Jean-Michel Courtault

Research in Economics and Management in France: A bibliometric study using the h-index 1

Keywords: JEL classification: A140 h-index, g-index, publication activity

This paper presents an application of new measures of research excellence, namely Hirsch's index (2005) and derived indexes. It gives a ranking of French departments of Economics, departments of Management and Business schools based on the quality of the academic environment offered by these institutions using these measures. It argues that, since the bulk of the research is done by a very small number of researchers, a greater concentration of the best researchers seems necessary for France to achieve international visibility in Economics and Management.

Introduction

A major concern of many countries is now to reform the universities and the research system. The economic success of the USA is attributed to its scientific supremacy. In order to direct reforms one needs tools that evaluate and measure the performances of institutions and researchers. Evaluation by peers is the traditional tool for evaluation but it is very costly. Recently, Great Britain through its

Research Excellence Framework (replacing the older Research Assessment Exercise

This paper presents an application of new measures of research excellence, namely Hirsch index and its derived indexes. According to [START_REF] Hirsch | An index to quantify an individual's scientific output[END_REF] a scientist has an index h if h of his p papers have received at least h citations each and his other p-h papers have received at most h citations each. These indexes allow us to measure individuals' as well as institutions' research achievements.

In particular, this paper gives a ranking of French departments of Economics, departments of Management and Business schools based on the quality of the academic environment offered by these institutions.

) supplements peer reviewing by quantitative information based on traditional bibliometric indexes such as the number of citations or the number of papers (Cf. J.-F. [START_REF] Méla | La Grande-Bretagne réévalue sa réforme de l'évaluation[END_REF]).

France is currently reforming its research and higher education system by imposing, among other things, minimum publication standards. This reform was initiated by two problems. In the innovation field, it is obvious one can only notice the failure of the policies of large projects in the Computer Science and the Biotechnologies areas. This failure is attributed to the top-down approach which is too centralized and which is followed in France. Although this approach has, in some other fields proved beneficial, such as Nuclear Plants, TGV, Airbus, Ariane… it is necessary for certain industries to have a bottom-up approach which is more decentralized (Cf. B. Coriat (2008)). In pure research, French scientists are much less cited than British and German ones (without taking the Americans into account): it is sufficient to look at the number of researchers highly cited per country on the ISI Web of Knowledge (158 for France, 258 for Japan, 259 for Germany, 376 for England, 4035 for the USA). If the number of citations measures the visibility or the originality of French research, it seems necessary to reform the system in such a way to incite the best researchers to stay in France or to come to France.

The Pécresse Act on governance of French universities, issued in august 2007, aims at increasing competition between research departments. It is therefore important to compute an empirical estimate of the quality of French research departments, since the incentives for academics to move from one department to a better one will increase. The incentives will be monetary bonuses and reductions in teaching loads. We concentrate on the possible evolution of research in Economics and Management.

A ranking of the best French business schools is periodically proposed by the business press but no such ranking exists for Universities. In those rankings, one takes into account the academic criteria (number of teachers with a PhD, quality of publications as measured by CNRS, etc.). The ranking takes into account also non-academic criteria (such as wage level for the first job) which are more interesting for students. We propose a ranking of French departments of Economics, departments of Management and Business schools based solely on the quality of the academic environment offered by these institutions as measured by the h-index of [START_REF] Hirsch | An index to quantify an individual's scientific output[END_REF]. Since there is a strong correlation between the academic ranking and the final ranking of the business schools, the absence of non academic criteria should not bias our own ranking. Moreover, non academic criteria, notably wages, are known to be overestimated.

We also aim to measure the influence of competition on research in business schools. This could give an idea of the possible evolution of research in Universities' departments of Economics and Management. There is at the moment a great reluctance among French Universities to accept the mere idea of competition. We argue that the already existing competition among Business Schools has boosted research.

Theoretical analysis

The h-index

Classic measures of the quantity and quality of an author's research are the number of papers and the total number of citations. However, an author may have numerous papers not cited, or one single paper often cited. Clearly, a better measurement of the cumulative impact of research output is needed. Jorge E. [START_REF] Hirsch | An index to quantify an individual's scientific output[END_REF] has proposed a new measure of scientific achievement. A scientist has an index h if h of his p papers have received at least h citations each and his other p-h papers have received at most h citations each. [START_REF] Glänzel | On the H-index: A mathematical approach to a new measure of publication activity and citation impact[END_REF] introduced the following alternative definition of the h-index as a solution to a maximization problem:

{ } { } j X j M a x h j p j ≥ = ∈ : , . . . , 2 , 1 (1)
where X j An optimal solution to (1) is such that X is the number of citations of the j-th most cited paper. If (1) has no solution we set h = 0.

h ≥ h and X h+1 < h + 1. Since X h
The h-index is a compound measure of productivity and quality, as measured by the number of citations received by the published papers. We assume that paper's citations are used as proxies for the paper's quality. Indeed, there is a good correlation between the prestige of a review and its impact factor (if a review is prestigious then more people will read it and cite extensively the papers published in it). If an author has a lot of citations it is presumably because his papers are published in prestigious journals. He may have also published high quality papers in lesser journals to shorten the publication delay or because of careless refereeing. Also we may assume that the number of publications is a proxy for scientific quality of a researcher. The h-index is neither entirely dependent on the number of citations nor on the number of publications. It synthesizes both measures as one needs to publish many papers with many citations each to have a high h-index. The higher the index, the greater the number of significant papers published by an author and the higher the significance of the papers. The underlying motivation being that none of the two measures taken is a decreasing sequence this means that h papers are cited at least h times each and p-h papers are cited at most h times (strictly less than h + 1). Hence, both definitions are equivalent.

separately can be considered reliable.

We may, alternately use [START_REF] Egghe | Theory and practice of the g-index[END_REF] g-index. It's a variation of the h-index where cumulative citations are used instead of citations alone. Given a set of articles ranked in decreasing order of the number of citations, the g-index is the largest number such that the top g articles received, together,

at least g 2 { }       ≥ = ∑ = ∈ j i i p j j X j M a x g 1 2 , . . . , 2 , 1 :
citations. The g-index is the solution to (2):

(2) Both the g-index and the h-index give all the weight to the most cited papers than to the lesser papers. However the g-index better takes into account the citation scores of the top articles than hindex so that the quality of a researcher depends mostly on his best papers.

One of the main attractions of the index is its relative robustness. That is the h-index does not vary greatly if the number of documents included (e.g. if we exclude books or book chapters and consider articles only or if we exclude older papers) changes significantly. Neither the h-index increases significantly if the total number of citations increases. In particular, the h-index does not depend on the less interesting (i.e. cited) papers an author has published and, once a paper has reached h citations, the extra number of citations does not increase the h-index. In particular, this means that the h-index does not give undue weight to review papers.

If a researcher having an h-index with value h tries to increase his/her index to h+1, he/she will often need to write more than one paper with h+1 citations (since he/she may not have had already h papers with h+1 citations). If one considers also that most papers ceased to be cited anymore after a relatively short lapse of time and that for any author the distribution of citations of his papers is very unequal, then a significant effort (both with respect to the quantity of papers published and their quality as measured by the number of citations received) has to be produced to increase one's h index.

Robustness may have two interpretations. First it means that a single paper cannot increase an index of scientific achievement by a big amount. Indeed, a single paper is not able to increase the h-index by more than 1 even if it has a lot of citations (even if the number of citations is greater than the initial h +1). In contrast, if we take the total number of citations as the index of scientific achievement, it is often the case that a single paper is able to increase the index by a considerable amount. Second, robustness means that, in order to increase an index, it is necessary to add a significant number of papers significantly cited. A sufficient condition to increase the h-index is that the number of the new papers is at least as great as the initial h+1 and the number of citations of at least h+1 of them is cited h+1 times. In contrast, if we take the number of publications as an index of scientific achievement, then it is possible to increase the index by the addition of a single paper, be it cited or not. [START_REF] Courtault | On the robustness of the h-index: a mathematical approach[END_REF] show that the h-index has an upper limit. As an author increases the number of his scientific production, the increase of his h-index is limited by the h-index of the new papers. That property is not shared by other indexes of scientific impact as the g-index of [START_REF] Egghe | Theory and practice of the g-index[END_REF]. As an author increases the number of his scientific production the increase of his h-index is limited by the h-index of the new papers. If we consider a set of 3 papers A with X 1 A = 4, X 2 A = 3 and X 3 A = 1 and the set of one paper B with X 1 B A significant number of papers significantly cited have to be published to increase the h-index.

= 8 then g(AUB) = 4 > g(A) + g(B) = 2 + 1 whereas the increase in the h-index is at most equal to the h-index of the new set of papers B,

h(AUB) = 3 ≤ h(A) + h(B) = 2 + 1.
This feature is particularly desirable for Social Sciences. Indeed the value of a social scientist cannot be fully evaluated with a single result, be it empirical (as with the discovery of a remedy for a fatal disease in the medical science) or theoretical (as with the proof of a famous mathematical conjecture in the mathematical science). Contrary to Nobel Prize philosophy, very few Nobel prizes in Economics were awarded for a single contribution (as the Black-Merton-Scholes formula for option pricing). Usually, they are awarded for several outstanding contributions, sometimes for their complete works, as it is the case with scholars who have initiated a new sub discipline.

The h-index is also a useful characterization when we try to compare the contributions of many scholars, since the evaluation process takes time. Researchers whose contributions are being evaluated earlier are not strongly disadvantaged relatively to researchers who are evaluated at the end of the investigation since the h-index does not depend significantly on the documents appearing after they were evaluated.

