Marc Deneire

published or not. The documents may come

INTRODUCTION

On March 7, 2007, the French General Inspection for General Affairs (IGAS) issued a report concerning a case of over radiation that that led to the death of four patients and to serious injuries in the case of twenty other patients, all of whom are expected to remain seriously handicapped for life. The report forcefully indicated that one of the main reasons for the incident was the use of the English version of a piece of software for which there is no available translation. This is probably only one of the most visible signs of the many dysfunctions that occur in the workplace everyday all over the world, because of an unreasoned and unreasonable use of English in the workplace. In 2005, an American medical equipment company, General Electric Medical System (GEMS), was condemned because of its attempts to impose English as a corporate language in a French subsidiary. Technicians complained that they had to install sophisticated medical equipment using documentation in English which they did not understand.

These incidents raise the question of the importance of English and of language in general in the information age. Whereas former industrial modes of production relied mainly on docile and silent bodies, the informational economy that came into full swing in the early 1990s mainly relies on computer and communication skills (Castells, 2000: 79-100).

Therefore, this "new age" has become a mixed blessing for those who do not master the "right language." While the general standard of living has continued to grow, the relative status of less educated classes in society has been downgraded and their personal dignity deeply affected one way or another. In 1982, John Gumperz already pointed to the structural importance of communication skills in modern society:

The role communicative skills play has thus been radically altered in our society. The ability to manage or adapt to diverse communicative situations has become essential and the ability to interact with people with whom one has no personal acquaintance is crucial to acquiring even a small measure of personal and social control. We have to talk in order to establish our rights and entitlements. When we are at work we often rely on interactive and persuasive skills to get things done. Communicative resources thus form an integral part of an individual's symbolic and social capital, and in our society this form of capital can be every bit as essential as real property resources were once considered to be. (4)(5) The global industrial restructuring of the past ten years fully illustrates what Gumperz was observing in the early 1980s. Manual work, including highly qualified work, has become extremely "cheap" on the job market as most such jobs are being exported off-shore to third world countries. Conversely, technological and communicative skills have been gaining ground on the market. However, only certain forms of knowledge and of communication skills are in a position to constitute symbolic and linguistic capital, that is, the forms of language associated with the "language of authority" (Bourdieu, 1991). In today's business world, that language is increasingly English 1 .

English has indeed become a marketable commodity, one that together with computer skills needs to be acquired through educational institutions that are themselves increasingly part of the market (see Pereiro, this issue). It is part of what in France is called "les industries de la langue," together with translation, the knowledge of expert systems, data banks, artificial intelligence, and other information technologies. It is measured through Cambridge examinations, TOEFL and TOEIC scores or other examinations such as the French CLES (Certification en langues de l'Enseignement Supérieur), each score giving access to further studies and/or specific jobs in international companies.

In this "New World Order," an important question is whether and how much the knowledge of English actually "pays off" For the language economist François Grin, the answer is clearly "yes." In Switzerland for example, the premium for knowing English ranges from 12 to 30 percent (Grin, 2001(Grin, , 2005). Yet, F. Grin insists on the fact that other factors need to be taken into account such as the social distribution of income as well as the symbolic, non-market impact of English on the workplace. It is precisely these other factors that we will try to account for in this study.

THE STUDY

Methodology

About 200 questionnaires were distributed among workers, employees, middle, and upper managers in large, medium-sized, and small companies in the North East of France as well as in public administrations. 96 were completed in full and included in this study. Further, eleven 20 minute interviews allowed us to gain better insight into the workplace and gave us the necessary tools to interpret our results.

Most people who have no competence in English turned our request down. On the basis of the 1999 French census (socio-economic profiles), I rate the population that is likely to have some competence in English at about 44 percent. I divided the sample using 2 criteria for the analysis: socio-economic group and the division between private/public sectors. The variable socio-economic group is based on professional occupation 2 . The lower socioeconomic group (LSEG) includes individuals who occupy the lower rank in the public sector 'Category C', as well as employees and workers who have a yearly renewable contract, and mainly manual workers in the private sector. The higher socio-economic group (HSEG)

includes public servants with managerial functions (Category A) in the public sector and managers, mainly business managers and engineers, in the private sector. Finally, the medium socio-economic group (MSEG) includes employees in the public sector (Category B) and employees, technicians, nurses, etc. in the private sector. Two categories seem to be overrepresented in our sample: the public sector as a whole, which represents 5 million out of a 25 million-people workforce and the higher socio-economic group in the private sector.

