Marylise Cottet
email: marylise.cottet@ens-lyon.fr

Florence Piola

Yves-François Le Lay

Soraya Rouifed

Anne Rivière-Honegger

How environmental managers perceive and approach the issue of invasive species: the case of Japanese knotweed s.l. (Rhône River, France)

Keywords: Discourse analyses, environmental managers, invasive species, Japanese knotweed s.l., management, perceptions

Studying the perceptions of stakeholders or interested parties is a good way to better understand behaviours and decisions. This is especially true for the management of invasive species such as Japanese knotweed s.l. This plant has spread widely in the Rhône basin, where significant financial resources have been devoted to its management. However, no control technique is recognized as being particularly effective. Many uncertainties remain and many documents have been produced by environmental managers to disseminate current knowledge about the plant and its management. This article aims at characterizing the perceptions that environmental managers have of Japanese knotweed s.l. A discourse analysis was conducted on the printed documentation produced about Japanese knotweeds.l. by environmental managers working along the Rhône River (France). The corpus was both qualitatively and quantitatively analysed. The results indicated a diversity of perceptions depending on the type of environmental managers involved, as well as the geographical areas and scales on which they acted. Whereas some focused on general knowledge relating to the origins and strategies of colonization, others emphasized the diversity and efficacy of the prospective eradication techniques.There is a real interest in implementing targeted actions to meet local issues. To do so, however, these issues must be better defined. This is a challenging task,as it must involve all types of stakeholders.

INTRODUCTION

When scientific uncertainties make management more difficult

Biological invasions, which are believed to be the result of global change, are of growing interest in the biological sciences (Vitouseket al. 1997) because of their potential effects on biodiversity. Invaded ecosystems are generally consideredasdisrupted areas,where native species are strongly threatened. Thus, the control of invasive species has become a priority for many countries, and several policy engagements have already been ratified and are being implemented [START_REF] Genovesi | European Strategy on Invasive Alien Species: Convention on the Conservation of European Wildlife and Habitats (Bern Convention)[END_REF][START_REF] Heywood | Code of Conduct on Horticulture and Invasive Alien Plants[END_REF]. Therefore, managers need to find efficient and feasible control methods (Delbartet al. 2012). However, in the case of particularly efficient invaders, managing such species presents a considerable challenge, and while ecological studies are numerous and can help to define successful management techniques [START_REF] Genovesi | Are we turning the tide? Eradications in times of crisis: how the global community is responding to biological invasions[END_REF], the complexity of the ecological processes involved in invasion often makes it difficult to develop effective control methods. Biological invasions pose not only a critical ecological issue but also an important social issue. The social dimensions of biological invasions were first considered in the early 2000s [START_REF] Mcneely | The Great Reshuffling: Human Dimensions of Invasive Alien Species[END_REF], following initiatives in environmental economics (Perringset al. 2000;[START_REF] Pimentel | Environmental and economic costs of nonindigenous species in the United States[END_REF]. If managers want to efficiently manage invasive species when defining a management strategy, they must also consider the social dimension of the issue (Binimeliset al. 2007;[START_REF] Gobster | Factors affecting people's responses to invasive species management[END_REF]). On the one hand, the impacts of biological invasions are many and not limited solely to ecological consequences. They may also affect market or non-market goods and services produced (or no longer produced)by invaded systems (Colauttiet al. 2006). On the other hand, what we define asa biological invasion (the degree to which the colonization of an area by a species becomes an invasion) as well as the management strategies we use to control them are widely influenced by human perceptions [START_REF] Mack | Motivations and consequences of the human dispersal of plants[END_REF].

Human perceptions, a key factor for managing biological invasions

When managing complex ecological processes, there are no standard rules defining what constitutes a good decision or action. Human perceptions that guide the definition of strategies rely sometimes on non-scientific criteria (Lévêqueet al. 2012). For example, Starfingeret al. (2003) showedthat when there are insufficient data available to tackle a specific issue, there is a tendency to believe in stories about the beneficial or noxious impact of an alien species. Focusing on human perceptions as well as human values associated with biological invasions is a good way to better understand behaviours and decisions. Furthermore, this focus may help to define more efficient management strategies (Vanderhoevenet al. 2011). This research direction is also legitimatized given the fact that there is a wide diversity of stakeholders or interested parties involved in the management of biological invasions, each of them having a specific perception regarding the issue and a specific point of view regarding the action to take (Simberloffet al. 2005). Human perceptions are at their most heterogeneous when addressing questions that are confounded by many scientific uncertainties [START_REF] Pahl-Wostl | The importance of social learning in restoring the multifunctionality of rivers and floodplains[END_REF], aseveryone tends to have his or her own perceptions depending on his or her personal experiences. With respect to invasive species, uncertainty is the norm (Williamson 1999;[START_REF] Horan | Biological pollution prevention strategies under ignorance: the case of invasive species[END_REF]. The definition of action relating to biological invasions must thereforetake into account this diversity of perceptions (Binimeliset al. 2007;Garcia Llorenteet al. 2008). This article aims to serve this objective by choosing an original angle: the study of managers' perceptions.

The choice of studying managers' perceptions

Many scientific studies until now have been interested in characterizing public perceptions of the control and eradication of invasive species (Simberloffet al. 2005;[START_REF] Hulme | Beyond control: wider implications for the management of biological invasions[END_REF]; [START_REF] Bardsley | Invasive species policy and climate change: social perceptions of environmental change in the Mediterranean[END_REF][START_REF] Bremner | Public attitudes to the management of invasive non-native species in Scotland[END_REF][START_REF] Andreu | Anassessment of stakeholder perceptions and management of noxious alien plants in Spain[END_REF], and public opposition has repeatedly caused delays in or the abandonment of control efforts [START_REF] Marshall | Considering the social dimension of invasive species: the case of buffel grass[END_REF][START_REF] Mcneely | Xenophobia or conservation: some human dimensions[END_REF]. Better knowledge of public opinion towards these actions is a first step in fostering public involvement and gaining social acceptance (Selgeet al. 2011).

Nevertheless, the perceptions of environmental managers require specific attention.

-Environmental managers are charged with defining and implementing environmental plans. They are the ones who have to act (or not)and who have to meet the challenge of overcoming uncertainties [START_REF] Liu | Incorporating uncertainty and social values in managing invasive alien species: a deliberative multi-criteria evaluation approach[END_REF]. How do these uncertainties influence their perceptions, decisions and behaviours towards the plant and its management? -They are also the ones in charge of producing and communicating information about invasive species to other stakeholders, tasks that require certain skills and principles (Jurinet al. 2010). Beyond the uncertaintiesthat they must address,when communicating on issues relating to a biological invasion, they face the challenge of correctly informing stakeholders about the invasive species. Indeed, the ambiguous and inconsistent use of terminology [START_REF] Richardson | Naturalization and invasion of alien plants: concepts and definitions[END_REF]Collauti and Richardson 2009)and the use of emotive and manipulative language [START_REF] Gobster | Invasive species as ecological threat[END_REF]Larson 2005;2008;[START_REF] Stromberg | Changing perceptions of change: the role of scientists in Tamarix and river management[END_REF]Selgeet al. 2011) have already been widely criticized.

The case of Japanese knotweed s.l

The importance of conducting case studies

Research conducted on invasive species is instructive. In particular, these studies show that different criteria influence human perceptionsregarding a biological invasion. Several researchers have listed influencing criteria (Garcia Llorenteet al. 2008;Selgeet al. 2011), and some have even proposed a model to explain human perceptions of biological invasions [START_REF] Gobster | Factors affecting people's responses to invasive species management[END_REF]. The following are among the influencing criteria identified:

-The impact caused by an invasive species appears to be one of the main structuring factors of human perceptions. These impacts can be negative or positive [START_REF] Shapiro | The Californian urban butterfly fauna is dependent on alien plants[END_REF]Bardsley and Edward-Jones 2006) and may affect ecosystems as well as social systems (Garcia Llorenteet al. 2008). The negative impacts of invasive species on ecosystems are found to be a strong motivation for their eradication [START_REF] Levine | Mechanisms underlying the impacts of exotic plant invasions[END_REF]Garcia Llorenteet al. 2008;Selgeet al. 2011). Nevertheless, ecological functions provided by species (Binimeliset al. 2007) or values associated with them (i.e., the aesthetic value of colourful plants such as purple loosestrife; the cultural value of feral pigs for particular ethnic groups, etc., Gobster 2011) may help to attenuate the observed negative impacts.

-Time can also influence human perceptions of a biological invasion such that the older the invasion, the less the species is identified as exotic, and the better it is valued [START_REF] Bardsley | Stakeholders' perceptions of the impacts of invasive exotic plant species in the Mediterranean region[END_REF]Garcia Llorenteet al. 2008;[START_REF] Gobster | Factors affecting people's responses to invasive species management[END_REF]). Moreover, Starfingeret al. (2003) showed that perceptions of an invasive species can evolve over time, according to the scientific data available.

-Personal factors add many variationsto the perceptions of an invasive species. Such factors include education, economic status, rural or urban residence, cultural or regional characteristics, house size and distance: the closer the invasion, the more concerned people are likely to be (Garcia Llorenteet al. 2008;[START_REF] Ehrenfeld | Ecosystem consequences of biological invasions[END_REF][START_REF] Gobster | Factors affecting people's responses to invasive species management[END_REF]. Knowledge and expertise also play a critical role in the way we consider issues linked to biological invasions. Consequently, different stakeholders will often have different perceptions of the issue.

