Vahagn Nersesyan
email: vahagn.nersesyan@math.uvsq.fr

Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional unbounded domain

Keywords: AMS subject classifications: 35Q30, 37A25, 93B05 Navier-Stokes system, unbounded domain, exponential mixing, controllability, asymptotic compactness

The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise, we establish uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is carried out by developing the controllability approach of the papers [Shi19, KNS18] and using the asymptotic compactness of the dynamics.

Introduction

The ergodicity of randomly forced 2D Navier-Stokes (NS) system has been widely studied in the literature in the case of bounded domains (see the papers [FM95, KS00, EMS01, BKL02] for the first results and the book [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF] and the reviews [START_REF] Flandoli | An introduction to 3D stochastic fluid dynamics[END_REF][START_REF] Debussche | Ergodicity results for the stochastic Navier-Stokes equations: an introduction[END_REF] for a detailed discussion of different methods and for further references). This paper is concerned with the ergodic behaviour of the NS system in an unbounded domain D in R 2 with smooth boundary ∂D: ∂ t u -ν∆u + u, ∇ u + ∇p = η(t, x), x ∈ D, (0.1) div u = 0, u| ∂D = 0, (0.2)

u(0) = u 0 . (0.3)
Here ν > 0 is the kinematic viscosity of the fluid, u = (u 1 (t, x), u 2 (t, x)) is the velocity field, p = p(t, x) is the pressure, and η is an external random force.

To have a suitable dissipativity property for solutions, we assume that D is a Poincaré domain, 1 i.e., there is a number λ 1 > 0 such that

D |v| 2 dx ≤ λ -1 1 D |∇v| 2 dx, v ∈ C ∞ 0 (D, R 2 ). (0.4)
The random force η is a process of the form

η(t, x) = ∞ k=1 I [k-1,k) (t)η k (t -k + 1, x), t ≥ 0, x ∈ D, (0.5)
where I [k-1,k) is the indicator function of the interval [k -1, k) and {η k } is a sequence of independent and identically distributed (i.i.d.) random variables in the space 2 E := L 2 ([0, 1], H). Moreover, the law of η k is assumed to be decomposable in the following sense.

Decomposability. There is an orthonormal basis {e j } in E such that

η k = ∞ j=1
b j ξ jk e j (0.6)

for some real-valued independent random variables ξ jk verifying |ξ jk | ≤ 1 and some positive numbers b j such that ∞ j=1 b 2 j < ∞. The law of the random variable ξ jk is absolutely continuous with respect to the Lebesgue measure and the corresponding density ρ j is C 1 -smooth and ρ j (0) > 0 for all j ≥ 1.

The restriction to integer times of the velocity field u t defines a family of Markov processes (u k , P u ) parametrised by the initial condition u 0 = u ∈ H. The associated Markov operators are denoted by P k and P * k . Recall that a measure µ ∈ P(H) is stationary for the family (u k , P u ) if P * 1 µ = µ. In this paper, we prove the following result.

Main Theorem. Under the above assumptions, the family (u k , P u ) has a unique stationary measure µ ∈ P(H). Moreover, it is exponentially mixing in the following sense: for any compact set H ⊂ H, there are numbers C > 0 and c > 0 such that

P * k λ -µ * L(H) ≤ Ce -ck , k ≥ 0 (0.7)
for any initial measure λ ∈ P(H) with supp λ ⊂ H. Here • * L(H) is the dual-Lipschitz metric defined by (0.11).

To the best of our knowledge, this is the first result that establishes uniqueness of stationary measure and exponential mixing for the NS system in an unbounded domain. In this unbounded setting there are at least two additional difficulties compared to the case of a bounded domain. First, the resolving operator of the equation is not compact and does not have a compact attracting set. The second and more important difficulty is related to the presence of a continuous component in the spectrum of the Stokes operator. Let us note that the existence of a stationary measure for the NS system perturbed by the random force (0.5) can be established by combining the Bogolyubov-Krylov argument and the asymptotic compactness of the dynamics. When the driving force is a white noise, existence results are obtained in [START_REF] Eckmann | Invariant measures for stochastic partial differential equations in unbounded domains[END_REF], for a real Ginzburg-Landau equation, in [START_REF] Rougemont | Space-time invariant measures, entropy, and dimension for stochastic Ginzburg-Landau equations[END_REF], for a complex Ginzburg-Landau equation, in [START_REF] Brzeźniak | Asymptotic compactness and sets for 2D stochastic Navier-Stokes equations on some unbounded domains[END_REF], for the NS system, and in [START_REF] Ekren | Existence of invariant measures for the stochastic damped Schrödinger equation[END_REF], for a damped Schrödinger equation.

The uniqueness of stationary measure and mixing properties for PDEs in unbounded domains have been studied previously only for the Burgers equation. The case of inviscid equation on the real line has been considered in the paper [START_REF] Bakhtin | Space-time stationary solutions for the Burgers equation[END_REF], under the assumption that the driving force is a space-time homogeneous Poisson point process. The proof is based on a combination of Lagrangian methods and first/last passage percolation theory. This result is generalised to the viscous case in the recent paper [START_REF] Bakhtin | Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation[END_REF], where the perturbation is a space-time homogeneous random kick force. In both papers, the stationary measure is space translation invariant. In the case of the NS system perturbed by a homogeneous noise, the uniqueness of a space-time translation invariant measure (and in some cases also the existence) remains an open problem.

The proof of the Main Theorem uses a controllability approach of the papers [START_REF] Shirikyan | Control and mixing for 2D Navier-Stokes equations with space-time localised noise[END_REF][START_REF] Shirikyan | Controllability implies mixing II. Convergence in the dual-Lipschitz metric[END_REF], where exponential mixing is established for the NS system with a space-time and boundary localised forcing. In [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF][START_REF] Kuksin | Mixing via controllability for randomly forced nonlinear dissipative PDEs[END_REF], methods of control theory are used to study a family of parabolic PDEs with a random perturbation that is highly degenerate in the Fourier space; see also the paper [START_REF] Kuksin | Exponential mixing for dissipative PDEs with bounded non-degenerate noise[END_REF], where a non-degenerate version of these results is presented. In [START_REF] Jakšić | Large deviations and entropy production in viscous fluid flows[END_REF], controllability is used to derive large deviations principle for the Lagrangian trajectories of the NS system.

The main novelty of the controllability argument we use here is that the deterministic operator is not supposed to be regularising to a space that is compactly embedded into the main phase space (see Theorem 1.1). That regularisation condition is replaced by two weaker properties: we assume that the nonlinear dynamics is asymptotically compact and the linearised operator can be decomposed into a sum of two operators one of which is dissipative and the other is compact (see (1.3)). It is proved that these weaker conditions are still sufficient for the exponential mixing. We make an essential use of the form of the nonlinearity to verify these properties for the NS system. The reader is referred to Section 1 for further discussion of the abstract controllability criterion and to Section 2 for the verification of the conditions of this criterion for the NS system.

Let us close this section with two remarks. Note that the convergence rate c in (0.7) depends on the initial compact H. Indeed, this is related to the fact that the NS system in the unbounded case does not have a compact invariant attracting set. Without going into the details, let us mention that the existence of such attracting set (and a uniform convergence rate) can be recovered if we take initial condition and forcing in weighted Sobolev spaces as in [START_REF] Babin | The attractor of a Navier-Stokes system in an unbounded channel-like domain[END_REF].

The second remark is about the NS system with the Ekman damping. By literally repeating the arguments of the proof of the Main Theorem, one can establish exponential mixing in the case of the whole space D = R 2 or arbitrary unbounded3 domain D ⊂ R 2 with smooth boundary for the following system:

∂ t u -ν∆u + au + u, ∇ u + ∇p = η(t, x), div u = 0, u| ∂D = 0,
where a > 0 is the damping parameter. The damping ensures a dissipativity property for the solutions without any assumption on the domain. We shall not discuss the details of this generalisation in this paper.

Notation

Let D ⊂ R 2 be an unbounded Poincaré domain with smooth boundary. In this paper, we use the following functional spaces.

C ∞ 0 (D, R 2 ) is the space of compactly supported smooth functions u : D → R 2 , and V = {u ∈ C ∞ 0 (D, R 2 ) : div u = 0}. (0.8)
H s (D, R 2 ) and L p (D, R 2 ) are the Sobolev and Lebesgue spaces on D. We consider the NS system in the usual functional spaces:

H = closure of V in L 2 (D, R 2 ), (0.9) V = closure of V in H 1 (D, R 2 ) (0.10)
endowed with the scalar products

u, v = D u • v dx, u, v 1 = D (∇u 1 • ∇v 1 + ∇u 2 • ∇v 2 ) dx
and the corresponding norms

• = •, • and • 1 = •, • 1 . The Poincaré in- equality (0.4) implies that • 1 is equivalent to the norm inherited from H 1 (D, R 2 ).
The dual of V with respect to the scalar product in H is denoted by V ′ .

Let X be a Polish space with metric d. B(X) denotes the Borel σ-algebra on X. C b (X) is the space of continuous functions g : X → R endowed with the supnorm g ∞ . When X is compact, we write C(X). L b (X) is the space of functions g ∈ C b (X) for which the following norm is finite:

g L(X) = g ∞ + sup u =v |g(u) -g(v)| d(u, v) .
P(X) is the set of Borel probability measures on X endowed with the metric

µ 1 -µ 2 * L(X) = sup g L(X) ≤1 | g, µ 1 -g, µ 2 |, µ 1 , µ 2 ∈ P(X), (0.11)
where g, µ = X g(u)µ(du).

Let E be a Banach space endowed with a norm • E , and let

J T = [0, T ]. L p (J T , E), 1 ≤ p < ∞ is the space of measurable functions u : J T → E such that u L p (JT ,E) = T 0 u(s) p E ds 1 p < ∞. C(J T , E) is the space of continuous functions u : J T → E with the norm u C(JT ,E) = sup t∈JT u(t) E .
L(E, Y ) is the space of bounded linear operators from E to another Banach space Y . We write

L(E) when E = Y . B E (a, R) is the closed ball in E of radius R centered at a. We write B E (R) when a = 0. D(η) is the law of E-valued random variable η.

