Chiên Van

Gérard Bui

Edmond Henry

Duchamp

Hoàn Quoc

Vincel Ngo

Ngoc Hoang

Vu Minh

Nguyen Dinh

Quoc Duchamp

Hoàn

Bui Van Chien

Gérard H E Duchamp
email: gheduchamp@gmail.com

Ngo Quoc Hoan

V Hoang

Ngoc Minh

Nguyen Dinh Vu
email: ndvu@math.ac.vn

Towards a Theory of Domains for Harmonic Functions and its Symbolic Counterpart

Keywords: theory of domains, harmonic sums, polylogarithms 7

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Riemann's zeta function is defined by the series

ζ (s) := ∑ n≥1 1 n s (1)
where s is a complex number. It is absolutely for ℜ(s) > 1 (for any s ∈ C, ℜ(s) stands for the real part of s).

It can be extended to a meromorphic function on the complex plane C with a single pole at s = 1 [30] [START_REF] Costermans | Noncommutative algebra, multiple harmonic sums and applications in discrete probability[END_REF] .). In fact, the story began with Euler's works to find the solution of Basel's problem. In these works, Euler proved that [14]

ζ (2) = ∑ n≥1 1 n 2 = π 2 6 . (2)
Moreover, for any s 1 , s 2 ∈ C such that ℜ(s 1 ) > 1 and ℜ(s 2 ) > 1, Euler gave an important identity as follows [START_REF] Van Chien | Families of eulerian functions involved in regularization of divergent polyzetas[END_REF] :

ζ (s 1 )ζ (s 2 ) = ζ (s 1 , s 2 ) + ζ (s 1 + s 2 ) + ζ (s 2 , s 1 ). (3)
where, for any s 1 , s 2 ∈ C such that ℜ(s 1 ) > 1 and ℜ(s 2 ) > 1,

ζ (s 1 , s 2 ) := ∑ n 1 >n 2 ≥1 1 n s 1 1 n s 2 2 . ( 4
)
The numbers ζ (s 1 , s 2 ) were called "double zeta values" at (s 1 , s 2 ). More generally, for any r ∈ N + and s 1 , . . . , s r ∈ C, we denote

ζ (s 1 , . . . , s r ) := ∑ n 1 >...>n r ≥1 1 n s 1 1 . . . n s r r . (5)
Then the results of K. Matsumoto [16] showed that the series ζ (s 1 , . . . , s r ) converges absolutely for s ∈ H r where H r := {s = (s 1 , . . . , s r ) ∈ C r |∀m = 1, . . . , r; ℜ(s 1 ) + . . . + ℜ(s m ) > m} .

In the convergent cases, ζ (s 1 , . . . , s r ) were called "polyzeta values" at multi-index s = (s 1 , . . . , s r ). Indeed s → ζ (s) is holomorphic on H r and has been extended to C r as a meromorphic function (see [17, 34]).

In fact, for any r-uplet (s 1 , . . . , s r ) ∈ N r + , r ∈ N + , the polyzeta ζ (s 1 , . . . , s r ) is also the limit at z = 1 of the polylogarithmic function, defined by: Li s 1 ,...,s r (z) := ∑ n 1 >...>n r >0 z n 1 n s [START_REF] Berndt | Ramanujan: Letters and Commentary[END_REF] 1 . . . n s r r (7) for any z ∈ C such that | z |< 1. It is easily seen that, for any s i ∈ N + , r > 1,

z d dz Li s 1 ,...,s r (z) = Li s 1 -1,...,s r (z) if s 1 > 1 (8)
(1z) d dz Li 1,s 2 ,...,s r (z) = Li s 2 ,...,s r (z) if r > 1 (9) and this formulas will be ended at the "seed" Li 1 (z) = log 1 1z .

Moreover, if X * is the free monoid of rank two (generators, or alphabet, X = {x 0 , x 1 } and neutral 1 X * ) then the polylogarithms indexed by a list (s 1 , . . . , s r ) ∈ N r + can be reindexed by the word x s 1 -1 0 x 1 . . . x s r -1 0

x 1 ∈ X * x 1 (10) In order to reverse the recursion introduced in Eqns. 8, we introduce two differential forms ω 0 (z) = z -1 dz and ω 1 (z) = (1 -z) -1 dz, (11) on Ω [START_REF] Duchamp | Harmonic sums and polylogarithms at negative multi-indices[END_REF] . We then get an integral representation [START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF] of the functions (7) as follows 11 [20] Li

w (z) =                      1 H (Ω ) if w = 1 X * z 0 ω 1 (s) Li u (s) if w = x 1 u z 1 ω 0 (s) Li u (s) if w = x 0 u and |u| x 1 = 0, i.e.w ∈ x * 0 z 0 ω 0 (s) Li u (s) if w = x 0 u and |u| x 1 > 0, i.e.w / ∈ x * 0 , (12)
the upper bound z belongs to Ω (we recall that

Ω = C \ (] -∞, 0] ∪ [1, +∞[
) is simply connected domain so that the intergrals, which can be proved to be convergent in all cases, depend only on their bounds). The the neutral element of the algebra of analytic functions H (Ω ), a constant function will be here denoted 1 H (Ω ) . This provides not only the analytic continuation of (7) to Ω but also extends the indexation to the whole alphabet X, allowing to study the complete generating series

L(z) = ∑ w∈X * Li w (z)w (13)
and show that it is the solution of the following first order noncommutative differential equation

d(S) = (ω 0 (z)x 0 + ω 1 (z)x 1 )S, (NCDE) lim z∈Ω ,z→0 S(z)e -x 0 log(z) = 1 H (Ω ) X , asymptotic initinial condition, (14)
where, for any S ∈ H (Ω ) X . Through term by term derivation, one gets [13]

d(S) = ∑ w∈X * d dz ( S | w )w. ( 15
)
9 Ω is the simply connected domain

C \ (] -∞, 0] ∪ [1, +∞[).
10 In here, we code the moves z d dz

(resp. (1 -z) d dz ) -or more precisely sections z 0 f (s) s ds (resp. z 0 f (s) 1 -s ds) -with x 0 (resp. x 1 )
. [START_REF] Duchamp | Towards a noncommutative Picard-Vessiot theory[END_REF] Given a word w ∈ X * , we note |w| x 1 the number of occurrences of x 1 within w.

This differential system allows to show that L is a ⊔⊔ -character [START_REF] Deneufchâtel | Independence of Hyperlogarithms over Function Fields via Algebraic Combinatorics[END_REF] [24], i.e.

∀u, v ∈ X * , L | u ⊔⊔ v = L | u L | v and L | 1 X * = 1 H (Ω ) . (16)
Note that, in what precedes, we used the pairing • | • between series and polynomials, classically defined by, for T ∈ k X and P ∈ k X

T | P = ∑ w∈X * T | w P | w , (17)
where, when w is a word, S | w stands for the coefficient of w in S and k any commutative ring (as here H (Ω )). With this at hand, we extend at once the indexation of Li from X * to C X by

Li P := ∑ w∈X * P | w Li w = ∑ n≥0 ∑ |w|=n P | w Li w . (18)
In [10], it has been established that the polylogarithm, well defined locally by (7), could be extended to some series (with conditions) by the last part of formula (18) where the polynomial P is replaced by some series. A complete theory of global domains was presented in [10], the present work concerns the whole project of extending H • [9, 19]. over stuffle subalgebras of rational power series on the alphabet Y , in particular the stars of letters and some explicit combinatorial consequences of this extension.

In fact, we focus on what happens in (well choosen) neighbourhoods of zero (see section 3), therefore, the aim of this work is manyfold. a) Use the extension to local Taylor expansions [START_REF] Drinfrl | On quasitriangular quasi-Hopf algebra and a group closely connected with Gal(Q/Q)[END_REF] as in (7) and the coefficients of their quotients by 1z, namely the harmonic sums, denoted H • and defined, for any w ∈ X * x 1 , as follows [START_REF] Euler | Variae observationes circa series infinitas[END_REF] ([22] see also related literature [4, 19])

Li w (z) 1 -z = ∑ N≥0 H π X (w) (N)z N , (19)
by a suitable theory of local domains which assures to carry over the computation of these Taylor coefficients and preserves the stuffle indentity, again true for polynomials over the alphabet Y = {y n } n≥1 , i.e. [START_REF] Euler | Meditationes circa singulare serierum genus[END_REF]

∀S, T

∈ C Y , H S T = H S H T and H 1 C Y = 1 C N , (20)
note that 1 C Y is identified with 1 Y * and 1 C N is the constant (to one) function [START_REF] Furusho | Desingularization of multiple zetafunctions of generalized Hurwitz-Lerch type[END_REF] N → C. This means that H

• : (C Y , , 1 Y * ) -→ (C{H w } w∈Y * , ×, 1) mapping any word w = y s 1 . . . y s r ∈ Y * to H w = H s 1 ,...,s r = ∑ N≥n 1 >...>n r >0 1 n s 1 1 . . . n s r r , (21)
is a (unital) morphism [START_REF] Goncharov | Multiple polylogarithms and mixed Tate motives[END_REF] . b) Extend these correspondences (i.e. Li • , H • ) to some series (over X and Y , respectively) in order to preserve the identity [START_REF] Hadamard | Théorème sur les séries entières[END_REF] [22] Li π X (S) (z)

1 -z ⊙ Li π X (T ) (z) 1 -z = Li π X (S T ) (z) 1 -z . ( 22
)
true for polynomials S, T ∈ C Y .

To this end, we use the explicit parametrization of the conc-characters obtained in [8, 10] and the fact that, under stuffle products, they form a group.

Polylogarithms: from global to local domains

Now we are facing the following constraint:

In order that the results given by symbolic computation reflect the reality with complex numbers (and analytic functions), we have to introduce some topology [START_REF] Hoffman | Multiple harmonic series[END_REF] . Let H (Ω ) = C ω (Ω ; C) be the algebra (for the pointwise product) of complex -valued functions which are holomorphic on Ω . Endowed with the topology of compact convergence [START_REF] Ngoc | Summations of polylogarithms via evaluation transform[END_REF] , it is a nuclear space [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF] . Definition 1. i) Let S ∈ C X be a series decomposed in its homogeneous (w.r.t. the length) components S n = ∑ |w|=n S | w w (so that S = ∑ n≥0 S n ) is in the domain of Li iff the family (Li S n ) n≥0 is summable in H (Ω ) in other words, due to the fact that the space is complete (see [31]), if one has

(∀W ∈ B H (Ω ) )(∃N)(∀n ≥ N)(∀k)( ∑ n≤ j≤n+k Li S j ∈ W ) . ( 23
)
where B H (Ω ) is the set of neighbourhoods of 0 in H (Ω ).

ii) The set of these series will be noted Dom(Li) and, for S ∈ Dom(Li), the sum ∑ n≥0 Li S n will be noted Li S .