The previous studies show that reflections about biological invasions cannot be exclusively conducted in a global framework, as human perceptions of biological invasions depend on which species is under consideration, which area has been invaded and which stakeholder is affected. Data, therefore, should be focused onthe effect of a single species over a single area on a single-type stakeholder.

The Japanese knotweed s.l. invasion: a preoccupation of environmental managers

2012

). In fact, some of them may even promote further invasion, as is the case for mowing interventions (Beerlinget al. 1994;[START_REF] Mchugh | A review of literature and field practices focused on the management and control of invasive knotweed[END_REF]), the most frequently used technique. Little is known about the response of the different taxa to the control methods. Moreover, as they invade similar habitats and as it is sometimes difficult to distinguish between them, they are often considered together. For these reasons and for convenience, in this paper, the term Japanese knotweed s.l. will be used to refer to the three taxa of interest: F. japonica, F.

sachalinensis, and F. x bohemica [START_REF] Bailey | Asexual spread versus sexual reproduction and evolution in Japanese Knotweed s.l. sets the stage for the -Battle of the Clones[END_REF].

Because of the uncertaintiesregarding management, there is no consensus with respect to the most effective way to control Japanese knotweed s.l. (Delbartet al. 2012) or evenwhetherit should be controlled. Ongoing ecological field research may produce knowledge likely to help define the best management plan for this plant [START_REF] Rouifed | Bases Scientifiques pour un Contrôle des Renouées Asiatiques : Performances du Complexe Hybride Fallopia en Réponse aux Contraintes Environnementales[END_REF]). Nevertheless, thisresearch must be completed with data relating to the field of human perceptions, especially to better understand the main issues associated with the management of Japanese knotweed s.l.and the best action to take from the perspective of the managers.

1.4.3.

A need for research relating to human perceptions of Japanese knotweed s.l.

Only a few studies have considered the social dimension of Japanese knotweed s.l. invasion. Herein, we aim to extend these initial data by characterizing the perceptions of environmental managers working along the Rhône River with Japanese knotweed s.l. By analysing what they have written about this plant, we aim to determine:

(1) What typesof environmental managers are actually concerned about Japanese knotweed s.l. and produce information about it?

(2) What are the perceived issues linked with the management of the plant?

(3) What strategies do the environmental managers recommend tomanagethis plant, and what arguments do they use to justify their choices?

MATERIALS AND METHODS

Study area

The Rhône River, one of the main Mediterranean rivers, originates at the Furka glacier in the Swiss Alps. The river is 812 km long, with more than 500 km located in France. The Rhône flows into the Mediterranean Sea, where it terminates in a very largedelta. Upstream of this delta, at Beaucaire, the mean river flow reaches 1700 m 3 /s [START_REF] Olivier | The Rhône River Basin[END_REF]. The Rhône River is a powerful unifying element of southern and eastern France because it crosses or delimits many administrative areas, including regional, infra-regional and local authorities.

Many environmental managers attached to these various geographical areas are facing the challenge of how to control Japanese knotweed s.l. as part of their mission.

While few data are available concerning the intensity of the Japanese knotweed s.l. invasion along this watercourse, according to the inventory conducted in 2001 for the regional water authority [START_REF] Boyer | Cartographie des renouées du Japon sur le réseau hydrographique du bassin Rhône Méditerranée Corse (hors Saône Doubs)[END_REF], the invasion is variable, with the upper regionexhibiting a far greater invasion than the lower region.

However, two sectors appear to be more specifically colonized. One, in the Upper Rhône, is the 40 km section downstream from the confluence with Les Usses River, and the second, in the Middle Rhône, is a 90 km section downstream from the city of Lyon. The Japanese knotweed s.l. invasion is of major concern to environmental managers and has led to numerous management plans, following the measures (2010 to 2015) defined by the regional water authority to -fight against invasive exotic species‖ (article OF6-C).

Constitution of the corpus

We aim to gather all documents relating to Japanese knotweed s.l. produced by managers involved in the management of this plant along the Rhône River in France. The preliminary analysis suggests that printed documents on this plant are relevant sources for studying stakeholders' perceptions, as each document published by a management structure is often collectively written and validated. The discourse it produces about a given issue is therefore assumed to correspond to management structure's perceptions regarding the issue.

To be as exhaustive as possible when collecting this documentation, a multi-step, rigorous collection strategy was established. As a first step, we conducted a general search of the Internet, using the scientific and common names of the plant, place names, and words such as ‗management', ‗control', and ‗management practices', in

French. As a second step, we contacted by telephone different types of management bodies whose responsibilities included issues related to the Rhône River. They were asked to send us by post or e-mail a copy of any documents relating to Japanese knotweed s.l. that they had produced or had in their possession. For some of these institutions, we went on-site to gather the available documents. This contact procedure was pursued at three geographical levels:

-at the local level: all "communautés de communes" (a regrouping of local authorities with jurisdiction on certain matters) or individual municipalities (when not collectively organized), as well as all local water associations;

-at the regional or infra-regionallevel: everyregional or infra-regionalauthority (-conseils régionaux et départementaux‖) as well as all decentralized state services (-direction régionale de l'environnement, de l'aménagement et du logement‖, -direction départementale des territoires‖);

-at the Rhone-Mediterranean basin level: the Rhône-Mediterranean Corsica water agency (-Agence de l'eau Rhône Méditerranée Corse‖).

Two criteria were used to justify the inclusion of a document in the corpus.First, it had to specifically focus on Japanese knotweed s.l.(not simply on invasive species); second, it must have been produced by an environmental manager who was responsible for the management of an area crossed or bordered by the Rhône River or one of its tributaries. We digitally scanned each document using OCR software (Omnipageprofessional©, Nuance Communications Inc., Burlington, Massachusetts, US) and built a database to associate metadata with each of these documents. These metadata included the following:

-the publication date. In the case that the publication was in draft (i.e., not final) form, the date planned for publication was entered into the database. As a result, the corpus includes some sources listed as published in 2013;

-the type of stakeholder producing the document;

-the nature of the document.

The collection of the documentation for the corpus ended in April 2012 and allowed usto gather 81 documents.

In spite of the systematic sampling, we may have missed certain documents. Nevertheless, since we have collected every document that the managers have archived or have heard about, we can consider that our collection is a large sample of the universe of relevant documents. One limit must however be underlined: we chose to gather only the printed documentation, which excludes online publications dealing with the question of Japanese knotweed s.l., such as websites, blogs… As the environmental managers are more and more inclined to use these media as a way for communicating, this exclusion may have biased our sample.

Corpus analysis: a statistical analysis of textual data

The corpus was both qualitatively and quantitatively analysed. A statistical analysis of textual data (Lebartet al. Iramuteq© reproduces the classification algorithm, described by Reinert (1983;[START_REF] Reinert | Une méthode de classification des énoncés d'un corpus présentée à l'aide d'une application[END_REF] and implemented in the Alceste© software, which has led to numerous publications [START_REF] Brochet | Wine descriptive language supports cognitive specificity of chemical senses[END_REF]Dransfieldet al. 2004;[START_REF] Parr | Representation of complexity in wine: influence of expertise[END_REF]. It corresponds to a top-down hierarchical classification based on five stages.

(1)

Segmentation. The corpus is segmented into two textual units: (a) the texts composing the corpusin our case, each document produced by environmental managers and qualified by the metadata defined above (publication date, type of stakeholder and nature of the document); (b) the text segments (through an iterative process, each text is cut into a number of segments, defined according to a number of words -40 by default).

(

) 2
Lemmatization. Using a grammatical dictionary, each verb, noun, adjective, etc., is reduced to its basic dictionary entry, named a lemma.

(3)

Production of a contingency table. The matrix crosses the reduced forms (in columns) and the text segments (in rows). Among the reduced forms, only the analysable forms (nouns, verbs, adjectives, adverbs, etc.) are retained; supplementary forms (prepositions, pronouns, conjunctions and auxiliary verbs, etc.) are excluded from the table. The presence or absence of each analysable reduced form within each text segment is specified at intersections of the matrix (respectively noted as 1 vs. 0).

(4)

Processing of a top-down hierarchical classification from the contingency table . Relying on an iterative algorithm, the software aims at defining classes thatmaximize the distance between two subsets (using χ² metrics). The iterative process stops when sub-classes are not significantly different. Specific forms of a class are then removed from the other class. This analysis is then repeated on the larger of the two classes, and so on, until the requested number of classes is reached (10 by default). Two independent analyses with different lengths of text segment are processed, and their results are statistically compared to test the stability of the classes.

(5)

Description of the classes. The number of text segments classified in each class is specified. For each class, a list of the reduced forms associated with it is created (the degree of association with the class is indicated by the χ² value). The modality of the metadata variables most associated with each class is also specified. These classes are then finally interpreted as -lexical worlds‖ (Rouré and Reinert 1993).

We also performedstatistical treatments relying on the co-occurrence analyses of specific lexical forms (Lebartet al. 1998). In particular, using Iramuteq©, we conducted similarity analyses. Based on graph theory, theseanalyses aim to study proximity and relationships between components (in our case, the lexical forms) of a set (in our case, the corpus) using a maximum tree [START_REF] Marchand | L'analyse de similitude appliquée aux corpus textuels: les primaires socialistes pour l'élection présidentielle française (septembre-octobre 2011)[END_REF]. The plot resulting from this analysis had specific properties: (a) the greater the occurrence of a lexical form, the greater the size of the characters; (b) the greater the co-occurrence between two lexical forms, the thicker the line that linked them.