Abstract criterion

Let H be a separable Hilbert space, and E be a separable Banach space. In this section, we consider a random dynamical system of the form

u k = S(u k-1 , η k ), k ≥ 1, ( 1
u 0 = u ∈ H (1.2)
and the vectors η i = ζ i , i = 1, . . . , k. For any set H ⊂ H, we define the set of attainability in time k ≥ 1:

A k (H) := {S k (u; ζ 1 , . . . , ζ k ) : u ∈ H, ζ 1 , . . . , ζ k ∈ K}
and the set of attainability in infinite time:

A(H) := ∪ ∞ k=1 A k (H) H .
The following five conditions are assumed to be satisfied for the mapping S and the measure ℓ.

(i) Regularity. The mapping S : H × E → H is twice continuously differentiable, and its derivatives are bounded on bounded subsets. Moreover, for any (u, η) ∈ H × K, the derivative (D u S)(u, η) can be represented as

(D u S)(u, η) = Ψ 1 + Ψ 2 (u, η), (1.3)
where the operators4 Ψ 1 , Ψ 2 (u, η) ∈ L(H) are such that

Ψ 1 L(H) =: κ < 1 (1.4)
and Ψ 2 (u, η) is compact.

(ii) Asymptotic compactness. For any bounded sequence {u n 0 } in H, any integers l n ≥ 1 such that l n → ∞, and any family

{ζ n m : m, n ≥ 1} ⊂ K, the sequence {S ln (u n 0 ; ζ n 1 , . . . , ζ n ln )} is precompact in H. (iii)
Approximate controllability to a point. There is a point û ∈ H with the following property: for any ε > 0 and any compact H in H, there is an integer n ≥ 1 such that, for any initial point u ∈ H, there are vectors

ζ 1 , . . . , ζ n ∈ K satisfying S n (u; ζ 1 , . . . , ζ n ) -û ≤ ε.
(1.5) (iv) Approximate controllability of the linearisation. For any u ∈ H and η ∈ K, the image of the linear mapping (D η S)(u, η) : E → H is dense in H.

(v) Decomposability. The set K is compact in E. Moreover, there are sequences of closed subspaces {F n } and {G n } in E satisfying the following properties:

• F n are finite-dimensional, F n ⊂ F n+1 for any n ≥ 1, and E = ∪ n F n .
• E is the direct sum of the spaces F n and G n , and the norms of the corresponding projections P n and Q n are bounded in n ≥ 1.

• ℓ is the product of projections P n * ℓ and Q n * ℓ for any n ≥ 1. Moreover, the measures P n * ℓ have C 1 -smooth densities with respect to the Lebesgue measure on F n .

System (1.1) defines a family of Markov processes (u k , P u ) in H parametrised by the initial condition (1.2). Let P k : C b (H) → C b (H) and P * k : P(H) → P(H) be the associated Markov operators.

Theorem 1.1. Under Conditions (i)-(v), the family (u k , P u ) has a unique stationary measure µ ∈ P(H). Moreover, for any compact set H in H, there are numbers C > 0 and c > 0 such that

P * k λ -µ * L(H) ≤ Ce -ck , k ≥ 1 (1.6)
for any initial measure λ ∈ P(H) with supp λ ⊂ H.

See the papers [Shi15, Shi19, KNS18, KNS19, KZ18, JNPS19] for related abstract criteria for uniqueness of stationary measure and mixing. The formulation of Theorem 1.1 is close to the results in [KNS19, JNPS19], but the proof is based on a theorem obtained in [START_REF] Shirikyan | Controllability implies mixing II. Convergence in the dual-Lipschitz metric[END_REF]. There are two main differences in our formulation. First, in Condition (i), we do not suppose that the mapping S (or its linearisation) takes values in a space that is compactly embedded into H. Instead, we assume that (D u S)(u, η) is a sum of dissipative and compact operators. The second difference is Condition (ii), which allows to recover some compactness properties5 for the dynamics. These two new conditions make Theorem 1.1 applicable to the NS system in unbounded domains. Moreover, this theorem can be applied to dissipative PDEs without parabolic regularisation (this will be addressed in a subsequent publication).

As in this paper the random perturbation is non-degenerate, the image of the mapping (D η S)(u, η) is assumed to be dense for any η in K; degenerate versions of this criterion can also be envisaged (cf. [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF][START_REF] Kuksin | Mixing via controllability for randomly forced nonlinear dissipative PDEs[END_REF]).

Proof of Theorem 1.1. Step 1: Existence of stationary measure. The set X := A(H) is compact in H. Indeed, it suffices to show that any sequence {v n } of the form

v n = S ln (u n 0 ; ζ n 1 , . . . , ζ n ln )
with some u n 0 ∈ H, ζ n 1 , . . . , ζ n ln ∈ K, and l n ≥ 1, is precompact in H. If the sequence {l n } is bounded, then v n ∈ A m (H) for all n ≥ 1, where m = max{l n }. The fact that A m (H) is compact (as image of a compact set by a continuous mapping) implies that {v n } is precompact. In the case l n → ∞, the conclusion follows from Condition (ii).

The compactness of X, combined with the invariance property S(X×K) ⊂ X and the usual Bogolyubov-Krylov argument (e.g., see Section 2.5 in [START_REF] Kuksin | Mathematics of Two-Dimensional Turbulence[END_REF]), implies the existence of a stationary measure µ ∈ P(X).

Step 2: Limit (1.6). According to Theorem 1.1 in [START_REF] Shirikyan | Controllability implies mixing II. Convergence in the dual-Lipschitz metric[END_REF], limit (1.6) will be established if we verify the following property.

Local stabilisation. Let D δ := {(u, u ′ ) ∈ X × X : u -u ′ ≤ δ}.
For any R 0 and any compact K ⊂ E, there is a finite-dimensional subspace E ⊂ E, and a continuous mapping

Φ : D δ × B E (R) → E, (u, u ′ , η) → η ′ ,
which is continuously differentiable in η and satisfies the inequalities

sup η∈BE (R) Φ(u, u ′ , η) E + D η Φ(u, u ′ , η) L(E) ≤ C u -u ′ , (1.7) sup η∈K S(u, η) -S(u ′ , η + Φ(u, u ′ , η)) ≤ q u -u ′ , (u, u ′ ) ∈ D δ (1.8)
for some positive constants C, δ, and q < 1. Let us show that Conditions (i), (iv), and (v) imply this local stabilisation property. We use a construction of approximate right inverse for linear operators from Section 2.2 in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF]. For any u ∈ X and η ∈ E, let A(u, η) : E → H be given by A(u, η) := (D η S)(u, η). Then G(u, η) := A(u, η) A(u, η) * : H → H is non-negative self-adjoint operator and Im(G(u, η)) is dense in H by Condition (iv). Thus (G(u, η) + γI) -1 is well defined for any γ > 0, and we have the limit

G(u, η)(G(u, η) + γI) -1 f → f as γ → 0 + , f ∈ H,
(1.9) by Lemma 2.4 in [START_REF] Kuksin | Exponential mixing for a class of dissipative PDEs with bounded degenerate noise[END_REF]. This shows that A(u, η) * (G(u, η) + γI) -1 is an approximate right inverse for A(u, η). We truncate it to obtain an operator with finite-dimensional image:

R M,γ (u, η) := P M A(u, η) * (G(u, η) + γI) -1 ,
where P M is a projection as in Condition (v) and M ≥ 1 and γ > 0 are parameters that will be chosen later. It is straightforward to see that

R M,γ (u, η) L(H,E) + (D η R M,γ )(u, η) L(H×E,E) ≤ C 1 (R, M, γ) (1.10)
for any u ∈ X and η ∈ B E (R), where the constant C 1 (R, M, γ) > 0 does not depend on (u, η). By the Taylor formula, for any u, u ′ ∈ X and η, η ′ ∈ B E (R), we have

S(u ′ , η ′ ) -S(u, η) = (D u S)(u, η)(u ′ -u) + (D η S)(u, η)(η ′ -η) + r(u, u ′ , η, η ′ ) (1.11) where r(u, u ′ , η, η ′ ) ≤ C 2 (R) u -u ′ 2 + η -η ′ 2 E .
(1.12)

The mapping Φ is defined by

Φ(u, u ′ , η) := -R M,γ (u, η)Ψ 2 (u, η)(u ′ -u).
Then (1.7) is verified due to (1.10) and

C 3 := sup (u,η)∈X×BE (R) Ψ 2 (u, η) L(H) + D η Ψ 2 (u, η) L(H×E,H) < ∞. (1.13)
The fact that C 3 is finite follows from the boundedness on X × B E (R) of the norms of the derivatives of S(u, η) and the representation (1.3). Assume that, for any ε > 0, we are able to find numbers M ≥ 1 and γ > 0 such that

(D η S)(u, η)R M,γ (u, η)Ψ 2 (u, η)g -Ψ 2 (u, η)g ≤ ε g (1.14)
for any u ∈ X, η ∈ K, and g ∈ H. Then combining (1.3), (1.4), and (1.11)-(1.14), we obtain

S(u, η) -S(u ′ , η + Φ(u, u ′ , η)) ≤ κ u ′ -u + ε u ′ -u + r(u, u ′ , η, η + Φ(u, u ′ , η)) ≤ (κ + ε + C 4 (ε, R) u ′ -u ) u ′ -u ≤ q u ′ -u
for any (u, u ′ ) ∈ D δ and η ∈ K, where ε > 0 and δ > 0 are sufficiently small and q < 1. This completes the proof of the local stabilisation and limit (1.6). Let us prove inequality (1.14). By (1.9), the continuous dependence of the operators (D η S)(u, η) and Ψ 2 (u, η) on (u, η), and a simple compactness argument, we find a large integer M ≥ 1 and a small number γ > 0 such that6

(D η S)(u, η)R M,γ (u, η)f -f ≤ ε
for any u ∈ X, η ∈ K, and f ∈ Ψ 2 (u, η)(B H (1)). Then inequality (1.14) follows by homogeneity.

Step 3: Uniqueness of stationary measure. To complete the proof, it remains to show the uniqueness of stationary measure in P(H). Assume that µ is the stationary measure supported7 in A({û}), and let λ 1 be any stationary measure for (u k , P u ) in P(H). As λ 1 (A(H)) = 1, there is a sequence

{H n } of compacts in H such that λ 1 (A(H n )) > 1 -1/n for any n ≥ 1.
By Condition (iii), we have A({û}) ⊂ A(H n ), and µ and

λ 1 /λ 1 (A(H n )) are stationary measures for (u k , P u ) in P(A(H n )). So µ = λ 1 /λ 1 (A(H n ))
by the uniqueness of stationary measure on X = A(H n ). Therefore,

λ 1 (Γ) = lim n→∞ λ 1 (Γ ∩ A(H n )) = lim n→∞ λ 1 (Γ ∩ A(H n ))/λ 1 (A(H n )) = µ(Γ)
for any Γ ∈ B(H). Thus λ 1 = µ.

