Eric Heyrman
email: eric.heyrman@inrets.fr

Hopes and limits of the new French policy: towards a fragmented accessibility?

With the enactment of the Equal Rights and Opportunities, Participation and Citizenship of People with Disabilities Act, France has reformed its accessibility policy. Within a period of ten years, collective transportation services must be made accessible. Specialized services are only a substitute to accessible transportation services. When it is technically infeasible to render existing conventional services accessible, transportation authorities must make specialized services available for people with disabilities within three years.

Admittedly the new law benefits from a favorable context. Indeed, people with disabilities no longer accept to have a special treatment. Furthermore, many technical solutions have already been developped. Lastly, the accessibility issue is now set on the institutional agenda of several organizations.

Nevertheless, future decrees may impact its effectiveness and change its objectives. Furthermore some shortcomings in the new legal framework can already be identified. It is to be feared that the priority order between accessible conventional services and specialized services may be reversed at a local level. Juxtaposed implementations in a same area without coordination and a differentiated implementation throughout the national territory are foreseeable. There are also structural forces such as financial and economic constraints which challenge the legitimacy of the new law.

Several tools for local actors should be set on the research agenda to avoid some of these likely outcomes.

INTRODUCTION

The French government enacted on February 11, 2005 a new law which has reformed its policy on people with disabilities. This act, named n°2005-102 for Equal Rights and Opportunities, Participation and Citizenship of People with Disabilities (Loi pour l'Egalité des Droits et des Chances, la Participation et la Citoyenneté des Personnes Handicapées), reaffirms the integration of people with disabilities in the daily life of society as a policy objective. Accessibility is one means to reach this goal. The law n°2005-102 contains several articles on accessibility. The main measure is: over the next ten years, public transport and all public areas, whether publicly or privately owned, must be made accessible in the accordance to the "everything for everyone" principle. Indeed, France had an accessibility policy elaborated in 1975. However, previous studies have shown a gap between that political will of integrating people with disabilities into society and its translation into concrete achievements at a local level (1, 2 and 3). In this paper, we try to apprehend the likely future effectiveness of the 2005-law. The effectiveness of a law is the difference between what is prescribed by law and what exists in reality [START_REF] Cohendet | effectivité et validité. Mélanges Pierre Avril[END_REF]. It is well known that a law does not strictly determine behaviors but instead restricts them [START_REF] Serverin | Sociologie du droit[END_REF]. Law is not inevitably a constraint. It is often a tool used by local actors. We rely on feedback from some foreign policies on accessibility to predict its future effectiveness.

In this paper, we present the main measures of the new French accessibility policy (I). Then we show that this policy has emerged in an international, social, technical and political context which would facilitate its implementation (II). Furthermore, the content of future decrees is a decisive factor in the policy's success, influencing its legitimacy, its coherence, its credibility and its acceptance (III). Lastly, we indicate its foreseeable limits and outcomes. A research agenda emerges from this study. It can be useful to tackle the raised shortcomings which concern many national policies (IV).

I-FRANCE HAS STATED A NEW AMBITION WITH A POLICY SHIFT

The law n°75-534 named Orientation Act in favor of People with Disabilities of June 30, 1975 said that these people are to remain in the ordinary living environment when possible. Since 1975, different regulatory texts have been published concerning establishments open to the public, residential buildings, places of work, transportation and more recently street infrastructures. The law n°2005-102 represents a shift in the national accessibility policy.

Indeed, the legal framework gradually adopted since 1975 was noticeable by an obligation to make accessible establishments open to the public and street infrastructures during new projects or when modifications were made. It also marked a provision for the right to mobility for people with disabilities. That right took the form of:

-specialized services, -the fitting of new travelling vehicles, -the adaptation of conventional collective transportation networks, -and the use of personal vehicles. The law n°2005-102 imposes an obligation to make the living environment accessible within a period of ten years. It marks the passage from a strategy of opportunity to a universal strategy with a deadline and from the right to mobility to an obligation to render transportation systems accessible.

According to the new law, "the travel chain" -including the built environment, street infrastructures, public areas, transportation systems and their intermodality -is organized to allow its accessibility on a whole to people with disabilities or to people with reduced mobility. In fact, the law n°2005-102 differentiates establishments open to the public, public transportation systems and street infrastructures.

Establishments open to the public: a legal framework with several exemptions

All new establishments open to the public must be accessible to all and namely to people with disabilities -whatever the type of impairment (in particular physical, sensory, cognitive or psychic impairment).

Existing establishments open to the public must be such as any person with disabilities can reach it, circulate and receive information which is disseminated in the parts open to the public. The information intended for the public must be spread by means adapted to various impairments.

Decrees will specify the accessibility requirements and the services that they must provide to people with disabilities according to the type and category of establishments open to the public. Existing establishments will have to fulfil these requirements within a timelimit as indicated in decrees, which will vary with the type and the category of establishment. But it cannot exceed a maximum period of ten years as from the enactment of the law n°2005-102. Some decrees will specify the particular exemptions which can be granted to establishments open to the public. The National Advisory Council of People with Disabilities (Conseil National Consultatif des Personnes Handicapées) will be consulted about the "exemptions" decree. These exemptions are granted after agreement of the Local Advisory Commission of Civil Protection, Safety and Accessibility (Commission Départementale Consultative de la Protection Civile, de la Sécurité et de l'Accessibilité). They must be accompanied by substitute measures for establishments open to the public which fulfil a "mission of public utility". All establishments open to the public in the travel chain will be affected by this measure (rail stations for instance).

Moreover, when construction work subject to a building permit is completed, the building owner must provide a document certifying compliance with accessibility regulations to the authority who has delivered the building permit. This certificate is established by a technical controller. The administrative authority can decide to close an establishment open to the public which does not meet accessibility requirements. Furthermore, the public authority who granted the subsidy will require a refund if the building owner is not able to provide this certificate.

Public transportation system: a legal framework based on master plans

Within a period of ten years as from the enactment of the law n°2005-102, collective transportation services must be accessible to people with disabilities and to people with reduced mobility. In the French policy, there are no specific deadlines for buses, trams and railway transportation systems.

Any vehicle purchased to replace the fleet or to extend the public transportation network must be accessible to people with disabilities or to people with reduced mobility.

Granting of a public subsidy to develop a collective transportation system is subordinated to the taking into account of accessibility: only accessible systems will be subsidized by the central State.

All transportation authorities and managers of main airports or main harbour stations have to prepare a master plan whose goal is to make accessible collective transportation services they are responsible for. This master plan must be published within a period of three years. It contains the time scheduling of accessibility measures which are distinguished by transportation modes.

When a proven technical infeasibility to make existing networks accessible arises, transportation services adapted to the needs of people with disabilities and people with reduced mobility must be made available for these persons. They are organized and financed by the transportation authority in charge of the conventional system within three years. The fare for the substitute transportation system should not be higher than the fare of the existing public transportation system.

Existing underground railway transportation systems and existing underground guided transportation systems are not subject to the ten-year-deadline provided that the relevant transportation authority designs an accessibility master plan and creates a substitute transportation system within three years.

Moreover, the mayor or the president of the "Inter-communal Structure" (see table 1) establishes a master plan of making street infrastructures and public installations accessible. It takes special care of walkways and parking areas.

Finally, the urban mobility master plans are the last tools put at the disposal of local authorities by the law n°2005-102 to make the city accessible. These plans have several goals. They have already aimed at solving mobility problems, promoting the least polluting and consuming energy modes of transportation and urban and social cohesion. Under the law n°2005-102 urban mobility master plans now integrate a particular appendix relating to accessibility. This appendix indicates measures to be implemented to improve the accessibility of public transportation systems and their timetable. The master plan of street infrastructures accessibility also forms an integral part of the urban mobility master plan. Associations of people with disabilities must be included during the preparation of urban mobility master plan if they want to.

Authority and traveller relations

Within a maximum period of three years, the online services of the State and local authorities must be accessible to people with disabilities. The services of transportation authorities and transportation operators are particularly concerned by this measure.

