Emmanuel Mignot
email: emmanuel.mignot@insa-lyon.fr

Wei Cai

Juan Ignacio Polanco

Cristian Escauriaza

Nicolas Riviere

Measurement of mass exchange processes and coefficients in a simplified open-channel lateral cavity connected to a main stream

Keywords: Mass exchange, Turbulent diffusion, Cavity, Exchange coefficient, Dye release method

Lateral cavities are major storage zones in riverine environments for which the mass exchanges with the main stream strongly impact the characteristics of the habitat in these dead zones. An experimental work is presented here with a controlled main stream and a connected openchannel lateral cavity to assess the processes responsible for these exchanges and to quantify the exchange capacities. In a first step, the measurements of passive scalar transport allow us to identify the physical processes involved in the exchange of mass from the main stream and its spreading within the cavity. In a second step, the quantitative mass exchange coefficient, representative of the exchange capacity, is measured for 28 flow and cavity configurations. The sensibility analysis to the governing parameters proposed by the dimensional analysis then reveals that changing the geometric aspect ratio of the cavity does not affect the exchange coefficient while increasing the normalized water depth or decreasing the Reynolds number of the main stream tend to increase this coefficient. Indeed, these parameters modify both the growth rate of the mixing layer width at the interface and the amplitude of the alternating transverse velocity across the interface, thus affecting the exchange capacities from the main stream to the cavity.

INTRODUCTION

Controlling the exchange of mass between a river and its connected storage zones is of primary importance for river management, given that in the storage zones the lower water velocities and higher local residence times enhance fish and vegetation biodiversity [START_REF] Jackson | A mean residence time relationship for lateral cavities in gravel-bed rivers and streams: Incorporating streambed roughness and cavity shape[END_REF].

Moreover, sediment and dissolved contaminant transferred from the main stream to the transient storage zones can be accumulated and modified due to biogeochemical reactions [START_REF] Ensign | In-channel transient storage and associated nutrient retention: Evidence from experimental manipulations[END_REF][START_REF] Argerich | Influence of transient storage on stream nutrient uptake based on substrata manipulation[END_REF]. Using former data from the literature, [START_REF] Jackson | A mean residence time relationship for lateral cavities in gravel-bed rivers and streams: Incorporating streambed roughness and cavity shape[END_REF] proposed empirical relationships relating the mean residence time of dissolved material in a cavity on the field as a function of several non-dimensional parameters of the flow and storage zone characteristics. These measurements of exchange capacities or residence times can represent mass exchange processes in conditions with complex shapes and bathymetry of the cavity. Nevertheless, due to obvious experimental restrictions, these field experiments do not allow to i) Investigate in detail the physical processes involved in the mass introduction within the storage zone; and ii) Vary the flow and cavity parameters in a controlled way in order to identify the impact of all governing parameters. To overcome these limitations, the present paper is based on an experimental approach in controlled and simplified conditions in order to gain information on the physics of the mass exchange processes and the impact of the non-dimensional parameters on the exchange capacity.

The selected storage zone geometry is an open-channel lateral cavity, which represents a major transient storage zone in riverine environments [START_REF] Jackson | A mean residence time relationship for lateral cavities in gravel-bed rivers and streams: Incorporating streambed roughness and cavity shape[END_REF].

Open-channel lateral cavities connected to rivers can be divided into (i) isolated cavities (as in the present case), (ii) cavities located between two consecutive groynes (as investigated by Uijttewaal et al., 2001[START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF] and (iii) a series of lateral cavities (see [START_REF] Erpicum | 2D numerical flow modeling in a macro-rough channel[END_REF] or [START_REF] Meile | Water-surface oscillations in channels with axi-symmetric cavities[END_REF]. Differences between these configurations mainly lie on the impact of the cavities located upstream and downstream from the studied one, and particularly in the direction of the main stream when approaching the cavity: for case (i), the upstream flow is parallel to the sidewalls and is not affected by any upstream mixing layer; for case (iii), the mean flow is parallel to the side walls but the flow is affected by intense turbulent structures generated in the mixing layer of the upstream dead zone (see [START_REF] Tuna | Self-sustained oscillations of shallow flow past sequential cavities[END_REF]; finally for case (ii), the transverse component of the main stream is directed towards the center of the main stream. The present paper is dedicated to lateral cavities of type (i), such as oxbows, which are examples of natural isolated lateral cavities, or fluvial or sea harbors (respectively connected to rivers or sea streams), which are examples of artificial isolated lateral cavities.

It appears from the literature that the main parameter governing the flow pattern in an isolated lateral cavity is the aspect ratio W/L of the cavity (see [START_REF] Mignot | Measurement of the flow pattern in open-channel lateral cavities with increasing aspect ratio[END_REF] where L is the length of the cavity in the direction parallel to the main stream and W is the cavity width along the crosswise direction, as sketched in Fig. 1. Most studies in the literature consider a low aspect ratio (W/L1) for which two main horizontal cells are aligned along the main stream axis for the lowest aspect ratios [START_REF] Sanjou | Hydrodynamic characteristics and related mass-transfer properties in open-channel flows with rectangular embayment zone[END_REF] or a single cell occupies almost the whole available space in the cavity for aspect ratios approaching 1 [START_REF] Mizumura | Flow in open channel embayments[END_REF]. For higher aspect ratios, [START_REF] Booij | Exchange of mass in harbors[END_REF] and [START_REF] Mignot | Measurement of the flow pattern in open-channel lateral cavities with increasing aspect ratio[END_REF] showed that for an aspect ratio W/L equal to 3, a second stable cell takes place on the side of the first one, aligned along the width of the cavity W, in a similar way as for [START_REF] Lecoz | Morphodynamics of the exit of a cutoff meander: experimental findings from field and laboratory studies[END_REF] on a physical model with a very long curved cavity.

For all cases, at the interface between the main stream and the cavity, a large velocity gradient forms [START_REF] Tuna | Shallow flow past a cavity: globally coupled oscillations as a function of depth[END_REF] which leads to a horizontal mixing layer that extends from the upstream to the downstream corner of the cavity. [START_REF] Mignot | Coherent turbulent structures at the mixinginterface of a square open-channel lateral cavity[END_REF] showed that the alternation of clockwise and counter-clockwise coherent turbulent structures, shed in the vicinity of the upstream corner and advected along the mixing layer towards the downstream corner, generates a periodic alternating transverse velocity across the interface. The mixing layer then transfers from the main stream towards the cavity: i) momentum, which maintains the recirculating motion of the flow in the cavity and ii) mass such as pollutants, nutrients and gases influencing the ecological equilibrium of the cavity, or fine sediments influencing the geomorphological budget.

The objectives of the present paper are, first, to gain information on the exchange processes of passive scalar such as pollutants between the main stream and the isolated open-channel lateral cavity and, second, to identify the main parameters influencing the exchange capacity. The first section describes the experimental set-up used in the present work and introduces the two methodologies available in the literature to quantify the exchange coefficient. This section then proposes a dimensional analysis applied to the mass exchange coefficient and summarizes the available data in the literature and the present experimental data set. The second and third sections are then devoted to the physical description of the exchange processes and the sensitivity analysis permitting to conclude on the parameters that influence the mass exchange capacity in the lateral cavity flow.

1 Methodology and literature data review

Experimental set-up

The experiments are performed in the channel intersection facility at the Laboratoire de Mécanique des Fluides et d'Acoustique at the Université de Lyon, France. The facility consists of a 4.9m long horizontal glass channel of b=0.3m wide rectangular shape section, which intersects a 2m wide (along W) similar glass side channel of length L=0.3m (see Fig. 1) forming a 90° angle (see [START_REF] Mignot | Impact of topographic obstacles on the discharge distribution in open-channel bifurcations[END_REF] for more details on the facility). The width of the cavity W can be continuously modified by displacing the movable wall that closes the side channel as sketched in Fig. 1. A honeycomb at the inlet of the upstream branch serves to stabilize and straighten the inlet flow and a downstream sharp crested weir is used to adjust the water depth h in the cavity region. The main stream discharge Q is measured within the pumping loop using a Promag 50 flowmeter (from Endress Hauser; accuracy 0.05L/s). The axis system (x, y) is centered on the upstream corner of the cavity (see Fig. 1) with the x axis parallel to the main stream axis and the y axis oriented from the main stream to the cavity and the instantaneous velocity components u and v are defined along the x and y axes respectively. For all flow configurations listed in Table 1, a maximum oscillation amplitude of the water depth equal to 3.3% is recorded using five wave probes distributed within the cavity area; the present configurations is then considered as "without seiching" (see [START_REF] Tuna | Shallow flow past a cavity: globally coupled oscillations as a function of depth[END_REF].

Two main methodologies are reported in the literature to estimate the mass exchange capacity between a main stream and a dead zone, which are replicated herein for the open-channel lateral cavity. The first methodology, named "dye release method", is based upon continuously releasing dye in the main stream and measuring its accumulation in the cavity. The second methodology, named "transverse velocity method", is based upon velocity measurements at the interface between the main stream and the cavity. Both methodologies derive from the 1D mass exchange model presented in the next section.

1D mass exchange model

The theoretical background for the dye release method derives from the 1D mass exchange model proposed by [START_REF] Valentine | Longitudinal dispersion with dead zones[END_REF] and Uijttewaal et al. (2001), which states that the mass of a passive scalar passing from the main stream towards the cavity evolves with time as a function of the exchanges across the interface between these two regions. This relation reads:

    ex ex MS C dM S V C C t dt  ( 1
)
where M is the mass of scalar located within the cavity at time t, S ex =Lh is the interface section between the main stream and the cavity (see Fig. 1), V ex is the characteristic, time and space averaged, exchange velocity across the interface, C MS is the approaching main stream concentration (here it is uniform and steady) and C C [kg.m -3 ] is the spatially averaged scalar concentration in the cavity at time t. Moreover, [START_REF] Valentine | Longitudinal dispersion with dead zones[END_REF] propose to relate V ex to the bulk velocity of the main stream U b =Q/(bh) as

V ex =kU b (2)
where k is the so-called non-dimensional exchange coefficient. The mass of scalar in the cavity at a given time is given by

WLh t C t M C ). ( ) (  (3)
where WLh is the volume of water in the cavity (constant with time). Thus Eq. 1 becomes:

  ) ( ) ( t C C W kU dt t dC C MS b c   (4)
Integrating Eq. 4 leads to an expression for the evolution of the concentration in time,

MS t W kU C C Ae t C b    ) ( (5)
with A being a constant. The initial concentration in the cavity is zero C C (t=0)=0, and the concentration in the main stream C MS remains constant with time. Then A=-C MS , and Eq. 5 can be written as follows:

t W kU MS C MS b e C t C C    ) ( (6)
The efficiency of the mass exchange is accounted for by the exchange coefficient k. Its value can be estimated from the time evolution of the concentration (Eq. 6) through the dye release method, as described in section 1.3. The coefficient k can also be directly estimated from the transverse velocity measurement (Eq. 2), as described in section 1.4 below.

