Magali Le Goff

Eric Bertin

Kirsten Martens

Giant fluctuations in the flow of fluidised soft glassy materials: an elasto-plastic modelling approach

In this work we study the rheological features of yield stress materials that exhibit nonhomogeneous steady flows and that are subjected to an additional mechanical noise. Using a mesoscale elasto-plastic model accounting for a viscosity bifurcation in the flow response to an external shear stress, we find that additional sources of noise can lead to a fluidisation effect. As we increase the noise intensity we evidence a transition from a non-monotonic to a monotonic rheology, associated with giant fluctuations of the macroscopic shear rate and long-time correlated dynamics. Although distinct noise models can lead to different rheological behaviours in the low stress regime, we show that the noise-induced transition from shear-localised to homogeneous flow at higher stresses appears very generic. The observed features in the dynamics can be interpreted as a result of an out-of-equilibrium phase transition, for which we estimate the critical exponents that appear to be independent of the specific choice of the noise implementation for the microscopic dynamics.

I. INTRODUCTION

Dense disordered materials such as emulsions, foams, colloidal suspensions, or granular materials exhibit rich rheological behaviours. These materials have in common that once deformed beyond their solid elastic regime they yield plastically towards a complex flow regime with a shear rate dependent viscosity [1,2]. The steady state flow behaviour of these yield stress fluids (YSF) can be described at the continuum level using empirical laws such as the Herschel-Bulkley relationship [3], or continuum descriptions, such as visco-elasto-plastic [4,5] and fluidity models [6,7]. While such descriptions account well for the average flow behaviour at a coarse grained scale, it has appeared that some flow features of YSF are dominated by giant fluctuations of the relevant rheological quantities on the macroscopic scale [8][9][10][11]. This can for example lead to non-local, strongly system-size dependent, transport coefficients for the material dynamics [12][13][14]. Accordingly, understanding the role of mechanical noise and its spatio-temporal features has not only attracted a strong fundamental interest [2] but is also of direct importance in rheological applications [1].

Part of the mechanical fluctuations in driven disordered materials are usually generated by the flow itself, for example resulting from the elastic response of the material to localised plastic events [15,16]. They are therefore very different in nature from thermally generated fluctuations [17] and must be incorporated into modelling approaches in a self-consistent manner [18,19]. Interestingly, flow-induced fluctuations can be associated in some cases with a self-fluidisation of the material, i.e. a decrease in shear stress with increasing shear rate. This leads to non-monotonic rheological constitutive curves [7,20,21], that can be associated with flow instabili- * magali.le-goff@univ-grenoble-alpes.fr ties, potentially leading to shear localisation, metastability and hysteresis [22]. In the case of granular materials, this self-fluidisation process finds its origin in sliding frictional contacts [22][23][24]. In non frictional YSFs, non monotonic flow curves can be explained by mechanisms such as inertia [25][26][27] or local softening following structural rearrangements [28,29].

Besides this self-generated mechanical noise there can be additional external sources of noise, which can be regarded as a first approximation as independent of the shear-induced one. This is for example the case of thermally induced [30,31] or strong vibration activated irreversible deformations [32][33][34][35]. Other mostly rateindependent fluctuations can also result from local processes such as coarsening in foams [36], or an internal activity [37][38][39]. One important aspect of such external noise sources is that they can induce a fluidisation of the system at small imposed external stresses.

An even more interesting case is the scenario where fluctuations result from the interplay of a self-fluidising and an external source of noise. Upon an increase of the external noise magnitude, it will dominate at some critical value over the self-fluidising noise. This scenario can induce a change from a non-monotonic constitutive behaviour to a simple monotonic one, due to the fluidisation effect of the external noise. The associated transition has been investigated in the context of nonequilibrium phase transitions, both experimentally in the case of frictional granular materials [23] and theoretically in a generic elasto-plastic model [40].

In this work, we consider two different models for an external fluidising noise, and show that although they lead to distinct rheological behaviours at low stress, both induce a transition from a non monotonic to a monotonic flow curve, associated with the transition from shear banded to homogeneous flow. We evidence that the competition between the endogenous noise and an external fluidising noise leads to giant fluctuations in the flow of soft glassy materials, that become relevant on the rheological scale. When interpreting the transition between the self-fluidised and the externally fluidised regimes upon increasing noise magnitude as a critical phenomenon [23,40], we find that critical exponents do not depend on the model of external noise. This suggests that this type of transition might be very generic, independent of the microscopic details in the underlying dynamics.

II. ELASTO-PLASTIC MODELS

A. Principle

For this study we chose to use coarse-grained elastoplastic models (EPMs), which provide a generic framework to describe the flow of soft glassy materials (for a review see Ref. [2] and references therein). In EPMs, the material first responds elastically to a global uniform driving (either by controlling the strain or the stress). The deformation or stress can then locally induce a rearrangement of particles [15], or plastic event, if a local threshold σ y is overcome. Such a plastic event causes a local relaxation of the stress and an elastic response of the surrounding material, and can trigger other local events. The resulting macroscopic plastic flow is thus the consequence of these sequences of local rearrangements. The particles rearrangements can be seen as plastic inclusions embedded in an elastic matrix, and are described as force quadrupoles (Eshelby problem). The elastic propagation kernel is thus described using the Eshelby propagator [41], with an asymptotic power-law decay ∼ 1/r d (d being the spatial dimension) and a quadrupolar symmetry.

B. Numerical model

The spatially-resolved models considered in this work are extended from previous versions used to describe both steady-state flows of YSFs using a shear-imposed protocol model [17,29,42,43], and transient (creep) flow using a stress-imposed protocol model [44].

We model an amorphous medium as a collection of mesoscopic blocks, each block being represented as a node (i,j) of a square lattice of size L × L (the lattice indices i,j represent the discretised coordinates along x and y directions respectively). The mesh size corresponds to the typical cluster size of rearranging particles undergoing a plastic rearrangement. These local plastic transformations are assumed to have the same geometry as the globally applied simple shear, i.e., we consider a scalar model. We decompose the total deformation of each node (i,j) into a local plastic strain γ pl ij , which is, in general, heterogeneous, and an elastic strain γ el ij .

Stress-imposed model

When controlling the global stress in the system as in [44], we also decompose the local stress into two parts,

σ ij = σ ext + σ int ij
, where σ ext is the externally applied uniform stress, and σ int ij describes the stress fluctuations resulting from the elastic interactions between regions undergoing plastic deformation (i.e. particle rearrangements), as described by:

σ int ij = µ i j G * ij,i j γ pl i j (1)
The interaction kernel, G * , is of Eshelby's type and is expressed, in Fourier space, as: G * (q) = -4

q 2
x q 2 y q 4 for q = 0 and G * (0) = 0 so that σ int ij describes the local stress fluctuations in a macroscopically stress-free state. Applying a macroscopic driving stress σ ext induces a uniform shift of the local stresses without altering internal fluctuations. The local dynamics is expressed as:

d dt γ pl ij = n ij σ ij µτ = n ij σ ext + σ int ij µτ (2)
with µ the elastic modulus, τ a mechanical relaxation time setting the time units of the model and

d dt γ pl ij = nij σij µτ
the strain rate produced by a plastic rearrangement occurring at a site (i, j). The state variable n ij , indicates whether the site deforms plastically (n ij = 1) or elastically (n ij = 0), and has its own stochastic dynamics that will be described below.

Strain-imposed model

Another widely used model consists in imposing the strain rate to the system, as in [17,29,42,43]. In this model, γ is the externally applied shear rate and we compute the local stress σ ij , evolving with the over-damped dynamics:

d dt σ ij = µ γ + µ i j G ij,i j d dt γ pl i j (3)
with

dγ pl ij dt = nij σij µτ .
The interaction kernel G is also of Eshelby's type [41], and G(0) = 0 is determined by the integral over the whole system of the elastic response. It is related to G * by: G = G * -1/L 2 , because unlike G * , G does not describe the stress fluctuations in a macroscopically stress-free state, but the full stress field instead.

Stochastic dynamics for the plastic activity

Besides the dynamics described in either Eq. (2) or Eq. (3) depending upon the driving protocol, each node alternates between a local plastic state (n ij = 1) and a local elastic state (n ij = 0). The stochastic rules, as described in [42], involve a rate of plastic activation 1/τ pl when the local stress exceeds a barrier |σ ij | > σ y (n ij : 0 → 1) and a rate 1/τ el for a plastic node turning elastic again (n ij : 1 → 0). We consider in this work that a fluidising noise induces additional plastic events (n ij : 0 → 1) with a "vibration rate" k vib . The three different types of transitions between elastic and plastic states are summarised below:

       n ij (t) : 0 1/τ pl → 1 if σ i > σ y n ij (t) : 0 k vib → 1 ∀σ i n ij (t) : 1 1/τ el → 0 (4)
We study two different models for the activation of plastic events by an external noise:

1. Model 1: Constant activation rate: k vib = 1/τ vib for any value of the local stress σ ij 2. Model 2: Arrhenius-like activation: k vib = k 0 e λ vib (σij -σy) with k 0 a prefactor kept constant in our study and λ vib controlling the magnitude of the noise.