Within three years as from the enactment of the law n°2005-102, transportation authorities must set up a procedure to register complaints for existing barriers which prevent people with reduced mobility from moving.

Creation of a new local commission for accessibility

A new commission named "Local Commission for Accessibility" (commission communale pour l'accessibilité) will be created in "Communes" (the smallest administrative area in France) or "Inter-Communal Structures" of more than 5000 inhabitants.

The representatives of the "Commune", those of associations of transportation users and those of associations of people with disabilities are ex officio members of this commission. The mayor chairs the commission and specifies the list of its members.

This commission draws up the state of accessibility of the built environment, street infrastructures, public areas and transportation systems existing in the "Commune".

It publishes an annual report presented at the city council and makes all useful proposals likely to improve the accessibility of the city.

Triennial national conference on disability policy

Article 3 of the law n°2005-102 provides that a national conference on the French disability policy is organized on the initiative of the Government every three years.

The public and private actors working in favor of people with disabilities are invited to this conference to discuss orientations and means of the national disability policy.

At the end of the national conference, the Government releases a report on the implementation of this policy. The National Advisory Council of People with Disabilities (Conseil National Consultatif des Personnes Handicapées) gives its opinion on this report. It is transmitted to the Senate and to the National Assembly. The evaluation of the national disability policy might be publicly debated in Parliament.

The Equal Rights and Opportunities, Participation and Citizenship for People with Disabilities Act lies down a deadline to make the living environment accessible. It also institutes new tools for planning and a new local commission is created. Decrees of application will be soon published to specify this ambition of a city accessible to people with disabilities.

DISCUSSION: IS THE 2005-LAW LIKELY TO HAVE A GOOD EFFECTIVENESS?

The law n°2005-102 has determined the objectives of the French accessibility policy. One cannot help wondering whether it will reach these goals or not. In this section, we argue that the 2005-law is more likely to make the living environment accessible than the 1975-law. However, the French accessibility policy is not completely defined. The content of future regulations could modify the legitimacy, the coherence, the credibility and the acceptance of this policy, which could affect its effectiveness. Lastly, some policy failures can already be predicted.

II-A favorable context to pass and implement this law

We use the John Kingdon's policy stream model in order to specify the conditions of the policy change [START_REF] Kingdon | Alternatives, and Public Policies[END_REF]. According to this model, three streams have to be explored: the problems stream, the solution stream and the politics stream. Kingdon argues that a reform window opens at the conjunction of the three streams. This is apparently the case for the accessibility policy.

A shift of prevailing model of disability

Since 1980s the prevailing medical model of disability has declined in favor of the social model of disability [START_REF] Oliver | The politics of disablement[END_REF].

People with disabilities no longer accept to have a special treatment. The first demonstration against inaccessible public areas took place in 1982 [START_REF] Larrouy | The Review of Disability Studies: an international journal[END_REF]. Since 1985, the main association of people with physical impairment (APF) has claimed for "accessibility of individual and collective transportation". Organizations of people with sensory impairment stayed in the background until late 1980s. The main association of people with cognitive disabilities (UNAPEI) expresses demands on accessibility since mid-1990s.

To illustrate this shift of prevailing model, we can notice that medias had stressed on allowance for people with disabilities in 1975 [START_REF] Sanchez | L'accessibilité et les associations dans les années soixante[END_REF]. In 2004 and 2005, they have emphasized on the accessibility of the living environment, compensation and administrative simplification. Furthermore the 1975-law was based on an individual model of disability (10 and 11). The 2005-law proclaims to be based on the social model of disability. Its third part is entitled "accessibility" whereas the 1975-law had only two articles on accessibility.

Moreover, France has now an ageing population. That was evidently not the case in 1975: French demography left the baby boom era. This has been used by accessibility advocates to promote "accessibility for everybody", "universal design" or "comfort for all" [START_REF] Heyrman | Les personnes à mobilité réduite : vers une mise en accessibilité globale des projets d'aménagement publics? L'exemple lyonnais[END_REF]. It has legitimated decisions made in favor of accessible transportation whose people with disabilities have benefited first.

Except medias implication correlated to the law making context, all these elements could remain accurate during the implementation of the new policy. So they should make it easier to implement the 2005-law.

Many technical solutions have already been developped

The previous accessibility policy encouraged local experiments and "city emulation". The State has got used to highligting exemplary achievements. As a result, many technical solutions have been designed since 1975. Indeed, all new vehicles can be designed as accessible to people with physical impairment. For instance, COST 322 Low Floor Buses and COST 335 Passengers' Accessibility of Heavy Rail System have validated technical solutions. A few ISO Standards have been released or are about to.

However, according to French policymakers, bridging the gap between platform and heavy rail vehicle floor (except for tramway systems) and making over-the-road passenger coaches accessible remain to be solved. Since associations of people with sensory impairment and those of people with cognitive disabilities have claimed for accessibility recently, all access problems of these disabilities have not been settled. These issues are currently set on the research agenda. Furthermore, carrying out research study in social sciences is now more common. This field had been neglected for a long time.

To be brief, almost all new vehicles are designed accessible… at least for people with physical impairment. Consequently, accessibility will be improved when transportation authorities buy new vehicles. Nowadays it is more difficult to buy a non-accessible vehicle than to purchase a low-floor one…

Several institutions have set the accessibility issue on their agenda

At an international level, the ECMT established in 1985 a working group on access and inclusion. Its role consists of bringing together Governments and experts to exchange ideas on transportation for people with reduced mobility. This group has produced many reports and recommendations which provide guidance on achieving barrier-free travel. Political support for this work has been given through the adoption of several formal resolutions by the Council of Ministers of the ECMT. This Council has encouraged its Member States to promote accessible transportation.

Furthermore, the United Nations adopted "the World Programme of Action concerning Disabled Persons" in 1982 and "the Standard Rules on Equalization of Opportunities for Persons with Disabilities" in 1993. The rule 5 is relating to accessibility. Moreover, on 15 February 2001, the Committee of Ministers of the Council of Europe adopted the Resolution "on the introduction of the principles of universal design into the curricula of all occupations working on the built environment".

These texts recommend that governments of member states draw up and implement national strategies applying their principles. Nevertheless, they are not coercive constraints.

The European Union has such a power. It addresses regulations, directives or decisions to the Member States. The European Parliament and the Council have adopted the "Directive 2001/85/EC relating to special provisions for vehicles used for the carriage of passengers comprising more than eight seats in addition to the driver's seat". This directive requires that all new urban buses to be low floor. Technical Specifications for Interoperability on Accessibility for Reduced Mobility will soon be passed. They will detail requirements on accessibility to infrastructure and rolling stock provided for in the Directive 2001/16/EC on interoperability of the trans-European conventional rail system. These directives are binding on the Member States as to their effect whilst leaving the choice of means of implementation to national administrations.

In a word, several international organizations encourage their member States to draw up national policies on transportation accessibility. We can notice a policy convergence amongst Western countries. Thus, the USA, the UK and Germany respectively passed the Americans with Disabilities Act in 1990, the Disability Discrimination Act in 1995 and the Equal Treatment for People with Disabilities Act (die BehindertenGleichstellungsGesetz) in 2002. They have similar objectives: they aim at a barrier-free living environment. These three laws require that all new vehicles and facilities be accessible. However their measures are quite unlike to render existing transportation systems accessible. According to the German policy, freedom from barriers as far as possible is a goal. But there is no deadline. The ADA requires that at least one car in each train having two or more cars be accessible within five years. Existing key stations in rapid rail, commuter rail and light rail systems must be readily accessible within three years. All existing Amtrak stations must be accessible by 2010 (49 CFR 37.55). As for the UK policy, it differentiates each type of transportation modes and attributes them a different deadline.

At a national level a new deal appeared when President Chirac was re-elected. In 2002 in a national address, the President made integration of people with disabilities into society one of three priorities for his second mandate. Taking this stand has made the policy shift easier. However, this national impulse will be likely to disappear in 2007 when a presidential election takes place. As a result, policy implementation stage would be likely to distinguish by a lack of involvement of national politicians.