Dye release method

The dye release method consists on the following steps:

-measuring the time evolution of averaged dye concentration in the cavity C C (t)

-plotting the evolution of the left-hand-side term of Eq.6 as a function of time -identifying the slope of the curve in order to estimate k using Eq.6. This k is noted k dye for "dye release method". The whole set-up is installed inside of a dark tent where a controlled and repeatable ambient light is set using two broadband light sources as sketched on Fig. 1. Previous to performing the experiments, the cavity is closed and consecutively filled with volumes of water with various controlled dye concentrations so that the water depth remains exactly equal to the water depth of the studied flow configuration, as indicated in Table 1, and the lightning (light intensity and spatial orientation) is kept similar during the whole calibration process (see [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF]. The dye used herein is red food color. A high-resolution photograph (4000-4000 pixels) of the cavity and intersection is taken from above the free surface for each concentration (see Fig. 1). A calibration curve of light intensity vs. dye concentration is then established for each pixel. Before starting the measurements, a steady-state clear-water flow is established in the main stream and the cavity. Then, a constant discharge of dye (two orders of magnitude lower than the water discharge) is injected at the upstream end of the inlet channel using a multipoint injection (see Fig. 1) so that complete mixing is achieved in the upstream channel before reaching the cavity. The initial time t=0, is set when the dye reaches the intersection. Similar photos under similar lightning as in the calibration process are taken along time. The light intensity at each pixel of each photo, along with all the calibration curves finally allow to estimate the concentration map of C C (x,y,t)/C MS at every time step (see Fig. 2).

The major limitations of such approach for estimating k (using Eq.6) are detailed hereafter. First the concentration field is assumed to be 2D, i.e. with no evolution along the vertical axis so that the estimated concentration at one pixel on the CCD sensor of the camera, which corresponds to the concentration of the top layer of the corresponding water column, is representative of the whole water column. Actually, the higher the concentration, the more opaque the water volume and thus the thinner this top-layer "seen" by the camera. Nevertheless, the uncertainty of the method (at least as it is applied herein) is not limited to the vertical non-uniformity of concentration, but rather to the non-uniformity of the whole water volume. Indeed, the light intensity recorded by the CCD sensor relates the light attenuation along its whole path, from the light sources, across the water volume and the free surface and finally to the camera (see Fig. 1). This light intensity is then compared with the light intensity in the calibration images, which are taken in water volumes with uniform concentrations. The second limitation of the method is then that we compare light attenuation in non-uniform concentration water during the mass-exchange process with uniform concentration water during the calibration process.

Transverse velocity method

The transverse velocity method consists in estimating the time and space averaged transverse exchange velocity V ex from a velocity measurement and then computing the exchange coefficient using Eq.2. [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF] proposed to evaluate V ex based on the measurement of transverse velocity along the main stream-cavity interface as:

  ex S ex ex S V dS t) z, v(x, 2 1 (7)
where |v(x,z,t)| is the absolute value of the instantaneous transverse velocity component (along the y axis, perpendicular to the main stream) and the overbar denotes time-averaging. The k coefficient obtained through this method is noted k veloc . Note that the value "2" in Eq. 7 derives from the fact that the absolute value of the transverse velocity component stands for the exchanges from the main stream to the cavity plus the exchanges from the cavity to the main stream, that is twice the exchanges represented by Eq.1.

The transverse velocity approach is performed here using 2D horizontal PIV data (as in [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF][START_REF] Tuna | Shallow flow past a cavity: globally coupled oscillations as a function of depth[END_REF]. A 40mW continuous laser coupled with a cylindrical lens generates a 1mm thick horizontal light layer at the desired laser elevation z L . Polyamid particles (diameter 50 m) are added to the water. Finally, a video camera is located above the cavity interface in order to record the particle motion over 3000 consecutive images of 1280x960 pixels each with a frequency of 30Hz and a spatial resolution of 0.3mm per pixel (see [START_REF] Mignot | Coherent turbulent structures at the mixinginterface of a square open-channel lateral cavity[END_REF] for more details). Then, the commercial software DaVis (from Lavision) allows to correct the image distortions, subtract the image background and finally compute the velocity field. It was verified that the maximum free-surface oscillations in the cavity remained negligibly small, with amplitudes smaller than 2% for all flows (lower than 0.5% for most flows). To estimate the uncertainty associated to the PIV measurements, we first verify that the Stokes number associated to the seeding particles remains low enough (St0.05). Then, as in [START_REF] Mignot | Analysis of flow separation using a local frame-axis: application to the open-channel bifurcation[END_REF], the uncertainty associated to the measured velocity  v is estimated as that of the Gaussian peak fitting process, assumed to be equal to the velocity associated to a displacement of 0.1 pixel, plus that the seeding concentration gradient which is assumed to be equal to that of the Gaussian peak fitting (see [START_REF] Jahanmiri | Particle image velocimetry: Fundamentals and its application[END_REF]. This gives  v 2 x 0.1 pixel x 0.3 mm/pixel x 30 Hz = 1.8 mm/s. Then the uncertainty associated to the exchange velocity V ex corresponds to

016 . 0 2   v Vex  
mm/s where

N N v v v     
with N=3000 velocity measurement samples in the present experiments.

Assuming a 2D-flow (with no variation along z), Eq.7 can be simplified as:

     L x x ex L V 0 L 2 dx t) , z z v(x, 2 1 (8)
and the exchange coefficient k veloc is finally obtained using Eq.2, using V ex2 instead of V ex . Note that the 2D flow assumption is supported by the measurements performed by [START_REF] Tuna | Shallow flow past a cavity: globally coupled oscillations as a function of depth[END_REF] and especially their figures 14 and 16 (left) without gravity standing waves, where the variation along z of the time-averaged absolute value of transverse velocity remains limited.

Discussion regarding the methods

The dye release method is a physically-based method in the sense that it is effectively based on exchange measurements of a given passive scalar. However, the scalar concentration is estimated here from a camera located above the free surface, recording for each pixel the light intensity transmitted from the whole water column. Note that [START_REF] Sanjou | Hydrodynamic characteristics and related mass-transfer properties in open-channel flows with rectangular embayment zone[END_REF] rather used LIF to measure the concentration field, with the limitation that their concentration is then estimated over a single horizontal plane. In both cases, C C (in Eq.6) is not a real volumetric averaged concentration but reduces to a concentration averaged over a portion of the cavity volume.

On the other hand, the transverse velocity method is simpler as it is based on velocity measurements; nevertheless, it is representative of the exchange processes only at early stages of mixing process, i.e. when the transfer is regulated by the dynamics of the shear layer. Tuna et al.

(2013) revealed that an estimation of V ex through measurements at a single elevation in the upper part of the water column (assuming V ex2 ~Vex ) leads to a fair estimate of this coefficient. Moreover, [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF] obtained an average error of 17% between the two techniques in a groyne field cavity (see their table 2).

Dimensional analysis and data set

The dimensional analysis applied to the mass exchange coefficient k in the present configuration is a simplification of the one proposed by [START_REF] Jackson | A mean residence time relationship for lateral cavities in gravel-bed rivers and streams: Incorporating streambed roughness and cavity shape[END_REF] as it is here applied to a horizontal, rectangular, smooth cavity connected to a subcritical, straight, steady, smooth main stream of rectangular cross-section, with no step at the interface, and all walls in the cavity and the main stream being vertical. Moreover, assuming that the boundary layers are fully developed, this dimensional analysis finally involves 9 variables that are the cavity width W and length L, the main stream width b, the roughness of the mainstream and cavity walls k s , the bulk velocity of the main stream U b , the water depth in the cavity and the mainstream h, the acceleration due to gravity g, and finally the fluid density  and dynamic viscosity . In the present case b=L, so that the number of variables reduces to 8. Choosing as length scale L, as time scale L/U b and as mass scale L 3 , the dimensional analysis reveals that 5 dimensionless parameters characterize the exchange coefficient,

i.e:

k=f(W/L, Fr, Re, h/L, k s /L) (9)
where Re and Fr are the upstream Reynolds and Froude numbers of the main stream, respectively.

The dimensionless roughness height k s /L can be discarded as, with glass walls, the flow regime in the main stream and the cavity is hydraulically smooth. Similarly, the effect of the Froude number

gh U Fr b 
is discarded as only subcritical main streams are considered herein, with a Froude number that does not exceed 0.25 (see Table 1). This assumption that the Froude number has a negligible influence is supported by the recent work from [START_REF] Karimpour | Mixing in shallow waters in high Froude number[END_REF]. Through Direct Numerical Simulations, the authors showed (in their figure 5a) that the growth rate of the mixing layer width (see Figure 10) remains about constant for convective Froude numbers (Fr c ) lower than 0.25 (and decreases for larger Fr c values), with Fr c ~Fr/2. As Fr<0.23 in Table 1 (i.e.

Fr c 0.12), the effect of variations of Fr on the characteristics of the mixing layer is assumed to be negligible. Note however that in realistic rivers, this Froude number can reach much larger values and thus should no longer be discarded from the dimensional analysis. The 3 remaining dimensionless parameters considered herein are then:

-W/L: the geometrical aspect ratio of the cavity Table 1 then comprises 3 series of experiments:

in Series 1, the main stream flow conditions remain constant and only the aspect ratio of the cavity W/L is varied, by regularly increasing its width W.

in Series 2, the dimensions of the cavity remain unchanged, so does the water depth in the whole domain; the bulk velocity in the main stream is gradually increased, leading to a corresponding increase of the Reynolds number Re.

in Series 3, the dimensions of the cavity remain unchanged, so does the bulk velocity of the main stream; the water depth is gradually increased, leading to a corresponding increase of the normalized water depth h/L and of the Reynolds number Re. In the literature, the exchange coefficient k between a main stream and a dead zone in laboratory conditions was measured for various configurations such as a groyne field (see for instance Uijttewaal et al., 2001 and[START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF], a closed cavity located below the main flow (as for [START_REF] Caton | Dispersion mechanism in a street Canyon[END_REF], a series of closed cavities below the main flow, or "street canyons" (as First, [START_REF] Booij | Exchange of mass in harbors[END_REF] used the temperature as an indicator to estimate k applying the "dye release method" for two flow configurations. However the temperature measurements were limited to a single location in the cavity so that C C (in Eq.6) was not spatially averaged.

Second, [START_REF] Altai | Retention time in a recirculating flow[END_REF] applied the dye release method by initially inserting dye in the cavity zone and measuring its concentration decay. No information is however given regarding the method used to estimate the dye concentration from the recorded images. These authors considered two square cavity dimensions with W/L=1 in both cases. For one cavity, they increased the main stream velocity, while for the other cavity they modified both the main stream velocity and water depth (see Table 2). However, the authors encountered no clear tendency of the k coefficient with these parameters.

Third, Sanjou & Nezu (2013) considered a narrow cavity with W/L=1/3 and applied the LIF method to measure the concentration field at one elevation near the surface region (z/h=0.94) for a single flow configuration. In this case, Cc is then averaged only over a single measured horizontal plane.