These activated events have the same properties as the ones induced by shear, i.e., they lead to a redistribution of stress in the system through the Eshelby propagator.

In the following, the values of stress, strain rate and time are respectively given in units of σ y , σ y /µτ and τ . We set τ pl = 1 and the restructuring time τ el = 10 is chosen large compared to the other timescales in the system in order to induce local softening. Long restructuring times lead to non monotonic flow curves [28] and are associated with permanent shear bands when imposing the shear rate in the system, as described in [29].

We study the influence of an external noise by varying the value of the vibration rate k vib , either varying τ vib = k -1 vib for model 1 (random activation), or λ vib for model 2 (Arrhenius-type activation) using both shear rate and stress controlled driving protocols, as they give access to different flow features in the case of non-monotonic flow curves. As we are interested in bulk quantities, we simulate the above elasto-plastic model using periodic boundary conditions in all directions. Large scale simulations of the elasto-plastic model are performed using a GPU-based parallel implementation [43].

III. AVERAGE FLOW FEATURES

A. Rheology: flow curves

We compute the average steady state shear stress as a function of the imposed shear rate (flow curve) for the two models of noise (Fig. 1), with magnitudes .

k vib = 3.3 • 10 -4
for model 1 (random activation) and λ -1 vib = 3.3•10 -2 for model 2 (Arrhenius) respectively. Let us first describe the generic features of the flow curves, irrespective of the model of noise. The system exhibits a fluid-like behaviour at low stress (and γ < 10 -3 ), while in the absence of noise, the elasto-plastic model considered in this work exhibits a finite yield stress at γ → 0 (upper dark blue curves in Fig. 2). This regime is followed by a stress-plateau (10 -3 < γ < 10 -1 ), associated with a shear banding instability (see Fig. 3). The decreasing part of the flow curve around γ = 10 -3 , reminiscent of the non-monotonic constitutive flow curve, is due to finite size effects and will not be discussed in detail here. The last regime ( γ > 10 -1 ) corresponds to a stable homogeneous flow.

While the global shape of the flow curve is the same, the rheological behaviour in the low stress regime differs between the two models of activation. The random activation rule (model 1) leads to a Newtonian behaviour (as shown in the log-log plot of the inset of Fig. 1), whereas the Arrhenius-like rule (model 2) leads to a logarithmiclike flow behaviour, reminiscent of experiments on vibrated granular media (see Fig. 1 in the work by Wortel et al. [23]). This can be understood simply from the activation rule. Assuming that the shear rate is proportional to the stress multiplied by the average number of active sites γ ∝ Σ n (which is reasonable in the case of a homogeneous flow [29]) and that, in the low stress regime, the mean number of active sites is proportional to the vibration rate k vib , then γ ∝ Σk vib . This leads to a linear rheological behaviour for model 1: Σ ∝ γτ vib , and a non linear rheology for model 2, with k vib = k 0 e λ vib (σij -σy) .

Flow curves for various noise magnitudes are depicted in Fig. 2 for the two models of activation (by varying ei-

B. Shear rate profiles

Fig. 3 depicts profiles of shear rate in steady state averaged over a strain window of 1000, for the two models of noise. In the low noise regime (non-monotonic flow curves) the system separates into two flowing regions (blue curves in Fig. 3), where the minimum and the maximum of the shear rate profile are determined by the boundaries of the stress plateau in Fig. 2. This is similar to the shear bands reported by Martens et al. [29] within a mesoscale elasto-plastic model in the absence of fluidising noise, leading to the coexistence of a flowing and an arrested region instead. In our model, the difference in shear rate between the two bands decreases as the magnitude of the noise is increased, until reaching a stable homogeneous flow regime as shown by the flat profiles (light purple curves in Fig. 3) corresponding to a monotonic flow curve.

The transition from a phase separated flow to a homogeneous flow can thus be characterised using the difference in shear rate between the two flowing bands. We define the order parameter of this transition as the logarithm of the ratio of shear rates in the two flowing bands: Order parameter = log( γfast γslow ) = S f ast -S slow (5) with S = log( γ).

Note that this transition can also be characterised using a stress-controlled protocol [40]. When imposing the stress to the system, the flow is necessarily homogeneous and hence the unstable (negative slope) part of the constitutive flow curve becomes inaccessible. By varying the initial conditions for the flow, the width of this unstable region can thus be characterised, giving access to the extent of the stress plateau, as done indirectly here by studying the shear rate profiles. As a consistency check, we will see in section V that the scaling of the order parameter near the transition is independent of the protocol used.

IV. GIANT SHEAR-RATE FLUCTUATIONS

We now investigate in more details the high noise regime corresponding to

k vib > k c vib (model 1) or λ -1 vib > λ c vib -1 (model 2)
, where the flow curve is monotonously increasing, so that the homogeneous flow is stable. In this regime, using shear-or stress-controlled protocols leads to the same average rheological behaviour. Interesting properties are rather to be found in shear-rate fluctuations (using a stress-controlled protocol), that we first describe qualitatively.

We measure the macroscopic shear rate in the system as a function of time in steady state, for two different values of k vib (for model 1)[Fig. 4], choosing stress values corresponding to the inflexion point of the flow curve. For large values of the noise magnitude (k vib = 5 • 10 -3 , lower red curve in Fig. 4), the fluctuations of shear rate γ are relatively small (variations of about 10 % of the mean value for a system size N = 512 2 ) and not correlated in time. When decreasing the noise towards its value at the transition between monotonic and non-monotonic flow curves (k vib = 1.32 • 10 -3 , see Fig. 2), the fluctuations of γ increase (50 %) and become correlated in time, as it can be seen on the upper (blue) curve of Fig. 4.

A. Rescaled shear-rate distributions

To perform a quantitative analysis, we compute the distributions of macroscopic shear rate in steady state for various system sizes N and noise magnitudes k vib , again for a stress value corresponding to the inflexion point of the flow curves. From the central limit theorem, one expects relative fluctuations of the shear rate to scale as 1/ √ N for large system size. Fig. 5 depicts the centred distributions of γ rescaled by √ N for different system sizes N . In this representation, curves collapse if relative fluctuations scale like 1/ √ N . For large noise magnitudes (Fig. 5(a), k vib = 5 • 10 -3 ), the data for all system sizes collapse onto the same curve, indicating that the shear rate fluctuations obey the central limit theorem. Unsurprisingly, the shear rate fluctuations in this regime follow a Gaussian distribution as shown by the fit in the inset of Fig. 5(a). As the noise magnitude is decreased (Fig. 5(bd)), the rescaled distributions widen and a systematic dependence with the system size appears. This indicates a deviation from the central limit theorem at the approach of the transition, associated with growing spatial correlations of the macroscopic shear rate. Moreover, the maximum system sizes for which finite size effects are observed in Fig. 5 give an estimate of the correlation length ξ in the system as the noise is varied (in Fig. 5 The increase of the correlation length when decreasing the noise indicates a possibly diverging length scale in the system at the transition, which is consistent with the existence of a critical point [23,40]. We show in the inset of Fig. 5(d) that the distribution for k vib = 1.32 • 10 -3 can be approximately collapsed by rescaling the shear rate by N x with x 0.275 (instead of 1/2 far from the transition). This shows that relative fluctuations of the shear rate decay approximately as 1/N 0.275 with system size, that is, much more slowly than the standard 1/ √ N scaling corresponding to the central limit theorem. This slower decay of relative fluctuations with system size can be described as the presence of giant shear-rate fluctuations. These giant shear-rate fluctuations are directly visible in an experimental context [23], and their characterisation is thus of interest.

Note that for a linear system size L (= √ N ) much larger than the correlation length of the system, the standard 1/ √ N scaling is recovered, together with a Gaussian shape of the distribution. This crossover is visible on Fig. 5

(b) and (c).

Instead of looking at the dependence of fluctuations on system size, one may also look at their dependence on stress. In Fig. 6, we depict the variance of S as a func- tion of the distance (in stress) to the inflexion point of the flow curve for various noise magnitudes (for a single system size, L = 128). It can be clearly seen that the variance at the inflexion point is maximal (for Σ = Σ i ) and increases monotonously as the noise magnitude is decreased towards its value at the transition. The same phenomenology is observed for model 2 (Fig. 6(b)). Enhanced fluctuations are thus observed close to the inflexion point, even for a fixed, intermediate system size.

B. Origin of non-standard fluctuations

The reason for the emergence of fluctuations with nontrivial statistics is actually the presence of a critical point, around which the system becomes correlated over a large length scale. Quite interestingly, non-trivial scalings are found close to the critical point not only for fluctuations, but also for average quantities, which may make some of the critical properties easier to measure. Both average quantities and fluctuations exhibit power-law scalings at the transition, and the critical point is described by a set of critical exponents, which are supposed not to depend on details of the system, but rather only on generic properties shared by a broad class of systems. For instance, one may expect that both model 1 and model 2 share the same critical properties. This statement, though, does not rest on any firm theoretical consideration, and should be tested numerically. This is one of the goals of the next section. In a previous work [40], we characterised in detail this critical point for the random activation model (model 1). In the next section, we perform a similar analysis for the Arrhenius activation model (model 2), and show that the scaling of both average quantities and fluctuations does not depend on the model of noise in the critical regime.