Several elements shown above might overcome this shortage. Furthermore, recent articles in the French national press have highlighted the French delay on accessibility.

France is seen to be out of step with other Western countries. Consequently, it has to implement an ambitious policy.

III-The unknown: the strategic content of future decrees

The law n°2005-102 makes provisions for publishing more than eighty decrees …within six months.

It is an important step since local actors usually wait for the publication of decrees to implement a law. Moreover, in his paper, Thoenig showed that decrees and circulars are more important, consistent and significant than law [START_REF] Thoenig | L'action publique. L'Harmattan[END_REF]. They specify the operational translation of public choices. They sometimes empty the content of a law or they sometimes change the objectives of a policy.

The first work of the Government and the central administration of the French Ministry of Transport and Public Works will be to give a definition of accessibility and people with reduced mobility. This definition reveals the ambition of integration of a society and of a time. This ambition takes shape into technical standards (14). The concept of accessibility has gradually included people with physical impairment, then people with sensory impairment and lately people with cognitive and psychic disabilities. Disability is eminently cultural (15 and 16). There are indications that the concept of accessibility will continue to evolve. Several decrees will be published to specify accessibility requirements: of new buildings, of existing residential buildings, of existing establishments open to the public, of vehicles of collective transportation services. These requirements of accessibility can take various forms. Thus, in the field of transportation there are two distinct procedures used by Western countries:

to define standards of performance, prescribing a minimal level of accessibility, or to impose an objective -i.e. providing a transportation service -whilst leaving the choice of means used to reach this obligation of result. On the one hand, the United States and the United Kingdom have privileged the first alternative. In the United States, the Congress has mandated the Department of Transportation to publish regulations specifying, amongst others, the horizontal and vertical gaps between platform and vehicle floor [START_REF] Simon | Factors influencing public transit operators' policies for accessible transportation[END_REF]. The United Kingdom has also standards of performance. Moreover, in the UK new vehicles must receive a license from the Department for Transport, certifying compliance with the DDA, before going into service [START_REF] Baker | Moving backwards: Canada's state of transportation accessibility in an international context[END_REF]. On the other hand, functional regulations are used by New Zealand to buy travelling vehicles of European, American and Asian technology freely (18 and 19). In New Zealand it is prohibited to refuse the access of vehicles or services open to the public because of disability.

The increasing practice of unfortunate candidates of calls for tenders (i.e. manufacturers) to sue the result of these calls would encourage policy makers to define detailed and precise standards of performance. Publishing regulations would increase the legal security of transportation authorities. Otherwise technical standards of accessibility would be defined by the judgments of the administrative courts.

The French Parliament also mandated the Government to specify the conditions to grant an exemption to the legal obligation to make residential buildings and establishments open to the public accessible. According to the 2005-law, there are three types of exemptions:

-when a proven technical infeasibility to make the building or the service accessible arises, -because of constraints relating to the conservation of the architectural heritage, -when there is manifest disproportion between improvements and their consequences.

The possibilities to depart from the rules should not be too many. Otherwise, organizations of people with disabilities would no longer believe in the politicians' discourses and in their goodwill. The credibility of the accessibility policy is at stake. On the other hand, if there were no exemptions, this policy woud be seen as an unrealistic policy by owners. Consequently, they would not apply the 2005-law which does not take their local constraints into account. Exemptions must exist to improve the acceptance and the credibility of the accessibility policy.

Moreover, regulations will have to define what collective transportation services are. We do currently not know which transportation services are really affected by the law n°2005-102. At the moment there is no legal text which precisely delimits the scope of these services. However the scrutiny of the law n°2005-102's parliamentary debates would encourage us to believe that this concept of "collective transportation" falls within the framework of the Duguit School of public service. As a consequence, the "collective transportation services" would include all the transportation services controlled and managed by transportation authorities. Taxis would be affected by the law n°2005-102 only if they were funded by a public body. Transportation systems managed by private organizations without public subsidise would not be concerned by the obligation to render collective transportation services accessible. Furthermore, decrees will clarify two principles of the law n°2005-102 which will deeply affect the action of local authorities. They will specify on the one hand what a technical infeasibility to make collective transportation services accessible is and on the other hand which organization is concretely responsible for asserting that it is technically infeasible to do so. The wide range of local situations would incite to choose a local organization. The law n°2005-102 of February 11, 2005 reinforces the powers of the Local Advisory Commission of Civil Protection, Safety and Accessibility. According to in force regulations and to the law n°2005-102, this commission has already been in charge of the exemptions to make establishments open to the public and street infrastructures accessible. A sensible and practical solution would be to centralize all of the exemptions of the accessibility policy into this commission (see table 2).

The law n°2005-102 imposes to make available transportation services adapted to the needs of people with disabilities within three years when it is infeasible to make conventional transportation services accessible. So "dial a ride" services are substitutes of totally accessible conventional transportation services. The language of the law n°2005-102 allows the Government to release regulations to specify the role allocated to these specialized services and their recipients.

Some transportation authorities have already managed such specialized services in France. For reasons of safety, liability and operation, society has preferred that certain people with severe disabilities do not use conventional public transportation services [START_REF] Imrie | Disability and discourses of mobility and movement[END_REF].

It is likely that attractiveness of these services and their demand will increase as their fare will decrease when the law n°2005-102 is applied. However the American example shows that the conditions of eligibility should be strict. In fact, American local authorities have often been reluctant to restrict use of specialized services to the disability conditions described in ADA rules …to avoid being prosecuted for discrimination (21 and 22). The results are an increase of the paratransit service use and a rise of their cost. The American local authorities are currently trying to shift people with disabilities able to use the accessible transportation system from the transport on demand to the conventional service.

Moreover, it is to be feared that French transportation authorities will develop paratransit services first since the legal deadline is three years from the enactment of the law n°2005-102. Then people with disabilities may get into the way of using these services. So their operating budget could rise. As a result, there could be no longer money to render conventional transportation services accessible.

Thus, the priority order between accessible conventional services and specialized services may be reversed at a local level. Therefore, the conditions of eligibility should be defined at a national level and be binding. These specialized services should be reserved for people who really need them to challenge the idea that paratransit is the best means to reach mobility of people with disabilities and to legitimate and give credibility to the accessibility policy. In the spirit of the law n°2005-102, these conditions would only depend on the functional aptitudes of the person -the real ones and those perceived by this person (23)and on the level of the transportation system accessibility.

IV-A research agenda to avoid foreseeable undesirable effects of the new legal framework

Publishing decrees is an important step and may modify and clarify a policy. However this stage only concerns fields for which the French Parliament has mandated the Government. According to articles 34 and 37 of the French Constitution, some measures of the new legal framework are already "definitive". Consequently, we can seek to anticipate likely policy failures and to tackle them by developing some specific tools. Three main risks jeopardize the policy objective of a barrier-free living environment.

1-Juxtaposed implementations in a same place

France has three levels of local authorities. It is divided into 26 "Régions", 100 "Départements" and more than 36000 "Communes". All these local authorities are transportation authorities (see table 1). Moreover, the State is in charge of long-distance railway services. As a result, at least three transportation authorities are concerned by a same territory. According to the law n°2005-102, all these authorities must make their collective transportation services accessible and elaborate an accessibility master plan.

How avoiding juxtaposed implementations in a same place? How coordinating the different access measures? Indeed, the 2005-law states that the travel chain includes "the built environment, street infrastructures, public areas, transportation systems and their intermodality". How treating the interface between two transportation networks? In fact, they are no hierarchical links between the different administrative levels of local government. French transportation authorities are not used to working together on the accessibility policy. The 2005-law does not create a coordinating committee.

In order to reach an accessible living environment, we suggest to expand the range of the missions of the Local Commission for Accessibility and its membership (see table 2). This commission would have two additional missions. On the one hand it would coordinate the programs of the different local actors. On the other hand it would manage the accessibility public policy at an urban area level. According to foreign studies, the Local Commission for Accessibility should include: associations of and for people with disabilities (24 and 25), representing the physical, sensory, cognitive and psychic disabilities (26, 27 and 28), associations of transport users, all transportation authorities present in the urban area, transportation operators which are responsible for the vocational training of their drivers (29) and for information to travellers, authorities in charge of street infrastructures, authorities in charge of parking policy. The Local Commission for Accessibility would be the place to enhance the city accessibility and to define together priorities and a coordinated program of measures. Formalization by contracts seems to be a relevant tool [START_REF] Gaudin | Gouverner par contrat -L'action publique en questions[END_REF].