Fourth, [START_REF] Tuna | Shallow flow past a cavity: globally coupled oscillations as a function of depth[END_REF] measured k veloc using PIV at 6 elevations for two flows: one without and one with seiching. They obtained two different k veloc profiles: from 0.026 (near the surface) to 0.03 (near the bed) without seiching and from 0.034 (near the surface) to 0.042 (at mid-elevation) with seiching (see Table 2).

The collected data from the literature does not permit to conclude regarding the impact of the dimensionless parameters sorted from the dimensional analysis (Eq. 9) on the mass exchange coefficient. The absence of a tendency is the starting motivation of the present work detailed in the next sections. 2

Impact of the geometrical aspect ratio W/L

The present section focuses on the impact of the geometrical aspect ratio of the cavity W/L on the mass exchange processes and mass exchange coefficient from the main stream to the cavity. The first subsection presents the mean velocity fields corresponding to the 5 aspect ratios of the cavity considered in Series 1 in Table 1 then the second subsection shows the time evolution of the 2D maps of passive scalar concentration in the cavity, and finally, the third subsection estimates the passive scalar exchange coefficients.

2D time-averaged velocity fields

In order to help understanding the mass exchange processes in the next sub-section, Fig. 3 first plots the time-averaged velocity field measured through 2D PIV (as described above) at the elevation z/h=0.71 for the 5 configurations from Series 1 in Table 1. To do so, the camera used to record the particle motion is placed above the cavity, the sampling frequency is reduced to 12.2Hz and the spatial resolution to about 1mm per pixel (see [START_REF] Mignot | Measurement of the flow pattern in open-channel lateral cavities with increasing aspect ratio[END_REF] for more details).

For a square cavity (W/L=1), Fig. 3 reveals that a single recirculation cell forms and occupies the whole available space in the cavity region, as previously shown for instance by [START_REF] Booij | Exchange of mass in harbors[END_REF], [START_REF] Kimura | Fundamental properties of flows in open channels with dead zone[END_REF] or [START_REF] Mizumura | Flow in open channel embayments[END_REF]. The typical velocity magnitude of the recirculation cell equals about 0.025m/s, that is about 0.15U b (as for [START_REF] Tuna | Shallow flow past a cavity: globally coupled oscillations as a function of depth[END_REF], for a slightly longer cavity W). For a slightly wider cavity (W/L=2), a second cell appears but remains confined in the downstream-external corner (x/L1, y/L2). As the width of the cavity W increases, the size of the second cell increases while the size of the first cell remains almost constant. For geometries with higher aspect ratios (W/L>4), the flow in the second cell becomes 3D and particularly complex (not detailed herein). Note that the streamlines remain unclosed due to limited but still existing vertical velocities.

For the purpose of the present paper, a major aspect of the velocity field is the velocity magnitude along the interface (at y=0) where mass exchanges occur through turbulent diffusion. This velocity slightly decreases for increasing W/L values: the typical velocity in the center of the interface (y=0 & x/L0.5) equals 0.0275m/s for W/L=1, and decreases to 0.0225m/s for W/L=2-3 and finally to 0.0175m/s for W/L=4-5. Thus the velocity gradient across the interface tends to increase, as W/L increases, from about U0.167-0.02750.14m/s for W/L=1 to U0.166-0.01750.15m/s for W/L=4-5. This corresponds to an increase of velocity gradient limited to about 7%. This provides a first hint that the exchange capacity should not vary much as W/L varies from 1 to 5.

Concentration maps

Fig. 2 plots the concentration maps measured through the dye release method, at selected times, for three configurations from Series 1: W/L=1, 3 and 5, while Fig. 4 sketches the physical processes involved in the mass exchange from the main flow to the cavity and within the cavity. At initial times (steps 1 to 3), the exchange processes are similar for all W/L configurations while for step 4, the processes differ for narrow (small W/L) and wide (high W/L) cavities.

Step 1. Dye enters in the cavity through turbulent diffusion in the mixing layer (arrows A on Fig. 4).

At time t=5s (Fig. 2), dye is already present on the cavity side (y>0) all along the interface (for

x/L=0 to 1). This turbulent diffusion process continues during the whole experiment.

Step 2. Dye entering the cavity during step 1 is transported through advection by the first recirculation cell of the cavity (arrow B, see Fig. 4) towards the downstream corner (x/L=1, y=0).

This low-velocity region (see Fig. 3) thus acts as a storage region where a high concentration is measured during the whole experiment for all geometries.

Step 3. The dye accumulating near the downstream corner of the cavity continues to be transported through advection by the first cell (arrow B) (i) along the downstream wall (along x/L=1), (ii) along the end wall (y/L=1) for W/L=1 geometry and along the end of the first cell (at y/L1.5-1.7) for the wider geometries (W/L=3 and 5, see for instance t=20s for W/L=3 geometry), and (iii) along the upstream wall (x=0) back to the interface region (y0).

Step 4. This step only applies to cavities with two cells (W/L=3 and 5) for which part of the scalar reaching the upstream wall (x=0) in step 3 is transferred to the second cell at the saddle point (y/L1.5 to 1.7 and x=0) through arrow D. The dye that enters the second cell is then advected in a counterclockwise motion (arrow B 2 ) and finally joins the first cell along the downstream wall (at y/L2, x/L=1).

Step 5. In the meantime of steps 3 and 4, turbulent diffusion (arrows C and C 2 ) occurs from the outer layer of each cell where the concentration is high to the center of the cells where the concentration is low. Dye then reaches the center of the first cell at t=20s to 35s for W/L=1 and at t=35s to 80s for W/L=3 and 5.

When comparing the scalar fields for the three geometries, it appears that at a given time, the dye distribution is much more homogenized in the square cavity than in the two wider ones (see for instance at t=80s). Two cumulative reasons can be listed for this effect. First, the time for the dye to rotate along the outer layer of the first cell and reach the upstream corner of the cavity ( xy0) is much higher for wide cavities than for the narrow one (see for instance the three concentration maps at t=20s); this is because (i) the width (along y axis) of the first cell is much shorter for the narrow cavity than for the wider ones and (ii) the typical velocity in the first cell is slightly higher for the narrow cavity than in the wider ones (see Fig. 3). The second reason is that for the wide cavities (W/L equal to 3 and 5), part of the dye from the first cell is transferred to the second cell.

Moreover, it appears that the exchange of scalar from the first cell towards the second cell is much more efficient for W/L=3, for which the second cell occupies the whole available space for x/L>1.7 than for W/L=5 for which the flow is 3-dimensional and complex (not shown here).

A B C

W/L=1

B C A D B2 C2 W/L>2

Exchange coefficients

Following the methodology applied by Uijttewaal et al. (2001) and [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF], the mean concentration C c (t) in Eq.6 (dye release method) is obtained by spatially averaging at each time the concentration maps such as plotted in Fig. 2 over the cavity area. Then, the relation [C MS -C C (t)]/C MS =f(tU b /W) is plotted in Fig. 5. According to Eq.6, the slope of the curve should be equal to -k/ln(10), permitting to estimate the k dye coefficient. This figure confirms that, at initial time, k dye is similar for the three cavities with increasing aspect ratio. Nevertheless, it appears that the absolute value of the slope decreases with time (as for Uijttewaal et al., 2001 or Sanjou and[START_REF] Sanjou | Hydrodynamic characteristics and related mass-transfer properties in open-channel flows with rectangular embayment zone[END_REF].

According to Uijttewaal et al. (2001), the fact that the measured data departs from the linear curve is due to additional phenomena within the cavity such as "exchange with the center of the gyre"

"where the mixing layer does not penetrate"; this exchange is "governed by small scale turbulent motion and is therefore slower". This might be true is the narrow cavities considered by these authors (W/L=0.3 & 0.7). However, in the present longer cavities (W/L1), we believe that this departure is mostly due to the uncertainty associated with the dye release method, detailed in section 1.3. It is thus difficult to estimate a constant k value from such plots.

The exchange coefficients can alternatively be obtained through the transverse velocity method (Eqs.8 and 2). The right-hand-side term of Eq.6 is then added to Fig. 5 as solid lines for the three configurations using the k values obtained through the transverse velocity method (k veloc ). The agreement between the spatially averaged concentrations (symbols) and the theoretical curve based on k veloc (solid lines) appears to be fair over the initial time but, as indicated above and reported in the literature, the measured concentration rapidly departs from this straight line (typically for t>50s). To summarize, the dye release method gives a nice overview of the processes involved in the dye transport across the mixing layer and within the cavity but estimations of exchange coefficients based on this method must be taken with caution. Consequently only data from the transverse velocity method will be used in the following section in order to assess the impact of the non-dimensional parameters on the exchange coefficient. Regarding the influence of the geometrical aspect ratio of the cavity on the exchange coefficient, Table 1 reveals that k veloc hardly evolves as a function of W/L as anticipated in section 2.1 when analyzing the 2D velocity fields. Fig. 6 plots the distribution of the mean absolute transverse velocity at the measured elevation (z/h=0.71) along the main stream / cavity interface, which averaged value equals k veloc as in Eq.8. The curves appear to be similar for all aspect ratio of the cavity and qualitatively in agreement with measurements from [START_REF] Tuna | Shallow flow past a cavity: globally coupled oscillations as a function of depth[END_REF] (in their figure 14): () vx is minimum near the upstream corner of the cavity and increases towards the downstream corner with a first maximum obtained near x/L=0.7 and a second one at the downstream corner (x/L=1). 3 Impact of the two other parameters: Re and h/L

The impact of the main stream Reynolds number Re on the exchange coefficient is estimated through the analysis of k veloc values (Eq.8) for all configurations from Series 2 (Table 1) with constant h/L and W/L parameters. These estimates are plotted in Fig. 7a. Similarly, the impact of h/L on the exchange coefficient is estimated through the analysis of all configurations from Series 3

(Table 1), with constant W/L but varying Re, and are plotted in Fig. 7b. 2) such variation does not affect the inflow (upstream from the cavity) but directly modifies the development of the mixing layer at the interface between the cavity and the mean flow.

To investigate the validity of each of these two assumptions, Figs.9a and 9b plot, for Series 2 and 3 respectively, the transverse (along y) profiles of mean streamwise (along x) velocity at the elevation z/h=0.71 at x/L=0.1, i.e. near the upstream corner of the cavity, which is assumed to be representative of the lateral boundary layer of the inflow (for x<0). On the one hand, Fig. 9a reveals that the Reynolds number Re hardly affects the mean streamwise velocity profile on the main stream side (y<0), which discards assumption # 1. On the other hand, Fig. 9b indicates that the combined effects of h/L and Re slightly affects the shape of the transverse boundary layer of the incoming flow: as h/L and Re increase, the maximum normalized streamwise velocity magnitude increases (see the arrow in Fig. 9b). Contrary to the previous case, this result does not completely discard assumption #1.