V. GENERIC CRITICAL POINT AT FINITE SHEAR AND VIBRATION RATES

We first study the evolution of average steady state quantities as the noise magnitude is varied in the critical regime by studying the scaling of the flow curves and of the order parameter. Then, we investigate the scaling of shear rate fluctuations.

In the following, the noise magnitude is designated by the relative distance to the critical point, which reads, for the two models of noise:

ε = k vib -k c vib k c vib (Model 1) (6) ε = λ -1 vib -λ c vib -1 λ c vib -1 (Model 2) ( 7
)
The stable flow (noise dominated) regime thus corresponds to ε > 0 and the phase separated regime to ε < 0.

We summarise below the list of scalings expected at the transition.

In the regime ε < 0, the order parameter S f ast -S slow vanishes as the noise magnitude is increased towards its critical value as a power law with an exponent β:

S f ast -S slow ∼ |ε| β for ε < 0 (8)
The critical point corresponds to the transition from a monotonic to a non-monotonic flow curve, and hence the slope of the flow curve at the inflexion point vanishes at the critical point. It can be interpreted as an inverse susceptibility χ, expected to scale as a power law of the distance to the critical point ε with an exponent γ:

χ ∼ ε -γ for ε > 0 (9)
At the critical point (ε = 0), the flow curve exhibits a zero slope and S is expected to vary as a power law of the imposed shear stress (after centring the flow curve using the coordinates of the critical point (S c , Σ c )) with an exponent 1/δ:

|S -S c | ∼ |Σ -Σ c | 1/δ for ε = 0 ( 10
)
The variance of S diverges at the critical point when varying the imposed stress:

∆S 2 ∼ |Σ -Σ c | -κ for ε = 0 (11)
The variance and the correlation time of S diverge as the critical point is approached from the high noise regime:

∆S 2 ∼ ε -γ * for ε > 0 (12) τ corr ∼ ε -µ for ε > 0 ( 13
)
A. Scaling of average quantities Scaling of the flow curves and susceptibility: Using a stress-controlled protocol, we compute the steady state flow curve in the stable flow regime (ε > 0), for the two models of fluidising noise and investigate the scaling of the stress Σ as a function of S = log( γ). The data for the two models can be well fitted to a Landau type expansion in the critical regime:

Σ = Σ i + a (S -S i ) 3 + b (S -S i ) ( 14
)
where a, b, S i and Σ i are fitting parameters shown in Appendix C (Fig. 12) for various values of ε and system sizes L. Using these fitting parameters, the data from the two models can be collapsed onto the same master curve, as shown in Fig. 7. a(ε) is roughly constant (Fig. 12(a)). The coordinates of the inflexion point (S i (ε),Σ i (ε)) evolve monotonously as the noise is varied, and describe the analogous of the super-critical liquidgas boundary in equilibrium phase transitions (Fig. 12(cd)). The prefactor b of the linear term in Eq. ( 14), (Fig. 12(b)), decreases linearly with ε and vanishes at the critical point. As it describes the slope of the flow curve at the inflexion point, it is interpreted as the inverse susceptibility b = 1/χ, which diverges at the critical point with an exponent γ 1 [Eq. ( 9)] (see Fig. 10). At the critical point, b = 0 and hence the flow curve ("critical isotherm") is well described by Eq. ( 10), with δ = 3.

Critical point location: To locate the critical value of noise, we fit the data of Fig. 12(b) to extract the value of k c vib and λ c vib for which b = 0 (diverging susceptibility). While significant finite size effects are observed for model 1 (random) (and discussed in detail in [40]), the finite size effects on the value of b remain within the error bars for model 2. We find k c vib = (1.35 ± 0.01)10 -3 for model 1 [40] and λ c vib -1 = (5.1 ± 0.2)10 -2 for model 2.

Order parameter: In Fig. 8, we investigate the scaling of the order parameter S f ast -S slow extracted from the shear rate profiles (Fig. 3) at the approach of the transition, in the regime ε < 0. The order parameter decreases as ε is decreased, and scales as a power law of the distance to the critical point (fit of the form S f ast -S slow = A|ε| β ). In the fit of Fig. 8,ε

= (k vib -k c vib )/k c vib for model 1, ε = (λ vib -λ c
vib )/λ c vib for model 2 and k c vib , λ c vib , β and A are free fitting parameters. We get β 1 = 0.58 ± 0.07 (model 1) and β 2 = 0.49 ± 0.04 (model 2), with k c vib = (1.21 ± 0.05) • 10 -3 and (λ c vib -1 = 4.95 ± 0.05) • 10 -2 . For the two models, the critical exponent β is close to 0.5, which is consistent with the Landau-type scaling in the regime ε > 0.

The critical noise magnitudes (k c vib , λ c vib -1 ) obtained from the fit for a system size L = 512, although they are not far from the previous estimate, are slightly underestimated. In fact, extracting the order parameter from the flow profiles is a difficult task near the critical point, as the coarsening time of the shear bands increases (and diverges for ε = 0). As a consequence, we cannot access steady state profiles in the critical regime. The configurations associated with the closest data points to the critical point in Fig. 8 are not coarsened yet, thus leading to large error bars and possibly explaining the slight underestimate of ∆S and hence an underestimate of the critical noise magnitudes. Using a stress-controlled protocol in our previous work [40], we were able to get accurate data close to the critical point and confirm the above scaling for model 1 (random), with β = 0.52±0.02. Finite size effects: Let us now recall some of the main results of a former finite size data analysis that we performed for the random activation model (model 1) [40], and that we will use to perform the finite size data collapse in this section. At the critical point, the correlation length of the system diverges as:

ζ ∼ ε -ν for ε > 0 (15)
Formally, we performed direct measurements of ζ for model 1 [40] and found ν ∈ [0.9, 1]. This was consistent with the finite size shift of the transition, which yielded a value of ν 1.

Using ν = 1, we show in Fig. 10(a) the data for the susceptibility χ = 1/b rescaled by a factor L -γ/ν as a function of the scaled distance to the critical point εL 1/ν . ε is expressed using the above value k c vib = (1.35 ± 0.01)10 -3 and refining the value of λ c vib -1 to get the best data collapse (λ c vib -1 = 5.15 • 10 -2 ). The ability to collapse the data for all system sizes for the two models of noise suggests that they share similar critical exponents for the susceptibility and the correlation length (γ 1 and ν 1). ). Data for model 1 (black empty data points) and model 2 (red solid points) and various system sizes (circles: L = 128, squares: L = 256, triangles: L = 512). Dashed lines: power-law guides to the eye with exponents 0.6 and 0.9.

B. Scaling of fluctuations

In equilibrium systems, fluctuations are related to average quantities like the susceptibility, and at a critical point, the divergences of both quantities are related. In this part we investigate the scaling of the fluctuations of S when varying the imposed stress and the noise magnitude in the critical regime.

Varying the imposed stress in the critical regime: As shown in Fig. 6, the variance of the order parameter increases as the stress approaches its value at the inflexion point of the flow curve, where it is maximal. In figure 9, we report, in a log-log plot, for the value of the noise closest to the critical point (ε 0), the value of the variance of S as a function of the distance to the inflexion point, for various system sizes. Let us point out that the scattering of data near the maximum is due to finite size effects, where the smallest values correspond to the smallest systems. We find that the variance of S varies as a power law of the distance to the critical stress, with exponents κ 1 = 0.82 ± 0.12 (model 1) and κ 2 = 0.73 ± 0.15 (model 2), that do not depend significantly on the noise model. From the Landau fit of Eq. ( 14), the susceptibility (χ = ∂S/∂Σ for ε = 0) varies as (|Σ -Σ c | 2/3 ), which is within the error bars of our estimate of κ. Note that a similar scaling is found for the correlation time of the fluctuations (see Appendix B, Fig. 11(b)).

While the extent of the power law spans two decades in the "shear dominated" regime (for Σ > Σ i , lower data points in Fig. 9(c)), the range is reduced in the "noise dominated" regime (Σ < Σ i , upper data points in Fig. 9(c)). This asymmetry is likely to be due to the different origins of mechanical noise in these two regimes, where it arises mainly due to activated events in the "noise dominated" regime.

Varying the noise magnitude ε: We now investigate the scaling of fluctuations at the inflexion point of the flow curve (Σ = Σ i ) when varying the noise magnitude ε. We compute the variance of the fluctuations and extract their correlation time from an exponential fit of the autocorrelation of S. A finite size data collapse is performed using ν = 1. We find, for the two models of noise, a power law increase of both the variance (Fig. 10(b)) and the correlation time (Fig. 10(c)) of the fluctuations when approaching the critical point, with the exponents γ * 0.9 and µ 1 respectively.

Let us now discuss the scaling of shear rate distributions depicted in Fig. 5(d) in the light of the critical exponents estimated in this section. In Fig. 5(d), the shear rate distributions at the transition could be approximately collapsed by rescaling the shear rate with a factor N x (N = L 2 ) and the best collapse was found with an exponent x = 0.275. In other words, the width of the distribution of shear rate scales as: ∆ γ ∼ L -2x ∼ L -0.55 . For sufficiently small fluctuations of the shear rate, the fluctuations of S = log( γ), ∆S, correspond approximately to relative fluctuations of γ, ∆ γ/ γc , with γc the critical shear rate (approximately independent of system size, see Appendix C). Hence similar scalings for ∆S and ∆ γ with the system size would be expected. From the data collapse of Fig. 5(d), the scaling for the shear rate variance is L 2 ∆ γ2 ∼ L 0.9 . This is consistent with the scaling of Fig. 10(b), where data collapse for the variance of S suggests a scaling form: L 2 ∆S 2 ∼ L γ * /ν , with γ * /ν = 0.9.