2-Differentiated implementation throughout the national territory

According to the law n°2005-102, the Local Commission for Accessibility draws up the state of accessibility in the territory of the "Commune". However, the French Parliament did not mandate the Government to publish a regulation on an evaluation methodology. Consequently, this methodology is likely to be defined at a local level by each local commission. Developing an evaluation methodology is an important step of the accessibility policy since it will define what is accessible or not. However, it would not be comprehensible that this definition of the "accessible city" varied from a city to another. Indeed French people have been far more ardently and tenaciously attached to equality than to freedom [START_REF] Gaudin | Gouverner par contrat -L'action publique en questions[END_REF]. So they prefer central norms to local bargaining of rules.

As a result, we suggest to elaborate an evaluation methodology at a national level. An ad hoc group would be created. It would include the national representatives of the members of local commissions for accessibility. These people would define an evaluation methodology and performance indicators of the accessibility policy. These indicators should have a double function. They would be a tool used by members of local commissions to manage the accessibility policy and they would constitute a means to inform people with disabilities of the accessibility level of their city. Consequently, these indicators should be easily comprehensible by all and have a significance shared by all actors as far as possible. This ad hoc group would rely on works of technical centers (for instance the Centre for the Study of Urban Planning, Transport and Public Facilities) and it would analyze foreign experiences to import and adapt them.

The appropriation of these methodology and performance indicators by their endusers -i.e. local actors -would constitute an important step. It is well known that national rules might be translated into local discourses and that their goals or significance might be (mis)interpreted [START_REF] Lascoumes | Normes juridiques et mise en oeuvre des politiques publiques[END_REF]. As a consequence, the evaluation methodology and performance indicators should be precisely defined to prevent this translation and improve the policy coherence.

Lastly, appropriating these indicators by local actors without misrepresenting them would make it possible to compile data truly comparable and to evaluate the national accessibility policy each three years as planned by the law n°2005-102.

3-Difference between "buildings exemptions" and "transportation exemptions": the consideration of the cost has been evaded

According to the law n°2005-102, exemptions to make an establishment open to the public accessible can be granted to the owner "when there is manifest disproportion between improvements and their consequences". The exemptions because of economic reasons are included into this type of exemptions. As for the transportation field the 2005-law holds that only technical exemptions can be granted. However, rail stations are considered as establishments "open to the public". Which types of exemptions apply to them? This incoherence in the law is likely to be settled at a local level. This is a risk of a spillover effect from the building accessibility policy to the transportation accessibility policy, more especially as no special subsidies for the accessibility policy have been planned by the Government.

Likewise the ADA [START_REF] Winter | Transit access for Americans: a proposal for the next stage of implementing the Americans with Disabilities Act of 1990[END_REF], the law n°2005-102 could be perceived as a costly mandate. For instance, the "Département of Eure-et-Loir" has assessed the cost to make their interurban transportation services accessible at more than 130 million dollars (110 million euros) [START_REF] Coliac | Analyses et propositions pour la mise en application des dispositions concernant l'accessibilité du cadre bâti et de la chaîne de déplacement[END_REF]. Its annual operating budget for transportation services is almost 70 million dollars (58 million euros) in 2005.

In order to fit with the way of thinking of transportation operators and transportation authorities, we suggest to develop a methodology of a cost benefit analysis. Unlike the United Kingdom, France is unused to assessing economic impacts of legal texts. Consequently, the capital and operating costs of rendering existing transportation services accessible might be overvalued. Furthermore, the benefits of accessibility are often forgotten. So we should identify and quantify costs, benefits and net benefice [START_REF] Oxley | Compliance costs for the bus industry. The impacts of the Disability Discrimination Act 1995, on local bus services[END_REF]. Likewise the USA (35), we suggest to monetize user and non-user benefits. We should specify the "total economic value" of accessible transportation services. This total economic value includes "use value", "option value", "bequest value" and "existence value". To be more accurate, option value represents an insurance value of accessibility in relation to the probability of becoming disabled (temporarily or permanently). Bequest value is the value of leaving something behind for the next generation. Existence value recognizes people's willingness to pay for such things as the extension of equal protection to different groups. By defining these figures, accessibility advocates would be able to meet the constraints and the way of thinking of transportation operators and of some local actors. Because of net benefits, the transportation accessibility policy would be legitimated. Moreover, these figures might be used to compare the economic efficiency of parallel paratransit services and the accessible conventional services ones.

CONCLUSION

France has recently reformed its policy in favor of people with disabilities. The law n°2005-102 spotlights the image that the French society gives to itself: it is supposed to be a friendly society which does not leave one of its members by the way side. One of the policy objectives is to make the living environment accessible within a period of ten years. Existing transportation services are also affected by this measure. As a result, France would be barrierfree in 2015.

Keep our distance from this statement! Some shortcomings in the legal framework can already be identified. For instance, there are no real sanctions for transportation services that fail to comply. Indeed there are no penal sanctions. Furthermore, a withdrawal of State funds took place in 2004. The French State has no longer allocated funds for transportation investments since that date. So the effects of financial sanctions would be quite limited: only local authorities will grant subsidies to improve local transportation systems.

The effectiveness of this law will depend on the "strength of the law" which is a cultural factor. Indeed, there is no doubt that the 2005-law is valid since it has been adopted by the French Parliament. But its legitimacy, its credibility, its coherence and its acceptance are at stake. There are structural forces such as financial and economic constraints which challenge its legitimacy.

In order to increase policy coherence, we suggest to clarify exemption cases and conditions of eligibility to specialized transportation services with the greatest precaution. Their definition might lead to a gap between national objectives and local achievements (or non-achievements).

Likewise most countries (19, 26 and 27), the French accessibility policy would be implemented by numerous local actors without coordinating their various programs. To prevent this, we put forward the creation of a coordinating committee which could be the Local Commission for Accessibility with additional missions and members. This commission would use a policy evaluation methodology and performance indicators defined at a national level. Local actors would be able to manage a coordinated policy using the same tools. Formalization by contracts would make up this plan of action.

Moreover, everywhere in the world, strong pressures are exerted on public budgets. Rendering existing conventional transportation services accessible is often seen to be costly and a burden. In order to convince opponents, we propose to value user and non-user benefits of accessible transportation services in monetary terms and to carry out cost-benefit analyses. Such a "legal value" would legitimate that the accessibility policy becomes one of the top priorities as giving a value to human life has justified road safety measures.

Developing policy evaluation methodology, defining performance indicators and giving a monetary value to accessibility emerge from this study. Since most countries have similar concerns, they should be set on the international research agenda. The Communes can work together and create an "Inter-communal structure". When a Commune has transferred a competence to the "Intercommunal structure", the legal obligation is also transferred.

2005-law obligations

Master plan of accessibility Master plan of accessibility Master plan of accessibility plan to make urban transport services accessible Master plan to make street infrastructures accessible for communes with more than 5000 inhabitants

The managers of airports and harbor stations are often the chamber of commerce and industry. But local authorities can also be managers of airports. The legal obligation is also to publish an access master plan.

Proposal:

It would be also in charge of the exemptions to make collective transportation services accessible.

Proposals:

All transportation authorities present in the urban area, transportation operators and authority in charge of parking policy would be members of this commission.

It would become the place to define an accessibility policy at an urban area level and to coordinate all master plans.

  TRB 2006 Annual Meeting CD-ROM Paper revised from original submittal.

TRB 2006

 2006 Annual Meeting CD-ROM Paper revised from original submittal.

TABLE 1 The

 1 Administrative Organization of France And New Legal Obligations

			Transport competences
		The State	Long-distance railway services
		Régions (France is	Regional railway services
		divided in 26 régions)
	Local authorities	Départements (France has 100 départements) Communes (France has	Urban transport services Interurban transport services School transport services
		more than 36000	Street infrastructures
		communes).