Moreover, Figs.10a and 10b plot, for Series 2 and 3 respectively, the streamwise evolution of the mixing layer width  computed as:

max / ) ( ) ( ) ( y x U x U x      (9)
with U the difference of outer velocities (between the main stream and the cavity) at a given x axis and the denominator is the maximum transverse (along y) gradient of mean streamwise (along x) velocity. Note that the estimation of  for x/L<0.1 is uncertain due to the complexity of the velocity field in the region surrounding the stagnation point around the upstream corner of the cavity. First, it appears from Figs.10a and 10b that the mixing layer width near the upstream corner of the cavity (x/L0.05-0.1) remains similar (/L0.09) for all configurations. This result confirms the observation from Fig. 9a (respectively Fig. 9b) that, in the ranges considered in Table 1, the Reynolds number (respectively the Reynolds number and the normalized water depth) does not affect (respectively have a limited impact) on the upstream boundary layer development.

Oppositely, Figs.10a and 10b reveal that further downstream (x/L>0.1), the mixing layer width increasing rate is strongly dependent on both parameters (Re and h/L): it decreases as Re increases in Fig. 10a and as h/L and Re decrease in Fig. 10b. This increasing rate, noted α herein, can be estimated through the following classical equation, simplified for the present case assuming that the typical outer velocity (noted with a tilde) on the mean flow side equals about U b and is nil on the cavity side:

    2 2 ~~    b b c U U U U dx d (10)
with U  the difference between the typical outer velocities on each side, and c U ~ their average value. Tendency curves are added as grey solid lines on Figs.10a and 10b for the minimum and maximum α values reported by [START_REF] Pope | Turbulent flows[END_REF] based on a review of the literature. The figures confirm that α is included in this range.

The fact that α decreases as Re increases (Fig. 10a) and as h/L and Re decrease (Fig. 10b) gives credit to hypothesis #2 for both series (2 and 3): Re and h/L directly affect the mixing layer development along the main stream / cavity interface and this is responsible for the decreasing exchange coefficients k values as Re increases or as h/L decreases (Table 1 and Fig. 7). Note that a decrease of α and thus of  indicates that the typical sizes of the coherent turbulent structures advected along the mixing layer are of reduced transverse (along y) extent (see [START_REF] Mignot | Coherent turbulent structures at the mixinginterface of a square open-channel lateral cavity[END_REF] and thus that the efficiency of passive scalar transfer through turbulent diffusion across the interface also decreases. The fact that decreasing the water depth tends to decrease the horizontal extension of the coherent turbulent structures was recently observed by [START_REF] Han | Shallow mixing layer downstream a sudden expansion[END_REF]. Figure 9 Transverse (along y) profiles of normalized mean streamwise (along x) velocity U measured across the cavity (y>0) / main stream (y<0) interface at z/h=0.71 and x/L=0.1 for series 2 (a) and series 3 (b) configurations from Table 1.

Figure 10 Streamwise evolution of the normalized mixing layer width /L for series 2 (a) and series 3 (b) configurations from see Table 1 along with tendency curves corresponding to Eq.10 with =0.06 and 0.11 (solid grey lines).

Discussions and conclusions

Comparison with tendencies available in the literature

The sensitivity analysis performed above permitted to identify the non-dimensional parameters that impact the coefficient of mass exchange between a simplified, wide (W/L>1) lateral cavity and a subcritical straight open-channel flow. We showed that the geometrical aspect ratio of the cavity does not affect this coefficient, but that an increase of the normalized water depth h/L or a decrease of the main stream Reynolds number Re tend to increase it. The aim of the present section is to compare these tendencies with the ones reported for cavities located within groyne fields, mostly by [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF], and in complex shape cavities and cavities located within groyne fields by [START_REF] Jackson | Flow structure and mean residence times of lateral cavities in open channel flows: influence of bed roughness and shape[END_REF]. Note that a section in the introduction discusses on the differences between isolated lateral cavities (as considered herein) and cavities located within groyne fields. First we checked (not shown here) that the fitting curve proposed by [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF], which relates the exchange coefficient to the so-called morphometric parameter R d does not apply to the present (Table 2) and literature (Table 1) cavity data. Moreover, applying the two empirical relations proposed by [START_REF] Jackson | Flow structure and mean residence times of lateral cavities in open channel flows: influence of bed roughness and shape[END_REF] to the present (respectively from the literature) cavity configurations lead to a typical difference in k of 42% (respectively 46%) using their equation 8 and 52% (respectively 29%) using their equation 9. None of these empirical relations appears to be appropriate to the simplified lateral cavity configuration.

Moreover, regarding the qualitative evolution of k:

-In Series 1 (Table 1), the variation of the W/L aspect ratio of the cavity appears not to affect the value of k, while [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF] and [START_REF] Jackson | Flow structure and mean residence times of lateral cavities in open channel flows: influence of bed roughness and shape[END_REF] investigations indicate that k decreases as W/L increases.

-In Series 2, k appears to decrease as the bulk velocity of the mean flow U b is increased, in agreement with equation ( 8) from [START_REF] Jackson | Flow structure and mean residence times of lateral cavities in open channel flows: influence of bed roughness and shape[END_REF], but not with equation ( 9) from [START_REF] Jackson | Flow structure and mean residence times of lateral cavities in open channel flows: influence of bed roughness and shape[END_REF] and the work from [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF], both of which predict no effect of U b on k.

-In Series 3, k appears to increase as the water depth increases while [START_REF] Weitbrecht | Experiments on mass exchange between groin fields and main stream rivers[END_REF] and [START_REF] Jackson | Flow structure and mean residence times of lateral cavities in open channel flows: influence of bed roughness and shape[END_REF] investigations predict that k decreases as the water depth increases.

These unsatisfactory agreements with the tendencies proposed in the literature confirm the statement made in the introduction that groyne and cavity flows essentially differ in terms of hydrodynamics and that the processes governing the mass exchange from the main stream to the dead zones is strongly dependent on the type of storage zone. Notably, present results highlight the influence of the mixing layer development on the exchange coefficient. Yet, this mixing layer is itself considerably influenced by the upstream groynes in the groyned field cases, as shown by [START_REF] Mccoy | Numerical investigation of flow hydrodynamics in a channel with a series of groynes[END_REF]

Conclusions

The main findings of the present work regarding mass exchange between a straight open-channel flow and its adjacent rectangular lateral cavity are finally listed below:

-The processes involved in the introduction and spreading of passive scalar in the cavity rely on the turbulent diffusion across the interface, the advection by the mean flow in the cavity and the turbulent diffusion within each recirculating cell and from one cell to the neighboring one (when existing).

-The use of the dye release method, useful for qualitative exchange analyses, should be taken with caution for estimating the mass exchange coefficient, even though its use during the very first stage of the mass exchange appears to be valid. On the other hand, the transverse velocity method should also be taken with caution as it is based on velocity measurements rather than concentrations; also, it is representative of the exchange processes only at early stages of mixing process, i.e. when the transfer is regulated by the dynamics of the shear layer.

-The dimensional analysis applied to this exchange coefficient brings to light three nondimensional parameters. Their impact on the exchange coefficient can be summarized as:

* the geometric aspect ratio of the cavity does not affect the exchange coefficient, at least as long as W/L remains larger than 1;

* increasing the normalized water depth tends to increase the exchange coefficient;

* increasing the Reynolds number of the main stream tends to decrease the exchange coefficient.

We showed that, in the present range of variation of these parameters, the boundary layer of the inflow is almost not affected by the variation of these parameters, but that the parameters directly affect the characteristics of the mixing layer at the interface between both flow regions, which in turn affect the mass exchange capacities between the main channel and the cavity.

Acknowledgments

This work was supported by ECOS-Conicyt C11E02 and Fondecyt 1130940 grants. We also acknowledge the funding from Conicyt/Fondap grant 15110017 and the PhD grant of W. Cai from the Chinese Scholarship Council.

Figure 1 .

 1 Figure 1. Scheme of the dye release method at initial time with a uniform concentration in the main stream and a null concentration in the cavity.

-

  h/L: the shallowness parameter -Re: the Reynolds number of the main stream, defined as Re=4U b Lh/[(L+2h)].

Figure 2 .Figure 3 .

 23 Figure 2. Time-evolution of dye concentration maps for configurations with W/L=1, 3 and 5.

Figure 4 .

 4 Figure 4. Scheme of the passive scalar transport processes for a cavity with a single cell (such as W/L=1) and two cells (W/L>2).

Figure 5 .

 5 Figure 5. Time-evolution of spatially averaged dye concentration in the cavity for W/L=1, 3 and 5 in Series 1 (symbols) along with the curves corresponding to the right hand side of Eq.6: y=exp(-kU b t/W) for W/L=1, 3 and 5 with two k values typical of k veloc measured for Series 1 inTable 1: k=0.025 and 0.027 (solid grey lines).

Figure 6 .

 6 Figure 6. Mean absolute transverse velocity component measured along the interface (y=0) between the main stream and the cavity for five values of W/L.

Figure 7 .

 7 Figure 7. Evolution of k veloc coefficient from Series 2 as a function of Re with fixed values of h/L (a) and from Series 3 as a function of h/L (symbols) with Reynolds numbers (solid line) increasing simultaneously (b).

Fig.7a reveals

  Fig.7a reveals that as Re increases from 1.4x10 4 to 3.6x10 4 , k decreases by about 20%. On the other hand, Fig.7b reveals that as both h/L and Re increase, k increases, and that as h/L is multiplied by 4, k increases by about 18%. Both graphs from Fig.7 allow to conclude that i) k increases as Re decreases, ii) k increases as h/L increases and iii) in the range of variation of h/L and Re considered in Series 3, the impact of h/L exceeds that of Re.