Note that, due to finite time limitations of our simulations in the critical region, the data for the correlation time of the fluctuations (Fig. 10(c)) is restricted to intermediate system sizes (L ≤ 512) and exhibit strong scattering (see Appendix B, Fig. 11(a)). An approximate collapse can still be performed with an exponent µ = 1.0 ± 0.2 and seems to be independent of the model of noise. As noted in [40], this value would correspond to a dynamic scaling exponent z = µ/ν ≈ 1, far from the equilibrium mean-field value z = 2 obtained for nonconserved scalar order parameters [45].

VI. CONCLUSION

In this work, we studied two models of a fluidising noise leading to a transition between a self-fluidised regime and an externally fluidised regime in the flow of soft glassy materials. Upon increase of the external noise amplitude we evidence the vanishing of shear bands, which is an indication of a transition from an unstable shearlocalised flow to a stable homogeneous flow. We find that a decrease of the fluidising noise amplitude in the homogeneous flow regime yields increasingly large and long lived fluctuations of the macroscopically measurable shear rate. In this scenario a single trajectory of the system can strongly differ from its average behaviour as given by the constitutive flow curve, thus leading to several difficulties for experimental characterisations of the flow behaviour.

Since the correlation time of the fluctuations becomes increasingly large at the transition point, averaging values in the steady state becomes tedious and care has to be taken in the data interpretation. This is a situation where conventional continuum descriptions of the flow tend to break down and call for more sophisticated modelling approaches that allow for incorporating the spatiotemporal features of the fluctuations in the rheologically relevant quantities.

Another important message regarding experimental and numerical studies is that, when using shear-imposed protocols, the flow can be heterogeneous due to the nonmonotonicity of the constitutive flow curve. In the vicinity of the transition, the time for the shear bands to coarsen becomes increasingly large, and the steady state of the system may be out of reach on any experimentally relevant timescale.

All of the above features can be accounted for in the framework of non-equilibrium phase transitions, as resulting from an out-of-equilibrium critical transition point. The properties of this critical point, studied in detail for the random activation model in a previous work [40] appear to remain unchanged for the Arrhenius activation model. While the rheology at low stress differs between the two models, we find a generic scaling of the flow curves in the critical regime that is well fitted by a Landau-type expansion, as in [23]. The scaling of the order parameter (power law scaling with an exponent β 0.5), as determined from the flow profiles in the phase coexistence regime, is consistent with the scaling of the flow curves in the stable flow regime. We also find power law scalings of the susceptibility and the shear rate fluctuations, with exponents being independent of the model of noise. In conclusion, the finite shear-rate critical point studied here in a minimal elastoplastic model suggests that a generic critical behaviour arises in systems combining a non-monotonic flow curve with a fluidisation process, irrespective of the detailed physical mechanisms at play. We discussed the consequences of competing selffluidisation and external fluidisation mechanisms on the rheology of soft glassy materials and in the following we would like to suggest situations where this phenomenon might be of importance. As shown experimentally, this transition can easily arise in frictional granular materials [23]. But one could also speculate about other systems where such mechanisms could be at play. The minimal ingredients for the emergence of this critical point in systems sheared at a finite strain rate are: (i) a microscopic mechanism at the origin of self-fluidisation (such as an intrinsic timescale for restructuration in the material [28]), and (ii) a source of mechanical noise independent of the flow. An example for a class of materials, that provides naturally various sources of additional mechanical noise with a fluidisation effect, are dense active systems, such as biological tissues [37][38][39]. Depending on the details of microscopic interactions, self-fluidisation could arise from a competition between intrinsic timescales in the system and shear; hence such systems would be good candidates for the emergence of critical dynamics accompanied by giant fluctuations as described in this work.

- 1 FIG. 1 .

 11 FIG. 1. Shear stress Σ as a function of the imposed shear rate γ for the two models of agitation with k vib = 3.3 • 10 -4 (model 1, black square symbols) and λ -1 vib = 3.3 • 10 -2 (model 2, red dots), for a system size L = 256. Inset: log-log plot showing the linear rheology regime at small stress for model 1..

FIG. 2 .

 2 FIG. 2. Shear stress Σ as a function of the imposed shear rate γ for various noise magnitude for (a) the random activation model (model 1, k vib ranging from 10 -5 to 3.3 • 10 -3 ) and (b) the Arrhenius activation model (model 2, λ -1 vib ranging from 2 • 10 -2 to 10 -1 ), for a system size L = 256. The upper curve in (a) and (b) is obtained in absence of noise.

FIG. 3 .

 3 FIG.3. Steady state shear rate profiles (computed over a strain window of 1000) for various values of the noise (for a system size L = 512) for (a) the random activation model (k -1 vib ranging from 3.3•10 -4 to 1.4•10 -3 ) and (b) for the Arrhenius activation model (λ -1 vib ranging from 2.5 • 10 -2 to 5 • 10 -2 ). .

FIG. 4 .

 4 FIG. 4. Macroscopic shear rate γ as a function of time for k vib = 1.325 • 10 -3 (top, blue curve) and k vib = 5 • 10 -3 (bottom, red curve) for a system size L = 512 (the y-axes of the two curves are shifted for readability).

  (b) 32 < ξ < 64 for k vib = 2 • 10 -3 , (c) 64 < ξ < 128 for k vib = 1.35 • 10 -3 and (d) ξ > 512 for k vib = 1.32 • 10 -3 ).

FIG. 5 .FIG. 6 .

 56 FIG. 5. Centred distributions of macroscopic shear rate γ computed for model 1 (random activation) at the inflexion point of the flow curve for various system sizes (from L = 32 to L = 512), with the x-axis rescaled by multiplying γ by the linear system size L = √ N and the y-axis by dividing P ( γ) by √ N , in the stable flow phase (monotonic flow curve), for decreasing values of the noise magnitude: (a) k vib = 5 • 10 -3 , (b) k vib = 2 • 10 -3 , (c) k vib = 1.35 • 10 -3 and (d) k vib = 1.32 • 10 -3 . Inset of (a): Lin-log plot of the distribution for L = 256 with Gaussian fit (red). Inset of (d): Finite size data collapse of the shear rate distributions using an exponent x = 0.275.

FIG. 7 .

 7 FIG. 7. Data collapse for the two models using the fitting parameters from the Landau expansion fit of the flow curve (a(ε), b(ε), Si(ε) and Σi(ε)) with u = b/a and v = b 3 /a. Data shown for a system size L = 128.

FIG. 8 .

 8 FIG. 8. Order parameter S f ast -S slow as a function of the scaled distance to the critical noise magnitude ε, λ c vib and k c vib being fitting parameters of the fit S f ast -S slow = A|ε| β (see text). Model 1: lower black square symbols (with dashed line fit). Model 2: Upper solid circles (with solid line fit). Error bars are estimated from the standard deviation of S in the bands.

9 FIG. 9 .

 99 FIG.9. Variance of S multiplied by the number of sites, N ∆S 2 as a function of |Σ -Σi| for values of the noise magnitude close to the critical value (ε 0). Data for model 1 (black empty data points) and model 2 (red solid points) and various system sizes (circles: L = 128, squares: L = 256, triangles: L = 512). Dashed lines: power-law guides to the eye with exponents 0.6 and 0.9.

1 FIG. 10 .

 110 FIG. 10. (a) Susceptibility χ = 1/b rescaled by L -γ/ν as a function of the scaled distance to the critical point εL 1/ν for model 1 (unfilled black symbols) and model 2 (solid red symbols). (b) Rescaled variance ∆S 2 vs. εL 1/ν . (c) Rescaled correlation time vs. εL 1/ν

6 FIG. 11 .

 611 FIG. 11. (a) Auto-correlation timescale τcorr as a function of Σ -Σi for various values of noise magnitude λ vib (model 2). (b) Log-log plot of the correlation time τcorr as a function of |Σ -Σc| for ε 0 ("critical isotherm") for the two models of noise.

Acknowledgments

KM and MLG acknowledge funding from the Centre Franco-Indien pour la Promotion de la Recherche Avancée (CEFIPRA) Grant No. 5604-1 (AMORPHOUS-MULTISCALE). KM acknowledges financial support of the French Agence Nationale de la Recherche (ANR), under grant ANR-14-CE32-0005 (FAPRES). Most of the computations were performed using the Froggy platform of the CIMENT infrastructure supported by the Rhône-Alpes region (GRANTCPER07-13 CIRA).

We perform simulations of the 2d elasto-plastic model using shear-and stress-controlled protocols.

Using a shear-imposed protocol, we measure the average steady-state stress in the system to compute the flow curve, by averaging over a strain window γ = 50. To compute the shear rate profiles in the shear banding regime, we average the profiles along the direction in which the flow is homogeneous and over a strain window γ = 1000.