Proof of the Main Theorem

In this section, we prove the Main Theorem by applying Theorem 1.1. We begin with a short discussion of the deterministic NS system, then turn to the verification of Conditions (i)-(v) in an appropriate functional setting.

Preliminaries

Applying the Leray projection 8 Π to Eq. (0.1), we eliminate the pressure term and consider the evolution system

u + νLu + B(u) = η, (2.1)
where L = -Π∆ is the Stokes operator and B(u

) = Π( u, ∇ u). Let us define a bilinear symmetric form [•, •] : V × V → R by [u, v] := u, v 1 - λ 1 2 u, v , u, v ∈ V.
The Poincaré inequality (see (0.4))

u 2 ≤ λ -1 1 u 2 1 , u ∈ V (2.2) implies that 1 2 u 2 1 ≤ [u] 2 := [u, u] ≤ u 2 1 . (2.3) Thus [•, •] defines a scalar product on V with norm [•] equivalent to • 1 .
Proposition 2.1. For any T > 0, u 0 ∈ H, and η ∈ L 2 (J T , H), there is a unique solution u ∈ C(J T , H) ∩ L 2 (J T , V ) of problem (2.1), (0.3). It satisfies the following inequalities

u(t) 2 ≤ e -νλ1t u 0 2 + ν -2 λ -1 1 η 2 L 2 (JT ,V ′ ) , t ∈ J T , (2.4)
u 2 L 2 (JT ,V ) ≤ ν -1 u 0 2 + ν -2 η 2 L 2 (JT ,V ′ ) , (2.5)
and the equality

u(t) 2 = e -νλ1t u 0 2 + 2 t 0 e -νλ1(t-s) η(s), u(s) -ν[u(s)] 2 ds, t ∈ J T .
(2.6)

Let us define the mapping

S t : H × L 2 (J T , H) → H, (u 0 , η) → u(t), t ∈ J T .
The following stability result holds.

8 That is the orthogonal projection in L 2 (D, R 2 ) onto H.

Proposition 2.2. Assume that the sequence {u n 0 } converges weakly to u 0 in H and the sequence {η n } converges strongly to η in L 2 (J T , H). Then

S t (u n 0 , η n ) ⇀ S t (u 0 , η) weakly in H for t ∈ J T , (2.7)
S • (u n 0 , η n ) ⇀ S • (u 0 , η) weakly in L 2 (J T , V ).
(2.8)

The proofs of these two propositions are carried out by standard methods and are given in the Appendix.

Let us now describe the functional setting in which Theorem 1.1 is applied. The space H is defined by (0.9), E := L 2 ([0, 1], H), and the mapping

S := S 1 : H × E → H, (u 0 , η) → u(1)
is the time-one shift along trajectories of Eq. (2.1). Then the restriction to integer times of the solution of (2.1), (0.3), (0.5) satisfies

u k = S(u k-1 , η k ), k ≥ 1, (2.9)
where {η k } is a sequence of i.i.d. random variables as in the decomposability condition in the Introduction. It is straightforward to see that Condition (v) satisfied.9 In the next three subsections, we check Conditions (i)-(iv).

Regularity condition

The smoothness of the mapping S : H × E → H and the boundedness of its derivatives on bounded subsets of H × E are proved using well-known methods (e.g., see Chapters I and VII in [START_REF] Babin | Attractors of Evolution Equations[END_REF] for the case of bounded domain D).

Let us consider the linearisation of Eq. (2.1) around the trajectory ũ(t) = S t (u, η) corresponding to an initial condition ũ(0) = u ∈ H and control η ∈ K:

ẇ + νLw + Q(ũ, w) = 0,
(2.10)

w(0) = w 0 , (2.11)
where

Q(a, b) = Π( a, ∇ b) + Π( b, ∇ a).
(2.12)

Then (D u S)(u, η)w 0 = w(1) for any w 0 ∈ H, and we can write w = v 1 + v 2 , where v 1 and v 2 are the solutions of the problems

v1 + νLv 1 = 0, v 1 (0) = w 0 , (2.13) v2 + νLv 2 + Q(ũ, w) = 0, v 2 (0) = 0.
(2.14)

The representation (1.3) holds with the linear operators

Ψ 1 : H → H, w 0 → v 1 (1), Ψ 2 (u, η) : H → H, w 0 → v 2 (1).
Inequality v 1 (t) ≤ e -νλ1t/2 w 0 implies (1.4) with κ := e -νλ1/2 . The following lemma (whose proof is given in the Appendix) completes the verification of Condition (i).

Lemma 2.3. For any (u, η) ∈ H × K, the operator Ψ 2 (u, η) : H → H is compact.

Asymptotic compactness

For any integer k ≥ 1 and any function η ∈ L 2 (J k , H) of the form (0.5) with some sequence {η n } ⊂ E, we write u(t) = S t (u 0 , η) =: S t (u 0 ; η 1 , . . . , η k ), t ∈ J k .

(2.15)

Proposition 2.4. For any bounded sequence {u n 0 } in H, any sequence of integers l n ≥ 1 such that l n → ∞, and any family {ζ n m : m, n

≥ 1} ⊂ K, the sequence v n = S ln (u n 0 ; ζ n 1 , . . . , ζ n ln ) is precompact in H.
Proof. Let us first explain the scheme of the proof. Using the dissipativity of the system, we show that {v n } is bounded in H. As H is a Hilbert space, there is a subsequence {v kn } such that v kn ⇀ w weakly in H.

( Step 1: Boundedness. Let us show that, for any bounded set H ⊂ H, the attainability set A(H) ⊂ H is bounded. Indeed, let

M 1 := sup u∈H u 2 < ∞.
For any u 0 ∈ H and ζ 1 , . . . , ζ k ∈ K, let u k := S k (u 0 ; ζ 1 , . . . , ζ k ). Then by inequality (2.4), we have

u k 2 ≤ κ u k-1 2 + M 2 ,
where κ := e -νλ1 < 1 and M

2 := ν -2 λ -1 1 sup η∈K η 2 L 2 ([0,1],V ′ ) . Iterating this, we get u k 2 ≤ κ k u 0 2 + M 2 (1 -κ) -1 ≤ M 1 + M 2 (1 -κ) -1 =: M.

This shows that sup

u∈A(H)

u 2 ≤ M.

Step

w 2 = u m (m) 2 = e -νλ1m w m 2 + 2 m 0 e -νλ1(m-s) ξ m (s), u m (s) -ν[u m (s)] 2 ds, (2.20)
where

u m (t) = S t (w m , ξ m ), ξ m (t) = m k=1 I [k-1,k) (t)ξ m k (t -k + 1), t ∈ J m .
On the other hand, again by (2.6), we have As m ≥ 1 is arbitrary, we arrive at (2.18).

Approximate controllability

By the decomposability assumption, we have 0 ∈ K. In view of (2.4), Condition (iii) is verified with û = 0, ζ 1 = . . . = ζ n = 0, and sufficiently large n ≥ 1.

To check Condition (iv), we consider the following linearisation of Eq. (2.1) around the same trajectory ũ as in Section 2.2:

ẇ + νLw + Q(ũ, w) = ζ(t),
(2.23)

w(0) = 0,
where Q is given by (2.12). Then where the constant C 3 > 0 does not depend on the numbers n, m, ε. Combining this with (3.7) and choosing n and m sufficiently large, we see that Φ ũ(w kn 0 ) -Φ ũ(w km 0 ) L 2 (J,H -1 ) < 2C 3 ε.

This shows that Φ ũ is compact. Thus the map Ψ 2 : H → H is compact as a composition of Φ ũ with some linear continuous map.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Regularity condition . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Asymptotic compactness . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Approximate controllability . . . . . . . . . . . . . . . . . . . . . 14

v kn 2 =+2 m 0 e 0 e 0 e 0 e

 20000 u n,m (m) 2 = e -νλ1m w n,m 2 -νλ1(m-s) η n,m (s), u n,m (s) -ν[u n,m (s)] 2 ds,(2.21)whereu n,m (t) = S t (w n,m , η n,m ), η n,m (t) = m k=1 I [k-1,k) (t)η n kn-m+k (t-k+1), t ∈ J m . Note that m -νλ1(m-s) η n,m (s), u n,m (s) ds → m -νλ1(m-s) ξ m (s), u m (s) ds, (2.22) as η n,m → ξ m strongly in L 2 (J m , H) and u n,m ⇀ u m weakly in L 2 (J m , V ). Since m 0 e -νλ1(m-s) [•] 2 ds 1/2is a norm in L 2 (J m , V ) which is equivalent to the original one, the following inequality holdsm 0 e -νλ1(m-s) [u m (s)] 2 ds ≤ lim infn→∞ m -νλ1(m-s) [u n,m (s)] 2 ds. Combining this with (2.20)-(2.22), we get lim sup n→∞ v kn 2 ≤ lim sup n→∞ e -νλ1m w n,m 2 + w 2 -e -νλ1m w m 2 ≤ e -νλ1m M + w 2 .

  (D η S)(u, η)ζ = w(1) for any ζ ∈ E. For any smooth function w 1 ∈ H, we can find a smooth function w : [0, 1] × D → R 2 such that w(0) = 0, w(1) = w 1 , and w(t) ∈ H for all t ∈ [0, 1]. Replacing w into Eq. (2.23), we find explicitly a control ζ ∈ E such that (D η S)(u, η)ζ = w 1 . This shows that the image of the mapping (D η S)(u, η) : E → H is dense in H for any (u, η) ∈ H ×K. Thus Conditions (i)-(v) are verified. Applying Theorem 1.1, we complete the proof of the Main Theorem.