Figure 8 :

 8 Figure 8: Streamwise evolution of the absolute value of the normalized transverse velocity component at the interface between the main stream and the cavity (y=0) for Series 2 (selected configurations, a) and Series 3 (selected configurations, b)

Table 1

 1

				. All measured configurations
		W(m) h (m) U b (m/s) W/L h/L	Fr	Re(x10 4 )	k veloc
		0.3	0.07	0.167	1	0.23	0.202	3.19	0.0252
	1 Series	0.6 0.9 1.2	0.07 0.07 0.07	0.167 0.167 0.167	2 3 4	0.23 0.23 0.23	0.202 0.202 0.202	3.19 3.19 3.19	0.0272 0.0275 0.0256
		1.5	0.07	0.167	5	0.23	0.202	3.19	0.0255
		0.3	0.07	0.071	1	0.23	0.086	1.36	0.0328
		0.3	0.07	0.083	1	0.23	0.101	1.59	0.0309
		0.3	0.07	0.095	1	0.23	0.115	1.82	0.0286
		0.3	0.07	0.107	1	0.23	0.129	2.05	0.029
	2 Series	0.3 0.3 0.3	0.07 0.07 0.07	0.119 0.131 0.143	1 1 1	0.23 0.23 0.23	0.144 0.158 0.172	2.27 2.50 2.73	0.0295 0.0278 0.0267
		0.3	0.07	0.155	1	0.23	0.187	2.95	0.0257
		0.3	0.07	0.167	1	0.23	0.201	3.18	0.0258
		0.3	0.07	0.179	1	0.23	0.215	3.41	0.026
		0.3	0.07	0.190	1	0.23	0.230	3.64	0.0264
		0.3	0.03	0.119	1	0.10	0.219	1.19	0.0279
		0.3	0.04	0.119	1	0.13	0.189	1.50	0.0258
		0.3	0.05	0.119	1	0.17	0.169	1.79	0.0281
		0.3	0.06	0.119	1	0.20	0.155	2.04	0.0263
	3	0.3	0.07	0.119	1	0.23	0.143	2.27	0.0283
	Series	0.3 0.3 0.3	0.08 0.09 0.10	0.119 0.119 0.119	1 1 1	0.27 0.30 0.33	0.134 0.126 0.120	2.48 2.68 2.86	0.0276 0.029 0.0276
		0.3	0.11	0.119	1	0.37	0.114	3.02	0.0293
		0.3	0.12	0.119	1	0.40	0.109	3.17	0.0304
		0.3	0.13	0.119	1	0.43	0.105	3.32	0.0305
		0.3	0.14	0.119	1	0.47	0.101	3.45	0.0304
	1.7 Data from the literature

  Gu (2001) respectively measured and computed the mass exchange coefficient k in a lateral cavity with an interface section S ex (see Eq.1) limited by a partial wall;[START_REF] Langendoen | Flow patterns and exchange of matter in tidal harbours[END_REF] considered an oscillating inflow;[START_REF] Muto | Velocity measurements in a straight open channel with a rectangular embayment[END_REF] included a step at the entrance of the lateral cavity;[START_REF] Jackson | A mean residence time relationship for lateral cavities in gravel-bed rivers and streams: Incorporating streambed roughness and cavity shape[END_REF] considered cavities with complex rough bathymetries and[START_REF] Jackson | Flow structure and mean residence times of lateral cavities in open channel flows: influence of bed roughness and shape[END_REF] considered a semi-circular and two conic-shaped cavities. These studies are then excluded from the data used for comparison in Table2.The remaining simple lateral cavity configurations (horizontal and rectangular isolated openchannel lateral cavity, with a steady main stream and no wall or step at the cavity entrance) available in the literature correspond to four studies, included in Table2, and detailed below.

[START_REF] Salizzoni | Street canyon ventilation and atmospheric turbulence[END_REF][START_REF] Markides | Flow and exchange processes in homogeneous urban street canyon geometries: an experimental study using particle image velocimetry[END_REF]

and finally a single lateral cavity as in the present configuration (corresponding references are detailed below).

Regarding studies of mass exchange in open-channel, single lateral cavities, about half of them exhibit specific geometrical or flow details that prohibit a direct comparison:

[START_REF] Li | Residence time in semi-enclosed water bodies[END_REF]

and Li &

Table 2 .

 2 Flow and geometrical configurations along with k estimations obtained from the literature data review.

			W (m)	L (m) U b (m/s) h (m)	W /L	h /L	Fr	k veloc	k dye
	Tuna	near-bed region:	0.457	0.305	0.24	0.0381	1.5	0.125	0.63	0.030
	et al .	surface region:	0.457	0.305	0.24	0.0381	1.5	0.125	0.63	0.026
	(2013)	surface region -seiching:	0.457	0.305	0.43	0.0381	1.5	0.125	1.16	0.035
			0.89	0.89	0.29	0.029	1	0.033	1.04		0.028
			0.89	0.89	0.16	0.05	1	0.056	0.32		0.014
			0.89	0.89	0.30	0.027	1	0.030	1.13		0.025
			0.89	0.89	0.13	0.06	1	0.067	0.21		0.019
		Altai et Chu (1997)	0.32	0.32	0.34	0.025	1	0.078	1.37		0.028
			0.32	0.32	0.30	0.025	1	0.078	1.21		0.014
			0.32	0.32	0.21	0.025	1	0.078	0.86		0.019
			0.32	0.32	0.14	0.025	1	0.078	0.58		0.013
			0.32	0.32	0.07	0.025	1	0.078	0.28		0.020
			1	1	0.50	0.11	1	0.110	0.46		0.020
		Booij (1989)	3	1	0.50	0.11	3	0.110	0.46		0.020
			3	1	0.50	0.11	3	0.110	0.46		0.014
		Sanjou & Nezu (2013)	0.1	0.3	0.12	0.053	0.33	0.177	0.23		0.014

Table 1

 1

: k=0.025 and 0.027 (solid grey lines).


Quentin Berger

Michele Salvi

Scaling of Sub-Ballistic 1D Random Walks Among Biased Random Conductances

Keywords: AMS subject classification (2010 MSC): 60K37, 60Fxx, 82D30 Random walk, random environment, limit theorems, conductance model, Mott random walk

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Introduction

We consider pX n q nPN , a discrete-time nearest-neighbor random walk in a random environment (RWRE) on Z, whose transition probabilities are determined by a random collection ω :" tω i u iPZ of positive numbers in r0, 1s sampled according to a stationary and ergodic measure P (we will call E the relative expectation). For a fixed realization of the environment ω, pX n q nPN starts at the origin and has transition probabilities P ω `Xn`1 " i `1 | X n " i ˘" ω i , P ω `Xn`1 " i ´1 | X n " i ˘" 1 ´ωi .

Necessary and sufficient conditions on P for the random walk to be transient, and to have a positive asymptotic velocity in the transient case, are well known. Let us summarize here some of the results that can be found in Zeitouni, [TZ04, Theorems 2.1.2 and 2.1.9], and were first proven by Solomon [START_REF] Solomon | Random walks in a random environment[END_REF] in the case of an i.i.d. environment ω:

Theorem A. Let ρ i :" 1´ω i ω i and assume that Erlog ρ 0 s is well defined. (i) The random walk pX n q nPN is P-a.s.: transient to the right if Erlog ρ 0 s ă 0; transient to the left if Erlog ρ 0 s ą 0; recurrent if Erlog ρ 0 s " 0. (ii) If Erlog ρ 0 s ă 0, then P-a.s. the velocity v :" lim nÑ8 X n {n exists P ω -a.s. and is equal to Er Ss ´1 P r0, `8q with S :" 1 ω 0

`ř8

i"1 1 ω ´i ś i´1 j"0 ρ ´j . One possibility for choosing the environment ω is to sample with measure P a shiftergodic sequence of positive random variables tc k u kPZ attached to the edges of Z -called conductances -and define for i P Z ω i :" c i c i´1 `ci .

(1.1)

We call the associated walk a random walk among random conductances (RWRC) and point out that pX n q nPN is reversible with respect to the measure πpkq " c k´1 `ck . In this case, the ρ i 's of Theorem A are given by ρ i " c i´1 c i and the formula for S simplifies to S " 1 `2 ř `8 i"1 c ´i c 0 . It is not hard to see that the RWRC on ergodic conductances has always asymptotic speed v " 0.

In this paper we consider conductances sampled in an i.i.d. manner, but we add an external force -or bias -, which "pushes" the random walk, say to the right. There are at least two natural choices for doing so, as discussed below. We will produce random walks pX n q nPN that are transient but have zero speed. Our aim is to identify the correct scaling for X n , i.e. find α P r0, 1s such that X n is typically of order n α`op1q . We stress that the case of an RWRE with i.i.d. ω i 's has been studied, for example in [KKS75, [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF][START_REF] Enriquez | Quenched limits for the fluctuations of transient random walks in random environment on Z 1[END_REF], and the picture is complete: the correct scaling has been identified as well as the scaling limits. However, in the case of i.i.d. conductances the ω i 's are not i.i.d. anymore, and the phenomenology is actually very different, see the discussion in Section 1.3.

The first way of tilting the conductances (see Section 1.1) is to multiply the k-th conductance by a factor e 2kδ , for some δ ą 0. We call the related process random walk among Biased i.i.d. Random Conductances (BiRC). This process has been studied in the ballistic regime in several cases: on Z (see [START_REF] Faggionato | Regularity of biased 1d random walks in random environment[END_REF]), on Z d with d 2 (see [START_REF] Berger | The speed of biased random walk among random conductances[END_REF][START_REF] Fribergh | Biased random walk in positive random conductances on Z d[END_REF]), on the percolation cluster of Z d with d 2 (see [START_REF] Berger | The speed of biased random walk on percolation clusters[END_REF][START_REF] Fribergh | Phase transition for the speed of the biased random walk on the supercritical percolation cluster[END_REF]), and on trees (see [START_REF] Gantert | Random walks on Galton-Watson trees with random conductances[END_REF] or [START_REF] Aïdékon | Speed of the biased random walk on a galton-watson tree[END_REF][START_REF] Ben Arous | Lyons-pemantle-peres monotonicity problem for high biases[END_REF]). Somewhat surprisingly, the behavior of sub-ballistic BiRC on Z d has been studied only in dimension d 2. In [START_REF] Fribergh | Biased random walk in positive random conductances on Z d[END_REF], Fribergh shows that the walk is ballistic if and only if the expectation of the conductances is finite and then identifies the right order of rescaling in the sub-ballistic regime, depending on the tail of the distribution of the conductances at `8 (finer results are given in [START_REF] Fribergh | Scaling limits for sub-ballistic biased random walks in random conductances[END_REF], under stronger assumptions). This represents one of the main differences with the one-dimensional model: as it will be clear from Theorem 1.1, in our case both the integrability of the conductance and the integrability of the inverse of the conductance play a role for determining the ballisticity of the walk and the right rescaling exponent. Roughly speaking, this is due to the fact that in higher dimension the walk will naturally go around traps generated by edges with a small conductance, whereas for d " 1 it is not possible to avoid them.

The second way of adding a bias (see Section 1.2) is inspired from Physics: the present work was motivated by the study of the sub-ballistic regime for the Mott variable-range hopping (see [START_REF] Faggionato | The velocity of 1d Mott variable range hopping with external field[END_REF][START_REF] Faggionato | Einstein relation and linear response in one-dimensional mott variable-range hopping[END_REF] and references therein), a model for the description of the movement of electrons in doped semi-conductors. The Mott walk is a long-range random walk on a random point process on R (we can see it as a walk on Z by projecting it); it may jump from its current position to any other site with a probability that decays exponentially in the distance and depends on some random energies associated to each point. One can then add an external electric field that induces a bias on the walk, but the bias depends this time on the distance between points. We consider here a simplification of this model, but we believe that our analysis captures the essence of the original one. We ignore the energies and allow the walk to jump only to its nearest-neighbors. We call it Range-1 Mott walk (R1M). We are not aware of any result for the sub-ballistic R1M in any dimension. Interestingly, we find that the behavior of the R1M is very different from that of the BiRC: the scaling of the walk depends this time on the distribution of the conductances but also on the intensity of the bias. 1.1. Biased i.i.d. Random Conductances (BiRC) with heavy tails. We take an i.i.d. sequence tc k u kPZ of random conductances under P and consider the biased random walk among these conductances. This corresponds to taking conductances

c δ k " e 2δk c k , (1.2)
where δ ą 0 is the bias intensity. Then, (1.1) and the ρ k 's appearing in Theorem A become

ω k " e δ c k e δ c k `e´δ c k´1 and ρ k " e ´2δ c k´1 c k . (1.3)
We find that Erlog ρ 0 s " ´2δ ă 0 so that Theorem A ensures that pX n q nPN is indeed transient to the right for any δ ą 0 and the asymptotic velocity is vpδq " 1

Er Ss with Er Ss " 1 `2Erc 0 sEr1{c 0 s e ´2δ 1 ´e´2δ .