However, this distribution reflects the knowledge of English in the global workforce where most upper managers hold university degrees and work internationally and where the public sector employs more "educated" individuals and fewer workers than the private sector (Pouget, 2006). Finally, the nature of our sample explains the male/female distribution (59.4 percent male and 40.6 female) as well as he global distribution where males are more likely to be in managerial positions.

Results

General competence in English

In our sample, 38 % claim to be functional in English (my rephrasing of "courant" or "fluent"). 46 % describe their English as "hesitant," and 16 % say that they know "a few words" in English. Considering our previous estimate concerning the English-knowing population (44 %), this means that about 15 % of the French population is "functional" in English 3 . This figure, based on self-report seems to be in agreement with other studies that empirically tested English proficiency. For example, on the basis of reading comprehension tests in Switzerland, Belgium, and France, Claude Piron rated at ten to fourteen percent the number of people who can comprehend simple texts written in everyday language (personal communication). I will further discuss these figures later in the paper. When looking at the distribution of English proficiency in our sample, we further note major differences in the distribution across categories.

Competence -General

Figure 2: General English proficiency by socio-economic category

When we look at the public sector (columns 1, 3, and 5) we note that proficiency is fairly evenly distributed. It is remarkably high for the lowest group for two reasons: first, because of the general higher level required for public sector workers, and second because this category includes a high number of highly educated underemployed individuals, either because they have not passed the highly competitive examinations to enter the administration or because they have accepted a lower position at a time when the job market for jobs in the public sector is rapidly shrinking (Pouget, 2006). Surprisingly, it is in the highest group (column 5) that proficiency seems to be the lowest. The age factor probably explains part of these results, as well as the fact that English is not usually used in the workplace. As a result, managers progressively "forget" the English they learned during their school years. 4 Differences are much more striking in the private sector where only a small part of the lowest category, mainly manual workers, is functional in English whereas the highest category displays medium or high proficiency, with the medium category in-between. Results seem to display an "English divide" in the private sector. This is a point we will further elaborate in our discussion.

English for Specific Purposes

Respondents were asked to rate their level of proficiency "in their own field" (dans votre domaine de spécialité). Previous studies show that the longer individuals learn a language, the more they remember it. For example, ten years after leaving school, 48 percent of the people who did not obtain a high-school diploma say that they don't remember anything of the language they learned; the figure is 31 percent for those with a high school diploma, and only 10 percent for those with a degree from higher education.

Competence -ESP

The demand for English Ambient discourse sometimes gives the impression that English proficiency is absolutely necessary to find a job in France. This position now also seems to be shared by educational institutions which were typically resistant to the "invasion of English" only a few years ago (see Pereiro, this issue). To investigate the "real" demand for English in the workplace, we In short, English is only required for international upper managers and for employees working in the tourist industry. Interestingly, English is not required for the many employees in the import/export sector when they do not have contacts with customers. Of course, job ads do not tell us who was actually hired. Advertisers may expect students from management schools to know English, and therefore may not feel the need to include it in their ads. At the same time, the mention "English desired" indicates that proficiency in English is not always available for other jobs. Thus the figures may be underrated for upper managers, but slightly overrated for other categories.

Discussion

Material from interviews, observation, and reactions in the press allow us to qualify these quantitative results. In our discussion, we will focus on people's responses concerning English proficiency, intercultural communication, and the anxiety that the introduction of English sometimes causes.

English proficiency

It is rather difficult to interpret competence in English when are based on self-report 5 .

For example, a respondent who claimed to be "fluent" in all of the skills admitted that her English was not good enough to function in an English-speaking country, but, she added: "the English I know enables me to communicate with foreigners within the limits of what I am required to know in my professional environment." Another person claimed to be fluent on the basis of high TOEIC results, but said that she would not be able to function in a meeting conducted in English. She also found articles from The Economist too difficult for her.

Conversely, a technical writer who reads specialized literature in English and writes instructions and users manuals said that she was "hesitant" because she did not speak like a British or an American "native speaker." Thus, respondents assess their level of proficiency on the basis of their needs, not according to academic or other external criteria. This may explain why in our sample, 69 percent were satisfied with their level of English, while only 19 percent were "more or less" satisfied, and only 12 percent not satisfied at all. Most respondents to the questionnaire thought that "it would be a good idea" to improve their English, but in the interviews, they came over as unwilling to spend much time and effort to doing so. In our sample, we did not find any negative reactions toward the spread of English.

Rather, people saw English as a tool to reach markets that would otherwise be unreachable.

English is seen as an important capital that companies need to acquire, especially at a time when intelligence and knowledge are perceived as essential for competitiveness. However, the perceived need for the acquisition of English was not very strong, either for individuals, or for the management which often has "other priorities." Beyond our sample, this seems to be the case in other parts of France. Indeed, in a survey conducted by the Paris Chamber of Commerce and Industry among 500 French companies (Foly and Maratier-Declety: 2006), 76 percent claimed that language was not a barrier for their activities and that there was therefore no need to worry about the "language question," either because they had bilingual individuals among their employees, or because free-lance translators were readily available.

In most companies, especially in small-and-medium-sized companies, this principle of realism applies. Most internal communication takes place in French (94 percent according the CCIP survey), and communication in English lies on the shoulders of a few individuals and/or translators. However, the situation is changing rapidly in large multinational companies that are increasingly adopting English following mergers and takeovers. For example, when the German multinational Hoechst (pharmacy, chemistry) merged with its French counterpart Rhône-Poulenc to form a new company (Aventis), the new company adopted English as a company language, even though both companies had actively promoted their respective languages in the past (Truchot, 2000). When Alcatel's CEO Serge Tchuruk tried to take over the American company, he readily pushed English to make it a corporate language, even in his Paris headquarters, in an effort to attract the favors of his American counterparts, claiming that: "We are not really a French company anymore" (Business Week, June 4, 2001). The idea that English has become the only international business language is now widely accepted in business circles, even though it often causes uproar in the political world 6 . In a study led by ETS among 26 French companies, 17 declared English as their corporate language and 6 English and French. Yet, the study indicates that the level of English remains "globally unsatisfactory." Indeed, the adoption of English as a corporate language in terms of image does not change much to the proficiency level of most employees and workers 7 . It does seem to create a language divide between an English-speaking management and a French-speaking workforce. In our interviews, we detected two types of resistance to the penetration of English in large companies: one linked to the imposition of an "American management model" and one that relates to the anxiety experienced by people whose competence is suddenly questioned.

Intercultural communication and the American model

Most scholars agree that there is no such thing as a "cultureless business culture." (Gimenez, 2001;Nickerson, 2005) Even when companies from closely related cultures start working together in English, accommodation to the other culture may lead to more misunderstanding and take a longer time than the acquisition of English (Louhiala-Salminen et al., 2005). When American companies settle in France, there seems to be little mutual accommodation though, the expectation being that the French workforce will adapt to American models of management. Large-scale studies conducted in France indicate that English plays an important role in the process. For example, many legal departments consider that the English version of a text must serve as a reference, even when it was first written in France (Gratiant, 2006). In our study, managers at a large company pointed to the fact that "English never comes alone"; it often comes with the introduction of management software (SAP and SCM: Systems, Applications, Products; Supply Management Chain) and high-tech IT architectures that profoundly modify management structures. In this way, monochronic approaches to management have been introduced through highly centralised information systems to overcome the problem of duplication of resources and information retention.

However, when introduced in a culture that is based on polychronic modes of operation and relations of trust, most of the crucial information may never appear on the system at all.

Indeed, many people try to avoid using a system which they do not fully understand. In this way the introduction of communication systems and of English that was supposed to ease communication and make relations smoother sometimes ended up increasing miscommunication and leading to conflict.

Anxiety and legal issues

The discrepancy between corporate language policy and real competence may be dysfunctional, especially when sensitive technology is involved. In our introduction we mentioned the extreme case that led to four casualties and twenty other victims. This follows another high profile case involving GEMS (General Electrics Medical Systems) near Paris.

Trade-unions reacted against the imposition of English on technicians, who, as a result of the lack of translated documentation, did not understand the instructions accompanying the medical equipment they were installing (scanners, mammography devices, etc.) They went to court on June 2004 and won their case in January 2005 on the basis of an article of the 1994 Toubon law that stipulates that "All documents that include obligations for workers or instructions that workers need to know for the execution of their job need to be made available in French." 8 As a result, GE was forced (1) to make the (existing) French version of their software (MS Office) available to their workers, (2) to provide French versions for all documents related to training, hygiene and safety, and (3) to translate all documents related to the products that GEMS produce. GEMS appealed the case, but the Court of Appeals confirmed the condemnation, ordering the company to pay 580,000 euros and 20,000 euros for any document that remaining untranslated beyond three months following the trial. Similar Many older managers put considerable effort into the learning of English, but have the impression that their English will never be good enough to "compete" with their younger colleagues and with "native speakers" of English. This leads to considerable levels of linguistic insecurity. As a result, they often prefer to remain silent, which often leads to frustration and resistance, 9 and to a waste of unique experience and expertise for the company. Unfortunately this situation is not likely to change in the near future. Indeed, in the words of an ELT instructor:

… companies tend to hire managers and high level technicians who know English already, and as a result, they cut costs related to ELT. In some cases, classes that used to be taken during working hours now need to be taken outside working hours. This is more and more the case following the introduction of a shorter work week, which means that many people and workers no longer attend. Finally, most companies have suppressed any classes that yield no immediate results. In the past, English was not just seen as the corporate language, but also as a tool for travelling and a passport to international communication. It represented a certain status and cultural enrichment ….

[Today],

immediate acquisition and application of skills is required. … As English is integrated into technical training, it has become a means rather than an end. As a result, the ELT instructor also needs to be a technician, and technicians who know no English at all are progressively disqualified … and later fired.

Conclusions and future directions.

Our study shows that the introduction and development of English in the workplace is accepted by most French managers and employees if it happens in a "reasonable" way, that is, if it is introduced to ease international communication and allows the expansion or survival of their company, and if new forms of English are allowed to develop internally for specific purposes. On the other hand, people resist English when it is imposed from above, especially when and where the use of French is more efficient, that is, in internal, intra-national situations, and when the use of English displaces or silences a well-qualified workforce. Both managers and employees also resist it when it is packaged with Anglo-American modes of corporate governance and culture. Our results demonstrate that these fears are justified.

Indeed, both quantitative and qualitative data illustrate an "English divide," in the private sector at least, along the lines of other types of fracture due to the change of paradigm in modes of production from an industrial to an information-based society where the mastery of language is becoming increasingly more important. Indeed, this divide contributes to a widening gap between the educated and the less educated, the computer literate and the computer-illiterate, between the young and the old, and between lower and upper socioeconomic groups 10 . In short it creates linguistic inequality.

The solution might come from globalisation, or more precisely, from successful glocalization, that is, from the search for a harmonious articulation between global and local needs for English. Indeed, as more and more off-shore outsourcing is taking place, both in the US and in Europe, Europeans are likely to communicate in English with non-native speakers more and more, a situation which they find "normal" and in which they feel much more comfortable as long as they are allowed to develop a language that fits their needs. If management allows this to happen, more multi-lingual, multi-cultural modes of operation may take place in the future. It is also possible that more products, including IT technology will be engineered by people belonging to different cultures, thereby making them more culturesensitive.

As a result, users might develop several concentric circles of influence in developing their English language competencies.

The first level would consist of exchanges with international colleagues in English. As a result, societies of NNS English speakers would create new optimized language entities adapted to their specific needs. English would constantly undergo modifications based on ever-changing and relatively unpredictable forms of interconnectedness. Although documents from headquarters may be influenced by NS practices and international constraints of legibility, local exchanges often deviate from what we now consider as "good English."

Cross-cultural structures and communication patterns would be adopted to meet the corporate needs of best business practices. As the goal is to respect cultural differences while maintaining a coherent core of back-office practices to optimize corporate resources and encourage internal growth, an inevitable development of satellite-system Englishes would probably be observed. Each group would decide on its local priorities and synergies while adhering to the common core through "globish" jargon and globalised business practices.

Official documents from corporate headquarters would act as a template.

A second circle would be formed by "laptop managers" who often travel and would be influenced by internal satellite groups as well as clients, customers and partners. This would create a combined NS/NNS group where higher levels of competency would be expected, especially in their adaptability to cross-cultural features. In the near future, adaptability to Asian-English accents and discourse in emerging markets such as China will add to the complexity of globalization.

Finally, a third circle would be formed by language and communication specialists who are in charge of guaranteeing the flow of information, for example by providing templates when needed, translation whenever necessary, and by proposing a terminology that is consistent with the industry at large.

This may be an optimistic view, but at a time when many American companies are bought out or merge with European and Asian companies, and in a period when many countries strongly resent America's unilateral modes of operation, it will probably be the market that determines not only the language that predominates, but also the many forms that it takes.

Figure 1 :

 1 Figure 1: Sampling distribution by socio-economic group and by sector

Figure 3 :

 3 Figure 3 : Competence in English for specific purposes per socio-economic group

Figure 4 :

 4 Figure 4: English proficiency and education (BEP = Vocational middle-school diploma;

  looked at job advertisements in the newspaper Le Monde as well as on the site of the French National Employment Agency (ANPE: Agence Nationale pour l'Emploi). The job ads section of Le Monde contains 2 subsections: one for jobs in the public sector and one for jobs in the private sector. Most jobs are upper-management jobs. Out of the 171 jobs advertised in the course of one month, 46 percent in the private sector required or "desired" proficiency in English. These include the ads written in English. However, in the public sector, only 10.3 percent required some knowledge of English. Here are the percentages we found on the site of the ANPE for other jobs on the basis of at least 20 ads per category Business upper manager(Ingénieur commercial)

  demands have been introduced at Alcatel, Euro-Disney, AXA (insurance company), and Europ Assistance among others.Beyond the obvious safety issue, trade unions argue in favour of "the right to work in French in France"(www.voxlatina.com, March 2, 2006) and point to the "discriminating effects of the supremacy of English in the workplace"(AEF, February 9, 2007). The comments around the GEMS case help us understand the concrete situation of people who suddenly feel out of place because of the imposition of English. For example, a 50 year-old documentation officer suddenly had to function in English when her company was bought out by GE. She reacts: "I know my job perfectly well, but I cannot express myself. It's as if I were gagged. The words, I need to decipher them … It makes me mad … the anxiety and the humiliation that many workers and employees endure because of the dictatorship of one language over the other." She also denounces "all these middle management smart alecs (petits chefs) who pretend to understand what is being said in meetings with the general management. But when they are asked what they have understood, they never come up with the same version."(Le Parisien: June 4, 2005). This reaction is fairly common among highly qualified middle and upper managers who say that they feel anxious, humiliated, incompetent, and tongue-tied following the introduction of English as a corporate language in their company.

It is important to note that not all sectors of the economy are being globalized, that is, sectors "with the capacity to work as a unit in real time, or chosen time, on a planetary scale" (Castells,

2000: 101-103). In fact, most governments make considerable efforts to protect some sectors -such as transportation, postal services,

SeeBredoux (2006) as well as the website of the "Observatoire des inégalités" for illustrations of these inequalities (http://www.inegalites.fr/). Studies in intercultural communication show that middle managers also express a high level of dissatisfaction concerning the power distance that this divide contributes to creating(Castel et al., 2007).

educationagainst globalization. Yet, they seem to be losing ground as global institutions such the European Union tend to qualify as "protectionist" many of the sectors that states consider to be of "national interest."

2 This nomenclature used by the National Institute for Statistics and Economic Studies (INSEE) seemed to be both too detailed and too imprecise to be adopted here. For example, elementary school teachers and secondary school teachers belong to two different categories ("catégories intermediaires" and "cadres") even though they require the same level of education and are very similar in terms of job profiles.

3 Much has been written concerning the poor level of English proficiency in France, when compared to other European countries (see Pereiro, this issue). Yet, one should be careful not to overrate proficiency in other countries. Indeed, even in Scandinavian countries, people may be excluded from meetings because of their lack of communicative proficiency in English (Louhiala-Salminen et al., 2005: 417). Similarly, Erling and Walton (2007: 36) note that some employees "avoid situations where English is used because of their poor level in English" in Germany. 4 All respondents in this category claim to have learned some English in school.

5 Some studies have tried to be more specific and asked questions about particular skills, but these are also based on self-report. In a 1996 survey by the National Institute of Statistics and Economic Studies (INSEE), 74.6 percent of the participants responded that they would not be able to participate in a conversation in English, 63.2 percent that they could not read a newspaper, 63.1 that they could not write a letter, and 69.7 percent that they could not hold a conversation over the phone. 6 In March 2006, Ernest-Antoine Seillière, then President of the French Mouvement des Entreprises de France (MEDEF) and leader of the European business lobby UNICE, addressed the 25 EU leaders in English. When President Chirac interrupted him to ask him why he was speaking English, Seillère explained: "I'm going to speak in English because that is the language of business." Chirac then stormed out of the room, followed by his team of 3 ministers and returned when E.-A. Seillère had finished speaking (Times, March 2006).

7 A 2003 OFEM report (Observatoire de la Formation de l'Emploi et des Métiers) indicates that English is the working language of 7 percent of French companies (based on a 501 companies survey).

8 Note that the 1994 Toubon law does not impose the use of French in the workplace, but it stipulates that a number of documents need to be available in French; these include work contracts, internal regulations, hygiene and safety standards, collective conventions and agreements, and all documents related to employees' duties and obligations). These obligations need to be complemented with a 1998 EU machines directive that enjoins 20 manufacturers to make instructions available in the language of the countries where machines are used (Foulon, 2005). 9 In one of the rare studies on the importance of silence in organizations, Milliken and Morrison (2003) analyse how failures to speak up often affect the lives of individuals and may lead to dysfunctional organizations. They note that: "Over time, the feeling of being unable to speak up about issues and concerns may result in a sense of helplessness as well as reduced job satisfaction, turnover, and other more long-lasting personal consequences."


Estelle Guyez
email: c.guyez@warwick.ac.uk

Jan-Bert Flor

Emil Hopfinger

Mixing by coherent vortices in a two-layer stratified fluid

Keywords: Mixing, stratification, laminar-turbulent

Here we present results of experiments on mixing at a density interface conducted in a Taylor-Couette device at Reynolds numbers such that the Taylor vortices are laminar to chaotic. The Laser Induced Fluorescence (LIF) technique has been used to obtain quantitative measurements of the spatio-temporal evolution of the density. It is shown that the entrainment rate and mixing efficiency are an order of magnitude smaller than in the turbulent vortex regime and the established entrainment laws do not apply in this laminar-chaotic vortex regime. The vertical transport in the homogeneous layers is smaller than the interfacial flux and several density interfaces appear at different heights. The interactions between the interfaces control the subsequent density flux, and hence the mixing efficiency.

Résumé :

Nous présentons ici les résultats d'expériences concernant le mélange au niveau d'une interface de densité, expériences réalisées au sein d'un dispositif Taylor-Couette à des nombres de Reynolds tels que les vortex de Taylor sont laminaires à chaotic. La technique de fluorescence induite par laser (LIF) a été utilisée afin d'obtenir des mesures quantitatives de l'évolution spatio-temporelle de la densité. On montre que la vitesse d'entrainement et l'efficacité du mélange sont un ordre de grandeur plus petit qu'en régime turbulent et que les lois d'entrainement établis ne s'appliquent plus dans ces régimes de vortex laminaire-chaotic. Le transport vertical dans les couches homogènes est plus faible que le flux interfacial, d'où l'apparition de plusieurs interfaces de densité à différentes hauteurs. Leurs interactions mutuelles modifient l'intensité du flux de densité, et donc l'efficacité du mélange.

Introduction

Mixing in stratified fluids has received considerable attention because of its importance in geophysical flows. These flows are generally turbulent. A fairly large number of experimental and theoretical studies have considered different aspects of turbulent mixing in stratified flows. [START_REF] Phillips | Turbulence in a strongly stratified fluid-is it unstable ?[END_REF] and [START_REF] Posmentier | The generation of salinity finestructure by vertical diffusion[END_REF] established a model predicting the variations of the mixing efficiency, that is expressed by the flux Richardson number Ri f , as a function of the buoyancy gradient, expressed in non-dimensional form by the Richardson number Ri o . The buoyancy flux first increases with buoyancy gradient and then, beyond a critical of Ri o the buoyancy gradient limits the vertical transport of density. This leads to a decrease in mixing efficiency and the formation of stronger density interfaces separated by homogeneous layers. [START_REF] Linden | Mixing in stratified fluids[END_REF], [START_REF] Hopfinger | On mixing across an interface in stably stratified fluid[END_REF] and [START_REF] Billant | Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid[END_REF] have found good agreement with this model predictions by studying mixing across a density interface by turbulence produced externally to the interface. A modified version of the Phillips-Posmentier model proposed by [START_REF] Balmforth | Dynamics of interfaces and layers in a stratified turbulent fluid[END_REF] suggests that the mixing efficiency should increase at large Ri o number to prevent interface sharpening without limit. The main feature of the [START_REF] Balmforth | Dynamics of interfaces and layers in a stratified turbulent fluid[END_REF] model is the assumption that the size of the turbulent vortices involved in mixing is not fixed; at large Ri0 energy is injected by the mixing device at smaller scales i.e. the eddy turnover time decreases at large Ri o . Experimental support of this model is provided by [START_REF] Guyez | Turbulent mixing at a stable density interface: the variation of the buoyancy flux-gradient relation J. Fluid Mech[END_REF], showing that the mixing efficiency indeed increases at very large Richardson number. The experiments also show a large variability in mixing efficiency for the same experimental conditions. Here we consider mixing at a density interface by low Reynolds number laminar-chaotic Taylor vortices. This regime is of interest as it can occur in many industrial processes. Most of the general purpose pharmaceutical companies and biologists face mixing problems for which turbulent flows can't be used as it may destroy the activated molecules (DNA, proteins, bloods ...). It is shown here that although the general mixing behaviour appears similar to that of turbulent Taylor vortices it is quantitatively quite different. The experimental setup is presented in Section 2. The results containing the spatio-temporal density evolution, the entrainment rate and the mixing efficiency are analysed in Section 3.

Experimental conditions

Experiments were conducted in a Taylor-Couette device where only the inner cylinder rotates. This device was previously used by [START_REF] Caton | Stability and bifurcations in stratified Taylor-Couette flow[END_REF] to study flow regimes in presence of a linear stratification. The radii of the inner and outer cylinders are respectively a = 4 cm and b = 5 cm giving a gap d = b -a = 1 cm. The Reynolds number based on the rotation speed Ω of the inner cylinder is Re = Ωad/ν, where ν is the fluid viscosity. In the experiments 364 < Re < 1025. The fluid depth in the gap is h = 56 cm. In order to reduce optical distortions, the device is placed inside a rectangular tank filled with water. The annular gap was filled with salt water below and fresh water above, with the interface at mid-height h/2. The initial interface thickness is closed to 20 mm and the buoyancy jump is:

∆B = g∆ρ/ρ, ( 1
)
where g is the gravitational acceleration, ∆ρ the difference of density between the layers and ρ the mean density of the flow. A sketch of a typical experiment is presented in Figure 1.

The spatio-temporal evolution of the density was measured by means of the Laser Induced Fluorescence (LIF) technique using a 2 mm thick, vertical laser light sheet of an adequate power.

The light emitted by a fluorescent dye (rhodamine 6G) is proportional to the dye concentration. Hence if a known concentration of dye is mixed with the salted water before filling the gap, then the intensity of the fluorescence will also be proportional to the concentration of salt. A camera (Jai CV-M4+CL, B&W, 1024*1380 pixels) records the intensity of the fluorescence in the space defined by the intersection of the laser plane and the fluid. It is fitted with a high-pass filter to only keep the fluorescent light. For all the experiments, the Richardson number Ri o can be measured at any density interface and at any instant of time:

Ri o = ∆B(t)d ϑ (t) U ϑ 2 (2)
d ϑ is the vortex height and U ϑ is the rms velocity of the Taylor vortices. Ri o evolves in time as flow regimes change. It is easily measured in these laminar-chaotic regimes as the separations between the vortices are highlighted by buoyancy gradients. For U ϑ , we use the data obtained by [START_REF] Wereley | Spatio-temporal character of non-wavy and wavy Taylor-Couette flow[END_REF] obtained in a Taylor-Couette flow of similar size. At each instant of time the values of Ri f and Ri 0 are then calculated: E is non-dimensional form of the entrainment rate U e , evaluated at a density interface localised at the height z, divided by U ϑ . The entrainment velocity is given by:

Ri f (t) = d ϑ (t) ∆B(t) U e (t) U ϑ 3 = E(t) Ri o (t) (3)
U e (z, t) = 1 ∆ρ(t) h z ∂ρ(z, t) ∂t dz (4)
3 Results and discussion:

Spatio-temporal evolution of the vertical density profile

The spatio-temporal evolution of the density depends on Reynolds number and on the initial buoyancy jump ∆B i . Density interfaces appear at all levels between the vortices at low and moderate Reynolds number (Re < 1000). The intensity of the density jump decreases with distance from the main interface. During a typical experiment, mixing occurs at different stages.

First, the main interface seem to control the mixing and has a stratification much higher (about five times) than that of the other interfaces. All secondary interfaces are issued from splitting of the main interface and finally occupy the whole fluid depth. The birth of a new interface is periodic and starts with a wavy excitation of the interfaces. When the main interface breaks, the mixing seems to be more disorganised as a large number of density interfaces appear within a height of about 10 cm. In this part of the flow, the interface positions change very rapidly (due to splitting and fusion phenomena) whereas, on the outside, the former interfaces are not disturbed. Gradually the flow reaches a more stable state. The vertical density profile is then similar to a stair case of steps of different heights. Moreover, it is observed that, for the same Reynolds number, a reduction of the initial buoyancy jump leads to the destruction of the main interface as soon as the cylinder begins to rotate. Between all vortices a stratified interface appears. These observations lead to some initial comments. When mixing occurs in the laminar-chaotic regime, the entrainment speed (at which the fluid is taken through a density gradient) is of the same order as the transfer speed (at which fluid is exchanged between homogeneous vortices). Thus the initial homogenous layers above and below the interface are not able to remain homogeneous, contrary to what is observed in the turbulent regimes [START_REF] Guyez | Turbulent mixing at a stable density interface: the variation of the buoyancy flux-gradient relation J. Fluid Mech[END_REF]). Nevertheless, when the initial interfacial buoyancy jump is large, the main interface is maintained until the stratification is small enough to be destroyed. In all the cases, the two-layer experiments and linearly stratified experiments lead to the formation of similar density profiles.