Of course criterium 23 is only a theoretical tool to establish properties of the Domain of Li. In further calculations (i.e. in practice), we will not use it but the stability of the domain under certain operations. [START_REF] Goncharov | Multiple polylogarithms and mixed Tate motives[END_REF] It can be proved that this morphism is into [22]. [START_REF] Hadamard | Théorème sur les séries entières[END_REF] Here ⊙ stands for the Hadamard product [18]. [START_REF] Hoffman | Multiple harmonic series[END_REF] Readers who are not keen on topology or functional analysis may skip the details of this section and hold its conclusions. [START_REF] Ngoc | Summations of polylogarithms via evaluation transform[END_REF] This topology is defined by the seminorms [START_REF] Ngoc | Differential Galois groups and noncommutative generating series of polylogarithms[END_REF] Space where commutatively convergent and absolutely convergent series are the same. This will allow the domain of the polylogarithm to be closed by shuffle products (i.e. the possiblity to compute legal polylogarithms through shuffle products).

p K ( f ) = sup s∈K | f (s)| (K ⊂ Ω is compact).
Example 1 ([20]). For example, the classical polylogarithms: dilogarithm Li 2 , trilogarithm Li 3 , etc... are defined and obtained through the coding (10) by

Li k (z) = ∑ n≥1 z n n k = Li x k-1 0 x 1 (z) = L(z) | x k-1 0 x 1
(where L(z) is as in Eq. 13) but, one can check that, for t ≥ 0 (real), the series

(tx 0 ) * x 1 belongs to Dom(Li • ) (see Def. 1.ii) iff 0 ≤ t < 1. In fact, in this case, Li (tx 0 ) * x 1 (z) = ∑ n≥1 z n n -t .
This opens the door of Hurwitz polyzetas [25].

The map Li • is now extended to a subdomain of C X , called Dom(Li • ) (see also [8, 10]).

Example 2. For any α, β ∈ C, (αx 0 ) * , (β x 1 ) * , and

(αx 0 + β x 1 ) * = (αx 0 ) * ⊔⊔ (β x 1 ) * . We have Li αx * 0 (z) = z α ; Li β x * 1 (z) = (1 -z) -β ; Li (αx 0 +β x 1 ) * (z) = z α (1 -z) -β where z ∈ Ω . Proposition 1. i) The domain Dom(Li) is a shuffle subalgebra of (C X , ⊔⊔ , 1 X * ).
ii) The extended polylogarithm

Li : Dom(Li) → H (Ω )
is a shuffle morphism,i.e. S, T ∈ Dom(Li), we still have

Li S ⊔⊔ T = Li S Li T and Li 1 X * = 1 H (Ω ) (24)
Proof. This proof has been done in [10].

The picture about Dom(Li) within the algebra (C X , ⊔⊔ , 1 X * ), the positioning of C rat X (rational series, see [2, 8, 10]) and shuffle subalegbras as, for example, A = C X ⊔⊔ C rat x 0 ⊔⊔ C rat x 1 read as follows:

Dom(Li) C rat X A C X
3 From Polylogarithms to Harmonic sums Definition of Dom(Li) has many merits [START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF] and can easily be adapted to arbitrary (open and connected) domains. However this definition, based on a global condition of a fixed domain Ω , does not provide a sufficiently clear interpretation of the stable symbolic computations around a point, in particular at z = 0. One needs to consider a sort of "symbolic local germ" worked out explicitely. Indeed, as the harmonic sums (or MZV [START_REF] Ngoc | On the solutions of universal differential equation with three singularities[END_REF] ) are the coefficients of the Taylor expansion at zero of the convergent polylogarithms divided by 1z, we only need to know locally these functions. In order to gain more indexing series and to describe the local situation at zero, we reshape and define a new domain of Li around zero to Dom loc (Li • ). The first step will be to characterize the polylogarithms having a removable singularity at zero The following Proposition helps us characterize their indices.

Proposition 2. Let P ∈ C X and f (z) = L | P = ∑ w∈X * P | w Li w .
1) The following conditions are equivalent i) f can be analytically extended around zero. ii

) P ∈ C X x 1 ⊕ C.1 X * . 2) In this case Ω itself 24 can be extended to Ω 1 = C \ (] -∞, -1] ∪ [1, +∞[).
Proof (Sketch). (ii) =⇒ (i) being straightforward, it remains to prove that (ii) =⇒ (i). Let then P ∈ C X such that f (z) = L | P has a removable singularity at zero. As a consequence of Radford's results [29], one can write down a basis of any free shuffle algebra in terms of Lyndon words. This implies that our polynomial reads

P = ∑ k≥0 α k (P k ⊔⊔ x ⊔⊔ k 0 ) with α k ∈ C, P k ∈ C X x 1 ⊕ C.1 X * (25)
the family (P k ) k≥0 being unique and finitely supported. Using ( 25) and ( 16), we get

Li P (z) = ∑ k≥0 α k Li P k (z) log(z) k
the result now follows easily using asymptotic scale x n log(x) m along the axis ]0, +∞[ (and for x → 0 + ).

The second step will be provided by the following Proposition which says that, for appropriate series, the Taylor coefficients behave nicely. ([S] n are the homogeneous components of S), we suppose that 0 < R ≤ 1 and that

Proposition 3. Let S ∈ C X x 1 ⊕ C1 X * such that S = ∑ n≥0 [S] n where [S] n = ∑ w∈X * ,|w|=n S | w w,
∑ n≥0 Li [S] n is unconditionally convergent (for the standard topol- ogy) within the open disk |z|< R 25 . Remarking that 1 1 -z ∑ n≥0 Li [S] n (z) is unconditionally convergent in the same disk, we set 1 1 -z ∑ n≥0 Li [S] n (z) = ∑ N≥0 a N z N .
Then, for all N ≥ 0,

∑ n≥0 H π Y ([S] n ) (N) = a N .
Proof. Let us recall that, for any w ∈ X * , the function

(1 -z) -1 Li w (z) is analytic in the open disk |z| < R. Moreover, one has 1 1 -z Li w (z) = ∑ N≥0 H π Y (w) (N)z N . Since [S] n = ∑ w∈X * ,|w|=n S | w w and (1 -z) -1 ∑ n≥0 Li [S]
n absolutely converges (for the standard topology [START_REF] Matthes | On the algebraic structure of iterated integrals of quasimodular forms[END_REF] .

) within the open disk D <R , one obtains, for all |z| < R

1 1 -z ∑ n≥0 Li [S] n (z) = 1 1 -z ∑ n≥0 ∑ w∈X * ,|w|=n S | w w Li w (z) = ∑ n≥0 ∑ w∈X * ,|w|=n S | w w Li w (z) 1 -z = ∑ n≥0 ∑ w∈X * ,|w|=n S | w w ∑ N≥0 H π Y (w) (N)z N ( * ) = ∑ N≥0 ∑ n≥0 ∑ w∈X * ,|w|=n S | w wH π Y (w) (N)z N = ∑ N≥0 H π Y ([S] n ) (N)z N .
( * ) being possible because ∑ w∈X * ,|w|=n is finite.

This implies that, for any N ≥ 0,

a N = ∑ n≥0 H π Y ([S] n ) (N).
To prepare the construction of the "symbolic local germ" around zero, let us set, in the same manner as in [8, 10],

Dom R (Li) := {S ∈ C X x 1 ⊕ C1 X * | ∑ n≥0 Li [S] n is unconditionally convergent in H (D <R )} (26)
and prove the following:

Proposition 4. With the notations as above, we have:

1. The map given by R → Dom R (Li) from ]0, 1] to 2 C X (the target is the set of subsets [START_REF] Montel | Lec ¸ons sur les familles normales de fonctions analytiques et leurs applications[END_REF] of C X ordered by inclusion) is strictly decreasing

2. Each Dom R (Li) is a shuffle (unital) subalgebra of C X . Proof. 1. For 0 < R 1 < R 2 ≤ 1 it is straightforward that Dom R 2 (Li) ⊂ Dom R 1 (Li).
Let us prove that the inclusion is strict. Take | z |< 1 and let us, be it finite or infinite, evaluate the sum

M(z) = ∑ n≥0 | Li [S] n (t) (z) |= ∑ n≥0 S(t) | x n 1 | Li x n 1 (z) |
then, by means of Lemma 1, with

x + 1 = x 1 x * 1 = x * 1 -1 and S(t) = ∑ m≥0 t m (x + 1 ) ⊔⊔ m , we have M(z) = ∑ n≥0 | S(t) | x n 1 | Li x n 1 (z) |= ∑ n≥0 ∑ m≥0 |t m (x + 1 ) ⊔⊔ m | x n 1 | Li x n 1 (z) | = ∑ m≥0 m!t m ∑ n≥0 S 2 (n, m) | Li x 1 (z) | n n! ≤ ∑ m≥0 m!t m ∑ n≥0 S 2 (n, m) Li n x 1 (| z |) n! , due to the fact that | Li x 1 (z) |≤ Li x 1 (| z |) (Taylor series with positive coefficients).
Finally, in view of equation ( 29), we get, on the one hand, for

| z |< (t + 1) -1 , M(z) ≤ ∑ m≥0 t m (e Li x 1 (|z|) -1) m = ∑ m≥0 t m ( | z | 1-| z | ) m = 1-| z | 1 -(t + 1) | z | .
This proves that, for all r ∈]0,

1 t + 1 [, ∑ n≥0 Li [S] n (t) (z) r < +∞.
On the other hand, if (t + 1) -1 ≤| z |< 1, one has M(|z|) = +∞, and the preceding calculation shows that, with t choosen such that

0 ≤ 1 R 2 -1 < t < 1 R 1 -1, we have S(t) ∈ Dom R 1 (Li) but S(t) / ∈ Dom R 2 (Li) whence, for 0 < R 1 < R 2 ≤ 1, Dom R 2 (Li) Dom R 1 (Li).

One has (proofs as in

[10]) (a) 1 X * ∈ Dom R (Li) (because 1 X * ∈ C X ) and Li 1 X * = 1 H (Ω ) . (b) Taking S, T ∈ Dom R (Li) we have, by absolute convergence, S ⊔⊔ T ∈ Dom R (Li). It is easily seen that S ⊔⊔ T ∈ C X x 1 ⊕ C1 X * and, moreover, that Li S Li T = Li S ⊔⊔ T 28 .
The combinatorial Lemma needed in the Theorem 1 is as follows [START_REF] Montel | Lec ¸ons sur les familles normales de fonctions analytiques et leurs applications[END_REF] For any set E, the set of its subsets is noted 2 E . [START_REF] Stanley | Enumerative Combinatorics[END_REF] Proof by absolute convergence as in [10].

Lemma 1. For a letter "a", one has

| (a + ) ⊔⊔ m | a n = m!S 2 (n, m) ( 27
∑ n≥0 m!S 2 (n, m) x n n! = (e x -1) m . ( 28
)
Proof. The expression (a + ) ⊔⊔ m is the specialization of

L m = a + 1 ⊔⊔ a + 2
∑ n≥0 m!S 2 (n, m) x n n! = (e x -1) m . (29)
In Theorem 1 below, we study, for series taken in C X x 1 ⊕ C.1 X * , the correspondence Li • to some H (D <R ), first (point 1) establishes its surjectivity (in a certain sense) and then (points 2 and 3) examine the relation between summability of the functions and that of their Taylor coefficients. For that, let us begin with a very general Lemma on sequences of Taylor series which adapts, for our needs, the notion of normal families as in [27]. Then I(τ) = / 0 and R(τ) ≥ R.

Proof. 1. The fact that I(τ) ⊂]0, +∞[ is straightforward from the Definition. If it exists

z 0 ∈ C such that ∑ n,N≥0 | a n,N z N 0 |< +∞ then, for all r ∈]0, |z 0 |[, we have ∑ n,N≥0 | a n,N r N |= ∑ n,N≥0 | a n,N z N 0 | r | z 0 | N ≤ ∑ n,N≥0 |a n,N z N 0 | < +∞
in particular I(τ) = / 0 and it is an interval of ]0, +∞[ with lower bound zero. (a) Take r ∈ I(τ) (hence r = 0) and N ∈ N, then we get the expected result as [START_REF] Riemann | Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie[END_REF] . We also have

r N ∑ n≥0 | a n,N |= ∑ n≥0 | a n,N r N |≤ ∑ n,N≥0 | a n,N r N |< +∞. (b) Again, take any r ∈ I(τ) and n ∈ N, then ∑ N≥0 | a n,N r N |< +∞ which proves that R(T n ) ≥ r, hence the result
| ∑ N≥0 a N r N |≤ ∑ N≥0 r N | ∑ n≥0 a n,N |≤ ∑ n,N≥0
| a n,N r N |< +∞ and this proves that R(T ) ≥ r, hence R(T ) ≥ R(τ). 2. Let 0 < r < r 1 < R and consider the path γ(t) = r 1 e 2iπt , we have

| a n,N |=| 1 2iπ γ T n (z) z N+1 dz |≤ 2π 2π r 1 T n K r N+1 1 ≤ T n K r N 1 with K = γ([0, 2π]), hence ∑ n,N≥0 | a n,N r N |≤ ∑ n,N≥0 | T n | K ( r r 1 ) N ≤ r 1 r 1 -r ∑ n≥0 T n K < +∞.
Remark 1. (i) First point says that every function analytic at zero can be represented around zero as Li S (z) for some S ∈ C x 1 . (ii) In point 2, the arithmetic functions H π Y (S) ∈ Q N , for S ∈ Dom(Li) are quickly defined (and in a way extending the old definition) and we draw a very important bound saying that, in this condition, for some r > 0 the array

H π Y ([S] n ) (N)r N n,N
converges (then, in particular, horizontally and vertically). (iii) Point 3 establishes the converse. [START_REF] Schaefer | Topological Vector Spaces[END_REF] Li S (z)

N -→ H π Y (S) (N) by
1 -z = ∑ N≥0 H π Y (S) (N)z N . ( 32
)
Then,

∀r ∈]0, R[, ∑ n,N≥0 | H π Y ([S] n ) (N)r N |< +∞. ( 33
)
In particular, for all N ∈ N, the series (of complex numbers),

∑ n≥0 H π Y ([S] n ) (N) con- verges absolutely to H π Y (S) (N). 3. Conversely, let Q ∈ C Y with Q = ∑ n≥0 Q n (decomposition by weights), we suppose that it exists r ∈]0, 1] such that ∑ n,N≥0 | H Q n (N)r N |< +∞, ( 34
)
in particular, for all N ∈ N, ∑ n≥0

H Q n (N) = ℓ(N) ∈ C unconditionally. Under such circumstances, π X (Q) ∈ Dom r (Li) and, for all z ∈ C, | z |≤ r, Li S (z) 1 -z = ∑ N≥0 ℓ(N)z N , (35)
Proof. 1. The fact that the series ( 31) is summable comes from the fact that ω(a N (-(-x 1 ) + ) ⊔⊔ N ) ≥ N. Now from the Lemma 1, we get

(S) n = ∑ N≥0 (a N (-(-x 1 ) + ) ⊔⊔ N ) n = (-1) N+n a N N!S 2 (n, N)x n 1 .
Then, with r = sup z∈K |z| (we have indeed r = ||Id|| K ) and taking into account that

Li x 1 K ≤ log(1/(1 -r)), we have ∑ n≥0 Li (S) n K ≤ ∑ n≥0 ∑ N≥0 | a N | N!S 2 (n, N) Li x n 1 K ≤ ∑ n≥0 ∑ N≥0 | a N | N!S 2 (n, N) Li x 1 n K n! ≤ ∑ N≥0 | a N | ∑ n≥0 N!S 2 (n, N) | Li x 1 | n K n! ≤ ∑ N≥0 | a N | (e log( 1 1-r ) -1) N = ∑ N≥0 | a N | r 1 -r N . Now if we suppose that r ≤ (B + 1) -1 , we have r(1 -r) -1 ≤ 1 B
and this shows that the last sum is finite. 2. This point and next point are consequences of Lemma 2. Now, considering the homogeneous decomposition S = ∑ n≥0

[S] n ∈ Dom R (Li). We first establish inequation (33). Let 0 < r < r 1 < R and consider the path γ(t) = r 1 e 2iπt , we have

| H π Y ([S] n ) (N) |=| 1 2iπ γ Li [S] n (z) (1 -z)z N+1 dz |≤ 2π 2π Li [S] n K (1 -r 1 )r N+1 1 , K = γ([0, 1]
) being the circle of center 0 and radius r 1 . Taking into account that, for

K ⊂ compact D <R , we have a decomposition ∑ n∈N | Li [S] n | K = M < +∞, we get ∑ n,N≥0 | H π Y ([S] n ) (N)r N | = ∑ n,N≥0 | H π Y ([S] n ) (N)r N 1 | ( r r 1 ) N = ∑ N≥0 ( r r 1 ) N ∑ n≥0 | H π Y ([S] n ) (N)r N 1 | ≤ ∑ N≥0 ( r r 1 ) N M (1 -r 1 )r 1 ≤ M (1 -r 1 )(r 1 -r) < +∞.
The series

∑ n≥0 Li [S] n (z) converges to Li S (z) in H (D <R ) (D <R is the open disk de- fined by |z| < R).
For any N ≥ 0, by Cauchy's formula, one has,

H π Y (S) (N) = 1 2iπ γ Li S (z) (1 -z)z N+1 dz = 1 2iπ γ ∑ n≥0 Li [S] n (z) (1 -z)z N+1 dz = 1 2iπ ∑ n≥0 γ Li [S] n (z) (1 -z)z N+1 dz = ∑ n≥0 H π Y ([S] n ) (N)
the exchange of sum and integral being due to the compact convergence. The absolute convergence comes from the fact that the convergence of Li S (z)

∑ n≥ Li [S] n (z) is uncon- ditional [31]. 3. Fixing N ∈ N, from inequation (34), we get ∑ n≥0 | H Q n (N) |< +∞ which proves the absolute convergence. Remark now that (π X (Q)) n = π X (Q n ) and π Y (π X (Q n )) = Q n , one has, for all | z |≤ r, | Li π X (Q n ) (z) |=| ∑ N∈N H Q n (N)z N |≤| ∑ N∈N H Q n (N)r N |, in other words Li π X (Q n ) D≤r ≤| ∑ N∈N H Q n (N)r N | and ∑ n∈N Li π X (Q n ) D≤r ≤| ∑ n,N∈N H Q n (N)r N |< +∞ which shows that π X (Q) ∈ Dom r (Li). The equation (35) is a consequence of point 2, taking S = π X (Q).
1 -z = ∑ N≥0 H π Y (S) (N)z N .

Applications

We remark that formula (7), i.e., Li s 1 ,...,s r (z

) := ∑ n 1 >...>n r >0 z n 1 n s 1 1 . . . n s r r
, still makes sense for |z| < 1 and (s 1 , . . . , s r ) ∈ C r so that we will freely use the list indexing to get index lists with s i ∈ Z for any i = 1, . . . , r and r ∈ N + . Recall that for any s 1 , . . . , s r ∈ N, we can present Li -s 1 ,...,-s r (z) as a polynomial of 1 1z with integer coefficients. Then, using (24) and (kx 1 ) * = [(x 1 ) * ] ⊔⊔ k , we get

1 (1 -z) k = Li (kx 1 ) * (z), ∀k ∈ N + , we obtain a polynomial P ∈ Dom(Li) ∩ C[x * 1 ] = C[x * 1 ]
such that Li -s 1 ,...,-s r = Li P (see [10]). Using Theorem 1, we have Li P (z)

1 -z = ∑ N≥0 H π Y (P) (N)z N .
This means that we can provide a class of elements of Dom(H • ) (as in Definition 2) relative to the set of indices of harmonic sums at negative integer multiindices. Here are some examples.

Example 3. For any |z| < 1, we have

Li x * 1 (z) = 1 1 -z ; Li x * 1 -1 X * (z) = z 1 -z = Li 0 (z) ; Li (2x 1 ) * -x * 1 (z) = z (1 -z) 2 = Li -1 (z); Li (2x 1 ) * -2x * 1 +1 X * (z) = z 2 (1 -z) 2 = Li 0,0 (z); Li 12(5x 1 ) * -33(4x 1 ) * +31(3x 1 ) * -11(2x 1 ) * +x * 1 (z) = z 4 + 7z 3 + 4z 2 (1 -z) 5 = Li -2,-1 (z);
Li 40(6x 1 ) * -132(5x 1 ) * +161(4x 1 ) * -87(3x 1 ) * +19(2x 1 ) * -x * 1 (z) =

z 5 + 14z 4 + 21z 3 + 4z 2 (1 -z) 6 = Li -2,-2 (z);
Li 1260(8x 1 ) * -5400(7x 1 ) * +9270(6x 1 ) * -8070(5x 1 ) * +3699(4x

(z) = z 7 + 34z 6 + 133z 5 + 100z 4 + 12z 3 (1 -z) 8 = Li -1,-2,-2 (z).
We can now state the Theorem 2. Let S, T ∈ Dom loc (Li), then S ⊔⊔ T ∈ Dom loc (Li), π X (π Y (S) π Y (T )) ∈ Dom loc (Li) and for all N ≥ 0,

Li S ⊔⊔ T = Li S Li T ; Li 1 X * = 1 H (Ω ) , (37)
H π Y (S) π Y (T ) (N) = H π Y (S) (N)H π Y (T ) (N). (38)
Li S (z)

1 -z ⊙ Li T (z) 1 -z = Li π X (π Y (S) π Y (T )) (z) 1 -z . ( 39
)
Proof. For equation (37), we get, from Lemma 4 that Dom loc (Li) is the union of an increasing set of shuffle subalgebras of C X . It is therefore a shuffle subalgebra of the latter.

For equation (38

), suppose S ∈ Dom R 1 (Li) (resp. T ∈ Dom R 2 (Li))
. By [18] and Theorem 1, one has Li S (z)

1 -z ⊙ Li T (z) 1 -z ∈ Dom R 1 R 2 (Li)
where ⊙ stands for the Hadamard product [18]. Hence, for |z| < R 1 R 2 , one has

f (z) = Li S (z) 1 -z ⊙ Li T (z) 1 -z = ∑ N≥0 H π Y (S) (N)H π Y (T ) (N)z N (40)
and, due to Theorem 1 point (32), for all N,

∑ p≥0 H π Y (S p ) (N) = H π Y (S) (N) and ∑ q≥0 H π Y (T q ) (N) =
H π Y (T ) (N) (absolute convergence) then, as the product of two absolutely convergent series is absolutely convergent (w.r.t. the Cauchy product), one has, for all N,

H π Y (S) (N)H π Y (T ) (N) = ∑ p≥0 H π Y (S p ) (N) ∑ q≥0 H π Y (T q ) (N) = ∑ p,q≥0 H π Y (S p ) (N)H π Y (T q ) (N) = ∑ n≥0 ∑ p+q=n H π Y (S p ) π Y (T q ) (N) = ∑ n≥0 H (π Y (S) π Y (T )) n (N). (41)
Remains to prove that condition of Theorem 1, i.e. inequation (34) is fulfilled. To this end, we use the well-known fact that if ∑ m≥0 c m z m has radius of convergence R > 0, then

∑ m≥0 | c m | z m has the same radius of convergence (use 1/R = lim sup m≥1 | c m | -m ),
then from the fact that S ∈ Dom R 1 (Li) (resp. T ∈ Dom R 2 (Li)), we have (33) for each of them and, using the Hadamard product of these expressions, we get

∀r ∈]0, R 1 .R 2 [, ∑ p,q,N≥0 |H π Y (S p ) (N)H π Y (T q ) (N) r N | < +∞, and this assures, for |z| < R 1 R 2 , the convergence of f (z) = ∑ n,N≥0 H (π Y (S) π Y (T )) n (N)z N (42) applying Theorem 1 point (3) to Q = π Y (S) π Y (T ) (with any r < R 1 R 2 ), we get π X (Q) = π X (π Y (S) π Y (T )) ∈ Dom loc (Li) and f (z) = ∑ N≥0 ∑ n≥0 H (π Y (S) π Y (T )) n (N) z N = Li π X (π Y (S) π Y (T )) (z) 1 -z .
hence we obtain (38).

As regards stuffle, the alphabet is Y = Y N + = {y s } s∈N + and is defined by the following recursion

u 1 Y * = 1 Y * u = u, ( 44
)
y s u y t v = y s (u y t v) + y t (y s u v) + y s+t (u v). ( 45
)
Be it for stuffle or shuffle, the noncommutative 32 polynomials equipped with this product form an associative commutative and unital algebra namely

(C X , ⊔⊔ , 1 X * ) (resp. (C Y , , 1 Y * )). Example 5.
As examples of characters, we have already seen

-Li • from (Dom loc (Li • ), ⊔⊔ , 1 X * ) to H (Ω ) -H • from (Dom(H • ), , 1 Y * ) to C N (arithmetic functions N → C)
In general, a character from a k-algebra 33 (A , * 1 , 1 A ) with values in (B, * 2 , 1 B ) is none other than a morphism between the k-algebras A and a commutative algebra [START_REF] Zhao | Analytic continuation of multiple zeta functions[END_REF] B. The algebra (A , * 1 , 1 A ) does not have to be commutative for example characters of (C X , conc, 1 X * )i.e. conc-characters -where all proved to be of the form

∑ x∈X α x x * (46)
i.e. Kleene stars of the plane [8, 10]. They are closed under shuffle and stuffle and endowed with these laws, they form a group. Expressions like the infinite sum within brackets in (46) (i.e. homogeneous series of degree 1) form a vector space noted C.Y . As a consequence, given P = ∑ (48) [START_REF] Wechsung | Functional Equations of Hyperlogarithms[END_REF] For concatenation. [START_REF] Lewin | Structural properties of polylogarithms[END_REF] Here we will use k = Q or C. [START_REF] Zhao | Analytic continuation of multiple zeta functions[END_REF] In this context all algebras are associative and unital. 35 Its inverse will be naturally noted π Umbra q .

This shows that if one sets, for z ∈ C and T ∈ C + [[x]], G(z) = (π Umbra Y (e zT -1)) * , we get a one-parameter stuffle group36 such that every coefficient is polynomial in z. Differentiating it we get

Conclusion

Noncommutative symbolic calculus allows to get identities easy to check and to implement. With some amount of complex and functional analysis, it is possible to bridge the gap between symbolic, functional and number theoretic worlds. This was the case already for polylogarithms, harmonic sums and polyzetas. This is the project of this paper and will be pursued in forthcoming works.

Lemma 2 . 30 )

 230 Let τ = (a n,N ) n,N≥0 be a double sequence of complex numbers. Setting I(τ) := {r ∈]0, +∞[| ∑ n,N≥0 |a n,N r N | < +∞}, one has 1. I(τ) is an interval of ]0, +∞[, it is not empty iff there exists z 0 ∈ C \ {0} such that ∑ n,N≥0 |a n,N z N 0 | < +∞ (In this case, we set R(τ) := sup(I(τ)), one has (a) For all N, the series ∑ n≥0 a n,N converges absolutely (in C). Let us note a N -with one subscript -its limit (b) For all n, the convergence radius of the Taylor series T n (z) = ∑ N≥0 a n,N z N is at least R(τ) and ∑ n∈N T n is summable for the standard topology of H (D <R(τ) ) with sum T (z) = ∑ n,N≥0 a N z N . 2. Conversely, we suppose that it exists R > 0 such that (a) Each Taylor series T n (z) = ∑ N≥0 a n,N z N converges in H (D <R ). (b) The series ∑ n∈N T n converges unconditionnally in H (D <R ).

Theorem 1 . 1 . 1 n

 111 Let T (z) = ∑ N≥0 a N z N be a Taylor series converging on some nonempty disk centered at zero i.e. such that lim sup N→+∞ | a N | = B < +∞, then the series S = ∑ N≥0 a N (-(-x 1 ) + ) ⊔⊔ N (31) is summable in C X (with sum in C x 1 ), S ∈ Dom R (Li) with R = (B + 1) -1 and Li S = T . 2. Let S ∈ Dom R (Li) and S = ∑ n≥0 [S] n (homogeneous decomposition), we define

Definition 2 .

 2 Now, we have have a better understanding of what can (and will) be the domain, Dom(H • ), of harmonic sums. We set Dom loc (Li) = 0<R≤1 Dom R (Li); Dom(H • ) = π Y (Dom loc (Li)) and, for S ∈ Dom loc (Li), Li S (z) = ∑ n≥0 Li [S] n (z) and

i≥1α

  i y i and Q = ∑ j≥1 β j y j , we know in advance that their stuffle is a conc-character i.e. of the form ( ∑ n≥1 c n y n ) * . Examining the effect of this stuffle on each letter (which suffices), we get the identity∑ i≥1 α i y i * ∑ j≥1 β j y j * = ∑ i≥1 α i y i + ∑ j≥1 β j y j + ∑ i, j≥1 α i β j y i+ j * (47)This suggests to take an auxiliary variable, say q, and code "the plane" C.Y , i.e. expressions like (46), in the style of Umbral calculus by π Umbra Y : ∑ n≥1 α n q n -→ ∑ n≥1 α n y n which is linear and bijective35 from C + [[q]] to C.Y . With this coding at hand and for S, T ∈ C + [[q]], identity (47) reads (π Umbra Y (S)) * (π Umbra Y (T )) * = (π Umbra Y ((1 + S)(1 + T) -1)) * .

  with the initial condition G(0) = 1 Y * integrates asG(z) = exp (zπ Umbra Y (T ))(50)where the exponential map for the stuffle product is defined, for anyP ∈ C Y such that P | 1 Y * = 0,is defined by exp (P) := 1 Y * + In particular, from (50), one gets, for k ≥ 1, (zy k ) * = exp -∑ n≥1 y nk (-z) n n .

  1 ) * -829(3x 1 ) * +71(2x 1 )

				-x * 1 (z)
	=	z 7 + 64z 6 + 424z 5 + 584z 4 + 179z 3 + 8z 2 (1 -z) 8	= Li -3,-3 (z);
	Li 10(6x 1 ) * -38(5x 1 ) * +55(4x 1 ) * -37(3x 1 ) * +11(2x 1 ) * -x * 1 (z) =	z 5 + 6z 4 + 3z 3 (1 -z) 6	= Li -1,0,-2 (z);
	Li 280(8x 1 ) * -1312(7x 1 ) * +2497(6x 1 ) * -2457(5x 1 ) * +1310(4x 1 ) * -358(3x 1 ) * +41(2x 1 ) * -x * 1

*

This research of Ngo Quoc Hoan is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2021.41

Here, the shuffle product is denoted by ⊔⊔ . It will be redefined in the section 5 .

Around zero.

Here, the conc-morphism π X :(C Y , conc, 1 Y * ) → (C X , conc, 1 X * ) is defined by π X (y n ) = x n-10 x 1 and π Y is its inverse on Im(π X ). See[START_REF] Van Chien | Families of eulerian functions involved in regularization of divergent polyzetas[END_REF][START_REF] Duchamp | Kleene stars of the plane, polylogarithms and symmetries[END_REF] for more details and a full Defi- nition of π Y .

[START_REF] Euler | Meditationes circa singulare serierum genus[END_REF] Here, stands for the stuffle product which will be recalled as in the section

5.[START_REF] Furusho | Desingularization of multiple zetafunctions of generalized Hurwitz-Lerch type[END_REF] In fact, it could be Q but we will use afterwards C-linear combinations.

As the fact that, due to special properties of H (Ω ) (it is a nuclear space[START_REF] Schaefer | Topological Vector Spaces[END_REF], see details in[START_REF] Van Chien | Families of eulerian functions involved in regularization of divergent polyzetas[END_REF]), one can show that Dom(Li) is closed by shuffle products.

Multiple Zeta Values.

The domain, for z of Li P .

With the definition given later[START_REF] Matthes | On the algebraic structure of iterated integrals of quasimodular forms[END_REF] this amounts to say that S ∈ C X x 1 ⊕ C1 X * ∩ Dom R (Li) .

For this topology, unconditional and absolute convergence coincide[START_REF] Schaefer | Topological Vector Spaces[END_REF].

See[START_REF] Stanley | Enumerative Combinatorics[END_REF], the twelvefold way, formula (1.94b)(pp. 74) for instance.

For a Taylor series T , we note R(T ) the radius of convergence of T .

i.e. G(z 1 + z 2 ) = G(z 1 ) G(z 2 ); G(0) = 1 Y * .

Thus, for any N ∈ N, for readability, below 1 stands for 1

n 0 2 ;

H π Y (12(5x 1 ) * -33(4x 1 ) * +31(3x 1 ) * -11(2x 1 ) * +x * 1 ) (N) =

H π Y (1260(8x 1 ) * -5400(7x 1 ) * +9270(6x 1 ) * -8070(5x 1 ) * +3699(4x 1 ) * -829(3x

Observe that, from this Definition, Theorem 2 will show us that Dom(H • ) is a stuffle subalgebra of C Y . Let us however remark that some series are not in this domain as shown below

However one can get unconditional convergence using a sommation by pairs (odd + even).

(ii) For all s ∈]1, +∞[, the series

Recall that, as in Example 3, for any s 1 , . . . , s r ∈ N, we can find an elements P ∈ Dom(Li) such that Li P (z)

Theorem 2 proves that H is a stuffle character on Dom(H). Then for any mixed multiindices s, we can find the elements

Example 4.

5 Some remarks about stuffle product and stuffle characters and their symbolic computations.

For the some reader's convenience, we recall here the Definitions of shuffle and stuffle products. As regards shuffle, the alphabet X is arbitrary and ⊔⊔ is defined by the following recursion (for a, b ∈ X and u, v ∈ X * ) u ⊔⊔ 1 X * = 1 X * ⊔⊔ u = u; au ⊔⊔ bv = a(u ⊔⊔ bv) + b(au ⊔⊔ v).

(43)


Celine Gaullier

Nicole Baran

Sylvie Dousset
email: sylvie.dousset@univ-lorraine.fr

Nicolas Devau

David Billet

Geraldine Kitzinger

Emeline Coisy

Céline Gaullier

Géraldine Kitzinger

Wetland hydrodynamics and mitigation of pesticides and their metabolites at pilot-scale

Keywords: constructed wetland, flow rate, hydraulic performance, pesticide, remediation, water tracer

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In order to protect and improve water quality [START_REF] Joce | the Council of 23 October 2000 establishing[END_REF], the use of Free Water Surface Constructed Wetlands (FWS CWs) with tile drains at about 1-m depth has emerged as a costeffective way of reducing contaminant loading from drainage waters [START_REF] Schultz | Design and placement of a multi-species riparian buffer strip system[END_REF][START_REF] Moore | Transport and fate of atrazine and lambda-cyhalothrin in an agricultural drainage ditch in the Mississippi Delta, USA. Agriculture[END_REF][START_REF] Tournebize | Codesign of constructed wetlands to mitigate pesticide pollution in a drained catch-basin: a solution to improve groundwater quality[END_REF]Vallée et al., 2015b). Their effectiveness in mitigating pesticide concentrations has been demonstrated in several studies [START_REF] Moore | Transport and fate of atrazine and lambda-cyhalothrin in an agricultural drainage ditch in the Mississippi Delta, USA. Agriculture[END_REF]Wilson et al., 2011;[START_REF] Elsaesser | Assessing the influence of vegetation on reduction of pesticide concentration in experimental surface flow constructed wetlands: Application of the toxic units approach[END_REF], but locally negative mitigation rates have been reported as well depending on molecules and hydrological year [START_REF] Passeport | Pesticide contamination interception strategy and removal efficiency in forest buffer and artificial wetland in a tile-drained agricultural watershed[END_REF]Maillard and Imfeld, 2014;Vallée et al., 2015b). A review of 47 studies covering 87 reveals average removals varying from 97 to 24% according to pesticide physicochemical parameters (Vymazal and Březinová, 2015). For both controlled and uncontrolled flow in CW, average retention efficiencies for pesticide are similar (32-39%) but the performance is highly variable dependent on hydrological conditions and seasons [START_REF] Tournebize | Implications for constructed wetlands to mitigate nitrate and pesticide pollution in agricultural drained watersheds[END_REF]. Indeed, several parameters are known to affect the hydraulic effectiveness of CWs, such as (i) the length-towidth ratio (LWR) of the wetland, and (ii) the hydraulic residence time (HRT), flow rate and water depth, but the influence of these hydraulic parameters on pesticide mitigation has rarely been studied [START_REF] Blankenberg | Pesticide retention in an experimental wetland treating non-point source pollution from agriculture runoff[END_REF][START_REF] Boutron | Effect of the submergence, the bed form geometry, and the speed of the surface water flow on the mitigation of pesticides in agricultural ditches: mitigation of pesticides in agricultural ditches[END_REF][START_REF] Elsaesser | Assessing the influence of vegetation on reduction of pesticide concentration in experimental surface flow constructed wetlands: Application of the toxic units approach[END_REF][START_REF] Haarstad | Pesticide retention in the watershed and in a small constructed wetland treating diffuse pollution[END_REF]Vallée et al., 2015a).

HRT is considered as one of the main factors influencing the mitigation effectiveness of CWs [START_REF] Stehle | Pesticide Risk Mitigation by Vegetated Treatment Systems: A Meta-Analysis[END_REF]Vyzamal and Březinová, 2015;[START_REF] Guo | Test study of the optimal design for hydraulic performance and treatment performance of free water surface flow constructed wetland[END_REF]; it is mainly controlled by the flow path through the wetland, inflow rate, and water depth. From field experiments, the influence of HRT on pesticide mitigation is not very clear (Maillard and Imfeld, 2014;Vallée et al., 2015b), but some studies carried out under controlled conditions reported an increased effectiveness with a longer HRT [START_REF] Elsaesser | Assessing the influence of vegetation on reduction of pesticide concentration in experimental surface flow constructed wetlands: Application of the toxic units approach[END_REF]Vallée et al., 2015a). These authors tested HRTs ranging from 4 hours to 7.4 days, corresponding to inflow rates ranging from 3.2 L.s -1 to 0.05 L.s -1 [START_REF] Holland | Effects of wetland depth and flow rate on residence time distribution characteristics[END_REF][START_REF] Elsaesser | Assessing the influence of vegetation on reduction of pesticide concentration in experimental surface flow constructed wetlands: Application of the toxic units approach[END_REF]Vallée et al., 2015a;[START_REF] Maillard | Dissipation of hydrological tracers and the herbicide S-metolachlor in batch and continuous-flow wetlands[END_REF], which can be assimilated to low flow rates in the field, where flow rates can range from 0.01 to 20 L/s [START_REF] Larson | The role of seepage in constructed wetlands receiving agricultural tile drainage[END_REF]Vallée et al., 2015b).

However, the effectiveness of high flow rates coupled with very short HRTs (a few hours) was rarely studied under controlled conditions, even though they may represent about 50% of drainage flow over one year in the field [START_REF] Tournebize | Pesticide de-contamination of surface waters as a wetland ecosystem service in agricultural landscapes[END_REF]Vallée et al., 2015b). In addition, flow rate and water depth both affect wetland hydraulics [START_REF] Holland | Effects of wetland depth and flow rate on residence time distribution characteristics[END_REF]) by changing the effective volume that can be used for pollutant mitigation, and the length-towidth ratio (LWR) seems to be a major factor in determining the effectiveness of a CW by controlling flow uniformity [START_REF] Thackston | Residence time distributions of shallow basins[END_REF][START_REF] Su | Optimal design for hydraulic efficiency performance of free-water-surface constructed wetlands[END_REF]; these authors showed that the mitigation effectiveness of a CW is considered as satisfactory for a LWR between 1.8 and 10.

Our study is one of the first to investigate the influence of highly contrasted flow rates on the flow patterns of hydraulic zones in CWs, and their impact on pesticide mitigation. For that purpose, laboratory experiments were carried out on two pilot-scale wetlands with different designs under variable hydrodynamic conditions, studying four pesticides-boscalid (BSC), cyproconazole (CYP), isoproturon (IPU) and dimethachlor (DMT)-with contrasted polarities. Such a controlled small-scale approach, compared to an in-situ study, is required for understanding the parameters and processes controlling pesticide mitigation in CWs. The experiments were conducted by associating two conservative tracers (chloride and bromide) with the four pesticides during a drainage period with alternating high-, low-and zero flow.

Material and Methods

Two laboratory pilot installations, both mimicking field CWs, were constructed in polyvinylchloride. Soils and sediments were sampled in field CWs, dried for one week at 24 °C, crushed and sieved at 2 mm, and then homogenized. Their main characteristics are provided in Table A.1. The first pilot reproduced a ditch-shaped CW at Manoncourt-sur-Seille (France) at 1:40 scale, using 22 kg of soil from this site deposited with a thickness of 3 cm and covered with 360 g of sediment over 0.5 cm thick. It was 225 cm long and 2.5 cm wide, with an enlarged section of 25 x 6.5 cm. The vegetation, grown from seeds in the soil, was dominated by Juncus effusus, J. bufonius and grasses on the edges of the pilot that not affected flow) (Fig. 1).

The second pilot simulated a CW formed by a succession of three ponds at Ville-sur-Illon (France) at 1:20 scale, using 55 kg of soil from the site deposited with a thickness of 5 cm and covered with 2 kg of sediment over 0.5 cm thick. It was 235 cm long and 20-30 cm wide, depending on the ponds, with an island (15 x 7 cm) at the beginning of the second pond, as in the field. The vegetation, again developed from seeds in the soil, was dominated by Juncus effusus, J. inflexus and J. articulatus on the pond edges (not affecting flow), and by Typha latifolia and Veronica becabunga inside the ponds (Fig. 1). The detailed dimensions of the pilots are given in Table A.2. Both ditch and pond CWs were chosen for their contrasted design and their simplicity of laboratory implementation, in order to reproduce their use for large-scale pesticide mitigation.

Before running the experiments over a 62-day period, both pilots were left during 8 months for stabilization and plant growth, and were watered by tap water. Operating conditions in the phytotronic room were 20±2 °C, with a day/night ratio of 14/10 h, a hygrometry of 70% and a brightness of 3100 to 3800 lx.

Selected pesticides

Four pesticides were selected on the basis of their actual field application, their detection in the water of existing CWs, and the diversity of their polarities (Octanol-water partition coefficient from 148 to 1230) or sorption coefficient KOC from 69 to 772 L.kg -1 ). The selected pesticides were the two fungicides boscalid

(2-chloro-N-(4'-chloro[1,1'-biphenyl]-2- yl)pyridine-3-carboxamide) and cyproconazole (α-(4-chlorophenyl)-α-(1-cyclopropylethyl)- 1H-1,2,4-triazole-1-ethanol), and the two herbicides isoproturon (N,N-dimethyl-N'-(4-(1- methylethyl)phenyl)urea) and dimethachlor (2-chloro-N-(2,6-dimethylphenyl)-N-(2- methoxyethyl)acetamide).
Their main properties are given in Table A.3. The analytical standards of these pesticides were purchased from Cluzeau (Sainte-Foy-La-Grande, France).

All four were analytical standard grade (>98 %) and are used both for spiking and confirmation. The analytical standards of the IPU and DMT metabolites (see Table A.4) were purchased from Restek (Lisses, France), Sigma (Lyon, France), Neochema (Bodenheim, Germany) and Techlab (Metz, France).

Simulated drainage, stagnation and tracing

In order to mimic real conditions, the experimental flow rates were determined based on in situ observations. In the field, flow-rate monitoring (with an ultra-sound flowmeter, 950-US 50 Hz, SIGMA) over one year indicated that 90% of the flow values varied from 0 to 4 L.s -1 and 10% varied from 4 to 10 L.s -1 for the ditch (D) CW. For the pond (P) CW, 70% of the flow values varied from 0 to 5 L.s -1 , 10% varied from 5 to 10 L.s -1 and 20% varied from 10 to 18 L.s -1 . In addition, a stagnation period (zero flow) also occurred after a drainage event.

Based on these field observations, we applied two distinct inflow rates to each pilot, calculated according to the pilot scales of 1:40 and 1:20. The high inflow rates were 7.0 mL.min -1 for the ditch pilot (D-HF) and 55.5 mL.min -1 for the pond pilot (P-HF), corresponding to field scales of 7.5 and 7.4 L.s -1 . These values are representative of the high and medium inflow rates defined for ditch CW and pond CW, respectively. Thus, for the high flow rate, the nominal HRTs (i.e. the wetland-volume/flow-rate ratio) were 4.0 and 2.8 h for ditch pilot and pond pilot, respectively. This agrees with the HRTs measured in the field at high flow (about 30 min to 6 h). The low inflow rates were 1.8 and 8.0 mL.min -1 for ditch pilot (D-LF) and pond pilot (P-LF), respectively, corresponding to field values of 1.9 and 1.1 L.s -1 , respectively. The nominal HRTs were 11.0 and 9.8 h for the ditch and pond pilots, respectively; these values have the same order of magnitude as the HRTs measured in the field at low flow (about 8 to 60 h). A drainage event was simulated during 12 h for a high inflow rate and during 36 h for a low inflow one, corresponding for each flow rate condition to at least three nominal HRTs in the pilot. Subsequently, a stagnation period (no-flow) was simulated for 50 days to represent a long in-situ period of stagnation (i.e. almost three months in summer). During this step, to balance evapotranspiration, the water depth was maintained at 1 cm and 1.5 cm, respectively for the ditch and the pond pilots, by adding distilled water twice a day.

Drainage inflow was injected continuously through Teflon tubing (AJD00017, Saint Gobain, France) using a peristaltic pump (Ismatec IPS-16). The water inlet and outlet were located in the middle of each pilot. The simulated drainage was divided into eight consecutive charge and discharge steps (S1 to S8) (Table 1). BSC, CYP, IPU and DMT were injected with a concentration of 200 µg.L -1 at the same time as water-tracer ions at 11.5±1.5 mg.L -1 during the S1 and S3 charge steps, and distilled water was injected during the S2, S4, S5 and S8 discharge steps. The tracers used were chloride (CaCl2) during the S1 step (high flow) and bromide (KBr) during the S3 step (low flow), for determining the mean (real) HRT and for measuring the hydraulic efficiency. No loss of ions and pesticides was observed in the inlet solution. Moreover, the first high inflow charge (S1) was done in a half-full pilot, i.e. in the field the CW was not dry when drainage started.

In addition, a brilliant blue (BB) dye tracer at 0.05 mg.L -1 was continuously injected after discharge S8. Pictures were automatically taking each minute for D-LF, and each 30 sec for D-HF, P-HF and P-LF, to monitor the water pathways.

Sampling strategy

During the drainage periods (S1 to S5 and S8), outlet effluents were continuously collected through a Teflon tube (AJD00017, Saint Gobain, France) with a peristaltic pump (Ismatec IPS-16); they were stored in polyethylene flasks using a fraction collector (Foxy 200 TM ). At high flow (S1-S2-S4), automatic samples were taken every 15 min during the first 3 h and then every hour for the following 9 h. At low flow (S3-S5-S8), automatic samples were taken every 30 min during the first 6 h and then every 3 h for the following 30 h, and over the next 6 days for the S8 steps. All samples were stored less than 12 h in the polyethylene flasks (pesticide loss from 1 to 5% in the flasks, determined in triplicate at 200 µg.L -1 ) and then stored in glass tubes. Moreover, all samples were individually analysed within 24 hours.

In addition, 5 mL water samples were collected manually inside the pilots in two specific zones (SZ) (Fig. 1), every 3 h at high flow (S1-S2-S4), and every 9 h at low flow (S3-S5-S8) for the two pilots (4 samples). During the S7 stagnation period, composite (6 x 1 mL) samples representative for the pilot were taken in the same place, daily for the first 10 days, and then after 12,14,16,18,20,26,32,38,44, 50 days of stagnation.

Finally, at the end of the experiment, after emptying the pilots, soil and plants were freeze-dried, manually crushed and homogenized, before bromide and pesticide extraction and analysis.

Pesticide analyses

For each water sample, 2 mL was placed in a PP 2 mL centrifuge tube and then centrifuged at 16,873 g for 5 min to eliminate any suspended particles (i.e. to investigate the mitigation on the dissolved phase only). Then, a 1 mL aliquot was transferred to an amber glass vial for pesticide analysis within 24 h after sampling (Method A). Finally, part of the centrifuged sample was frozen at -20 °C to analyse for IPU and DMT metabolites (methods B and C) and analysis within 2 months.

Method A was performed by liquid chromatography (LC) using an Ultimate 3000

RSLC system and a Diode Array Detector [START_REF] Gaullier | Is pesticide sorption by constructed wetland sediments governed by water level and water dynamics?[END_REF]

Ionic tracer analysis

Each water sample was filtered through a 0.22 µm cellulose acetate filter and then a 1 mL aliquot was transferred to a polypropylene vial for analysis with a Dionex ICS-3000 ionchromatographic instrument (Sunnyvale, CA, USA), using Chromeleon® software (Version 6.80) to determine Cl -and Br -concentrations. The analytical method is described in the Appendices. In addition, at the end of the experiment the bromide tracer was extracted from 10 g of sediment with a CaCl2 solution (10 mL -10 -4 M) and from 0.8 g of plant material with distilled water (80 mL), by stirring for 40 min, and then analysed.

Data analysis 2.7.1. Water transport parameters

The residence time distribution (RTD) function, E(t), is defined mathematically such that E(t)×dt is the fraction, at the outlet, of fluid that has spent between t and t+dt time in the flow system [START_REF] Adeosun | Numerical and experimental studies of mixing characteristics in a T-junction microchannel using residence-time distribution[END_REF]. In order to compare the RTDs for our four studied conditions (D-HF, D-LF, P-HF and P-LF), each RTD must be normalized. The normalized hydraulic retention time (HRT) (tϴ), was defined as the ratio between the mean HRT (tmean)

and the nominal HRT (tn) [START_REF] Fan | Application of computational fluid dynamic to model the hydraulic performance of subsurface flow wetlands[END_REF]. In order to estimate the hydraulic performance of the pilots, the normalized HRT (tϴ), the normalized variance (σϴ), the number of cells (N) and the hydraulic efficiency (λ) were determined. The detailed calculation is given in the Appendices. The hydraulic performance is considered as good when all these parameters are optimized, such as when the water is transported from inlet to outlet throughout the entire volume system, limiting short-circuits.

Restitution of tracer ion and pesticide mitigation rate

The restitution percentage of each tracer was calculated according to Equation (1):

= × 100 (1)
where Qin is the total amount injected (mg) and Qout is the total amount outlet (mg) defined by Equation (2).

= ∑

(2)

where C(t) (mg.L -1 ) is the outlet concentration at each time step and V(t) (L) is the outlet volume at each time step.

The effectiveness (%) of the pilots to mitigate pesticides was also calculated with Eq.

(1) and ( 2), replacing the ion tracer by pesticides mass.

Dissipation and degradation of pesticides

Pesticide dissipation kinetics in water during the stagnation period were described using a second-order model before calculating the associated half-live DT50 was calculated (see Appendices).

The dissipation of pesticides can partly be due to their degradation. The rate of degradation (ratio between the molar concentrations of metabolites and parent molecule) was estimated for each metabolite during (i) the drainage period and ii) the stagnation period.

Statistical analysis

First, the hydraulic variables used for defining the drainage period in the two pilot systems were converted into categorical (discrete) variables. The semi-quantitative values measured for the volume-and hydraulic-efficiency-variables were divided into four ranks; another three ranks were defined for the flow-rate variable, and HRT and the LTW values were aggregated into two more ranks (Table A.5). The ranking groups were defined beforehand.

Of the variables used for describing the pesticide properties (DT50 water, Kd, Pmax, discharge) except the Teq variable, the quantitative values were also transformed into categorical (discrete) variables. To this end, a unimodal distribution was calculated for each variable after centring and scaling all data. After that, first and third quartiles as well as the median were defined. Based on these values, four ranks were defined for each variable. The first corresponds to values below the first quartile, the second is for values ranging between the first quartile and the median, the third rank consists of values ranging between the median and the third quartile, and the fourth rank aggregates the values above the third quartile. For the Teq variable, five ranks are considered.

Multiple correspondence analysis (MCA) then ordered the 16 observations according to the hydraulic and pesticide variables used for their definition. The Φ 2 distance-based matrix was calculated from re-organized data according to a Burt table. The results were displayed as a symmetric plot showing both variables and observations along the first two principal axes.

Based on the Φ 2 distance-based matrix associated with Ward's method for aggregating data, an agglomerative hierarchical principal-component clustering (HCPC) was calculated and the first four principal components then were used for constructing the associated dendrogram.

The dendrogram tree was then cut into several groups based on the higher relative loss of inertia criteria, allowing the aggregation of observations into classes.

Finally, a partial least squares discriminant analysis (PLS-DA) discriminated observations among the groups defined from the HCPC analysis. To this end, multivariate analysis computed the best discriminant functions for differentiating objects in groups while maximizing variance between treatments, though minimizing intra-variance for which the first four main axes from the MCA calculations were used. This analysis is based on a partial least squares regression algorithm that searches for latent variables with maximum covariance and represents the relevant sources of data variability with linear combinations of the original variables [START_REF] Ballabio | Classification tools in chemistry. Part 1: linear models[END_REF]. The results were presented as a correlation circle of variables, distributing the sixteen observations along the two discriminant axes. A confusion matrix, comparing the a priori (real) and a posteriori (calculated) classification of the observations, was calculated as well with the cross-validation technique. All statistical tests were carried out using FactoMineR and ade4 R-software.

Results and discussion

Influence of pilot design parameters and flow rate on hydrodynamic

Table 2 shows the water-transport parameters derived from RTD data. For high flow (HF) or low flow (LF), both ditch and pond have quite similar tmean and vmean values, indicating that water-residence time and water velocity were roughly the same for both pilots. However, σ²ϴ was lower for ditch than pond and N was higher, indicating that the ditch behaved more like a plug-flow, whereas the pond behaved as a mixing system [START_REF] Fan | Application of computational fluid dynamic to model the hydraulic performance of subsurface flow wetlands[END_REF]. In addition, σ was greater for pond than ditch, indicating a greater dispersion of water in the pond compared to the ditch, probably toward the borders, due to its larger width. Moreover, the hydraulic efficiency (λ) was higher for the ditch than for the pond. Thus, all parameters indicated that the hydraulic performance was higher for ditch than pond, regardless of flow rate. This could be due to the morphology and dimensions of both pilots: as the length-to-width ratio for ditch is five times higher than for pond, an optimal water-flow path is induced in the ditch pilot as suggested by other authors [START_REF] Thackston | Residence time distributions of shallow basins[END_REF][START_REF] Haarstad | Pesticide retention in the watershed and in a small constructed wetland treating diffuse pollution[END_REF][START_REF] Su | Optimal design for hydraulic efficiency performance of free-water-surface constructed wetlands[END_REF].

For both pilots, the tmean at LF (22 h) is higher than that at HF (4-5 h), indicating that the water spends a longer time in the pilot when the flow rate is lower (Table 2). In addition, for both pilots, the hydraulic efficiency λ and the effective volume (ev) were higher at LF than at HF. Moreover, ev was >1, probably indicating a storage zone in both pilots, especially at LF and probably due to greater dispersion of water. This assumption was confirmed by (i) a greater σ at LF than at HF, (ii) a more dispersed progression front of the BB dye at LF than at HF, and (iii) similar ionic-tracer concentrations at the outlets and in specific zones at HF, which are different at LF (Fig.

A.1). Thus, water-transport parameters indicated that the hydraulic performance was higher at LF than at HF for both pilots, which might be due to distinct water-transport mechanisms inside the pilot, depending on flow rate. At HF, the transport was dominated by convection in the main channel, while at LF the transport was also governed by hydrodynamic water dispersion (including molecular diffusion and kinematic dispersion) from the main channel into the whole pilot volume, convection still occurring in the main channel [START_REF] Gaullier | Is pesticide sorption by constructed wetland sediments governed by water level and water dynamics?[END_REF][START_REF] Holland | Effects of wetland depth and flow rate on residence time distribution characteristics[END_REF].

Influence of hydrodynamics on pesticide mitigation

Tracer recoveries were over 90% at the end of the experiments (Fig. 2). Regardless of the studied flow rate and pesticides, the pesticide-export rate at the outlet (charge + discharge)

was lower for the ditch (49.1 to 86.9%) than for the pond (78.8 to 99.4%) (Fig. 2), indicating a better mitigation through the ditch. Its higher mitigation could be due to the lower water depth (at LF 1 cm and 1.5 cm in ditch and pond, respectively, and at HF 1.5 cm and 3 cm in the same pilots), which enhances the surface contact with sediments and increase the adsorption [START_REF] Gaullier | Is pesticide sorption by constructed wetland sediments governed by water level and water dynamics?[END_REF]. Regardless of the studied pilots and pesticides, the exported rates at the outlet (charge + discharge) for the four pesticides were lower at LF (49.1 to 87.1%) than at HF (66.2 to 99.4%) (Fig. 2). This result agrees with those reported under controlled conditions by other authors [START_REF] Boutron | Effect of the submergence, the bed form geometry, and the speed of the surface water flow on the mitigation of pesticides in agricultural ditches: mitigation of pesticides in agricultural ditches[END_REF][START_REF] Moore | Aqueous pesticide mitigation efficiency of Typha latifolia (L.), Leersia oryzoides (L.) Sw., and Sparganium americanum Nutt[END_REF]Vallée et al., 2015a). Thus, pesticide mitigation is higher at LF than at HF, probably associated with a longer residence time as well as with water-transport mechanisms governed by hydrodynamic dispersion and convection at LF, whereas at HF only convection is the main process of water transport.

For both flow rates, during charge phases (S1 and S3), the exported rate of pesticides was lower than that of ionic tracers in the ditch, indicating pesticide retention and/or degradation inside the pilots. According to their mitigation, pesticides can be classified as BSC > CYP > DMT ≥ IPU, which is consistent with their adsorption coefficient and suggests that retention could be a major process [START_REF] Gaullier | Is pesticide sorption by constructed wetland sediments governed by water level and water dynamics?[END_REF]Vymazal andBřezinová, 2015, Tournebize et al. 2017). Note, however, that the mitigation we observed is lower than those mentioned by Tounebize et al. (2017). According to their classification IPU and DMT belong to group I (average of 25% of removal efficiency, BSC to group II -49% and CYP to group III -51 %). In the pond, however, the exported rate of pesticides was almost similar to that of ionic tracers, indicating that pesticides travelled freely in the water with few interactions with substrates except for BSC having the highest adsorption coefficient.. Taking into account the discharge phases (S2 and S4+S5), the exported rate was similar for all pesticides and both pilots during HF conditions. During LF, however, it was higher for pond than for ditch.

Moreover, BSC and CYP had higher exported rates than DMT and IPU, especially during LF discharge (S5), as previously shown in a pilot-scale experiment by Vallée et al. (2015a). Thus, for HF, the pesticides not adsorbed on sediment during charge step (S1) are released during the following discharge step (S2) due to convective flux. At LF, however, the rate of exported pesticides could be due to both convective and dispersive flux and pesticide desorption from sediment and vegetation, especially for BSC and CYP. Indeed, the amount of released ionic tracer (i.e. not adsorbed) was lower than that of released BSC and CYP, especially for the ditch pilot, indicating that desorption reactions may occur. In addition, a higher percentage of metabolites was recovered at LF (0.5-1.4% of initial IPU and 0.4-0.9% of initial DMT) than at HF (0.2-0.6% of initial IPU and 0.03-0.13% of initial DMT) (Fig. A.2). A low flow-rate highlights differences that are related to molecular properties, probably due to the longer retention time that also promotes slower processes, such as degradation or desorption.

Finally, the HCPC test aggregated the sixteen observations in four groups corresponding to the four studied conditions: pond-HF (obs. 1-4), pond-LF (obs. 5-8), ditch-HF (obs. 9-12) and ditch-LF (obs.13-16) (Figs 3). The PLS-DA statistical test (fig 3D )     highlights the important factors in pesticide mitigation. The first (F1=57.27%) and second (F2=35.18%) discriminant axes describe with a high degree of confidence (F1+F2 represent 90% of variance) the distribution of the four classes determined from the sixteen observations (Fig. 3). The four groups determined from HCPC analysis are all separated by discriminant axis 1, which was mainly based on the first principal axis calculated from the MCA that is itself based on the four hydraulic parameters of flow rate, water depth, pilot volume and hydraulic efficiency. In addition, the groups are separated by the second discriminant axis, calculated based on the third principal MCA-derived axes. This third principal component is mainly related to pilot volume and hydraulic efficiency. Observations 1 to 4 and 13 to 16 are located in the upper part of the scatter plot and are discriminated from observations 5 to 12 according to the second discriminant (Fig. 3). These results show that the observations with higher pilot volume and hydraulic efficiency (observations 1-4 and 13-16) are clearly discriminated from the others. The MCA results show that hydraulic variables contributed more strongly to the principal axes than the pesticide-related variables, indicating that the classes calculated from HCPC are strongly dependent upon the hydraulic variables. The high variance associated with the PLS-DA confirms that hydraulic variables (flow rate, water depth, volume and hydraulic efficiency) are suitable for group observations. The group showing higher pesticide mitigation is associated with lower flow rates, water depth and volume. These results agree with observations by others authors [START_REF] Holland | Effects of wetland depth and flow rate on residence time distribution characteristics[END_REF][START_REF] Boutron | Effect of the submergence, the bed form geometry, and the speed of the surface water flow on the mitigation of pesticides in agricultural ditches: mitigation of pesticides in agricultural ditches[END_REF][START_REF] Moore | Aqueous pesticide mitigation efficiency of Typha latifolia (L.), Leersia oryzoides (L.) Sw., and Sparganium americanum Nutt[END_REF], even if the results were contradictory for flow rates [START_REF] Holland | Effects of wetland depth and flow rate on residence time distribution characteristics[END_REF][START_REF] Moore | Aqueous pesticide mitigation efficiency of Typha latifolia (L.), Leersia oryzoides (L.) Sw., and Sparganium americanum Nutt[END_REF]. Flow rate and volume conditions were different for a similar HRT (due to pilot morphology), indicating that hydraulic parameters have a stronger influence on pesticide mitigation than the HRT. Combining MCA and PLS-DA analyses

shows that pesticide-related variables (Kd, DT50-water) contribute mainly to intragroup variance. In addition, the small size of the data set implies that one observation is associated with only one pesticide. Therefore, the difference of pesticide mitigation according to their physico-chemical properties could not be correctly determined from our dataset. One way to overcome this limit would be to repeat part of the experiments, especially the LF experiments where mitigation is higher, several times to obtain multiple observations associated with each tested pesticides.

Mitigation of pesticides during a no-drainage period followed by discharge

Regardless of pilot and molecules, the water DT50 values, ranging from 0.5 to 1.2 days showed fast dissipation of pesticides from the aqueous phase, as earlier shown for IPU and BSC (Vallée et al., 2015a) (fig. 4). The DT50 values are generally lower or equivalent in the pond than in the ditch (0.5 vs. 0.8 for BSC, 0.6 vs. 1.1 for CYP, 0.8 vs. 1.2 for IPU and 0.7 vs. 0.7 for DMT). Consequently, after 50 days of stagnation, less than 1% and 0.5% of the four pesticides remained in the ditch and pond waters, respectively. The decrease in IPU and DMT concentrations was accompanied by a simultaneous increase in metabolite concentrations (Fig. 4B and4C) during the first ten days of stagnation, testifying to a degradation of these molecules during the experiment.

After 50 days of stagnation, the two IPU-metabolite concentrations in water were less than LOQ, whereas a small amount of DMT metabolites remained in water notably DMT-OA. For the two pilots, at least 5% of the IPU was degraded into metabolites after 50 days, in accordance with IPU degradation rates determined elsewhere in water: 1.6-2.2% (160-200 days) [START_REF] Rönnefahrt | Comparison of the fate of isoproturon in small and large-scale water/sediment systems[END_REF] and3.5-35% (50 days) (Vandermeeren et al., 2016). The concentration peak of the metabolite MD-IPU was 9 and 4.5 times higher than that of DD-IPU (for ditch and pond, respectively), indicating that IPU is preferentially degraded into MD-IPU as earlier shown [START_REF] Rönnefahrt | Comparison of the fate of isoproturon in small and large-scale water/sediment systems[END_REF]. Regarding the degradation of DMT in the ditch, about 17.5% was degraded into DMT-OA, CGA 42443, DMT-ESA, CGA 39981 or CGA 373464, and in the pond about 35% was degraded into DMT-OA, CGA 39981 (higher amount), CGA 42443, DMT-ESA or SYN 528702 (Fig. 4C). We noticed that the ratios between DMT metabolites are pilot-dependant with a prevalence of DMT-OA and CGA 39981 vs. DMT-ESA in the pond during the first days while the three metabolites are similar concentrations in the ditch. As our study is the first to provide information on DMT metabolites, we could not compare our results to previous data.

During the last 7-day step with low-flow discharge (S8) following the stagnation period, the four pesticides were quantified at higher concentrations than at the end of stagnation step (S7), indicating that they may have been desorbed from sediment and/or plants. As observed during drainage and in both pilots, BSC and CYP are more released into water (0.9-1.8% of the initial injected amount in S6) than IPU and DMT (0.2-0.3% of the initial injected amount in S6) due to their higher desorption. Compared to the previous steps, the pesticideexportation rate measured during the S8 step may be considered as negligible, even though the four pesticides are present in exported water. This could indicate that renewed flow after a long period of stagnation (i.e. between two drainage seasons in the field) causes a small release of these four pesticides.

Global pesticide mitigation

In the field, a drainage season is composed of successive periods of high-, low-and zero flow.

In order to determine the global pesticide mitigation during a complete drainage season, a mass balance considered the amounts of pesticides injected during the charge steps (S1, S3

and S6), and the amount of pesticides and metabolites exported in water during drainage periods (from S1 to S5 and S8) and adsorbed onto sediment (Fig. 2). Over the 62 days of the experiment, the mitigation ranged from 35.2 to 45.3% for the ditch, and from 7.9 to 14.8% for the pond, with earlier reported mitigation values of 24% for IPU and of 38-67% for BSC (Vallée et al., 2015a). The pesticides from the aqueous phase were exported at outlet mainly during the first steps of HF and LF (S1 to S5, vs. 1.8% for S8). In addition, less than 1% of IPU and DMT metabolites were exported in water at the outlets of both pilots from S1 to S8

(BSC and CYP metabolites were not analysed). From 0.01 to 4.6% of pesticide residues were extracted from sediments; in terms of concentrations, the most recovered was BSC, IPU and DMT were least recovered, and CYP had an intermediate recovery, agreeing with their adsorption coefficients [START_REF] Gaullier | Is pesticide sorption by constructed wetland sediments governed by water level and water dynamics?[END_REF]Vallée et al., 2015aVallée et al., , 2014)). We can thus conclude that pesticide mitigation may result from: (i) adsorption onto sediment as previously shown by other authors for BSC [START_REF] Papaevangelou | Treatment of agricultural equipment rinsing water containing a fungicide in pilot-scale horizontal subsurface flow constructed wetlands[END_REF] and IPU [START_REF] Durst | Transport of pesticides and artificial tracers in verticalflow lab-scale wetlands: combined reactive transport of pesticides and tracers[END_REF] (no data for CYP and DMT); (ii) degradation as previously shown for IPU and DMT, [START_REF] Durst | Transport of pesticides and artificial tracers in verticalflow lab-scale wetlands: combined reactive transport of pesticides and tracers[END_REF]; and (iii) other processes such as plant or biofilm absorption, hydrolysis, photolysis or volatilization, not evaluated in this study but mentioned by [START_REF] Rönnefahrt | Comparison of the fate of isoproturon in small and large-scale water/sediment systems[END_REF][START_REF] Rose | The effect of vegetation on pesticide dissipation from ponded treatment wetlands: Quantification using a simple model[END_REF].

Conclusions

Our mesocosm experiments demonstrated that pesticide mitigation in CWs is linked to: i) hydrodynamics and in particular flow rates; ii) CW morphology, in this case a ditch and a succession of ponds; and iii) the physico-chemical properties of the pesticide. For the studied flow rates involving similar hydraulic residence times in the two pilots, i.e. high flow and low flow of about 5 h and 24 h, respectively, the ditch seems more effective for mitigating pesticides than the ponds. This is probably due to a lower water depth in the ditch. For both pilots, the mitigation was higher at low flow than at high flow, regardless of the pesticides. In addition to the kinetic aspects of adsorption, again the difference in water depth in each pilot, depending on the flow regime, can be another factor affecting mitigation. At low flow, a higher production of metabolites indicates enhanced degradation, leading to even better mitigation. Finally, no-flow periods led to (almost) total mitigation of both pesticides and their metabolites produced in the water phase.

These laboratory results obtained in controlled conditions highlight how the hydraulic regime that will exist in situ and the characteristics of the different drainage episodes will have an impact on the rate of mitigation.

Tables and Figures captions

Table 1: Simulated hydraulic parameters and characteristics of each step (S0 to S8) Distributions of the 16 observations (A) and contributions of the variables (B) obtained from MCA analysis applied on results listed in Table A.5. Based on MCA results, a classification tree (C) obtained with the ascending hierarchical principal component clustering. Groups issued (pond-HF, 1-4; pond-LF, 5-8; ditch-HF, 9-12; ditch-LF, 13-16) from classification tree are used to develop a PLS-DA model (D). These groups correspond also to the intensity migration rate: low (0.6-9%), middle (12.6-20.8%), high (13-33.7%), very high (43.3-50.8%).

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Experimental ditch pilot (A) and pond pilot (B) and location of specific zones (SZ)

  Figure 1Experimental ditch pilot (A) and pond pilot (B)

Figure 3

 3 Figure 3

Figure 4

 4 Figure 4

  The time scale, i.e. long term vs. specific episodes of drainage considered to evaluate the effectiveness of the CWs could thus lead to contrasting results. Geological Survey), University of Lorraine, the FEDER-FSE Lorraine et Massif des Vosges (Fonds Européens de DEveloppement Regional -Fond Social Européen) and the Zone Atelier du Bassin de la Moselle. The authors thank the Rhin-Meuse Water Agency and the BRGM for the fellowship. The authors also thank C. Coureau (BRGM, Orléans) for her helpful laboratory work and the farmers R. Cherrier and F.X. Schott (Chambre Régionale d'Agriculture du Grand Est) for field access. Dr. H.M. Kluijver edited the final English version of the MS.

Table 2 :

 2 Water transport parameters of the pilots

Acknowledgements

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