  .1) where S : H × E → H is a continuous mapping and {η k } is a sequence of i.i.d. random variables in E. Let K ⊂ E be the support of the law ℓ := D(η k ). For any sequence {ζ k } in K, let us denote by S k (u; ζ 1 , . . . , ζ k ) the trajectory of (1.1) corresponding to the initial condition

  2: Proof of (2.18). From the previous step it follows that {v n } is bounded in H. Let {v kn } be a subsequence verifying (2.16). It is of the formv kn = S ℓn (u n 0 ; η n 1 , . . . , η n ℓn )for some vectors η n 1 , . . . , η n ℓn ∈ K and integers ℓ n ≥ 1 such that ℓ n → ∞. Using the boundedness of the set A({u n 0 , n ≥ 1}) and passing to a subsequence if necessary (applying the diagonal process), we can assume that

	w n,m := S ℓn-m (u n 0 ; η n 1 , . . . , η n ℓn-m ) ⇀ w m weakly in H	(2.19)
	From (2.16) we derive the equality
	w = S m (w m ; ξ m 1 , . . . , ξ m m ) for any m ≥ 1.
	This and (2.6) imply that

for any m ≥ 1 and some w m ∈ H. Using the compactness of K and again passing to a subsequence if necessary, we can assume that

η n ℓn-i → ξ m m-i strongly in E, i = 0, . . . m

-1. Combining this with (2.19) and Proposition 2.2, we get v kn = S m (w n,m ; η n ℓn-m+1 , . . . , η n ℓn ) ⇀ S m (w m ; ξ m 1 , . . . , ξ m m ) weakly in H.

I.e., domain D which does not necessarily satisfy the Poincaré inequality.

Ψ 1 does not depend on (u, η).

The asymptotic compactness is a well-known property in the study of the attractors for deterministic PDEs (e.g., see[START_REF] Ladyzhenskaya | Attractors for semigroups and evolution equations[END_REF][START_REF] Rosa | The global attractor for the 2D Navier-Stokes flow on some unbounded domains[END_REF]).

Here we use also the boundedness of the projections P M (see Condition (v)), which implies the limit P M → I as M → ∞ in the operator topology.

Using Condition (iii), it is easy to see that supp µ = A({û}).

Note that K := supp D(η k ) is compact in E, since it is contained in a Hilbert cube.

Acknowledgments

The author thanks Armen Shirikyan for helpful discussions and suggestions. This research was supported by Agence Nationale de la Recherche grant ANR-17-CE40-0006-02 and CNRS PICS grant Fluctuation theorems in stochastic systems.

Appendix

3.1 Proof of Proposition 2.1

The existence and uniqueness of solution is proved, e.g., in Chapter III of [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF].

Here we give a formal derivation of inequalities (2.4) and (2.5) and equality (2.6).

Taking the scalar product in H of Eq. (2.1) with 2u and using the identity

Combining this with the Poincaré inequality, we obtain (2.4). From (3.1) we also derive the inequality

which implies (2.5). To prove (2.6), we rewrite the equality in (3.1) in the form

and apply the variation of constants formula.

Proof of Proposition 2.2

By Proposition 2.1, the sequence

) and numbers α 0 = α 1 = 2, and using the diagonal process, we find a subsequence {u kn } such that

for any R > 0. Passing to the limit in the equation for u kn (t), we conclude that u(t) is the solution S t (u 0 , η), t ∈ J T . Moreover, by the uniqueness of the limit, it is easy to see that limits (3.4)-(3.6) hold for the full sequence {u n }. This proves (2.8).

Let us take any ϕ ∈ V (see (0.8)). By inequality (2.4), we have

and by inequality (3.3),

for any t ∈ J T and τ ∈ (0, T -t). Thus the Arzelà-Ascoli theorem implies that

uniformly in t ∈ J T . Using the fact that V is dense in H, we get (2.7).

Proof of Lemma 2.3

Let us set J := [0, 1]. By Proposition 2.1, we have ũ = S • (u, η) ∈ L 2 (J, V ) for any u ∈ H and η ∈ K. Using standard methods, one can show that the mapping Ψ : w 0 → w (i.e., the resolving operator of problem (2.10), (2.11)) is continuous from

). Let us show that the linear mapping Φ ũ : H → L 2 (J, H -1 ), w 0 → ũ, ∇ w + w, ∇ ũ is compact. Indeed, let {w n 0 } be a bounded sequence in H. Then the sequence {Ψ(w n 0 )} is bounded in X . As in the previous subsection, we apply Theorem 2.1 of Chapter III in [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF] with spaces

) and numbers α 0 = α 1 = 2, and use the diagonal process to find a subsequence {Ψ(w kn 0 )} converging strongly in L 2 (J, L 2 (D R )) for any R > 0. On the other hand, for any ε > 0, there is a number R > 0 and a smooth function ϕ : J × D → R 2 with supp ϕ(t, •) ⊂ D R for any t ∈ J and ũ -ϕ L 2 (J,V ) < ε.

(3.7)

Using the boundedness of the sequence {Φ ũ(w kn 0 )} in L 2 (J, V ) and the estimate a

we obtain Φ ũ(w kn 0 ) -Φ ũ(w km 0 ) L 2 (J,H -1 ) ≤ Φ ϕ (w kn 0 ) -Φ ϕ (w km 0 ) L 2 (J,H -1 ) + Φ ũ-ϕ (w kn 0 ) -Φ ũ-ϕ (w km 0 ) L 2 (J,H -1 ) ≤ C 3 Ψ(w kn 0 ) -Ψ(w km 0 )

1/2 L 2 (J,L 2 (DR))

+ C 3 ũ -ϕ L 2 (J,V ) ,


Michal Swierczewski

Alexis Chenneviere

Lay-Theng Lee

Plinio Maroni

Thomas Bürgi

Nanomechanical and Structural Study of Au38 Nanocluster Langmuir-Blodgett Films Using Bimodal Atomic Force Microscopy and X-Ray Reflectivity

come   L'archive ouverte pluridisciplinaire


Viviane David
email: viviane.david@mnhn.fr

Anne Mouget

Yannick Perrot

Loïc Le Goff

Pierre Thiriet

Ndague Diogoul

Eric Feunteun

Anthony Acou

Patrice Brehmer

Insights from a multibeam echosounder to survey pelagic fish shoals and their spatio-temporal distribution in ultra-shallow waters

Keywords: Pelagic fish, multibeam, echosounder, ultra-shallow waters, coastal ecosystems

INTRODUCTION

Shallow coastal waters, and in particular ultra-shallow waters (< 20m), provide many biological and ecological functions for fish species at different stages and throughout their life cycle [START_REF] Beck | The Identification, Conservation, and Management of Estuarine and Marine Nurseries for Fish and InvertebratesA better understanding of the habitats that serve as nurseries for marine species and the factors that create site-specific variability in nursery quality will improve conservation and management of these areas[END_REF][START_REF] Franco | Use of shallow water habitats by fish assemblages in a Mediterranean coastal lagoon[END_REF][START_REF] Pihl | Habitat use by Fishes in Estuaries and Other Brackish Areas[END_REF]. Numerous fish species (sardines, European anchovies, Atlantic mackerels, etc.) are also known to migrate to coastal and estuarine areas for various reasons which can vary extensively between fish species, e.g., for spawning, feeding reasons or environmental conditions [START_REF] Bellier | Historical fluctuations in spawning location of anchovy (Engraulis encrasicolus) and sardine (Sardina pilchardus) in the Bay of Biscay during 1967-73 and 2000-2004[END_REF][START_REF] Morais | Diversity of anchovy migration patterns in an European temperate estuary and in its adjacent coastal area: Implications for fishery management[END_REF][START_REF] Nøttestad | Feeding strategy of mackerel in the Norwegian Sea relative to currents, temperature, and prey[END_REF]). In addition, shallow waters are subjected to many anthropogenic pressures, such as urbanization of coasts, fisheries, pollution, physical disturbance of habitats, etc. [START_REF] Jackson | Historical overfishing and the recent collapse of coastal ecosystems[END_REF][START_REF] Halpern | Evaluating and ranking the vulnerability of global marine ecosystems to anthropogenic threats[END_REF][START_REF] Dauvin | The main characteristics, problems, and prospects for Western European coastal seas[END_REF]). To study marine ecosystems, acoustic methods are often effective as they allow rapid sampling with high spatial and temporal coverage and are non-invasive and non-extractive techniques to observe several marine organisms [START_REF] Simmonds | Fisheries Acoustics: Theory and Practice[END_REF][START_REF] Benoit-Bird | Ecological Insights from Pelagic Habitats Acquired Using Active Acoustic Techniques[END_REF]. However, ultra-shallow waters (< 20m) are less regularly acoustically surveyed because scientific monitoring of pelagic fish populations using active acoustic methods are usually conducted by research vessels which are restricted to work in ultra-shallow areas for safety reasons (Brehmer et al. 2006b). Consequently, knowledge on the pelagic fish spatio-temporal distributions in ultra-shallow waters are still scarce although crucial for effective management and conservation of marine and coastal ecosystems.

Split-beam echosounders associated with trawl catches are widely used to assess pelagic fish population distributions and abundances in order to estimate fish stocks in offshore waters [START_REF] Doray | Pelagic fish stock assessment by acoustic methods at Ifremer[END_REF], 2018[START_REF] Demer | Prediction and confirmation of seasonal migration of Pacific sardine (Sardinops sagax) in the California Current Ecosystem[END_REF]). However, the use of multibeam echosounders has gained increasing attention [START_REF] Melvin | Multibeam Acoustic Detection of Fish and Water Column Targets at High-Flow Sites[END_REF], Brehmer et al. 2019a[START_REF] Lamouret | Measuring fish activities as additional environmental data during a hydrographic survey with a multi-beam echo sounder[END_REF]. They have been developed to operate with frequencies ranging from 12 up to >500 kHz allowing them to be used in different water depths from shallow to deep waters [START_REF] Colbo | A review of oceanographic applications of water column data from multibeam echosounders[END_REF], and could transmit either continuous wave (CW) signals or frequency modulated (FM) signals. Multibeam echosounders sample a larger volume of waters compared to split-beam echosounders and should provide more information about the distribution and movement of targets in the field of view [START_REF] Gerlotto | From two dimensions to three: the use of multibeam sonar for a new approach in fisheries acoustics[END_REF][START_REF] Trenkel | The new fisheries multibeam echosounder ME70: description and expected contribution to fisheries research[END_REF]. In particular, in ultra-shallow waters (< 20m), where the sampled volume is restricted by the bottom, sampling a higher volume is interesting. Multibeam echosounders have thus been used to survey schooling fish on artificial and natural reefs [START_REF] Holland | Fine-scale spatial and diel dynamics of zooplanktivorous fish on temperate rocky and artificial reefs[END_REF]. They could also limit the bias in fish biomass estimations due to avoidance reaction to boats [START_REF] Reid | Report on Echo Trace Classification[END_REF]. This could be particularly relevant in shallow waters where the avoidance reaction are supposed to be higher than in deeper areas due to the noise and visual effects of the vessel [START_REF] Draštík | Fish avoidance of acoustic survey boat in shallow waters[END_REF], Brehmer et al. 2019b). Finally, multibeam echosounders allow 3-D observations of the entire fish shoals which could have implications for behaviour studies [START_REF] Reid | Report on Echo Trace Classification[END_REF] as well as help for species identification [START_REF] Guillard | Three-dimensional internal spatial structure of young-of-the-year pelagic freshwater fish provides evidence for the identification of fish school species[END_REF].