Using a stress-controlled protocol, we analyse timeseries of the average shear rate in the system (of average duration T = 2.10 6 for L = 128 and L = 256, T = 6.10 5 for L = 512 and T = 10 5 for L = 1024, corresponding to strains ranging from γ = 2000 to 4.10 4 ). We compute S = log( γ), as well as its variance ∆S 2 .

Appendix B: Temporal autocorrelation

To extract the correlation time τ corr , we fit the autocorrelation function C(τ ) = ∆S(t + τ )∆S(t) to an exponential and extract the characteristic time τ corr from the fit. We report in Fig. 11(a) the correlation time as a function of the distance (in stress) to the inflexion point of the flow curve for various values of λ vib (model 2), which exhibits a sharp peak at the critical point. We report in a log-log plot the correlation time as a function of the absolute distance to the inflexion point |Σ -Σ i |, for a noise magnitude close to the critical point (ε 0), for the two models of noise. Although the data is more noisy than for the variance of the fluctuations (see main text, Fig. 6), we can see there is a power-law scaling of the correlation time as well, with an exponent close to that of the variance.


Jean Guillard

Daily migration cycles of fish populations in a tropical estuary ž / Sine-Saloum, Senegal using a horizontal-directed split-beam transducer and multibeam sonar

Keywords: Estuary, Split-beam, Multibeam, Senegal, Shallow waters

The tropical Sine-Saloum estuary Senegal is an ecologically important area for fish population dynamics. Daily cycles of fish movements in a tidal channel were observed using both split-beam and multibeam sonar. A trap-net was set and lifted regularly and simultaneously with the acquisition of acoustic data. The relationship between the two sets of data was satisfactory when acoustic records for single targets were separated from those containing fish schools. The heterogeneity of fish spatial and temporal distributions is examined, as well as the rapidity of changes of fish population density in an environment subject to tides. The consequence of this on sampling strategies is discussed.

Introduction

This study carried out in the Sine-Saloum estuary Ž . Senegal was part of a research programme looking at the use of acoustic methods in shallow waters. The first part of the study was carried out in the coastal Ž . zone of Senegal [START_REF] Guillard | Spatial structures and behavior of fish populations in a Senegalese costal area[END_REF] . The second part was carried out in an estuary, where the environmental conditions are better, for this kind of approach, than in the sea.

Many research programmes have been carried out in the Sine-Saloum estuary in order to understand estuarine and lagoon environments, and how they Ž function ecologically and economically Diouf,

) INRA, 75 avenue de Corzent, BP 511, 74203 Thonon, France. Tel.: q33 4 50 26 78 00; fax: q33 4 50 26 07 60; e-mail: guillard@thonon.inra.fr . 1996 . Estuaries are an interface between continental and marine environments and often have very high levels of production. They are important places for the development, growth and reproduction of many Ž . marine fish species [START_REF] Odum | The detritus-based food web of an Ž . estuarine mangrove community[END_REF] . The rich and diverse Sine-Saloum estuary is inhabited by various categories of estuarine, marine and freshwa-Ž . ter fish [START_REF] Albaret | Diversite des poissons des lagunes ét des estuaires ouest-africains[END_REF] and provides most of the animal protein for the local populations. In the estuary, fish catches have fluctuated between 48 582 Ž . t in 1978[START_REF] Bousso | L'exploitation des stocks dans «l'estuaire» et les bolons du Sine-Saloum[END_REF] , to the present stable yield Ž . of 10 000 tryr [START_REF] Diouf | Contribution a l'elaboration d'un Plan d'Ac-`tion Forestier. Peche et aquaculture continentales[END_REF] . Over 111 species Ž have been catalogued for this estuarine system Di-. ouf, 1996 . The Bandiala estuarine branch is richest in fish species because of its proximity to the sea, its Ž diversity of habitat, stable salinity rarely ) 40 mg. y1 . l and its relatively calm waters. Its fish fauna is dominated in number and weight by three clupeid 0165-7836r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.

Ž . PII S 0 1 6 5 -7 8 3 6 9 8 0 0 0 5 6 -3 species Sardinella maderensis, Ethmalosa fimbriata, Ž . and Ilishia africana [START_REF] Diouf | Les peuplements de poissons des milieux estuariens de l'Afrique de l'Ouest: l'exemple de l'estuaire du Sine-saloum[END_REF] .

Ž . The Sine-Saloum estuary complex Fig. 1a is made up of three main and many subsidiary branches called «bolons». These tidal channels surround islands over a total surface area of more than 800 km 2 . The downstream part of the estuarine complex is colonised by extensive mangrove that declines upstream. Unusually, the average duration of the rising Ž . Ž . tide 7 h is longer than that of the ebb tide 5.5 h ; this results from penetration of coastal waters due to shallow estuarine slopes and a combination of low freshwater flows and very high evaporation levels. There also exists a salinity gradient along the estuar-Ž . ine complex [START_REF] Diouf | Les peuplements de poissons des milieux estuariens de l'Afrique de l'Ouest: l'exemple de l'estuaire du Sine-saloum[END_REF] Earlier studies had shown the need for sampling to be concentrated in a small spatial area for short time periods if the distribution in time and space of Ž fish populations was to be understood Guillard, . 1994 . The structure and size of the Diotto bolon Ž . within Bandiala Fig. 1a appeared to be suitable for Ž . this sampling strategy. According to Diouf 1996 , the Diotto bolon was situated in an intermediate ecological zone with homogeneous spatial units Ž which were also functional units e.g., the zone directly influenced by the sea, one of the three main branches, the so-called central zone whose characteristics were the same as those of the closed waters in . the upper part of the bolon . The tidal channel was rich in fish, was shallow but had deep holes. There was an abundant mangrove presence on both banks.

The main aim was to assess the feasibility of simultaneous acoustic and fish sampling at the same site in order to determine the spatial distribution of fish in a small branch of the sea, as well as the dynamics of fish movements over a tidal cycle of 24 h. Using a fixed point for acoustic sampling avoided the problem of fish escaping from a moving boat and might provide information about fish migration in Ž rivers and estuaries Benech and Le Hong Chuong,

´.

1993; [START_REF] Steig | Monitoring fish movement patterns in a reservoir using horizontally scanning split-beam techniques[END_REF][START_REF] Thorne | Experiences with shallow water acoustics[END_REF]

Methods and material

Site

The Diotto bolon was chosen and its position is set as shown in Fig. 1a. At each run, environmental Ž parameters were recorded hour, season, direction . and strength of the tide, Secchi Disk .

Acoustic equipment

For a description of how acoustic sampling of fish works, the reader is referred to MacLennan and Ž . Simmonds 1992 . Two kinds of acoustic sonar was applied in the present study.

The SIMRAD EY-500 split-beam echosounder was used at a fixed location with its sound beam Ž emitting horizontally across the Diotto bolon Fig.

. 1b from the circular transducer, 2 m below the surface, 120 kHz frequency, with a total beam angle of 9.18 at y3 dB, emitted pulse lengths of 0.1 and 0.3 ms. The ping rate was set at 6 pings per second in order to gain maximal information about the targets. An IBM-PC compatible computer controlled the echosounder, and data were stored in an external Ž . hard disk. The amplification function TVG was set Ž . Ž . at 20 log R and 40 log R .

Calibrations were carried out both in an experi-Ž . mental basin IFREMER, Brest France , and during each series of acoustic runs, using a copper calibration sphere and the standard protocol as recom-Ž . mended by [START_REF] Foote | Calibration of Acoustic Instruments for Fish Density Estimation: a Practical Guide[END_REF] . Acoustic data were analysed by means of the analysis programmes SIM-RAD EP-500 and by visual counting on the echograms. Parameters for discrimination of single fish targets were set with a threshold fixed at y50 dB, using the tracking menu of the EP-500. For the Ž acoustic biomass calculations Area back-scattering . coefficient, S using 20 log R, the detection thresha old was fixed at y55 dB. This was accompanied by a visual counting on the echogram of the number of individual targets, and shoals. The shoals were classified according to their size on the echogram: small, medium, or large, and allocated an arbitrary index on a scale between 1 and 10. During analysis, acoustic runs with or without shoals were treated separately. Noise level varied between y60 to y30 dB.

The second instrument was a multibeam Reson Ž . Seabat 6012 sonar system [START_REF] Soria | Analysis of influence on śpatial behaviour of fish schools using a multi-beam sonar and consequences on biomass estimates by echosounder[END_REF] which was used in combination with the above system during the preliminary series of runs. This had a frequency of 455 kHz, a pulse length of 0.06 ms and was made up of 60 transducers, each with a beam angle of 1.58 at y3 dB, thus covering a volume of Ž . 908 vertically and 158 horizontally Fig. 2a . The TVG was set at 20 log R, the gain adjusted to 50 m, and the image generated was the resultant of smoothing 3 or 4 emissions-receptions. The sonar was Ž . connected to a video screen Fig. 2b and a recorder which stored signals as video images in two dimensions. The sonar head of transducers was placed at 2 m below the water surface in such a way as to record in the vertical plane of 908. This sonar system, therefore, samples the water volume «extensively». No calibration procedure for this system was available at the time of the runs.