(1.4)

Hence, vpδq ą 0 if and only if Erc 0 s ă `8 and Er1{c 0 s ă `8. We realize that the zero velocity regime can occur for two different reasons: either Erc 0 s " `8, i.e. c 0 has some heavy tail at `8; either Er1{c 0 s " `8, i.e. c 0 has some heavy tail at 0. This reflects the fact that a slowdown of the random walk is usually due to the presence of traps, i.e. regions of the space where it tends to spend the majority of the time, of two different kind (see Figure 1). The first kind of trap is produced by an edge tk, k `1u with conductance c k " M " 0 surrounded by two edges with conductances of Op1q: in this case the walk will tend to jump back and forth from point k to k `1 several times (roughly a geometric number of times of mean M ) before hopefully escaping to the right. The other kind of trap is represented by an edge tk, k `1u with a conductance ε ! 1: here the walk will attempt to jump to k `1 every time it finds itself on k, but it will manage to do it only with, roughly, probability ε. We therefore need to make some assumption on the tail of c 0 at `8 and 0 to understand the asymptotic behavior of pX n q nPN : we suppose that there are some α 8 , α 0 P r0, `8s such that lim

k ´1 k k `1 k `2 k ´1 k k `1 k `2 Op1q M Op1q Ó Ó Ó Op1q ε Ó Ó
M Ñ`8 log Ppc 0 ą M q log M " ´α8 , lim εÑ0 log Ppc 0 ă εq log ε " α 0 . (1.5)
We denote ᾱ :" minpα 0 , α 8 q, so that if the random walk is sub-ballistic then necessarily ᾱ P r0, 1s.

Theorem 1.1. Let δ ą 0 and assume that (1.5) holds with ᾱ " minpα 0 , α 8 q 1. Then we have lim nÑ8 log X n log n " ᾱ P b P ω ´a.s.

We stress that, in order to find a finer scaling for X n , the assumption (1.5) on the conductances needs to be strengthened. A natural way of doing so is to assume that there are slowly varying functions L 8 p¨q and L 0 p¨q such that

Ppc 0 ą M q M Ñ8
" L 8 pM qM ´α8 , and P `c0 ă ε ˘εÑ0 " L 0 pεqε α 0 .

(1.6)

In [START_REF] Fribergh | Scaling limits for sub-ballistic biased random walks in random conductances[END_REF], the authors consider the dimension d 2, and use this assumption (for the tail at `8, since only large conductance traps exist in that case) to show that the properly rescaled walk converges in law to the inverse of a stable subordinator. In dimension 1, the situation is fairly complicated due to the existence of several types of trap. When α 0 ‰ α 8 then the deepest traps are of one of the types presented in Figure 1. But when α 0 " α 8 , a third kind of trap may be predominant (depending on the slowly varying functions L 8 and L 0 ), as a combination of two consecutive edges, one with high and the other with small conductance -we refer to Section 1.3 for further discussion. This more subtle landscape is beyond the scope of this paper, and we postpone its study to a subsequent work.

1.2. The range-one Mott hopping model (R1M) with an external field. We consider a random sequence of positive random variables pZ i q iPZ that are independent and identically distributed. The pZ i q iPZ determine the position of points tx i u iPZ on R such that x 0 :" 0, x k "

ř k´1 i"0 Z i if k 1 and x k " ´řk i"1 Z ´i if k ´1.
The range-one Mott random walk p Ỹn q nPN is the process starting in zero and jumping from point x k to point x k`1 with probability equal to e ´Zk e ´Zk `e´Z k´1 and to x k´1 with the remaining probability. We can also see this walk as the RWRC on the set of points tx k u with conductance associated to the edge tx k , x k`1 u equal to c k :" e ´Zk . We introduce an external electrical field of intensity λ P p0, 1q. From a physical point of view, this will result in a modification of the conductances which are given now by

c λ k :" c k ¨eλpx k `xk`1 q .
(1.7)

We will stick to the symbol c λ k when dealing with the conductances of the biased R1M, while we will use c δ k for the BiRC and c ‹ k when dealing with both models at the same time. It will be convenient to look at a projection of p Ỹn q nPN on Z: we let pY n q n 0 be the random walk on Z such that Ỹn :" x Yn . Studying the asymptotics of the two chains is equivalent, since in the transient case we have the relation Ỹn Yn Ñ ErZ 0 s as n Ñ 8 by the ergodic theorem.

For k P Z, the probability of jumping from k to k `1 for pY n q nPN and the ρ k 's appearing in Theorem A are, respectively,

ω k " e ´p1´λqZ k e ´p1´λqZ k `e´p1`λqZ k´1 and ρ k " e ´p1`λqZ k´1 `p1´λqZ k . (1.8)
It follows that Erlog ρ 0 s " ´2λErZ 0 s ă 0, so that part (i) of Theorem A guarantees that pX n q nPN is transient to the right for any λ ą 0. Part (ii) of the theorem and a straightforward computation also show that the limiting velocity is vpλq " 1

Er Ss , with Er Ss " 1 `2Ere p1´λqZ 0 s Ere ´p1`λqZ 0 s 1 ´Ere ´2λZ 0 s .

(1.9)

Hence, vpλq ą 0 if and only if Ere p1´λqZ 0 s ă `8 (note that this condition is equivalent to the condition for ballisticity for the full-range Mott random walk, cfr. [FGS18, Theorem 1]). We define λ c :" inf λ : Ere p1´λqZ 0 s ă `8( , (1.10) so that vpλq " 0 if λ ă λ c and vpλq ą 0 if λ ą λ c . Notice that from the definition of λ c it follows that PpZ 0 ą tq " e ´p1´λcq t p1`op1qq . Interestingly, by (1.9) we see that v is continuous at λ " λ c if and only if Ere p1´λcqZ 0 s " `8.

Theorem 1.2. For any λ λ c , define αpλq :" 1´λc 1´λ . Then,

lim nÑ8 log Y n log n " αpλq P b P ω ´a.s.
We stress once more that a notable difference with Theorem 1.1 is that here the scaling αp¨q depends on the intensity of the bias λ, whereas ᾱ does not depend on δ.

1.3. Discussion about the traps and products of heavy-tail random variables. In order to give a mathematical sense to the notion of traps, Sinai introduced in [START_REF] Yakov | The limiting behavior of a one-dimensional random walk in a random medium[END_REF] the potential function V , canonically associated to the environment ω. It can be defined as V pkq :" ř k i"1 log ρ i for k P Z `(for k P Z ´add a minus sign, and V p0q " 0) and it is a powerful and intuitive tool, since "valleys" in the potential landscape correspond to traps for the random walk. The notion of valleys as traps has then been extended in [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF] to the case of transient random walks on Z (say to `8). Since in this case Erlog ρ 0 s ă 0, one has that the potential has a negative drift (V pkq Ñ ´8) and traps correspond to portions where V increases by a large amount. In particular, when the ω i 's are i.i.d., the potential is a random walk with independent increments and the valleys are associated to its excursions: they are large regions with a large deviation behavior of the sum of log ρ i , see the discussion in Section 3 of [START_REF] Enriquez | Limit laws for transient random walks in random environment on Z[END_REF]. This is in sharp contrast with our framework: in the BiRC (an analogous reasoning can be made for the R1M), the increments of V are strongly correlated, and as a matter of fact we have V pkq " log c 0 ´2δk ´log c k , for k ě 1. Valleys, i.e. large increases in the potential, will then be caused by isolated large values of V pkq ´V pk ´1q " log ρ k . For us, it is therefore crucial to understand the tail of the distribution of ρ 0 " e ´2δ c ´1{c 0 .

The following general result goes in this direction and will be useful throughout the paper. From now on, when treating the two models at the same time, we will use α for ᾱ or αpλq as defined in Theorems 1.1 and 1.2 respectively. Lemma 1.3. Let X, Y be two independent, positive, random variables. Assume that there is some α ą 0 such that, as t Ñ 8, PpX ą tq " t ´α`op1q and PpY ą tq t ´α`op1q , i.e. Y has a lighter tail than X. Then we have that P `XY ą t ˘" t ´α`op1q as t Ñ 8 .

Proof For the lower bound, let y 0 " infty : PpY yq 1{2u; then PpXY ą tq PpY y 0 qP `X ą t{y 0 ˘" t ´α`op1q .

For the upper bound, we fix some ε P p0, αq, and we write

PpXY ą tq P `Y ą t ˘`E " P `X ą t{Y | Y ˘1tY tu ‰ t ´α`op1q `cε t ´α`ε E " Y α´ε 1 tY tu ‰ ,
where we used the fact that there is a constant c ε such that PpX ą xq c ε x ´α`ε for all x 1. Then, because of our assumption on Y , we easily get that for all ε ą 0, ErY α´ε s ă `8. Therefore, we get that there is a constant C ε such that PpXY ą tq t ´α`op1q `Cε t ´α`ε , so that PpXY ą tq t ´α`op1q , since ε is arbitrary.

Thanks to this lemma, we are able to obtain the tail of ρ 0 : we have that for both models Ppρ 0 ą tq " t ´α`op1q .

(1.11) Indeed, for the BiRC we have that ρ 0 is the product of two independent random variables: X :" e ´2δ c ´1, which has a tail PpX ą tq " t ´α8`op1q , and Y :" 1{c 0 , which has a tail PpY ą tq " t ´α0 `op1q (we might have to exchange the role of X and Y to properly apply Lemma 1.3). Analogously, for the R1M, ρ 0 is the product of X :" e p1´λqZ 0 which has a tail PpX ą tq " PpZ 0 ą 1 1´λ log tq " exp ´1´λc 1´λ log t p1 `op1qq ( " t ´αpλq`op1q , and Y :" e ´p1`λqZ ´1 which is not larger than 1 (and therefore satisfies PpY ą tq t ´αpλq`op1q ). Note that this proof gives an indication on the easiest way to create a large trap, i.e. a large log ρ i : the tail of ρ 0 comes from having either a large c ´1 (with c 0 of order one), or a small c 0 (with c ´1 of order one), depending on which one is the easier.

