Hélène Bouscasse

Iragaël Joly

Jean Peyhardi

Estimating travel mode choice including rail in regional area, based on a new family of regression models

. In logit models used by economists, the link function can be decomposed into the reference ratio of probabilities and a cumulative distribution function (cdf). Alternative cdfs (Student, Cauchy, Gumbel, Gompertz, Laplace, Normal) can be used. These cdfs differ in their symmetry (symetric or asymetric distributions) and in their tails (heavy or thin tails), each allowing a different distribution of behaviors. We test the statistic and economic implications of changing the cdf. First, we investigate the goodness-of-fit indicators (AIC, BIC, Mc-Fadden R 2 ). Then, we compare estimated parameters in terms of sign and significativity. And finally, we look at behavioural outputs such as value of time and demand elasticities. We use a recent stated preferences survey conducted by the author in the Rhône-Alpes Région (France). Its specificity is to specifically address the question of mode choice (rail, coach and car) in a regional context. Attributes include travel time, cost and comfort. We also investigate the cross effect of travel time and comfort. Comparisons between cdfs are made on the basis of three models, including only attributes variables or only individual variables or both. Our results show that the different cdfs provide quite similar results. But, in our estimations, the logistic cdf never ranks among the best options. In terms of significance and sign of coefficients, parameters' estimation are globally the same even if some special features can be noticed. Looking at time equivalence of comfort, we notice that in the model without individual variables, the cdf has a major influence on outputs. In particular, the Student cdf provides very consistent results while some other cdfs (e.g. Gompertz, Logistic, Normal) are extreme.

Introduction

Mode choice modeling is a recurring challenge, broadly discussed in the transportation economics literature. It is the support of the development of discrete choice models: conditional logit [START_REF] Mcfadden | The measurement of urban travel demand[END_REF], nested logit (Ben-Akiva, 1 1974), mixed logit [START_REF] Mcfadden | Mixed mnl models for discrete response[END_REF] were all developed and illustrated with mode choice applications. While developing these models, the authors proved their compatibility with economics theory, namely the random utility models [START_REF] Thurstone | A law of comparative judgment[END_REF]. Thus, they are used to explore the determinants of mode choice behavior and to derive economic concepts such as willingness-topay, value of time, elasticities or market shares. For example, [START_REF] Wardman | Consumer benefits and demand impacts of regular train timetables[END_REF] uses a multinomial logit to calculate the rail demand elasticities of a general cost function, a clockfaced index and a timetable's memorability index. [START_REF] Yanez | Inclusion of latent variables in mixed logit models: modelling and forecasting[END_REF] calculate subjective value of time, wait and walk with an integrated choice latent variables model . They also predict changes in market shares following an artificial increase of individual income.

Multiple variations of the above mentioned models can be found in the literature. In particular, a significant body of literature on discrete choice model deals with heterogeneity in behaviour and preferences [START_REF] Fiebig | The generalized multinomial logit model: accounting for scale and coefficient heterogeneity[END_REF][START_REF] Hensher | Applied Choice Analysis[END_REF]. They focus on changes in error terms specification. For example, HEV models specify errors which are not independently distributed. In mixed logit, parameters are random since the error terms are divided into two parts with one following any distribution specified by the author. Latent class models authorizes variations of coefficients between groups.

For the multinomial logit, the conditional logit and the mixed logit model, the link function between the probabilities and the predictors is the multinomial logit function (canonikal link function). It can be remarked that this function is defined using the logistic cdf. Alternative cdfs were proposed by [START_REF] Peyhardi | A new specification of generalized linear models for categorical responses[END_REF], e.g. Student, Cauchy or Laplace cdf. These cdf differ in their symmetry (symetric or asymetric distributions) and in their tails (heavy or thin tails), each allowing a different distribution of behaviors. For example, heavy tails distributions, such as the Cauchy or Student cdf, permit a greater dispersion ofthe response behavior to changes in predictors. Our aim is to test and compare these new specifications with the logistic cdf and to address the economic implications of such models on a real and recent database collected by the author. Beyond explanatory consequences and potential differences in significance of predictors, we test to which extent these new models affect mode choice and behavioral outputs like value of time, equivalent between time and comfort as well as elasticities.

In this paper, we use a dataset that explores the determinants explaining train as a mode choice. Train is a relevant alternative for some suburban trips but mainly for interurban trips. We also believe that to understand the determinants of rail use, it is important to study the behavior of train users as well as non-train users such as car users and coach users. Nevertheless, mode choice models including rail have previously been studied either on a single specific rail corridor (Kottenhoff, 1995;[START_REF] Ben-Akiva | Comparing ridership attraction of rail and bus[END_REF]Börjesson and Eliasson, 2011) or interviewing only train users [START_REF] Wardman | Consumer benefits and demand impacts of regular train timetables[END_REF][START_REF] Richter | Modelling mode choice in passenger transport with integrated hierarchical information integration[END_REF]. One exception is [START_REF] Arentze | Travelers' preferences in multimodal networks: design and results of a comprehensive series of choice experiments[END_REF] who conduct four stated preferences survey (SP) in the Netherlands, interviewing all types of users. In three surveys, rail alternatives are proposed. But there is no car alternative, except for medium distances (20 kms). Yet it seems that, in SP surveys, offering alternatives to the train is a condition for understanding mode choice. We do not know of any paper that examines regional trips, offering differentiated travel modes to travelers. We intend to fill this gap with the data analyzed in this paper. [START_REF] Ben-Akiva | Comparing ridership attraction of rail and bus[END_REF] provides a comprehensive list of factors which are important in a choice of a mode between train and coach but which are usually not specified in the models. It includes for example, reliability, comfort (e.g. seating availability) or safety from accidents. For these specific characteristics of train, stated preferences data may be preferred over revealed preference data. Indeed, in a real choice world the variables may not vary sufficiently (e.g. frequency) ; there may be strong correlations between variables of interest (e.g. time and cost) ; it is not possible to directly evaluate demand under conditions which do not yet exist (e.g. a new timetable with clockfaceness) ; variables must be expressed in objective units (such as euros for cost) which is difficult for some secondary variables such as seat design. Neverthess, stated preferences data also have drawbacks [START_REF] Wardman | A comparison of revealed preference and stated preference models of travel behaviour[END_REF]. Stated preferences may not correspond to actual preferences due to systematic bias responses or because of difficulties in carrying out the choice task.

In our survey, we study comfort which is solely addressed in the literature. Comfort, and more precisely the possibility of working or resting during the trip, is often seen as an asset of rail trips. But this advantage is being challenged in trains where seating availability is not guaranteed. [START_REF] Richter | Modelling mode choice in passenger transport with integrated hierarchical information integration[END_REF] find out that seat availability is the most important component of comfort in trains. Most papers in which values of time are differentiated according to the mode show that value of time is greater for car trips than for public transport trips. [START_REF] Fosgerau | Between-modedifferences in the value of travel time: Self-selection or strategic behaviour?[END_REF] investigate the reasons explaining this result: strategic bias responses, self-selection (car users having higher values of time) or linked to comfort (travel time is seen less as a waste of time if mode choice is the train). There may thus be a link between comfort and travel time: the longer the trip, the more important the guarantee of a seating availability. For more literature on travel time use, see [START_REF] Lyons | Comparing rail passengers' travel time use in great britain between 2004 and 2010[END_REF][START_REF] Joly | Intensive travel time: an obligation or a choice?[END_REF].

The paper is organized as follows. Section 2 presents the new specification in the context of discrete choice models. Section 3 presents the comparison method of the estimations with the different specifications and presents the data. Results are discussed in section 4. Section 5 concludes.