Analysis of signals from the multibeam sonar was carried out by playing the video tapes and by visual observation of echoes on the screen. Only echoes from shoals were analysed. The distance from the transducer and the depth and length of each shoal on the screen are measured. A density index was allocated by the operator, to each shoal determined, using a scale from 1 to 50 arbitrary units as a function of detected shoal size.

Combined acoustic and fish sampling

During a 24-h cycle, both the split-beam and multibeam sonar, fixed at each side of a boat, emitted continuously in a horizontal orientation for peri-Ž ods of 2.5-3 h. At the same time, a purse-seine 250 . m long, 20 m height with a mesh size of 14 mm was Ž . set as a net-trap to block the bolon Fig. 1b and was lifted after 3 h. Fish caught were identified, sorted, Ž . measured as fork length and counted, during which time sonar emissions were stopped to avoid record- ing perturbations due to boat movement and trap Ž . lifting [START_REF] Kubecka | Echo counting or echo integration for fish biomass assessment in shallow wa[END_REF] The trap was set again for another 2.5-3 h and acoustic sampling re-started. In this way, two sets of samples were achieved for each tidal cycle. The periods of slack tide were not sampled. Datasets were reduced to 1 h duration in July due to technical problems.

When working in the tropics, problems arise resulting in loss or deliberate elimination of acoustic or fish capture data. High temperatures and high humidities can cause computer failure for as long as 20-30 min, thus preventing continuous acoustic data Ž . recording. As recorded earlier [START_REF] Guillard | Utilisation de l'echosondage dans l'etude des ´Ž . populations de poisson du Sine-saloum Senegal[END_REF] , the speed of pelagic fish movement is such that the two ( ) techniques cannot be compared. There were times when the seine was not completely closed, so that fish escaped, and fish capture was underestimated. Such incomplete samples were not included in the comparative analysis for which only 78% or 22 out of 28 field sampling data was available.

Results

Species and number of fish captured

Only a few species dominated the catch in the Ž . Bandiala Table 1, Fig. 3a , as also reported by Ž . Diouf 1996 . The three main species caught were Ž pelagic clupeids Ethmalosa fimbriata, Ilischia . africana, and Sardinella maderensis together with a variety of fish species, shrimps, crabs, and various floating objects such as drift wood and leaves. The number of fish caught varied greatly on different Ž . dates and time in 24 h Fig. 3b . For the main species captured, their size structures were generally monomodal and characteristic of juveniles. For ex-Ž . ample, the bonga shad E. fimbriata caught belong to the same size class at a given date, and were Ž . always juveniles Table 2 .

Acoustic results

Each time the fish trap was set and operated for 2.5-3 h, the split-beam sounder was emitting horizontally and perpendicularly across the bolon. Several kinds of echograms appeared which represented Ž . different situations Fig. 4 . In Fig. 4D, numerous Pseudotolithus brachygnathus PBR Ž . Fig. 3. a Relative numbers of the main fish species captured by Ž the net-trap in the Diotto bolon see Table 1 for key to species . Ž . names . b Total number of fish per 3-h exposure captured throughout 24 h on four sampling dates.

targets were detected during the same cycle. The acoustic data for runs like this were eliminated from the dataset because the environmental noise level was more than y30 dB and masked fish passages. Such echograms may be caused by concentrations of Use of split-beam sonar enabled the direction of movement to be determined in fish hit by 3-4 pings Ž . Fig. 4F . When working with relatively short ranges and with a narrow beam, the direction of movement could be detected in only about 30% of the targets. This was not possible for schools. Nevertheless, tracking showed that less than 10% of single targets moved against the current.

Comparison of acoustic and fish capture data per 3-h samplings

Ž

When fish capture data by number and by weight . per 3-h sample were compared with the area backscattering coefficient per 3 h of all detected targets, there was no significant correlation. Therefore, captured fish were compared with data collected from counting in the 3-h-long echograms. A comparison between number of fish caught per 3 h and the number of single fish targets counted on the 3-h-long echogram was possible in 8 samples Ž . 36% of the total . As one of the eight points differed significantly from the rest, it was omitted from the analysis and a significant correlation was obtained Ž . with the remaining seven points Fig. 5a . The rejected data point corresponded to a very large catch of bonga shad with 1250 fish.

A comparison between number of fish captured per 3 h and number of counted shoals on the 3-h-long echogram was based upon those echograms containing both shoals and single fish targets. When all the Ž . 12 samples were plotted Fig. 5b , a non-significant negative relationship was obtained between fish caught per 3 h and the detected number of fish shoals per 3 h. In Fig. 5b, three points caused this negative relationship and, without these, the relation-Ž . ship became positive and significant Fig. 5c . These three data points correspond to large catches of Ž . bonga shad 6440 and 12200 fishr3 h and sar-Ž . Ž . dinella S. maderensis once 3440 fishr3 h .

Fish distribution in time and space

Counting per 3-h split-beam echogram permitted counts of single targets and shoals in time sequences of 7 min. Fig. 6 shows how fish and shoal number per 7 min varied through 24 h but reveals no clear pattern with time of day and tides.

Comparison of split-beam and multibeam acoustic data

Discussion

Because of the very few number of data points, no significant relationship between fishing data and environmental data was obtained. Nevertheless, cer-Ž . tain generalisations can be made: 1 there is a strong Ž . heterogeneity between fishing operations; 2 fewer fish were caught on the ebb tide, than on the rising Ž . tide; 3 fewer fish were caught during the hot Ž . season than during the cold one; 4 Sardinella were mostly captured at night and bonga shad were mainly caught on high tides and towards the end of the cold Ž . season; 5 the average size of fish caught was small and were juvenile fish, thus conserving the ecologi-Ž . cal functions of the estuary [START_REF] Diouf | Les peuplements de poissons des milieux estuariens de l'Afrique de l'Ouest: l'exemple de l'estuaire du Sine-saloum[END_REF] Ž . As Diouf 1996 also found, captures of the main species are independent of the hour at which the net is set. However, when the quantity of fish caught by species or as a total is considered, our results do not confirm this suggestion but neither do they contradict it.

Although 20 log R data was recorded, echo integration analysis was not possible due to the high Ž quantities of suspended particles in the bolon with . Secchi Disk depths of 0.5 and 1 m , the speed of fish movement across the sonar beam, and integration of high bottomrsurface noise levels into the acoustic data. Targets counting on the echograms overcame some of these limitations.

Despite the special problems associated with acoustic work in the tropics, significant relationships were obtained between the number of single targets counted, the number of shoals detected and the number of fish captured. All the removed data points were associated with large catches of small pelagic fish species; that is when a large shoal was captured, Ž . but not recorded acoustically for various reason: 1 the shoal was captured while the echosounder was Ž . not operating; 2 the echosounder was inactivated Ž . to avoid boat noise during the 10-15 min period while the seine-net was being lifted and a shoal may Ž . have passed; 3 a shoal may have passed too near the transducer to be detected or too near the opposite bank. Two or more split-beam transducers might cover a greater volume of water or a properly calibrated multibeam sonar might solve some problems, since it can sample the whole water section. The weak correlation between the split-beam and multibeam data may also be associated with some of these sampling problems. The multibeam sonar seems well-adapted for acoustic work in shallow waters as shown by Gerlotto, Hernandez and Linares and by Gonzales and Gerlotto in this volume.

Conclusion

The use of acoustic techniques in this environment has produced some information about the ( ) quickly changing fish populations in time and space. Echosounders are capable of generating large datasets. These can be analysed in relation to variable environmental parameters. This is particularly valuable for environments as variable as estuaries and for studies on the movements of clupeid fish, which require a sampling regime with short time scales. In estuaries, the influence of the tidal cycle upon fish movement requires a well-planned sampling strategy, as well as knowledge of the physiology and behaviour of the fish species.

  a Map of the Sine-Saloum estuary partial and the Bandiala branch showing the geographical position of the sampling site. b The cross-section of the Diotto bolon shows the fixed position of the transducer with its horizontal sound beam and the position of the net-trap during a rising and ebbing tide.

  Diagram of the zone sampled by the Reson multibeam sonar in vertical plane a and its image projected on the video screen b .

  Examples of echograms from the split-beam echosounder, showing A single fish; B small shoal of fish; C: larger shoals of fish; Ž .Ž . Ž . Ž . D environmental noise; E an enlargement of A ; F direction of fish movement by EP-500 tracking techniques. zooplankton, fish larvae and suspended particles detected at 120 kHz frequency. Similar observations Ž have already been recorded in the estuary[START_REF] Levenez | Acoustique en petits fonds. Approche experimentale dans l'estuaire du Sine-Saloum[END_REF]

Fig. 5 .

 5 Fig. 5. Relationship between fish detected acoustically and fish Ž . captured in the Diotto bolon. a Significant regression of number Ž . of single fish targets on number of fish captured by net ns 7 ; Ž . b Relationship between number of fish schools and number of Ž . Ž . fish captured, using all the datasets ns12 ; c Significant regression of number of schools and number of fish captured Ž . resulting from removal of 3 data points in b .