TABLE 2

 2 Proposals to Improve the Legal Framework

	Local advisory commission of civil protection,	(New) local commission for accessibility
	safety and accessibility (already settled)
	According to the previous regulation, its members	According to the 2005-law, its members are:
	are:	-The commune
	-The representative of the State in the	-Associations of people with disabilities
	"Département" (the "Prefect")	-Associations of users
	-Local services of the central State
	-Representatives of local authorities
	-Representatives of associations of people with
	disabilities
	It is managed by the "Prefect".	It is managed by the mayor.
	It gives an opinion which has an obligatory statute in	It draws up the state of accessibility of the living
	building permit process.	environment.
	It is in charge of the exemptions to make	It publishes an annual report.
	establishments open to the public and street	It makes all useful proposals likely to enhance
	infrastructures accessible.	accessibility.

TRB 2006 Annual Meeting CD-ROM Paper revised from original submittal.

-Iwarsson, S. and A. Stahl. Accessibility, usability and universal design -positioning and definition of concepts describing person-environment relationships. Disability and rehabilitation, Vol 25, n°2, 2003, pp. 57-66. TRB 2006 Annual Meeting CD-ROM Paper revised from original submittal.

-De Boer, E. Introducing and sustaining accessible transport. Social and physical challenges. Transportation Research Record: the journal of the Transportation Research Board, n°1885, 2004, pp. 15-20. TRB 2006 Annual Meeting CD-ROM Paper revised from original submittal.

LIST OF TABLES


Andrea G Filippini

Maria Kazolea

Mario Ricchiuto

A Flexible 2D Nonlinear Approach for Nonlinear Wave Propagation, Breaking and Run up

Keywords: Green-Nagdhi equations, unstructured meshes, Finite Volume, Finite Element, wave breaking

We present a hybrid solution strategy for the numerical solution of the two-dimensional (2D) partial differential equations of Green-Nagdhi (GN), which simulates fully nonlinear, weakly dispersive free surface waves. We re-write the standard form of the equations by splitting the original system in its elliptic and hyperbolic parts, through the definition of a new variable, accounting for the dispersive effects and having the role of a non-hydrostatic pressure gradient in the shallow water equations. We consider a two-step solution procedure. In the first step we compute a source term by inverting the elliptic coercive operator associated to the dispersive effects; then in a hyperbolic step we evolve the flow variables by using the non-linear shallow water equations, with all non-hydrostatic effects accounted by the source computed in the elliptic phase. The advantages of this procedure are firstly that the GN equations are used for propagation and shoaling, while locally reverting to the nonlinear shallow water equations to model energy dissipation in breaking regions. Secondly and from the numerical point of view, this strategy allows each step to be solved with an appropriate numerical method on arbitrary unstructured meshes. We propose a hybrid finite element (FE) finite volume (FV) scheme, where the elliptic part of the system is discretized by means of the continuous Galerkin FE method and the hyperbolic part is discretized using a third-order node-centred finite volume (FV) technique. The performance of the numerical model obtained is extensively validated against experimental measurements from a series of relevant benchmark problems.

INTRODUCTION

Accurate simulations of water wave's propagation and non-linear wave transformations is of fundamental importance to marine and coastal engineering. Over the last decades, significant efforts in the development of depth averaged models have been made in order to provide the means of accurately predicting near-shore wave processes such as shoaling and runup, diffraction, refraction and harmonic interaction. One of the most applied depth averaged models is the Non-linear Shallow Water Equations (NSWE) which are able to model important aspects of wave propagation phenomena, the general characteristics of the ru-nup process, and the wave breaking with broken waves represented as shocks, but they are not appropriate for deeper waters and shoaling since they neglect all the dispersive effects that play a very important role. In order to take dispersive effects in to account we must keep the O(µ 2 ) terms from the full water waves equations, which where neglected in the derivation of the NSWE. µ is the shallowness parameter defined as water depth to wavelength ratio h 0 /L. This leads to the Green-Naghdi (GN) equations [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF] known also as Serre equations. The range of validity of the model may vary as much as far the non-linearity parameter (defined as the ratio of wave amplitude to water depth A/h 0 ) is concerned, but it requires the shallowness parameter µ to be small.

In this work we use the improved GN system of equations in the form proposed in Bonetton et al. (2011). This formulation has been recovered by adding some terms of O(µ 2 ) to the momentum equation in order to improve the frequency dispersion description of the original GN model, using a tuning parameter α. The two dimensional form of the system can be written in the following form:

h t + ∇ • (hu) = 0 (1) (I + αT c ) q t + ∇ • q ⊗ q h + gh∇η + g α h∇η + hQ 1 (u) = 0
where the operator Q(•) is:

Q(•) = 2 3h ∇ h 3 ∇(•) 1 • ∇ ⊥ (•) 2 + (∇ • (•)) 2 + +h 2 ∇(•) 1 • ∇ ⊥ (•) 2 + (∇ • (•)) 2 ∇b + + 1 2h ∇ h 2 (•) • (•) • ∇ ∇b + (•) • (•) • ∇ ∇b ∇b. (2)
We denote h(x, t) = h 0 + η(x, t)b(x) the total water depth, where η(x, t) the free surface elevation with respect to the water rest state h 0 , b(x) the topography variation and u(x, t) the flow velocity. (•) 1 and (•) 2 indicates respectively the first and second component of the vector (•) and ∇ ⊥ states for the normal gradient operator.

The operator T c (•) plays a key role, as its inversion is necessary to be able to obtain evolution equations for the physical variables. T c (•) can be written in compact form involving two operators S 1 (•), S 2 (•) and their adjoints S 1 (•), S 2 (•), as:

T c (•) = S 1 hS 1 (•) h + S 2 hS 2 (•) h (3)
where

S 1 (•) = h √ 3 ∇ • (•) - √ 3 2 ∇b • (•), S 2 = 1 2 ∇ • (•). (4)
Note that this formulation is essential to show the coercivity of the operator (I + αT c ) see [START_REF] Filippini | A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up[END_REF] and referenced therein, for further details. Note also that when α = 1 we retrieve the original GN equations.

DISCRETIZATION STRATEGY

To numerically solve (1), we rewrite the system of the two dimensional enhanced GN equations as:

(I + αT c ) φ = W -R, (5)
h t + ∇ • q = 0, (6)
q t + ∇ • q ⊗ q h + gh∇η = φ (7)
by splitting the original system in its elliptic and hyperbolic parts, through the definition of the new variable φ = [φ x , φ y ] T . Φ accounts for the dispersive effects and has the role of a non-hydrostatic pressure gradient in the Shallow water equations. W = T c (gh∇η) and R = hQ(u).

In this work we solve ( 5)-( 7) using a hybrid Finite Element (FE)-Finite volume (FV) scheme where the elliptic part of the system is discretized by means of the continuous Galerkin FE method. The hyperbolic part of the system is discretized by the two dimensional node centred FV formulation used in [START_REF] Kazolea | An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations[END_REF].

Elliptic Phase

We can prove the coercivity of the operator (I + αT c ) [START_REF] Filippini | Free surface flow simulation in estuarine and coastal environments: numerical development and application on unstructured mehes[END_REF] via the variational form of T . This property descends on the self-adjoint nature of the operator T c and is of primary importance to insure the invertibility of the matrix derived from the discretization (I + αT c ), which is a necessary condition for the discrete equation to be solved. The result of the elliptic equation will not be used inside the hyperbolic part in nodes where h i ≤ wd h ( wd h is a very small threshold value), and simply φ i = (0 , 0) T is set, so we can define a specific (per unit depth) dispersive correction. ψ such that: φ = hψ. We can, thus, formulate the elliptic step using the unknown ψ. The definitions of the operator T c and of S 1 (•) and S 2 (•), lead to the following variational form of the elliptic equation:

Ω h ν • ψ + α Ω S 1 (ν) h S 1 (ψ) + α Ω S 2 (ν) h S 2 (ψ) = RHS , (8)
being RHS a compact form to write the variational formulation of the right hand side of the equation, and being ν the vector of components respectively (ϕ i , 0) T or (0 , ϕ i ) T (with ϕ i the standard linear Lagrange basis functions), if the equation in the x or y direction is taken into account. ψ i = (0 , 0) T is imposed when h i ≤ wd h . The use of ψ as an unknown in the FE phase leads to a symmetric positive definite bilinear form on the left hand side of the variational equation ( 8), namely for the matrix of the differential operator (h + αhT nc ). Defining T c (•) = hT nc ( • h ) it holds that (I + αT c )φ = (h + αhT nc )ψ. The elliptic part of the system is, thus, solved for ψ. The value of φ, to use in the conservative hyperbolic equations, are then recovered nodally a posteriori by reverting the definition φ = hψ.

Let Ω h denote, an unstructured triangulation of the spatial domain Ω, with h the reference element size and with K denoting the generic element of area |K|. For a node i ∈ Ω h , let K i is the set of elements containing the node i. We consider piecewise linear continuous approximations η h and q h with standard piecewise linear continuous Lagrange bases. Denoting with (•) x and (•) y the component of the vectorial quantity (•) along the x-axis and y-axis respectively, we define Ψ = [Ψ x , Ψ y ] T , with Ψ x = [ψ x 1 (t), ψ x 2 (t), ..., ψ x N (t)] T and Ψ y = [ψ y 1 (t), ψ y 2 (t), ..., ψ y N (t)] T , and

U = [U, V] T , being U = [u 1 (t), u 2 (t), ..., u N (t)] T and V = [v 1 (t), v 2 (t), ..., v N (t)] T .
The discrete form of the elliptic equation (5) will thus read:

(M G H + αT)Ψ =W -R , (9)
W =T δ , (10)
R =Q (h, U) , (11)
with δ an approximation of g∇η. The matrix M G H is the four block matrix with zero off-diagonal terms and two equal blocks M G H on the main diagonal. M G H is the symmetric positive definite matrix whose entries are represented by M G H i, j = Ω h h ϕ i ϕ j .

The operator T(h h , b h ) is a matrix differential operator of order two, acting on two dimensional vectors, which can be written as:

T = T 1 1 T 1 2 T 2 1 T 2 2 .
We call T m,n i, j the element (i, j) belonging to the block (m, n) of the matrix T. Its fully discrete expression is obtained evaluating, with approximate numerical quadrature over each mesh element and the hypothesis of piecewise linear variations of all the quantities involved, the following series of integrals (obtained by the assumption of periodic boundary conditions):

T m,n i, j (h h , b h ) = 1 3 Ω h ∂ Xm (ϕ i ) h 3 h ∂ Xn (ϕ j ) - 1 2 Ω h ϕ i h 2 h ∂ Xn (ϕ j ) ∂ Xm (b h ) + - 1 2 Ω h ∂ Xm (ϕ i ) h 2 h ∂ Xn (b h ) ϕ j + Ω h ϕ i h h ∂ Xn (b h ) ϕ j ∂ Xm (b h ) ,
where {X} = {x, y} and m = 1, 2 and n = 1, 2.

Proceeding similarly, the full discrete form for the operator

Q[h h , u h , b h , δ bh ]
is found by evaluating, with a quadrature approximation over each triangle of the mesh and with the hypothesis of piecewise linear variation of the quantities involved, the following series of integrals:

Q[h h , u h , b h , δ bh ] = - 2 3 Ω h ∇ϕ i h 3 h ∇u h • ∇ ⊥ v h + ∇ • u h 2 + + Ω h ϕ i h 2 h ∇u h • ∇ ⊥ v h + ∇ • u h 2 ∇b h + - 1 2 Ω h ∇ϕ i h 2 h u 2 h ∂ x δ x bh + v 2 h ∂ y δ y b h + 2 u h v h ∂ x δ y bh + + Ω h ϕ i h u 2 h ∂ x δ x bh + v 2 h ∂ y δ y b h + 2 u h v h ∂ x δ y bh ∇b h . (12)
Following the linear dispersion analysis performed on the onedimensional scheme in [START_REF] Filippini | A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up[END_REF] and of the resulting optimum configuration that have been found to minimize the dispersion error, we use non-lumped mass matrix in equation ( 9) and for the reconstruction of the auxiliary variables δ h and δ bh .

M G δ h i = Ω h ϕ i g h h ∇η h and M G δ bh i = Ω h ϕ i ∇b h .
Note that the additional linear system is solved just once at the beginning of the simulation. Finally, the complete discrete linear system is solved by making use of the standard functionalities of the MUMPS algebraic library to factorize the (M G H + αT) matrix in any time step of the simulation.

Hyperbolic Phase

For the discretization of the hyperbolic part of the system ( 5)-( 7) we use the node-centered FV technique developed and validated in [START_REF] Kazolea | An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations[END_REF]. In this finite volume framework, the triangulation of Ω used for the finite element discretization of the elliptic phase provides what we refer to as mesh. In the node-centered discretization, a median-dual partition is used to generate non-overlapping control volumes, covering the entire computational domain. We denote by C i the median dual cell obtained by joining the gravity centers of the triangles in K i with the midpoints of the edges meeting in the node i, being K i the set of triangles of the mesh containing i . We can thus write:

|C i | = K∈K i |K| 3 .
The interface belonging to adjacent nodes i, j is named ∂C i j and is composed by the union of two segments connecting the barycenters of the two triangles satisfying K i, j with the midpoint of the edge i j . The boundary of the median dual cell of i can thus be defined as:

∂C i = j∈K i ∂C i j
. Moreover, we define r i j the vector connecting nodes i and j, while the normal and the area associated to the interface i j are:

n i j = 1 2 K i, j n K i j , |C i j | = K i, j |C K i j | with |C K i j | = |K| 6
For simplicity we rewrite the system of conservation laws ( 6)-( 7) as

U t + ∇ • H(U) = S b + Φ. ( 13
)
The FV integration over each computational cell C i leads to the semidiscrete form of the scheme as:

∂U i ∂t + 1 |C i | j∈K i ∂C i j Fn x + Gn y = 1 |C i | j∈K i C i j S b + Φ , (14)
where Φ = C i j Φ, is computed by exact integration over C i of the piecewise linear polynomial φ h , obtained from the elliptic phase. n = [n x , ny ] T is the unitary outward vector normal to the boundary of the computational domain and U i is the volume averaged value of U at a given time. F and G are the numerical flux vectors across each internal and boundary face. The numerical fluxes are evaluated solving a Riemann problem at cell interfaces using the approximate Riemann solver of Roe (1981). To reach higher-order spatial accuracy an extension of the MUSCL methodology of [START_REF] Van Leer | Towards the ultimate conservative difference scheme III. Upstream Centered Finite Difference Schemes for Ideal Compressible Flow[END_REF] is used. Each component of the physical variables and bed topography, b, is extrapolated using solution gradients obtained using a combination of centered and upwind gradients. In this way a third-order well-balanced scheme is obtained.

To obtain a well-balanced FV scheme, an upwind discretization approach for the bed topography source term is adopted to satisfy the so-called Cproperty in hydrostatic (flow at rest) conditions To this end, the topography source term, S b must be linearized in the same way and evaluated in the same Roe-average states as the flux terms. More details on this and on the wet/dry fronts special considerations that are needed t to accurately model transition between wet and dry areas and maintain the high-order spatial accuracy and mass conservation can be found in [START_REF] Kazolea | An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations[END_REF] and [START_REF] Filippini | Free surface flow simulation in estuarine and coastal environments: numerical development and application on unstructured mehes[END_REF].

Interface with FE

The FV scheme evolves the average solution U i over the dual cells C i but the FE elliptic phase needs to know the values of the variables in the nodes os the mesh. When the mesh is symmetric the two values coincide. This is not the case when unstructured meshes are used. The nodal values are thus reconstructed at every time step as:

w h i = w i -(∇w) i • r i (15)
where r i is the distance between node i and the center of gravity of the dual cell C i .

TIME INTEGRATION AND BOUNDARY CONDITIONS

Concerning the time discretization the third order strong stability preserving (SSP) Runge-Kutta (RK) scheme was adopted under the usual CFL stability restriction. In proximity of dry areas and in general when the water depth is low, the friction effect must be taken in to account. The friction term τ = -C f uu/2 is added to the momentum equation as a source term. C f can be a constant number or it can be define by a low which account for the local bottom steepness and water depth. The friction term discretized in a semi-implict approach, since the explicit treatment of this term may impose a severe time step restriction in the presence of dry areas. We use the semi-implicit approach of Chertock et al. ( 2015) which conserves the high-order accuracy of the initial scheme when friction becomes dominating.

In this work only fully reflective and absorbing boundary conditions have been used. φ • n = 0 , ∂ n φ t = 0 . While the first condition simply seems reasonable and can be derived from the momentum equation, considering that u • n = 0, the second one, which forces the tangent component of φ to be constant in the normal direction with respect to the wall, seems artificial. Note that this condition can be obtained by requiring that ∇ ∧ Φ on the wall boundaries (with ∇∧ indicating the curl operator), and using the fact that ∂ t (φ • n) = 0 along these boundaries. The imposition is made by directly modifying the right hand side of (9) and the related entries of the matrix operator (M G H + αT). A strong imposition of the reflective wall boundary is also applied in the hyperbolic phase through the condition: u • n = 0. Absorbing boundaries are also applied in order to dissipate completely the energy of the incoming waves, trying at the same time to eliminate any non-physical reflection. This kind of boundaries requires the definition of a sponge layer in which the surface elevation and the momentum are damped by multiplying their values by a coefficient (see [START_REF] Kazolea | An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations[END_REF] and [START_REF] Filippini | Free surface flow simulation in estuarine and coastal environments: numerical development and application on unstructured mehes[END_REF] for details).

WAVE BREAKING

A hybrid strategy for wave breaking treatment is implemented in the scheme. We first estimate the location of breaking waves using explicit criteria, applying the NLSW equations to solve the flow in the flagged cells and the GN ones elsewhere. Following the work of [START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq type equations[END_REF] we use the combination of the two above phase-resolving criteria for the triggering mechanism The values of γ and φ c are depending on the type of the breaker. The first criterion flags for breaking when η t is positive, since breaking starts on the front face of the wave, while the second criterion, acting complementary to the first, is useful for the detection of hydraulic jumps. In this work the value of φ c = 30 o is used. Moreover, the estimation of the Froude number of the wave is used to established when to switch of the breaking and to detect non-breaking bores. A practical implementation of the breaking mechanism can be found in [START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq type equations[END_REF] and [START_REF] Filippini | Free surface flow simulation in estuarine and coastal environments: numerical development and application on unstructured mehes[END_REF].

NUMERICAL RESULTS

2D solitary wave propagation

The accuracy of the two-dimensional scheme proposed for the GN equations is verified by performing a convergence analysis on a solitary wave propagation problem. A solitary wave of amplitude 0.2m is left propagating for t = 1s inside a computational domain of [0, 70] × [0, 0.8]m, characterised by a value of still water depth of h 0 = 1m In order to measure the rate of convergence of the numerical solution to the exact one, we compute the relative error on the total water depth

E L 2 (h) = ||h num -h ex || 2 /||h ex || 2
, where h num is the numerical solution and h ex is the analytical one. The numerical model used triangular grids consisting of triangles with side length of h K = [0.2, 0.1, 0.05, 0.025]m. Figure 1 shows the results obtained, together with the slopes 2.5 and 3 as references, confirming the expected order of accuracy for the numerical scheme.

log 10 (h K ) -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 log 10 ||err|| 2 -10 -9 -8 -7 -6 -5 -4 -3 -2 FV3 slope 2.5 slope 3
Fig. 1 Grid convergence for the propagation of a 2D solitary wave

Wave diffraction over a semi-circular shoal

We consider here the reproduction of the tests carried out in [START_REF] Whalin | The limit of applicability of linear wave refraction theory in a convergence zone[END_REF] involving the study of the focusing effect induced by a semicircular shoal on wave trains of different periods. The experiments carried out in a wave tank 6.096m wide and 25.6m long, its middle portion consisted in a semicircular shoal leading the water depth to decrease from h 0 = 0.4572m (at the wave maker) to 0.1524m at the end of the tank. The bottom topography is described by the equation:

z(x, y) =              0
x ≤ 10.67 -Λ(y), (10.67 -Λ(y)x)/25 10.67 -Λ(y) < x < 18.29 -Λ(y), 0.3048

x ≥ 18.29 -Λ(y),

where Λ(y) = 6.096yy 2 . The depth h is obtained as h = h 0z. This test has become a standard test case for 2D dispersive numerical models to test nonlinear refraction and diffraction, we refer for example [START_REF] Walkley | A finite element method for the two-dimensional extended Boussinesq equations[END_REF], [START_REF] Kazolea | An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations[END_REF], [START_REF] Ricchiuto | Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries[END_REF], [START_REF] Lannes | A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations[END_REF] among others.

Two cases are considered here with wave trains characterised by [START_REF] Whalin | The limit of applicability of linear wave refraction theory in a convergence zone[END_REF] (a) T = 1s, A = 0.0195m, h 0 /λ = 0.306 ;

(b) T = 2s, A = 0.0075m, h 0 /λ = 0.117 ;

where T denotes the period, A the wave's amplitude and λ the wavelength. For both cases, the harmonic analysis of free surface elevation measurements taken along the tank centreline are available, and are used to verify the capabilities of a model to reproduce nonlinear refraction and diffraction. The computational domain is [-10, 36] × [0, 6.096]m . Periodic waves are generated using the internal wave generator of [START_REF] Wei | Generation of waves in Boussinesq models using a source function method[END_REF], centred at x = 4m. Sponge layers of 6m length are placed at both ends of the domain, while reflective boundary conditions are imposed along the top and bottom boundaries. Following [START_REF] Walkley | A finite element method for the two-dimensional extended Boussinesq equations[END_REF] and [START_REF] Kazolea | An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations[END_REF] for case (b) a relatively sparse triangular grid was used, consisting of equilateral triangles with side length of 0.1m, leading to a mesh of N=28,151 while a more refined mesh of side length 0.05m and N=56,211 nodes had to be used for case (a). The CFL number used was set equal to 0.5 for all cases.

Figures 2 and5 give an illustration of the fully developed 3D free surface elevation for case (a) and (b) respectively, while figures 3 and 6 depicts the free surface elevations at the centreline at the final time of the simulation. The incoming waves are linear in the deeper portion of the tank, but as they propagate onto the topography they become steeper due to shoaling. After the focusing, wave energy gradually spreads out due to diffraction. Figure 4 compares the spatial evolution of the first and second harmonics with the experimental data. In order to make sure that a steady periodic state is obtained, we started sampling the solution after at least 15 periods of oscillations. Then we performed a Discrete Fourier Transform on the time series of the free surface elevation along the centreline, measured over one period of the main incoming wave. It can be observed that both the first and second harmonics increase in magnitude in the focal zone and the numerical harmonics are consistent with the laboratory data but slightly underestimate it. Case (b) has a weaker dispersive degree but a higher relative nonlinearity. The incoming waves are linear, but after the focusing on the shoal, higher harmonics become significant due to non-linear effects and the energy transfer in to the second and third harmonics can de seen in figure 7.The results are consistent and in very good agreement with the experimental data.