2 Generalized linear models for categorical data

Logit models

The family of logit models is a particular subset of generalized linear models [START_REF] Nelder | Generalized linear models[END_REF] which are composed of a random response variable associated to a probability distribution, a set of predictors which are deterministic and a link function which describes the relation between the predictors and the expectation of the response variable. Such models are characterized by the distribution of the response variable Y , the linear predictor η and the link function between the expectation E(Y ) and η.

Let Y i denote the response variable corresponding to the choice of individual i (with alternatives j = 1, . . . , J), x i denote the vector of individual variables and ω i,j denote the vector of alternative-specific variables. In the following, we will suppress the individual subscript i without loss of generality. Luce's choice axiom [START_REF] Luce | Individual Choice Behavior: A Theoretical Analysis[END_REF] and the principle of random utility maximization lead to the logit defined by

π j = exp(η j ) 1 + J-1 k=1 exp(η k ) (1)
where π j = P (Y = j). Depending on the form of the linear predictors η j , we obtain different logit models:

• η j = α j + x T δ j . Individual characteristics x are used with J -1 different slopes δ j . This is the classical multinomial logit model.

• η j = α j + ωT j γ where ωj = ω jω J . Choice characteristics ω j are used with common slope γ. This is the conditional logit model introduced by [START_REF] Mcfadden | The measurement of urban travel demand[END_REF].

• η j = α j + x T δ j + ωT j γ. This is a combination of the two previous parametrizations.

A new specification

To introduce the new specification, let us remark that the equation ( 1) is equivalent to: π j π j + π J = exp(η j ) 1 + exp(η j )

, j = 1, . . . , J -1.

(2)

More generally, all the classical regression models for categorical responses [START_REF] Tutz | Regression for Categorical Data[END_REF][START_REF] Agresti | Categorical Data Analysis[END_REF] share the generic equation

r j (π) = F (η j ), j = 1, . . . , J -1,
where r is a differentiable and invertible map between the simplex ∆ = {π ∈ (0, 1) J-1 | J-1 j=1 π j < 1} and an open subset of the hypercube (0, 1) J-1 , π is the vector of probabilities (π 1 , . . . , π J-1 ) T and F is a continuous and strictly increasing cdf [START_REF] Peyhardi | A new specification of generalized linear models for categorical responses[END_REF].

The ratio of probabilities r is chosen according to the ordinal or nominal nature of the response variable. The reference ratio (see the left part of the equation ( 2)) is appropriate in the context of qualitative choices because there is no natural ordering among the different alternatives. Therefore, this part of the equation is conserved. Now, remark that the right part of the equation ( 2) is the logistic cdf. The main idea of our methodology is to replace the logistic cdf by other continuous and strictly increasing cdfs. The proposed models are thus described by the equations

π j π j + π J = F (η j ), j = 1, . . . , J -1.
where F is either the logistic, normal, Laplace, Gumbel, Gompertz or the Cauchy cdf. The heavy tails of the Cauchy distribution or the asymmetry of the Gumbel and Gompertz distributions may markedly improve the model fit.

Since all these cdfs are strictly increasing functions with η j , the parameters can still be easily interpreted (Equation ( 3)). For instance, if the slope parameter of the alternatives cost is significantly negative then the proportion of alternative j is decreasing when its cost is increasing.

π j π J = F 1 -F (η j ) (3)
In the particular case of the logistic cdf, we have :

π j π J = exp(η j ) (4)
This family of models stays easily estimated using the Fisher scoring algorithm.

The scores and the Fisher's information matrix computation are a few changed by the use of different cdf F ; see [START_REF] Peyhardi | A new specification of generalized linear models for categorical responses[END_REF] for more details. In the same way, the elasticities of a single alternative specific characteristic ω are given by

∂ ln π j ∂ ln ω k = γω k d k (1 -π k ) , j = k, -π k , j = k, (5)
where

d k = f (η k )/[F (η k ){1 -F (η k )}],
f denoting the derivative of F , i.e. the associated probability density function. For the particular case of logit model (i.e. F is the logistic cdf) we have f = F (1-F ) (i.e. d k = 1 for all alternatives k) and ( 5) is the classical result of elasticities computation for logit model, given in [START_REF] Greene | A latent class model for discrete choice analysis: contrasts with mixed logit[END_REF] for instance. One can remark that the IIA property is conserved since d k is not depending on alternative j.

3 Methods, data and models

Methods for comparing models

Our objective is to compare the different specifications of the model, in terms of explanation and in terms of prediction. We test 7 seven cdfs: Cauchy, Gompertz, Gumbel, Laplace, Logistic, Normal and Student.

Comparison of models is a classical task. For example, [START_REF] Greene | A latent class model for discrete choice analysis: contrasts with mixed logit[END_REF] compare latent-class model to mixed logit and multinomial logit. To do so, they use Log-Likelihood, pseudo-R2, estimates of Value Of Time (VOT) kernel density for choice probabilities (mixed and latent-class model), direct elasticities of time and cost as well as probabilities profiles of mixed and latentclass models. [START_REF] Yanez | Inclusion of latent variables in mixed logit models: modelling and forecasting[END_REF] uses likelihood ratio tests to compare the fit of a multinomial logit and two hybrid choice models. They also compare predictions of market shares and so-called "subjective value of time". [START_REF] Fiebig | The generalized multinomial logit model: accounting for scale and coefficient heterogeneity[END_REF] propose a systematic comparison of multinomial logit, scale multinomial logit, mixed logit and generalized multinomial logit with AIC, BIC and CAIC indicators. They also compare the willingness-to-pay between models and face them to the comparison of the percent of individual choosing an alternative versus another facing a given price change.

The methods to be used depend on the objectives (e.g. AIC is preferred to BIC to predict an outcome), on the comparability of the models (LR tests require the models to be nested) and on the nature of the data (comparisons on VOT or elasticities show which estimates are in the range of plausible values but do not point to a "better" model if data are real and not simulated).

The 7 models we propose are comparable in terms of goodness-of-fit: McFadden R2, Loglikelihood, AIC and BIC. The three lasts indicators will offer the same results since all models have the same number of parameters except the one with the Student cdf which requires the estimation of only one additional parameter, the degrees of freedom.

As a second step, we focus on the outputs of the models: significativity of parameters, elasticities, value of time and equivalence between time and comfort.

With this approach, we can grasp the scopes of values and estimate the impact of choosing one model over the others. Even if the scales of parameters are partly normalized to provide comparable estimations, a residual scale effect preclude a direct comparison of estimated parameters . We thus focus on a comparison of significativity and on the relation linking coefficients of a model. Indeed, the parameters' estimations should correspond to identical preference structures between models [START_REF] Viney | Empirical investigation of experimental design properties of discrete choice experiments in health care[END_REF]. This linear structure is frequently illustrated plotting estimates of coefficients from two models [START_REF] Flynn | Estimating preferences for a dermatology consultation using best-worst scaling: comparison of various methods of analysis[END_REF][START_REF] Kerr | Efficient design for willingness to pay in choice experiments: evidence from the field[END_REF]. We also propose a comparison of significativity. Concerning elasticities and equivalences cost-time-comfort, there is no scale effects, but the formulas integrate derivates of utility with respect to the attributes (see part2.2).

Data

We conducted a choice experiment (web and face-to-face) among 1, 774 inhabitants of the Rhône-Alpes region, in February and March 2015. Each respondent had to choose between three travels modes: train, coach and car. In the experiment, alternatives are described in terms of mode (train, coach and car), cost, time, probability and time delay, frequency, clock-face timetable and comfort.