Hirsch's linear model

In his original paper [START_REF] Hirsch | An index to quantify an individual's scientific output[END_REF] illustrated his h-index with a particular model of publications and citations (which we dubbed linear model): let an author publish each year p papers each being cited c times each year. What will be the total number of citations and the h-index of this author after n years? (The table corresponds to p=1)

c 0 … 0 0 Year 2 c c … 0 0 …. … … … … … Year n-1 c c … c 0 Year n c c … c c X nc t (n-1)c … 2c c
The most cited paper is the first published paper. The least cited paper is the last published. More generally, the j th c j n X j ) 1 ( -+ = most cited paper has citations, with j = 1, …, n. Assume 0 < h < n. Then, h is uniquely defined by:

c 1 c c n h c 1 1 c n 1 h X h X 1 h h + + ≤ < + - ⇔    + < ≥ + (3)
There is only one integer h that verifies this inequality, since the difference between the boundaries of the inequality is equal to 1 and the interval is open on the left. Hence, h is equal to

c c n c n c h + ≈       + + = 1 1 ) 1 ( ( 4
)
if n is large enough.

The total number of citations of the author is:

c 2 h ) c 1 ( 2 c n 2 c ) 1 n ( n X 2 2 2 n 1 j j + ≈ ≈ + = ∑ = (5)
where we used (4) to replace n. [START_REF] Hirsch | An index to quantify an individual's scientific output[END_REF] generalizes ( 4) and ( 5) for p>1. He assumes pc > 0 and c/p < n. Then, for n large enough:

n m n p c c h =         + ≈ 1 (6) ∑ = =         + ≈ n p j j a h h p c p c X 1 2 2 2 2 1 (7)
To sum up, the h-index combines quality with quantity of publications. It is easy to compute and may be used to evaluate individuals as well as journals or institutions. The h-index is robust. More papers do not imply directly a higher h-index. It is a measurement of scientific achievement and it does not depend solely on a few important papers.

Empirical analysis

The data set

We make use of four lists of researchers working in a French academic institution: Professors and lecturers in Economics who hold a position in a French university, Professors and lecturers in Management who hold a position in a French university, Permanent Professors and lecturers in Economics and Management who hold a job in a business school, Members of a CNRS (French equivalent for NSF) laboratory.

Note that a member of the teaching staff of a university is not always a member of a research center accredited to CNRS; a business school is not part of an university; some CNRS laboratory members are researchers only and do not teach; some CNRS laboratory members teach in institutions which are neither universities nor business schools (engineering schools, statistics schools..) and some of them are neither economists nor researchers in Management (they could be mathematicians, for instance).

Our observations are drawn from two different databases: Google Scholar and Scopus. Google scholar has a broader coverage but includes gray literature as well. Both have limited coverage of pre-1990 publications.

For each of these data bases, we calculate for approximately 6000 researchers a number of bibliometric indicators such as the total number of documents (articles, books and working papers) produced, the total number of citations received, the length of their scientific career, the h-index and the g-index 2 . For each query in the database, it is necessary to remove false positives due to namesakes, and to avoid false negatives, typically married women who may be listed with their married name or their maiden name, in the list or in the database.

Test of Hirsch's linear model

One could think that empirical studies using the h-index should take into account the length of academic career. For example, in his linear model Hirsch's shows that h-index increases linearly with time (Cf. Equation ( 6)). The objective of this section is to find whether we need or not to take into account the length of academic career. The results show that h-index does not depend on the length of one's career.

Equation ( 6) and ( 7) are tested on our data set with regressions:

ε β α + + = 2 1 1 h c i t (8) ε n β α h 2 2 + + = ( 9
)
where cit is the total number of citations of an author in the database, h is the h-index of the author, and n is the length of career of the author (time since his first publication 3 2 Harzing's Publish or Perish software (downloadable at http://www.harzing.com/resources.htm) gives these indicators for Google Scholar whereas we had to compute them for Scopus.

). The regression results

are the following (Table 2):

3

For Google Scholar we used instead (Age -30) as the data given by Harzing's program is usually very poor. We could not do such calculations for permanent professors of Business Schools as we do not have their age. Relation (8) between citations and the h-index holds in our data set, but not relation ( 9). Hirsch's linear model is rejected by the data. The hypothesis that papers are cited equally and regularly is clearly false. The fact that h-index is slightly correlated with the length of the career of a researcher instead of being strongly correlated may be due to the fact that data bases do not record well publications prior 1995. It also may mean that younger researchers are more productive than older ones 4 . Standard errors in brackets. Standard errors are heteroskedastic-consistent estimates (see [START_REF] White | A heteroskedasticity-consistent covariance matrix, and a direct test for heteroskedasticity[END_REF].

Pareto's Distribution

The distribution of the h-index follows approximately Pareto's law. A similar result was found in [START_REF] Combes | La production d'articles de recherche en économie en France[END_REF] using another index of scientific achievement. We estimated the function:

4

If we consider that the output of a researcher rises and then declines, a quadratic regression would improve the fit of the model. Indeed such a regression improves the fit.

( )

L o L o g h L o g N + α - = (10)
where N(h) is the number of researchers with an h-index equal to or greater than a given value h.

The smaller the coefficient α and the greater the inequality of the distribution.

Standard errors in brackets

The inequality of distribution is greater in Business Schools. There seems to be a specialization within each Business Schools where some people do more research while others do more teaching or other administrative work. There is also a similar specialization among Business Schools as we will see later.

Pareto's Law indicates that the bulk of the research is done by a very small number of researchers.

According to Google Scholar, about 5% of researchers (the elite in the sense of [START_REF] Allais | Classes Sociales et Civilisations[END_REF]) of any discipline realizes a quarter of the documents (articles, books or working papers) and collects half of the citations. According to Scopus, about 5% of researchers of any discipline publish half of the total number of papers and collect 80% of the citations.

Those inequalities of distribution suggest that a sound human resource policy has to manage those 5% top researchers differently from the others 95%. Adjustment of teaching duties according to one's research output may be an important part of a contribution -retribution system that works well.

Our Scopus data indicates that half of the researchers in Economics departments, three quarter in Management departments and two third in Business Schools do not publish peer-reviewed papers.

However we argue that it is still necessary that many researchers devote a significant portion of their time to research, even if it cannot be expected that any one of them will make great discoveries. The bulk of researchers do a useful job by distinguishing the important contributions with their citations.

Stochastic Dominance

Since one does not interact solely with the best researchers in a department, it is useful to have a global idea of the entire distribution of the quality of researchers in that department. Table 8, in the annex, contains the entire distribution of h-index as extracted from our databases. The following proposition helps analyzing this data.

Proposition 5

Let F and G be two distributions functions where the random variable x can take n values x [START_REF] Fishburn | Stochastic dominance on unidimensional grids[END_REF]):

1 < x 2 < …< x n F ≥ . FSD G ⇔ D 1 (x i ) ≥ 0 for i = 1, …, n where D 1 (x i ) = G(x i ) -F(x i F ≥ ) SSD G ⇔ D 2 (x i ) ≥ 0 for i = 1, …, n where D 2 (x i ) = Σ D 1 (x j ) × (x j+1 -x j
where ≥

) ≥ 0 for i = 1, …, n-1 FSD (respectively ≥ SSD A distribution of the h-index F first-order stochastically dominates another distribution G if the proportion of people whose h-index is smaller than or equal to a given value x

) stands for First (respectively Second) order Stochastically Dominates.

i is smaller for F than G for every value x i

We found that distribution of h-index for the faculty of Economics first-order (second-order)

. Everybody, who has a utility function increasing with respect to the h-index, will have greater expected utility under F than under G. Similarly, any individual who has a utility function increasing and concave with respect to the h-index will have a greater expected utility under distribution F which second-order stochastically dominates distribution G.

stochastically dominates the distribution of h-index for the faculty of Management (Business Schools), respectively. The first-order dominance holds true whether we use Google Scholar or Scopus data bases whereas second-order dominance does not hold with Scopus. research. Among the elite professors (the top 5%) of Business Schools 43% for Google Scholar (38% for Scopus) belong to the Economy-Finance Department or Logistics. Another possibility for improvement of research in Management Departments could be to increase the link between research and promotion.

Ranking analysis

We used the results to classify the departments of Economics and Management of universities, Business Schools and CNRS laboratories. For this purpose, we use another measure based on the hindex (respectively, g-index) which we denote h°h (respectively, g°g). Contrary to other studies using the h-index applied to institutions (Scientific Journals in particular) our h°h is not calculated as the h-index of the set of all the papers published by the members of this institution but rather is calculated in the following way: for each institution we can rank the staff in descending order of h (or g) and then calculate the h-index of this set (or the g°g).

An institution with rank h°h is such that h of its faculty has an h-index greater or equal to h and the rest has an h-index smaller than h+1. The higher the index the higher the number of important scientists this institution hosts and the greater the importance of those scientists. We may consider that an institution with higher h°h offers a better scientific environment. We prefer this ranking to ordinary h-index applied to all papers published by the staff of an institution for practical and philosophical reasons. On the practical side we should have to run Boolean research for the 6000 researchers, not an easy task especially with Scopus. On the philosophical side although Boolean research would have the advantage of avoiding double counting of papers written by coauthors belonging to the same institution it would be a measure of the h-index of the best two researchers of any institution. Therefore it would not be a very robust measure of the quality of academic environment of an institution especially in a time where best researchers should be more inclined to move. The main advantage of our h°h index is that it does not depend in an essential way on the most important scholar working in an institution. The g°g index tries to capture this influence.

The top 10 Economics departments are the intersection of the best departments according to Google Scholar ranking (h°h ≥ 5) and Scopus ranking (h°h ≥ 3). Departments which did not verify both conditions were eliminated from the final list. The best department of Economics according to both rankings is Toulouse 1 (hardly a surprise). The second best department is more difficult to ascertain even though Paris 1 and Aix-Marseille 2 are good candidates. Half of the best departments are concentrated in Paris (Paris 1, 2 and 9) or near Paris (Paris 10, 13 and Cergy). We also found that the distribution of h-index of economists of the Paris region second-order stochastically dominates that of non-Paris economists.

It is quite surprising that the best department of Economics is not hosted by a single University of the Paris area, contrary to most other disciplines (in the hard sciences and social sciences alike).

These findings explain the creation of the Paris School of Economics which aims to consolidate the leading research departments of the Paris region in a single institution, regardless of the University/ non university status of the departments.

The top 7 Management departments are the intersection of the best departments according to Google Scholar ranking (h°h ≥ 4) and Scopus ranking (h°h ≥ 2). Departments which did not verify both conditions were eliminated from the final list. The best department of Management according to both rankings is Paris 9 (hardly a surprise too, although its supremacy is not so obvious according to Scopus). Contrary to Economics, most of the best Management departments are not located in Paris. Paris 9 seems to have absorbed the best researchers in Management in the Paris Region whereas this is not the case for Paris 1 with economists, in spite of its superiority in numbers.

More than half of the best departments of Management are in Universities where the department of Economics is among the best. However, researchers in Economics are not used to work with researchers in Management as a single united research team (and often they do not wish to). This prevents them to put forward competitive trainings against those of Business schools.

The top 9 Business Schools are the intersection of the best Business Schools according to Google Scholar ranking (h°h ≥ 4) and Scopus ranking (h°h ≥ 3). Business Schools which did not verify both conditions were eliminated from the final list. There was a natural grouping (by market forces) of the best researchers from business schools towards a few institutions. We may consider that there is a small number of business schools that make high-level research while other schools are mostly teaching institutions. It is likely that we will witness the same phenomenon for the departments of Economics and Management of Universities except that this phenomenon is not entirely spontaneous but organized by the State.

One of the main levers to achieve that goal is through CNRS accreditations.

There is a risk for the State of being captured by special interest groups as it needs assessment that it cannot ensure itself. History shows that the State has often encouraged the creation of monopolies rather than competition 6 6 Cf. [START_REF] Lever | Grande et petite histoire de la Comédie Française[END_REF] shows this for Theater in France.

. If some concentration of the best researchers seems necessary for France to achieve international visibility in economics it is essential that the academic market remains contestable. Indeed, the recent emergence of Economics departments such as Toulouse 1 and Aix-Marseille 2 shows that the creation of institutions of the highest level is based on few individuals, both researchers of international level and "Scientific entrepreneurs". If the university had not been contestable in France thirty years ago Laffont in Toulouse and Gérard-Varet in Marseille could not have created important international research centers.

The top 9 CNRS Laboratories are the intersection of the best CNRS Laboratories according to Google Scholar ranking (h°h ≥ 8) and Scopus ranking (h°h ≥ 4). CNRS Laboratories which did not verify both conditions were eliminated from the final list. The best CNRS laboratory is either PSE or GREMAQ, the former being affiliated with Paris School of Economics and the latter with University of Toulouse 1. Note that CES affiliated with Paris 1 is better than GREQAM affiliated by Aix Marseille 2 whereas we saw that the departments of Economics of these universities are equivalent. The difference there is due to mathematicians.

Indeed, a great part of the best scholars of CES are specialized in mathematical economics, in particular decision theory and general equilibrium theory. Although mathematical economics is not considered as central as it used to be (and as a consequence does not attract as many citations as the Economics of Contracts, for example, where there are so many active researchers) and although Paris 1 mathematicians do not generally specialize in the more popular field of mathematical economics (as Econometric Theory, Game Theory or Mathematical Finance) they nevertheless gained international recognition far beyond their economists' colleagues. In all fairness to Paris 1 economists it should be said that their doctoral training has contributed more than any other doctoral training to turn out a considerable number of very able economists who contributed to disseminate all over French universities main stream economics. Indeed prior 1970, economics taught in French universities were almost exclusively heterodox or nationally oriented. Although, not without merits (indeed many heterodox themes, such as irrationality, are now included in mainstream economics) this teaching tended to isolate French universities' economists even more.

Isolation was already very important as most people did their research exclusively in French.

Although more and more economists publish nowadays in English there are still many French economic journals.

  Half of the best CNRS laboratories are not affiliated with Universities but with a School either Business School or Engineering Schools. This shows the great importance of the engineerseconomists school in French economic research. Prior 1960 almost no research was performed in Economics and Management departments of French universities 7 . On the contrary there were a few engineers who did research internationally recognized such as Allais, Boiteux, Debreu, Divisia, Gibrat, Malinvaud, Massé, Roy, Rueff. Some important results bear their names. It is not clear whether their followers have been as successful as them.ConclusionWill the Paris School of Economics and Toulouse School of Economics succeed to emulate Harvard or other top US universities? Indeed, the ambition of the Paris School of Economics according to its former director Piketty is to "build a center of excellence at the level of the best institutions in the world economy (Harvard, LSE, MIT, Princeton, etc.) based on mobilization and strengthening of Parisian potential research". The Economics department of Harvard has an (Google Scholar based) h°h of 26, Berkeley 21, Yale 17. If we consider all researchers in Management and Economics working in France we get 22 (22 people at least have an h-index greater or equal than 22). We have to concur with Dreze and Estevan (2006): « It is thus clear that emulating Harvard and other top US universities is not within reach for Europe 8 Moreover, too big a concentration might not be desirable as the example of the Cowles foundation , a fact that I do not regard as dramatic. As suggested above, our eyes should rather be directed to Michigan or Cornell. » 7 The teaching of economics in French universities became independent of the Law curriculum in 1959 only. Annex Calculations were run during October 2007 until February 2008 for Google Scholar and June 2008 until November 2008 for Scopus.

Table 1 : Hirsch's linear model for p=1

 1

	Citations	1 st	Paper	2 nd	Paper	….	(n-1) th	Paper	n th Paper
	Year 1

Table 2 : Test of equations (8) and (9)

 2

	Database		Google scholar				Scopus
	Data set	Department of	Business	Department of	Department of	Business	Department of
			Economics	school	Management	Economics	school	Management
	Dependent	Citations	h	Citations	Citations	h	Citations	h	Citations	h	Citations	h
	variable
	Constant	-1.84	2.44	-10.74	-3.40	1.39	1.17	1.07	-5.26	1.17	2.65	0.49
			(1.34)	(0.13)	(8.06)	(1.43)	(0.08)	(1.54)	(0.13)	(12.80)	(0.24)	(1.15)	(0.12)
	h 2	4.31	-	7.03	4.89	-	3.24	-	5.36	-	3.52	-
			(0.17)		(0.90)	(0.32)		(0.27)		(1.07)		(0.31)
	n	-	-0.016	-	-	0.003	-	0.093	-	0.18	-	0.07
				(0.006)			(0.005)		(0.016)		(0.03)		(0.02)
	R	2	0.93	0.003	0.81	0.93	0.0002	0.875	0.10	0.77	0.19	0.93	0.10

Table 3 : Test of equation (10) -Dependent variable: logN(h)

 3

	Database		Google scholar			Scopus
	Data set	Department of	Business	Department of	Department of	Business	Department of
			Economics	school	Management	Economics	school	Management
	Constant	3,694	3,212	3,411	7,281	6,951	5,329
			(0.13)	(0.08)	(0.12)	(0.35)	(0.29)	(0.19)
	Logh	-2,176	-1,601	-2,406	-2,374	-1,946	-1,964
			(0.12)	(0.08)	(0.13)	(0.18)	(0.13)	(0.10)
	R	2	0,933	0,957	0,967	0,936	0,935	0,978

Inefficiency of Management departments as measured by Google Scholar Research

  If we build the production frontier for research with two outputs, documents per researcher and citations per researcher. Research output is efficient in Business Schools and Economics Departments, both with respect to Google Scholar and Scopus data bases. Although Economics departments produce more research than Business schools they receive less citations, indicating that research in Economics Departments might be of a worse quality or less central.However, inefficiency of research in Management Departments is strong. With a better management of research it should be possible to double (according to Google Scholar) the number of documents per researcher as well as the number of citations per researchers. According to Scopus, this inefficiency is even stronger, reflecting the fact that most part of research in Management Departments is written in French. It should be possible to multiply the number of papers and citations by 3.6.

	The fraction of
	researchers having a low h-index is greater in Management departments than in Economics
	departments (for any value of the h-index). Hence, if you have a utility function increasing with
	respect to the quality of research then you should prefer working/studying in a department of
	Economics instead of Management. Second degree stochastic dominance reflects the greater
	inequality of research in Business Schools.
	University Professors are better researchers than University Lecturers. Distribution of h-index for Indeed, for each level of the h-index x i D , we have 1EconLect (x i ) -D 1EconProf (x i ) > D 1BusLect (x i ) -D 1BusProf (x ) > 0 Management Economics Business Schools 16 9 Graph 1: Documents/Faculty Citations/Faculty 52 22 78

Professors of Economics first-order stochastically dominates the distribution of h-index for

Lecturers. This is true both for faculty of Economics and Management and with both data bases. This tends to prove that research is a criterion for promotion in Universities. However this criterion for promotion is stronger among the faculty of Economics than with the faculty of

Management. i

This means that first-order stochastic dominance between Professors and lecturers is stronger in Economics than in Management departments. This would tend to prove that University Lecturers in Economics who are doing more research than others tend to become University Professors more easily than University Lecturers in Management. in Management Departments strongly needs to be reorganized. The reorganization could go through the assignment of economists and mathematicians on Management positions as the Quantitative Management Techniques can be taught by economists and mathematicians as well as managers. That's partly how Business Schools have significantly improved their performance in

Table 4 : Economics departments

 4

				Google Ranking			Scopus Ranking
	Universities	Faculty	Documents	Citations	h°h	g°g	Papers	Citations	h°h	g°g
	Toulouse 1	64	3068	20866	10	22	487	3233	7	12
	Paris 1	112	3001	11404	8	18	294	1078	4	7
	Paris 10	55	1638	6879	7	15	198	805	4	6
	Aix-Marseille 2	49	1036	3222	6	12	221	770	5	7
	Strasbourg 1	30	818	3161	6	12	122	672	4	7
	Paris 9	59	917	3785	5	11	222	1443	4	8
	Lille 2	15	362	1965	5	10	70	360	3	6
	Paris 2	38	600	1720	5	10	108	256	3	5
	Paris 13	36	651	2044	5	10	59	267	3	4
	Cergy	22	460	1039	5	8	81	195	3	4

Table 5 : Management departments

 5

				Google Ranking			Scopus Ranking
	Universities	Faculty	Documents	Citations	h°h	g°g	Papers	Citations	h°h	g°g
	Paris 9	63	1026	4954	6	14	58	348	2	5
	Grenoble 2	76	689	1725	5	9	35	136	2	3
	Toulouse 1	39	512	2700	4	11	62	406	2	5
	Aix-Marseille 3	46	577	1851	4	10	60	337	2	5
	Aix-Marseille 2	24	294	946	4	8	57	560	2	5
	Paris 1	63	622	1331	4	8	32	82	2	3
	Rennes 1	53	406	901	4	7	19	28	2	2

Table 6 : Business Schools

 6 The best Business School according to both rankings is INSEAD. What is more surprising is that it ranks even higher than the department of Economics of Toulouse 1. Although most Business Schools did not have active research departments until recently, except INSEAD and ESSEC, they caught up and can now rival with the best Economics departments. This leads to an increasing competition, since jobs in Business schools are financially more attractive than in universities.When we consider the entire distribution of the h°h-index for Economics and Management departments we observe that they are log normally distributed whereas the same distribution for Business Schools follows Pareto's Law. That is, the vast majority of Economics and Management departments makes research at an average level whereas a great number of Business Schools do not have active research at all and a few of them are doing very well.

	Business Schools	Permanent		Google Ranking			Scopus Ranking
		Professors	Documents	Citations	h°h	g°g	Papers	Citations	h°h	g°g
	INSEAD	121	4420	76270	13	36	1200	20160	9	19
	HEC	106	2031	13948	9	20	404	2369	5	9
	ESSEC	106	2003	10200	7	18	403	2351	5	10
	Edhec Lille Nice	74	913	10168	6	15	129	5081	4	9
	ESC Toulouse	64	733	4510	5	11	118	712	3	6
	EM Lyon	76	581	2184	5	10	92	531	4	6
	ESCP-EAP	113	772	2143	5	9	115	619	3	6
	Grenoble EM	81	626	1481	4	9	134	441	3	6
	Ceram Sophia-	50	260	640	4	7	53	250	3	4
	Antipolis

Table 7 : CNRS Laboratories

 7

	Laboratories	Researchers		Google Ranking				Scopus Ranking
			Documents	Citations	h°h g°g Papers	Citations	h°h g°g
	UMR8545 PSE	56	5150	51461	14	31	638	5927	7	14
	UMR5604 GREMAQ	62	4447	52017	13	28	670	6499	8	15
	UMR8174 CES	164	4853	18987	11	21	1032	4821	7	12
	UMR2773 GRECSTA	37	2327	22325	10	23	238	1714	5	10
	UMR2959 GREGHEC	70	2033	15470	9	21	434	2762	6	10
	UMR6579 GREQAM	51	2040	15179	9	20	452	2604	5	10
	UMR7166 EconomiX	94	2268	7709	8	16	262	699	4	5
	UMR7176 PREG	41	1313	6038	8	15	273	1049	5	8
	UMR7522 BETA	86	1621	6343	8	15	294	1286	4	8

Table 8

 8 Distribution of the h index

			Google Scholar			Scopus
	h	Economists	Management	Business Schools	Economists	Management	Business Schools
	0	515	562	797	1207	1396	1237
	1	454	453	351	343	138	167
	2	354	294	174	132	44	82
	3	200	137	93	57	15	49
	4	112	74	51	37	2	27
	5	51	39	41	28	0	16
	6	36	17	30	6	1	19
	7	24	8	21	7	1	11
	8	18	6	19	2	1	11
	9	10	5	12	2	0	5
	10	12	1	7	2	1	2
	11	12	3	11	3	1	0

See also[START_REF] Courtault | Characterization of Stochastic Dominance for Discrete Random Variables[END_REF].

Needless to say for France alone and in particular in the field of Economics and Management.


F Moro

T Faug

H Bellot

F Ousset

Large mobility of dry snow avalanches: insights from small-scale laboratory tests on granular avalanches of bidisperse materials

Keywords: snow, avalanche dynamics, granular flows, binary mixtures, experiments, mobility

This paper reports small-scale laboratory tests on granular avalanches of bidisperse materials made of fine particles and larger ones. These experiments were motivated by a recent study on the rheology of dense flowing snow which provided evidence for relevant similarities in flow behavior between bidisperse granular materials and dry cold snow [Rognon and others, J. Rheol., 52, 3 (2008)]. The mass proportion of fine particles in the initial binary mixture was systematically varied at constant initial released volume, and we measured the resulting flow depth, the avalanche front velocity and the final avalanche runout. In particular, we show that the avalanche mobility is largely increased, about 40% in our tests, when the mass proportion in fine particles reaches a critical value, around 0.25 in our tests. The avalanche deposit is shallow and lengthened for this critical mass proportion in fine particles. The experimental results are interpreted by the existence of different avalanche mobility regimes on the basis on an heuristic model previously reported in literature. Finally, we discuss their possible implications for the dynamics of full-scale dry snow avalanches.

Introduction

Snow is a cohesive granular material and granular flows are often considered to model dense avalanches of dry and cold snow [START_REF] Hutter | Avalanche dynamics. Hydrology of disasters[END_REF][START_REF] Issler | Experimental information on the dynamics of dry-snow avalanches. Dynamic response of granular and porous materials under large and catastrophic deformation[END_REF]Naaim and others, 2004). Dry flowing snow is a granular polydisperse material as corroborated by recent investigations on some deposits of full-scale snow avalanches [START_REF] Bartelt | Instruments and Methods. Granulometric investigations of snow avalanches[END_REF]. Recent systematic experiments on a decametric snow chute has evidenced the fact that dry, cold and dense flowing snow can be seen as a mixture of snow grains, with a diameter typically smaller than 1 mm, and of larger snow aggregates with a maximum size close to the flow thickness (Rognon and others, 2008). These experiments have been combined with discrete element simulations of bidisperse granular materials (Rognon and others, 2007). It has been shown that dry, cold and dense flowing snow exhibits a behavior steming from specific properties of bidisperse granular flows (Rognon and others, 2008). First, Rognon and others (2008) showed that steady and uniform flows are possible in a range of slope between two limit angles θ min and θ max . Furthermore, when approaching the critical angle θ min , the velocity abruptly drops to zero. These features are reminiscent of the behavior of granulars materials [START_REF] Gdr-Midi | On dense granular flows[END_REF][START_REF] Forterre | Dense granular flows[END_REF]. Second, velocity profiles measured by Rognon and others (2008) suggest that the snow flow is divided into two layers: a highly sheared basal layer of small thickness and made of snow grains is surmounted by a thick and less sheared layer made of snow 2 aggregates. It corresponds to a behavior close to the behavior of bidisperse granular mixtures (Rognon and others, 2007). These recent results suggest that assimilating dry, cold and dense flowing snow to a bidisperse granular fluid is a rather good approximation. Many previous small-scale laboratory tests to model snow avalanches have been carried out using monodisperse granular flows to predict avalanche trajectories [START_REF] Savage | Gravity flow of cohesionless granular materials in chutes and channels[END_REF][START_REF] Savage | The motion of finite mass of granular material down rough incline[END_REF] or, more recently, to study the influence of obstacles (Tai and others, 2001;Hákonardóttir and others, 2003;Faug andothers, 2004, 2008).

In this paper we investigate experimentally flows of bidisperse granular mixtures in order to quantify the influence of the bidispersity on the avalanche dynamics. The thickness and the flow velocity during the propagation phase, as well as the maximum runout distance and the final shape of the deposit in the runout zone, are studied here. Our results show that these macroscopic flow descriptors, of crucial interest for hazard zoning and risk management, are largely influenced by the bidispersity. In particular, it exists a mass proportion in fine particles for which the runout distance is maximum. This result demonstrates that bidispersity may be a good candidate to contribute to enhance the mobility of granular snow avalanches, and more generally, of geophysical granular flows.

First, we describe the experimental set-up and procedure with a focus on an advanced experimental technique to catch the shape of the granular mass. The second part presents the results concerning the influence of the binary mixture composition on (i) the granular deposit in the runout zone and (ii) the flow characteristics in the upstream flume. The main result is the observation of an enhanced mobility at a given mass proportion in fine parti-cles inside the granular mixture. Then, we try to highlight various mobility regimes depending on the mass proportion in fine particles, on the basis of an heuristic model recently proposed by Philipps and others (2006). Finally, the results and their possible implications for snow avalanche dynamics are discussed.

Experimental device and procedure

The experimental device we used corresponds to a geometry typically used to study snow avalanches on reduced models (Hákonardóttir and others, 2003;Faug andothers, 2003, 2008), and more generally geophysical flows (Hutter and others, 1995). It consists of a system of two inclined planes including: (i) a reservoir to store the material and equipped with a gate to release it, (ii) a steep inclined plane equipped with a flume inside which the granular material propagates, (iii) a downstream plane inclined at a lower slope angle where the granular mass decelerates, spreads and finally stops. A typical granular deposit is thus generated. The slope angle of the flume corresponding to the propagation zone was fixed at 35 • and the one corresponding to the runout zone was fixed at 6 • for the whole experiments. Fig. 1 shows a sketch of the experimental device with the associated instrumentation.

As already mentioned above, most of the previous studies available in literature deal with monodisperse granular flows. Some recent studies tackle the issue of bidisperse granular flows and the associated enhanced mobility using 2D (channelized flows) small-scale laboratory tests (Phillips and others, 2006) or 2D discrete numerical simulations (Linares-Guerrero and others, 2007). Philipps and others (2006) performed experiments with granular sys-tems made of rectangular (rafter with typical size 75 mm × 25 mm × 2.2 mm) or spherical (gravel of diameter 10-20 mm) coarse particles mixed with fine spherical particles (about 100 µm in size). Their bimodal granular systems are characterized by a few number of coarse (rafter or gravel) particles.

In this paper, we analyse 3D granular avalanches involving a great number of fine and large spherical particles using advanced experimental techniques.

It gives us access to a large number of measures to characterize the 3D (lateral extension) avalanche deposit: maximum length, location of the center of mass, spreading length above the center of mass, mean width, flow depth profile of the deposit, location of the maximum depth in the main flow direction.

Thus, the tests presented here are full 3D flows and the observed enhanced mobility can be characterized by additional parameters that were not studied in (Phillips and others, 2006;Linares-Guerrero and others, 2007). The bidisperse granular materials consists of glass beads of diameter d 2 =1 mm mixed homogeneously (before releasing the whole granular mass) with glass beads of diameter d 1 =200 µm. The value d 1 is a mean value: a slight polydispersity characterizes the fine particles whose diameter ranges from 150 to 250 µm.

The ratio D = d 2 /d 1 is then between 4 and 6.66 being a value sufficiently large for the occurrence of segregation processes. Indeed, it is known that segregation starts to occur above 20% in polydispersity [START_REF] Hirshfeld | Molecular dynamics studies of grain segregation in sheared flow[END_REF]Berton and others, 2003) which corresponds to a ratio D = d 2 /d 1 typically higher than 1.2. The mass initially released was kept constant and equal to m=10 kg. The channel roughness was smooth (wooden surface) relatively to the roughness of the runout zone made of sandpaper glued on the inclined plane. We will come back to this point when giving an attempt to analyse the typical values of the ratios d 1 /λ r and d 2 /λ r with regard to the recent results on bidisperse granular media (Goujon andothers, 2003, 2007), λ r being the typical size of the roughness asperities (sandpaper). Finally the only initial varying parameter in the experiments is the mass proportion, called r, in fine particles of diameter d 1 :

r = m 1 m = m 1 m 1 + m 2 , ( 1
)
where m 1 is the mass of fine particles and m 2 is the mass of larger particles. We carried out many laboratory tests by varying the ratio r between the value 0 (glass beads of diameter d 2 only) and 1 (glass beads of diameter

d 1 only).
A laser sheet which generates a line parallel to the channel walls and centered on this latter (see the red line in Fig. 1) is projected on the inclined plane. When the granular mass propagates on the channel, this laser line is deflected. Thanks to a video camera placed normal to the channel bottom, we can visualize the shifted laser line (video camera n • 1 on Fig. 1). The deviation is directly proportional to the flow thickness. A calibration phase using various rectangular objects of known thickness allows us to determine the proportionality coefficient. Then, the thickness at each time t and at each point X (in the centerline of the channel) can be easily derived. Fig. 2 gives a depth-profile in time obtained at various abscissa of the channel in the investigation zone of the laser sheet. This latter is filmed by the video camera n • 1, has a length of 1 m and is defined on Fig. 1. Let us note that the zone inside which we measured flow thicknesses is smaller than 1 m. The order of magnitude is 55 cm, which corresponds to the zone between the locations X 1 and X 5 defined on Fig. 1. In this investigated zone and at a given location X, the typical depth-profile in time presents a maximum with a plateau more or less pronounced. We observe that the value of this maximum varies with the location X in the channel but this variation is very weak in the ending part of the channel. The typical values of gradients of maximum flow-depth in the ending part of the channel are around 1 mm (see Fig. 2). The precision of depth measurements has been estimated to 1 mm.

We systematically defined h as the mean value of the three maxima obtained at locations X 3 , X 4 and X 5 , in the bottom part of the channel, excluding the maxima obtained at locations X 1 and X 2 in the top part. Locations X i=1,2,3,4,5 are defined on Fig. 1. It is then possible to analyse the influence of the ratio r on the flow-depth h. These results are presented and discussed in the following section.

The laser line allows us also to determine the locations of the front x f in the channel and, consequently, the propagation speed of the granular front (Lagrangian front speed), in want of the knowledge of the depth averaged flow speed of the granular avalanche core at a given location X (Eulerian speed).

The point corresponding to the front position is the point downstream which the laser line is not shifted and upstream which it is shifted. Fig. 3 gives an example of the front position x f versus the time. This graph allows us to calculate the front velocity, called u, and defined by the slope of the straight line (u = dx f /dt). The precision on front speed was estimated to 0.1 m s -1 .

The results about the influence of the ratio r on the front velocity u are presented and discussed in the following section.

Parallel fringes, alternatively black and white, were projected on the gran-ular mass on the runout zone with the help of a projector. Similarly to the shifted laser line technique, the fringes network is shifted by the granular mass and the deviation is proportional to the deposit thickness. A video camera placed normal to the inclined plane (video camera n • 2 on Fig. 1) allows us to visualize the shifted fringes network (Fig. 4b). The calculation consists in determining with accuracy the phase difference ∆ϕ(x, y) between the reference signal, without the granular mass, and the object signal, in presence of the granular mass which distorts the fringes. The reference signal can be modelled by the following sinusoidal signal:

I ref (x, y) = Ī(x, y) + Ī(x, y)γ(x, y) cos(ϕ 0 + 2πf 0 x), (2)
where Ī(x, y) is the mean intensity of the signal, γ(x, y) the lightness contrast at the point location (x, y), f 0 is the frequency linked to interfringe and ϕ 0 the origin phase. The object signal can be modelled by the following sinusoidal signal:

I object (x, y) = Ī(x, y) + Ī(x, y)γ(x, y) cos(ϕ 0 + 2πf 0 x + ∆ϕ(x, y)), (3)
The phase difference ∆ϕ(x, y) between the two signals is directly proportional to the local deviation of fringes δ(x, y):

∆ϕ(x, y) = 2πf 0 δ(x, y) (4)
It is possible to derive theoretically the proportionality coefficient relating the local deviation of the fringes and the object height. For this aim, the geometrical and optical features of the system formed by the projector, the surface of the inclined plane and the video camera have to be considered (Sansoni and others, 1999;[START_REF] Vincent | Recalage et fusion pour reconstruire des surfaces 3D obtenues par projection de franges. Application à un système de vision dédié à l'orthopédie[END_REF]. However, in order to avoid errors linked to the determination of these geometrical features and to optics of instruments, we used an empirical calibration thanks to rectangular objects of known height to deduce the deposit thickness at each point (x, y) of the inclined plane. This experimental technique using fringes projection is now currently used to study flows in small-scale laboratory models [START_REF] Pouliquen | Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane[END_REF][START_REF] Cochard | Tracking the free surface of time-dependent flows: Image processing for the dam-break problem[END_REF].

Several methods exist to have access to the phase difference [START_REF] Surrel | Fringe analysis[END_REF][START_REF] Vincent | Recalage et fusion pour reconstruire des surfaces 3D obtenues par projection de franges. Application à un système de vision dédié à l'orthopédie[END_REF]. Some methods are based on a single image projection of the fringes network as the Fourier transform method (global approach). Other methods, more accurate, are based on the principle of phase shifting which consists in projecting several images shifted one to each other (local approach). This second approach is not adapted to dynamic problems with rapid changes in time but it brings an increased precision for static (or quasistatic) problems. In the experiments presented here, we use this second approach by projecting four grids shifted of π/2. The displacement of fringes by the granular mass is interpreted as a fringes phase modulation and can be expressed according to the four intensities I i (x):

φ(x) = arctan I 4 (x) -I 2 (x) I 1 (x) -I 3 (x) , (5)
where the four shifted images (φ

1 = 0, φ 2 = π/2, φ 3 = π, φ 4 = 3π/2)
are modelled by a sinusoidal signal:

I i (x, y) = Īi (x, y) + Īi (x, y) cos(φ(x) + ∆φ i (x)
). The thickness is then directly proportional to the phase modulation φ(x) and is obtained by calibration. The calibration procedure consists in projecting the four reference shifted images (φ 1 = 0, φ 2 = π/2, φ 3 = π, φ 4 = 3π/2) on various rectangular objects of known height. Fig. 4 shows the projected fringes on a deposit as well as the numerical result for the deposit shape in three dimensions. The precision varies according to the image quality and is generally smaller than 500 µm.

Experimental tests consisted in releasing a mass of 10 kg and changing the ratio r at each test. We tested values of r ranging from to 0 to 65% with a 1% step. For values greater than 65%, we did tests with the following values: 80% and 100%. In the following section, we present results concerning the influence of the ratio r on (i) the deposit morphology and (ii) the flow dynamics inside the ending part of the flume.

Results

Evidence of segregation

The qualitative observation of deposits, as displayed in Figs. 5a and5b, shows (i) the presence of fine particles at the base of the deposits, and (ii) the presence of coarse particles mostly at the front and, to a less extent, at the sides. These observations, previously discussed by Pudasaini and Hutter (2007, p. 63, figure 2.6), are the proof of size segregation effects, during the propagation phase and during the stopping phase, where the fine particles tend to fall (percolation) and large grains tend to go toward the surface.

The present paper does not tackle the strickly speaking segregation process, largely studied for relatively dense granular flows [START_REF] Savage | Particle size segregation in inclined chute flow of dry cohesionless granular solids[END_REF][START_REF] Ottino | Mixing and segregation of granular materials[END_REF][START_REF] Félix | Evidence of two effects in the size segregation process in dry granular media[END_REF][START_REF] Gray | Particle-size segregation and diffusive remixing in shallow granular avalanches[END_REF] but it rather analyses the consequences of the segregation process on the granular avalanche, during its propagation and stopping phases. Our experiments show, which is described in detail in this section, that several features of the granular avalanche are influenced by the ratio r: (i) the velocity and the thickness during the accelerating propagation phase, (ii) the final length of the deposit in the runout zone, (iii) the thickness of the final deposit (maximum value and mean value along the main flow direction in the

x-axis direction), (iv) the location where the deposit thickness is maximum. in mass of fine particles) is increased of nearly 40% in comparison with a monodisperse mixture of large particles. Let us note that L/L 2 is slightly smaller than 1 when r = 1, which illustrates that the runout with fine particles only is slightly shorter than the runout with large grains only. This observation shows that large particles go further than small ones, for fixed roughness conditions. We can introduce here the critical diameter d c defined by Goujon and others (2007), and which depends on the typical size λ r of the downstream plane roughness and its compacity C:

d C = π 6C √ 3 λ r ≈ 0.3 λ r C . (6)
For monodisperse dry granular flows down inclined planes, there exist a diameter d C of particles for which the basal friction is maximum: minimum in mean velocity, maximum in deposit thickness and minimum in deposit length (Goujon and others, 2003). However the difference between the two monodisperse deposits is not so large.

We may think it is caused by the very low value of the runout zone slope angle (6 • ) in our experiments, which allows the large grains to come at rest.

Goujon and others (2007) investigated flows at higher slope angles on which grains were less stable and they obtained a larger difference between the two monodisperse deposits.

Thanks to the fringes projection technique, we could determine with accuracy the position of the center of mass x G (calculated from the beginning of the runout zone: x = 0 in Fig. 1). x G /x G2 (Fig. 6) displays a more pronounced maximum (value around 1.5) than L/L 2 . This implies that the center of mass is more sensitive than the deposit front to the proportion in fine particles. Let us consider a simple model for which the total maximum runout is the sum of the position of the center of mass (x G ) and a spreading length above the center of mass (x spreading ), as previously suggested in (Faug and others, 2004):

L = x G + x spreading . (7)
A similar approach was been recently proposed in [START_REF] Staron | Understanding how volume influence the mobility of dry debris flows?[END_REF] where the runout of a gravity-driven granular flow is decomposed in two contributions: a so-called sliding along the topography and a so-called spreading of the unconsolidated mass. From this simple model, we can conclude that the segregation process acts differently on the displacement of the total mass (sliding) and on the way the unconsolidated mass spreads beyond the center of mass. Indeed, the spreading length is less sensitive to the ratio r than the position of the center of mass, as shown by Fig. 6.

In Fig. 6 is also shown the maximum width being relatively independent of the ratio r. This result tends to show that the segregation acts mainly in the prevailing direction of the avalanche propagation (simple shear flow). A similar effect has been also observed by Zanuttigh and Di Paolo (2006).

The fringes projection technique allows us to study other effects on the deposit shape: the mean depth-profile, H, and the maximum height, H max , as well as the point location where the height is maximum,

x[H max ].
The height H is defined by:

H = S S , ( 8
)
where S is the area of the transverse section of the deposit at a given abscissa x and S is the deposit width at the abscissa x. The height H corresponds to the mean value in the transverse y-axis direction (Fig. 1). If we neglect the changes in volume fraction (we will come back to this point later in the paper), we can assume that the volume is constant because we kept the total mass m constant. As the deposit width is independent of the ratio r, it results that the average depth along the whole profile, H x , is, in first approximation, inversely proportional to the deposit length L. This result is displayed in insert in Fig. 8.

In Fig. 6 is finally displayed the position of the maximum height, x[H max ].

The results are more scattered because this parameter is more sensitive to experimental conditions. However, the curve (see the gray trend line reported in Fig. 6) shows that this parameter displays a behavior similar to the one observed for L and x G . Let us remark that the maximum is very pronounced since the value of x[H max ] is nearly double when r = r crit , in comparison with large particles only. This result indicates that the deposit morphology is very sensitive to r, and therefore to segregation effects.

3.2.2. Existence of a critical proportion in fine particles for which the friction is minimum?

Rognon and others (2007) studied, on the basis of discrete numerical simulations, velocity profiles of 2D bidisperse steady flows of disks with varying the ratio S g as well as the size ratio D = d 2 /d 1 . S g was defined as the surface proportion of large disks (Rognon and others, 2007):

S g = (n 2 d 2 2 )/(n 2 d 2 2 + n 1 d 2 1 ),
where n 2 is the number of large grains and n 1 the number of small grains.

They observed an increase in thickness of the top layer made of large grains when increasing S g , and for S g = 3/4, the basal layer of fine grains was shown to disappear. By analogy, we can define the dimensionless number V g representing the volume proportion of large grains (1 mm glass beads):

V g = (n 2 d 3 2 )/(n 2 d 3 2 + n 1 d 3 1
). Since large and small particles has the same density (glass beads), the volume proportion of large grains is equal to the mass ratio between the large grains and all the grains: V g = 1 -r.

According to Fig. 6, the various parameters defined previously L/L 2 ,

x G /x G2 , x spreading /x spreading2 , x[H max ]/x[H max2 ] increase when decreasing r (when increasing V g ) and reach a maximum when r is around 0.2-0.3 (V g around 0.7-0.8). These parameters then decrease to tend toward 1, extreme value for which there is no more fine grains (n 1 = 0). These results appear to be compatible with the observations by Rognon and others (2007). In this decreasing part of the curves, everything acts as if there was no enough fine grains to form a homogeneous basal layer allowing the optimal lubrication of the whole granular mass mainly made of large grains. In order to develop segregation processes that are able to contribute to avalanche lengthening, it is necessary to exceed a given proportion in fine particles inside the mixture.

Conversely, this proportion should be not too high in order to maintain the efficiency of the segregation processes. It is really a challenge to find the critical value theoretically. However we can provide a qualitative physical explanation to the critical condition. Below r = 0.2 -0.3 (above V g = 0.7 -0.8), the lubrication can act but is not optimal. The optimum is probably reached when a homogeneous layer of fine particles is formed at the base of the flow and, simultaneously, no -or a few-fine particles are present inside the voids between the large particles. Above r = 0.2 -0.3 (below V g = 0.7 -0.8), as the basal layer of fine particles is saturated, the added fine particles start to fill the voids between the large particles. This process offers additional contact surfaces for interparticle friction (the velocity of the large particles is decreased) which counteracts the basal lubrication.

These mechasnisms can be interpreted by the existence of a critical proportion for which the friction is minimum. The observed maxima on the curves correspond to this minimum friction. These results are in agreement with the results front 2D discrete numerical simulations showing that the friction coefficient versus the proportion in fine particles displays a minimum (see figure 11 in (Linares-Guerrero and others, 2007)).

Volume of the avalanche granular deposit

The fringes projection technique combined with phase shifting allowed us to access to the final volume of the deposit. This volume is also a function of r as shown in Fig. 8. In spite of weak variations, it illustrates that the assumption made previously on the volume fraction is not correct. Since we kept the total mass constant (m=10 kg), we can deduce the volume fraction evolution:

φ(r) = m ρ P V (r) . (9)
At constant total mass, the volume fraction is inversely proportional to the volume. Given that a minimum is shown in the volume variation, the volume fraction displays a maximum. However, this results is non trivial to interpret since the initial volume is not perfectly controlled. Rearrangement mechanisms, and consequent variations in volume fraction, may have taken place before the avalanche release, when setting the granular mixture up inside the reservoir. Indeed, it is known that the compacity of a bidisperse granular in a static state depends on the proportion of large grains in the mixture [START_REF] Ben Aim | Etude de la texture des empilements de grains. Application à la détermination de la perméabilité des mélanges binaires en régime moléculaire[END_REF][START_REF] Dodds | The porosity and contact points in multicomponent random sphere packings calculated by a simple statistical geometric model[END_REF]. Adding particles of different size (larger or smaller), even in small amount, to a monodisperse medium, modifies and generally increases the solid fraction when the size ratio D is sufficiently large. This increase in compacity is explained by two main mechanisms (Ben Aim, 1970): (i) the insertion mechanism for which small particles fill the voids inside a granular skeleton mainly made of large grains (small r) and (ii) the substitution mechanism for which a large grain replaces a cluster of fine grains and the openings between these fine grains inside a matrix mainly made of fine grains (large r). The mean volume fraction of the deposits depends strongly on the segregation processes during the propagation phase. It may also depend on the initial granular rearrangements and resulting initial compacities (difficult to control) of the granular mixture put in place inside the reservoir. However, the behavior observed in Fig. 8 remains difficult to interpret and cannot be explained simply by initial rearrangements. It probably suggests inherent mechanisms occurring during the flow and stopping phases which need further investigations in the future.

As a first approximation, we can relate the total volume of the deposit to its length L, its width and its mean thickness H x by the following equation:

V = αLH x = m ρ P φ(r) , ( 10
)
where α is a shape coefficient linked to the deposit. The results show that α is constant (close to 3/4) whatever the value of the ratio r. This result is in agreement with the observations showing, for each test, a mean shape of the deposit similar to the one given in Fig. 4a. Let us note that all the deposits observed during our tests, whatever the r value, are uniforms (see Fig. 5a) in the sense that we do not observe digitation mechanism, similarly to the fingering features analysed by other authors (Pouliquen and others, 1997;[START_REF] Pouliquen | Segregation induced instabilities of granular fronts[END_REF], and which can be also observed for some natural snow avalanches.

In this section, we showed the influence of the segregation process on the characteristics of the final deposit. We also observed important effects in the propagation zone on the flow depth and velocity. This tends to show that segregation processes can occur at very short times once the avalanche has been initiated. We discuss this question in the following section.

Flow in the channel

We remind here that the channel is smooth (wooden bottom), which implies that the typical size, λ c , of the channel roughness is very small in comparison with the diameter d 2 (λ d 2 ) and also d 1 (λ d 1 ). Contrary to the runout plane, complex effects due to coupling between the roughness and the flowing binary mixture (Goujon andothers, 2003, 2007) have not to be considered for the propagation phase in the channel. Fig. 9 shows the mean flow depth h (defined in section 2) in the channel versus the ratio r. The flow depth is relatively constant when the mass proportion in fine particles is weak. It decreases afterwards when the ratio r reaches a critical value around 0.2-0.3. Then, the flow depth remains relatively constant and, finally, slightly increases at high r values.

In Fig. 9 is also shown the mean front velocity in the channel (defined in section 2) versus the ratio r. The speed increases when the mass proportion in fine particles increases, before reaching a plateau when the ratio r reaches a critical value. The transition is observed around 0.2-0.3, which corresponds to a order of magnitude similar to the value obtained for h. Let us note that, in spite of the weak number of available data, the speed tends to decrease at high r values.

The Froude number of the free-surface flows is often considered as an important similitude criterion which is rather easy to estimate with a prior knowledge of the flow depth h and the velocity u. The Froude number is defined as:

F r = u √ gh . ( 11
)
We reported the value of the Froude number versus the ratio r in Fig. 9.

The Froude number ranges from 5 to 9 in our tests with granular materials.

It corresponds to typical values for rapid snow avalanches [START_REF] Issler | Experimental information on the dynamics of dry-snow avalanches. Dynamic response of granular and porous materials under large and catastrophic deformation[END_REF]. The

Froude number is shown to increase when increasing the mass proportion in small grains inside the mixture and it reaches a relatively constant value when the ratio is above a critical value around 0.3. Let us note that above a certain value of r (around 0.6-0.7), the Froude number decreases. This observation results from the decrease in velocity and the increase in flow depth. Segregation generates thinner and more and more rapid channelized flows. These results are the proof that the segregation and the induced lubrication can be active even if the bottom surface is smooth. The trapping of fine particles is limited here. The large velocity gradients (between the basal layer of fine particles and the large particles above) needed to lubricate the flow are more due to the high inclined slope of the channel. This latter makes the large particles (high inertia) above the small moving ones (r = r crit ) flowing down faster than the small ones alone (r = 1). These results are compatible with the fact that the granular avalanche covers a greater distance and the deposits are thinner and lengthened thereafter in the downstream runout zone.

Data interpretation

On the basis of the study by Philipps and others (2006), we can interpret experimental data by distinguishing three mobility regimes identified in Fig. 10. Fig. 10 gives the deposit length, L, normalized by the deposit length, L 2 , obtained for r = 0 (large grains only) versus the mass proportion in fine particles, r.

The regime (I) is characterized by a mixture made of a majority of large grains with a few fine particles (0 < r < r I-II , where r I-II is the proportion corresponding to the transition toward to the regime (II) defined below).

In the regime (I), the fine particles, located at the base of the flow (as a consequence of segregation effects in the channel), can play the role of a rolling plane for the incoming larger grains. The limit between regimes (I) The regime (III) corresponds to a mixture made of a dominant part of fine particles with a few large grains (r III-II < r < 1, where r III-II is the proportion corresponding to the transition toward to the regime (II) defined below). As soon as the proportion in fine particles becomes dominant, their effect becomes more complex and they do not play the simple role of lubricating the mass of large grains. In the regime (III), we expect that the interparticle friction is the main source of energy dissipation, which limits the avalanche mobility. A simple idea to model this regime is to consider that dissipations by friction are directly proportional to the surfaces involved in friction contacts (Phillips and others, 2006). Note that this simple approach does not take into account the dissipations due to collisions which are likely to be important inside the basal sheared layer made of fine particles with fluctuating velocities. The effect of substituting a large grain for the same mass of fine particles is to reduce the contact surfaces availables for interactions, which leads to a region weakly effective in terms of friction dissipations in comparison with regions where fine particles are free to interact. By comparing a situation with a binary mixture (mixture A) and a situation with a monodisperse mixture exclusively made of fine particles (mixture B), we can estimate the ratio R f between the energies dissipated by friction in each case (Phillips and others, 2006):

R f = n 1 S 1 F + n 2 S 2 F nS 1 F , (12)
where S 1 = d 2 1 (respectively S 2 = d 2 2 for the large grains) is the surface of the fine particle of diameter d 1 (respectively d 2 for the large grains). The product n 1 S 1 F is the total dissipation by friction in the mixture B, where F is the friction force by surface unit. For the mixture A, at constant surface unit force F , the fine particles but also the large grains contribute to friction dissipations, which gives the following total dissipation:

n 1 S 1 F + n 2 S 2 F .
Some calculations allow us to derive the ratio R f as a function of the diameters ratio D, the density ratio χ = ρ 2 /ρ 1 and the mass proportion in fine particles r:

R f = 1 + 1-r χrD 1 + 1-r χrD 3 . ( 13
)
Considering that χ = 1 (in our experiments), and that the value of D is greater than 1, the ratio R f is necessarily smaller than 1. This prediction is in agreement with the experimental observation showing that adding large grains in a mixture of fine particles (by keeping the total mass constant) leads to an increase in avalanche runout. We compared the prediction of this heuristic model to data on deposit length, L, normalized by the deposit length, L 2 , for r = 0 (large grains only). It is crucial to notice that in Fig. 10 is reported the ratio R f multiplied by the ratio (L 1 /L 2 ) because the data are normalized by the reference value for r = 0. Yet, by construction, R f is defined with regard to a reference situation for r = 1. We do not have many data for high r values corresponding to the regime (III) but we can observe a good agreement between the data and the heuristic model. Figs.

6, 9 and 11 are relatively flat for values r greater than 0.7. Thus the limit between the regimes (III) and (II) is r III-II = 0.7 in agreement with the value proposed by Phillips and others (2006).

The regime (II) is a transition regime (r I-II < r < r III-II ) where involved mechanisms are more complex than in regimes (I) and (III). The regime (I) shows that adding fine particles in a mixture of monodisperse large grains tends to lubricate the mass of large grains. This induces an increased mobility. The regime (III) shows that adding large grains to a mixture of monodisperse fine particles tends to reduce the contact surfaces avaiblable for energy dissipation by interparticle friction. This induces an increased mobility. The transition between both regimes is necessarily characterized by the existence of a maximum. In the regime (II), we observe a maximum in avalanche runout (L and x G show a maximum) correlated to a minimum in deposit height (H max and H x show a minimum). At the critical value r crit , deposits are thin and lengthened. We reported the ratio -1 = L/H x versus the mass proportion of fine particles in Fig. 11. -1 is defined as the ratio between the deposit length L and the mean deposit height H x . This ratio, which characterizes the deposit extension, exhibits a maximum at the critical value r crit . We called it -1 to refer to the ratio = H x /L, classically used in depth-averaged equations models describing shallow avalanche flows ( << 1). The critical value r crit for which the mobility is maximum is around 0.25 in our experiments and, to our knowledge, it does not exist any model to predict it. We will come back to this value in conclusion.

Discussion and conclusion

The experiments described above show the strong influence of the binary mixture composition on the averaged characteristics of the granular avalanche. Our 3D experiments are in good agreement with the experimental results about 2D, channelized, flows in (Phillips and others, 2006) and the 2D discrete numerical results in (Linares-Guerrero and others, 2007) concerning the maximum runout. They bring new results on the influence of the binary mixture composition on (i) the final shape of the full 3D deposit (maximum flow depth, the location of this latter on the main flow direction, mean width) and (ii) the dynamics of the 2D, channelized, incident flow.

Apart from these results largely discussed in the paper, we close the paper by analysing the possible implications of these results on avalanches of dense and dry snow.

Dynamics of the incoming flow and runout deposit

We analysed the influence of the mass proportion in fine particles on the deposit in the runout zone and on the dynamics of the incoming flow. Re-sults show a strong influence of the mass proportion in fine particles inside the binary mixture on (i) the flow dynamics that can be characterized by the Froude number of the incoming flow F r (diamond symbols in Fig. 9) and (ii)

the shape of the deposit that can be characterized by the ratio -1 = L/H x (Fig. 11). Then, a question arises: can we relate the ratio -1 to the Froude number F r? Fig. 12 shows the ratio -1 , which will be named "deposit lengthening coefficient", versus the Froude number. No correlation is found between these two dimensionless numbers. On the curve displayed in Fig.

12, we distinguished data for which r is greater than r crit on the one hand from the data for which r is smaller than r crit on the other hand. First, two distinct groups appear. The "deposit lengthening coefficient" ranges from 40 to 100 for both groups but with different values of the Froude number of the incoming flows: F r ≈ 5 -7 for r < r crit (group A in Fig. 12) and F r ≈ 7 -9 for r > r crit (group B in Fig. 12). Second, in each group, a same Froude number can give a very distinct avalanche extension (see, in Fig. 12, the points A 1 and A 2 on the one hand and the points B 1 and B 2 in the other hand).

This result highlights that not taking into account an internal property of the flowing material (the granulometry here) and the induced mechanism (the segregation here) can lead to an apparent macroscopic behavior being not intuitive. It points at a crucial implication for dry snow avalanches. Indeed, when operating avalanche zoning or designing a protection dam in avalanche engineering, it is necessary to calculate the maximum avalanche extension on the basis of average models: either the simplified analytical models (see, for example, the method proposed by (Salm and others, 1990) and largely used in avalanche engineering) or the advanced numerical models based on shallow depth-averaged equations (for example (Naaim and others, 2004)).

These models are based on the basal effective friction coefficient, µ, which has been to be estimated as a function of the snow type in particular. The friction coefficient µ drives the runout distance when using these analytical or numerical models. Given that the small-scale experiments presented in this paper show the strong influence of the granulometry on the avalanche runout (a same Froude number but a different r can lead to two different values of ), µ should be chosen by taking into account the snow granulometry. Future research should be devoted to the determination of the friction coefficient as a function of the proportion in small snow grains and the mean diameter of the large aggregates, the ratio between the larger aggregates and the smaller grains, and also the particle shapes.

Influence of the diameters ratio and size effect

Our study at small-scale shows that segregation induces an increased mobility if the fine particles proportion is at a critical value (0.25 in our experiments). The ratio D = d 2 /d 1 was fixed between 4 (d min 1 =250 µm) and 6.66 (d max 1 =150 µm) in our tests. Linares-Guerrero and others (2007) have analysed the effect of this ratio. They have shown that the higher the difference between the diameter of large and small particles, the more important the effect of segregation is. We have to keep in mind that all the effects observed would have been amplified if we would have had a larger D, and conversely, softened if we would have had a smaller D. The mobility will be more and more enhanced when the ratio D will be higher. We can see here a kind of size effect. Indeed, if the avalanche volume is larger, its flow depth is larger and the maximum size of snow aggregates may be also larger. The idea suggested here relies on the statement that the initial largest blocks and chunks can keep their size, or some of the blocks or chunks may grow, during the flow propagation leading to snow aggregates of large size. It is a challenge to investigate such a statement by well-targeted snow experiments in the future. However, if this statement is true, it may result in a large ratio between the typical size of snow aggregates and the size of small snow grains, and consequently amplified segregation effects. In this sense, we think that the segregation mechanism may be a relevant candidate to enhance the mobility of major dry and dense snow avalanches. It could provide arguments to explain the field measurements showing a decrease of the effective friction as a function of the volume generally observed for geophysical flows [START_REF] Legros | The mobility of long-runout landslides[END_REF], and also for snow avalanches [START_REF] Ancey | Monte carlo calibration of avalanches described as coulomb fluid flows[END_REF]. If we consider a snow avalanche made of snow aggregates of typically 10 cm in size (d 2 =0.1m [START_REF] Bartelt | Instruments and Methods. Granulometric investigations of snow avalanches[END_REF]) and millimetric snow grains (d 1 =10 -3 m), then the ratio D can reach a value extremely high equal to 100! It would result a very low value of the effective friction coefficient if the proportion in fine particles is close to the value corresponding to a minimum (for r = r crit ). Coming back to our small-scale experiments, the critical proportion is r crit = 0.25 for 4 < D < 6.66. From the preliminary results of Linares-Guerrero and others (2007, figure 12), the ratio r crit is likely to decrease when D increases. Thus, it can be an additional amplication factor. At high D values, a weak proportion of fine particles (small r crit ) is sufficient to produce a largely enhanced mobility. In parallel to researches already mentioned in conclusion of the previous section, it appears crucial to systematically investigate the granulometry of snow avalanches deposits when full-scale avalanches are surveyed following the example of Issler and others (2008). A recent study showed that the granular-size distributions in avalanche snow deposits is nearly log-normal [START_REF] Bartelt | Instruments and Methods. Granulometric investigations of snow avalanches[END_REF].

Further investigations on polydisperse granular materials are also a future challenge to corroborate or not the conclusions reported here for simple binary granular mixtures. The segregation mechanism can also take place in many other geophysical flows where the flowing material is polydisperse with the presence of fine particles, with a large difference in diameter between fine and large particles, and with various shape particles. Alternatively black and white fringes were projected on the bottom of the downstream plane to visualize, thanks to a digital camera (n • 2), the boundaries of the granular mass in the runout zone (see Figure 4).

3. 2 .

 2 Fig. 6 shows various features of the granular avalanche deposit versus the mass proportion in fine particles r: length L, position x G of the center of mass, spreading length x spreading above the center of mass, maximum width , maximum height H max , mean depth along the centerline of the deposit H x and position x[H max ] of the maximum height. Each dimension x has been normalized by the corresponding dimension, x 2 , obtained with the monodisperse mixture of beads with diameter d 2 =1 mm. In the following, the results are systematically normalized by the values obtained with the monodisperse mixture made of beads with diameter d 2 =1 mm (reference values with subscript 2). In Fig. 6, for the dimensions L and , are given the results obtained from the fringes projection technique combined with phase shifting as well as the results from the direct measure with a ruler. Similar results are obtained with both methods, which validates our implemention of the fringes projection technique. The deposit length is very sensitive to the ratio r. It displays a maximum around r = 0.2 -0.3 and this maximum reaches the value 1.4. It shows

Fig. 7

 7 Fig.7shows examples of mean depth-profiles obtained from various values of the ratio r. The segregation is shown to decrease the thickness of the deposits. At the critical value of the ratio r (around 0.2-0.3), the deposits are lengthened and thinner. At values greater and smaller than the critical ratio, the deposits are thicker and shorter. Fig. 6 also shows the maximum deposit height, H max , as well as the mean depth along the centerline of the deposit, called H x . Both H max and H x are very sensitive to r. The height reduction is maximum (around 35-40%) at the critical ratio r crit =0.2-0.3.

  and (II) is close to the maximum observed on Fig. 10. It is then taken identical to the critical value r crit : r I-II = r crit = 0.25. Philipps and others (2006) proposed a similar value based on their experiments.

Figure 1 :

 1 Figure 1: Sketch of the experimental set-up and instrumentation. The channel width was w c =0.30 m, its length was L c =1.50 m and its slope inclination was fixed at θ c =35 • . A laser sheet was projected on the channel bottom to visualize, thanks to a digital camera(n • 1), the displacement of the granular mass (flow depth h and front velocity u) in the flowing zone. The locations x 1 was at 55 cm from the reservoir exit and the distance between two successive positions x i was around 13.75 cm. The width of the downstream plane was w r =1.20 m, its length was L r =2.50 m and its slope inclination was θ r =6 • .

Figure 2 :

 2 Figure 2: Flow-depth temporal profiles measured thanks to the shifted laser line at different channel locations. Example for r = 0.26: the height h retained is the mean value of the maxima at locationsX 3 [h(X 3 ) =10 mm], X 4 [h(X 4 ) =9.3 mm] and X 5 [h(X 5 ) =8.8 mm],which gives h ≈ 9.37 mm.

Figure 3 :

 3 Figure 3: Granular front position x f in the channel versus the time. Example for r = 0.26. The slope of the straight line allows us to estimate the mean front speed (u) in the investigated flow zone where the granular mass deflects the laser line (see figure 1).

Figure 4 :

 4 Figure 4: (a) Photograph of a granular deposit in the runout zone, (b) image of a fringes network projected on a deposit on the runout zone and (c) example of a numerical deposit obtained by the phase shifting method (color levels indicate the value of the deposit thickness, from the low heights in blue color to the larger ones in red color (the numerical values given on axis correspond to pixels).

Figure 5 :

 5 Figure 5: (top) Visualization of the frontier above which no fine particles are present. The area A is a location where only large particles of diameter d 2 =1 mm are present. The area B is a location where we can find both fine and large particles with a majority of fine particles, of diameter d 1 , located at the base of the deposit. The frontier between the areas A and B is visible on the photograph and appears to be white colored, which provide evidence of the segregation operating inside the bidisperse granular material. (bottom)Some material has been removed from the deposit to show the evidence of segregation processes: a bed of fine particles is located at the base of the flow while the larger ones are above and go further (in the main flow direction and also laterally).

Figure 6 :

 6 Figure 6: Various features of the granular avalanche deposit versus the mass proportion r in fine particles: length L (triangle symbols), position x G of the center of mass (circle symbols), spreading length x spreading above the center of mass (diamond symbols), maximum width (square symbol), maximum height H max (times symbol), mean depth along the whole profile H x (cross symbols) and position x[H max ] of the maximum height (star symbols). Each dimension, x, has been normalized by the corresponding dimension, x 2 ,obtained with the monodisperse mixture of beads with diameter d 2 =1 mm (r = 0). Note that the results from the direct measurements with a ruler are also shown for L/L 2 (red line) and / 2 (blue line) to validate the fringes projection technique. Since large and fine particles have the same density, the mass ratio r is equal to 1 -V g , where V g is the volume proportion of large grains.

Figure 7 :

 7 Figure 7: Evolution of the deposit shape as a function of the mass proportion, r, in fine particles: mean depth profile (H) for various mass proportions in fine particles: r = 0, r = 0.104, r = 0.2015, r = 0.299, r = 0.403, r = 0.5005, r = 0.6045 (see Figure 1 for the definition of x = 0 corresponding to the flume exit).

Figure 11 :

 11 Figure 11: Ratio of the deposit length to the mean deposit height, -1 = L/H x , versus the mass proportion r in fine particles of diameter d 1 .

  The diameter d 2 is nearly ten times greater than the diameter d c , which implies that the 1 mm in diameter beads can not be trapped easily in the roughness (d 2 /λ r ∼ 4). On the other hand, the diameter d 1 is less than twice greater than d c (d 1 /λ r ∼ 0.8), which implies that fine particles of 200 µm in diameter are more easily trapped. It results, at constant initial mass and slope angles, a runout greater for the large grains of diameter d 2 .

The value of d c is equal to 0.42λ r for a typical roughness compacity of 0.72, mean value between the maximum compacities in two dimensions (C 2D ≈ 0.64) and three dimensions (C 3D ≈ 0.8). The mean value of λ r is 250 µm (corresponding to the norm FEPA for the sandpaper P60 used in our tests), which gives the value d c ∼105 µm.

Acknowledgments

The authors acknowledge financial support from the French National Research Agency (ANR-MONHA). The authors thank Mohamed Naaim for very fruitful discussions, and T. Faug is grateful to C. Schlosser for her help to initiate the preliminary experiments on bidisperse granular avalanches. We would like to thank to an anonymous reviewer for the constructive criticisms which helped to improve the quality of the paper.