Such a graph was used to analyse the relationships between all lexical forms associated with a class, which resulted from the top-down hierarchical analysis. 11

3. RESULTS

Who produces information about Japanese knotweed s.l., and in what form?

The corpus was composed of 81 documents (195,006 words total) published between 1998 and 2013. However, the production of information about Japanese knotweed s.l.was marginal until 2006 (Fig1A). That year marked a strong increase in the number of documents produced, and the number continued to increase in the following years.

Diverse types of environmental managers were involved in this production (Fig1A). Some were attached to the regional water authority, to local or regional authorities or state services, to environmental protection associations, to environment consultancy firms, to building companies operating on the river, to regional conservation bodies (conservatoires botaniquesrégionaux), or to local water associations. Thus, both public and private stakeholders contributed to the production of information about Japanese knotweed s.l. Managers working within local water associations were the main stakeholders diffusing information about Japanese knotweed s.l.as they alone produced almost half of the documentation. However, their contribution to the production was relatively late and occurred mainly from 2010 onward. In other words, the strong increase in production observed over these last few years was mainly due to this one group of stakeholders. Nonetheless, local and regional authorities, state services, and environmental consultancy firms also had a significant input.

The documentation produced was quite diverse, consisting of management plans, syntheses of technical meetings, management guides, identification guides, information leaflets and specialized journal articles (see Table 1 for a description of the documents). The various documents addressed diverse target groups in that some -management guides and synthesis studies -were intended for environmental managers who already confronted ecological invasions and thus focused on management methods. Others -specialized journal and information leaflet -addressedthe general public with potentially no knowledge of the issueand thus provided informationabout the plant, its colonization and its impacts.These two uses were roughly equally shared if we consider the number of produced documents per category (Fig1B).

Following a spatial analysis, we observed that the production of documents relating to Japanese knotweed s.l.

also turned out to be diverse (Fig2), with the number of documents produced upstream being far higher than the number produced downstream. This longitudinal division of the river seemed to correspond to the intensity of the invasion [START_REF] Boyer | Cartographie des renouées du Japon sur le réseau hydrographique du bassin Rhône Méditerranée Corse (hors Saône Doubs)[END_REF]. That is, the more invaded the area was, the more documentation on the plant being produced by environmental managers.

3.2.

What is said about Japanese knotweed s.l.

Main components of the discourses

The top-down hierarchical clustering classified97.7 % of the text segments within 5 lexical classes (Fig3a). The clustering tree marked a first segmentation dividing textual segments into two parts, each having a semantic unity.The first regrouped classes 1 and 2(including 13.2 % and 24.2 % of the textual segments, respectively), and focused on the fight against Japanese knotweed s.l. The second regrouped classes 3, 4 and 5 (including 19.6 %, 18.7 % and 24.3 % of the textual segments, respectively), and involved information on the invasion (mechanisms of dispersion) as well as strategies for managing the plant. The most significant textual segments for each class are presented inTable 2.

Class 3 (Fig3b) brought together all available strategies and resources for managing Japanese knotweed s.l. On the one hand, it listed all possible actions aimed at controlling the plant: it mentioned both preventive measures aimed at limiting its expansion, such as awareness actions (-prevention‖, -awareness‖, -communication‖) or follow-up actions (-cartography‖, diagnostic‖, -monitoring‖, -inventory‖), and restorative measures when the plant is already established (-eradication‖, fight‖, -techniques‖, -method‖, -means‖). The need to acquire knowledge also held a significant position within this class (-information‖, -knowledge‖, -to know‖). On the other hand, this class took stock of all stakeholders concerned with the invasion of Japanese knotweeds.l. in that it referred both to management experts (-manager‖, -Rhône-Méditerranée-Corse‖ [water agency], -local‖ or -field‖ ‖stakeholder‖, -authority‖), to economic actors (-company‖), and to the general public (-public‖). Many geographical areas and many spatial scales involved in the management of the plant were also listed, including -basin‖, -Saône‖, -Rhône‖, -regional‖, -région‖ (or regional authority), and -département‖ (or local authority).

This inventory responds to the concerns of developing specialized stakeholdernetworks to better control Japanese knotweeds.l. (-network‖, -work[ing]‖ -group‖).

Classes 5 (Fig3d) and 4 (Fig3c) comprised knowledge relating to the dispersal mechanisms of Japanese knotweed s.l.Class 5 described the origins of the introduction of the species (-to introduce‖, -exotic‖, -Europe‖, -Asia‖, -origin‖, -ornamental‖, -century‖, -human‖ -activity‖), its strategies of colonization and adaptation (-hybrid‖, -competition‖, -reproduction‖, -pioneer‖), including its preferred environments for developing (-wetland‖, -railway‖ -track‖, -road‖, -side‖) and finally, problems it could cause (-ecological‖-impact‖, -nuisance‖, -biodiversity‖, -monospecific‖, -ecosystem‖ -functioning‖, -bank‖ -erosion‖). This class appeared to more generally describe certain dispersal mechanisms common to invasive species (-giant hogweed‖, -ragweed‖, -animal‖). Class 4 was more specifically focused on watercourses, described as one of the main affected environments but also one of the main means of dispersal.

Eradication was at the core of classes 1 (Fig3e) and 2 (Fig3f). Class 2 listed a considerable number of the techniques usedeither today or in the pastto fight Japanese knotweed s.l. (-to mow‖, -to cut‖, -chemical‖ -treatment‖, -uprooting‖, -tarpaulin‖). Actions specifically focused on the main reproductive organs of the plant: its aerial (-stalk‖), or its underground parts (-rhizome‖)as both are strongly involved in its reproductive capabilities in vegetative propagation. Therefore, a significant place in this class was given to the physiology of the plant. The experimental component of these actions was strongly present, as the effectiveness of the eradication techniques used wasassessed experimentally (-protocol‖, -experimentation‖, -plot‖, -counting‖, -m²‖). Finally, the lexicon relating to seasons and seasonality was very much present (-end‖, -beginning‖ -May‖, -June‖, -season‖, -period‖, -month‖), as was the lexicon linked to repetition (-to repeat‖, -times‖, -repetitive‖). This finding indicated, on the one hand, that environmental managers were used to relying on a seasonal calendar to implement actions against Japanese knotweed s.l., and on the other hand, that their interventions must necessarily be repeated. Class 1 focused on a specific technique that was experimentally used to eradicate the plant. This technique involved mechanical action aimed at crushing the rhizomes into small enough particles (using construction machinery) to impede their vegetative reproduction. This class insisted on the effectiveness of this method (-complete‖ -mortality‖, -to achieve‖ -result‖, -efficiency‖) and mentioned the necessary preventive measures to implement in such experimental trials so as not to further disperse the plant (-cleaning‖, -to clean‖, -caterpillar‖ -machine‖).

A technical discourse responding to a wish for effective action

This classification shows that more than one-half of the discourses were dedicated to actions aimed at controlling Japanese knotweed s.l., and more than one-third specifically focused on restorative methods. The discourses of environmental managers appeared to strongly promote the implementation of actions to eradicate the plant. This willingness to take action against Japanese knotweed s.l. was specifically obvious when we considered the -efficiency‖ lemma, cited as many as 121 times in the corpus. Table 3, presenting its co-occurring lemmas, highlights the fact that the -efficiency‖ lemma relates, above all, to the actions implemented for controlling this plant, as this wish to be -more efficient" (36 co-occurrences) led environmental managers to be creative about the methods used to destroy the plant. Figure 4 summarizes the new techniques proposed (and when) and how their uses have evolved over time. We observed that while many techniques were tested, some were only very marginally cited in the documentation produced by environmental managers (thermal actions, grazing, etc.), and others had sometimes been abandoned (chemical treatments). On the other hand, other techniques were often mentioned.Mowing, for example, was the most cited method for controlling the plant. Uprooting was also well represented. It can be concluded that the information produced about Japanese knotweed s.l.was highly technical.

Issues mentioned for controlling this plant

In the documentation produced by environmental managers, actions against Japanese knotweed s.l.were strongly motivated by its impacts:there were 228 occurrences of the -impact‖ lemma present in the corpus. However, when we considered its co-occurring lemmas (Table 4), discourses on their possible impacts appeared to be rather weakly explained. While -environments‖ (n=30), -ecosystems‖ (n=16) and -landscapes‖ (n=11) were said to be affected by Japanese knotweed s.l., the ways in which they were affected were unclear. The only impacts concretely qualified, albeit with very poor frequency, were related to the loss of biological diversity (-biodiversity‖, n=11; -diversity‖, n=7, -monospecific, n=4) or landscape diversity (-homogenization‖, n=2;

-standardization‖, n=2). Few references were made to the role of foliage in reducing biodiversity (-foliage‖, n=4; -shade‖, n=3). Reduced access (-accessibility‖, n=4), and bank erosion (-undercutting‖, n=2), both induced by the plant, were only marginally mentioned.

Does every type of environmental manager have similar views about Japanese knotweed s.l.?

A diversity of arguments focusing on certain themes

The distribution of textual segments within the different classes of the top-down hierarchical classification, according to the type of environmental manager who produced them, provided information about the diversity of proposals relating to Japanese knotweed s.l. Each type of stakeholder was inclined to use certain classes of discourse. The probability of occurrence of classes, for each type of stakeholder, is particularly high a low in respect to a hypothesis of independence (Pearson's residuals; tables5a and 5b).

-Class 3, which describes all strategies and stakeholders involved in the management of Japanese knotweed s.l., was significantly more represented in the discourse of regional conservation bodies compared to other stakeholders.

-Class 5, which describes origins, colonization strategies and the potential impact of Japanese knotweed s.l. invasion, was greatly more represented in the discourse of the regional water authority, whereas less represented in the discourse of consultancy firms.

-Class 4, which describes the role of watercourses in dispersing the plant, was more represented in the discourse of consultancy firms but also, to a far lesser extent, of environmental protection associations.

It was on the contrary less represented in the discourse of other stakeholders.Class 2, which describes all the techniques used to eradicate Japanese knotweed s.l., was generally more represented in the discourse of local or regional authorities or state services, oflocal water associations, and, to a lesser extent, ofenvironmental protection associations. Conversely, it was greatly less represented in the discourse of the regional water agency.

-Class 1, which is related to mechanical meansof crushing rhizomes, was more represented in the discourse of consultancy firms and, to a lesser extent,of building companies operating on the river. This class is less represented in the discourses of other stakeholders, particularly the regional water agency.

These results show that each group had a specific position when approaching the issue of Japanese knotweed s.l.

Some types of stakeholders, such as the regional water agency, tended to emphasize all knowledge relating to the Japanese knotweed s.l. invasion processes. On the other hand, other types of stakeholders, such as local or regional authorities, state services, local water associations and environmental protection associations, were more likely to diffuse information relating to the techniques used for eradicating the plant. This was also the case for consultancy firms, who more specifically mentioned mechanical crushing techniques. Regional conservation bodies appeared to take a particular stand,as they gave a more general view of the issue, mentioning both strategies and the stakeholders involved in the management of Japanese knotweed s.l.

Degrees of emotionalism linked to the Japanese knotweed s.l. invasion

Four terms that were frequently usedto qualify the Japanese knotweed s.l. invasion were -impact‖, -nuisance‖, -risk‖ and -menace‖, all of which have a negative connotation. Nevertheless, they were associated in the French language (the original language of the corpus), with different significations, thus revealing information about the way the environmental managers perceived the invasion.The -nuisance‖ and -impact‖ terms (in this corpus, systematically considered as negative) characterize the invasion in a tangible, measurable way.Other terms, such as -risk‖ or -menace‖, on the other hand, refer to the invasionin a more intangible way: both evoke a potential danger. The former supposes that ecosystems and/or societies (often not specified further) were vulnerable to this invasion. The latter clearly evoked the danger associated with the invasion, as the termswere related to the vocabulary of fear and had a strong emotive connotation.

The different types of environmental managers seemed to have different uses for some of these terms (Fig5). In particular, the terms -impact‖ and -nuisance‖ were more represented in the discourse of the regional water agency compared to the whole population of managers. Conversely, the term -menace‖was more represented in the discourse of local water associations. Thus, the Japanese knotweed s.l. invasion appeared to generate strong emotions, specifically among this group of environmental managers.

DISCUSSION

Discourses promote an integrated action andfocus on the eradication stage

This study indicates that environmental managers of the Rhone River strongly want to take action against Japanese knotweed s.l. This attitude may be encouraged by the many policy engagements aimed at controlling and even eradicating priority invasive species (Delbartet al. 2012). As there is heavy pressure being placed on the environmental managers who are held responsible for maintaining environmental quality for the future,the uncertainties linked with the management of Japanese knotweed s.l. may have contributed, in several ways, to the willingness to take action. Most invasions are spreading too fast and too unpredictably to do anything other than respond immediately [START_REF] Sims | When is a -wait and see‖ approach to invasive species justified?[END_REF]. Thus, the -wait and see‖ approach appears to be, in many cases, inappropriate, thereby justifying the motivation for environmental managers to act. Moreover, behavioural studies havedemonstrated that the threat of fearsome risks (those that induce strong emotional response, such as fear and anxiety) activate certain cognitive mechanisms that push people towards action [START_REF] Loewenstein | The role of affect in decision making[END_REF]Sunstein and Zeckauser 2011). Such reactions are thought to be more frequent in uncertain situations [START_REF] Patt | Action bias and environmental decisions[END_REF]. The emotional response of environmental managers to a Japanese knotweed s.l.invasiontangible in their discourses-may explain their willingness to take action against the plant. This tendency may also be reinforced by uncertainties relating to the effectiveness of the proposed experimentalmethods of control. The absence of visible results and the need for repeated interventions encourage managers to experiment with other methods. This series of successive failures and the lack of control over the plant may have resulted in frustration among the network of environmental managers [START_REF] Allison | The paradox of invasive species in ecological restoration: do restorationists worry about them too much or too little?[END_REF], thus encouraging them to pursue other strategies.

Which actions are considered by environmental managers? Their discourses tackle a wide diversity of modalities of intervention and promote an integrated approach towards environmental management:

-Theytake into account every stakeholder (managers, economic actors, thepublic) and every territorial scale (local, regional, catchment) concerned with Japanese knotweed s.l.

-They consider different actions depending on the invasion stage.During introduction stage, knowledge acquisition and information campaigns are recommended; during colonisation stage, follow-up actions are proposed; andduring establishment stage, eradication measures are advanced.

Nevertheless, the discourses of environmental managers focus, above all, on possible actions when the plant is already established. They have tested over time a large variety of techniques aimed at stopping or at least slowing down colonization. The information produced about Japanese knotweed s.l. is very technical, and environmental managers mention all of their successes and failures, sometimes describing in detail the way they implemented actions and sometimes recommending certain procedures for increased efficiency. We believe that the development of discoursesand consequently of actionsrelating tothe invasion prevention and monitoring stages are beneficial, and we argue that it is best to act at the earliest possible stage of the invasion, that is, before it is too advanced to be reversed [START_REF] Boyer | L'invasion des cours d'eau par les renouées du Japon s.l. : réflexions et propositions pour des stratégies de lutte efficaces[END_REF]. In the Rhône River case, for instance, we have found that uninvaded areas, such asthe downstream section of the river, produce little information about the Japanese knotweed s.l. invasion. Conversely, the more heavily invaded an area is, the more documentation the environmental managers produce about these plants, and therefore, the more they worry about the presence of the plants. An efficient management system should consider implementing action before this stage.

Varying positions of environmental managers

Japanese knotweed s.l. has led to widely differing proposals, according to the type of environmental manager making the proposal. In particular, there appears to be a strong difference of approach between those managers who define policy at a regional level (regional water agency) and focus on the knowledge available and those who implement policy at a regional or local level (local or regional authorities, state services, local water associations) and focus on action. Managers implementing environmental policies arehighly interested in control and eradication techniques and the effectiveness of these techniques. They are also the ones more frequently using emotional language to characterize the Japanese knotweed s.l. invasion. This use of language may reflect their strong and increasing need for success in managing this issue given that since 2006, the majority of information produced about Japanese knotweed s.l. has come from these stakeholders. The pressure resulting from management policies that expect quick results in managing invasive species may be a more sensitive issue for managers in the field, as they directly experience the success or failure of their intervention and feel more directly responsible for it. From another perspective, the emotional component of their discourses may not be a sign of a greater preoccupation but may only translate their willingness to bring the Japanese knotweed s.l. issue to the attention of funding bodies (including the regional water agency) to obtain funds to act against the spread of this plant. The use of controversial language in the field of invasion biology has already been widely observed, sometimes because its terminology connotes nativism, racism or xenophobia [START_REF] Subramaniam | The aliens have landed! Reflections on the rhetoric of biological invasions[END_REF][START_REF] O'brien | Exotic invasion, nativism and ecological restoration: on the persistence of a contentious debate[END_REF], sometimes because it is militaristic [START_REF] Davis | Elton and the dissociation of invasionecology from the rest of ecology[END_REF]Larson, 2005). These criticisms have led some authors to advocate that scientists use more neutral terms for introduced species (Larson, 2005;[START_REF] Davis | Invasion biology[END_REF]. The results of our study may suggest that environmental managers do the same and be more attentive to the terminology used. We doubt that this evolution of language, if indeed possible [START_REF] Larson | Embodied realism and invasive species in: de[END_REF][START_REF] Simberloff | Invasionalmeltdown six years later-important phenomenon, unfortunate metaphor, or both?‖[END_REF], will lead to a consensus in the way that Japanese knotweed s.l. is perceived and has to be managed: those attitudes appear to be more firmly embedded and to be influenced by heterogeneity of worldviews about relation (and definition) of human society and biota [START_REF] Simberloff | Nature, natives, nativism and management: worldviews underlying controversies in invasion biology[END_REF]. Nevertheless, such an effort would enable sharing of views and discussion about actions to implement on a more neutral basis.

The information provided by the regional water agency (the managers defining policy at a regional level) uses less emotive terms. Moreover, this agencysays little about the available methods for controlling the plant but rather focuses on the diffusion of knowledge related to the origins of the Japanese knotweed s.l. invasion, its colonization strategies and the nuisance it creates. The position of the regional water agency appears to be more reflective, or at least more dedicated, to the diffusion of information related to the Japanese knotweed s.l.

invasion. Nevertheless, the attitude of the agency is far from disassociated with action,as the control of invasive species is part of the programmed measures (2010-2015) that it defined and funded 1 . However, as the information it produced did not focus on a way to achieve this objective, a gap exists between the two types of managers: environmental managers who implement management policies need effective methods to efficiently control the plant, as required by the regional water agency, but the agency only diffuses general knowledge related to colonization processes. Misunderstandings and frustrations may result from these heterogeneous positions. The communication strategy of the regional water agency may be more successful if their general information was complemented with more technical detailsregarding methods of eradication. However, even if the information produced by the regional water agency does not meet the expectations of environmental managers who have to implement these management projects, the efforts aimed at diffusing general knowledge regarding the origins, mechanisms and impacts of Japanese knotweed s.l. colonization should be maintained.

1 Article OF6-C requiring to -fight against invasive exotic species‖.

Towards a more targeted management

In theenvironmental managers' discourses, the management of Japanese knotweed s.l.isapproachedfrom a global perspective. Recommendations for action define general rules regardless of the socio-economic and environmental context of the area under consideration. These rules flow from the premise that ecosystems are bounded by frontiers within which are found, either exclusively or primarily, native species. This premise, however,is far from the reality, and defining such a reference state is delicate if not impossible [START_REF] Dufour | From the myth of a lost paradise to targeted river restoration: forget natural references and focus on human benefits[END_REF]. This perception of a stable and balanced nature has long been called into question by academic ecologists [START_REF] Simberloff | The -balance of Nature‖evolution of a panchreston[END_REF]. According to this author, the idea of a balance of nature lives on especially among conservationists and environmentalists. This may explain why this point of view is salient in the discourse of environmental managers.

The efficiency of management may be improved ifa targeted strategy for each identifiable plant area was defined. Such a definition should then be the result of a wide collective reflection relating to local issueslinked withactionagainst invasive species. In view of thisidea, environmental managers may have interest in clarifying the local issues linked with action (or inaction) against the plant in each given context andin defining the priorities related to its control. There are questions that must be addressed. For example, are there ecologicalissues (detrimental to other species)?aestheticissues (size of plant and visual place in the landscape)?securityissues (decrease in visibility)?economic issues (bank erosion)?Are ecological or geomorphological characteristics of the invaded area favourable to a rapid dispersion of the plant? The efficiency of themanagement may benefit from a more specific, spatially heterogeneous action that considers local issues [START_REF] Epanchin-Niell | Controlling established invaders: integrating economics and spread dynamics to determine optimal management[END_REF]. Certain studies have already considered this approach and merit a specific attention [START_REF] Filippi | Plantes invasives en région méditerranéenne : quelles restrictions d'utilisation préconiser pour les jardins et les espaces verts ?[END_REF].

CONCLUSION

This article studied the documentation relating to Japanese knotweed s.l. produced by environmental managers working along the Rhône River and characterized their views on this invasive plant. The results indicatedthat there isa gap between the proposals of stakeholders defining management plans at a regional level and those of stakeholders who are implementing actions locally. Whereas the former werefocused on providing general knowledge about invasion processes, the latter werefocused on listing technical methods for controlling and eradicating the plants and providing information regarding the effectiveness of these methods. These different approaches result, each in their own way, from uncertainties related to biological invasions and may lead to misunderstandings among stakeholders. Nevertheless, they all agreed on one point: the need to take action against Japanese knotweed s.l. While there wasan interest in conducting targeted actions to meet local issues, to do so, the issues must be better defined. This is a challenging task that must involve all types of stakeholders including environmental managers, scientists, association members, users and the public. Uncertainties relating to environmental management can only be overcome if management projects result from political projects that have been collectively discussed and validated. 3 List of lemmas co-occurring with the -efficiency‖ lemma (co-frequency threshold = 3; distance = 10 lexical forms before and after the considered lemma)

Table 4 List of lemmas co-occurring with the -impact‖ lemma (co-frequency threshold = 3; distance = 10 lexical forms before and after the lemma under consideration)

Table 5 Values assessing the association between the textual segments and the classes of the top-down

  1998) was performed using the open source software Iramuteq© (Ratinaud and Dejean 2009), which was recently developed (2008) and which isregularly updated. It relies on R software (R Core team, 2013) as well as the python language.

Fig1Fig2Fig3

  Fig1 Number of documents produced (a) over time according to the type of manager; (b) according to the document's category and the type of manager Fig2 Area under study showing the number and the repartition of the documents produced by each type of stakeholder Fig3 Results of the top-down hierarchical clustering of the textual segments: (a) segmentation tree indicating the proportion of textual segments classified in each class; (b-f) characterization of the 5 classes using similarity analyses (note: as each graph was independently produced, the size of the character cannot be compared from one graph to another).

Fig4Fig5

  Fig4 Techniques cited over time to control Japanese knotweed s.l.Fig5Occurrence of different words cited to characterize Japanese knotweed s.l. invasion according to the type of stakeholder

  hierarchical classification according to the stakeholder groups having produced them; (a) metric derived from the Chi 2 method ([observed value -expected value] / expected value ); (b) proportion of the classified textual segments

  [START_REF] Child | Japanese Knotweed Manual: The Management and Control of an Invasive Alien Weed (Fallopia Japonica)[END_REF] conducted a contingent valuation study to assess the socially acceptable cost of controlling the plant. Apart from this study, only factual and incomplete data relating to the human perceptions of Japanese knotweed s.l. are available. For instance, Vanderhoevenet al. (2011) determined that, in Belgium, Japanese knotweed s.l.is one of the species mentioned by horticulture professionals and nature reserve managers during surveys when asked which species would become a problem in the next few years. Further research on human perceptions of Japanese knotweed s.l. is therefore required.

Table 1

 1 Description of document's categories that compose the textual corpus

Table 2

 2 List of the most significant textual segments composing each class resulting from the top-down hierarchical clustering

Table

ACKNOWLEDGEMENTS

We thank the anonymous referees for their helpful comments. We are grateful to the regional water agency (in partnership with the LTER-ZABR) and the IngECOTechprogramme (INEE-CNRS) for funding and supporting this research. We extend our appreciation to Céline Cordani, who collected the data during a master 1 internship and also participated in the first discussions, to HervéTronchère for his help with figure 2, and to John Stella for his valuable reading and comments. We finally wish to thank all environmental managers who took the time to gather the documentation included in our analysis corpus.

LIST OF FIGURES


Microtearing turbulence: magnetic braiding and disruption limit

Marie-Christine Firpo 1

Laboratoire de Physique des Plasmas, CNRS -Ecole Polytechnique, 91128 Palaiseau cedex, France

A realistic reduced model involving a large poloidal spectrum of microtearing modes is used to probe the existence of some stochasticity of magnetic field lines. Stochasticity is shown to occur even for the low values of the magnetic perturbation δB/B devoted to magnetic turbulence that have been experimentally measured. Because the diffusion coefficient may strongly depend on the radial (or magnetic-flux) coordinate, being very low near some resonant surfaces, and because its evaluation implicitly makes a normal diffusion hypothesis, one turns to another indicator appropriate to diagnose the confinement: the mean residence time of magnetic field lines. Their computation in the microturbulence frame points to the existence of a disruption limit, namely of a critical order of magnitude of δB/B above which stochasticity is no longer benign yet leads to a macroscopic loss of confinement in some tens to hundred of electron toroidal excursions. Since the level of magnetic turbulence δB/B has been measured to grow with the plasma electron density this would also be a density limit.

I. INTRODUCTION AND OBJECTIVES

In the tokamak terminology, magnetic microturbulence refers to the simultaneous excitation of a large spectrum of magnetic modes with poloidal mode numbers m ≫ 1. It might however be thought that the smallness of the relative magnitude of the magnetic perturbation δB/B devoted to magnetic turbulence as measured in some tokamak experiments 1,2 makes the question of the stochasticity of the magnetic field lines rather irrelevant. Yet, several recent results coming from gyrokinetic simulations have unveiled the ubiquity of the magnetic stochasticity under physical parameters relevant to fusion-oriented tokamak plasmas. In particular, electromagnetic numerical simulations in the conditions of tokamak ion temperature gradient turbulence 3 have demonstrated that the magnetic field could be stochastic even at very low plasma pressure. Concurrently, another study 8 aiming at unveiling the nature and mechanisms behind turbulent transport in tokamak plasmas based on ab initio gyrokinetic simulations demonstrated that heat transport is dominated by the electron magnetic component and pointed to microtearing modes as a possible candidate to explain turbulent transport in tokamak plasmas. Moreover, other nonlinear gyrokinetic numerical simulations of microtearing mode turbulence using the experimental parameters from a high-β discharge of the spherical torus NSTX experiment 9 indicated that the transport is almost entirely electromagnetic as a result of the electrons diffusing in the stochastic magnetic field. Lastly, electromagnetic gyrokinetic simulations of plasma microturbulence 10 have also supported the nearubiquitous character of magnetic stochasticity.

Experimentally, there is increasing evidence of the microtearing nature of magnetic turbulence. Very recently, the first direct experimental verification of microtearing mode turbulence in the core region of tokamak plasmas has been reported in the JIPPT-IIU tokamak 2 , where local magnetic fluctuations were measured using a heavy ion beam probe. In the ASDEX-Upgrade, edge measurements using electron cyclotron emission imaging just detected also that the temperature fluctuations have features similar to microtearing modes 11 . Accordingly, there is now a growing body of evidence both from numerical gyrokinetic simulations and novel experimental diagnostic approaches that magnetic fluctuations come under the form of microtearing modes that control electron heat transport.

Gyrokinetic numerical simulations are certainly useful since they should eventually provide a fine-scale selfconsistent picture of tokamak plasma dynamics, yet they are highly numerically-demanding and challenging. It remains therefore desirable to continue to build and develop reduced and versatile models to improve the understanding of the puzzling and critical tokamak transport issue and to test paths towards the improvement of the confinement. In order to reduce the complexity of the system formed by the strong equilibrium magnetic field and the assemblies of charged particles selfconsistently interacting with turbulent electromagnetic perturbations, one may adopt a non self-consistent test particle approach under suitable assumptions for the electromagnetic field. A more fundamental approach is to focus on the magnetic structure of the confinement. Indeed the Maxwell equation divB = 0 forms a closed and exact equation enabling to isolate the effect of the magnetic field and study its properties. This does not mean at all that one neglects the electric field or assumes the nullity of the electric perturbations. This does not mean either that the trajectories of charged particles in a spatially integrable magnetic field are integrable since the threedegrees of freedom Hamiltonian from which derive the particle equations of motion may a priori allow chaotic motions. Indeed in the case where the (integrable) magnetic structure possesses an X-point, as in the divertor scheme, charged particles experience a chaotic motion in the vicinity of the magnetic separatrices 12 . This is just a reduction of the picture. It was notably shown by Cary and Littlejohn in a seminal paper 13 that the zero divergence of the magnetic field, which may be interpreted as a condition of phase space conservation, makes the equations of magnetic field lines derive from a Hamiltonian system.

This line of research connected years ago to the introduction of the paradigmatic standard map 14 that had a considerable importance in the understanding of the onset and phase-space manifestations of Hamiltonian chaos. This captured the universal behaviour of area-preserving maps with divided phase space when integrable islands of stability are surrounded by a chaotic component. Yet, this model is not meant to address the quantitative transport issues posed by magnetic confinement fusion. Using the present computing capacities and the improved knowledge of the magnetic turbulence coming from experimental measurements and recent gyrokinetic results, it has become possible to consider more realistic models of magnetic field lines under microturbulence conditions and compute their transport properties. The microturbulence magnetic framework used in the present study will be exposed in Section II. Some numerical results obtained within such microtearing turbulence models having a large spectrum of poloidal modes will be presented. It will be shown that a value of the magnitude of the magnetic perturbation as low as 10 -5 does not imply that magnetic field lines are close to integrability.

The problem of the disruptions in toroidal devices for magnetic confinement fusion must be tamed to ensure the viability of industrial fusion reactors. Contrarily to macroscopic dramatic phenomena, such as the sawtooth crash, that may be related to the onset of the chaos of magnetic field lines due to some subset of longwavelength modes with different helicities (See e.g. [4][START_REF] Igochine | ASDEX Upgrade Team[END_REF][6][7] ), the consequences of the loss of the integrity of magnetic surfaces in the presence of plasma microturbulence have been observed in some gyrokinetic numerical simulations 3 to be normally benign or moderate. Yet, there is certainly some limit in the braiding of the magnetic field lines above which magnetic confinement breaks and a disruption occurs. Section III will be devoted to the exploration of disruption limits associated to magnetic microturbulence. This study will involve the evaluation of some probability distribution functions (pdf) of magnetic confinement times. These are obtained through rather demanding numerical simulations within the reduced magnetic turbulence model. Yet, it is useful to point out the benefit of using a reduced approach by noting that the numerical cost of analogous simulations within a full gyrokinetic frame would have been much higher and possibly prohibitive. A short discussion concludes the work.

II. THE HAMILTONIAN MICROTEARING TURBULENCE FRAMEWORK

A. Presentation

The canonical representation of an arbitrary magnetic field in a toroidal magnetic confinement device is written in Eq. (1). Let us consider a set of variables (ρ, θ, ϕ)

where ρ is a radius-like variable that vanishes along the magnetic axis. Then, there exist 15 two single-valued functions ψ(ρ, θ, ϕ) and Φ(ρ, θ, ϕ), such that the magnetic field can be written in the form B = ∇ψ × ∇θ + ∇ϕ × ∇Φ.

(

Eliminating the variable ρ by using Φ(ψ, θ, ϕ), the equations for the magnetic field lines read

dψ dϕ = B.∇ψ B.∇ϕ = - ∂Φ ∂θ , ( 2
)
dθ dϕ = B.∇θ B.∇ϕ = ∂Φ ∂ψ . ( 3
)
In this representation, Φ plays the role of a generically one-and-a-half degrees of freedom Hamiltonian [START_REF] Boozer | [END_REF][17][18] , the toroidal angle ϕ is a time-like variable and ψ is the momentum conjugated to the poloidal angle θ. In a nonsteady state, the field-line Hamiltonian depends also on the real time t, that plays the role of a parameter, but not of a canonical variable, and will be noted Φ t . In tokamaks, the equilibrium configuration is axisymmetric. The equilibrium magnetic field B 0 (r) has its field lines spiralling on perfect nested magnetic surfaces around the magnetic axis and these are defined by a constant ψ. This amounts to say that the magnetic fieldline Hamiltonian associated to B 0 is integrable, being Φ 0 (ψ). It identifies with the poloidal magnetic flux and ψ with the toroidal magnetic flux. From the equilibrium poloidal flux Φ 0 (ψ), one defines the very important quantity called the safety factor profile q(ψ), or equivalently its inverse, the winding profile w (ψ), through

dΦ 0 (ψ) dψ = 1 q (ψ) = w(ψ). ( 4
)
Yet, axisymmetry is bound to be broken by external effects, such as the toroidal ripple in the magnetic field, which appears in tokamaks due to the finite number of toroidal magnetic field coils, or as the edge magnetic perturbations due to additional coils serving to control particle and heat loads on tokamak walls 19 or by some intrinsic instability phenomena such as MHD activity or microturbulence. The generic form of the magnetic fieldline Hamiltonian reads then

Φ t (ψ, θ, ϕ) = Φ 0 (ψ) + δΦ t (ψ, θ, ϕ). ( 5
)
The order of magnitude of the relative perturbation |δB|/B 0 quantifies the deviation from axisymmetry and integrability. Using the poloidal and toroidal periodicities, this can be Fourier decomposed as

δΦ(ψ, θ, ϕ) = ∑ m,n ε m,n δΦ mn (ψ) cos(mθ-nϕ+χ mn ), (6)
where from now on, the time dependence index has been dropped to simplify the notations. A rough model for the behavior of microtearing waveforms was proposed by Stix in 28 . In the present work, a more refined model of microtearing turbulence described in 27 will be used. This is an empirical experimentally-based model taking

δΦ mn (ψ) ≡ [ cosh ( 1 ∆ ln ψ ψ mn )] -m ∆ 2 . ( 7
)
The (m, n) waveform is maximal on the rational surface ψ mn , defined as the value of the toroidal magnetic flux for which q(ψ mn ) = m/n. The physically realistic value of the parameter ∆ is about 0.01. This ensures the correct minimally-smooth peaking of the waveforms. In numerical simulations, the poloidal modes with 1 < m < 40 have been retained and the associated toroidal modes are such that the resonances (m, n) are present. Figure 1 shows these modes. A standard winding profile is used with

w(ψ) = 0.2(2 -ψ)(ψ 2 -2ψ + 2), (8)
corresponding to a monotonously growing q-profile. It should be noted that the present model considers the whole spectrum of magnetic modes, not the only the part with the largest m modes that may be properly defined as the turbulent part. This is consistent with the experimental reality where relatively large scale magnetohydrodynamic-like modes coexist with small scale perturbations.

B. The Hamiltonian for a single mode: estimation of the resonance widths

Let us first characterize the Hamiltonian model in the case of a single arbitrary mode (m 0 , n 0 ). Using a canonical transform with the generating function F 2 (θ, Ψ, ϕ) = (m 0 θ -n 0 ϕ)Ψ, that amounts to moving to the (m 0 , n 0 ) wave frame, the single mode Hamiltonian reads Φ(Ψ, Θ, ϕ) = Φ(ψ, θ, ϕ)

+ ∂F 2 /∂ϕ with ψ = m 0 Ψ and Θ = m 0 θ -n 0 ϕ. This yields Φ(Ψ, Θ) = Φ 0 (m 0 Ψ) -n 0 Ψ + ε m0,n0 δΦ m0n0 (m 0 Ψ) cos Θ.
(9) Figure 2 and Figure 3 represent respectively the perturbed Hamiltonian waveform and the topology of the Hamiltonian phase space for the same one-wave integrable case. The fixed points of the Hamiltonian ( 9 FIG. 2. Perturbed Hamiltonian waveform δΦm 0 n 0 for the mode m0 = 33, n0 = 8 and for the parameter ∆ = 0.01. The q-profile considered here is the same as in Figure 3. FIG. 3. Phase space portrait of magnetic field lines in the case where a single microtearing mode is retained, displayed in the wave frame in the (Θ, Ψ) space. Here this is the mode m0 = 33, n0 = 8. The q-profile used in the figure is q(ψ) = 1.5 + 3.2ψ which gives ψm 0 =33,n 0 =8 ≃ 0.82 that corresponds, in the new wave frame variables, to Ψ ≃ 0.248. The amplitude of the mode is εm 0 ,n 0 = 5.10 -4 .

are given by Θ = 0 or π. For Θ = π, the action ψ of the fixed points satisfies

0 = 1 q(ψ) - n 0 m 0 +ε m0,n0 m 0 2ψ δΦ m0n0 (ψ) tanh ( 1 ∆ ln ψ ψ m0n0
) .

(10) This equations admits an obvious solution ψ = ψ m0n0 corresponding to an elliptic fixed point and it admits two other roots ψ + > ψ m0n0 and ψ -< ψ m0n0 in the case where the q-profile is monotonously increasing that correspond to hyperbolic fixed points. We can then estimate the resonance width associated to the (m 0 , n 0 ) single wave Hamiltonian. The largest toroidal flux corresponding to the upper half-width of the resonance, denoted by ψ+ , may be estimated from the energy conservation on the upper separatrix so that Φ(ψ + /m 0 , π) = Φ( ψ+ /m 0 , 0). Under the assumption that ψ+ ≃ ψ + , this yields

ψ+ -ψ + ∼ ε m0,n0 δΦ m0n0 ( ψ+ ) + δΦ m0n0 (ψ + ) n0 m0 -1 q( ψ+ )
. (11) Finally, for the consistent hypotheses ensuring the smallness of ψ+ -ψ + , namely ε m0,n0 sufficiently small and a non-vanishing magnetic shear at the rational surface, it is possible to further simplify Eq. ( 11) by making ψ + ≃ ψ m0n0 , δΦ m0n0 ( ψ+ ) ≃ δΦ m0n0 (ψ + ) ≃ 1, and by using

q ( ψ+ ) -m 0 /n 0 ≃ q ′ (m 0 /n 0 ) (ψ + -ψ m0n0 ).
The expression of the resonance half-width, δψ m0n0 = ψ+ -ψ m0n0 , amounts then to

δψ m0n0 = ( 2ε m0,n0 |w ′ (ψ m0n0 )| ) 1/2 = m 0 n 0 ( 2ε m0,n0 q ′ (ψ m0n0 ) ) 1/2 .
(12) This is the classical expression of the microtearing resonance half-width.

Unless otherwise specified, we define ε such that ε m,n ≡ εn/m, so that the mode amplitudes ε m,n are indeed all of the same order ε. The expression of the resonance half-widths reduces then to δψ m0n0 = [2εq(ψ m0n0 )/q ′ (ψ m0n0 )]

1/2 . The magnetic field perturbation δB mn relates to ε m,n through

δB mn B 0ϕ = 1 2 √ 2 mε m,n A √ ψ mn , ( 13
)
where A denotes the tokamak aspect ratio (A = R 0 /a).

Although the knowledge of the magnetic spectrum would be necessary to make quantitative estimations, which remains a really challenging task due to the experimental difficulties to measure internal magnetic perturbations, the objective of the present study is more qualitative than quantitative so that we shall mainly retain for our purposes that δB/B ∝ ε.

C. Characterization of the local magnetic stochasticity

The previous expression of the microtearing resonance half-widths may serve to estimate the local Chirikov parameter that quantifies the local chaoticity of the Hamiltonian magnetic field lines. The expression of the Chirikov parameter relative to any couple of modes (m, n) and (m ′ , n ′ ) reads The segments between the ψmn and their associated ψ m ′ n ′ are marked but are only wide enough to be visible about the resonances associated to the integer values of q. The q-profile used in the calculations corresponds to Eq. ( 8).

s (m,n),(m ′ ,n ′ ) = δψ mn + δψ m ′ n ′ |ψ mn -ψ m ′ n ′ | .
there are local minima of stochasticity in the close vicinity of the resonances associated to integer values of the q-profile while there are local maxima of stochasticity just in the periphery of these zones. This result is valid for q-profiles having a non-vanishing magnetic shear, that is a condition for the validity of Eq. ( 12), and is consistent with experimental results indicating that transport barriers preferentially form about surfaces where q is an integer.

Another consequence of the results shown on Figure 4 is that, apart from those neighborhoods of the q = integer and low rational surfaces, some mid scale stochasticity is realized even at extremely low values of the relative magnetic perturbation amplitude ε, in our case even at ε ∼ 10 -7 -10 -6 . Experimentally, ε has been reported 1 to be of the order of 10 -5 to 10 -4 (in the L mode) in Tore Supra and, very recently, to be of order 10 -4 in the JIPPT-IIU tokamak plasmas 2 . Therefore, within the microtearing turbulence frame, the stochasticity of the magnetic field lines under experimentally relevant values of ε is generic. This is in agreement with the seminal Stix's analysis 28 and the aforementioned recent gyrokinetic results 3,8-10 .

D. Numerical integration of the equations for the magnetic field lines

The Hamiltonian dynamics associated to the equations ( 2)-( 3) has been integrated using a symplectic integrator with toroidal angle 'time'-step 2π/3200.

Figures 5 and6 represent the radial-type component, ψ, of different magnetic field lines as a function of the toroidal angle ϕ for the microturbulence model for two values of the relative magnetic perturbation, namely ε = 2 • 10 -5 and ε = 10 -4 . The non-integrable character of the magnetic field lines is quite manifest from the figures.

However, whereas the behavior of the magnetic field lines looks stochastic for ε = 10 -4 , this is much weaker for ε = 2 • 10 -5 where the magnetic field trajectories correlate with the (m, n) resonances. As the magnetic braiding gets smaller, the stochasticity in the ψ-domain may become locally quite low so that the resonances are not fully disaggregated behaving as cantori. As apparent on the left plot of Figure 5, magnetic field lines can be reflected by remnants or almost intact resonances, possibly of a high order, which induces some memory effect.

In Figure 4, the relative minimal values of the two-wave overlap Chirikov parameter divided by ε 1/2 are close to 124 about the (4, 1) resonance and to 133 about the (3, 1) resonance. Being obtained for ε = 2•10 -5 , Figure 5 corresponds then to a situation where the Chirikov parameter about the resonances associated to the integer values of q, where the chaos is minimal, is about 0.6, so noticeably below 1. One could therefore be surprised to observe some (at least partial) breaking of the KAM tori about the integer surfaces.

However, it must be emphasized that the classical criterion for chaos -that the Chirikov overlap parameter s 1,2 between two resonances waves be larger than one -is a crude one and many refinements have been proposed [START_REF] Meiss | Differential Dynamical Systems[END_REF] . In particular, even if we had a Hamiltonian system with just two waves, taking into account the secondary resonances arising from the nonlinear beating between the two primary resonances would reduce the effective distance between resonances [START_REF] Lichtenberg | Regular and Chaotic Motion[END_REF] . A rule of thumb is that connected chaos occurs when s 1,2 ≃ 2/3 instead of 1 due to this mechanism. The most sophisticated treatment for the paradigmatic case of two primary resonances consists in taking into account the creation of infinitely many secondary resonances and using renormalization theory to compute the chaos threshold [START_REF] Escande | [END_REF][START_REF] Mackay | Renormalization in area-preserving maps, Singapore[END_REF] . Since this effect of secondary resonances couples here to the existence of a spectrum of primary modes, it is not surprising to observe the onset of chaos for values of the Chirikov overlap parameter below 2/3. Moreover there is some source for additional chaos since the resonances are not purely pendulum sinusoidal waves. This is apparent in the phase space portrait of a microtearing mode that has two close x-points instead of one in the pendulum sinusoidal case (see Figure 3).

The situation of Figure 5 with ε = 2 • 10 -5 is certainly above but very close to the threshold for connected chaos since one can observe that two magnetic field lines remain contained between the (3, 1) and (4, 1) resonance radial locations whereas the other two magnetic field lines happen to cross these. For low enough values of ε, the Chirikov parameter between the magnetic resonances eventually becomes small enough in macroscopic domains of the ψ-space impeding (normal) diffusion.

III. MAGNETIC CONFINEMENT TIMES

A. The magnetic microturbulence picture

In Sect. II D, from the integration of the magnetic field line trajectories at experimentally relevant values of the relative magnetic perturbation parameter ε emerges the picture of the microturbulent magnetic transport. Some locally stochastic domains of the phase space are separated by partial transport barriers (cantori) limiting the large scale magnetic diffusion. This suggests that characterizing the transport through a magnetic diffusion coefficient, D m , may not be a universally appropriate approach. Indeed, the presence of cantori should induce a strong locality of the diffusion coefficient: namely D m depends on the action ψ, with locally almost vanishing minima about the ψ mn associated to the cantori. Moreover, the diffusive nature of the transport is unclear. Computing a diffusion coefficient implicitly means that the diffusion is normal, neither sub-nor super-diffusive. An alternative physical indicator, that is a natural indicator in the fusion context where Lawson's criterion is central, would rather be given by the characteristic time spent by the magnetic field lines to visit the phase space, that may be viewed as a confinement time. The benefit of this indicator is that it remains valid even in the absence of (normal) diffusion.

B. First exit times

A physically meaningful indicator of the magnetic confinement for fusion applications is the residence time of the magnetic field lines inside some inner volume of the device. Indeed, in magnetic confinement devices, electrons are strongly magnetized, so that a breakdown of the magnetic confinement may result in the termination of the electron confinement, or in other words, in a disruption.

To probe the magnetic transport properties within the microturbulence framework, extensive numerical simulations have been performed. Starting from magnetic field lines in the core of the tokamak, more precisely such that ψ(ϕ = 0) be in the range [0.05; 0.1], the number of toroidal turns ϕ exit /(2π) after which ψ(ϕ exit ) becomes larger than 0.9 was computed for a large number of field lines and at different values of the stochasticity parameter ε. Figure 7 represents the distribution of the first exit times ϕ exit in units of toroidal turns for two values of ε. In Figure 8 is plotted for some values of ε above 10 -4 the mean exit times computed by launching as many as some tens of thousand initial conditions for the magnetic field lines in the tokamak core. Figure 7 validates the magnetic microturbulence picture depicted in Sect. III A.

For values of ε roughly below 10 -4 , the values of the exit times become very large, in the sense that it becomes numerically too demanding to compute them. Moreover as ε decreases, there is an increasingly growing fraction of the trajectories that never attain the tokamak border, getting glued about low order resonances. As ε increases, the local Chirikov parameter eventually raises above one in the whole ψ domain and the conditions for the usual diffusion in ψ space are fully satisfied. Indeed, for the range of ε values considered in Fig. 8, the mean exit times scale with ε as ε -2 . This is the scaling expected from a quasilinear diffusion approach [START_REF] Rechester | [END_REF] , since D QL ∝ (δB/B) 2 and δB/B ∝ ε.

The Brownian nature of the behaviour of magnetic field lines may be assessed by comparing the numerically obtained distribution of first exit times with its prediction in the case of Brownian motion. The first passage time distribution of Brownian motion with positive drift takes the form

f (τ ; µ, λ) = ( λ 2πτ 3 ) 1/2 exp [ - λ 2µ 2 τ (τ -µ) 2 ] . (14)
It can be seen on Figure 9 that the distribution of the numerical exit times in the case ϵ = 10 -3 is fairly well approximated by a function of the form (14). The next Section will be devoted to some consequences for tokamak physics of the present results.

IV. INDICATIONS FOR A DISRUPTION LIMIT DUE TO MAGNETIC MICROTURBULENCE

A. Preliminary justification for a static frame

The magnetic model presented in Section II may be made time-dependent to account for the time variations of field perturbations in real plasmas. Typical fluctuation frequencies have been reported to be in the range 10 4 to 10 5 Hz. In this Section, our interest will lie however on disruptive-type phenomena taking place on the time scale of tens to hundreds of electron toroidal excursions. For such a short time scale, it is possible to neglect the diffusion effects induced by phase and restrict to a static approach.

Moreover, it is a well-known experimental fact that phase-locking between the rotating modes is one 'favorable' ingredient for disruptions [34][35][36] . Considering that modes rotate in phase is equivalent in the frame of the modes to consider constant phases. One just proceeds to a Galilean transform to the wave frame. Disruptions have also been observed to take place with slowly rotating modes. Then, with regard to the brevity of the disruptions, treating the disruption scenario in a static frame can be a good approximation.

In any case, the static situation should be the more deleterious, limit scenario for the loss of confinement since one can easily imagine that some differential poloidal rotation in the modes contrarily makes the field lines spend more time moving in the poloidal direction which increases their mean exit times.

B. Magnetic turbulence-driven disruption limit

In tokamak physics, there exist some dramatically swift phenomena taking place on the lapse of time of some tens to a couple of hundreds of microseconds. These are the collapse phases of sawteeth, edge localized modes and disruptions. All these phenomena may be interpreted as transient, benign or severe losses of the magnetic confinement and are mediated by the lighter charged particles of the plasma, namely by the electrons. In a tokamak such as JET, the electron thermal velocity at a temperature of 10 keV is about 4.10 7 m.s -1 so that the characteristic time spent by electrons to make a toroidal excursion is 0.5µs. Consequently, a lapse of time of 100µs amounts to 200 electron toroidal excursions. It is quite instructive to see on Figure 8 that 200 toroidal excursions is the order of magnitude of the qualitative threshold mean exit time separating a quick loss of the magnetic confinement for ε > ε th from a much better magnetic confinement provided the perturbation parameter ε becomes some fraction of ε th . From the Figure, one may retain the value 5.10 -4 for ε th . The magnetic perturbation δB/B is of order ε with the spectrum of modes used here. Its value associated to ε = 5.10 -4 could be derived using Eq. ( 13).

Looking at Figure 8 comes the impression of a threshold effect in the behaviour of the mean exit time as a function of ε. We shall here examine its origin.

In the regime where the resonance overlap is sufficient to ensure the complete stochasticity of the magnetic field lines and the validity of the quasilinear approximation, the mean exit 'times' ⟨ϕ exit ⟩ scales as ε -2 : there exists some constant C > 0 such that ⟨ϕ exit ⟩ = C 2 ε -2 . Converting the toroidal angle to a real time τ through the use of the electron parallel velocity as we have just done and using the fact that δB/B is proportional to ε, this means that the characteristic exit time scales with respect to the relative magnetic perturbation as ⟨τ exit ⟩ ∝ (δB/B) -2 .

Consequently, all other things being equal, a small change in the magnetic perturbation ∆(δB/B) is associated to a much larger change in the mean exit time ∆⟨τ exit ⟩ since

∆⟨τ exit ⟩ ⟨τ exit⟩ = -2 ( δB B ) -1 ∆ ( δB B ) , ( 15
)
where δB/B is a (very) small parameter. From this it follows that a relatively small change in the magnetic perturbation is associated to a much larger relative variation in the mean confinement time. This sensitivity effect is beneficial when the magnitude of the magnetic perturbation is reduced since the confinement time may drastically increase but this effect is deleterious when the magnetic perturbation is increased, by intrinsic or extrinsic routes, as the confinement time may drop to the point where a macroscopic (major) disruption occurs and confinement is lost.

V. DISCUSSION

In the present study, the magnetic perturbation δB/B, or equivalently the parameter ε, is a control parameter which may take arbitrary values. In the framework of magnetic microturbulence, magnetic field lines become stochastic in a macroscopic fraction of the tokamak volume when the Chirikov parameter computed for any two resonant waves (m, n) and (m ′ , n ′ ) is roughly above 2/3 [START_REF] Meiss | Differential Dynamical Systems[END_REF] . This has been shown to take place for the values of δB/B that have been experimentally measured by cross polarization scattering in Tore Supra 1 or by using the poloidal sweep of a heavy ion probing beam recently in JIPPT-IIU 2 . These tokamaks therefore operate in regimes where the stochastic hypothesis necessary to derive a Fokker-Planck diffusive approach, like the quasilinear theory, is satisfied. Consistently, both the Tore Supra and JIPPT-IIU teams have verified that the experimentally measured levels of magnetic perturbation were compatible with their experimentally measured electron heat diffusivities χ e when using the quasilinear expression for χ e [START_REF] Rechester | [END_REF] .

Predicted here is the existence of some qualitative threshold on δB/B in this stochastic regime above which the mean confinement exit times become too low to be sustainable by the tokamak plasma. The sensitivity in δB/B is reflected by Eq. ( 15). One may wonder whether this disruptive-like limit may actually be reached. Indeed, once there is an overall stochasticity of magnetic field lines, there is some self-healing process coming into play in the sense that the stochasticity of magnetic field lines quenches the thermal gradients that are a source of instability for the microtearing modes and may regulate the magnetic microturbulence. However, collisionality may remain as a source of destabilization of microtearing modes.

These results should be put in perspective with the recent achievements on runaway electron suppression above some magnetic turbulence threshold reported by Zeng et al. 37 . These experimental measurements done on TEX-TOR have shown that magnetic turbulence is mainly contributed from the background plasma and that its level does strongly dependent on the toroidal magnetic field and plasma density. Combining the two scaling laws obtained by the authors comes the experimental scaling for the level of magnetic turbulence

δB B ∝ B -3 √ n e , ( 16
)
where n e stands for the line averaged central density.

It follows from Eq. ( 16) that the previous thresholdlike limit on magnetic turbulence for disruption should also be a density limit and that a high magnetic field is beneficial.

FIG. 1 .

 1 FIG.1. The (m, n) modes retained in this study are within the limits imposed by the minimal and maximal q-values defining respectively the lower and upper lines.

Figure 4 FIG. 4 .

 44 Figure4represents the local maximal Chirikov parameter associated to the modes (m, n). More precisely, for each resonant surface ψ mn is plotted the maximal Chirikov parameter s(ψ mn ) = max (m ′ ,n ′ ) s (m,n),(m ′ ,n ′ ) divided by √ ε that the mode (m, n) realizes with some other mode (m ′ , n ′ ). The extent of the local maximal stochasticity zone is represented by an horizontal segment between each couple (ψ mn , ψ m ′ n ′ ). From Figure4, it is clear that

FIG. 5 .FIG. 6 .

 56 FIG. 5. (Left) Four magnetic field line traces ψ(ϕ) for the magnetic microturbulence model with ε = 2 • 10 -5 . (Right) Focus on one of the magnetic field line traces ψ(ϕ) being transiently exploring a narrow ψ-domain below ψ31 ≃ 0.5816. The locations in the ψ-space of some secondary resonances are indicated.

FIG. 7 . 2 FIG. 8 .

 728 FIG. 7. Probability density function (PDF) of the first exit times for two different values of the stochasticity parameter ε in the magnetic microturbulence model: (left) for ε = 3.10 -4 , (right) for ε = 5.10 -4 .

FIG. 9 .

 9 FIG.9. Probability density function of the first exit times for ε = 10 -3 in the magnetic microturbulence model (histogram and plain line continuous representation). In dotted line is superimposed the first passage time distribution of Brownian motion with positive drift given in Eq. (14) having the same mean and variance as the data.

  )

				∆ 33,8 Ψ
				1
				0.8
				0.6
				0.4
				0.2
						Ψ
	0.5	0.6	0.7	0.8	0.9	1

ACKNOWLEDGMENTS

Preliminary discussions with S.S. Abdullaev and D. Constantinescu and assistance on parallel computing by A.F. Lifschitz are gratefully acknowledged. This work was carried out within the framework the French Research Federation for Fusion Studies.