To avoid a cognitive burden, attributes describing the journey are split into three exercises. In exercise 3, on which we focus here, modes vary according to time, cost and comfort (Table 1). Respondents had to answer to four choices questions in exercise 3, leading to a database with 6,373 observations since some rare respondents did'nt answer all four questions.

Respondents described a reference journey in terms of time, cost, purpose, origin and destination... This reference journey is then used to personalize the choice questions and minimize the well-known hypothetical bias. Levels of attributes are also presented in Table 1. Levels of time and cost attributes are pivoted around the values collected for a reference journey. One of the three alternatives was systematically a status quo alternative with the mode, travel time and travel cost identical to the reference journey.

To improve the efficiency of the design, a Bayesian efficient design was implemented. A priori weights of attributes were taken from the literature and adjusted during the pilot tests. Degree of freedom 4 Results

Seven specifications are estimated and compared on three models (S1, S2 and S3). Models are estimated using the sofware "Statiskit" developed by Pierre Fernique and Jean Peyhardi, in C++ with an interface in Python. For the logistic distribution, we compared and validated estimations' results with the one provided by the package "Mlogit" in R. The Student distribution depends on the degrees of freedom (df) chosen. Several Student distributions were tested to select the one that maximizes the log-likelihood which appears to be empirically generally concave. As shown in 2 and the zoom 3, it is the Student distribution with 2 degrees of liberty which meet that criterion for the S3 model (and df=0.35 for the S1 and df=1.5 for S2; 4 to 7 ). Note that for model S3, the specification with the Gompertz cdf did not converge.

Comparison of models

In this subsection, we compare the models considering the usual goodness of fit indicators (

Estimation of models

Before comparing estimations between models, we can draw some conclusions on the effects of the included variables on the probability to choose the coach or the car option relative to the train option. All significant effects are consistent with theory, in the final model S3 including alternative and individual variables. Some exceptions appear in S1 and S2 models.

As expected, time and cost are associated to significant and negative effects.

Comfort is significant and positive. In model S3, the crossed effect of time and comfort positive and globally significant, indicating the propensity to accept longer time in comfortable situation. In model S1, it is either non significant or significant and negative which is contrary to intuition. Except with the Cauchy cdf in model 1, car and coach are associated to a significantly smaller baseline utility than train.

Age, high income, using coach and car in the reference journey, having already used a car alternative for the reference journey and level of accessibility to a car all increase the probability to choose the coach or the car options in the SP survey. Population density at destination, regular trip and frequent use of modes other than car, all decrease the probability to choose the alternatives to train. Other variables have adverse effects. For example, density at the origin has no influence on the coach choice's probability but has a positive effect on car choice which is surprising. Indeed, our expectation was a positive effect: high density being a motive to take public transports. On the contrary, having an imperative at destination and having already used a public transport alternative for the reference journey decrease the probability of choosing the car option.

Comparisons of estimations are done on the sign of coefficients, their significativity and the structure of preferences (figures 8 to 22).

First, we can notice that the signs of coefficients are the same whatever the chosen cdf and the model investigated. Exception is, as mentioned before, in the model S1, for the car intercept and for the cross-variable (Time*Comfort). Indeed, the car intercept in model S1 with the Cauchy cdf is positive whereas it is negative in all other situations. The cross-variable is significantly negative in model S1 (Gompertz, Logistic, Normal) and significantly positive in model S3 (all cdfs taken together).

Second, significativity of variables can vary greatly between estimations. But we can notice some patterns. In model S1, the only differences in significativity concerns the cross variable (time*comfort) which is either not significant, either highly significant. In model S2, some variables' significativity is stable over specifications (e.g., car intercept) and others are highly heterogeneous. Some individual variables have thus higher significativity depending on the model. For example, age associated with the car option is significant at the 0.1% level with the Cauchy or Laplace cdf and not significant with the Gompertz cdf. In model S3, the estimations done with the Cauchy cdf seem to be atypical. The car intercept is not significant, the density at destination has a highly significant negative effect. These differences in terms of variables significance show that the choice of the distribution matters if the model is used for explanatory purposes.

The general structure of the preferences seems to be captured almost equivalently by the different model specifications. Figures 8 to 22 illustrate relatively stable patterns of the estimates. The Cauchy and the Student give estimates closer to the logistic ones than the Normal, the Gumbel and Laplace distributions. 2013) finds that users are ready to spend between 13 and 51 more minutes in the train to avoid a standing position in comparison to a seated position.

We also notice that choosing one cdf over the others has strong implications in terms of prediction and thus in terms of transport policy which commonly uses these kinds of values to calculate a generalized travel cost or time function. Indeed, even considering only model S3, VOT may be increased by up to 16% depending on the chosen cdf and equivalent time of comfort by up to 38%. This result has strong implications.

Elasticities

As a first step, elasticities are presented without probabilities' weighting (Table 9 andTable 10). Weighted elasticities are available from authors upon request.

Own-elasticities measure how the probability of choosing alternative i is influenced by increasing time or cost of the same alternative i. Cross-elasticities measure how the probability of choosing alternative j is influenced by increasing time or cost of alternative i (∀j = i).

Probability of choosing the train and coach options is much more sensitive to their own cost and time than probability to choose the car option. It is hard to find a specific pattern for cross-elasticities, which may be due to the IIA property.

Comparison between cdfs show that there is a strong heterogeneity, especially for models S1, models S3 being more homogeneous. As for VOT and equivalent time of comfort, we notice that with some distributions, misspecification has more impacts. For example, own-time-elasticities with the Normal or the Gumbel cdf are very different in model S1 and in model S3. For a same type of model (S1, S2 or S3), our results suggest that changing the cdf does not alter the estimated sign of the covariables effects on the choice probability (apart one exception in model S1), but can lead to variations in the significance level of the coefficients. Whereas the coefficients can not be compared in absolute value between models, we can observe relatively stable structure of the preferences (or weight of covariables in the decision).

Models S1 and S3 are voluntary misspecified since important variables are withdrawn. We show that some cdfs can encounter this misspecification and provide behavioral outputs consistent with literature or expectations. In model S1, the Student (0.35) cdf largely outpeforms the other cdfs. Even if the model is misspecified, the Student (0.35) cdf captures the effects of time and comfort. In model S2, it is less clear but we can notice that the Laplace, Cauchy and Student (1.5) cdfs stand out.

Conclusions drawn with model S3 are consistent with expectations. Analysis of mode choice between rail, coach and car indicates usual significant negative effects of time and monetary cost. The comfort attribute is a significant and positive determinant of mode choice. We also find a crossed effect with travel time. Comfort increases the propensity to accept higher travel time. Estimated value of time are around 12 euros / hour if a seated position is guaranteed, and around 14 euros / hour if it is not. Our results suggest that for example substituting the Logistic cdf for the Cauchy cdf induces a reduction in value of time (with comfort = 1) of approximately 10% or 1.1 euro / hour. Thus, even with a model that we believe to be correctly specified, changing the cdf has operational impacts. Impacts are also in terms of explanation, since the significant explanatory variables differ from one model to the other. In terms of goodness-of-fit indicators, changing the cdf is associated with a low premium. Yet, we notice that, with our data, the logit function is never associated with the best fit indicators.

