,588,239 (3,013.7 ± 4,770.2)	437,395 (830 ± 1,602.1)	7,438,380 (14,114.6 ± 15,124.5)	691,399 (1,319.5 ± 2,263.9)	5,274,333 (13,386.6 ± 24,467.6)	8,802,809 (21,418 ± 23,679.7)	290,140 (1,458 ± 4,538.5)	3,463,460 (7,141.2 ± 16,896.2)	5,304,179 (11,119.9 ± 13,724.7)	127,187 (247.9 ± 650.7)	526,809 (1,011.1 ± 1,029)	213,181 (405.3 ± 522)	210,868 (1,270.3 ± 3,706.3)	11,518 (22.6 ± 26.2)	16,035 (31.9 ± 68)	67,829 (130.4 ± 174.1)	232,544 (446.3 ± 485.7)	1,956,200 (4,347.1 ± 6,678.4)	1,723,259 (4,351.7 ± 7,281.1)	233,881 (449.8 ± 484.9)	4,594 (10.3 ± 26.6)	16,436 (31.7 ± 38.3)	4,327 (9.5 ± 11.6)	19,822 (42.3 ± 67.1)	308,176 (581.5 ± 537.3)
	B.chemolithoautotroph bact01	B.photolithoautotroph bact01	P.photoautotroph euka02	B.heterotroph bact01	F.saprotroph (litter) fung02	F.saprotroph (other) fung02	F.saprotroph (soil) fung02	F.saprotroph (wood) fung02	Earthworms olig01	Enchytraeidae olig01	I.detritivore inse01	F.arbuscular_mycorrhizal fung02	F.ectomycorrhizal fung02	F.root_endophyte fung02	B.phytoparasite bact01	F.phytoparasite fung02	I.phytophagous inse01	N.herbivore euka02	P.phytoparasite euka02	N.bacterivore euka02	P.bacterivore euka02	C.epigeic coll01	C.euedaphic_hemiedaphic coll01	M.fungivore euka02	N.fungivore euka02	P.fungivore euka02	Rotifera euka02	Tardigrada euka02	P.omnivore euka02
	Autotrophs			Decomposers					Detritivores			Plant mutualists			Phytophagous					Bacterivores		Fungivores					Omnivores

Trophic position Trophic group DNA marker Total reads (per sample) Total families/orders (per sample) Total MOTUs (per sample) Total Shannon diversity (per sample)

	56.5 (1.3 ± 0.5)	9.4 (3.6 ± 2.2)	7.9 (1.6 ± 0.8)	61.1 (11.2 ± 5.5)	112.1 (34.2 ± 19.7)	45.5 (4.9 ± 2.6)	93.5 (5.7 ± 3.5)
	142 (2 ± 1.3)	336 (13.8 ± 6.8)	19 (1.8 ± 1)	334 (15.8 ± 8.4)	4,329 (142.7 ± 80.9)	238 (8.7 ± 5.2)	371 (8.4 ± 5.4)
	20 (1.7 ± 0.8)	32 (6.1 ± 2.6)	7 (1.6 ± 0.8)	4 (1.6 ± 0.7)	78 (18.7 ± 4.9)	12 (3.6 ± 1.5)	10 (3.1 ± 1.2)
	207,077 (1,327.4 ± 4,557.6)	492,098 (928.5 ± 1,055.5)	3,344 (8.8 ± 14.8)	36,815 (69.5 ± 75.8)	536,202 (1,029.2 ± 1,234.2)	74,803 (147.5 ± 256.1)	20,162 (38.8 ± 60.2)
	inse01	euka02	euka02	euka02	bact01	fung02	euka02
	I.predator	M.predator	N.predator	P.protistivore	B.zooparasite	F.zooparasite	P.zooparasite
	Predators				Zooparasites

Abbreviation: N/A: not applicable.

Funding information Grant/ Award Number: ANR-10-LAB-56, ANR-15-IDEX-02 and ANR-16-CE02-0009

DATA AVA I L A B I L I T Y S TAT E M E N T

The data supporting the results of this study are publicly available in the DRYAD repository https://doi.org/10.5061/dryad.q573n 5tm6.

ACK N OWLED G EM ENTS

This project was possible thanks to the many people that participated in the field work and in the laboratory procedures, including students, interns, postdoctoral fellows, field assistants, park managers and rangers. We thank Cindy Arnoldi for conducting

CO N FLI C T O F I NTE R E S T

The authors declare that they have no conflict of interest.