Fish shoals refer to aggregated social assemblages, and can display synchronized and/or polarized behaviors [START_REF] Pitcher | Functions of Shoaling Behaviour in Teleosts[END_REF]. The ability of fish to form shoals depends on many factors including species-specifics features such as behaviour, physiology and/or biology, and the environment. For example, fishes could form shoals for physiological reasons (i.e. feeding and reproduction), due to interactions with other species (i.e. anti-predator response), or environmental conditions (i.e. diurnal cycles, seasons) [START_REF] Bahri | Spatial structure of coastal pelagic schools descriptors in the Mediterranean Sea[END_REF]. Consequently, surveying fish shoals by a multibeam echosounder could help to better understand population spatio-temporal distributions as well as ecological interactions with their environment [START_REF] Benoit-Bird | Ecological Insights from Pelagic Habitats Acquired Using Active Acoustic Techniques[END_REF][START_REF] Innangi | High resolution 3-D shapes of fish schools: A new method to use the water column backscatter from hydrographic MultiBeam Echo Sounders[END_REF]. Especially, the marine habitat features (rocky substrate, sediments, etc.) should be better investigated as they would likely influence the fish shoal distributions [START_REF] Lamouret | Measuring fish activities as additional environmental data during a hydrographic survey with a multi-beam echo sounder[END_REF][START_REF] Lamouret | Coupling the map of marine habitats and fish accumulation zones : a three-dimensional spatial approach for the management of halieutic ressources[END_REF]. In addition, analyzing the distribution of a species along environmental or anthropic gradients can help identifying geographic areas considered suitable, or on the opposite, unsuitable for the presence of this species [START_REF] Guisan | Predicting species distribution: offering more than simple habitat models[END_REF], which is necessary for the management of the marine environment.

Our study was made in the framework of the context of the Marine Strategy Framework Directive (MSFD) which requires the Member States of the European Union to put measures in place to achieve and maintain good environmental status of all marine waters [START_REF] Long | The Marine Strategy Framework Directive: A New European Approach to the Regulation of the Marine Environment, Marine Natural Resources and Marine Ecological Services[END_REF], including shallow coastal waters. This study focuses on shallow water areas in French Brittany which are poorly known. We used a multibeam echosounder to estimate the spatio-temporal distributions of fish shoals from shallow waters to ultra-shallow water areas (< 20m) in two French Britany areas at different seasons (spring, summer, autumn) in 2020. Fish shoal detections and characteristics were analysed related to environmental gradients. Finally, we concluded on the insight of the multibeam echosounder to detect pelagic fish shoals in ultrashallow waters and assess and their characteristics and spatio-temporal distributions.

MATERIAL AND METHODS

Survey area description

Data were collected in two coastal areas near Brest and Douarnenez located in the North of the Bay of Biscay (called Bay of Biscay area), and near Roscoff, in the Celtic Sea (called Celtic Sea area) (Figure 2a). In the Bay of Biscay area, two surveys were conducted in early summer 2020 and again in early autumn 2020. The transects designed for this area covered a total of 99.0 nautical miles (nmi) and the same transects were made during each season (See supplementary information, Figure S1a). In the Celtic Sea area, three surveys were conducted in spring 2020, summer 2020 and autumn 2020. The defined transects for this area covered a total of 63.7 nautical miles and the same transects were made during each season (See supplementary information, Figure S1b).

The transects have been designed to cover a range of 5 to 70 m bathymetry and different seafloor substrate types in order to assess the impact of environmental factors on pelagic fish distributions. A sampling effort was made in the ultra-shallow water areas (< 20m) as they are not traditionally surveyed. Surveys were made with two small vessels, the Albert Lucas and Neomysis, dedicated respectively for the study sites (in the North of Bay of Biscay and in the Celtic Sea) by the French Oceanographic Fleet (https://www.flotteoceanographique.fr/Nosmoyens/Navires-engins-et-equipements-mobiles/Navires-de-station). Due the availability of these vessels, surveys were made only during the day. The speed of the vessels was set to 5 knots.

Recorded data

Water column was sampled along the transects by a Simrad M3 multibeam echosounder (Kongsberg M3 Sonar, 922-20007011), side mounted and used in vertical beaming. We used a multibeam echosounder with a high frequency (500 kHz, 128 beams in imaging mode, 120° swathe, 1.6° angular resolution, 30° vertical beam widths, detection up to 150 m) and a relatively short pulse duration (200 µs) giving a vertical resolution of 15 cm. The "Time Variable Gain" was adjusted in the M3 software following the manufacturer's recommandations. In order to have absolute backscattering strength levels, the multibeam echosounder was previously calibrated in situ with a 22-mm tungsten sphere [START_REF] Foote | Protocols for calibrating multibeam sonar[END_REF][START_REF] Perrot | Efficient multibeam sonar calibration and performance evaluation[END_REF]. This operation was carried out at quayside in quiet weather. The calibration coefficients were calculated as in [START_REF] Cochrane | Quantification of a multibeam sonar for fisheries assessment applications[END_REF]. Hence, when the sphere's echo was stable, the difference between the theoretical and measured responses of the calibration sphere was estimated. In addition, due to the difficulty of positioning the sphere in all beams, the equivalent two-way beam angle was calculated theoretically from the geometry of the elementary sensors of the transmitting and receiving antennas. For fish shoals, by making empirical comparisons with results from another calibrated split-beam echosouder (Simrad EK80, not presented in this study), our calibration was relevant.

Seawater temperature, salinity and dissolved oxygen were measured with Conductivity-Temperature-Depth (CTD) (SeaBird SBE 19plus V2) casts at the extremities of the transects (See Supplementary Information Figures S4 andS6). Data on the seafloor substrates came from the Naval Hydrographic and Oceanographic Service (SHOM) in which the substrate were classified in different categories dependent on the grain size (rock, stone: > 20 mm, gravel: 2-20 mm, sand: 0.05-2 mm and mud: < 0.05 mm).

Acoustic data processing

For data processing, we adapted the Matecho open-source tool which is implemented in Matlab [START_REF] Perrot | Matecho : an open-source tool for processing fisheries acoustics data[END_REF]. Matecho is an automated tool which allows fish shoal extractions.

Routine analyses have been developed to import the multibeam data into Matecho and convert them into a HDF5 file format, detect the seafloor to create a bottom line which can be manually corrected as well as remove the noise coming from potential interferences. In particular, the bottom depth is estimated for each ping by Matecho only for the beam perpendicular to the bottom. Indeed, a disadvantage of using multibeam echosounder is the bottom echo, which reverberates in all beams at distances larger than the depth. Consequently, the analysed volume of water excludes the area where the bottom echo was reverberated (Figure 1). The shoal number in this area was nevertheless negligible. After applying the algorithms of bottom and noise detections on multibeam data, all filter results were manually checked using the graphical interface of Matecho. When necessary, we manually corrected the bottom line, as usually made for all acoustic data.

The automatic extraction process of shoals in Matecho for multibeam data is the same than for split-beam data [START_REF] Perrot | Matecho : an open-source tool for processing fisheries acoustics data[END_REF]. It is based on the Movies3D algorithm [START_REF] Trenkel | Overview of recent progress in fisheries acoustics made by Ifremer with examples from the Bay of Biscay[END_REF]) and used a threshold for the (i) volume backscattering strength S v in decibels, (ii) a maximum along-ping-axis integration distance in m and (iii) a depth integration distances in m. For our analyses, these parameters were set to -55 dB for the S v threshold, 0.61 m for the maximum integration length for the distance and 0.6 m for the maximum integration length for the depth. This high S v threshold was chosen to facilitate automatic extraction of fish shoals and exclude macrozooplankton [START_REF] Petitgas | A method for the identification and characterization of clusters of schools along the transect lines of fisheries-acoustic surveys[END_REF][START_REF] Brehmer | Schooling behaviour of small pelagic fish: phenotypic expression of independent stimuli[END_REF]). In addition, the extractions were made up to 0.3 m from the bottom. Compared to the former version of Matecho, a visualization of the beam fan was added (Figure 1) as well as a 3-D view of the shoals. Hence, all extracted fish shoals have been visually checked using the graphical interface.

To keep only the fish shoals and avoid big individual fish, only the extractions with a minimum volume of 1 m 3 were kept for the analyses. Number of shoal detections were recorded and corrected by the sampling effort per study sites (transects of 97.4 and 63.7 nmi for the Bay of Biscay and Celtic Sea areas, respectively). The fish shoal descriptors automatically calculated are the maximal height in m (along the depth-axis), maximal width in m (along the athwart distance), total length in m (along the ping distance), surface in m 2 , volume in m 3 , percentage of holes under the threshold [START_REF] Guillard | Three dimensional characteristics of young-ofyear pelagic fish schools in lake[END_REF][START_REF] Long | The Marine Strategy Framework Directive: A New European Approach to the Regulation of the Marine Environment, Marine Natural Resources and Marine Ecological Services[END_REF][START_REF] Paramo | A three-dimensional approach to school typology using vertical scanning multibeam sonar[END_REF], mean acoustic volume backscattering strength S v in dB [START_REF] Maclennan | A consistent approach to definitions and symbols in fisheries acoustics[END_REF] and mean distance from the bottom in m [START_REF] Scalabrin | Acoustic detection of the spatial and temporal distribution of fish shoals in the Bay of Biscay[END_REF], expressed in percentage to the bottom depth. In particular, the percentage of or holes was calculated by dividing the number of samples under the extraction threshold in dB by the total number of samples from the shoals and multiplied it by 100. Other information are also provided for each shoal like the GPS coordinates (GPS with a circular error probable of 3m), the sampling time and the vessel speed. Finally, the environmental factors such as the bottom depth (given by Matecho) and the seafloor substrate types (from the SHOM data) where the shoals were detected were added.