Globally, testing a variety of cdfs allowed us to point out the one that best fit the data. If behaviour are heterogeneous, it is interesting to select cdfs with heavy tails. Some behaviour patterns, such as high value of time for some types of users (e.g. car users or travelers with a business purpose), may be preferably modeled with asymetric cdfs. These assumptions still have to be tested on simulated data sets but we can already state that playing on the form of the cdf allows more flexibility and a better fit to the data. Further research on the cdf influence is thus needed to evaluate the potentials gains of the different specifications. This first evaluation is conditional to our data and need to be extended to data from different fields, survey types, etc. Evaluations and comparisons have to be done including the well established and well known models in the literature (as for example, the probit model, nested and mixed logit).

Figure 1 :

 1 Figure 1: Exemple of choice question for exercise 3

Figure 2 :

 2 Figure 2: Log-likelihood depending on the degrees of freedom -S3

Figure 4 :Figure 6 :Figure 7 :

 467 Figure 3: Log-likelihood depending on the degrees of freedom (zoom) -S3

Figure 8 :

 8 Figure 8: Coefficients Comparison -S1 -Cauchy : Logistic

FigureFigure 22 :

 22 Figure 9: Coefficients Comparison -S1 -Gumbel : Logistic

Table 1 :

 1 Attributes by exercise

		Variations unit	and	Exercise 1	Exercise 2	Exercise 3
	Mode			Train, train, car	Train, train, car	Train, coach, car
	Travel time	-30%,0%,+30% (minutes)		X	X	X
	Travel cost	-30%,0%,+30% (eu-ros/travel)	X	X	X
	Frequency	In number of trains or coach/hour	X	X	-
	Time delay	In minutes		X	-	-
	Probability of delay	In %		X	-	-
	Clockfaceness	Yes (1) or No(0)		-	X	-
	Comfort	Guarantee of a seat (1) or not (0)	-	-	X

Table 2 :

 2 Along modal alternative-specific constants (ASC), we include variables describing the alternatives (time, cost, comfort and time * comfort) as well as individual variables. Individual variables are related to socio-economic characteristics (age, income), spatial characteristics (% of the origin and destination municipalities in high density area), journey characteristics (type of user (car or coach user for the reference trip), frequency of the trip, imperative schedule) and general mobility (access to car, frequency of use of modes alternative to car). Summary statistics for alternative-specific variables are in table 2. For coach and car, comfort is set to 1 since seating position is guaranteed. Summary statistics for individual variables are in table3. Concerning the choice made by respondents, 29.6 % chose the train alternative, 21% chose the coach alternative and 49,4% chose the car alternative. Descriptive statistics on alternative-specific variables

		Mean S.D. Min	Max
	CostA	8.88	7.56 0.63	62.00
	CostB	8.90	7.80 0.46	78.00
	CostC	9.99	8.64 0.38	62.00
	TimeA	69.89 51.86 7.00 325.00
	TimeB	70.22 53.23 7.00 325.00
	TimeC	57.24 36.89 4.00 330.00
	ComfortA	0.50	0.50 0.00	1.00
	TimeA*ComfortA	37.16 53.04 0.00 325.00

Table 3 :

 3 Descriptive statistics on individual variables

	Label Definition	Mean	S.D. Min Max
	age In years	47.57 15.76 19.00 90.00
	size hh Size of the household	2.63	1.27	1.00	8.00
	nb car number of cars available in the household	1.69	0.72	1.00	5.00
	access car nb car/size hh	0.76	0.38	0.12	4.00
	income h 1 : Income above 4 000 euros	0.28	0.45	0.00	1.00
	dep dens	% of the origin municipality in high density area	0.74	0.40	0.00	1.00
	arr dens	% of the destination municipality in high den-sity area	0.86	0.30	0.00	1.00
	type coach 1 : user of coach for the reference trip	0.03	0.16	0.00	1.00
	type car 1 : user of car for the reference trip	0.54	0.50	0.00	1.00
	regular 1 : regular trip	0.28	0.45	0.00	1.00
	imperative 1 : imperative schedule at destination	0.45	0.50	0.00	1.00
	alt pt	1 : car user who had already used public trans-ports to do the reference trip	0.15	0.35	0.00	1.00
	alt car	1 : train or coach user who had already used car to do the reference trip	0.26	0.44	0.00	1.00
	freq alt 1 : frequency of use of modes other than car	0.41	0.49	0.00	1.00

  table 4): Log-likelihood, AIC, BIC, and McFadden R 2 .For all distributions we focus on the rank of the models in terms of AIC, BIC, and McFadden R 2 . In model S1, all indicators converge in a common finding: the Student cdf performs the best. It is followed by the Cauchy cdf. In model S2, the Laplace cdf performs best. Cauchy and Student (1.5) follow depending on the criteria. In model S3, the Gumbel cdf ranks first, followed by Student (2) and Laplace cdfs.

Table 4 :

 4 Comparison of fit indicators

	Cauchy Gompertz Gumbel Laplace Logistic Normal Student (0.35)

Table 5 :

 5 Estimation of models S1 Value of Time and equivalence time of comfortDue to the introduction of the cross variable (Time*Comfort), values of time (VOT) differ depending on the guarantee to have a seat in the train. VOT should be greater when seat is not guaranteed since travel time can't be used to rest or to work. Values depend also greatly on the model (S1 or S3) and on the chosen cdf (Table 8). For model S1, if seat is not guaranteed, VOT ranges from 4.44 to 7.42 euros per hour. In the same conditions, it ranges from 12.53 to 14.56 euros per hour in model S3. If seating position is guaranteed, then VOT is comprised between 5.95 and 7.2 euros par hour (model S1) or between 11.47 and 12.66 (model S3). Minimum and maximum values are not always observed with the same cdf. But can notice that the smallest ratio between VOT estimated with models S1 and VOT estimated with model S3 are observed with the Student and Cauchy cdfs which are the best performing model regarding likelihood based indicators for model S1.Despite very heterogeneous across models and cdfs, VOT found in this paper are in line with literature. Literature's VOT also prove to be very heterogeneous depending on mode, travel purpose, type of survey...[START_REF] Abrantes | Meta-analysis of uk values of travel time: An update[END_REF][START_REF] Wardman | Meta-analysis of post 1994 values of non work travel time savings[END_REF]. For example, in a stated preference survey,[START_REF] Arentze | Travelers' preferences in multimodal networks: design and results of a comprehensive series of choice experiments[END_REF] find values between 14.4 and 17.4 euros per hour for train travels. Even if we do not have a great confidence to the values found in model S1, we can notice that they are not totally inconsistent with the french rail context. A stated preference survey done by the french rail operator (RFF) in 2013 provide values between 3.9 and 7.7 euros per hour.Concerning comfort, we convert it in minutes to get a time equivalent. First of all, we notice that most values found with models S1 are totally inconsistent. Cauchy and Student cdfs are the only one providing believable values. The values found with the Student distribution in model S1 are even very close to the values found with models S3. This result tends to support that, even if the model is misspecified, the Student distribution captures well the effects of time and comfort. We now focus on equivalence time of comfort in models S3. For a short travel time of 30 minutes (corresponding to the first quartile of the sample), having a seated position guaranteed is equivalent to 14.51 to 20.07 minutes of travel time. For medium travel time of 60 minutes, it ranges from 17.05 to 23.17 minutes. And for a long travel time of 60 minutes (corresponding to the last quartile of the sample), seated position is equivalent to 19.58 to 26.27 minutes of travel time. In the literature, values of comfort are rare. RFF (