Remark 1.4. In the case where c 0 has a regularly varying tail at 0 and `8, as in (1.6), [START_REF] Embrechts | On closure and factorization properties of subexponential and related distributions[END_REF] gives that ρ 0 " c ´1{c 0 has a regularly varying tail with exponent ᾱ " minpα 0 , α 8 q, and [Cli86, Corollary 5] provides the sharp behavior of Ppρ 0 ą tq as t Ñ 8. Let us stress here that in the case α 0 " α 8 , the main contribution to Ppρ 0 ą tq will come from a combination of having c ´1 large and c 0 small, therefore suggesting a new type of trap, different from those of Figure 1. As an example, if Ppc 0 ą tq " t ´ᾱ and Ppc 0 ă εq " ε ´ᾱ in (1.6), a straightforward calculation gives that P `ρ0 ą t ˘" plog tqt ´ᾱ , and the main contribution comes from all possibilities of having c ´1 -t a and c 0 -t a´1 with a P r0, 1s.

2. Proof of Theorems 1.1 and 1.2

A fundamental preliminary result. A central tool for the study of the walks are the hitting times

T n :" inftk , X k " nu .

(2.1)

We will also use the notation T n for the hitting times of the R1M pY n q nPN . Understanding the behavior of T n is the key to finding the right scaling for X n (respectively, Y n ). In fact, the following proposition, in loose terms, shows that the increasing map n Þ Ñ T n is the inverse of the map k Þ Ñ X k , up to an error of at most a constant times log n. Corollary 2.2 shows the relation between the asymptotics of T n and that of the position of the walker.

Proposition 2.1. There exists a constant C ą 0 such that, for P-almost every ω we have that, P ω -a.s., there exists n 0 P N such that, for all n n 0 ,

inf k 0 X Tn`k n ´C log n .
The same result holds for the R1M process pY n q nPN .

Proof We set f pnq :" n ´C log n, with C ą 0 to be determined later on. We call A n :" tW k ă f pnq , for some k T n u, where W k can be taken equal to X k or Y k , and control

P ω pA n q P ω n pτ f pnq ă `8q " lim M Ñ8 C eff ptnu Ø tf pnquq C eff ptnu Ø tf pnqu Y tM uq c ‹ f pnq 8 ÿ j"n 1 c ‹ j . (2.2)
where P ω n is the law of the random walk in random environment ω, starting from n. Here τ j is the first time the walk hits j P N, while C eff pA Ø Bq indicates the effective conductance between set A and set B, see [START_REF] Lyons | Probability on trees and networks[END_REF] for a formal definition. For the equality in (2.2) we have used the well known formula for walks among conductances (see, e.g., [LP16, Exercise 2.36])

P ω k pτ A ă τ B q " C eff ptku Ø Aq C eff ptku Ø A Y Bq , (2.3)
and then we have used the explicit expression of C eff for conductances in series and in parallel (detailed later on in formulas (2.8) and (2.9)).

In the BiRC model case c ‹ j " c δ j and (2.2) becomes P ω pA n q e ´2δpn´f pnqq c f pnq 8 ÿ j"0 e ´2δj 1 c n`j ": e ´2δpn´f pnqq K n pωq .

We use Lemma 1.3 with X " c f pnq and Y " ř 8 j"0 e ´2δj {c n`j (by Lemma 2.5 we have PpY ą tq " t ´α0 `op1q ), or the other way around if α 8 α 0 : we obtain PpK n ą tq " t ´ᾱ`op1q . By a Borel-Cantelli argument it follows that for any ε ą 0, P-a.s., there exists an m 0 " m 0 pωq such that K m m p1`εq{ ᾱ for each m m 0 . As a consequence, Pa.s. there exists n 0 such that, for each n n 0 , we have P ω pA n q n p1`εq{ ᾱe ´2δpn´f pnqq " n p1`εq{ ᾱ´2δC . Since this probability is summable for C big enough, again by a Borel-Cantelli argument we have that A n happens only a finite number of times and the claim follows for pX n q nPN .

For the R1M, one has c ‹ j " c λ j " e ´Zj `λpx j `xj`1 q in (2.2) and therefore P ω pA n q e ´Zfpnq `λpx f pnq `xfpnq`1 q 8 ÿ j"n 1 e ´Zj `λpx j `xj`1 q 8 ÿ j"n e Z j ´λpx j ´xfpnq q . Now, P-a.s. there exists n 0 " n 0 pωq such that, for each n n 0 and j n, we have x j ´xfpnq pj ´f pnqqErZ 0 s{2 and also Z j pj ´f pnqqλErZ 0 s{4. It follows that P ω pA n q n ´CλErZ 0 s{4 , and for C big enough this probability is summable. It follows that A n happens only a finite number of times and the claim follows for pY n q nPN , too.

An easy and important consequence of Proposition 2.1 is the following corollary, which says that in order to prove Theorems 1.1-1.2, we simply need to focus on T n , and prove

lim nÑ8 log T n log n " 1 α P b P ω ´a.s. (2.4)
for both models (recall that α indicates either ᾱ or αpλq).

Corollary 2.2. For P-almost every ω we have that, P ω -a.s.,

lim nÑ8 log T n log n " 1 a ðñ lim nÑ8 log X n log n " a (2.5)
with a : Ω ˆpZ d q bN Ñ R `. The same result holds for the R1M process pY n q nPN .

Proof Throughout the proof we will use the fact that T n Ñ 8 almost surely as n Ñ 8, which comes from the transience to the right of the walk. As in [TZ04, Lemma 2.1.17],

for n P N we define k n to be the unique integer such that T kn n ă T kn`1 and note that k n Ñ 8. We first prove "ñ". By Proposition 2.1 we have that, for n big enough, 2.2. Lower bound in (2.4). Firstly we show that there are many traps deeper than n 1{α´ε between 0 and n with high probability; secondly we show how to use this fact to conclude that lim nÑ8 log T n { log n 1{α.

k n ´C log k n X n ă k n `1. Hence,
Step 1: There are many deep traps. From the tail probability of ρ 0 given in (1.11) and the fact that the ρ k have only a range-two dependence, we get for both models the following lemma, as a standard application of the Borel-Cantelli lemma.

Lemma 2.3. For every ε ą 0 we get that, P-a.s., there exists some n 0 pε, ωq such that, for any n n 0 the events

A n " A n pεq :" ! Card 1 k n : ρ k ą n 1{α´ε ( 2{ε
)
are verified.

Proof We may reduce to a sequence of i.i.d. random variables ρ k by separately proving the statement for the sequences of i.i.d. r.v.'s pρ 2i q iPN and pρ 2i`1 q iPN . In the following, we therefore assume that pρ k q k 1 is a sequence of i.i.d. r.v.'s which satisfy (1.11). By the independence of the ρ k we have that

PpA c n q P ´D 1 k 1 , . . . , k 2{ε n : ρ k ď n 1{α´ε for all k R tk 1 , . . . , k 2{ε u ¯ n 2{ε P `ρ0 n 1{α´ε ˘n´2{ε n 2{ε exp ! ´n 2 P `ρ0 ą n 1{α´ε ˘) ,
the last inequality holding for n ě 4{ε, using also that p1 ´xq n{2 ď e ´nx{2 . Then, we use (1.11) to get that n Ppρ 0 ą n 1{α´ε q " n ´αε`op1q . Therefore, PpA c n q exp ´n´αε`op1q ( , which is summable; the conclusion follows by Borel-Cantelli's lemma.

Step 2: Deep traps slowdown the walk. We are now ready to prove the lower bound in (2.4): we show that, for both models, lim inf

nÑ8 log T n log n 1 α P b P ω ´a.s. (2.6)
Let us fix ε ą 0 (we assume for simplicity that 2{ε is an integer). Thanks to Lemma 2.3, it will be sufficient to control P ω pT n n 1{α´2ε q only on the event A n that there are at least 2{ε points k such that ρ k n 1{α´ε : let us denote them by k 1 , . . . k 2{ε . On A n we have that

P ω `Tn n 1{α´2ε ˘ 2{ε ź i"1 P ω `Gk i n 1{α´2ε ˘,
where G k are geometric random variables with respective parameter p k " ω k 1{ρ k . In fact, every time the walk is in k, it tries to overjump the edge tk, k`1u and has a probability p k of succeeding: G k represents the random number of attempts the random walk has to make before crossing tk, k `1u for the first time. Since we have that p k i 1{ρ k i ă n ´1{α`ε for all i " 1, ..., 2{ε, we get that, provided that n is large enough,

P ω `Gk i n 1{α´2ε ˘" 1 ´`1 ´pk i ˘n1{α´2ε 2n ´ε .
Therefore, on the event A n and for n big enough, we get that

P ω `Tn n 1{α´2ε ˘ 4 1{ε n ´2 .
Since by Lemma 2.3 we know that A n is realized for n big enough P-a.s., an application of Borel-Cantelli's lemma gives that P ´a.s., P ω ´a.s. one eventually has T n ą n 1{α´2ε .

This proves that for any ε ą 0, a.s. log T n { log n ě 1{α ´2ε for n large enough, and (2.6) follows.

2.3. Upper bound in (2.4). Also for the upper bound we divide the proof in two steps. First we show that, P-a.s., E ω rT n s n 1{α`ε as n Ñ 8; then we use this estimate to obtain a bound on P ω pT n ą n 1{α`ε{2 q which in turn gives the upper bound in (2.4).

Step 1: Estimates on E ω rT n s. A central tool of our analysis will be the following representation for the expectation of the hitting times (cfr. [Bov06, Formula (3.22)])

E ω rT n s " 1 C eff pt0u Ø tnuq ÿ kăn πpkqP ω k pτ 0 ă τ n q (2.7)
where πpkq :" c δ k´1 `cδ k is a reversible measure for pX n q nPN (analogously, πpkq :" c λ k´1 `cλ k for the R1M), τ j denotes the first time the walk hits point j P Z and C eff is the effective conductance introduced after (2.2). We notice that in (2.7) the quantity P ω k pτ 0 ă τ n q is equal to 1 for k 0, while for 0 ă k ă n we can use (2.3). Moreover C eff is very easy to handle in our case: for i ă k ă j we have C eff ptku Ø tjuq " Spk, j ´1q ´1

(2.8)

C eff ptku Ø tiu Y tjuq " Spi, k ´1q ´1 `Spk, j ´1q ´1 , (2.9)

where Spi, jq :" ř j "i 1 c ‹ . For the first formula we have used that we have conductances in series, while for the second formula we have two sequences of conductances-in-series that are in parallel. Therefore, using (2.8) and (2.9), we get that for both our models Hence, by an application of Borel-Cantelli lemma, we get that, P ω -a.s., there exists some k 0 " k 0 pωq 1 such that T 2 k `2k´1 ˘1{α`2ε for all k ě k 0 . We therefore conclude by observing that T n is increasing, so that for any n such that k n :" tlog 2 nu ě k 0 we get that T n ď T 2 kn`1 ď `2kn ˘1{α`2ε ď n 1 α `2ε for all n ě n 0 pωq :" 2 k 0 . This proves that for any ε ą 0, one eventually has log T n { log n ď 1{α `2ε a.s., and (2.19) follows.

E ω rT n s " 1

Figure 1 .

 1 Figure 1. On the l.h.s. a trap of the BiRC due to the presence of a large conductance, c k " M " 1, surrounded by two conductances of Op1q. On the r.h.s. a trap generated by a small conductance c k " ε ! 1 preceded by a conductance of Op1q.