Ž

  A comparison of the split-beam data shoals . counted on echograms and the multibeam data Ž . shoals counted on video screen was possible, despite the difference in sampled volumes and in sonar frequencies. A positive linear correlation was obtained: X s 1.30 q 0.47Y R s 0.659, with a risk of 5%, Ž n s 25, February 1995 , . where X is the number of shoals detected by the Number of fish schools left abscissa and single fish targets right abscissa detected acoustically in the Diotto bolon at different times within 24 h on February 9, 1995. split-beam system and Y the number of shoals detected by the multibeam sonar.

Table 1

 1 Main species of fish captured by the net-trap in the Diotto bolon

	Family	Species	Codes
	Clupeidae	Ethmalosa fimbriata	EFI
		Ilischia africana	IAF
		Sardinella aurita	SAU
		Sardinella maderensis	SEB
	Polynemidae	Galeiodus decadactylus	GDE
	Gerreidae	Gerres nigri	GNI
	Mugilidae	Mugil spp. and Liza spp.	MUG
	Haemulidae	Pomadasys jubelini	PJU
	Sciaenidae

Acknowledgements

I would like to thank the personnel of the CRODT Ž . Centre de Recherche de Dakar-Thiaroye who welcomed me, particularly Mr. D. Toure, the Director, and B. Samb. A kind thought for I. Sow who was present at each campaign, for his technical competencies, and his knowledge of estuarine fish. Thank you also to C. Tendeng, skipper, for his patience, E. Ž . Charles-Dominique and G. Vidy ORSTOM for allowing us to use the boat. I am grateful to the anonymous referees and Dr. A. Duncan for improving the manuscript.


L ' Écluse De Beaucaire-Vallabrègues

J Guillard

B Colon

NOTE BRÈVE ESTIMATION HYDROACOUSTIQUE DU NOMBRE DE POISSONS MIGRATEURS FRANCHISSANT

Keywords: hydroacoustique, index de réflexion, Alosa fallax, Mugil cephalus, Rhône, migrateur, écluse hydroacoustic, target strength, Alosa fallax, Mugil cephalus, Rhone, migratory fish, lock

Utilisées fréquemment en mer et en lac pour échantillonner les poissons, les techniques hydroacoustiques se développent de plus en plus dans les milieux de faible profondeur. Le nombre de poissons qui franchissent l'écluse de Beaucaire-Vallabrègues, premier obstacle rencontré sur le Rhône, est estimé à l'aide d'un sondeur SIMRAD EY500, fréquence 70 kHz, à faisceaux partagés (split-beam). Deux transducteurs échantillonnent une tranche d'eau évaluée à environ 35 % de la section de l'écluse. Le nombre de poissons franchissant l'écluse est calculé pour chaque éclusée dite à poissons. Les caractéristiques de franchissement des poissons (dates, position, durée de sortie) sont observées. Les analyses actuelles ne permettent pas de faire la distinction entre les deux principales espèces qui franchissent l'obstacle, l'alose (Alosa fallax) et le mulet (Mugil cephalus).

INTRODUCTION

Dès les années trente, le Rhône, fleuve possédant un fort potentiel hydraulique et un caractère impétueux, a été aménagé par la CNR (Compagnie Nationale du Rhône). Des barrages hydroélectriques ont été construits pour la production d'électricité et des écluses ont été mises en place pour le passage des bateaux ; les endiguements réalisés ont protégé les terres contre les crues du fleuve.

Ces différents aménagements ont modifié le fonctionnement de l'hydrosystème.

Parmi toutes les populations de poissons présentes, les poissons migrateurs tels que l'esturgeon (Acipenser sturio), la lamproie marine (Petromyson marimus) et l'alose (Alosa fallax) (Figure 1a) furent les plus touchées. L'alose est devenue l'espèce cible de ce programme de restauration (LE CORRE et al., 1997).

Pour permettre aux poissons migrateurs de rejoindre leurs lieux de fraie, des passes à poissons ont été installées. Cependant, la mauvaise conception générale de ces passes et la difficulté de les améliorer les rendent souvent inefficaces. Parallèlement, on s'est rendu compte que les écluses permettaient le franchissement des barrages par les poissons (MENELLA et MAZENS, 1997). Outre l'alose, l'écluse est franchie en grand [START_REF] Guillard | Etude de la migration des aloses. Utilisation des techniques acoustiques[END_REF], qui a permis d'effectuer en 1997 un suivi quotidien (GUILLARD et COLON, 1997). Utilisé depuis longtemps pour la détection des poissons en mer et en lac [START_REF] Marchal | La détection acoustique dans l'étude des peuplements pisciaires[END_REF][START_REF] Brandt | Acoustic assessment of fish abundance and distribution[END_REF], le développement de cette technique pour l'étude des milieux de faible profondeur évolue rapidement [START_REF] Guillard | L'hydroacoustique, méthode d'étude de la distribution spatiale et de l'abondance des peuplements pisciaires lacustres[END_REF]. En particulier, on utilise la détection horizontale en position fixe ou mobile afin d'étudier les répartitions spatio-temporelles des poissons en rivières, estuaires... [START_REF] Lévesque F | Application de l'hydroacoustique en milieu fluvial : exemple de l'étude de la montaison du saumon atlantique (Salmo salar L.) par sondage à double faisceau dans la rivière Moisie, Québec[END_REF][START_REF] Kubecka | Patchiness of longitudinal fish distributions in a river as revealed by a continuous hydroacoustic survey[END_REF][START_REF] Stables | Acoustic measurement of trout distributions in Spada Lake, Washington, using traditionary transducers[END_REF][START_REF] Steig | Monitoring fish movement patterns in a reservoir using horizontally scanning split-beam techniques[END_REF][START_REF] Gauthier | Evaluation of a variable angle scanning method to estimate relative abundance and distribution of fish using a single-beam echosounder in shallow lakes[END_REF].

L'objectif de cette étude était de suivre quotidiennement le franchissement de l'écluse de Beaucaire-Vallabrègues par les poissons migrateurs, et d'optimiser le mode de fonctionnement de l'écluse de navigation en tant que passe à poissons.

MATÉRIEL ET MÉTHODES

Les éclusées dites à poissons

Une éclusée à poissons est l'utilisation de l'écluse en tant que passe à poissons. Sur l'écluse de navigation du barrage hydroélectrique de Beaucaire-Vallabrègues, le fonctionnement est le suivant (Figure 1c) :

-sas vide et portes aval ouvertes, une vanne de remplissage d'un débit de 60 m 3 .s 1 est ouverte afin de créer un courant d'attrait dans l'écluse qui permet aux poissons de trouver l'entrée et d'y pénétrer : c'est l'attrait aval ; -au bout de 20 minutes, l'attrait aval terminé, la vanne de remplissage et les portes aval sont fermées ; -le remplissage du sas se fait ensuite avec les vannes de remplissage de 60 nf.s 1 ; -le remplissage du sas terminé, la porte amont est ouverte et un courant d'attrait est créé pendant 20 minutes pour que les poissons sortent du sas : c'est l'attrait amont. Ce courant est de 5 m 3 .s 1 en ouvrant les 2 vantelles de la porte, ou il est nul (0 m 3 .s 1 ) si les deux vantelles restent fermées. Ces manoeuvres sont analogues à celles permettant le passage des bateaux, mis à part les courants d'attrait. Elles sont effectuées entre 2 et 5 fois par jour pendant la période de migration, c'est-à-dire de mi-avril à fin juin.

L'écluse de Beaucaire-Vallabrègues possède une porte baissante en amont, qui peut être abaissée totalement ou partiellement. Un abaissement partiel (Figure 1c) permet un comptage visuel du nombre de poissons sortant de l'écluse. Le principe est le même, sauf que lorsque le dénivelé entre le niveau amont et le niveau du sas atteint 0,20 m, la porte amont de l'écluse est baissée de 0,30 m en dessous du niveau amont. Il y a alors formation d'une vague que les poissons franchissent, ce qui permet un dénombrement visuel. Ce mode de comptage est efficace mais très lourd à gérer et mettre en place car il nécessite une équipe d'au moins 4 personnes.

L'hydroacoustique

Cette technique est basée sur les propriétés du son dans l'eau. Un son émis dans l'eau, à une certaine fréquence, se propage sous forme d'ondes à une vitesse dépendante des caractéristiques du liquide : température, salinité, pression. Les ondes émises sont réfléchies par toute cible rencontrée : le fond, la surface, les poissons... Les échos ainsi produits sont réceptionnés par le transducteur, qui fonctionne en émission-réception [START_REF] Maclennan | Fisheries acoustics. Fish and fisheries série[END_REF]. Pour réaliser cette étude, un sondeur SIMRAD EY500 (SIMRAD, 1996 a) à faisceaux partagés (split-beam) [START_REF] Maclennan | Fisheries acoustics. Fish and fisheries série[END_REF] En utilisant le logiciel EP500, on peut piloter en direct et à distance, avec un modem relié au réseau téléphonique, le sondeur EY500. Dans le protocole d'une éclusée à poissons, l'éclusier appelle la Station d'Hydrobiologie Lacustre de Thonon lorsqu'il lance le remplissage de l'écluse. On connecte alors les deux ordinateurs distants afin de déclencher les émissions et les acquisitions. Cette méthode évite la permanence d'une personne sur le site de Beaucaire-Vallabrègues, et permet d'obtenir des données à chaque éclusée.