			Cauchy	Gompertz Gumbel	Laplace	Logistic	Normal	Student (0.35)
	Intercepts	Coach Car	-0.608 *** -1.16 *** 0.173 *** -0.617 *** 0.164 *** -0.497 *** -0.403 *** -0.67 *** 0.108 *** 0.176 ***	-0.413 *** -1.223 *** 0.104 *** 0.338 ***
	Alternative-specific variables	Time Cost Comfort Time*Comfort 0.0003 -0.021 *** -0.004 *** -0.01 *** -0.195 *** -0.072 *** -0.099 *** -0.106 *** -0.147 *** -0.079 *** -0.496 *** -0.01 *** -0.012 *** -0.006 *** -0.061 *** 0.606 *** 0.598 *** 0.49 *** 0.433 *** 0.796 *** 0.511 *** 1.089 *** -0.004 *** -0.0001 -0.001 -0.003 ** -0.002 *** 0.003
	*** = significant at the 0.1 % level
	** = significant at the 1 % level
	* = significant at the 5 % level
	. = significant at the 10 % level

*** = significant at the 0.1 % level ** = significant at the 1 % level * = significant at the 5 % level . = significant at the 10 % level 4.3

Table 9 :

 9 Elasticities for models S1

	Own-time-elasticities	Cauchy Gompertz Gumbel Laplace Logistic Normal Student (0.35)
	Train	-1.329	-0.227	-0.702	-1.013	-0.583	-0.461	-2.973
	Coach	-1.156	-0.179	-0.944	-0.885	-0.665	-0.547	-1.345
	Car	-0.583	-0.115	-0.551	-0.551	-0.37381 -0.314	-0.624
	Own-cost-elasticities	Cauchy Gompertz Gumbel Laplace Logistic Normal Student (0.35)
	Train	-1.636	-0.677	-0.884	-1.377	-0.946	-0.803	-3.128
	Coach	-1.380	-0.520	-1.196	-1.170	-1.057	-0.934	-1.377
	Car	-0.953	-0.456	-1.004	-0.877	-0.827	-0.746	-0.842
	Cross-time elasticities	Cauchy Gompertz Gumbel Laplace Logistic Normal Student (0.35)
	Train	0.525	0.103	0.298	0.419	0.252	0.205	1.172
	Coach	0.340	0.046	0.218	0.251	0.174	0.139	0.447
	Car	0.446	0.112	0.340	0.340	0.310	0.266	0.431
	Cross-cost elasticities	Cauchy Gompertz Gumbel Laplace Logistic Normal Student (0.35)
	Train	0.570	0.274	0.33462 0.500	0.355	0.310	1.131
	Coach	0.361	0.128	0.24522 0.296	0.248	0.215	0.417
	Car	0.679	0.431	0.55179 0.647	0.637	0.589	0.552
	5 Conclusion
	Based on the work of Peyhardi et al. (2015), we estimate and compare to the
	multinomial logit six models with different cdfs (Cauchy, Gumbel, Gompertz
	Laplace, Normal and Student). This work is done on three different types of

Figure 17: Coefficients Comparison -S2 -Student : Logistic

Acknowledgments: This work would not have been possible without the funding of the survey by the Regional Rhôone-Alpes Board and its partners.

Acknowledgments: This work would not have been possible without the funding of the declared preferences survey by the Regional Rhône-Alpes Board and its partners


Marie Couliou

Romain Monchaux

Growth dynamics of turbulent spots in plane Couette flow

We experimentally and numerically investigate the temporal aspects of turbulent spots spreading in a plane Couette flow for transitional Reynolds numbers between 300 and 450. Spot growth rate, spot advection rate and large-scale flow intensity are measured as a function of time and Reynolds number. All these quantities show similar dynamics clarifying the role played by large-scale flows in the advection of the turbulent spot. The contributions of each possible growth mechanism: growth induced by large scale advection or growth by destabilization, are discussed for the different stages of the spot growth. A scenario which gathers all these elements is providing a better understanding of the growth dynamics of turbulent spots in plane Couette flow that should possibly apply to other extended shear flows.

Introduction

Transition to turbulence in wall-bounded shear flows often occurs through subcritical scenarios when their natural control parameter, the Reynolds number (Re) is increased. This is the case of plane Couette flow (PCF), Hagen-Poiseuille flow, Taylor-Couette flow (when the outer cylinder is rotating) and plane Poiseuille flow. In all these cases, the laminar profile is linearly stable for Re < Re c but it is actually observed to be nonlinearly destabilized by finite amplitude perturbations above Re g < Re c . Below this threshold Re g , any perturbation brought to the laminar profile eventually vanishes. Above, the flow is highly sensitive to the perturbation nature and intensity. The ultimately observed state results from the competition between coexisting turbulent and laminar areas. When Re is larger than Re t > Re g , this final state consists of homogeneous featureless turbulence, while for Re g < Re < Re t , it involves both laminar and turbulent areas. This mixed state can be unpredictably evolving or steady and organized -as in the turbulent spiral formerly observed by [START_REF] Coles | Transition in circular Couette flow[END_REF] and [START_REF] Van Atta | Exporatory measurements in spiral turbulence[END_REF] in Taylor-Couette flow and later characterized at a much larger aspect ratio by [START_REF] Prigent | Large-Scale Finite-Wavelength Modulation within Turbulent Shear Flows[END_REF]. In both cases, the laminar-turbulent interfaces are found to be oblique, a so far unexplained feature shared by all such patterns now observed in all the extended "subcritical" flows cited abovesee [START_REF] Prigent | Transition to Versus from Turbulence in Subcritical Couette Flows[END_REF], [START_REF] Barkley | Computational study of turbulent laminar patterns in couette flow[END_REF] and [START_REF] Duguet | Formation of turbulent patterns near the onset of transition in plane Couette flow[END_REF] for plane Couette flow and [START_REF] Hashimoto | An experimental study on turbulent-stripe structure in transitional channel flow[END_REF] and [START_REF] Tuckerman | Turbulent-laminar patterns in plane poiseuille flow[END_REF] for plane Poiseuille flow. These patterns are even observed in Waleffe flow, a shear flow without walls, as recently proven by [START_REF] Chantry | Turbulent-laminar patterns in shear flows without walls[END_REF].

More basically, the simple development of turbulence within a formerly laminar state is far from being fully understood. From the early observations by [START_REF] Emmons | The laminar-turbulent transition in a boundary layer. part1[END_REF] in boundary layers, it obviously appears that localized perturbations trigger the growth of turbulent spots that consist of simply connected turbulent regions immersed in an otherwise laminar flow and bounded by fronts that move in order to expand the turbulent area provided Re is large enough. Turbulence in these kinds of transitional flows consists of long streamwise velocity streaks associated with counter-rotating vortices that are stacked together along the spanwise direction. As the wavelength of the streaky pattern is narrowly distributed around a given value λ s , the spanwise growth of a turbulent spot corresponds to the nucleation of new streaks. The mechanisms at work to achieve this expansion have not yet been all revealed in spite of many experimental and numerical studies. Shedding light on the growth process of turbulent spots can also be a key point in understanding the formation and sustainment of organized patterns. In the case of plane Couette flow, for which no mean advection is present, turbulent spot growth can be split into two phases: in the first stage of their development, they grow in a more or less symmetric fashion leading to diamond-like shape as seen in figure 1 and then, when their spatial extension is large enough, they start being distorted and tend to form oblique branches. As first evidenced numerically in the pioneering work by [START_REF] Lundbladh | Direct simulation of turbulent spots in plane couette flow[END_REF], the spot remains centro-symmetric all along the first phase. The flow and pattern symmetries are detailed in an extensive review [START_REF] Barkley | Mean flow of turbulent-laminar patterns in plane couette flow[END_REF]. Figure 1 gives an illustration of the growth of such a spot in our experiment around a localized perturbation. The spot is indeed elliptic, the principal axis along the streamwise direction being longer than the one along the spanwise direction. As time passes, both principal axes tend to have the same length and the spot takes the form of a more or less regular diamond whose sides form an angle with respect to the streamwise direction. This angle is prefiguring the organized pattern orientation. This dynamic results from growth rates that are significantly larger in the spanwise than in the streamwise direction as shown successively by the two experimental works by [START_REF] Tillmark | Experiments on transition in plane Couette flow[END_REF] and by [START_REF] Dauchot | Finite amplitude perturbation and spots growth mechanism in plane Couette flow[END_REF] confirming the numerical investigation by [START_REF] Lundbladh | Direct simulation of turbulent spots in plane couette flow[END_REF]. These researchers focused on global growth rates along both relevant directions and found in first approximation that they were time independent but increasing functions of the Reynolds number as can be seen in figure 2 that gathers their time-average results of spanwise growth rate s a function of Re.