For species allocation, we tried to capture fish using longlines (having five hooks with a shank of 3 cm) to identify the species responsible of observed detections in the multibeam echosounder's field of view. The species, total fish length, and weight of each individual fish was recorded before releasing it into the water. Probability of error in the shoal identification was reduced as shoals were observed at the same localization and fishing identification operation was done directly after the detection with the multibeam echosounder. The places to do the species allocation were chosen based on the acoustic observations (numerous and/or large detections).

Statistical analysis

All statistical analyses were made with the R software (R version 3.6.2) (R Development Core Team 2019) and the significance threshold was set to 0.05. The number of detected shoals was analysed in function of the seafloor substrate types and the bathymetry, taking into account the frequency in which the different categories of seafloor substrates as well as the bathymetric ranges (< 10, 10-20, 20-30, 30-40, 40-50, 50-60 and 60-70 m) were encountered along the entire transects for both study sites.

Test correlations were made using the Spearman methods to analyse the correlations between fish shoal descriptors (mean acoustic volume backscattering strength S v as well as the percentage of holes, their maximum width, maximum height, length, surface area and volume and the distance from the bottom in percentage). Fish shoal descriptors were also analysed in function of the environmental variables (bottom depth, season) using linear models ("lm" function). The seafloor substrate type was excluded from the model as it was correlated to the bathymetry (see Supplementary Information, Figure S2). The tested equation was:

Eq. 1

With the different fish shoal descriptors, to are the regression coefficients, Bathymetry and Season the environmental variables and the error independent and normally distributed. The maximal width, maximal height, length, surface and volume were log-transformed for the analyses. For each shoal descriptor, the fitting model was chosen based upon the Aikake Information Criterion (AIC) selection (Akaike 1974). Two models were considered different if their Δ AIC was higher than 2, in which case the lowest AIC defined the best model. However, if the Δ AIC was lower than 2, we used the Akaike weight [START_REF] Anderson | Null Hypothesis Testing: Problems, Prevalence, and an Alternative[END_REF]. In addition, we also made multiple pairwise comparisons using t-test using the Bonferroni correction to further analyse the differences between seasons.

For those shoals where species was identified based on longline sampling, 3-D morphological characteristics of the fish shoals (maximal height, maximal width, length, surface, volume) as well as the percentage of holes, the mean acoustic volume backscattering strength S v and the distance from bottom in percentage were compared for the same species and for the different areas where they were caught. In addition, we also compared the environmental factors (seafloor substrate, bottom depth). Finally, we performed a Multiple Factor Analysis (package R "FactoMineR") [START_REF] Pagès | Multiple Factor Analysis by Example Using R[END_REF]) which allows to take into account both quantitative (fish shoal descriptors and bathymetry) and qualitative data (seafloor substrate type, season and study site) to compare the shoal groups identified by the species to the other shoals.

RESULTS

Physical-chemical parameters

For the Bay of Biscay area, mean surface (1-m depth) water temperature, salinity and dissolved oxygen were respectively 16.7 ± 0.90°C, 34.3 ± 1.08 and 8.33 ± 0.66 mg L -1 in early summer and 17.8 ± 0.83°C, 35.1 ± 0.40 and 7.86 ± 0.32 mg L -1 in early autumn (1-m depth) (see Supplementary information, Figures S3 andS4). The mean surface water temperature and salinity were significantly higher in early autumn compared to early summer 2020 whereas the dissolved oxygen was significantly lower (ANOVA, p-value < 0.05). Water temperature decreased with depths, i.e. the mean temperature difference being below 1.5°C between surface and bottom for both seasons. Dissolved oxygen decreased within the water column with depths (mean difference of 0.9 and 1.2 for both early summer and autumn respectively).

For the Celtic Sea area, mean surface water temperature, salinity and dissolved oxygen were respectively 14.6 ± 0.67°C, 34.8 ± 0.13 and 7.67 ± 0.34 mg L -1 in spring, 16.0 ± 0.75°C, 34.7 ± 0.42 and 7.0 ± 0.7 8mg L -1 in summer and 14.9 ± 0.52°C, 34.2 ± 1.89 and 6.45 ± 0.22 mg L -1 in autumn (1-m depth) (Supplementary information, Figures S5 andS6). Mean surface water temperature were significantly higher in summer compared to spring and autumn 2020 (Pairwise comparisons using t tests, p-values < 0.05) whereas no difference was found between spring and autumn 2020. No statistical differences were found for the mean surface water salinity between the seasons. Dissolved oxygen significantly decreased throughout the seasons (Pairwise comparisons using t tests, p-values < 0.05). Temperature slightly decreased with depths, i.e. the mean temperature difference being below 0.3 °C between surface and bottom for all seasons in Celtic Sea area and dissolved oxygen was more homogeneous throughout the water column.

For both study sites, water temperature was higher close to the coast with a maximal difference of 3.2°C in summer for the Bay of Biscay area (see Supplementary information, Figures S4 andS6). Salinity did not vary with bathymetry, except in some particular locations near the coast where surface salinity was lower due to freshwater arrivals and was homogeneous throughout the water column (mean salinity differences below 1 between the surface and bottom).

Acoustic results

Shoal number and characteristics for the Bay of Biscay area

In Bay of Biscay area, 14.3 and 5.38 shoals nmi -1 having a minimum volume of 1 m 3 were detected in summer and autumn 2020, respectively (Figure 2b,c). The shoals were mainly located where the seafloor was composed of mud (21 %) or fine sand (20 %) in early summer 2020. However, shoals were mainly located above sand (31 %) and sand mixed with mud (24 %) in autumn 2020 (Figure 3a). In addition, the shoals were detected in similar bathymetry during summer and autumn, except for depths under 10 m where 26 % of the detected shoals where observed in summer compared to 3 % in autumn (Supplementary Information, Figure S7a).

According to the AIC criterions and Akaike weights, the chosen models explaining respectively the mean acoustic volume backscattering strength S v of the shoals, the percentage of holes, the distance from bottom in percentage and the maximal height included an interaction effect between bathymetry and season (Table S2 given the AIC values and Akaike weights per models in Supplementary Information). In particular, the acoustic energy decreased with increasing bathymetry (Figures 4 andS8) and the acoustic energy was significantly higher in summer than autumn (p-value < 0.05) (see Supplementary Information, Table S1). The models explaining the length, maximal surface area and volume included bathymetry and season without an interaction effect and the model explaining the maximal width included only the season effect (Table S2).

Shoals in shallow waters were significantly larger (vertically and horizontally elongated with larger surface areas and volumes) compared to shoals in deeper areas (Figures 4 andS8).

The maximal width, maximal height, length, maximal surface and volumes were all positively correlated (> 66.7, Spearman method, p-value <0.05). They were also significantly closer to the bottom in deeper areas compared to shallower areas as seen with the distance from bottom in percentage. Shoals in shallow waters had also less holes as shown by the percentage of holes which was inversely correlated to the acoustic volume backscattering strength S v of the shoals (-93.1, Spearman method, p-value <0.05). Finally, shoals were significantly smaller (smaller length, maximal surface area, and volume) in autumn than in summer (Figure 5).

Shoal number and characteristics for the Celtic Sea area

For the Celtic Sea area, 10.3, 6.8 and 11.0 shoals nmi -1 having a minimum volume of 1 m 3 were detected in spring, summer and autumn 2020, respectively (Figure 2d,e,f). Shoals were mainly located above sand (46 %) and sand mixed with gravel (17 %) in spring 2020 whereas they were mainly located above gravel mixed with stone (59 %) and fine sand (17 %) in summer 2020 and above fine sand (28 %) and rock (19 %) in autumn 2020 (Figure 3b). In addition, 41 % of the detected shoals were observed in depths <10 m in summer compared to 0% in spring and autumn (Supplementary Information, Figure S7b).

According to the AIC criterions and Akaike weights, the chosen models explaining mean acoustic volume backscattering strength S v of the shoals, the percentage of holes and the distance from bottom in percentage included an interaction effect between bathymetry and season (See Table S3 given the AIC values and Akaike weights per models in Supplementary Information). In particular, the mean acoustic volume backscattering strength S v was significantly higher in autumn than in spring and summer but no differences were found between spring and summer (pairwise comparisons using t tests, p-values < 0.05). The models explaining the length, maximal height and maximal width included bathymetry and season without an interaction effect and the models explaining the maximal surface and volume included only the season effect (Table S3).

Shoals in shallower areas were significantly larger (length, width and height) compared to the shoals in deeper areas (Figures S9 to S11). The maximal width, maximal height, length, maximal surface and volumes were all positively correlated (> 55.1, Spearman method, pvalue <0.05). Shoals in shallow waters had also less holes as shown by the percentage of holes which was inversely correlated to the acoustic volume backscattering strength S v of the shoals (-93.2, Spearman method, p-value <0.05). The position in the water column varied with seasons as they were closer to the bottom in deeper areas compared to shallower areas in summer whereas they were closer to the surface in deeper areas in spring and autumn. Finally, shoals were significantly smaller (smaller maximal width, maximal surface area and volume) in autumn than in summer and spring (Figure 5) but no differences were found between summer and spring (pairwise comparisons using t tests, p-values < 0.05). The length and maximal width were also significantly lower in autumn compared to spring and summer and in summer compared to spring for the length whereas it was higher in summer compared to spring for the maximal width (pairwise comparisons using t tests, p-values < 0.05).

Observations of coastal Atlantic mackerels

Fish measurements

Overall, we succeeded to fish in few places (Figure S1 for the localization of the fishing sites on both study sites). A total of 35 Atlantic mackerels (S. scombrus) were caught in the Bay of Biscay area, specifically in the Douarnenez bay, during the summer survey in 2020.

Total lengths ranged from 15.0 cm to 41.5 cm (mean: 28.0 cm and standard-deviation (SD):

5.3) and wet weights ranged from 23.0 to 500.0 g (mean: 212.9 g and SD: 96.6) (Supplementary Information, Figure S12a). Fishing was not successful during the autumn survey.