Calibration

La calibration du sondeur et des deux transducteurs est effectuée à l'aide d'une boule de cuivre étalon selon la méthode standard préconisée par [START_REF] Foote | Calibration of acoustic instruments for fish density estimation : a practical guide[END_REF]. D'après les caractéristiques des transducteurs, le volume d'eau échantillonné est égal à environ 35 % du volume de la section de l'écluse. Cette valeur a été vérifiée expérimentalement, en prenant en compte les zones non prospectées par le sondeur [START_REF] Maclennan | Fisheries acoustics. Fish and fisheries série[END_REF], près du transducteur et du mur opposé (Figure 2).

RÉSULTATS Évolution du nombre de poissons franchissant l'écluse pendant la période d'étude (15 avril -30 juin 1997)

Afin d'étudier l'évolution saisonnière du nombre de poissons franchissant l'écluse, le nombre de poissons détectés au m 2 par éclusée est pondéré par le nombre d'éclusées réalisées pendant la journée. Nous avons ainsi un nombre moyen de poissons migrateurs par m 2 et par éclusée pour chaque jour, moyenne par semaine pour une meilleure vision de révolution (Figure 4a).

Évolution quotidienne

De la même façon que pour l'évolution mensuelle, le nombre de poissons au m 2 a été pondéré par le nombre d'éclusées réalisées aux différentes heures de la journée (Figure 4b).

Répartition des poissons

La répartition verticale des poissons (Figure 5a) ne diffère pas selon le débit d'attrait de sortie. D'après ces résultats (Figure 5b), les poissons ont tendance à sortir sur les bords de l'écluse, sauf dans les éclusées à abaissement total où les poissons passent plus dans l'axe de l'écluse. On remarque aussi que le débit d'attrait influence peu la répartition des poissons dans la largeur de l'écluse.

Temps de sortie des poissons

Pour étudier le temps de sortie, on calcule le pourcentage de poissons détectés pendant une plage de temps par rapport au nombre total de poissons ayant franchi l'écluse (Figure 5c). Dans les cas d'abaissement total de la porte, 90 % des poissons sortent pendant les 10 premières minutes, alors que lors des comptages visuels les poissons sortent plus lentement. [START_REF] Maclennan | Fisheries acoustics. Fish and fisheries série[END_REF][START_REF] Soule | Evidence of bias in estimates of target strength obtained with a split-beam echo-sounder[END_REF][START_REF] Zhao | Some pitfalls of short-range standard calibration[END_REF] sur la signification des index de réflexion, en particulier aux faibles distances, n'ont pas été soulevés ici, car on utilise uniquement le comptage des cibles et le sens de déplacement.

DISCUSSION

La distinction de groupe de poissons de taille très différente est souvent possible [START_REF] Brandt | Acoustic assessment of fish abundance and distribution[END_REF] mais n'a pas permis dans notre cas de différencier les aloses de l'autre espèce, le mulet, qui franchit également l'écluse en très grand nombre. Ses

  nombre par une autre espèce, le mulet (Mugil cephalus). En 1991, la D.I.R.E.N. (Direction Régionale pour l'Environnement) et le S.N.R.S. (Service de Navigation Rhône-Saône) ont établi un Schéma de Vocation Piscicole du Rhône, ayant pour conclusion la réhabilitation des poissons migrateurs dans le Bas-Rhône, jusqu'à l'Ardèche, et ses affluents. Après acceptation de ce schéma par le Comité de Bassin Rhône-Méditerranée-Corse, un plan quinquennal a été instauré, le plan Migrateur Rhône-Méditerranée (MRM). Lobjectif de ce schéma est de rétablir la circulation des aloses dans le Bas-Rhône et ses affluents en leur facilitant le franchissement des 3 premiers barrages et des différents seuils. Le retour des poissons migrateurs dans ces zones est un symbole de restauration du fleuve, car ils sont à la fois des indicateurs de la qualité écologique et de la continuité des systèmes fluviaux (MENELLA et MAZENS, 1997). Depuis 1993, l'association MRM organise des campagnes d'études sur le site de Beaucaire-Vallabrègues (Figure 1b), premier obstacle rencontré par les poissons migrateurs. Suite aux expérimentations réalisées en 1995 (JOUFFRE, 1997) montrant la faisabilité des techniques d'échantillonnage acoustique, la CNR s'est équipée d'un matériel hydroacoustique et l'a mis à disposition de MRM. En 1996, une étude préliminaire a été réalisée (

  Figure 1 a -L'alose présente dans le Rhône (Alosa fallax). b -Situation géographique de l'écluse de Beaucaire-Vallabrègues. c -Principe de franchissement de l'écluse par les poissons migrateurs.

Figure 1 a-

 1 Figure 1 a -The shad in the river Rhone (Alosa fallax). b -Geographical position of Beaucaire-Vallabrègues lock. c -Principal of lock's cross over by the migratory fish.

Figure 3

 3 Figure 3Example of an echogram, and direction of a target.

  Figure 4 a -Evolution du nombre de poissons franchissant l'écluse ; les dates indiquées sur les graphiques correspondent au premier jour de la semaine. b -Evolution journalière du nombre de poissons franchissant l'écluse.

  Figure 5 a -Répartition des poissons dans la profondeur de l'écluse, b -Répartition des poissons dans la largeur de l'écluse, c -Temps de sortie des poissons hors de l'écluse.

Figure 5 a-

 5 Figure 5 a -Fish distribution in the lock (depth), b -Fish distribution in the lock (width), c -Delay to go out of the lock.

  caractéristiques morphométriques sont trop peu différentes de celles de l'alose. L'étude d'autres paramètres tels que la vitesse de franchissement et le positionnement n'ont pas permis jusqu'à présent de faire de distinction. Les prochains suivis, plus étalés dans le temps, de mars à juillet, permettraient d'obtenir des données lorsque les aloses ne sont pas présentes, afin d'essayer de dégager les caractéristiques comportementales du franchissement par les mulets.CONCLUSIONL'année 1997 a permis d'avoir un système de détection acoustique des poissons franchissant l'écluse de Beaucaire-Vallabrègues efficace pendant 3 mois (du 15 avril au 30 juin 1997). Le pic du nombre de poissons qui franchissent l'obstacle a pu être observé, et un certain nombre de caractéristiques comportementales de ces poissons ont été mises en avant. Lors des éclusées à poissons, 90 % des poissons sortent de l'écluse, en abaissement total, au bout de 10 minutes, alors qu'en abaissement partiel le franchissement est plus long. Dans la pratique, le temps de l'attrait amont pourrait donc être réduit, ce qui simplifiera la mise en place des éclusées. Aucune différence significative n'a été constatée entre le débit d'attrait à 5 m 3 .s ' et l'absence de débit. Le dispositif expérimental utilisé a montré ses avantages mais aussi ses limites : bonne description de la dynamique de franchissement, mise en évidence du temps de sortie des poissons, automatisation de l'acquisition des données. Les limites sont une vue partielle du milieu qui implique que l'on ne dénombre qu'une partie des poissons franchissant l'écluse, mais surtout la non differentiation des deux principales espèces qui franchissent l'écluse. La possibilité de pouvoir accumuler un grand nombre d'observations devrait permettre de dégager les paramètres comportementaux qui différencieront les deux espèces. Les indices d'abondance de l'espèce cible qui en découleront pourront alors mesurer l'évolution du nombre de poissons migrateurs franchissant l'écluse au cours des prochaines années.

  majorité près de la surface et sur les bords. Le passage sur les bords correspond aux observations du comportement de ce poisson qui migre en suivant les berges. Lors d'un comptage visuel, cette sortie latérale des poissons est modifiée au profit d'une répartition au centre de l'écluse. Selon le mode d'abaissement, le comportement des poissons est différent, d'autant plus qu'en abaissement partiel les poissons passent forcément très en surface. La comparaison des données provenant des deux types de comptage est donc difficile et ne pourra être éventuellement réalisable qu'avec un grand nombre de données.L'étude du temps de sortie montre une décroissance rapide du nombre de poissons sortant de l'écluse : 90 % des poissons détectés sont sortis au bout de 10 minutes pour les éclusées à abaissement total de la porte. Ce résultat permettra de modifier le protocole des éclusées en réduisant leur durée à 10 minutes, au lieu des 20 minutes actuelles. Lors des comptages visuels, le franchissement est plus lent. Ce résultat confirme qu'une éclusée à comptage visuel doit être plus longue qu'une éclusée à abaissement total de la porte.L'utilisation des index de réflexion pour étudier la taille des poissons, bien que fréquemment cités dans la littérature[START_REF] Maclennan | Fisheries acoustics. Fish and fisheries série[END_REF], n'est pas retenue. Ce paramètre varie beaucoup selon le positionnement du poisson, et en détection horizontale cette variabilité augmente fortement. Il est donc peu recommandé d'utiliser les mesures d'index de réflexion[START_REF] Gauthier | Evaluation of a variable angle scanning method to estimate relative abundance and distribution of fish using a single-beam echosounder in shallow lakes[END_REF], en particulier lorsque l'on observe des poissons en mouvement. Les nombreux problèmes soulevés dans la littérature