A Laouar

A Guerziz

A Boussaha
email: boussahaaicha@yahoo.fr

Mous

A Novel Scheme of Dispersive Propagation Model for Earthquake Generated Tsunami

Keywords: Alternating directions scheme, Numerical dispersion-correction, Shallow water

In this study, we interested in simulating an irregular wave trains (IWTs) propagation generated by tsunami source. There didn't discuss in the literature, for this we will explain the e¤ects of ocean viscosity by virtue of mathematical aspect. Weakly dissipative terms are introduced into linear shallow water equations (LSWqs) modi…ed system . We employ the third-order alternating directions implicit (ADI) predictor corrector (PC) scheme for spatial derivatives with three-time level. As a result, both numerical weakly dissipation and linear dispersion are kept with good precision. Subsequently, we shall show that even oceanic propagation of dispersive waves can be quite e¢ ciently modeled using mathematically well-founded LSWqs modi…ed scheme. The convergence, stability and consistency criteria of the scheme are also studied. The results are compared with the work of Madsen et al.[12].

Introduction

Tsunamis are ocean water surface waves generated by undersea earthquakes, landslides, volcanic eruptions or even meteoric impact on water surfaces. A tsunami moves in any direction of the source region , where huge quantities of water are displaced. These may cause serious damage in coastal areas. To minimize such damage, a proper monitoring, forecasting and warning system needs to be found [START_REF] Warming | The modi…ed equation approach to the stability and accuracy analysis of …nite di¤erence methods[END_REF][START_REF] Yang | Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part I: Formulation,implementation and numerical properties[END_REF][START_REF] Yoon | Dispersion-Correction …nite di¤erence model for simulation of transoceanic tsunamis[END_REF]. Thus, the development of an accurate and e¢ cient numerical model for computing all aspects of tsunami propagation is indispensable. The tsunamis grow as they approach shallow water and smash onto the shore. The models for the propagation of surface waves over shallow inviscid ‡uid layer are Boussinesq's equations [START_REF] Boussinesq | Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal[END_REF]. A perturbation method to solve the Laplace equation in the bulk is developed by Boussinesq [START_REF] Antonopoulos | Numerical solution of the'classical'Boussinesq system[END_REF][START_REF] Boussinesq | Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal[END_REF]. He arrived at a generalized wave equation that contains dispersion in addition to the standard terms [START_REF] Boussaha | Etude numérique de quelques problèmes issus de la mécanique des ‡uides: ondes longues de gravité et propagation d'un tsunami[END_REF][START_REF] Boussaha | A New modi…ed scheme for linear shallow water equations with distant propagation of irregular wave trains tsunami dispersion type for inviscid and weakly viscous ‡uids[END_REF]. Boussinesq equations are one of the best choices for governing equations to construct numerical models for far-…eld tsunamis as they can take into account the dispersion e¤ects. Analytical and numerical studies with improved characteristics of the Boussinesq models have received considerable attention of scientists. However, the Boussinesq numerical models [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF][START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowlyvarying bathymetry[END_REF][START_REF] Madsen | On the solitary wave paradigm for tsunamis[END_REF], require a small mesh size to suppress numerical dispersion errors. This consumes huge amounts of computer resources due to the implicit nature of the solution technique to deal with dispersion terms. Thus, the Boussinesq model is not preferred for the simulation of the far-…eld tsunamis, and shallow-water equations are generally employed instead [START_REF] Yang | Second-order coupling of numerical and physical wave tanks for 2D irregular waves. Part I: Formulation,implementation and numerical properties[END_REF]. This subject has been studied by several authors; so we refer for examples to: Daripa and Hua [START_REF] Daripa | A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: Filtering and regularization techniques[END_REF] considered an ill-posed Boussinesq equation which arises in shallow water waves and nonlinear lattices. Bratsos et al. [START_REF] Bratsos | Linearized numerical schemes for the Boussinesq equation[END_REF] applied two di¤erent linearized schemes to the parametric …nite di¤erence scheme concerning the numerical solution of the Boussinesq equation. The nonlinear term of the equation is substituted by an appropriate value at the …rst linearized scheme. Antonopoulos et al. [START_REF] Antonopoulos | Numerical solution of the'classical'Boussinesq system[END_REF][START_REF] Antonopoulos | Numerical solution of Boussinesq systems of the Bona-Smith family[END_REF] considered the one-parameter family of Bona-Smith systems, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a channel. Mitsotakis [START_REF] Mitsotakis | Boussinesq systems in two space dimensions over avariable bottom for the generation and propagation of tsunami waves[END_REF] applied the standard Galerkin-…nite element method to the simpli…ed Boussinesq model of surface water wave theory over a variable bottom with homogeneous Dirichlet boundary conditions. They studied the generation and propagation of tsunami waves. The tsunami waves generated by Boussinesq model are compared by the linearized Euler equations. Dougalis et al. [START_REF] Dougalis | On some Boussinesq systems in two space dimensions: theory and numerical analysis[END_REF] considered a three-parameter family of Boussinesq type systems in two space dimensions. The three-dimensional Euler equations are approximated by these systems, and consist of three nonlinear dispersive wave equations that describe two-way propagation of long surface waves of small amplitude in ideal ‡uids over a horizontal bottom [START_REF] Boussaha | Etude numérique de quelques problèmes issus de la mécanique des ‡uides: ondes longues de gravité et propagation d'un tsunami[END_REF]. Simulating irregular wave trains (IWTs) propagation generated by tsunami source didn't discuss in the literature, for this we will explain the e¤ects of ocean viscosity by virtue of mathematical aspect. So in this study, weakly dissipative terms are introduced into linear shallow water equations (LSWqs) modi…ed system [START_REF] Boussaha | Etude numérique de quelques problèmes issus de la mécanique des ‡uides: ondes longues de gravité et propagation d'un tsunami[END_REF][START_REF] Boussaha | A New modi…ed scheme for linear shallow water equations with distant propagation of irregular wave trains tsunami dispersion type for inviscid and weakly viscous ‡uids[END_REF][START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF]. We employ the third-order alternating directions implicit (ADI) predictor corrector (PC) scheme for spatial derivatives with three-time level. As a result, both numerical weakly dissipation and linear dispersion are kept with good precision. Subsequently, we shall show that even oceanic propagation of dispersive waves can be quite e¢ ciently modeled using mathematically well-founded LSWqs modi…ed scheme. The convergence, stability and consistency criteria of the scheme are also studied. The results are compared with the work of Madsen et al. [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF]. -The incompressibility condition div u = 0;

Phenomenon Description and hypothesises

(

-The irrotational ‡ow u = r :

where potential of the velocity vector u = (u; v; w) t :

-kinematic condition @ @t + r :r = 0 with z = (x; y; t) :

-Dynamic conditions @ @t + 1 2 jr j 2 + g = 0 with z = (x; y; t) :

-The impermeability of the bottom

r :rh = 0 with z = h (x; y) : (5)

Some Models

We present and comment some known models by beginning with the linear shallow water equations (LSWqs) (see [START_REF] Boussaha | Etude numérique de quelques problèmes issus de la mécanique des ‡uides: ondes longues de gravité et propagation d'un tsunami[END_REF]):

@ @t + @M @x + @N @y = 0; (6)
@M @t + gh @ @x = 0;

@N @t + gh @ @y = 0;

where M (= uh) and N ( = vh) are the horizontal discharge ‡ow along the x and y directions. Such that h is a large enough constant depth , u and v are the average speeds calculated on the depth h of the horizontal directions x and y respectively

Remark 1 The numerical solution of the system ( 6) -( 8) lacks precisely because of the neglect of the physical dispersion. Since this system does not take into account the Coriolis acceleration and is therefore suitable for the propagation of the tsunami in short distances in a limited time (t hours)

The second model is the classical linear Boussinesq equations (CLBqs) including the coriolis force [START_REF] Antonopoulos | Numerical solution of the'classical'Boussinesq system[END_REF] and describing the propagation of distant tsunami:

@ @t + @M @x + @N @y = 0; (9)
@M @t + gh @ @x = h 2 3 @ 3 M @t@x 2 + @ 3 N @t@x@y ;

@N @t + gh @ @y = h 2 3 @ 3 N @t@y 2 + @ 3 M @t@x@y :

The right-hand side terms of Eqs.( 10)-( 11) represent the frequency dispersion.

Remark 2 Note that the system ( 9)-( 11) re ‡ects the acceleration Coriolis and describes the spread of a distant tsunami. But the presence of terms of physical dispersion represent downside for the numerical solution. Thus, the convergence of the used algorithms is relatively slow (see [START_REF] Boussinesq | Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal[END_REF] ).

The improved Boussinesq equations (IBqs): Madsen et al. [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF][START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowlyvarying bathymetry[END_REF] proposed a new form of the CLBqs in order to improve their dispersion characteristics. This model can simulate the propagation of irregular wave train (IWT) travelling from deep water to shallow water. @ @t + @M @x + @N @y = 0:

@M @t + gh @ @x = B + 1 3 h 2 @ 3 M @t @x 2 + @ 3 N @t @x @y + Bgh 3 @ 3 @x 3 + @ 2 @x @y 2 :

(13)

@N @t + gh @ @y = B + 1 3 h 2 @ 3 N @t @y 2 + @ 3 M @t @x @y + Bgh 3 @ 3 @y 3 + @ 2 @x 2 @y : ( 14
)
Where B is a curve …tting parameter (see, [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF]). The system ( 12)-( 14) is called the improved Boussinesq equations (IBqs).

Free surface equation: substitution from ( 13)-( 14) into ( 12) and eliminate M and N , yields:

@ 2 @t 2 gh @ 2 @x 2 + @ 2 @y 2 = Bgh 3 @ 4 @x 4 + 2 @ 4 @x 2 @y 2 + @ 4 @y 4 ; +h 2 B + 1 3 @ 4 @t 2 @x 2 + @ 4 @t 2 @y 2 : (15)
Eq.( 15) is called ILBq. The frequency dispersion given by the right-hand side terms of eq.( 15) may cause serious numerical di¢ culty in practice because of higher order derivatives. An alternative way is to solve a set of lower order partial di¤erential equations, it is the LSWqs. The numerical dispersion induced by the numerical scheme can be manipulated to represent the physical frequency dispersion of the ILBq. [START_REF] Mitsotakis | Boussinesq systems in two space dimensions over avariable bottom for the generation and propagation of tsunami waves[END_REF].

Mathematical Formulation and Discretization of the Problem

The main idea being proposed in this study, weakly dissipative terms are introduced into LSWqs modi…ed system ( 7)- [START_REF] Dougalis | On some Boussinesq systems in two space dimensions: theory and numerical analysis[END_REF]. In fact, the numerical dispersion plays the same role that the dispersion physical in the improved Boussinesq system of equations ( 12)-( 14); As the explicit analytical solution dimensional Boussinesq problem is almost impossible to obtain, we will then solve a problem approached. As also the simulation of irregular wave trains is little discussed in the literature [START_REF] Harter | The e¤ect of surface tension on trapped modes in water-wave problems[END_REF] , we will analyze the e¤ects of viscosity on this type of dispersion. We propose simulations step of dispersing a tsunami a¤ected by low dissipation. It focuses on the spread dispersive a non-viscous ‡uid and that of a low viscosity ( including the e¤ect a low kinematic viscosity ).

Following the approach suggested by Warming and Hyett in [START_REF] Warming | The modi…ed equation approach to the stability and accuracy analysis of …nite di¤erence methods[END_REF] the Taylor series expansions of the variables , M and N at the point represented by (k; i; j) are applied to Eqs.( 7) and [START_REF] Dougalis | On some Boussinesq systems in two space dimensions: theory and numerical analysis[END_REF]. Using the …nite di¤erence method, where the domain is discretized in a regular grid x y with a …nite number of nodes spaced x and y,the time axis is discretized in regular steps t, and the derivatives are replaced by di¤erences over small intervals. The equations ( 7) and ( 8) discretized by the scheme of ADI method [START_REF] Gunn | A General Formulation of Alternating Direction Methods, Part I.parabolic and hyperbolic problems[END_REF] give (see …gure 2.):

M k+ 1 2 i+ 1 2 ;j M k i+ 1 2 ;j t 2 + gh k+ 1 2 i+1;j k+ 1 2 i;j x = o ( t; x) ; (16)
N k+1 i;j+ 1 2 N k+ 1 2 i;j+ 1 2 t 2 + gh k+1 i;j+1 k+1 i;j y = o ( t; y) : (17)
M

k+ 1 2 i+ 1 2 ;j M k i+ 1 2 ;j t 2 = @M x i+ 1 2 ; y j ; t k @t + t 2 @ 2 M x i+ 1 2 ; y j ; t k @t 2 +; + ( t) 2 24 @ 3 M x i+ 1 2 ; y j ; t k @t 3 + o t 3 ; (18)
k+ 1 2 i+1;j k+ 1 2 i;j x = @ x i ; y j ; t k+ 1 2 @x + x 2 @ 2 x i ; y j ; t k+ 1 2 @x 2 +; + ( x) 2 6 @ 3 x i ; y j ; t k+ 1 2 @x 3 + o x 3 : (19)
In order to transform the system of equations ( 6)-(8) in a system similar to that of improved Boussinesq system ( 12)-( 14) and to make a comparison between them , we will proceed as follows: we replace ( 18) and ( 19) in ( 16), and the equation ( 7) is used to get the term @ 3 @x @t 2 ; afterwards, we apply the convection equation @M @t c @M @x = 0;

where c = p gh is the speed.

To obtain @ 3 @x @y 2 instead of the term @ 3 M @t 3 one hand, and on the other hand, to eliminate the terms @ 2 M @t 2 and @ 2 @x 2 because these terms do not appear in the improved system of Boussinesq ( 12)- [START_REF] Madsen | On the solitary wave paradigm for tsunamis[END_REF]. By analogy, the procedure is the same for the variable y: Then the problem is posed as follows:

A New Modi…ed Scheme for Linear Shallow Water

For inviscid ‡uid, the new system of equations can be written as follows: @ @t + @M @x + @N @y = o ( t; x; y) :

@M @t + gh @ @x + gh 2

1 2 @ 3 @x 3 + 2 3
@ 2 @x @y 2 + 3 @ 2 @t 2 @x ;

= o t 3 ; x 3 ; x y 2 ; t 2 x :

@N @t + gh @ @y + gh 2

1 2 @ 3 @y 3 + 2 3
@ 2 @x 2 @y + 3 @ 2 @t 2 @y ;

= o t 3 ; y 3 ; x 2 y; t 2 y : (23)

Free Surface Equation

To derive a resulting modi…ed equation, some higher time derivatives are replaced by the corresponding spatial derivatives and the volume ‡ux components, M and N , are eliminated. After a lengthy algebra, a modi…ed equation for is obtained as follow:

@ 2 @t 2 c 2 @ 2 @x 2 + @ 2 @y 2 c 2 ( x) 2 12 1 + 1 C 2 r @ 4 @x 4 + 2 @ 4 @x 2 @y 2 + @ 4 @y 4 ; + (1 + 1 2 ) c 2 ( x) 2 6 @ 4 @x 2 @y 2 c 2 ( t) 2 4
(1 3 ) @ 4 @x 2 @t 2 + @ 4 @y 2 @t 2 ; = O x 3 ; x 2 t; t 2 x; t 3 :

(24

)
where C r = c t= x is called Courant-Friedrichs-Lewis number:

Initial and Boundary Conditions @M @t c @M @x = 0; on @ (t) ;

@N @t c @N @y = 0; on @ (t) : (26)

(x; y; 0) = h;
(27) M (x; y; 0) = N (x; y; 0) = 0:

(28)

Determination of Dispersion-Correction parameters

Comparing Eq.( 24)with the ILBq [START_REF] Mitsotakis | Boussinesq systems in two space dimensions over avariable bottom for the generation and propagation of tsunami waves[END_REF], these equations are seen to be identical as long as the following relations are satis…ed:

1 = C 2 r 12Bh 2 x 2 1;
(29)

2 = 1 + 1; (30)
and

3 = 1 B + 1 3 h 2 c 2 t 2 : (31)
The value of B is not limited to the values discussed in [START_REF] Harter | The e¤ect of surface tension on trapped modes in water-wave problems[END_REF]. To verify the stability condition of the current ADI scheme, the value of B can be chosen as

(C 2 r 2) x 2 12h 2 B g t 2 12h ( 32
)
The following relation is obtained if B is unique

( x) 2 = 4h 2 2gh( t) 2 : (33)
Note that the (33) gives a relationship between the spatial grid size x and time step size t and permits to modify easly the mesh grid of domain (t):

A New Modi…ed Scheme for Weakly Viscous ‡uid

Now we are going to introduce weak dissipative e¤ects directly into LSWqs which will be modi…ed to ILBqs [START_REF] Boussaha | Etude numérique de quelques problèmes issus de la mécanique des ‡uides: ondes longues de gravité et propagation d'un tsunami[END_REF][START_REF] Boussaha | A New modi…ed scheme for linear shallow water equations with distant propagation of irregular wave trains tsunami dispersion type for inviscid and weakly viscous ‡uids[END_REF]. We put 2 ( @ 2 @x 2 + @ 2 @y 2 ); 2 @ 2 M @x 2 and 2 @ 2 N @x 2

into Eqs, respectively. The new set of equations with the above additional expressions describes a PDEs problem that we shall restrict our study to the non-advective term, numerical linear dispersion and weak dissipation terms.

The new proposed scheme is given by the following procedure:

@ @t + @M @x + @N @y 2 @ 2 @x 2 + @ 2 @y 2 = o ( t; x; y) ; (34)
@M @t + gh @ @x + gh 2 1 2 @ 3 @x 3 + 2 3 @ 2 @x @y 2 + 3 @ 2 @t 2 @x 2 h @ 2 M @x 2 =; o t 3 ; x 3 ; x y 2 ; t 2 x ; (35)
@N @t + gh @ @y + gh 2

1 2 @ 3 @y 3 + 2 3 @ 2 @x 2 @y + 3 @ 2 @t 2 @y 2 h @ 2 N @y 2 =; = o t 3 ; y 3 ; x 2 y; t 2 y : (36)

Dispersion-dissipation scheme

We employ the third-order ADI predictor corrector (PC) scheme [START_REF] Gunn | A General Formulation of Alternating Direction Methods, Part I.parabolic and hyperbolic problems[END_REF] for spatial derivatives with three-time level. As a result, both numerical weakly dissipation and linear dispersion are kept with good precision.

The x -sweep equations, in which

k+ 1 2 i;j
and M k+ 1 2 i+ 1 2 ;j are the unknown variables: Preditor stage

k+ 1 2 i;j k i;j t 2 + M k+ 1 2 i+ 1 2 ;j M k+ 1 2 i 1 2 ;j x + N k i;j+ 1 2 N k i;j 1 2 y ; = 2 ( x)
+ k i;j i ; = 2 h ( x) 2 M k+ 1 2 i+ 3 2 ;j 2M k+ 1 2 i+ 1 2 ;j + M k+ 1 2 i 1 2 ;j : (38)
The y-sweep equations, in which k+1 i;j and M k+1 i;j

+ k i;j i ; = 2 h ( y) 2 N k+1 i;j+ 3 2 2N k+1 i;j+ 1 2 + N k+1 i;j 1 2 : (40)
A mesh of more delicate is considered, taking into account the e¤ects of both numerical linear dispersion and weak dissipation e¤ects. Then, from Eqs.(37)-(40)

we get

A x k+ 1 2 = U x k + V x N k ; (41)
M k+ 1 2 = E x h Z x k + W x k+ 1 2 + S x k+1 i ; (42)
A y k+1 = U y k+ 1 2 + V y M k+ 1 2 ; (43)
N k+1 = E y h Z y k+ 1 2 + W y k+1 + S x k+1 i : (44)
where k = h k 1;1 ; :::; The relative error (rerr) is de…ned by following relationship:

rerr = P x i=1 P y j=1 [ (i;j) ref (i;j)] 2 P x i=1 P y j=1 [ ref (i;j)] 2
where, ref (i; j) is calculated at x = 1/1000 as a reference among the proposed solutions.

Figure 3 and Table 1, show that the error decreases gradually as x approaches zero. Then, the error between the approximate solution and the exact solution decreases corresponding to the theoretical convergence of order O(( x) 3 ; ( x)( t) 2 ; ( x) 2 ( t)), (see section:Convergence of the Scheme).The free surface pro…le for inviscid and weakly viscous ‡uid.

.

13

Comparison the free surface pro…le obtained and the numerical scheme of [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF] 5

Conclusion and remarks

We have applied a novel scheme of dispersive propagation model for earthquake generated tsunami. It was also shown that simulations including both dispersive long distant tsunami propagation as well as weak dissipation are well achieved. As a bene…t, our work can model the earthquake source ‡exibly in the propagation model. It is a useful numerical solution when existing theories are not available to …nd exact solution; this last is ideal and sometimes is not compatible with the physical phenomenon. Also experimental study is unable to achieve natural pro…le of solution because the inability to employ natural values. In this model, we proposed the introducing an additional term to the novel modi…ed scheme. This latter is comparatively has some limitations, i.e., the grid size has to be equal in both horizontal directions and check stability criteria 33. In other words, our approach calculates the dispersion-correction factors instead of choosing spatial grid size and step size again to mimic the frequency dispersion of the ILBqs, [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF].

Figure 1 .

 1 Figure 1. Tsunami elevation sea Denoting by the free surface with z = (x; y; t) and h (x; y) the depth of water.

A x and A y are the discretization matrices.

The rank of the matrix is di¤erent of unknown numbers, then the solution of (41)-(47) system is not unique. For that, we support the work Harter [START_REF] Harter | The e¤ect of surface tension on trapped modes in water-wave problems[END_REF] that a general uniqueness proof is now known to be unobtainable.

Convergence, Stability and Consistency

The schemet ADI de…ned by Eqs .(37)-( 40) is stable (see, [START_REF] Boussaha | Etude numérique de quelques problèmes issus de la mécanique des ‡uides: ondes longues de gravité et propagation d'un tsunami[END_REF][START_REF] Boussaha | A New modi…ed scheme for linear shallow water equations with distant propagation of irregular wave trains tsunami dispersion type for inviscid and weakly viscous ‡uids[END_REF]. By obtaining the local discretization error and the well-known classical theorem [START_REF] Gunn | A General Formulation of Alternating Direction Methods, Part I.parabolic and hyperbolic problems[END_REF], we concluded that,the scheme is convergent.

The solution of the Eqs. ( 37)-(40) can be written in the following Fourier forms, (see [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowlyvarying bathymetry[END_REF]):

In the stability analysis, the ampli…cation factor, j t j, should be less than or equal to unity.(see [START_REF] Boussaha | Etude numérique de quelques problèmes issus de la mécanique des ‡uides: ondes longues de gravité et propagation d'un tsunami[END_REF][START_REF] Boussaha | A New modi…ed scheme for linear shallow water equations with distant propagation of irregular wave trains tsunami dispersion type for inviscid and weakly viscous ‡uids[END_REF])

TThe intermediate ADI solution introduces an added complication, thus we can either combine separate estimates of the local discretization errors of the predictor and corrector steps, [START_REF] Gunn | A General Formulation of Alternating Direction Methods, Part I.parabolic and hyperbolic problems[END_REF].

Numerical Results and Discussion

The goal of this numerical study is recognition of mathematical aspect when non local terms give a more realistic pro…le to dispersive free surface of tsunami. Surpass the obstacle that appears in the experimental when we introduce delicate values of , i.e., values of in nature.There are some time intervals for which the solution is Bichromatic Wave (BW). These intervals intersect only at the points x [0; 15m], so that BW for this mode is established only at these values.

Free Surface Pro…le

Following this application, our simulation focuses at waves dispersive a¤ected by weak dissipation. The choice of the kinematic viscosity for the practical simulations of water waves is not obvious. However, in various experimental and theoretical studies, researchers independently concluded that, a value of = 10 3 m 2 =s …ts very well available data (see, [START_REF] Dutykh | Derivation of dissipative Boussinesq equationsusing the Dirichlet-to-Neumann operator approach[END_REF]). The most recent experimental study of plunging breakers con…rms this value again. Consequently, we retain this value for our numerical illustrations as well. Then, according to the criteria provided in [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics[END_REF] and satisfying Eq.(32), it is advisable to choose B = 1=21. Firstly, we study tendency of numerical solution convergence.


Blanche Segrestin

Kevin Levillain

Armand Hatchuel

PURPOSE-DRIVEN CORPORATIONS: HOW CORPORATE LAW REORDERS THE FIELD OF CORPORATE GOVERNANCE

Keywords: Corporate law, purpose-driven corporation, corporate governance, agency theory, stewardship, stakeholder

In a number of States, new legal "benefit" or "purpose" corporate forms are introduced to promote stakeholder-oriented companies. If it is too early to know empirically if this legal framework will work, we can build upon what we know on corporate governance to predict whether it is likely to work or not. In this paper, we present a mapping of the theories on corporate governance and derive some tentative predictions for purpose-driven corporations.

We find that theories make diverging predictions, and in a paradoxical way: agency theory is seen as a shareholder-oriented theory, but it supports the new legal forms while stakeholder theory does not. This reordering of the field reveals that theories may overlook the possibility of a legal change. Our work thus contributes to suggest an empirical test of the theories with the purpose-driven companies, but it also opens further avenues to reappraise the theories of corporate governance.

PURPOSE-DRIVEN CORPORATIONS: HOW CORPORATE LAW REORDERS THE FIELD OF CORPORATE GOVERNANCE

In the past few years, several new legal corporate forms have emerged in the United States.

In 2010, Maryland was the first state to create the "Benefit Corporation" (BC). In 2012, California introduced simultaneously the "Flexible Purpose Corporation" (relabelled "Social Purpose Corporation", SPC) and the "Benefit Corporation" (BC). Today, in the United States alone, the BC has been adopted in 31 states [START_REF] Hiller | The Benefit Corporation and Corporate Social Responsibility[END_REF] and is still spreading. These new corporate forms present themselves as profit-with-purpose corporations (hereafter PPCs).

Although they are very similar to conventional corporations, their charters stipulate that in addition to profit, they will pursue the goal of a general positive impact or a specific purpose (charitable, environmental, etc.). The duties of directors and reporting procedures are modified accordingly.

Compared to traditional business corporations, these different legal forms are designed to allow socially or environmentally beneficial businesses and to overcome the limitations of normal corporations. Normal (for-profit) corporations are indeed expected to maximize shareholder value: when they pursue social or environmental objectives, entrepreneurs or managers can fear to be sacked by shareholders, they can fear the eventuality of takeover or a prosecution for "breach of fiduciary duties" (Mac Cormac and Haney 2012). Now that PPCs Our findings are threefold: we find 1) that the theories have much to say on PPCs, but 2) that their predictions were diverging, and 3) in a paradoxical way. Agency theory is seen as a shareholder-oriented theory, but it appears to support the new legal forms while stakeholder theory does not. These findings, we argue, have important implications:

-First they pave the way to a grounded and detailed agenda for empirical research on the profit-with-purposes corporations incorporated with the new legal forms. They can also be useful for managers and policy makers to prevent or correct eventual negative impacts.

-Second, the findings call for a theoretical reappraisal in corporate governance. Future empirical results on PPCs will validate or invalidate the existing theories and their predications. But more fundamentally, the deep reorder of the field reveals that the theoretical foundations may not be robust with a law change. We conclude with further avenues for research.

The article proceeds as follows. We first begin by mapping the literature on corporate governance, with a particular focus on the relation to corporate law issue. Second, we present rapidly the new purpose-driven corporate forms. Third, we examine what can be expected from the theories on corporate governance. Fourth and finally, we discuss how a new legal framework re-orders the field of knowledge on corporate governance and open new avenues for further research.

I. A FIRST MAPPING OF THE LITERATURE ON CORPORATE GOVERNANCE

Traditionally, corporate governance refers to "the ways in which suppliers of finance to corporations assure themselves of getting a return on their investment » Shleifer and Vishny (Shleifer and Vishny 1997: 737). But this conception has been expanded to see « governance as the determination of the broad uses to which organizational resources will be deployed and the resolution of conflicts among the myriad participants in organizations » (Daily, Dalton et al. 2003: 371). Clearly, corporate law is a decisive element on governance, as it refers not only to leadership systems, but to « managerial control protocols, property rights, decision rights, and other practices that give organizations their authority and mandates for action » [START_REF] Tihanyi | RETHINKING GOVERNANCE IN MANAGEMENT RESEARCH[END_REF]. Cioffi goes as far as defining corporate governance « as a 'nexus of institutions' defined by company law, financial market regulation, and labour law » (Cioffi 2000: 574).

The literature has intensively debated the economic influence of corporate law [START_REF] Porta | Investor Protection and Corporate Valuation[END_REF][START_REF] Fligstein | Law and Corporate Governance[END_REF][START_REF] Roe | A CAPITAL MARKET, CORPORATE LAW APPROACH TO CREDITOR CONDUCT[END_REF] Similarly, the possible reforms of corporate law have been at the core of research on corporate governance since its beginning inception [START_REF] Berle | CORPORATE POWERS AS POWERS IN TRUST[END_REF][START_REF] Berle | The Modern Corporation and Private Property[END_REF][START_REF] Dodd | For who are corporate managers trustees ?[END_REF] (see [START_REF] Wells | The Cycles of Corporate Social Responsibility: An Historical Retrospective for the Twenty-First Century[END_REF] for a historical review). But most of the time, the theories of corporate governance -such as agency theory, team production theory or stewardship theory-are established for a given corporate law and do not integrate corporate law as a contingency variable.

In order to discuss how existing theories are transposed in the new purpose-driven corporate framework, we now review the different analytical approaches in corporate governance. Our ambition is neither to map extensively the immense and diversified field of corporate governance, not to give a detailed analysis of the origins and developments of each approach.

Instead, we look for a comprehensive map of main theoretical propositions: what is considered as good governance (which criteria is used)? And how can good governance be designed or obtained? Are they some legal, cultural, organizational features or parameters to take into account and to what extent is corporate law appropriate?

To simplify, we identify 6 fundamental approaches. The first one, the agency theory (1), is tremendously pregnant in corporate governance and it is considered as advocating a shareholder-orientation. Conversely, stewardship (2) and stakeholder theory (3) challenge fundamental assumptions of agency theory to provide alternative analysis of governance and substitute other interpretations of law. We gather in the stakeholder theory the CSR approach and the team production for sake of clarity. Stewardship theory is grounded on psychology and stakeholder theory is a management-based theory. Other approaches will focus respectively on politics (4) or cognitive aspects (5) instead of purely economic dimensions.

Table 1 summarizes the mapping.

I.1. Agency Theory

General assumptions and objective function of governance. Regarding governance, the economic literature has long focused exclusively on the asymmetrical relationship between owners and managers. This relationship is modelled as an agency relationship where managers are the agents mandated by the owners-principals [START_REF] Eisenhardt | Agency Theory: An Assessment and Review[END_REF]. Assuming that individuals are opportunist, the governance is then concerned by the means principals can use to make sure managers maximize shareholder value and not their own surplus.

Conditions and means of corporate governance. One important concept in this theory is the notion of residual claim, which was first introduced by Alchian and Demsetz [START_REF] Alchian | Production, Information costs and economic organization[END_REF]. Compared to the other constituencies of the firm, shareowners are legitimate principals because they are the only ones to have no guarantees on their rewards: while other parties have all retributions ex ante defined contractually, dividends can only be paid if the net results, after the payment of all other contractual obligations by the corporations, are positive.

According the Alchian and Demsetz, the corporation is the best governance configuration because shareowners because it has the best allocation of incentives. The owners control the other parties and have incentives, as residual claimants, to invest in efficient monitoring of the other parties.

The efficacy of governance thus depends centrally on the control. And the control is all the more effective that controllers have incentives.

Interpretation of corporate law and function of the board. This framework not only alleges the supremacy of corporation over other governance configuration (not-for profit organizations, employee-owned companies, cooperative, …), but it also provides an economic interpretation of corporate law: The corporation is the legal contract between owners and their mandated managers. And the board is the tool to control managerial activity.

Further variables of performance. This general framework has then been largely developed and sophisticated [START_REF] Jensen | Theory of the Firm: Managerial Behavior, Agency Costs and Ownership Structure[END_REF]Meckling 1976, Tirole 2001). Different parameters have been put forward to achieve better performances, which are not substitute but rather complementary means [START_REF] Misangyi | SUBSTITUTES OR COMPLEMENTS? A CONFIGURATIONAL EXAMINATION OF CORPORATE GOVERNANCE MECHANISMS[END_REF] :

-Information and systems of reporting are to be deployed to improve control [START_REF] Armstrong | The role of information and financial reporting in corporate governance and debt contracting[END_REF]. The role of financial reporting is supposed to alleviate information asymmetries and to improve monitoring and bonding mechanisms.

-Incentives and remunerations. Different means to curb the preferences of agents and align them on stakeholders' interests are also promoted, such as incentive packages, stock options [START_REF] Garen | Executive compensation and principal-agent theory[END_REF][START_REF] Cyert | Corporate Governance, Takeovers, and Top-Management Compensation: Theory and Evidence[END_REF][START_REF] Dittmann | Sticks or carrots? Optimal CEO compensation when managers are loss averse[END_REF][START_REF] Frydmann | Executive Compensation: A New View from a Long-Run Perspective, 1936-2005[END_REF].

-Shareholders' profile and board composition. Oversight by board can also be extended if it is well composed (Pearce Ii and Zahra 1991): the profile of directors has been highly studied [START_REF] Johnson | Boards of Directors: A Review and Research Agenda[END_REF]. For instance, shareowners are supposed more vulnerable to managers' opportunist if they are more dispersed [START_REF] Armour | How do legal rules evolve? Evidence from a cross-country comparison of shareholder, creditor and worker protection[END_REF][START_REF] Gelter | Taming or Protecting the Modern Corporation? Shareholder-Stakeholder Debates in a Comparative Light[END_REF]. This can explain why, especially in the US, corporate law has continuously tried to reinforce shareowners' protection.

Similarly, the communication and the joint action of shareholders has been favoured to facilitated shareholder activism and therefore to discipline managers. Dalton reported that an early 1990s regulatory change by the SEC made it significantly easier for institutional investors, in particular, to engage in activist efforts 2 .

Limitations. Overall, agency theory has had a huge normative impact. The literature has showed that it has produced unexpected effects, such as the phenomenon of "CEO-board only" [START_REF] Joseph | THE STRUCTURAL ELABORATION OF BOARD INDEPENDENCE: EXECUTIVE POWER, INSTITUTIONAL LOGICS, AND THE ADOPTION OF CEO[END_REF]: the rise of board independence has led to a supermajority 2 Prior to the regulatory change, shareholders were prohibited from discussing corporate matters with more than ten shareholders or shareholder groups without prior SEC approval (Jensen, 1993). This rule was relaxed, permitting shareholders holding less than 5 percent of outstanding shares-with no vested interest in the issue being discussed and not seeking proxy voting authority-to freely communicate with other shareholders (Jensen, 1993;[START_REF] Daily | CORPORATE GOVERNANCE: DECADES OF DIALOGUE AND DATA[END_REF].

of outside directors in U.S. corporate boards and even to a rising number of corporations in which the CEO is the only insider in the board. This can be problematic, as other directors cannot anymore challenge CEO on internal affairs. Agency theory needs to be complemented by other approaches, including social or organizational dimensions [START_REF] Wiersema | CEO Dismissal: The role of investment analysts[END_REF][START_REF] Aguilera | An Organizational Approach to Comparative Corporate Governance: Costs, Contingencies, and Complementarities[END_REF]. And this is all the more necessary that agency theory has highly normative effect, although it is heavily criticized [START_REF] Aguilera | Law and Finance": Inaccurate, Incomplete, and Important[END_REF].

We know for instance that empirical researches are not conclusive: according to recent metaanalyses, firms following agency theory rule (separation of position of CEO and chair, improved incentives, control mechanisms, independent directors, etc.) do not outperform the others [START_REF] Daily | CORPORATE GOVERNANCE: DECADES OF DIALOGUE AND DATA[END_REF][START_REF] Aguilera | An Organizational Approach to Comparative Corporate Governance: Costs, Contingencies, and Complementarities[END_REF]. Quite on the contrary, many research documents the important social and environmental costs of shareholder governance [START_REF] Margolis | Misery Loves Companies: Rethinking Social Initiatives by Business[END_REF]Walsh 2003, Dalton, Hitt et al. 2007).

The literature on corporate governance has thus often emerged in reaction to agency theory and to challenge its fundamental assumptions.

I.2-Stewardship theory

General assumptions and objective function of governance. Based on psychology and sociological approaches, the stewardship theory challenges the underlying "model of the man" of the agency theory. It is instead of a purely self-interested and opportunistic behaviour, it contends CEO often behaves genuinely for the sake of their principals and to be a goo steward of corporate assets [START_REF] Donaldson | Stewardship Theory or Agency Theory: CEO Governance and Shareholder Returns[END_REF]. Where agency theory is focused on the in-built conflict of interest between owner and manager, stewardship theory shifts from an issue of control to an organizational logic: the issue becomes whether or not the organisation structure helps the executive to formulate and implement plans for high corporate performance (Donaldson 1985).

In this perspective, the objective function of governance is not to maximize principals' interest but to pursue longer-term and beneficial collective ends.

Conditions and means of corporate governance. Interestingly, stewardship doesn't question the position of agent of CEO. It rather insists on the condition for the actors, be they principals or agents, to adopt a steward attitude instead of a opportunist behaviour. The question then becomes less the control than the possibility of a organizational commitment: organizational commitment refers to the affective identification of individuals with the organization or the endorsement of its values, the personal costs of exit or the feeling of moral obligation [START_REF] Moriarty | Compensation Ethics and Organizational Commitment[END_REF]. The performance of governance depends of the possibility of a "convenantal relationship" between the parties [START_REF] Hernandez | TOWARD AN UNDERSTANDING OF THE PSYCHOLOGY OF STEWARDSHIP[END_REF].

Interpretation of corporate law and function of the board.

As mentioned, CEOs are still considered as mandated by the shareholders and the stewardship theory questions neither the legitimacy of the corporate law nor its interpretation by the agency theory. The role of the board is therefore still the control of management.

Further variables of performance. The literature has identified several parameters likely to support governance:

-Most of stewardship recent developments focus on the psychology and personality of CEO. A symposium on "responsible leadership" in Academy of Perspective has been interestingly published in this perspective in 2014 [START_REF] Filatotchev | AND CORPORATE SOCIAL RESPONSIBILITY: ORGANIZATIONAL EFFICIENCY VERSUS ORGANIZATIONAL LEGITIMACY?[END_REF].

Not surprisingly, cultural and psychological factors, as well as management philosophy are key to the development of stewardship [START_REF] Davis | Toward a stewardship theory of management[END_REF]).

-Dual position is considered positively by stewardship theory as it allows managerial discretion [START_REF] Donaldson | Stewardship Theory or Agency Theory: CEO Governance and Shareholder Returns[END_REF]. But the global outcome of the relationships will depend not only on the attitude of the CEO but also of the attitude of the board of directors. Therefore, more or less cooperative games can take place, depending on the attitude (agent versus steward) of directors and managers [START_REF] Davis | Toward a stewardship theory of management[END_REF]).

-Recently Hernandez adds to the stewardship theory by identifying organizational antecedents to stewardship. He found that control and rewards structures play a significant role in the individual attitude, if they are oriented to collective and organizational goals [START_REF] Hernandez | TOWARD AN UNDERSTANDING OF THE PSYCHOLOGY OF STEWARDSHIP[END_REF]. He further argues that some variables, such as psychological ownership or purpose-oriented organizational identity also contribute to stewardship.

I.3-Stakeholder Theory and Team Production Theory

General assumptions and objective function of governance. This trend of research has a long history and various ramifications. But the corner stone is that the focus on the exclusive relationship between stockowners and managers is challenged. As a « new-wealth-creatingteam » [START_REF] Kaufman | A team production model of corporate governance[END_REF], the firm needs to build upon "team-specific assets, invested by shareholders, managers, employees, and others who hope to profit from team production » [START_REF] Gabrielsson | Understanding the Leadership Role of the Board Chairperson Through a Team Production Approach[END_REF]. Stakeholder theory came from the conviction that management's role necessarily includes [START_REF] Friedman | The Social Responsibility of Business is to Increase its Profits[END_REF] interests and claims of nonstockholding groups (Mitchell, Agle et al. 1997: 855). As Jones (1980: 59-60) argues, "corporations have an obligation to constituent groups in society other than stockholders and beyond that prescribed by law or union contract indicating that a stake may go beyond mere ownership".

The capacity to create value for all its constituencies is considered for the firm to be both a condition of its sustainable existence and its raison d'être [START_REF] Clarkson | A stakeholder framework for analysing and evaluating corporate social performance[END_REF][START_REF] Post | Managing the Extended Enterprise: The New Stakeholder View[END_REF][START_REF] Harrison | Managing for stakeholders, stakeholder utility functions, and competitive advantage[END_REF]. The sustainability of the firm is the key objective function.

Conditions and means of corporate governance. The capacity of a firm to generate sustainable wealth over time is determined by its relationships with critical stakeholders [START_REF] Post | Managing the Extended Enterprise: The New Stakeholder View[END_REF], and thus one of the critical issue is to allocate collective outputs and rewards in a way that is legitimate enough for the parties to collaborate. And as all these constituencies put assets at risks without any certitude about the returns they will have, the model of agency theory needs to be broadened: the simple agency relationship is to be substituted by a multiple agency one, Hill and Jones have proposed a "stakeholder-agency" [START_REF] Hill | Stakeholder-agency theory[END_REF]. Others put emphasis on property rights that are necessarily distributed as soon as we consider human capital: it is crucial for the firm to build mechanisms and systems that encourage the development and safeguarding of its human capital-based capabilities [START_REF] Mahoney | ADVANCING THE HUMAN CAPITAL PERSPECTIVE ON VALUE CREATION BY JOINING CAPABILITIES AND GOVERNANCE APPROACHES[END_REF].

One branch of literature is concerned by the correlation between corporate social performance and economic performance. A large body of research studies the positive impact of Corporate Social Responsibility, for instance [START_REF] Vilanova | Exploring the nature of the relationship between CSR and competitiveness[END_REF][START_REF] Harjoto | The Causal Effect of Corporate Governance on Corporate Social Responsibility[END_REF][START_REF] Jayachandran | Product and environmental social performance: Varying effect on firm performance[END_REF]. Direct effect can be difficult to establish, but other indirect impact, such as impact on financial analysts (and then on market value) may be more sensible [START_REF] Harjoto | Legal vs. Normative CSR: Differential Impact on Analyst Dispersion, Stock Return Volatility, Cost of Capital, and Firm Value[END_REF].

Another line of research focuses on the possibility for managers to engage the different parties in the collective venture. In this respect, the allocation of returns appears as critical. If managers are stewards or bona fide experts, the question becomes how can the parties invest in the firm with no guarantees regarding how final results are allocated among the parties?

The role of governance is thus to allow specific investments (meaning that they are difficult to recover once committed to the project) from different stakeholders, by organizing a fair and legitimate distribution of the collective gains.

Interpretation of corporate law and function of the board. The Team Production Theory developed by Blair and Stout has provided a new view of corporate law. They have first contested the idea that shareholders should be the considered the "owners" of the firm or its profits [START_REF] Blair | A Team Production Theory of Corporate Law?[END_REF]Stout 1999, Lan and[START_REF] Lan | Rethinking agency theory: the view from law[END_REF]. In law, the shareholders only own their shares. It is the corporation, as a legal entity, which owns the corporate capital [START_REF] Bainbridge | Director primacy and shareholder disempowerment[END_REF]. The standard shareholder agency approach therefore gives a misleading interpretation by not taking the corporation into account.

In their view, the corporation is a new legal person introduced by law to allow various specific investments: it may be impossible for team members to draft explicit contracts that protect them from each other's opportunism. To agree on a procedure they all consider to be fair, they form a public corporation, i.e. a legal person and the owner of the corporate assets and results. Thus, the corporation is the principal and the residual claimant, not the shareowners.

The board monitors team members' efforts and determines how each can best be rewarded for past contributions, as well as be incentivized for future contributions, in the process also guarding against mutual opportunism among the parties" (Lan & Heracleous, 2010: 300). The board is a "mediating hierarch", a third party capable of arbitration between the stakeholders [START_REF] Blair | A Team Production Theory of Corporate Law?[END_REF]. Consequently, rather than being in charge of controlling the managers, the board of directors' key role is to conduct arbitration between the stakeholders and solve potential conflicts of interests [START_REF] Hillman | Boards of directors and firm performance: integrating agency and resource dependance perspectives[END_REF]Dalziel 2003, Lan and[START_REF] Lan | Rethinking agency theory: the view from law[END_REF]. It is an internal court of appeals to resolve disputes that may arise among the team members.

Further variables of performance.

There are mainly two important dimensions to secure for the sake of the corporate sustainability. The first one is the information of shareholders. In this model, the neutrality of the board is key. The premise is that it would in the long-term interest of shareowners to pay attention to social aspects. But it implies that shareowners have a good understanding of what conditions the firm long-term performances. CSR in practices has developed a series of tools, such as social reporting, evaluation, indexes, etc. to increase companies' social awareness. The issue is to institutionalize in business strategies and practices the notion of "enlightened shareholder value" [START_REF] Queen | Enlightened Shareholder Maximization: Is this Strategy Achievable?[END_REF]. The second dimension is the neutrality of the board and to make sure directors are "benevolent" and act in the best interest of the corporation. According to Blair and Stout, corporate is already featured in this way, with the fiduciary duties (duty of loyalty, duty of care, "best interest of the corporation"), and the influence of shareowners would be in practice limited over the board.

However any measure to improve directors' neutrality, such as impacts on their reputation, etc., is potentially important for good corporate governance.

Limitations. CSR has made considerable progress over the past decades. However, limitations are put forward at different levels. More and more studies show that there is no clear link between social and economic performance. After the first meta-analysis of Margolis and Walsh in 2003, other survey showed that the literature was rather inconclusive on the strategic interest to engage in CSR. In other words, the business case of social engagement is not established [START_REF] Margolis | Misery Loves Companies: Rethinking Social Initiatives by Business[END_REF]Walsh 2003, Salzmann, Ionescu-Somers et al. 2005). Quite on the contrary, business companies take into account their stakeholders when it constitutes a business opportunity. But beyond the "zone of opportunity", and when trade-offs are necessary, CSR appears trapped in "a logical trap" [START_REF] Karnani | CSR Stuck in a Logical Trap[END_REF].

Second, different authors contest the neutrality of corporate law. Shareowners are given a great power to influence over directors, as directors are appointed exclusively by shareowners [START_REF] Greenwood | Discussing Corporate Misbehavior[END_REF][START_REF] Greenfield | The Failure of Corporate Law: Fundamental Flaws and Progressive Possibilities Chicago[END_REF][START_REF] Mayer | Firm Commitment. Why the corporation is failing us and how to restore trust in it ?[END_REF]). The board can, of course, always try to reconcile the different parties' interests but, as [START_REF] Kaufman | A team production model of corporate governance[END_REF] point out, directors are legally accountable to the firm's shareholders [START_REF] Kaufman | A team production model of corporate governance[END_REF]. Therefore corporate does not really "insulate" the board from shareholders' pressure [START_REF] Millon | New game plan or business as usual? A critique of the team production model of corporate law[END_REF]. And in these conditions, studies look on how corporate governance can be adapted to facilitate CSR [START_REF] Filatotchev | AND CORPORATE SOCIAL RESPONSIBILITY: ORGANIZATIONAL EFFICIENCY VERSUS ORGANIZATIONAL LEGITIMACY?[END_REF][START_REF] Devinney | Corporate Social Responsibility and Corporate Governance: Comparative Perspectives[END_REF].

I.4-Stakeholders politics

General assumptions and objective function of governance. Another line of research can be labelled as "stakeholders politics" to point out the power, capacity of influence and potential conflicts among parties. Following salience theory [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF], some stakeholders, even if they do to have more legal rights, "count" more than others when they have the power to influence the firm's behaviour. And it is established that the power of the parties has a direct effect on their capacity to capture part of the collective rents [START_REF] Coff | When Competitive Advantage Doesn't Lead to Performance: The Resource-Based View and Stakeholder Bargaining Power[END_REF].

In terms of corporate governance, a political approach leads to consider ethical standards and to look for justice among parties.

Conditions and means of corporate governance.

Two main issues will be critical in corporate governance: balance of power and managerial discretion. Balance can be achieved by different means, either by protecting the more vulnerable parties or by giving them means to defend their interests. But conversely, ethical issue requires to protect managerial discretion against the constrained imposed by stakeholders [START_REF] Phillips | Strategy, stakeholders and managerial discretion[END_REF]): stakeholders oftent act to constrain managers when they find their interests opposed [START_REF] Frooman | Stakeholder Influence Strategies[END_REF], including with protests and civil lawsuits.

Interpretation of corporate law and function of the board. Apart from labour law which is specifically devoted to the protections of employees, corporate law has long be considered as a way to defend or represent different corporate constituencies. In a political approach, the corporation has a representation function: directors represent and defend the interest of different stakeholders.

Further variables of performance and limitations. Since the famous debate between M. Dodd [START_REF] Dodd | For who are corporate managers trustees ?[END_REF] and A. Berle [START_REF] Berle | CORPORATE POWERS AS POWERS IN TRUST[END_REF], the debate on the scope of fiduciary duties of managers has kept vivid. « For who are corporate managers trustees »? As noted by [START_REF] Marens | Getting real: stakeholder theory, managerial practice and the general irrelevance of fiduciary duties owed to shareholders[END_REF], business ethicists have extensively debated over the last years the implications for stakeholder theory of the fiduciary obligations [START_REF] Teubner | Corporate Fiduciary Duties and their Beneficiaries[END_REF][START_REF] Marens | Getting real: stakeholder theory, managerial practice and the general irrelevance of fiduciary duties owed to shareholders[END_REF][START_REF] Sandberg | Socially Responsible Investment and Fiduciary Duty: Putting the Freshfields Report into Perspective[END_REF][START_REF] Sandberg | Re-)Interpreting Fiduciary Duty to Justify Socially Responsible Investment for Pension Funds?[END_REF]: they can indeed be interpreted either as a legal obligation to the shareowners [START_REF] Kaufman | Managers' double fiduciary duty: to stakeholders and to freedom[END_REF], or as the foundations for a professional leeway in favour of a stewardship approach [START_REF] Lipton | The Inconvenient Truth About Corporate Governance: Some Thoughts On Vice-Chancellor Strine's Essay[END_REF].

Wells reports than many proposals have been formulated historically to change the composition of the board and include representative from different stakeholders. In the 60's, "public interest directors" were also suggested. But such reforms were left aside and heavily criticized [START_REF] Pound | Don't Replace Democracy with Bureaucracy. (cover story)[END_REF][START_REF] Pound | The Promise of the Governed Corporation[END_REF]: even if the relevant constituencies could be listed and represented, opening the board could limit managers' accountability [START_REF] Boatright | Fiduciary duties and the shareholder-management relation: or, what's so special about shareholders?[END_REF][START_REF] Keay | Tackling the Issue of the Corporate Objective: An Analysis of the United Kingdom's 'Enlightened Shareholder Value Approach[END_REF][START_REF] Keay | Stakeholder theory in corporate law: has it got what it takes?[END_REF]. Constituency statutes were better received as they do not change the board but allow directors to take into consideration the interests of constituencies other than stockholders [START_REF] Bainbridge | Interpreting nonshareholder constituency statutes[END_REF][START_REF] Orts | Beyond Shareholders: Interpreting Corporate Constituency Statutes[END_REF]. They thus extend the range of permissible concerns by boards of directors to other parties, including employees, creditors, suppliers, customers, and local communities [START_REF] Donaldson | The stakeholder theory of the corporation: concepts, evidence and implications[END_REF]. Other proposals are still under consideration, such as the concept of "demarchy" [START_REF] Zeitoun | LEARNING FROM ANCIENT ATHENS: DEMARCHY AND CORPORATE GOVERNANCE[END_REF], the notion of "double-layered agency" (Raelin and Bondy 2013).

I.5-Resource Based View and corporate governance.

General assumptions and objective function of governance. The mapping of the field of corporate governance wouldn't be complete without including a reference to RBV. This approach views firms as collective and organized processes to create value and are concerned by the capacity to generate new business opportunities and innovative strategies [START_REF] Castanias | Managerial Resources and Rents[END_REF][START_REF] Zahra | How do threshold firms sustain corporate entrepreneurship? The role of boards and absorptive capacity[END_REF]. In this view, firm performance depends on its strategic resources [START_REF] Barney | Firm Resources and Sustained Competitive Advantage[END_REF]. Here, the function of governance is to meet the conditions of value creation and innovation. Firms would thus exist in order for their principals-to-be to capture value (i.e. profit from) their appropriable value creative advantages [START_REF] Pitelis | The (new) nature and essence of the firm[END_REF].

Conditions and means of corporate governance. The performance of the company depends critically on the Top-Management-Team (Hambrick andMason 1984, Finkelstein and[START_REF] Finkelstein | Top-Management-Team Tenure and Organizational Outcomes: The Moderating Role of Managerial Discretion[END_REF]) and the managers' capability to design strategies and rally different participants in a joint venture. Alternatively, the access to critical resources is strategic.

Governance should secure corporate competencies, and facilitate absorptive capacity and organisational learning.

Interpretation of corporate law and function of the board.

The cognitive view has lead to a theory of the firm based on knowledge rather than on opportunism: Conner and Prahalad for instance propose that hierarchy is institutionalized to manage organizational learning processes [START_REF] Conner | A Resource-based Theory of the Firm : Knowledge Versus Opportunism[END_REF]. The role of the board has also been revisited with the cognitive lenses: the primary role of the board is resource provision [START_REF] Hambrick | THE QUAD MODEL FOR IDENTIFYING A CORPORATE DIRECTOR'S POTENTIAL FOR EFFECTIVE MONITORING: TOWARD A NEW THEORY OF BOARD SUFFICIENCY[END_REF], directors being experts and brokers [START_REF] Hillman | Boards of directors and firm performance: integrating agency and resource dependance perspectives[END_REF].

Further variables of performance and limitations. Zahra et al. 2000 have found that the commitment of the company's top executives were critical for corporate entrepreneurship [START_REF] Zahra | How do threshold firms sustain corporate entrepreneurship? The role of boards and absorptive capacity[END_REF]. But the composition of the board is also important as the board needs to gather directors who combine independence, motivation with expertise [START_REF] Hambrick | THE QUAD MODEL FOR IDENTIFYING A CORPORATE DIRECTOR'S POTENTIAL FOR EFFECTIVE MONITORING: TOWARD A NEW THEORY OF BOARD SUFFICIENCY[END_REF]. And a new body of literature also suggests that "managerial governance" could build upon external resources. Starbuck for instance proposes that the interplay with universities could be a critical lever in competition driven by innovation capabilities [START_REF] Starbuck | WHY CORPORATE GOVERNANCE DESERVES SERIOUS AND CREATIVE THOUGHT[END_REF].

INSERT TABLE 1 HERE

II-NEW LEGAL FORMS OF PPCS: MOTIVATIONS AND FEATURES

BCs, as well as FPCs, are very similar to traditional corporations. Both of these new entities will be taxed the same as for-profit corporations under current tax law. The difference is that the charters or bylaws of these corporations stipulate a purpose different from, and in addition to, profit [START_REF] Hiller | The Benefit Corporation and Corporate Social Responsibility[END_REF][START_REF] Hemphill | The Benefit Corporation: Corporate Governance and the Forprofit Social Entrepreneur[END_REF][START_REF] Nass | The Viability of Benefit Corporations: An Argument for Greater Transparency and Accountability[END_REF]. This differs from general corporations, which are allowed to form for any lawful purpose, but have no explicit purpose requirement.

Several explanations of the creation of a new legal framework, can be put forward:

-The rigidity of the legal separation between for-profit and non-profit organizations [START_REF] Battilana | In search of the hybrid ideal[END_REF][START_REF] Ebrahim | The governance of social enterprises: Mission drift and accountability challenges in hybrid organizations[END_REF][START_REF] Rawhouser | Benefit Corporation Legislation and the Emergence of a Social Hybrid Category[END_REF]; -The fear that fiduciary duties create obstacles for corporate leaders who set objectives other than shareholders' interests (Mac Cormac 2007 , Clark andBabson 2012, Mac Cormac and Haney 2012 ); -And, more generally, the precarious position of corporate commitments or missions with regard to possible changes in shareholders or company directors [START_REF] Sandberg | Socially Responsible Investment and Fiduciary Duty: Putting the Freshfields Report into Perspective[END_REF]) (Prior, 2014).

So far, the literature has focused on why such new legal framework was needed, and in which conditions a law could pass [START_REF] Rawhouser | Benefit Corporation Legislation and the Emergence of a Social Hybrid Category[END_REF]. Some authors have analysed why some companies considered a change of status [START_REF] Haigh | Hybrid Organizations as Shape-Shifters: ALTERING LEGAL STRUCTURE FOR STRATEGIC GAIN[END_REF].

But the overall efficacy of the law remains to be studied: does it effectively dissipate the fear of breach of fiduciary duties for corporate leaders? And are corporate missions better protected and hence more credible with the new law? Before addressing these questions, we first characterize PPCs. For the sake of simplicity, we focus here exclusively on Benefit corporations and Social Purpose Corporations (SPC or FPC). To summarize, PPCs present 3 main features: a purpose is specified, it commits the corporation through a lock-in mechanism and special accountability is required.

II.1-Principle of purpose

The corporation must first and foremost adopt a particular purpose (or several), whether generic or contingent, in addition to profit. There are different types of purpose. For instance, the law requires that the special purpose of the FPC be: "(1) One or more charitable or public purpose activities that a non-profit public benefit corporation is authorized to carry out; or (2) The purpose of promoting positive short-term or long-term effects of, or minimizing adverse short-term or long-term effects of, the flexible purpose corporation's activities upon any of the following: (a) The flexible purpose corporation's employees, suppliers, customers, and creditors; (b) The community and society; or (c) The environment."

In contrast, the mission of the Benefit Corporation is expressed as a "general public benefit". More precisely, BCs are required to have the purpose of creating a "general public benefit" and are allowed to identify one or more "specific public benefit" purposes. In California, the general public benefit is defined as a "material positive impact on society and the environment, taken as a whole, as assessed against a third-party standard, from the business and operations of a benefit corporation." In addition, the law requires that directors take into account the impact of their decisions on the firm's significant stakeholders. A company may also designate a specific public benefit, in addition to its general public benefit purpose. Compared to SPCs, a specific public benefit can only be in addition to the commitment to a general public benefit.

II.2-Mission-lock principle

Profit-with-purpose corporations must adopt a lock-in mechanism for their mission. For instance, both FPCs and BCs require a positive vote by two-thirds of each class of shares in order to adopt, change, or repeal the Flexible Purpose or Benefit Corporation statute. Once enacted, a flexible purpose is thus a mid-or long-term commitment by the corporation.

Conversely, statutes may grant rights to shareholders that want to break away from the mission, for example, by ensuring that these shareholders can sell their shares at a fair value when they leave ("dissenters' right"). Significantly, this commitment also concerns potential buyers of the corporation: for FPCs and BCs, the same super majority is required to authorize an acquisition or a merger of an FPC or Benefit Corporation if the acquirer refuses to adopt the same corporation form, with the same purposes.

II.3-Accountability principles

Profit-with-purpose corporations can adopt special procedures to evaluate the management of their purpose. To attain the appropriate level of transparency, both BCs and FPCs must publish the following results in an annual report made accessible to the public:

-The Benefit Corporation requires a third-party evaluation standard. The annual evaluation of the corporation's results regarding its mission must follow that standard.

-FPCs' reports must also include a discussion of the strategies followed during the year, the investments made, and the future strategies to be deployed for the year to come. Additionally, local reports may be required for significant decisions involving a compromise between profit and purpose. The FPC also requires operational objectives to be derived from the special purpose, so that the directors' actions can be evaluated against these objectives.

III-THE PREDICTIONS OF THE LITERATURE ON PPCS' FORTUNE

How the change introduced in law by PPCs can affect corporate governance? We examine here how the features of PPCs affect the table mapping the literature.

III.1-Which parameters does the new law impact?

We can make the following observations: PPCs' features have no impact on the way 1) board is insulated, 2) parties are represented in the board, or 3) resources are provided by the board. Conversely, PPCs' law clearly changes on critical parameters (see table 2): 1) It impacts the accountability of managers: managers are still accountable exclusively to shareowners, but they account on the purpose or on the benefits the company pursues. a. Information systems and reporting procedures are modified accordingly to integrate measures of the impact of the business activities and the assessment of the purpose' achievement.

b. Control procedures are oriented toward an explicit and presumably collective cause.

2) It impacts shareowners' behaviour and profile. Shareowners are committed contractually to a cause: with the lock-in mechanism, their engagement is lasting and credible in the long-run. Thus PPCs impact the profile of directors of the board and limit the possibility of non-cooperative games between steward-managers and agentboards.

3) The organizational identity is also impacted once a cause or a purpose is clearly marked up in the constitutive documents of the corporation. 4) Finally, we expect that PPCs impact the balance of powers between stakeholders: depending on the purpose of the PPCs, some communities will be better protected and their relative influence will be altered.

We can now turn into a more detailed analysis of the potential effects of these changes.

III.2-Will the new law be efficient? Predictions from the theories

Agency theory and stewardship theory: converging support to PPCs. As agency theory focuses on control mechanism, and as the main change introduced in the law is the transformation of control by shareowners, agency theory will most probably predicts a strong effect of the new law. The effect can first be to strengthen the position of residual claimants of shareowners. Stakeholder theory justly contests this position because other parties take risks and as they have exclusive control rights, shareowners can even transfer their risks upon other parties. But, with the new legal corporate forms, shareowners commit themselves to demand to their agent-managers to pursue general public benefit or special social purposes: in these conditions, other parties are given more contractual guaranties while shareowners accept to be paid only if there is a positive residual outcome.

Agency and stewardship theories thus converge: shareowners and directors are bound to be stewards, with the mission lock-in mechanisms; and the organizational structure (control procedures) and identity favours managers' stewardship. Therefore we can formulate the following proposition.

P1 -Agency theory and Stewardship Theory converge to predict that the new legal framework of PPCs will be efficient (meaning PPCs will be effectively more responsible companies).

Stakeholder theory and Team production theory: negative expectations for PPCs.

Following stakeholder theory or CSR, the PPCs will have no effect on the information of shareowners: for enlightened shareowners, the contractual commitment will be useless as they are already concerned by the social and environmental impact of their corporate activities.

And for the other, PPCs will have no incentive and no effect.

Simultaneously, as team production theory insists on the insulation of the board as a critical feature of corporate law, the board could not be influenced by the demand of shareowners to pursue one general or specific social benefit. On the contrary, according to this theory, corporate is designed to ensure that the board of directors serve the corporate welfare (the survival of the corporations) beyond the changing and contingent demands from the parties.

Alternatively, the introduction of a purpose in the charters of the corporation could further limit the latitude of the board to allocate the collective results in a fair way. Because it imposes to favour such cause or such community, isn't the new law likely to introduce new constrains for results allocation? We can expect, if we follow the team production theory, that the investment of the different parties is hindered by the stipulation of a social purpose.

Hence our second proposition: P2 -Stakeholder theory and Team Production Theory converge to predict that the new legal framework of PPCs will be inefficient (meaning that it will dissuade responsible companies).

III.3-What side effects can we anticipate from the theories?

The screening of the literature further reveals possible difficulties that the new law may create. Further analysis is necessary to identify the possible consequences, and here we aim only at illustrating some possible upshots that would be specific to PPCs (we do not come back on the already noticed problems of hybrid organizations for instance).

As for agency theory, the stipulation of a purpose doesn't necessarily create a risk of unaccountability: Unlike constituency statutes, PPCs do not give their directors and officers carte blanche. The specification of a purpose comes with the stipulation of specific control and monitoring procedures: BCs must publish an evaluation of their action and their impact based on set criteria determined externally. And for SPCs, the law requires a public report on the evaluation of the management of purposes (Management Discussion & Analysis). However, one question emerges concerning the managerial remuneration. Agency theory calls for incentive packages to align managers' interests on shareowners' ones. Will PPCs conserve this principle or will they intentionally condition the retribution to the fulfillment of the social purpose? Stakeholders politics will be also inevitably modified by a commitment on a purpose. With the new legal framework, non-stockholder parties will not be represented in the board. But the hypothesis of PPCs is that their interests can be defended and preserved by shareholders' commitment. This of course deserves further inquiry. And one critical issue will be to what extent can the parties be heard or bring suit against a management team if the strategy is not in line with the purpose. SPCs so far do not give this legal possibility to non-stockholders parties. So one can still question the possibility in these corporations to practically protect non-stockholder parties may still be an issue. We may then expect that additional guarantees may be asked or that the parties targeted by the purpose of the corporation call for an information/control right or organize themselves to be heard.

Conversely, if the purpose gives effective guarantees to some parties, then shareowners may be threatened by their opportunism. Following the team production theory, team members may be all opportunism and all of them can try to exploit the purpose to their own ends. The commitment towards certain parties may move the balance between parties and give too much power to the latter. Then, one can wonder whether the legal commitment of shareowners is sufficient, or whether the commitment of the most powerful parties would not be necessary.

One possible consequence of PPCs could be the development of other means of engagement for the different constituencies around a common purpose.

Finally, regarding RBV, the introduction of a purpose may make even harder the ideal composition of the board. It may be more difficult to find directors that are simultaneously available, motivated by the purpose and competent in the business. As suggested by Starbuck, one can expect that other or extended forms of control or governance emerge, beyond the board of directors.

The analysis would need to be deepened. But at this stage, we believe that existing theories on corporate can help managers and policy makers to identify pitfalls or risks. And further empirical research should investigate if they actually take place and how managers do to cope with them.

IV-DISCUSSION AND FURTHER RESEARCH

The objective of our paper was to explore what could be expected with the new corporate forms of PPCs from what we on corporate governance. Our analysis shows that several proposals can be derived from existing theories. We have formulated two main proposals: P1 -Agency theory and Stewardship Theory converge to predict that the new legal framework of PPCs will be efficient (meaning PPCs will be effectively more responsible companies). P2 -Stakeholder theory and Team Production Theory converge to predict that the new legal framework of PPCs will be inefficient (meaning that it will dissuade responsible companies). These predictions are diverging, which means that further empirical observations will be necessary to say which one are true in their predictions and which ones are wrong.

In this discussion, we would like to discuss further the theoretical implications of our results. As we have already mentioned, the success or the failure of new corporate forms will challenge, based on our predictions, either agency theory or stakeholder theory. But at this stage, our findings already turn upside down our theoretical representation of corporate governance.

IV. 1-The paradox of PPCs: Agency theory for, stakeholder theory against

PPCs reappraise the question of the adequacy of corporate law with stakeholder management. This question goes back to the call Donaldson and Preston addressed in their influential article (1995): "it remains to implement in law the sanctions, rules, and precedents that support the stakeholder conception of the corporation; in short, it remains to develop the legal version of the stakeholder model" (Donaldson and Preston 1995): 87). Since this call, many different reforms have been proposed or experimented, to diversify the board of directors, to reinforce the so-called "Business Judgement Rule" (BJR) [START_REF] Bainbridge | The Business Judgment Rule as Abstention Doctrine[END_REF], or to allow directors to take into consideration interests of consistuencies other than stockholders [START_REF] Orts | Beyond Shareholders: Interpreting Corporate Constituency Statutes[END_REF])… But so far, none of our theories (agency, stewards, stakeholder, team production, …) had anticipated the introduction of PPCs.

Furthermore, the predictions of these theories regarding the efficacy of PPCs are highly unexpected and surprising. Agency theory appears as the cornerstone of shareowners-oriented governance. A tremendous body of research has tried for decades to challenge it both theoretically and empirically. One would have expect that the theoretical model of shareholder primacy will work against PPCs, as stakeholder-oriented forms of corporations.

And similarly, one would have normally suppose that stakeholder theory would have supported the new legal framework. Yet, paradoxically, the exact contrary happens: agency theory brings support to the new law, while stakeholder theory tends to condemn it. This surprising result highlights how sensible the theories are to corporate law. In our view, it reveals some weaknesses in both theories.

IV.2-A re-opening of theories of control and engagement

In our view, the former paradox is due to the way theories have been developed. Agency theory is based on a theory of control, whereas all the other theories have criticized the over-emphasis on control to invest other governance issue. As a result, agency theory supports PPCs as the related law reinvest the control in a stakeholder perspective. In our view, this reveals a gap in stakeholder theory and calls for a new and extended theory of control in governance.

Similarly, stakeholder theory and team production theory consider the firm as invested by many different parties. But they do not explain why PPCs exist and they do not see how PPCs can allow new companies. According to these theories, the normal corporate law would be sufficient to engage multiple constituencies in collective ventures. But, without presuming of their effectiveness, the very existence of PPCs tend to demonstrate that additional commitments may be necessary for parties to invest jointly in new ventures. We can assume that some investments are made possible, which would not have been possible in the normal corporate forms. In our view, this reveals another research gap, as the conditions of engagement are not fully understood.

These new avenues for research confirm that our theories of corporate governance would need to integrate corporate law as a critical contingency variable. So far, they have been developed in a given legal context. We may now need to re-elaborate them to overcome legal contingencies.

  have a legal existence, the question for scholars is: to what extent does this new legal framework impact corporate stakeholder-orientation or corporate social responsibility? Is it efficient and does it actually allow genuinely responsible and socially positive businesses? Does it induce positive or negative side effects on corporate governance? In this paper, we would like to address the question of the potential efficacy of the new legal corporate forms by drawing upon the literature on corporate governance: what is to be expected from what we know? And what can existing theories predict regarding the effects of these new legal forms? The paper is an attempt to use what we know on corporate governance to predict the effect of the new legal framework. It proposes to map the literature on corporate governance, with a special attention to how it relates to corporate law, with the intent to derive tentative predictions about the efficacy and effects of the new corporate framework.

  1

A law protecting better shareowners could basically favour financial investments and support economics. But this "legal hypothesis" according to which common law countries would have a competitive advantage is challenged. Some authors argue that law derived from political issues more than economics ones. And more importantly, the economic impacts of law are not really demonstrated. While having well-defined property rights is generally positive for economic growth, « the literature is less clear on whether one particular system of organizing financial markets is superior to any other and even suggests that banks, stock markets, and bond markets all generate comparable economic growth. Finally, there is little evidence that labour laws that provide more extensive worker rights and welfare provisions inhibit economic growth » Fligstein, N. and J. Choo,

2005, "Law and Corporate Governance." Annual Review of Law and Sociology Science Vol. 1: pp. 61-84..

Access to critical resources

Interpretation of corporate law

The board makes managers accountable toward shareowners

-idem

The board is insulated to be a mediating third part

The board allows a representation of the parties

The board provides resources

Further variables and recommendations