A total of 56 Atlantic mackerels (S. scombrus) have been caught in the Celtic Sea area, specifically in the Lannion bay, during the summer survey in 2020. Total lengths ranged from 10.8 to 34.3 cm (mean: 24.5 cm and standard-deviation (SD): 6.2) and wet weights ranged from 15.0 to 340.0 g (mean: 152.4 g and SD: 91.6) (Supplementary Information, Figure S12b). The Atlantic mackerels caught in the Celtic Sea area were significantly smaller than the Atlantic mackerels caught in the Bay of Biscay area (t-test, p-value < 0.05). Fishing was not successful during the spring and autumn surveys.

Atlantic mackerels shoals

A total of 19 and 24 shoals were observed in the area where Atlantic mackerels (S.

scombrus) were caught in the Bay of Biscay and Celtic Sea areas, respectively. In the Bay of Biscay area, these shoals were mainly located above fine sand (87 %) and all these shoals were found in depths of 10-20 m (87 %). In the Celtic Sea area, these shoals were also mainly located above fine sand (99 %) and in depths of 10-20 m (97%). Finally, we observed that the shoals were static which facilitated their catch and allowed a relevant species allocation.

Examples of echotraces of presumed Atlantic mackerel shoals can also be visualized in Supplementary Information (Figures S13 andS14). General shape of the echotraces were the same for both study areas. No statistical differences were found between the minimum and maximum width, minimum and maximum height, total length, maximum surface, volume and distance from bottom in percentage between the Atlantic mackerel echotraces found in both study areas (t-test, p-value > 0.05) (see Supplementary Information Figure S15). However, the mean acoustic volume backscattering strength S v was significantly higher and the percentage of holes significantly lower for the echotraces in the Celtic Sea area compared to the Bay of Biscay area (p-value < 0.05), and these two variables were significantly inversely correlated (-93.8, Spearman test, p-value < 0.05).

The Multiple Factor Analysis indicated that the 3-D morphological characteristics of the shoal (maximal height, maximal width, length, surface and volume) fell out along the first axis and represented 16% of the total variance, whereas the variables related to the position of the shoal in the water column and the environmental variables fell out along the second axis and represented 13% of the total variance (Figure 6). All presumed shoals of Atlantic mackerels from both study areas were grouped together and the second axis was more discriminating since the distribution of the data had a low variability on this axis.

DISCUSSION

Pelagic fish shoal descriptors according to environmental gradients

Overall, fish shoals observed on the whole surveys are probably due to different fish species. Indeed, small fish species as European sprats (Sprattus sprattus), smelts (Atherina For both study sites and for all seasons, the mean acoustic volume backscattering strength S v linearly decreased with deeper depths. Similar results was found by Brehmer et al. (2006b) in the Mediterranean Sea showing larger acoustic densities where bathymetry was lowest (< 20m) compared to deeper areas. In addition, shoals were also significantly larger in shallow waters compared to deeper areas as seen with several 3-D shoal morphological variables (maximal width, maximal height, length, maximal surface and volume) depending on the study sites. Furthermore, detections of shoals in shallow areas with a bathymetry under 10 m were important during summer (26 % and 41 % for the Bay of Biscay and Celtic Sea areas respectively). Consequently, these results confirm the importance of surveying pelagic fish populations as they could count for important biomass in shallow waters, which are not surveyed by the conventional research vessels in order to enhance a better management of pelagic fish populations.

Different assumptions could be made to explain the decrease of the fish shoal descriptors with deeper depths. Firstly, the efficiency of which an individual organism scatters sound varies with its acoustic properties (e.g., swim bladder, backbone) as well as with its length, shape and behaviour (Benoit-Bird & Lawson 2016). Hence, species and sizes of the individuals could have varied with bathymetry as the acoustic energy was significantly impacted. In addition, shoal in shallow water areas had higher acoustic energy and a lower percentage of holes compared to deeper areas structures, suggesting that they were denser.

Shoals were also closer to the bottom with increasing depths, especially in the North of Bay of Biscay. Shoal structure could vary to exhibit an anti-predator response, especially during daytime as they could be spotted more easily by predators [START_REF] Gerlotto | Gregariousness and school behaviour of pelagic fish : impact of the acoustics evaluation and fisheries[END_REF][START_REF] Nøttestad | Pretty patterns but a simple strategy: predator-prey interactions between juvenile herring and Atlantic puffins observed with multibeam sonar[END_REF]). Here, the shoals in ultra-shallow water areas could be more vulnerable to visual predators. However, shoal structure could also result from a predatory behaviour. For example, small pelagic fish could form schools for foraging purpose, especially during daytime [START_REF] Gerlotto | Gregariousness and school behaviour of pelagic fish : impact of the acoustics evaluation and fisheries[END_REF][START_REF] Brehmer | Schooling behaviour of small pelagic fish: phenotypic expression of independent stimuli[END_REF]). The influence of the water column depth on the shoal formation was also seen in [START_REF] Kaltenberg | Diel behavior of sardine and anchovy schools in the California Current System[END_REF]. Indeed, in shallow water areas, individuals may less disperse vertically which facilitates organization of individuals into shoals. However, in larger depths, the diel vertical migrations of zooplankton with a pattern of ascent at dusk and descent at dawn could explain that the shoals were mainly observed near the bottom during the day [START_REF] Kaltenberg | Diel behavior of sardine and anchovy schools in the California Current System[END_REF].

Our study focused on pelagic shoals located in the water column, consequently shoals close to the bottom (under 0.3 m from bottom) were not considered. We observed that pelagic shoals were detected above all type of substrates but were predominant over sand and mud.

Preference of fish schools for seabed with finer granulometry such as mud was observed for small pelagic fish species [START_REF] Sulli | Distribution and spatial structure of pelagic fish schools in relation to the nature of the seabed in the Sicily Straits (Central Mediterranean)[END_REF]) and gravel and sand bottoms were also suggested as preferred habitats for shoals in several studies [START_REF] Maravelias | Seabed substrate, water depth and zooplankton as determinants of the prespawning spatial aggregation of North Atlantic herring[END_REF][START_REF] Manik | Quantifying Sea Bottom Surface Backscattering Strength and Identifying Bottom Fish Habitat by Quantitative Echo Sounder[END_REF]). In addition, shoals were found in all bathymetric ranges but numerous shoals were detected in areas with bathymetry under < 10 m only during summer, which could then represent suitable areas during this period. Overall, understanding the combined environmental factors, which structure the fish species distributions, is challenging as species-environmental relationships can be remarkably complex in shallow waters [START_REF] Beck | The Identification, Conservation, and Management of Estuarine and Marine Nurseries for Fish and InvertebratesA better understanding of the habitats that serve as nurseries for marine species and the factors that create site-specific variability in nursery quality will improve conservation and management of these areas[END_REF][START_REF] Staveley | Seascape structure and complexity influence temperate seagrass fish assemblage composition[END_REF]). In the future, the development of more autonomous techniques to directly classify the seabed and analyse fish shoals using multibeam echosounders could improve the understanding of the distribution patterns of fish in different shallow-water habitats [START_REF] Lamouret | Measuring fish activities as additional environmental data during a hydrographic survey with a multi-beam echo sounder[END_REF]. Other environmental variables could influence the fish distribution in shallow waters. Especially, we observed that surface sea water temperature was higher in shallower areas close to the coast compared to deeper areas, particularly during summer in both study areas. Finally, additional data like chlorophyll-a concentration would be relevant to analyse at small-scale to study the pelagic fish distribution [START_REF] Diogoul | On the robustness of an eastern boundary upwelling ecosystem exposed to multiple stressors[END_REF].

Data showed a seasonal variability as the mean morphological characteristics of the shoals (width, height, length, surface and volume) significantly decreased in autumn compared to spring and summer 2020 for both study sites. The motivation to form shoals could vary considerably between species and relies on different reasons (reproduction, feeding, size of the individuals) [START_REF] Freon | Dynamics of Pelagic Fish Distribution and Behaviour: Effects on Fisheries and Stock Assessment[END_REF][START_REF] Pavlov | Patterns and mechanisms of schooling behavior in fish: A review[END_REF] which could vary seasonally. Indeed, at the beginning of the reproductive season, clupeid fishes such as Atlantic herring (Clupea harengus) are known to regularly mass in large shoals for spawning at specific times and locations which could be due to initial conditions on population density and external stimuli such as light level [START_REF] Makris | Critical Population Density Triggers Rapid Formation of Vast Oceanic Fish Shoals[END_REF]. For Atlantic mackerels in Celtic Sea and west of Ireland, [START_REF] Dawson | Changes in Western mackerel (Scomber scombrus) spawning stock composition during the spawning season[END_REF] observed that the size of the individuals in populations decreased throughout the spawning season and, at the end of the spawning season, predominantly young fish remain in the populations. For European sardine in Atlantic, spatiotemporal variability in school morphology was related to length and age of individuals of the populations [START_REF] Muiño R (ramón | Sardine (Sardina pilchardus Walbaum) characterisation off the Spanish Atlantic coast[END_REF].

Changes in the fish communities according to the bathymetry and with the seasons of the surveys could also have occurred as interactions between those two variables were found for several fish shoal descriptors, especially for the mean acoustic volume backscattering strength (S v ). Indeed, several pelagic species are known to exhibit migration patterns. For example, distribution of European sea bass changes seasonally as they migrate between feeding and spawning grounds [START_REF] Quayle | Observations of the Behaviour of European Sea Bass (Dicentrarchus labrax) in the North Sea. In: Tagging and Tracking of Marine Animals with Electronic Devices. Reviews: Methods and Technologies in Fish Biology and Fisheries[END_REF]. Furthermore, migratory species such as anchovies, sprats, sardines, mackerels often exhibit schooling behaviour [START_REF] Pavlov | Patterns and mechanisms of schooling behavior in fish: A review[END_REF] and their migration were suggested to be driven by abiotic parameters like sea surface temperatures [START_REF] Berge | First Records of Atlantic Mackerel (Scomber scombrus) from the Svalbard Archipelago, Norway, with Possible Explanations for the Extensions of Its Distribution[END_REF][START_REF] Diankha | Round sardinella (Sardinella aurita) and anchovy (Engraulis encrasicolus) abundance as related to temperature in the senegalese waters[END_REF][START_REF] Twatwa | Characterising and comparing the spawning habitats of anchovy Engraulis encrasicolus and sardine Sardinops sagax in the southern Benguela upwelling ecosystem[END_REF]. However, the effects of environmental factors on their migration at small local scales, especially in ultrashallow water areas are still poorly understood. Brehmer et al. (2006a) suggested that decrease of temperature and salinity could have influenced the migration of several fish species including D. labrax in two lagoons in the Mediterranean Sea during the autumn migration period. Here, we also observed significant seasonal variations in several abiotic parameters (sea water temperature, salinities and dissolved oxygen). Especially, the dissolved oxygen decreased throughout the seasons for both study sites which could have also impacted the spatial distribution of fish [START_REF] Roman | Interactive Effects of Hypoxia and Temperature on Coastal Pelagic Zooplankton and Fish[END_REF].