  We fix some ε ą 0 and we use Markov inequality (with the pαpλq´εq-th moment) to obtain Ere p1´λc´εp1´λqqZ 0 sK αpλq´ε is a finite constant for each ε ą 0. The geometric sum is also finite if we choose η small enough. It follows that PpY ą tq t ´αpλq`ε`op1q for arbitrary ε ą 0, and this concludes the proof.Let us fix ε ą 0. Thanks to Markov inequality and Proposition 2.4-(2.11), we get that, P-a.s., for n large enough, we have P ω `Tn ą pn{2q 1{α`2ε ˘ pn{2q ´1{α´2ε E ω rT n s 2 1{α`2ε n ´ε .

	PpY ą tq	ÿ	`t´1 Ke η ˘αpλq´ε	E "´e p1´λqZ ´2λpZ 0 `¨¨¨`Z ´1q ¯αpλq´ε ı
			1
		t ´αpλq`ε B	ÿ	e ´2λpαpλq´εqZ 0 " ´eηpαpλq´εq E	ı¯
						1
	where B " Step 2: Conclusion of the argument. We finally prove that
			lim sup nÑ8	log T n log n	1 α	P b P ω ´a.s.	(2.19)
	Sp0, n ´1q ´1 ´ÿ k 0	πpkq	`ÿ 0ăkăn	πpkq	Sp0, k ´1q ´1 Sp0, k ´1q ´1 `Spk, n ´1q	´1 "
	ÿ k 0 pc ‹ k´1	`c‹ k qSp0, n ´1q	`ÿ 0ăkăn pc ‹ k´1	`c‹ k qSpk, n ´1q .	(2.10)

This formula allows us to prove the following.

Proposition 2.4. For any β ą 1{α, we have that

P `Eω rT n s n β ˘ n 1´αβ`op1q .

M. Salvi was financially supported by the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie Grant agreement No 656047.

As a consequence, for every ε ą 0, we get that, P-a.s., there exists some n 0 pε, ωq such that, for any n n 0 , E ω rT n s n 1 α `ε`op1q .

(2.11)

Proof The second part of the lemma comes as an easy consequence of the first part. Indeed, we have PpE ω rT n s n 1{α`ε q n ´αε`op1q , so an application of Borel-Cantelli lemma gives that, P-a.s., E ω rT 2 k s p2 k q 1{α`ε for k k 0 pωq. Then, since E ω rT n s is non-decreasing, we can set k n " tlog 2 nu and see that

for all n 2 k 0 pωq ": n 0 pωq, P-a.s.

We therefore focus on the first part of the lemma. We start from equation (2.10), which admits as an easy upper bound

Hence we get

) and we want to control the two probabilities. As a preliminary, we give an estimate on the tail of the two random variables of interest. We postpone the proof of the lemma to the end of Step 1.

Lemma 2.5. Set X :"

We have: (i) For the BiRC P pX ą tq " t ´α8`op1q and PpY ą tq " t ´α0 `op1q as t Ñ 8 .

(2.13) (ii) For the R1M (recall that αpλq " p1 ´λc q{p1 ´λq)

P pX ą tq t ´αpλq`op1q and PpY ą tq " t ´αpλq`op1q as t Ñ 8 .

(2.14)

We can deal now with the first term (Term A) and second term (Term B) in the r.h.s. of (2.12). Term A. We pull out the term " 0 and k " 0 in A, so that we can write

with X and Y defined as in Lemma 2.5. Hence, we get that

(2.15)

Using Lemma 1.3 (note that X, Y and c ‹ 0 are mutually independent), we therefore get that PpA n β {4q n ´αβ`op1q . Term B. For the term B, we pull out the terms " k, so that we can write

Hence, setting V k :" c ‹ k ř ąk 1 c ‹ , and for n so large that n β {4 ´n n β {5, we get that

(2.16)

Note that the V k 's are not independent, but they have the same distribution as V 0 " c ‹ 0 Y . Moreover, Lemma 2.5, combined with Lemma 1.3 (possibly exchanging the roles of c ‹ 0 and Y ) gives that PpV 0 ą tq " P `c‹ 0 Y ą t ˘" t ´α`op1q as t Ñ 8.

(2.17) Now, going back to (2.16), we split the sum of the V k 's into two parts, writing

For the first term, we notice that the event is realized if and only if one of the indicator function is non-zero. Using a union bound, the first term is therefore bounded by

For the second term, we simply use Markov's inequality to get

where we used that ErV 0 1 tV 0 ătu s " t 1´α`op1q as t Ñ 8, thanks to (2.17).

Plugging the last two estimates in (2.18) and going back to (2.16) proves that PpB ě n β {4q " n 1´αβ`op1q . This and (2.15), inserted into (2.12), conclude the proof.

Proof of Lemma 2.5 For item piq, we focus on PpX ą tq, since the tail of Y can be found in a similar way. On the one hand, since X ě c δ ´1, we simply have that PpX ą tq Ppc 0 ą tq " t ´α8`op1q . On the other hand, X " ř kď´1 c δ k ą t implies that there exists some j ě 1 such that c δ ´j ą δe ´δj t, so that a union bound gives

Ppc 0 ą δe δj tq " 8 ÿ j"1 pδe δj tq ´α8`op1q " t ´α8`op1q .

For item piiq, for X, we get that by Markov inequality and the independence of the Z i 's

since Ere ´λZ 0 s is clearly smaller than 1 and αpλq ă 1. We now turn to the tail of Y . First, we have the lower bound PpY ą tq P `ep1´λqZ 1 ´2λZ 2 ą t ˘ PpZ 2 ě bqPpe p1´λqZ 1 ą te 2λb q " t ´αpλq`op1q .

For the first inequality we have restricted the sum in Y to the first summand. For the second inequality, we used the independence of the Z i 's, and we used some constant b ą 0 such that PpZ 2 ď bq ą 0. Finally, for the last identity, we just used that for a ą 0, PpZ 1 ą a log tq " t ´ap1´λcq`op1q , see (1.10) and below.

For the upper bound, we take a constant η ą 0 to be determined later, and call K :" e ´η{p1 ´e´η q. We notice that Y " ř ě1 1{c λ ą t implies that there exists at least some 1 such that 1{c λ ą t e ´η {K. A union bound therefore gives PpY ą tq ÿ 1 P `ep1´λqZ ´2λpZ 0 `¨¨¨`Z ´1q ą t e ´ η {K ˘.


Maria Elisa Ruizechartea

Isaure Chauvot De Beauchêne

David W Ritchie

EROS: A Protein Docking Algorithm Using a Quaternion -Ball Representation for Exhaustive and Accelerated Exploration of 3D Rotational Space

Proteins are involved in many essential cellular processes of living organisms. Proteins form macro complexes joining themselves to other proteins to carry out these processes. Therefore, to know the 3D structures of such complexes is of biomedical interest.

Proteinprotein docking algorithms aim to predict how two proteins interact with each other to form a 3D complex. Docking algorithms need to fulfill two main tasks:

(1) sampling all the possible relative positions of the two proteins and (2) computing the interaction energy at each position to find the minimum energy (= the best solution). Obtaining the interaction energy is a computationally expensive task. We are developing a new algorithm based on the ATTRACT coarsegrained forcefield [1] and using a quaternion ball representation to accelerate the search of the 3D rotational space.

Maria Elisa Ruiz Echartea is supported by a doctoral bursary (CORDIS) from Inria Nancy Grand Est.

1. Zacharias, M. Protein Science, 12, 12711282 (2003).

( , , ) α β γ

Based on a preliminary study, we found that a large number of protein interfaces contain at least one pair of atoms at almost their optimal distance according to an empirical force field. Therefore, our idea is to perform a series of restricted docking searches in which a pair of surface beads from each protein is placed at the coordinate origin (see Fig. 4). The 3D rotational space is represented by a ball contained inside a cube. Any point π within the ball may be mapped to an Euler rotation defined by the three Euler π rotation angles, ( , , ). Points within the ball may be represented as a unit α β γ π quaternion, Q = (cos( /2), sin( /2)u), where u is a unit vector from the origin.

θ θ

A series of sample rotations is generated by dividing the cube containing the ball into π 8 subcubes, and recursively subdividing each such cube into 8 smaller cubes until a given threshold cube dimension is reached. Each cell contains a subspace of similar rotations of the rotations space, since they are near each other.

2.The Quaternion π-Ball

Conclusions

Introduction

The set of all possible sample rotations from the ball cube centres π Rc and their radii  may be collected as a set of nodes in a 3D "search tree" data structure.

Each node can be tested to know if such subspace leads to forbidden orientations (i.e. leading to atomic clashes). If yes, the node leads to a forbidden orientation so it is labeled along with its descendants as forbidden.

Application to Protein Docking

Input

The docking process Output The rotational space can be labeled in order to identify unfavorable rotations before computing any energy. Thus, the labeled ball can be used as a rotational map avoiding to compute π energies at useless rotations. Therefore, the exploration of the rotational space is done is an efficient way. The exploration is exhaustive since the complete rotational space is analyzed and labeled. The RootMeanSquare Deviation (RMSD) is a measure of the average distance between the atoms of two superimposed proteins. In practice, to evaluate the efficiency of a docking algorithm, the RMSD is used to evaluate the difference between a calculated and an experimentally determined 3D structure. Above, we show the rank and RMSD of some of the solutions found by our algorithm.

Optimal distance

Then, the tree is walked to label as forbidden those rotations of the search space that intersect some orientation of the list. If a bead comes close enough to receptor bead a to always give a clash, then that rotation and all the rotations in the cube's descendants are labeled as forbidden (see Fig. 6).

In the last stage, the tree is walked to compute energies in order to find the best solutions. The ball is used π as a rotational map. Therefore, many trial rotations are discarded and the rotational search space is pruned to achieve an improvement in efficiency.

In our experiments we use a minimum cell radius of 7.5 degrees, which divides the ball π into 299,593 cells. We observe that about 96% of these are forbidden, leaving only 4% of the search space where interaction energies have to be computed.

In order to detect the bead pairs that can lead to steric clashes upon ligand rotation, we compute the differences between the distances. Thus, for all (a, b) in (Receptor, Ligand):

If |Ra -Lb | > σab, where σab is the clashing distance for atom types a and b, see Fig. 3 and5, then beads a and b will never clash else we have to examine in more detail if the beads might overlap or not (see Fig. 5 and6).

= rMin

Results

Rc

  2π

Fig 1 .

 1 Fig 1. The ball representation. π

  Fig 5. Two beads aligned to find if they are clashing.

Fig 3 .

 3 Fig 3. ATTRACT force field describing the interaction energy for a given pair of beads.

Fig 4 .

 4 Fig 4. Example of an initial starting position for a 3D rotational search.

  cause a steric clash.Case 2) > + ∆s => No rotation within ω β cube s can cause a steric clash.

Fig

  Fig 6. Possible cases when a rotation moves a bead.

Acknowledgements

Divide and Conquer