The second phase is illustrated in figure 1 at times larger than 400. Specific studies of this second phase are scarce. In very large domains, [START_REF] Duguet | Formation of turbulent patterns near the onset of transition in plane Couette flow[END_REF] nevertheless present the typical evolution of a diamond shaped spot turning progressively into a turbulent pattern made of several adjacent bands with various orientations that are similar to the more steady patterns described above (see in particular their figures 2 and 4). In domains of moderate extension (say when the spanwise direction is 40 times the distance between the walls), this second phase starts when the spot size becomes sizable with the domain spanwise extension and consists mostly in a reorganization of the turbulent area into an inclined pattern. In the following we will focus on the first phase.

Understanding the mechanisms responsible for the spot growth requires taking into account all the features of the spot during this first phase. From the numerical simulations of PCF by [START_REF] Lundbladh | Direct simulation of turbulent spots in plane couette flow[END_REF] and by [START_REF] Duguet | Oblique laminar-turbulent interfaces in plane shear flows[END_REF] in larger domains, from the model flows studied by [START_REF] Lagha | Modeling of plane Couette flow. i. Large scale flow around turbulent spots[END_REF] and [START_REF] Schumacher | Evolution of turbulent spots in a parallel shear flow[END_REF] and from our own experimental study [START_REF] Couliou | Large scale flows in transitional plane couette flow: a key ingredient of the spot growth mechanism[END_REF], it appears that large-scale flows are always present around the growing spot. Their origin can be explained by the laminar-turbulent coexistence and more particularly by the streamwise flow rate mismatch occurring in the overhang region where the laminar flow on one wall faces turbulent flow on the other wall [START_REF] Coles | Transition in circular Couette flow[END_REF][START_REF] Lundbladh | Direct simulation of turbulent spots in plane couette flow[END_REF]. Large scale flows actual shape depends on the wall normal plane of observation, but they are globally oriented in the spanwise direction and outward at the spot spanwise tips and in the streamwise direction and inward at the spot streamwise tips. Experimental works already cited also report the existence of waves at the spot spanwise tips. These waves may correspond to the advection of streamwise vortices that are actually visible on the snapshots presented in figure 1 and that have been observed from visualizations in spanwise-wall normal plans by [START_REF] Hegseth | Turbulent spots in plane Couette flow[END_REF]. Both these features are thought to [START_REF] Dauchot | Finite amplitude perturbation and spots growth mechanism in plane Couette flow[END_REF] (filled circles), our experiment (filled squares with errobar) and our numerical simulation (filled stars with errobar). Note that σ z sf is defined in the present article as the temporal average of half the total spot spanwise growth rate, or equivalently as the average of both spanwise front velocities. Other authors' data have been obtained graphically from their publications and scaled to comply with the present definition of σ z sf .

be relevant in understanding the spot spanwise growth dynamics. Following the work of [START_REF] Gad-El-Hak | On the growth of turbulent regions in laminar boundary layers[END_REF] on the boundary layer, the spot spanwise growth is usually attributed to local instabilities of the modified laminar profile at the spanwise laminarturbulent interface [START_REF] Hegseth | Turbulent spots in plane Couette flow[END_REF][START_REF] Dauchot | Finite amplitude perturbation and spots growth mechanism in plane Couette flow[END_REF]. [START_REF] Schumacher | Evolution of turbulent spots in a parallel shear flow[END_REF] emphasize the fact that suppressing the large scale recirculation in the parallel flow of interest in their work inhibits the spot growth. The same conclusion may also hold for plane Couette flow. Trying to select the most meaningful features, [START_REF] Duguet | Stochastic and deterministic motion of a laminar-turbulent front in a spanwisely extended Couette flow[END_REF] have studied the growth of turbulent spots in narrow domains for which the periodicity in the streamwise direction is achieved at a wavelength shorter than the typical spot streamwise extent. By doing so, the large-scale recirculation is suppressed and the authors found a stochastic growth resulting from the competition between streak retreat and generation events. The corresponding growth rates are found to be one order of magnitude lower than those usually measured in domains extended in both directions. It is nevertheless difficult to know if this stochastic growth is actually active in such large domains since the stochastic growth rate could be smaller than the measurement uncertainties on the total growth rate. The role of large-scale flows can primarily be seen as a disturbance to the stable laminar profile around the turbulent spot. The temporal evolution of the large-scale flows is quite slow as compared to the rapid turbulent fluctuations and happens on the same order of time scales as those associated to the spot growth as shown for example in our experimental study [START_REF] Couliou | Large scale flows in transitional plane couette flow: a key ingredient of the spot growth mechanism[END_REF] or a recent model by [START_REF] Manneville | Towards a model of large scale dynamics in transitional wall-bounded flows[END_REF]. As a result, the large-scale flows may induce modifications to the laminar flow around the turbulent spot that are strong enough to turn this profile unstable and steady enough to allow the instability to develop. The local corresponding mechanism is the growth by destabilization we alluded to above.

A quantitative study of this mechanism has been undertaken in plane Poiseuille flow by [START_REF] Henningson | Turbulent spots in channel flows[END_REF], but the growth rate was found too small to fully explain the experimentally observed rapid expansion of turbulent spots. If relevant, the two mechanisms we have just described are not sufficient and should be only part of the growth process. In a recent article [START_REF] Couliou | Spreading of turbulence in plane couette flow[END_REF], we have shown the existence of two localizations for the generations of new streaks occurring during the spot growth. We have attributed these two localizations to two different mechanisms.

The first one is local, happens at the spanwise tips of the growing spot and results in the nucleation of new streaks at the laminar-turbulent interface, outside of the turbulent region. It could result from instabilities of the modified laminar flow as explained above or from the stochastic process revealed by [START_REF] Duguet | Stochastic and deterministic motion of a laminar-turbulent front in a spanwisely extended Couette flow[END_REF]. The second mechanism was actually shown by our study. It occurs globally and is induced by an advection due to the large scale flows present around the spot that tend to spread it in the spanwise direction. A first consequence of this spreading is a temporary widening of the streaks constituting the turbulent spot before a new streak is nucleated by some wavelength instability inside the turbulent phase to restore the favoured state consisting in more or less equally spaced streaks.

The spot growth process results from the interplay between several mechanisms and involves large-scale flows. Isolating each mechanism and quantifying its contribution to the total growth rate now seems possible. In the present article, the temporal aspects of spot spreading is investigated in order to quantify to which extend the two identified growth mechanisms are present and interact to achieve the time-averaged results which we have presented in our ealier work [START_REF] Couliou | Spreading of turbulence in plane couette flow[END_REF]. To do so, we analyse experimental and numerical data which acquisition and post-processing will be described in section 2. The main results consist of growth rates, front velocities and large-scale flow measurements that are gathered in section 3. Discussion of these results are given in section 4 before we summarize our findings and identify perspectives of this work in section 5.

Methods

In this section, we present our experimental setup, the numerical code used to perform the direct numerical simulations (DNS), the protocols used to trigger the turbulent spots, the measurement and visualisation systems and the various post-processing techniques applied to our data.

Experimental setup

As sketched in figure 3 (left), an endless 0.25 mm thick plastic belt is stretched between two cylinders. One of them is connected to a brush-less servo-motor which drives the system through a gear-reducer of ratio 9. This motor can achieve a maximal speed of 2500 rpm. Speed and acceleration are controlled to an accuracy of 2/1000. Four smaller cylinders guide the belt and enable to adjust the gap between the two walls moving in opposite directions at an accuracy of 0.05 mm. All the results presented here corresponds to a gap of 2 h = 7.5 mm. This set-up is immersed in a tank filled with water. The water temperature is monitored by a thermo-couple with a 0.5 • C accuracy. The Reynolds number is defined as Re = U h/ν, where ν is the kinematic viscosity of the fluid, U the belt velocity and h the half-gap between the two faces of the belt. The uncertainties on the belt velocity, the gap width and the temperature lead to an Re-uncertainty of 5. The x, y, z-directions are respectively the streamwise, wall-normal and spanwise directions From previous experiments not detailed here, the critical Reynolds numbers have been obtained as R g 305 and R t 390, which are consistent with previous experiments and well-resolved direct numerical simulations.

Experimental protocol

The flow is permanently perturbed by a bead positioned close to the y = 0.5 h plane at (x, z) = (0, 0) corresponding to the center of the experiment. The bead is maintained by means of a thin wire stretched between two stalks placed between and aligned with the guiding and main cylinders as illustrated in figure 4. This type of permanent perturbation was used by [START_REF] Bottin | Intermittency in a locally forced plane couette flow[END_REF]. The wire diameter is φ wire /h = 0.1 and the bead diameter is φ bead /h = 0.8. As explained in [START_REF] Bottin | Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow[END_REF], a thin wire parallel to the streamwise direction does not influence the flow since it does not induce any wake formation.The bead diameter does not strongly affect the qualitative results and provided that Re > Re g , no intermittency is observed and the bead is rather only triggering turbulent spots. We studied the influence of the stalks and wire on the transition to turbulence in our system when the bead is not present: their influence is hardly noticeable when considering critical Reynolds numbers and transition time to turbulence. When turbulence develops in a formerly laminar domain, it always occurs first around the bead. A typical experiment is a step consisting in a sudden increase of Re from 0 to a final Reynolds number Re f . The acceleration used is such that the belt reaches its final velocity in less than 0.3 s, i.e. less than 10 h/U , which is always very short as compared to the spot development time scales. For each kind of experiment (final Re value Re f , presence of the bead or not . . . ) we have performed from 5 to 10 independent realisations. Unless otherwise specified, results presented here are ensemble averaged over these different realisations.

PIV measurements and visualisations

Velocity measurements have been achieved by Particle Image Velocimetry (PIV). The Dantec system consists of a dual pulse Laser (Nd:YAG, 2 × 135 mJ, 4 ns, 532 nm) and a CCD camera (FlowSenseEO,4 Mpx). Image pairs are acquired at a rate of 5 Hz. The PIV Laser is mounted above the test section on a linear traverse such that velocities over different y planes can be measured with a spatial resolution of about 1.3 mm along y. The laser sheet produced from a cylindrical lens gives 2D velocity fields (U x ,U z ) in the streamwise/spanwise planes. The flow is seeded with particles coated with Rhodamine, the diameter of which d p is in the range = 1 -20 µm and that diffuse light at a shifted wavelength. A filter mounted on the camera lens is used to keep only wavelengths around this shifted wavelength, getting rid of unwanted light reflections on the vessel and plastic belt. The 1948 × 2048 pixels 2 observation window corresponds to physical sizes of 305 × 310 mm 2 . An adaptive cross-correlation processing is applied to an initial interrogation area of 64 × 64 pixels 2 followed by a final interrogation area of 32 × 32 pixels 2 with 50% overlap. The corresponding spatial resolution is 2.45 mm, i.e. 0.6 h. Each instantaneous velocity field is further filtered by removing values exceeding 1.2 U . Resulting outliers are replaced using local median filters.

Alternatively, visualisations are performed over the entire area where a plane Couette flow is achieved. To that purpose, the flow is seeded with Iriodin particles and lighted with a Laser diode combined with a polygon mirror rotating at high frequency to produce a laser sheet which illuminates the flow along a plane at a given y. Iriodin particles are micron-sized platelets which tend to align with the flow stream. These platelets respond to local flow changes and reveal laminar and turbulent regions: the turbulent state is indicated by quick fluctuations in the reflected light intensity while the laminar state corresponds to more uniform light reflections. Images at a resolution of about 2200×1100 pixels 2 , corresponding to an observation window of 800×400 mm 2 (exactly fitting the area where the laminar Couette profile is achieved), are captured by a PCO sCMOS Camera provided by PhotonLine. Figure 3(right) illustrates typical full size visualisations with both laminar and turbulent phases coexisting.

Growth of turbulent spots in the whole flow are obtained from visualisations and will be presented in 3.2. PIV measurements are used to study large-scale flows; associated results will be presented in section 3.4. For practical reasons, it is not possible to perform PIV measurements and visualisations at the same time. As a consequence, from the experimental data, it will not be possible to match the time evolution of large-scale flows and of spot growth. We will thus mostly rely on the numerical data for such comparisons.

Direct numerical simulations

DNS of a plane Couette flow are computed using the Channelflow software written by John F. Gibson [START_REF] Gibson | Channelflow: A spectral Navier-Stokes simulator in C++[END_REF][START_REF] Gibson | Visualizing the geometry of state space in plane Couette flow[END_REF]. The software solves Navier-Stockes equations numerically by using pseudo-spectral methods for the spatial discretisation with a Fourier decomposition in the (x,z) directions and Chebyshev polynomials in y direction. The boundary conditions are periodic in the (x,z) directions and no-slip conditions are imposed at the walls i.e at y = ±1. Numerical simulations are performed in a domain of size (L x ,L y ,L z )=(180,2,80) h. In order to resolve all the relevant modes of turbulent flow in the Re range studied, the numerical resolution is (768,33,384) dealiased modes in the (x,y,z) directions. According to [START_REF] Philip | From temporal to spatiotemporal dynamics in transitional plane couette flow[END_REF] this is sufficient to observe pattern formation. A time-step of 0.01 was used resulting in a CFL number less than 0.6. A pertubation consisting of four pairs of counter-rotating vortices like that used by [START_REF] Lundbladh | Direct simulation of turbulent spots in plane couette flow[END_REF] is introduced as an initial disturbance to trigger turbulent spots. As for the experiment, five realisations of the DNS have been performed for each Re value and ensemble-averaged results are presented. Note that for the DNS, a slight change in the perturbation amplitude (typically 0.1%) is introduced to achieve some variability between realisations.

Space time Diagrams and front detection

A large part of the results presented in the next section are obtained from space-time diagrams drawn from both numerical and experimental data. They are obtained from the dimensionless streamwise velocity component U x /U extracted on the line (x, y) = (0, 0) in the numerical case and from the x = 0 line of light intensity acquired by our visualisation system in the experimental case. To strengthen the detected structures and in order to improve the signal-to-noise ratio, sliding averages are applied along the x coordinates over 3 samples and along the time coordinate over 5 samples. Typical examples of numerical and experimental space-time diagrams are provided in figure 5.