Complete mixing then takes a very long time as low density gradients result in very low density fluxes. -n emerges. In Figure 2a the evolution of the flux Richardson number as a function of the Ri o is shown. For all Reynolds numbers Re < 1025, these curves exhibit the general bell shape form in accordance with the Phillips-Posmentier model. However, the detailed behaviour differs from that obtained with turbulent vortices; the fall-off beyond maximum mixing efficiency is more rapid (see figure 2a) and there is a strong dependency on Reynolds number. There is no increase in mixing efficiency at large Ri o primarily because the Reynolds number based on an eddy scale smaller than d ϑ is too small. Maximum mixing efficiency is reached when the Richardson number approaches 30. On the spatio-temporal images, maximum Ri f corresponds to the moment when the stratification of the main interfaces decrease very rapidly. The maximum of Ri f for Re = 1025 is still about ten times smaller than in the turbulent regime [START_REF] Guyez | Turbulent mixing at a stable density interface: the variation of the buoyancy flux-gradient relation J. Fluid Mech[END_REF]). A possible reason is the small scales disappear progressively in the laminar regimes. In order to produce mixing in the laminar-chaotic regime the vortices have to displace buoyant fluid elements over a height equal to their seize. The buoyancy flux is then limited by the kinetic energy available on the eddy scale. This is much less than in the turbulent case. Also the homogenisation inside the vortices due to diffusion lasts longer as there no small scales and hence less stretching of the fluid.

Entrainment rate and mixing efficiency

Another interesting result appears when we compare the curves of mixing efficiency measured for different interfaces in the same experiment (Figure 3). The value of maximum mixing efficiency changes with the interfaces considered and is highest for the interface with largest buoyancy jump. This unexpected behaviour can be linked to the other observation. When representing the time evolution of the stratification, it appears that the stratification of the main interface does not decrease randomly. It tries to maintain its value during a multiple of the rotation speed of the inner cylinder (Figure 3). Here, the dips between 6 and 4 rad/s correspond to a change of interface. Indeed, the higher value of the stratification is linked to different interfaces as the mixing time increases. The coupling of the main interface and the rotation speed is only observed when several density interfaces are present at the same time, namely at low Reynolds number. The stratification of the other interfaces is not coupled to the rotation speed. It is supposed that the density flux across these other interfaces adjusts itself in such a way that the stratification of the main interface can be kept at a specified value. The coupling between the stratification and the rotation speed is then a parameter that should be taken into account for the determination of the mixing efficiency. Further investigation are needed to better understand what are the mechanisms which lead to this coupling.

Conclusion:

Below a certain Reynolds number when eddies are laminar to chaotic, mixing is strongly Reynolds number dependent. The mixing efficiency is considerably less than in the turbulent regime because the energy available for mixing is much less. Furthermore, the absence of small scale structures increases the time needed for the homogenisation inside the vortices. Several density interfaces develop above and below the main interface as the exchange of fluid between eddies is reduced [START_REF] Akonur | Three-dimensional velocity field for wavy Taylor-Couette flow[END_REF]). This is demonstrated by the spatio-temporal evolution of the density that shows a stair case density profile. The same density structure is obtained when the initial density stratification is linear [START_REF] Caton | Stability and bifurcations in stratified Taylor-Couette flow[END_REF]). An interesting result is that the Ri f -Ri o relation is affected by the presence of surrounding density interfaces. Maximum of efficiency is measured at the interface of largest buoyancy jump; the others exhibit lower value of Ri f -max . This behaviour seems to be a consequence of the adjustment of the secondary interfaces to allow a coupling between the stratification of the main interface and the rotation speed of the inner cylinder. The complete destruction of the stratification is then very long which is consistent with Janiaud et al. (2000) who found that in these regimes, the effective diffusion coefficient is very small.

Figure 1 :

 1 Figure 1: Left : Sketch of the experimental setup -Right : Spatio-temporal evolution of the density (blue for the fresh water and red for the salted water) at low Reynolds number. The time evolution t is measured in term of the energy W used to rotate the cylinder during the corresponding time : W = M Ωt, M being the torque applied by the cylinder on the water.

Figure 2 :

 2 Figure 2: Variation of the Entainment speed E and Richardson flux number Ri f as a function of the Global Richardson number Ri o , measured at most stratified interfaces during experiments at Re = 455, •Re = 727 and Re = 1025. The initial buoyancy jump is ∆B i = 0.35 kg/m 3 .

Figure 3 :

 3 Figure 3: Left : Evolution of the Richardson flux number as a function of the Richardson number, measured at three different interfaces during an experiment at Re = 727 and ∆B i = 0.35 kg/m 3 . Right : Stratification of the main interface versus the energy used W (= M Ωt).

ème Congrès Français de Mécanique Grenoble, 27-31 août 2007

Acknowledgements

The authors acknowledge the help of Pierre Carecchio and Joseph Virone for their technical assistance. E. Guyez acknowledges a DGA fellowship. This work has been supported by SHOM grant n o : 02.87.028.00.470.29.25.