Solitary wave run up on a plane beach

The next test case is one of the most intensively studied problems in long-wave modelling. The solitary wave-run-up on a plane beach. [START_REF] Synolakis | The run-up of solitary waves[END_REF] carried out laboratory experiments for incident solitary waves of multiple relative amplitudes over a planar beach with a slope 1 : 19.85. Performing this test, we want to asses the ability of our model to describe propagation, shoreline motions, breaking and run-up. Detailed description of the test case, along with the initial conditions, can be found in [START_REF] Synolakis | The run-up of solitary waves[END_REF]), [START_REF] Tonelli | Hybrid finite volume -finite difference scheme for 2DH improved Boussinesq equations[END_REF] The solitary wave, util time t g/h = 15 , propagates to the shore and shoaling over the inclined bathymetry. The breaking procedure starts around t g/h = 15 and the experimental wave, breaks around t g/h = 20. The numerical solution is represented like a bore storing the water spilled from the breaking wave behind the front. The numerical model reduce to NSWE during the breaking event, which approximate the turbulent breaking wave by a flow discontinuity hence, the computed front face becomes steeper. The actual wave is not discontinuous but contains air bubbles and turbulence. The bore collapse on the slope at time t g/h = 25. Because of volume conservation in all models, the computed solution fully recovers until the water reaches the maximum run-up point around t g/h = 45. A good agreement with the experimental data is observed for times t g/h = 60 and t g/h = 60. (2014) and in [START_REF] Filippini | Free surface flow simulation in estuarine and coastal environments: numerical development and application on unstructured mehes[END_REF]. Nine wave gauges were placed into the basin in order to measure the variation of the free surface elevation: gauges 1, 2, 3, 7 were located at y = 0m and x = 7.5, 13, 21, 25m gauges 4, 5, 6, 8 were located at y = 5m and x = 7.5, 13, 21, 25m, while gauge 10 have been set at y = 10m and x = 25m.

Compared to the experimental case, the computational domain has been extended from x = 0m to x = 5m in order to be able to completely contain the initial solitary wave. It has been discretized by means of a non-uniform unstructured grid, adapted to the bed curvature, as shown in 10, and characterised by reference maximum and minimum size respectively: max(h K ) = 0.3m and min(h K ) = 0.125m. A solitary wave of amplitude A = 0.39m, corresponding to = 0.5, is initially placed in x = 0 and wall reflecting boundary conditions are imposed in each boundary of the domain. We used a Manning coefficient N m = 0.0014 for representing bed roughness. A CFL number of 0.5 was used, together with γ = 0.6 for the breaking detection criterion.

Figure 11 shows the computed free water surface at different time instants. With red colour we denote the time evolution of the breaking regions detected by the criteria of the breaking mechanism. As the solitary wave propagates towards the beach it shoal, increases its steepness and nonlinearity, up to reaching a breaking point at t = 5 on the center line of the domain, when it reaches the apex of the triangular shelf. At t = 6.5s the central part of the wave has completely overtopped the concrete cone, while on the two sides, the surge continues to shoal, diffracting around the base of the cone. By t = 8.5s, the refracted and diffracted waves collide on the lee side of the shelf. After t = 9s, the water starts to withdraw from the con top and a bore-front forms, from the combined waves after the diffraction, and propagates on the shelf behind the cone and then onshore. After t = 15s, a new bore is creates from the the drawn-down of the water and collides with the refracted waves.

The next figure 12 plots the computed free surface time series on the wave gauges 2, 3, 7 and 9 against the experimental data. The arrival of the first incoming wave is correctly captured in gauge 2, as it is for the refracted and diffracted waves at the lee side of the cone, as can be seen from gauge 3, except for the minimum of water height registered at t = 8.5s. A slight deviation from the measurements is displayed after t = 35s, maybe due to late arrival of the reflected waves from the extended left wall boundary. The signal at the gauges located at the north side of the cone indicates that wave shoaling, breaking and propagation on the shelf is accurately predicted, together with the complex nonlinear interaction between diffracted and refracted waves.

CONCLUSIONS

In this work a fully nonlinear-weakly dispersive unstructured grid wave model is presented. The model is a first unstructured grid generalisation of the fully nonlinear weakly dispersive models based on the seminal work of Wei andKirby (1995), andWei et al. (1995). The underlying model of partial differential equations if the enhanced fully nonlinear -weakly dispersive Green-Naghdi system proposed by [START_REF] Bonneton | Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes[END_REF]. Following [START_REF] Filippini | A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up[END_REF], the system is decomposed in the hyperbolic shallow water equations plus an algebraic correction satisfying an auxiliary system of partial differential equations. These equations are elliptic and defined by a self-adjoint operator, as observed in Alvarez-Samaniego and Lannes (2008). We exploit this property to construct a hybrid finite element-finite volume scheme. In particular, the self adjoint nature of the elliptic operator is used to construct a coercive continuous finite element approximation providing the algebraic source term to add to a classical hyperbolic finite volume solver. Following [START_REF] Filippini | A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up[END_REF], and in the spirit of Wei and Kirby (1995), we have used a second order approximation of this elliptic high order correction. For the hyperbolic solver, we have instead used a non-dispersive, higher order MUSCL approach. The dispersion properties of the resulting scheme are shown in [START_REF] Filippini | A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up[END_REF] to be comparable to those of a fully fourth order finite difference approximation. Wave breaking is embedded in the model simply locally neglecting the dispersive correction, and following the detection criteria of [START_REF] Kazolea | Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq type equations[END_REF]. The results show an accurate capturing of highly dispersive wave transformation, as well as an accurate resolution of complex interactions involving wave breaking and run-up. This work opens the door to a large number of model improvements such as e.g.

• the study of optimized strategy to solve the assemble and solve the block symmetric linear systems obtained in the elliptic phase;

• the use of simplified variants of the Green-Naghdi equations, such as those proposed by [START_REF] Lannes | A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations[END_REF] which would allow to obtain constant in time matrices for the dispersive terms;

• the study of the benefits of higher orders for the elliptic phase in terms of error reduction at constant CPU time;

• the use of other schemes in the hyperbolic phase, such as e.g. the successful Discontinuous Galerkin method;

• the combination of our model with static and dynamic mesh adaptation techniques tailored to these flows;

• the extension to spherical coordinate systems;

Initial application and benchmarking of this model are being carried out in the framework of the French research program TANDEM (http://www-tandem.cea.fr), and will be pushed further in the near future.

•

  the surface variation criterion: |η t | ≥ γ gh with γ ∈ [0.35, 0.65] • the local slope angle criterion: ||∇η|| ≥ tan φ c with φ c the critical angle value.

Fig. 2

 2 Fig. 2 Shoaling of regular waves, case (a): perspective view of the free surface

Fig. 4

 4 Fig. 4 Shoaling of regular waves, case(a): comparison of the computed and experimental results for the wave amplitudes for the first and second harmonics along the centreline

  Solitary wave propagation over a three dimensional reef Swigler et Lynett (2011) performed laboratory experiments at the O.H. Hinsdale Wave Research Laboratory of Oregon State University to study the specific phenomena which occurs when a tsunami like wave approaches the coast, including shoaling, refraction, breaking and run-up. The basins 48.8m long 26.5m wide and 2.1m deep. An extensive description of the set-up of the problem can be found in Kazolea et al.

Fig. 5

 5 Fig. 5 Shoaling of regular waves, case (b): perspective view of the free surface

Fig. 7

 7 Fig. 7 Shoaling of regular waves, case(b): comparison of the computed and experimental results for the wave amplitudes for the first and second harmonics along the centreline

Fig. 8

 8 Fig. 8 Solitary wave run-up on a plane beach: snapshots of the free surface elevation for dimensionless times t g/h = 15, 20, 25, 45 (from top to bottom).

Fig. 9

 9 Fig. 9 Solitary wave run-up on a plane beach: snapshots of the free surface elevation for dimensionless times t g/h = 60, 80 (from top to bottom).

Fig. 10

 10 Fig. 10 Solitary wave propagation over a three dimensional reef: Close up view of the adapted mesh used for the computation. Fig. 11 Solitary wave propagation over a three dimensional reef: computed free surface solution with friction at times t =3.5, 5.5, 6.5, 8.5, 9.5, 11.5s (from top to bottom). The red area represents the region where wave breaking is detected and the NLSW equations are solved.

Fig. 12

 12 Fig. 12 Solitary wave propagation over a three dimensional reef: computed time series of the free surface elevation on gauges positions

ACKNOWLEDGMENTS

Work partially funded by the TANDEM contract, reference ANR-11-RSNR-0023-01 of the French Programme Investissements d' Avenir.