Atlantic mackerel shoals in ultra-shallow waters

During the surveys, Atlantic mackerels were observed in both the Bay of Biscay and Celtic Sea areas during summer in ultra-shallow waters. The Atlantic mackerels were concentrated at a narrow range of depth (10-20m) near the coast and above the same substrate (fine sand) for both study areas. They were caught at the end of their spawning period [START_REF] Dawson | Changes in Western mackerel (Scomber scombrus) spawning stock composition during the spawning season[END_REF]) which is consistent as we observed both juvenile Atlantic mackerels (less than 28.7 cm) and adults (higher than 28.7 cm) [START_REF] Froese | [END_REF]. Indeed, Atlantic mackerels are known to spawn from January to July along the continental shelf edge from Portuguese to Scottish waters and in the North Sea. In autumn, less shoal detections were observed in this same areas. This could suggest that the shoals dispersed or move toward deeper areas, or that the populations were lower at this season as mackerels migrate toward the North Sea and Norwegian Sea after the spawning season [START_REF] Reid | Cross-shelf processes north of Scotland in relation to the southerly migration of Western mackerel[END_REF][START_REF] Bruge | Thermal Niche Tracking and Future Distribution of Atlantic Mackerel Spawning in Response to Ocean Warming[END_REF].

Furthermore, the 3-D morphological characteristics of the mackerel shoals were similar between the two study sites as no significant differences were found for the width, height, length, surface and volume of the shoals. All presumed shoals of Atlantic mackerels were grouped together according to the Multiple Factor Analysis highlighting the similarities of the shoals between both study areas. The position of the shoal in the column water as well as the environmental variables were the most discriminant variables. However, Atlantic mackerel shoals were denser in the Celtic Sea area compared to the Bay of Biscay area as the S v was higher and the percentages of holes inside the detections lower. As mackerels were significantly smaller in Celtic Sea than in the North of Bay of Biscay, the higher energetic acoustic values could not be directly related to their size. However, fish abundances could have been larger in Celtic Sea. Another possible explanation is that avoidance reaction to the boat could have been higher in the Celtic Sea areas as the boat was larger and louder compared to the one used in the North of Bay of Biscay [START_REF] Draštík | Fish avoidance of acoustic survey boat in shallow waters[END_REF], Brehmer et al. 2019b). Hence, the same boat should be used for further surveys to avoid this bias. In addition, as longlines used to catch mackerels does not provide exhaustive view of the shoals, the use of alternative methods such as underwater cameras [START_REF] Langlois | Cost-efficient sampling of fish assemblages: comparison of baited video stations and diver video transects[END_REF][START_REF] Mallet | Underwater video techniques for observing coastal marine biodiversity: A review of sixty years of publications (1952-2012)[END_REF], Brehmer et al. 2019a) would be highly interested in shallow waters where visibility is improved compared to larger depth.

Atlantic mackerels are supposed to be highly sensitive to environmental conditions such as sea water temperature [START_REF] Jansen | Migration and Fisheries of North East Atlantic Mackerel (Scomber scombrus) in Autumn and Winter[END_REF][START_REF] Hughes | Changes in the spatial distribution of spawning activity by north-east Atlantic mackerel in warming seas: 1977-2010[END_REF]). In our study, mean sea surface temperature during summer was 16.7°C in the North of Bay of Biscay and 16.0°C in the Celtic Sea respectively which was consistent to the fact that they could be present in temperatures up to 15°C, although they are supposed to prefer areas with temperatures between 9 and 13°C [START_REF] Olafsdottir | Geographical expansion of Northeast Atlantic mackerel (Scomber scombrus) in the Nordic Seas from 2007 to 2016 was primarily driven by stock size and constrained by low temperatures[END_REF]. Observations of mackerel schools during the summer feeding migration between the surface and a depth of 40 m were suggested to be due to temperature preferences as well as their foraging strategy and food availability [START_REF] Godø | Behaviour of mackerel schools during summer feeding migration in the Norwegian Sea, as observed from fishing vessel sonars[END_REF][START_REF] Nøttestad | Feeding strategy of mackerel in the Norwegian Sea relative to currents, temperature, and prey[END_REF]. Especially, shoals were found static in both areas and Atlantic mackerels, as they were supposed to reduce their swimming speed when they reach the most productive area [START_REF] Nøttestad | Feeding strategy of mackerel in the Norwegian Sea relative to currents, temperature, and prey[END_REF]. Overall, our data could help understanding the Atlantic mackerels population dynamics in ultra-shallow waters where they are not traditionally surveyed (Brehmer et al. 2006b) which is important for conservation. Indeed, observations of shifted distributions of the mackerel populations in response to climate change were already observed [START_REF] Berge | First Records of Atlantic Mackerel (Scomber scombrus) from the Svalbard Archipelago, Norway, with Possible Explanations for the Extensions of Its Distribution[END_REF][START_REF] Bruge | Thermal Niche Tracking and Future Distribution of Atlantic Mackerel Spawning in Response to Ocean Warming[END_REF].

CONCLUSION

Overall, our study showed the interest of surveying fish populations in shallow waters areas and especially ultra-shallow waters (< 20m) using a multibeam echosounder. Especially, numerous shoals were detected in ultra-shallow waters and could account for important biomass. Seasonal variations were also observed on the maximal surfaces and volumes of shoals as they significantly decreased in autumn for both study sites. In addition, observations of Atlantic mackerels in ultra-shallow waters in summer were made in both study sites and exhibit similar shoal characteristics, which could be related to their feeding strategy. Finally, it should be noted that the first surveys in both study sites were made when the restrictions due to the COVID-19 pandemic were being lifted. Especially, the lock down may have resulted in reduced anthropogenic pressures and especially fisheries pressure. Hence, a longterm survey over several years would be necessary to investigate if the fish shoal descriptors during this period were similar to other years. Finally, the multibeam echosounder is a workable and relevant tool for setting up long-term surveying of pelagic fish in ultra-shallow waters such as coastal and estuarine areas which are potentially exposed to high anthropogenic pressures. We thus suggest the interest of this method to better assess pelagic fish ecological status and enable conservation management.

ACKNOWLEDGMENTS

This project was funded by the measure 80 of the FEAMP agreement (PFEA800219DM0530003). The authors wish to thank all people who contributed to the surveys including the crew members of the Albert Lucas (Univ Brest, IRD, CNRS) and Neomysis (MNHN). We also thank Gildas Roudaut (IRD, UMR Lemar), Fabrice Roubaud (IRD, US Imago) and Laurent Berger (Ifremer) for their help and the Naval Hydrographic and Oceanographic Service (SHOM) for their data (data.shom.fr). We also thank Eva Russier for her exchange with local fishermen and Gaëtan Richard for his helpful proofreading. We thank administrative supports of the MNHN (Dinard and Paris) and the IRD Delegation Regional

West. Finally, we thank the anonymous reviewers for their helpful suggestions of improvements.

FIGURES AND TABLES

  Insights from a multibeam echosounder to survey pelagic fish shoals and their spatio-temporal distribution in ultra-shallow waters Viviane David, Anne Mouget, Yannick Perrot, Loïc Le Goff, Pierre Thiriet, Ndague Diogoul, Eric Feunteun, Anthony Acou, Patrice Brehmer To cite this version: Viviane David, Anne Mouget, Yannick Perrot, Loïc Le Goff, Pierre Thiriet, et al.. Insights from a multibeam echosounder to survey pelagic fish shoals and their spatio-temporal distribution in ultra-shallow waters. Estuarine, Coastal and Shelf Science, 2021, 264, pp.107705. 10.1016/j.ecss.2021.107705. hal-03831015

  sp.) and sand eels (Ammodytes tobianus) are frequent in the studied areas[START_REF] O'connell | The Biology of the Lesser Sand-Eel Ammodytes tobianus L. In the Galway Bay Area[END_REF][START_REF] Limborg | Genetic population structure of European sprat Sprattus sprattus: differentiation across a steep environmental gradient in a small pelagic fish[END_REF]) and account for important biomass. These small pelagic fish form inshore shoals and are important prey for top predators such as the European shag (Phalacrocorax aristotelis) or the Northern gannet (Morus bassanus)[START_REF] Harris | The Importance of the Lesser Sandeel Ammodytes marinus in the Diet of the Shag Phalacrocorax aristotelis[END_REF][START_REF] Hamer | Foraging ranges, diets and feeding locations of gannets Morus bassanus in the North Sea: evidence from satellite telemetry[END_REF]. In addition, from personal exchanges with local fishermen, European sea bass (Dicentrarchus labrax), Atlantic horse mackerel (Trachurus trachurus), black seabream (Spondyliosoma cantharus), pollack (Pollachius pollachius), Atlantic mackerels, pout whiting (Trisopterus luscus) and Ballan wrasse (Labrus bergylta) are frequently present in these areas in French Britany. In our study, Atlantic mackerels were indeed observed during the summer in both the Bay of Biscay and Celtic Sea areas in shallow waters.

Figure 1 .

 1 Figure 1. Screenshot of the Matecho software (Perrot et al. 2018), an automated tool which allows

Figure 2 .

 2 Figure 2. Detected shoals observed by the multibeam echosounder during the surveys in

Figure 3 .

 3 Figure 3. Frequency of occurrence of detected shoals according to seafloor substrate type in

Figure 4 .

 4 Figure 4. Fish shoal characteristics as a function of the seafloor depth (m) for the shoal

Figure 5 .

 5 Figure 5. Fish shoal characteristics as a function of the seasons. The top panels represent the

Figure 6 .

 6 Figure 6. Multiple Factor Analysis taking into account both quantitative and qualitative data: