domain: nlin phys

text:
Sergey Gavrilyuk
email: sergey.gavrilyuk@univ-amu.fr

Henrik Kalisch
email: henrik.kalisch@math.uib.no

Zahra Khorsand
email: zahra.khorsand@math.uib.no

A Kinematic Conservation Law in Free Surface Flow

The Green-Naghdi system is used to model highly nonlinear weakly dispersive waves propagating at the surface of a shallow layer of a perfect fluid. The system has three associated conservation laws which describe the conservation of mass, momentum, and energy due to the surface wave motion. In addition, the system features a fourth conservation law which is the main focus of this note. It will be shown how this fourth conservation law can be interpreted in terms of a concrete kinematic quantity connected to the evolution of the tangent velocity at the free surface. The equation for the tangent velocity is first derived for the full Euler equations in both two and three dimensional flows, and in both cases, it gives rise to an approximate balance law in the Green-Naghdi theory which turns out to be identical to the fourth conservation law for this system. An analogous equation valid along each contact surface in the fluid bulk (and not only on the free surface) was also derived in both two and three dimensions.

Introduction

The surface water wave problem considered here concerns the motion of an inviscid incompressible fluid with a free surface and over an impenetrable rigid bottom. The mathematical description of this motion involves the Euler equations for perfect fluids coupled with freesurface boundary conditions. Solving these equations presents a rather complex mathematical problem, and in many instances, it can be expedient to use an approximate model for the description of the free surface and the motion of the fluid below. The classical simplifications of this problem are the linear theory which may be used for waves of small amplitude, and the shallow water theory, which may be used for long waves. A third class of approximations can be found in the so-called Boussinesq scaling which requires a certain relationship between the amplitude and wavelength of the waves to be described.

The Green-Naghdi system was derived as a long-wave model for surface water waves which are long, but which may not necessarily have small amplitude. In fact, the Green-Naghdi system was first found in the special case of one-dimensional waves over a flat bottom [START_REF] Serre | Contribution à l'étude des écoulements permanents et variables dans les canaux[END_REF], in which case it can be thought of as a higher-order nonlinear shallow-water system. The system was later extended to two-dimensional surface waves [START_REF] Su | Korteweg-de Vries equation and generalizations, III. Derivation of the Korteweg-de Vries equation and Burgers equation[END_REF], and put into the general context of fluid sheets by [START_REF] Green | On the theory of water waves[END_REF][START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF]. If l is a characteristic wavelength and h 0 is the the mean water depth, we define the dimensionless small parameter β = h 2 0 /l 2 . The Green-Naghdi equations are obtained by depth-averaging the Euler system and keeping in the resulting set of equations only first order terms in β; without making any assumptions on the amplitude of the waves.

While the studies mentioned so far give a formal justification of the Green-Naghdi system as a wave model accommodating larger amplitude waves, a mathematical justification of this property was given for the Green-Naghdi and some related systems in [START_REF] Lannes | The Water Waves Problem[END_REF][START_REF] Lannes | Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation[END_REF][START_REF] Li | A shallow-water approximation to the full water wave problem[END_REF][START_REF] Makarenko | A second long-wave approximation in the Cauchy-Poisson problem[END_REF][START_REF] Saut | Well-posedness on large time for a modified full dispersion system of surface waves[END_REF]. Recent years have seen increased activity in both the study of modeling properties of the Green-Naghdi system ( [START_REF] Barros | Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties[END_REF][START_REF] Barthélemy | Nonlinear shallow water theories for coastal waves[END_REF][START_REF] Bonneton | Recent advances in Serre-Green-Naghdi modelling for wave transformation, breaking and runup processes[END_REF][START_REF] Carter | The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations[END_REF][START_REF] Dias | On the fully-nonlinear shallow-water generalized Serre equations[END_REF][START_REF] El | Unsteady undular bores in fully nonlinear shallowwater theory[END_REF][START_REF] Gavrilyuk | Generalized vorticity for bubbly liquid and dispersive shallow water equations[END_REF][START_REF] Khorsand | Particle trajectories in the Serre equations[END_REF][START_REF] Li | Linear stability of solitary waves of the Green-Naghdi equations[END_REF]) and in the development of numerical discretization techniques, such as in [START_REF] Cienfuegos | A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations, Part I: model development and analysis[END_REF][START_REF] Dutykh | Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations[END_REF][START_REF] Li | High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model[END_REF][START_REF] Métayer | A numerical scheme for the Green-Naghdi model[END_REF].

The Green-Naghdi system is usually written in terms of the average horizontal velocity ū(t, x) and the total flow depth h(t, x) = h 0 + η(t, x), and take the form

h t + (hū) x = 0, (1)
ūt + ūū x + gh x -1 3h

∂ ∂x h 3 ūxt + ūū xx -ū2 x = 0. ( 2
)
Note that if higher-order terms in [START_REF] Ali | On the formulation of mass, momentum and energy conservation in the KdV equation[END_REF] are discarded, the system reduces to the classical shallowwater system. On the other hand, considering waves of infinitesimally small amplitude, one obtains the linearization of the Green-Naghdi system, which is identical to the linearized classical Boussinesq system [START_REF] Peregrine | Long waves on beaches[END_REF]. Smooth solutions of (1), (2) also satisfy the following conservation laws:

∂ ∂t {h} + ∂ ∂x hū = 0, (3)
∂ ∂t hū + ∂ ∂x 1 2 gh 2 + hū 2 - 1 3 h 3 ūxt + 1 3 h 3 ū2 x - 1 3 h 3 ūū xx = 0, (4)
∂ ∂t

1 2 h gh + ū2 + 1 3 h 2 ū2 x + ∂ ∂x hū gh + 1 2 ū2 + 1 2 h 2 ū2 x -1 3 h 2 (ū xt + ūū xx ) = 0, (5)
∂ ∂t ū -hh x ūx - 1 3 h 2 ūxx + ∂ ∂x gh + 1 2 ū2 -hh x ūū x -1 2 h 2 ū2 x -1 3 h 2 ūū xx = 0. (6)
The conservation law (3) is easily seen to describe mass conservation. It can also be shown that (4) represents momentum conservation and (5) represents the energy conservation in the Green-Naghdi approximation [START_REF] Kalisch | Mechanical balance laws for the Green-Naghdi system and applications to undular bores, shoaling and wave loading[END_REF]. The last conservation law ( 6) is not easily interpreted. Some authors have ascribed the conservation of angular momentum to this balance equation [START_REF] Carter | The kinematics and stability of solitary and cnoidal wave solutions of the Serre equations[END_REF]. On the other hand, it was shown in [START_REF] Miles | Weakly dispersive nonlinear gravity waves[END_REF] that it originates from the particle relabeling symmetry for a corresponding approximate Lagrangian of the Euler equations. The invariance of the integral of the density ū-hh x ūx -

1 3 h 2 ūxx = ū+ 1 3h ∂ ∂x h 2 Dh
Dt , where D Dt = ∂ ∂t + ū ∂ ∂x over the real line has also been linked to the irrotationality assumption in the context of two-layer flow [START_REF] Choi | Fully nonlinear internal waves in a two-fluid system[END_REF].

In the present work, the focus will be on giving a precise physical meaning to the quantities appearing in the fourth conservation law. As will come to light, the density appearing in the conservation law ( 6) is proportional to the tangential fluid velocity at the free interface, while the flux can be interpreted as an approximation of the energy per unit mass of fluid. The conservation law derives from a corresponding balance law which holds exactly for the surface water-wave problem for the full Euler equations, and can be written in Lagrangian variables as

∂ u• ∂x ∂s ∂t + ∂ ∂s p ρ + gh - |u| 2 2 = 0. ( 7
)
Here u denotes the fluid velocity, x(s, t) is the fluid particle motion, p is the pressure, and ρ is the density. The variable s featuring in this balance law denotes a parametrization of an arc lying entirely in the fluid domain or in the free surface. In the particular case where the arc is located in the free surface, the term u • ∂x ∂s can be interpreted as the tangent velocity along the free surface, multiplied by the local element of arclength. The balance law [START_REF] Bjørkavåg | Wave breaking in Boussinesq models for undular bores[END_REF] provides the basis for the conservation law [START_REF] Benjamin | Hamiltonian structure, symmetries and conservation laws for water waves[END_REF] in the Green-Naghdi approximation. This conservation law actually holds even for the three-dimensional water-wave problem, but in this case, an additional term enters the corresponding identity in the Green-Naghdi approximation, so that the identity corresponding to (6) in three dimensions is not a pure conservation law.

The disposition of the present paper is as follows. In the next section, we describe how to obtain the balance equation [START_REF] Bjørkavåg | Wave breaking in Boussinesq models for undular bores[END_REF] in the context of the full Euler equations. Then, the Green-Naghdi approximation is considered, and the associate approximate density and flux are obtained using a method introduced in [START_REF] Ali | Mechanical balance laws for Boussinesq models of surface water waves[END_REF]. In Section 4, the three-dimensional case is considered, and Section 5 contains a brief discussion.

2 Kinematic balance laws in two-dimensional flows

2D balance law for perfect fluids

The motion of a homogeneous inviscid and incompressible fluid with a free surface over a flat bottom can be described by the Euler equations with appropriate boundary conditions. The unknowns are the surface elevation η(t, x), and the horizontal and vertical fluid velocity components u(t, x, z) and w(t, x, z), respectively. The problem may be posed on a domain Ω t = {(x, z)|0 < z < h 0 + η(t, x)} which extends to infinity in the positive and negative xdirection. The two-dimensional Euler equations are

∂u ∂t + (u • ∇)u + 1 ρ ∇p = g, in Ω t , (8)
∇ • u = 0, in Ω t , (9)
where u = (u, w) T represents the velocity field, and g = (0, -g) T is the gravitational acceleration. The free-surface boundary conditions are given by requiring the pressure to be equal to atmospheric pressure at the surface, i.e. p = p atm if surface tension effects are neglected, and the kinematic boundary condition η t + uη x = w for z = h 0 + η(t, x). For the purposes of this paper, we work with the gauge pressure, so that we may take p atm = 0. Following Zakharov [START_REF] Zakharov | Benney equations and quasiclassical approximation in the method of the inverse problem[END_REF], we introduce semi-Lagrangian coordinates (x, λ). The change of variables (x, z) → (x, λ) is given by the function where Φ (t, x, λ) is the solution of the equation

z = Φ (t, x, λ) , L 0 L t x A (0) x A (t) x B (0) x B (t ) x z x 0 ( s) x( s , t)
∂Φ ∂t + u ∂Φ ∂x = w,
with the initial condition Φ t=0 = λh 0 + λη 0 (x), where η 0 (x) is the initial disturbance of the free surface. At each time t, this change of variables represents a foliation of the domain in (x, z) variables by contact lines indexed by 0 ≤ λ ≤ 1, and where each line corresponds to λ = const in the (x, λ)-plane. The free surface z = h 0 + η(t, x) corresponds to the value λ = 1. It can be shown that in semi-Lagrangian coordinates, the material derivative is defined only in terms of the horizontal velocity, and the curvilinear domain occupied by the fluid in the plane {(x, z), 0 < z < h 0 + η(t, x)} maps to a strip in the plane {(x, λ), 0 < λ < 1}. Let L λ t be a finite material arc lying entirely in one of the contact lines defined above, starting at a point x A (t) = (x A (t), z A (t)) T , and ending at another point x B (t) = (x B (t), z B (t)) T . The arc corresponding to the value λ = 1 lies entirely in the free surface, and will be denoted simply by L t .

Parametrizing the initial arc L λ 0 using the real parameter s, we obtain the description of the arc L λ t in the form

x = ϕ(t, s, λ) = (ϕ x (t, s, λ) , ϕ z (t, s, λ)) T , s A ≤ s ≤ s B ,
where x = ϕ(t, s, λ) is the solution of the Cauchy problem

dx dt = u(t, x), x| t=0 = x 0 (s, λ), x 0 (s, λ) ∈ L λ 0 .
Using the Euler equations ( 8), [START_REF] Bonneton | A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model[END_REF] one can obtain the time evolution of the total drift γ λ along L λ t in the form

dγ λ dt = d dt L λ t u • dx = |u| 2 2 - p ρ -gz x B (t) x A (t) . ( 10
)
Observing that

L λ t u•dx = L λ 0 u • ∂x ∂s ds,
we see that [START_REF] Borluk | Particle dynamics in the KdV approximation[END_REF] can also be written in local form as [START_REF] Bjørkavåg | Wave breaking in Boussinesq models for undular bores[END_REF] along the material curve L λ t . In fact, the previous computation is nothing but the standard proof of Kelvin's circulation theorem, but applied on a contour which is not necessarily closed. Taking a contour embedded in the material line L λ t , we obtain from [START_REF] Borluk | Particle dynamics in the KdV approximation[END_REF] the conservation law in the Lagrangian coordinates

∂ u • ∂ϕ ∂s ∂t + ∂ ∂s p ρ + gz - |u| 2 2 = 0. ( 11
)
Introducing K λ (t, x) defined along L λ t by u(t, s, λ)• ∂ϕ(t, s, λ) ∂s = K λ (t, x) ∂ϕ x (t, s, λ) ∂s , (12)
we obtain the following conservation law in the Eulerian coordinates (t, x) along L λ t :

∂K λ ∂t + ∂ ∂x u (t, x, λ) K λ + p (t, x, λ) ρ + gz (t, x, λ) - 1 2 (u 2 (t, x, λ) + w 2 (t, x, λ)) = 0 (13)
In order to understand the meaning of K λ , we let

n λ = -∂z(t,x,λ) ∂x , 1
T be the unit normal to L λ t , scaled by the element of arclength of L λ t . By the same token, let τ λ = 1, ∂z(t,x,λ) ∂x T be the unit tangent vector to L λ t , scaled by the element of arclength. It can then be shown that ( 12) is equivalent to defnining

K λ = u (t, x, λ) • τ λ L λ t .
At the free surface z = z (t, x, 1) = h(t, x), the pressure vanishes, and we obtain the simpler conservation law

∂K ∂t + ∂ ∂x uK + gh - 1 2 (u 2 + w 2 ) = 0. ( 14
)
This is an exact conservation law representing the evolution of the tangent velocity K along the free surface. While this conservation law can be related to the densities found in [START_REF] Benjamin | Hamiltonian structure, symmetries and conservation laws for water waves[END_REF] in the case of potential flow (since then K = ∂ x φ(t, x, η(t, x)), it is clear from the preceding discussion that ( 14) holds even in the case of rotational flows. In fact, the equation for K corresponds to the equation ( 9) in [START_REF] Castro | Well-posedness and shallow water stability for a new Hamiltonian formulation of the water waves equations with vorticity[END_REF] where a new Hamiltonian formulation of the water waves equations in the presence of vorticity was proposed. To the best of our knowledge, the general equation ( 13) valid for each material curve L λ t is new, and its derivation has a clear geometrical meaning. In the case where the conservation law is evaluated at the free surface, a geometric interpretation was also provided in [START_REF] Bardos | Mathematics for 2d interfaces, Singularities in Mechanics: Formation, Propagation and Microscopic Description[END_REF].

2D balance law in the Green-Naghdi approximation

In the case of potential motion the exact balance law ( 14) can be related to the fourth conservation law [START_REF] Benjamin | Hamiltonian structure, symmetries and conservation laws for water waves[END_REF] satisfied by solutions to the Green-Naghdi system (1)-( 2) by truncating the resulting set of dimensionless equations at second order in β. Indeed, we recall the scaling which is used in the derivation of the system (1)- [START_REF] Ali | On the formulation of mass, momentum and energy conservation in the KdV equation[END_REF]. Define the non-dimensional variables

x = x l , z = z h 0 , t = c 0 l t, h = h h 0 , ũ = u c 0 , w = w √ βc 0 , P = P ρgh 0 . ( 15
)
Using this scaling, the quantity K = u + wh x | z=h(t,x) is given in non-dimensional variables as

K t, x = ũ + β wh x,
and the conservation law ( 14) turns into

∂ K ∂ t + ∂ ∂ x K ũ + h - 1 2 (ũ 2 + β w2 ) = 0.
In the case of potential flows, the expressions of the horizontal and vertical velocity are as follows (see, for example, [START_REF] Barthélemy | Nonlinear shallow water theories for coastal waves[END_REF]):

ũ (x, z, t) = ũ + β h2 6 - z2 2 ũxx + O(β 2 ), w(x, z, t) = -z ũx + O(β).
Substituting the expressions for ũ and w evaluated at the free surface into the formula for K t, x yields

∂ ∂ t ũ - β 3 h2 ũxx -β hh x ũx + ∂ ∂ x 1 2 ũ2 - β 3 h2 ũũ xx -β hh x ũũ x + h - β 2 h2 ũ2 x = O(β 2 ).
Using similar reasoning as in [START_REF] Ali | Mechanical balance laws for Boussinesq models of surface water waves[END_REF][START_REF] Ali | On the formulation of mass, momentum and energy conservation in the KdV equation[END_REF], we define the non-dimensional quantities K and qK by

K = ũ - β 3 h2 ũxx -β hh x ũx and qK = 1 2 ũ2 -β 3 h2 ũũ xx -β hh x ũũ x + h - β 2 h2 ũ2 x .
Then the balance is

∂ K ∂ t + ∂ qK ∂ x = O(β 2 ).
Using the natural scalings K = c 0 K and q K = c 2 0 qK , the dimensional forms of these quantities are given by

K = ū - 1 3 h 2 ūxx -hh x ūx , q K = gh + 1 2 ū2 - 1 3 h 2 ūū xx -hh x ūū x - 1 2 h 2 ū2 x ,
respectively. While the preceding analysis shows that the conservation law

∂K ∂t + ∂q K ∂x = 0, (16)
holds approximately to second order in β, it can be seen immediately that the density K and the flux q K are the same as the respective quantities in [START_REF] Benjamin | Hamiltonian structure, symmetries and conservation laws for water waves[END_REF], so that ( 16) is an exact identity for solutions of the Green-Naghdi system (1)-( 2). The asymptotic expansion for K can also be found in (5.17) in [START_REF] Lannes | The Water Waves Problem[END_REF].

It is interesting to see what happens in the shallow-water approximation where the balance

∂ K ∂ t + ∂ qK ∂ x = O(β)
only holds to the order of β. In this case, we obtain

K = ū
for the density and

q K = gh + 1 2 ū2
for the flux. Thus in this case the equation ∂K ∂t + ∂q K ∂x = 0 also holds exactly, and in fact it is just the second equation in the shallow-water system. It can be seen clearly from this form that this is a balance law relating horizontal velocity energy density per unit mass. In the shallowwater theory, the horizontal velocity is taken to be uniform throughout the fluid column, so that the simple form for K and q K is achieved. However, the quadratic dependence of the horizontal velocity gives a more refined representation which may also be used advantageously for kinematic studies of the fluid motion, such as in [START_REF] Bjørkavåg | Wave breaking in Boussinesq models for undular bores[END_REF][START_REF] Borluk | Particle dynamics in the KdV approximation[END_REF][START_REF] Khorsand | Particle trajectories in the Serre equations[END_REF].

In the derivation of the Green-Naghdi equations ( 3)-( 4), such as summarized in the appendix, no hypothesis about the flow potentiality was used, except that it needs to be assumed that the vorticity be small. Since the conservation laws (4), ( 5) and ( 6) are exact consequences of the Green-Nagdhi system, these formulas also hold in the case of rotational flow. However, in order to show the asymptotic equivalence of ( 14) and ( 6), the horizontal velocity needed to be assumed to vary quadratically with the depth, and this in turn depends on the flow being irrotational.

To obtain an asymptotic link between ( 14) and ( 6) in the general case remains an open problem. Indeed, the vorticity distribution in the fluid layer will certainly influence the average velocity and the surface velocity. In this case, the conservation law (6) might correspond to the equation ( 13) with some λ depending on the vorticity distribution.

3 Kinematic balance law in three dimensions

3D balance law for perfect fluids

The Euler equation ( 8)-( 9) are written in general form, and may also be interpreted to hold in three dimensions, for a velocity field u = (u, v, w) T , a pressure p(t, x, y, z) and body forcing g = (0, 0, -g) T . In this case, the kinematic free surface boundary condition is given by η t + uη x + vη y = w at z = η(t, x, y). Note that for the purpose of this section, we define the horizontal spatial coordinates, velocity vector field, gradient and material derivative by

x H = (x, y) T , u H = (u, v) T , ∇ H = ∂ ∂x , ∂ ∂y T , and
D H Dt = ∂ ∂t + u H • ∇ H ,
respectively. An analogous vector law for the evolution of the tangent velocity can also be derived in three-dimensional case. Let Σ t be defined by z = h(t, x H ), be the free surface, which is assumed can be projected onto the horizontal plane (Oxy). As before, we will introduce a special Lagrangian coordinate 0 ≤ λ ≤ 1 corresponding to the foliation of the fluid domain by the material surfaces Σ λ t . The free surface Σ t will correspond to the value λ = 1. Such a foliation can be achieved by solving the Cauchy problem for the equation

∂Φ ∂t + u H • ∇ H Φ = w,
with the initial condition Φ| t=0 = λh 0 + λη 0 (x H ), and making then the change of variables (x H , z) → (x H , λ), z = Φ(t, x H , λ).

Then we introduce the Lagrangian coordinates s 1 , s 2 which may be taken as the (x, y) coordinates of a particle at time t = 0. Each surface Σ λ t is thus parametrized by the same parameters s 1 , s 2 . The change of variables is assumed to be invertible: s i = s i (t, x H , λ), for i = 1, 2. The Lagrangian coordinates s i are conserved along the particle trajectories in the Eulerian coordinates. This implies that the transport equation for the Lagrangian coordinates

s i reduces to ∂ ∂t + u H • ∇ H s i = 0, i = 1, 2, (17)
in the coordinates (x H , λ). Let

e i = ∇ H s i , i = 1, 2,
be a local cobasis at each surface Σ λ t (we call it cobasis because e i are gradients). Taking the horizontal gradient ∇ H of ( 17) yields

e i t + ∇ H (e i • u H ) = 0. ( 18
)
Since the matrix ∂e i ∂x H is symmetric, (18) can be written in the form

D H Dt e i t = e i t + ∂e i ∂x H u H = - ∂u H ∂x H T e i . (19)
In analogy with the two-dimensional case, one may take the drift along each s i -curve containing in the contact surface Σ λ t to obtain conservation laws analogous to [START_REF] Bjørkavåg | Wave breaking in Boussinesq models for undular bores[END_REF] in the form

∂ u • ∂ϕ ∂s i ∂t + ∂ ∂s i p ρ + gz - |u| 2 2 = 0. ( 20
)
where ϕ(t, s 1 , s 2 , λ) is the motion of particles along the surfaces Σ λ t . Defining

K λ i = u(t, s 1 , s 2 , λ) • ∂ϕ(t, s 1 , s 2 , λ) ∂s i
in analogy with the two-dimensional case, we can define the tangent component of the velocity vector at Σ λ t as

K λ = K λ 1 e 1 + K λ 2 e 2 .
Taking the partial derivative of K λ (t, s 1 , s 2 ) with respect to time at fixed Lagrangian coordinates s 1 , s 2 we obtain

∂K λ t, s 1 , s 2 ∂t = ∂K λ 1 ∂t e 1 + ∂K λ 2 ∂t e 2 + K λ 1 ∂e 1 (t, s 1 , s 2 , λ) ∂t + K λ 2 ∂e 2 (t, s 1 , s 2 , λ) ∂t = - ∂ ∂s 1 p ρ + gz - |u| 2 2 e 1 - ∂ ∂s 2 p ρ + gz - |u| 2 2 e 2 + K λ 1 ∂e 1 (t, s 1 , s 2 , λ) ∂t + K λ 2 ∂e 2 (t, s 1 , s 2 , λ) ∂t .
Converting to Eulerian coordinates, we obtain

D H Dt K λ (t, x H ) = -∇ H p ρ + gz - |u| 2 2 + K λ 1 D H Dt e 1 (t, x H , λ) + K λ 2 D H Dt e 2 (t, x H , λ).
Using [START_REF] Dutykh | Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations[END_REF], we may write the last relation as

D H Dt K λ (t, x H ) = -∇ H p ρ + gz - |u| 2 2 - ∂u H ∂x H T K λ .
This equation may also be written as

∂K λ (t, x H ) ∂t + ∂K λ ∂x H - ∂K λ ∂x H T u H + ∂K λ ∂x H T u H + ∂u H ∂x H T K λ +∇ H p ρ +gz - |u| 2 2 = 0,
or equivalently as

∂K λ (t, x H ) ∂t + curl(K λ ) × u H + ∇ H K λ •u H + p ρ + gz - |u| 2 2 = 0. ( 21
)
Equation ( 21) holds on the material surface Σ λ t , and is the equivalent of ( 13) in the threedimensional setting.

At the free surface z = h(t, x H ) equation ( 21) can be writen in a simpler form. Indeed, since at the free surface p = 0 and

h t + u H •∇ H h = w,
where u H is evaluated at the free surface and w is the vertical velocity evaluated at the free surface, (21) may be rewritten in the form

∂K ∂t + curl(K) × u H + ∇ H K • u H + gh - 1 2 |u H | 2 + D H h Dt 2 = 0, ( 22
)
which is analogous to [START_REF] Castro | Well-posedness and shallow water stability for a new Hamiltonian formulation of the water waves equations with vorticity[END_REF]. Since K is a vector in the (x, y)-plane, the vector curl(K) has only one non-zero component and is orthogonal to the (x, y)-plane. Equation ( 22) is exactly the equation proposed in [START_REF] Castro | Well-posedness and shallow water stability for a new Hamiltonian formulation of the water waves equations with vorticity[END_REF] (see equation [START_REF] Bonneton | A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model[END_REF] in that paper written by using different notations). Two observations are in order. First of all, equation ( 22) holds in the case of non-zero vorticity as well as the case of potential flow. In the latter case, we have curl

(K) × u H = 0, since curl(K) × u H = curl(u)| z=h(t,x H ) • N u ⊥ H , N = (-∇ H h, 1) T , u ⊥ H = (-v, u) T ,
so that ( 22) is a proper conservation law in the case of potential motion.

Secondly, it will be convenient to express K in terms of the velocity field in Eulerian coordinates. For this purpose, note the following computation:

K = u H • ∂x H ∂s 1 + w ∂h ∂s 1 ∇ H s 1 + u H • ∂x H ∂s 2 + w ∂h ∂s 2 ∇ H s 2 = ∇ H s 1 ⊗ ∂x H ∂s 1 + ∇ H s 2 ⊗ ∂x H ∂s 2 u H + w ∂h ∂s 1 ∇ H s 1 + ∂h ∂s 2 ∇ H s 2 . = u H + w∇ H h.

3D balance law in the Green-Naghdi approximation

Next, we consider the approximation of ( 22) in the Green-Naghdi regime. Recall that the Green-Naghdi equations for three-dimensional flows are written in terms of the depth-averaged horizontal velocity ū and the total flow depth h as

η t + ∇ H • (hū) = 0, ( 23
) ūt + T (h, ūt )} + ū • ∇ H ū + ∇ H η + Q(h, ū) = 0, ( 24
)
where

T (h, ūt ) = - 1 3h ∇ H (h 3 ∇ H • ūt ), Q(h, ū) = -1 3h ∇ H h 3 ū • ∇ H (∇ H • ū) -(∇ H • ū) 2 .
In order to obtain an unbiased representation, we write the non-dimensionalized version of ( 22) in terms of the principal unknown variables of the Green-Naghdi equation. Thus in non-dimensional form, [START_REF] Green | On the theory of water waves[END_REF] becomes

∂ ∂ t ũH + β w ∇H h + curl ũH + β w ∇H h × ũH + ∇H ũH + β( w ∇H h) • ũH + h - 1 2 |ũ H | 2 + β w2 = 0. ( 25
)
Note that since the velocity fields are now approximated, the term curl(K) × u H may not evaluate to zero even for irrotational flow, and therefore needs to included. Similar to the two-dimensional case, in the case of potential flow, the non-dimensional horizontal velocity field ũH , and the vertical velocity component w are given in the Green-Naghdi approximation at any location (x H , z) in the fluid by the following expressions in terms of the depth averaged velocity:

ũH (x, z, t) = ũ + β h2 6 - z2 2 ∆H ũ + O(β 2 ), (26) w
(x H , z, t) = -z ∇ H • ũ + O(β). (27)
Moreover, in the Appendix, the following important relation is established:

∆H ũ = ∇H ( ∇H • ũ) + O(β).
This relation holds as long as irrotational solutions of Euler's equations are considered. We now look at the three terms in equation [START_REF] Israwi | Large time existence for 1d Green-Naghdi equations[END_REF] individually. The first term can be written as

∂ ∂ t ũH + β w ∇H h = ∂ ∂ t ũ - β 3 h2 ∆ũ -β h( ∇H • ũ)∇ H h + O(β 2 ), which suggests defining K = ũ - β 3 h ∇H h3 div H ũ , (28) 10
With this definition, the second term can be written as

curl ũH + β ∇H h × ũH = ∂ ∂ ỹ (ũ + β wh x)ṽ -∂ ∂ x (ṽ + β wh ỹ)ṽ ∂ ∂ x (ṽ + β wh ỹ)ũ -∂ ∂ ỹ (ũ + β wh x)ũ =   ∂ ∂ ỹ K1 -∂ ∂ x K2 ũ ∂ ∂ x K2 -∂ ∂ ỹ K1 ṽ   + β h2 3   ∂ ṽ ∂ x -∂ ũ ∂ ỹ ∆H ṽ ∂ ũ ∂ ỹ -∂ ṽ ∂ x ∆H ũ   + O(β 2 ) =   ∂ ∂ ỹ K1 -∂ ∂ x K2 ũ ∂ ∂ x K2 -∂ ∂ ỹ K1 ṽ   + O(β 2 ),
where we have used the fact (shown in the appendix) that

∂ ũ ∂ ỹ -∂ ṽ ∂ x = O(β).
Finally, the last term in [START_REF] Israwi | Large time existence for 1d Green-Naghdi equations[END_REF] can be written as

∇H ũ -β 3 h2 ∆ ũ -β h(∇ H • ũ)∇ H h + O(β 2 ) • ũ -β h2 3 ∆H ũ + O(β 2 ) + h - 1 2 | ũ| 2 -2β h2 3 ũ • ∆ũ + β| h( ∇H • ũ)| 2 + O(β 2 ) = ∇H K • ũ + h - 1 2 | ũ| 2 + β| h( ∇H • ũ)| 2 + O(β 2 ).
Thus with the definition of K as in [START_REF] Kim | A derivation of the Green-Naghdi equations for irrotational flows[END_REF], the relation ( 25) can be written as

∂ ∂ t K + curl( K) × ũ + ∇H K • ũ + h - 1 2 β h2 | ∇H • ũ| 2 = O(β 2 ). ( 29
)
This equation is the approximate form of [START_REF] Gavrilyuk | Generalized vorticity for bubbly liquid and dispersive shallow water equations[END_REF] which is valid to the same order of approximation as the Green-Naghdi system. However, as in the two-dimensional case, it appears that the equation is actually an exact consequence of the Green-Naghdi equations, and the O(β 2 ) term in (29) may be replaced by zero. Then, using the equation ( 23), the balance law (29) may be written in dimensional variables as

∂K ∂t + curl(K) × ū + ∇ H K • ū + gh - 1 2 |ū| 2 + D H h Dt 2 = 0, (30)
where [START_REF] Kim | A derivation of the Green-Naghdi equations for irrotational flows[END_REF], and the material derivative in this context is defined as

K = ū -1 3h ∇ H h 3 ∇ H • ū is the dimensional form of
D Dt = ∂ ∂t + ū • ∇ H .

Discussion

In both potential and rotational flows, solutions of the water wave problem satisfy the exact balance equation

∂ u • ∂ϕ ∂s i ∂t + ∂ ∂s i p ρ + gz - |u| 2 2 = 0,
where s i are Lagrangian coordinates, and ϕ denotes an ensemble of particle paths. This equation holds both in two and in three dimensions. In two dimensions, the above relation gives rise to the differential conservation equation ( 14) involving the tangential velocity along the free surface which we denoted by K. In three dimensions, the somewhat more complicated equation ( 21) appears. These conservation equations have been studied in the Green-Naghdi approximation which assumes that surface waves are long when compared to the undisturbed depth of the fluid, and where terms of order higher than β, where β = h 2 0 /l 2 is the long-wave parameter, are disregarded. In the case of potential flow, it was found that these equations give rise to corresponding approximate balance laws which have the same form as the exact conservation equations ( 6) and ( 28) of the Green-Naghdi equations in two and three dimensions, respectively. In this respect, we have completed the physical interpretation of the conservation laws connected with the Green-Naghdi equations in the case of potential flow.

To summarize the conservation laws in the three-dimensional setting, note that solutions of the Green-Naghdi equations admit conservation of mass, given by

∂h ∂t + ∇ H • (hū) = 0.
Conservation of momentum is represented by

∂ ∂t {hū} + ∇ • hū ⊗ ū+ g h 2 2 + 1 3 h 2 D 2 h Dt 2 I = 0,
where

D 2 Dt 2 = ∂ ∂t + ū • ∇ H ∂ ∂t + ū • ∇ H
, and I is the identity matrix. Conservation of energy in the three-dimensional Green-Naghdi system is represented by the following conservation law:

∂ ∂t h |ū| 2 2 + E + ∇ H • ū h |ū| 2 2 + E + p = 0,
where

E = gh 2 2 + h 6 Dh Dt 2 .
Finally, the equations admit the exact consequence [START_REF] Lannes | Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation[END_REF]:

∂K ∂t + curl(K) × ū + ∇ H K•ū + gh - 1 2 |ū| 2 + Dh Dt 2 = 0, with K = ū -1 3h ∇ H h 3 ∇ H • ū = ū + 1 3h ∇ H h 2 Dh
Dt . This conservation law is obtained as an approximation of a general exact balance law [START_REF] Green | On the theory of water waves[END_REF] at the free surface for the full Euler equations in the case of potential motions. Since both ( 22) and ( 30) also hold in the the case of rotational flow, a natural question is if the asymptotic equivalence can be extended to this more general case. One possible approach would be to use ideas from [START_REF] Castro | Well-posedness and shallow water stability for a new Hamiltonian formulation of the water waves equations with vorticity[END_REF] where a fully nonlinear long-wave system is derived in the case of non-potential flow. In this approximation, the velocity field below the surface is governed by an additional evolution equation, and this equation could be the basis for finding an approximation to the velocity field in the rotational case. However, this is beyond the scope of the present article.

The importance of the variable K for numerical purposes is demonstrated in [START_REF] Li | High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model[END_REF][START_REF] Métayer | A numerical scheme for the Green-Naghdi model[END_REF]. Analogous equations can also be obtained in the case of multi-layer flows, such as for instance in [START_REF] Barros | Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties[END_REF] and [START_REF] Percival | A Euler-Poincaré framework for the multilayer Green-Naghdi equations[END_REF].

Appendix

Derivation of the 2D Green-Naghdi equations

The derivation of the Green-Naghdi equations in two spatial dimensions can be found in [START_REF] Barthélemy | Nonlinear shallow water theories for coastal waves[END_REF] in the case of potential flows. The derivation in the case of non-potential flows can be found, in particular, in [START_REF] Barros | Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties[END_REF] where a variational approach was used based on the Hamilton principle. However, a direct averaging the Euler equations can also be done. Here we will indicate some of the key steps for the vorticity case. In order to understand the approximation of ( 14) in the Green-Naghdi theory, we recall the scaling which is used in the derivation of the system (1)-( 2). Using the non-dimensionalization (15) from Section 2.2, and averaging the continuity equation ( 9) over the local depth of the fluid leads to the continuity equation ( 1), where the average horizontal velocity ũ is defined by

ũ = 1 h h 0 ũdz.
Using the boundary conditions, the continuity equation and depth averaged values, the momentum equation In particular, it implies (see details in [START_REF] Barros | Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties[END_REF]) that 1) ).

ũt + ũũ x + ηx + β h ∂ ∂ x h 0 zΓ(x, z, t) dz = - 1 h ∂ ∂ x h 0 ũ2 -ũ2 dz
|ũ -ũ| = O(β min(s,1) ), h 0 ũ2 -ũ2 dz = O(β 2 min(s,
Since the principal term of w is in the form w(x, z, t) = -z ∂ ũ ∂

x + O(β min(s,1) ), and therefore, Γ(x, z, t) = -z ũx t + ũũ xx -ũ2

x + O(β min(s,1) ), we find in the case s > 1/2 the equation

ũt + ũũ x + ηx - β 3 h ∂ ∂ x h3 ũx t + ũũ xx -ũ2 x = O(β 2 min(s,1) )
valid up to order β. So, the derivation of the equation for the average velocity does not require the potentiality hypothesis, the vorticity should only to be quite small.

Derivation of the 3D Green-Naghdi equations

In the three-dimensional case we use the additional scalings ỹ = y l and ṽ = v c 0 , so that we can write the horizontal velocity in the form ũH = 1 c 0 u H . Averaging over the local depth of the fluid leads to the continuity equation

ηt + ∇H • ( hũ ) = 0, (31)
where ũ = ( ũ, ṽ) is the averaged non-dimensional horizontal velocity: ū = ũ = 1 (ũṽ) 2 -( ṽ) 2 dz, appear, where Γ(x, ỹ, z, t) = wt + ũ wx + ṽ wỹ + w wz .

As in 2D case, no need to suppose that the flow is potential. The only hypothesis is that the dimensionless vertical gradient of the horizontal velocity ũH is initially small (see details in [START_REF] Barros | Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties[END_REF] ) :

∂ũ H ∂ z t=0 = O(β s ), s > 1/2.
Then (see [START_REF] Barros | Dispersive nonlinear waves in two-layer flows with free surface. I. Model derivation and general properties[END_REF]) :

ũH -ũH = O(β min(s,1) ),

where T (h, V) = -

1 3h ∇H (h 3 ∇H • V), Q(h, V) = - 1 3h ∇H h 3 V • ∇H ( ∇H • V) -( ∇H • V) 2 .

Alternative representation of the horizontal velocity

An alternative representation of the horizontal velocity field can be obtained by using the relation ∆H ũ = ∇H ∇ H • ũ + O(β), which holds for potential flows. Indeed, note that

∇H ∇H • ũ = ∂ ∂ x ∂ ũ ∂ x + ∂ ṽ ∂ ỹ , ∂ ∂ ỹ ∂ ũ ∂ x + ∂ ṽ ∂ ỹ = ∂ 2 ũ ∂ x2 + ∂ 2 ũ ∂ ỹ2 - ∂ ∂ ỹ ∂ ũ ∂ ỹ - ∂ ṽ ∂ x , ∂ 2 ṽ ∂ x2 + ∂ 2 ṽ ∂ ỹ2 + ∂ ∂ x ∂ ũ ∂ ỹ - ∂ ṽ ∂ x = ∆H ũ + ∂ ω ∂y , - ∂ ω ∂x ,
where we have introduced the the vorticity of mean horizontal velocity:

ω = ∂ ṽ ∂ x - ∂ ũ ∂ ỹ .
Let us estimate ω under the condition that the instantaneous flow is irrotational: v x -u y = 0.

We Hence, the vorticity of mean horizontal velocity can be seen to be of the order of β. As a consequence, one can also use the following approximate formula for the horizontal velocity field at the free surface in terms of the average horizontal velocity:

ũH (x, h, t) = ũ -β h2 3 ∇H ∇H • ũ + O(β 2 ).

Figure 1 :

 1 Figure 1: This figure indicates the time evolution of an arc L 0 which lies entirely in the free surface.

  appears, where Γ(x, z, t) = wt + ũ wx + w wz is the vertical acceleration. The estimation of the correlation h 0 ũ2 -ũ2 dz is now needed. In the two-dimensional case, the vorticity has only one scalar component ω which satisfies the equation ωt + ũω x + w ωz = 0, where ω = ũz -β wx . This implies the following equation for Ω = ũz : Ωt + ũ Ωx + w Ωz = O(β). If, initially, Ω t=0 = O(β s ), s > 0, then for any time Ω = O(β min(s,1) ).

h h 0

 0 ũH dz. Using the boundary conditions, the continuity equation and depth averaged values, the momentum equations ũt + ( ũ • ∇H ) ũ + ηx + β h ∂ ∂ x h 0 zΓ(x, ỹ, z, t) dz =

h 0 ũ2- 0 (ũṽ) 2 - 2 +

 0022 ũ2 dz = O(β 2 min(s,1) ),h ũ2 = O(β 2 min(s,1) ), h 0 ṽ2 -ṽ2 dz = O(β 2 min(s,1) ). and that the vertical velocity becomesw(x, z, t) = -z ∇H • ũ + O(β min(s,1) ). Therefore, Γ(x, ỹ, z, t) = -z D Dt ∇H • ũ -∇H • ũ O(β min(s,1) • ∇H ,and we find the equation ũt + β T ( h, ũt ) + ũ • ∇H ũ + ∇H η + βQ( h, ũ) = O(β 2 min(s,1) ),

  Now from[START_REF] Kalisch | Mechanical balance laws for the Green-Naghdi system and applications to undular bores, shoaling and wave loading[END_REF] and[START_REF] Khorsand | Particle trajectories in the Serre equations[END_REF], we see that| ũ -ũ (t, x, y, h(t, x, y))| = O (β) ,and| ṽ -ṽ (t, x, y, h(t, x, y))| = O (β) .

	have
	ω =	∂ ∂ x	1 h h 0	ṽdz -	∂ ∂ ỹ	1 h h 0	ũdz
	=	hx h ṽ -	hỹ h ũ +	1 h h 0	(ṽ

x -ũỹ ) dz + hx h ṽ t, x, ỹ, h( t, x, ỹ) -hỹ h ũ t, x, ỹ, h( t, x, ỹ) = -hx h ṽ -ṽ t, x, ỹ, h( t, x, ỹ) + hỹ h ũ -ũ t, x, ỹ, h( t, x, ỹ) .

Acknowledgments. The autors thank reviewers for valuables comments and suggestions. SG was partially supported by ANR BoND, France. HK and ZK acknowledge support by the Research Council of Norway.


MED12, OCEANIC COMPONENT FOR THE MODELING OF THE REGIONAL MEDITERRANEAN EARTH SYSTEM

Introduction

MED12 [START_REF] Brossier | The Mediterranean response to different space-time resolution atmospheric forcings using perpetual mode sensitivity simulations[END_REF], 2012a;[START_REF] Beuvier | Spreading of the Western Mediterranean Deep Water after winter 2005: time-scales and deep cyclone transport[END_REF] is the new regional configuration of the Mediterranean Sea of the ocean general circulation model NEMO [Madec and the NEMO Team;[START_REF] Madec | NEMO ocean engine, Note du Pôle de modélisation[END_REF]. The MED12 grid has been extracted from the 1/12°global operational Mercator model [START_REF] Lellouche | Evaluation of global monitoring and forecasting systems at Mercator Océan[END_REF]]. The development of MED12 is made in the continuity of the evolution of the French modeling of the Mediterranean Sea, following OPAMED16 [START_REF] Béranger | Seasonal variability of water transport through the Straits of Gibraltar, Sicily and Corsica, derived from a high-resolution model of the Mediterranean circulation[END_REF], OPAMED8 [START_REF] Somot | Transient climate change scenario simulation of the Mediterranean Sea for the twenty-first century using a high-resolution ocean circulation model[END_REF] and NEMOMED8 [START_REF] Beuvier | Modeling the Mediterranean Sea interannual variability during 1961-2000: Focus on the Eastern Mediterranean Transient[END_REF]. The MED12 domain covers the entire Mediterranean Sea and a part of the Atlantic Ocean, until 11°W. MED12 does not cover the Black Sea. With a horizontal resolution of 1/12°, MED12 is an eddy-resolving model in the major part of the domain. The vertical resolution uses 50 unevenly spaced vertical levels with the partial-cells parameterization for the bottom layer. The time step is 12 minutes.

The bathymetry comes from the 10 th version of the Mercator-LEGOS bathymetry at a resolution of 30''x30'', composed of the merging between the GEBCO-08 database, the MEDIMAP bathymetry [Medimap [START_REF] Group | Morpho-bathymetry of the Mediterranean Sea[END_REF] and the Ifremer bathymetry of the Gulf of Lions [START_REF] Berné | Le golfe du Lion -Carte morpho-bathymétrique[END_REF]. The initial states come from the MEDATLAS-II climatology [MEDAR/ [START_REF] Group | MEDAR/MEDATLAS 2002 Database, Mediterranean and Black Sea Database of Temperature Salinity and Bio -chemical Parameters[END_REF] in the Mediterranean domain and from the climatology of [START_REF] Levitus | Warming of the world ocean, 1955-2003[END_REF] in the Atlantic domain. MED12 includes a new parameterization of the Atlantic inputs modelled through damping of 3Dtemperature, 3D-salinity and sea surface height climatologies. The sea surface height climatology is built with GLORYS-1 data [START_REF] Ferry | Mercator Global Eddy Permitting Ocean Reanalysis GLORYS1V1: Description and Results[END_REF] to conserve the ocean volume in a regional configuration using the filtered sea surface parameterization of NEMO. The bottom turbulent kinetic energy background in the bottom friction parameterization is a 2D field corresponding to the mean tidal energy computed from a tidal model [START_REF] Lyard | Modeling the global ocean tides: modern insights from FES2004[END_REF]. The river runoffs of [START_REF] Ludwig | River discharges of water and nutrients to the Mediterranean and Black Sea: Major drivers for ecosystem changes during past and future decades?[END_REF] and the Black Sea inputs are included following the method of [START_REF] Beuvier | Modeling the Mediterranean Sea interannual variability during 1961-2000: Focus on the Eastern Mediterranean Transient[END_REF]. More details can also be found in [START_REF] Beuvier | Spreading of the Western Mediterranean Deep Water after winter 2005: time-scales and deep cyclone transport[END_REF].

The MED12 model has been developed in the context of the SiMED and MORCE projects, in collaboration with Mercator Ocean teams and national laboratories. Several ongoing projects aim at coupling MED12 with atmospheric models and biogeochemical models. In the following, we present first results obtained with a free ocean simulation and with an ocean-atmosphere coupled simulation.

Ocean modeling

A 53-year simulation from 1958 to 2011, called hereafter MED12-long [START_REF] Beuvier | Modeling the long-term variability of circulation and water masses in the Mediteranean Sea : impacts of the ocean-atmosphere exchanges[END_REF], was carried out. For the initial state, 3D temperature and salinity initial fields are weighted by a low-pass filtering with a time window of ten years of the MEDATLAS data covering the 1955-1965 period. The simulation starts in October 1958 with an ocean at rest. The atmospheric forcing is ARPERA [START_REF] Herrmann | Relevance of ERA40 dynamical downscaling for modeling deep convection in the Mediterranean Sea, Geophys[END_REF], a dynamical downscaling of the ERA40 reanalysis [Simons and Gibson, 2000] by the ARPEGE-Climate model [START_REF] Déqué | High resolution climate simulation over Europe[END_REF].

The long-term variability of the Mediterranean Sea is illustrated by interannual variations of heat and salt contents between 1958 and 2011, expressed here by 3D-average temperature and salinity, T3D and S3D respectively (Figure 1). The model is compared with the two gridded climatologies IN3 [START_REF] Ingleby | Quality control of ocean temperature and salinity profiles -Historical and real-time data[END_REF] and MEDATLAS [START_REF] Rixen | The Western Mediterranean Deep Water: A proxy for climate change[END_REF]. The T3D and S3D of the whole water column (Figures 1a and1b) are in good agreement with observations. The T3D is a little too high by 0.1°C in MED12-long while the S3D trend is well simulated, but with a lack of interannual S3D variations. In the surface layer (0-150m), the T3D is very well reproduced (Figure 1c), mainly thanks to the retroaction term relaxing the model sea surface temperature towards the "observed" sea surface temperature of [START_REF] Reynolds | An Improved In Situ and Satellite SST Analysis for Climate[END_REF]. However, values of the surface layer S3D are too low by 0.04 to 0.28 psu (Figure 1d). In the intermediate layer (Figures 1e and1f), values of both T3D and S3D are too high (+0.2°C for T3D, +0.02 to +0.08 psu for S3D). Nevertheless, the T3D interannual variations of the intermediate layer are well captured. In the bottom layer (Figures 1g and1h), the T3D and S3D trends are a little higher than in the climatological datasets (+0.08°C, +0.03 psu) and no interannual variations are simulated. However, a large uncertainty exists for this layer, as both gridded datasets may over-estimate deep temperature and salinity interannual variations because of under-sampling (more pronounced for salinity). 600m-bottom, g and h), and, for differents datasets: the MED12-long simulation (blue solid line), the interannual gridded database of [START_REF] Rixen | The Western Mediterranean Deep Water: A proxy for climate change[END_REF] (black dashed lines, with the ±1 standard deviation interval in grey) and the EN3 interannual gridded database [START_REF] Ingleby | Quality control of ocean temperature and salinity profiles -Historical and real-time data[END_REF] (green solid line). The S3D scales are five times lower than the T3D allowing the representation of the same impact on potential density. This MED12 configuration is devoted to be used soon in the reanalysis of the Mediterranean circulation driven by ALADIN-Climate atmospheric forcing from CNRM (horizontal resolution of 12 km). At the same time, new versions of Mediterranean configurations are being developed, using more 75 vertical z-levels or using a higher horizontal resolution MED36 (1/36°, horizontal resolution of 2.5 km).

Ocean-Atmosphere Coupling

MED12 is the ocean compartment in the MORCE (Model of the Regional Coupled Earth system) system [START_REF] Drobinski | Model of the Regional Coupled Earth system (MORCE): application to process and climate studies in vulnerable regions[END_REF]. MED12 is coupled to the Weather Research and Forecasting (WRF) atmospheric model [START_REF] Skamarock | A description of the Advanced Research WRF Version[END_REF]. The WRF domain covers the Mediterranean basin [28°N-50°N, 13°W-42°E] with a 20-km horizontal resolution and has 28 sigma-levels in the vertical. Initial and lateral conditions are taken from the ERA-interim reanalysis [START_REF] Simons | ERA-interim: New ECMWF reanalysis products from 1989 onwards[END_REF] provided every 6 hours with a 0.75° resolution and a nudging towards the driving fields is applied with a coefficient of 5.10 -5 s -1 for temperature, humidity and velocity components above the planetary boundary layer. The complete set of physical parameterizations chosen for the WRF model is detailed in Lebeaupin Brossier et al. [2012b].

Companion uncoupled and coupled simulations have been done. In the first one, MED12 is driven in surface with a 3-hourly frequency, by WRF fluxes extracted from the dynamical downscaling of the ERA-interim reanalysis obtained with WRF alone. And the coupled simulation runs with two-way interactive exchanges between the two compartment-models managed by the OASIS coupler [START_REF] Valcke | OASIS3 User Guide (prism_2-5)[END_REF]. The exchanged variables are the sea surface temperature, the heat flux (solar Q s and non-solar Q ns ), the water flux (F w =E-P) and the momentum flux (tau). The coupling frequency is 3 hours. Both simulations run from January 1989 to December 2008 and start with an ocean at rest.

Focusing on the extreme 2005 deep convection event in the Gulf of Lions, the comparison between the uncoupled and coupled simulations is illustrated by the horizontal and vertical distributions of the convection in Figure 3. Deeper mixed layer is formed in the southern part of the Gulf of Lions (around 4°E-41°N) in the uncoupled simulation (Figure 3a) while the deep convection area is extended farther eastwards in the coupled simulation (Figure 3b). In particular, a larger turbocline depth is found in the Tyrrhenian sub-basin (Bonifacio gyre area) and in the Ligurian subbasin. In the Ligurian sub-basin, the chronology of the convection events driven by strong winds is the same in the two simulations. But the saltier and colder intermediate layer is rapidly mixed on 15 th February 2005 in the coupled simulation and then the mixed layer reaches the sea bottom of the Ligurian sub-basin (2200 m). In the uncoupled simulation, the turbocline depth reached only 450 m depth on 8 th March 2005. The difference in the characteristics of intermediate and deep layers is the ocean response to the coupled mode, and is an integrated signature of the air-sea flux modulation by the coupling (for example, larger heat loss over the Ligurian and Bonifacio cyclonic gyres in winter 2005, Figure 3).

The coupling of MED12 with other atmospheric models is currently in development (COSMO-CLM at Frankfurt University, ALADIN-Climate at Météo-France), to be part of multiple-compartment regional climate systems over the Mediterranean region and to further investigate the interannual variability of strong air-sea exchanges that trigger in particular the deep ocean convection.

Conclusion

MED12, oceanic regional model based on the NEMO code, is an efficient numeric tool, dedicated to the study of complex oceanic processes and adapted to the specificities of the Mediterranean Sea. Driven by interannual atmospheric forcings, rivers and Black Sea freshwater inputs, exchanges with the Atlantic Ocean, for long-term runs, MED12 allows assessing the variability of processes such as the open sea convection. MED12 is also coupled to several atmospheric models in order to study the interactions between atmospheric and oceanic fine scale processes on long periods of time. Finally, note that coupling with biogeochemical models (e.g. Eco3M or PISCES) is underway. Added to the coupling of atmospheric models, including hydrological and vegetation models, MED12 is an important modeling component that fully completes several Regional Earth System configurations, and is thus a key model involved in several current projects that investigate the Mediterranean Sea climate system.

  Figure1: Mean potential temperature T3D (a, c, e and g) and mean salinity S3D(b, d, f and h) for different layers of the Mediterranean Sea: the whole water column (0 m-bottom, a and b), the upper layer (0-150 m, c and d), the intermediate layer (150-600 m, e f), and the deep layer (600 m-bottom, g and h), and, for differents datasets: the MED12-long simulation (blue solid line), the interannual gridded database of[START_REF] Rixen | The Western Mediterranean Deep Water: A proxy for climate change[END_REF] (black dashed lines, with the ±1 standard deviation interval in grey) and the EN3 interannual gridded database[START_REF] Ingleby | Quality control of ocean temperature and salinity profiles -Historical and real-time data[END_REF] (green solid line). The S3D scales are five times lower than the T3D allowing the representation of the same impact on potential density.

  component for the modeling of the regional Mediterranean earth system Intermediate and deep convection regularly occur in specific areas of the Mediterranean Sea as illustrated in Figure2. The turbocline depth is used to show the convection depth as it is a better criterion available in NEMO for the low stratified Mediterranean Sea than the criteria on the thermocline or the pycnocline depth. The 53-year mean of the annual turbocline depth maxima (Figure2a) highlights well known areas of intermediate or deep convection, such as the Gulf of Lions (around 42°N -5°E), the Rhodes Gyre in the Levantine sub-basin (around 34°N -26°E), the southern Adriatic sub-basin and the Aegean sub-basin. A new area is also captured in the northern Ionian sub-basin. However, the values for the Adriatic sub-basin are not deep enough, barely reaching 300 m depth in average whereas observations show values reaching a maximum of about 1200 m depth during several years. Interannual variations (Figure2b) are relatively high in the Gulf of Lions (standard deviation of 800 m). Interannual variations are also relatively high in the Levantine sub-basin (standard deviation of 500 m) and in the northern Ionian basin. According to the ratio between the standard deviation and the 53-year mean of turbocline depth (Figure2c), values over 120% at the north-eastern and the south-western extremities of the convection area in the Gulf of Lions, indicate the maximum spatial extent of the deep water formation that can be reached during strong events. In the same way, high values in the Levantine area are relative to formation of Levantine Deep Water. And values around 80% in the northern Ionian sub-basin show intermediate water formation during very strong winters.

  Figure2: (a) Average of the yearly maximum turbocline depth over the period 1958-2011 (in meters, contours every 250 m), (b) standard deviation of the yearly maximum turbocline depth around the average in (a) (in meters, contours every 200 m), and (c) ratio (in %) between (b) and (a).

  Figure3: Top panels: Depth of the turbocline for 28 th February 2005 (in m, color) and monthly mean heat loss (W m -2 , dashed lines every 75 W m -2 ). Bottom panels: for the period January to March 2005 at 7.8°E-43.4°N between 0 and 2000 m in the Ligurian sub-basin, Howmöller diagrams of temperature profiles (°C, color), salinity profiles (psu, purple contours every 0.1 psu) and turbocline depth (in m, white contour) -in the (a) uncoupled and (b) coupled experiments.

Acknowledgements

This work was granted access to the HPC resources of IDRIS (Institut du Développement et des Ressources en Informatique Scientifique) of the Centre National de la Recherche Scientifique (CNRS) under allocation 2010 and 2011 (project number 010227) made by Grand Equipement National de Calcul Intensif (GENCI). This work is a contribution to the HyMeX international program on the study of the water cycle in the Mediterranean Sea.


Sylvie Masse

Guillaume Laurent

T Coradin

Solid state NMR and diatoms: probing the interfaces

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

  (t cp =3 ms) While a lot of work is still needed to fully understand diatoms frustule interface, solid-state NMR appears to be a powerful toolbox. Indeed, this technique is able to selectively probe either mobile or rigid species at a very local scale. Varying methods, species proximity can be checked NMR experiments allow to probe another part of the sample: the rigid one. Playing with the selected nuclei, spatial proximities can be assessed. 1. Tesson, B., Masse, S., Laurent, G., Maquet, J., Livage, J., Martin-Jézéquel, V. and Coradin, T. Contribution of multi-nuclear solid state NMR to the characterization of the Thalassiosira pseudonana diatom cell wall. Anal Bioanal Chem 390, 1889-1898 (2008). 2. Hatté, C., Hodgins, G., Jull, A. J. T., Bishop, B. & Tesson, B. Marine chronology based on 14 C dating on diatoms proteins. Marine Chemistry 109, 143-151 (


Thibaud Coradin

Correlating biological methods to assess Escherichia coli bacteria viability in silica gels

Correlating biological methods to assess

The viability of Escherichia coli bacteria entrapped in silica gels can be monitored using the Alamar Blue assay and ATP-metry, two complementary methods that correlate with the traditional plate count technique while avoiding its current limitations related to cell growth conditions.

Assessment of bacterial cell viability is routinely performed using the plate count technique. This technique relies on the spreading of a cell suspension on a solid nutrient media that is kept in usual optimal growth conditions, i.e. at 37°C over 24 h for common bacteria. After this delay, each initial living cell has grown into a colony so that viability is given by the number of colony forming units (CFU) per mL of the starting suspension. However, there are two main situations where the plate count technique cannot be applied. The first one concerns cells whose growth conditions are uncommon, if not unknown. 1 The second possibility is that cells have entered the so-called viable but non-culturable (VBNC) state where the cells are alive but have a very low metabolic activity and thus cannot be grown in usual conditions. 2 Bacterial cells usually enter the VBNC state as a response to a stress situation. 3 This is usually the case for Escherichia coli bacteria trapped in silica gels as they are unable to divide in the mineral network. 4,5 In these conditions, their viability may not be reliably assessed by the plate count technique after extraction from the gels since this method excludes non culturable cells. Alternative methods must be found to investigate the immobilized cells, such as monitoring complex metabolisms 4,6 or genetic responses via bioluminescent reporters 5,7 However, none of these approaches can be directly correlated to bacteria viability.

To address this problem, we have here adapted and evaluated two methods, the Alamar Blue test and ATP-metry measurements to silica-immobilized E. coli bacteria, and compared them to the plate count technique. The sol-gel procedure followed the previously-described aqueous route in the presence of glycerol. 4 However only sodium silicate solutions (0.8 M) were used as silica source and addition of silica nanoparticles was not performed to favor gel redispersion. Full experimental details for encapsulation and cell viability determination protocols are provided as ESI. All experiments were performed at least in triplicate.

As a reference, the CFU of the initial bacteria suspension (CFU) 0 was determined by plate counting. The same technique was used to determine the number of viable cells in silica gels at different ageing time (CFU) t just after encapsulation (t = 15 min) and after 1, 7 and 15 days. The viability rate was then calculated as 100x(CFU) t /(CFU) 0 . As shown in Fig. 1, a slow decay in cell viability is observed with time, reaching ca. 50 % after 2 weeks.

Viability (%) Time

The Alamar Blue assay is based on the reduction of the resazurin blue dye into a colourless hydroresorufin molecule, as a result of an oxygen-consuming metabolic pathway and therefore mainly due to cell respiration. 8 It is widely used to assess viability of mammalian cells but was also applied to bacteria and yeasts. 9 In a first step, a calibration curve was established between the cell number in a bacterial suspension, as determined by the plate count technique, and the reduction rate (in %). As shown in Figure 2, a linear correlation is obtained, with a slope of 3.7 ± 0.4 10 -7 %.mL.CFU -1 . The same procedure was then applied to bacteria recovered from gel resuspension after 15 min to 15 days ageing. A linear correlation is also obtained between cell number and % reduction, with a slope of 3.6 ± 0.5 10 -7 %.mL.CFU , i.e. similar to the reference bacterial suspension. This indicates that the aerobic respiration of bacteria is not significantly perturbed as a result of encapsulation. It also becomes possible to use the calibration curve to obtain the number of viable cells from the Alamar Blue test and to calculate the corresponding viability rate. As shown in Fig. 1, a very good correlation is obtained between this method and the plate count technique.

Fig.2 Calibration curve of the Alamar blue reduction as a function of cell density for bacteria suspension and encapsulated cells. Error bars correspond to standard deviation

In a step forward, we evaluated the ATP-metry method, based on the bioluminescent reaction associated with the simultaneous ATP hydrolysis and luciferin oxydation catalyzed by the luciferase enzyme. 10 Depending on the protocols, intracellular ATP, indicative of cell metabolism, extracellular ATP, assessing cell lysis or both can be measured. 11 ATP-metry is widely used for microbial assessment of drinking water as well as in toxicology studies. 12 To avoid the use of the plate count technique, the number of viable cells was estimated from the Alamar Blue test described above; the concentration of intracellular ATP was calculated for 10 6 CFU. As shown in Figure 3, that ATP production was constant over time for encapsulated bacteria and higher than the mean value obtained for the initial bacteria suspension. Since this increase is observed just after encapsulation and does not vary upon ageing, it suggests that the sol-gel reaction itself has an impact on the cell metabolic activity. It is interesting to note that the nearly two-fold increase in ATP production upon encapsulation correlates well with recent demonstration of enhanced fluorescent production in silica-immobilized transformed bacteria. 7b This effect appears irreversible in the conditions of these experiments. However, it must be noticed that ATP-metry applied to bacterial suspension showed high standard deviation (> ± 1 nM) so that care must be taken when drawing conclusions. In this situation, it is not strictly possible to correlate these responses with viability rates taking the bacteria suspension as a reference. However, the evolution of cell viability with time can still be monitored using the t = 15 min gel sample as a reference. As shown in Fig 4, it then becomes possible to establish a good correlation between plate count, Alamar Blue and ATP-metry measurements. Noticeably, at each time point, there is no statistical difference in relative activity determined by the three methods.

Time

Hence, the two methods appear highly complementary as the Alamar Blue provides a direct indication on cell viability whereas the ATP-metry reveals a modification of the cell activity. In the present case, where the initial sodium silicate concentration (0.8 M) was significantly higher than in our previous procedure (0.4 M) 4 in order to compensate for the lack of silica nanoparticles, the observed decrease in viability and increase in ATP production can be attributed to the high ionic strength. 13 This hypothesis is strengthened by SEM observation of the encapsulated bacteria (Fig. 5) showing large intracellular polyphosphate granules that have been reported in microbial cells as a consequence of osmotic pressure. 14

Conclusions

Due to their simplicity, robustness and limited experimental time (max. 2 h vs. 24 h for plate counting), the Alamar Blue and ATP-metry methods appear very useful for future investigations of sol-gel cell encapsulation. They can be applied to many different organisms in oxic or anoxic conditions, 9,12 opening the route to inter-species comparison studies. This would be particularly interesting for different prokaryotic phyla, such as Archea or cyanobacteria cells, not only targeting applications for energy production or environmental remediation, 15 but also for more fundamental issues related to fossilization processes. 16

  Escherichia coli bacteria viability in silica gels M. Blondeau, a,b R. Brayner, c F. Guyot d and T. Coradin* a,b

Fig. 1 .

 1 Fig. 1. Evolution of bacteria viability within silica gels using the plate count and Alamar blue techniques. Error bars correspond to standard deviation.

Fig. 3

 3 Fig. 3 Evolution of ATP production by encapsulated bacteria. The yellow dashed line corresponds to ATP production by bacteria suspension. Error bars correspond to standard deviation. Yellow dashed line indicates mean value for bacteria suspension.

Fig. 4

 4 Fig. 4 Comparison of the relative viability of encapsulated bacteria (using t = 15 min as 100 %) as determined by the plate count, Alamar blue and ATP--metry. Error bars correspond to standard deviation.

Fig. 5

 5 Fig. 5 SEM--FEG image of encapsulated E. coli bacteria. White arrows indicate polyphosphate granules that appear dark under secondary electron imaging. Scale bar = 1 µm

Acknowledgments

M.B. thanks the Région Ile-de-France for funding in the framework of C'Nano IdF. C'Nano-IdF is the nanoscience competence center of Paris Region, supported by CNRS, CEA, MESR and Région Ile-de-France.

Notes and references