Laminar-turbulent fronts are manually tracked in the experiment and automatically detected from the numerical data by defining two possible states: the laminar one which corresponds to the area where U x = 0 and the turbulent state which corresponds to the remaining area. We focus on the spanwise expansion on line x = 0 and one can note that at this location the large-scale flow is mainly oriented toward z and is consequently not that visible from U x .

Large-scale flows

As reported by many authors (see section 1), large-scale flows develop around growing turbulent spots in shear flows. They are easily identified by studying the two-dimensional spatial power spectrum of either velocity component in the (x, z) plane. These power spectra show an obvious scale separation between two peaks; one occurs around the wavelength λ 4 -5 h corresponding to turbulent streaks and another one, associated with large-scale flows, is situated around λ = 40 h [START_REF] Duguet | Oblique laminar-turbulent interfaces in plane shear flows[END_REF][START_REF] Tuckerman | Statistical analysis of the transition to turbulent-laminar banded patterns in plane Couette flow[END_REF][START_REF] Couliou | Large scale flows in transitional plane couette flow: a key ingredient of the spot growth mechanism[END_REF]. In this work, we will use two quantities associated with large-scale flows: A LSF z = kE z (λ = 40 h)/U 2 that represents the U z premultiplied power spectra amplitude at λ = 40 h and M LSF z = max( U z (x = 0, y = 0, z), z), the maximal amplitude of U z which is the spanwise velocity component filtered above the cutoff wavelength λ cut = 24 h on the line x = 0, y = 0. More details can be found in the work by [START_REF] Couliou | Large scale flows in transitional plane couette flow: a key ingredient of the spot growth mechanism[END_REF].

Results

General features

Growth time scales and region definition

Due to our numerical and experimental protocols, the growth of the triggered spot differs slightly in the two cases presented in figure 5. In the numerics, a turbulent zone is established very quickly (few h/U ) from the initial perturbation that is already large, of the order of 5h. The spot growth in our experiment takes place in two phases as can be seen in the space-time diagram of figure 5 (right). During a first phase lasting t p h/U (t p being about 210 in the displayed example), structures develop around the bead. Then an actual spot, with an interior made of turbulent fluctuations like the ones displayed in any numerical or experimental study of such spots, grows until the end of the acquisition. In the experimental case, as our step protocol starts at Re = 0, a time t c is required for the laminar profile to establish itself. An important point is to know how the time t c compares to t p associated with the development of the perturbation before turbulence is triggered. According to measurements performed by Tillmark and reported in [START_REF] Tillmark | Experiments on transition in plane Couette flow[END_REF], the linear Couette velocity profile establishes in a time t c 0.25 Re h/U . In our experiment, this corresponds to a time of the order of 100 h/U which is short enough as compared to 210 h/U when the spot is triggered.

The experimental space-time diagram presented in figure 5(right) allows us to define three distinct zones during the growth of a turbulent spot. The region connected to the bead is turbulent and displays fluctuating disorganized structures. Far away from the bead, the flow is laminar which results in roughly homogeneous dark areas. Between these two regions, long coherent inclined structures are visible. They correspond to the waves evoked by many authors as explained in the introduction (see section 1) and it has been argued that these waves correspond to the advection of streamwise vortices by the large-scale flows present at the spot spanwise tips [START_REF] Couliou | Spreading of turbulence in plane couette flow[END_REF]. A similar distinction is actually also observed on the numerical space-time diagram.

From fronts to spreading rate

The spanwise fronts of the turbulent spot are tracked to measure the spot front velocity. From the individual fronts detected manually from the experiment and automatically from the DNS as explained in section 2.5 we obtain scatter plots such as those presented in figure 6(a) and 6(b) that provide the front position as a function of time for each experiment or simulation. In the experimental case, the time at which the spot starts growing may vary from one realization to another. To obtain the scatter plot presented in figure 6(a), we have translated each front so that they coincide when they have reached z = ±16 h. Note that when Re < 320, it may happen that no spot grows in some cases. The results presented in the following are obtained by gathering fronts associated with spots that eventually grow. In both numerical and experimental scatter plots, gathering from 5 to 10 independent realizations, the dispersion between different spot fronts is small enough such that an average front can be obviously distinguished. The position of this average front is obtained from cubic spline interpolation with a smoothing parameter µ. This cubic spline is further differentiated to obtain the average front velocity, i.e. half the average spot spreading rate, as a function of time at a given Re. In practice, we perform this velocity calculation by regularly varying the smoothing parameter µ from 0.5 to 0.9. We thus obtain several time velocity signals that slightly differ from one another. The ensemble average of these velocity signals is used as the spot front velocity and their ensemble standard deviation as the corresponding error bar. Figure 6(c) presents such a velocity signal and its error bar for Re = 365. We can also appreciate on this figure the relative symmetry of both fronts (top and bottom of the experiment) that is representative of the symmetry obtained at any Re. Note that the numerical front velocity signals are not found to be more symmetric than the experimental ones, even though the numerical boundary conditions are perfectly symmetric. Consequences are twofold: the top-bottom symmetry is fairly preserved in our experimental setup and the remaining asymmetry in the growth dynamics has to be attributed to the dynamics itself and not to the boundary conditions.

Velocities of edge vortices

From experimental space-time diagrams as the one presented in figure 5(b), the edge vortices are manually detected as straight lines. Their individual velocity is then estimated as the corresponding slope. For a given Re, all detected velocities of the vorticies are gathered in a scatter plot as done in figure 6 for spot fronts. It appears that, as time passes, the velocity increases linearly and a linear fit of the scatter plot is thus performed.

Quantities of interest and relation to growth mechanisms

To study the spreading dynamics, we consider four quantities: σ z sf , the velocity of the spot front, σ z adv , the velocity of the vortices, σ z loc defined as σ z sf -σ z adv and the ratio σ z loc /σ z sf . All these quantities depend on time and Re. Their time averages have been studied by [START_REF] Couliou | Large scale flows in transitional plane couette flow: a key ingredient of the spot growth mechanism[END_REF] to reveal the two growth mechanisms described in the introduction. σ z sf corresponds to the spot spreading rate in the spanwise direction. σ z adv is associated with the global growth induced by large-scale flows and corresponds to the rate of streak nucleations inside the turbulent spot. From its definition, if only these two mechanisms are at work, σ z loc measures the importance of the local mechanism and corresponds to the rate of streak nucleations outside the turbulent region. σ z loc /σ z sf simply quantifies the relative importance of both mechanisms and can be compared to the fraction of nucleations occurring outside or inside the turbulent spot, which can be obtained from the numerical data. Next we present the time dependence of these four quantities in our experiments and in our simulations. In the sequel, X will denote the time average of the quantity X. The different quantities defined above are recalled in table 1.

Influence of box size and reproducibility

Figure 7(a) illustrates the spot fronts for three numerical simulations performed at the same Reynolds number Re = 380 but with increasing box sizes at constant aspect ratio. The darkest lines correspond to the spot fronts for the largest boxes. The brightest lines correspond to a smaller box than that used here and the darker to a larger box. It appears that the influence of the box size is negligible at the beginning of the growth, but becomes significant at later times when effect of the periodic boundary conditions on the spot front is not negligible any more. In order to avoid any bias from this size effect, we have decided to study the spot growth dynamics before the box size influences the spot development. More specifically, the recording of the front position should be stopped when it reaches ±36h with our box size. The same restriction has been applied to our experimental data.

Figure 7(b) illustrates the average spot fronts for the six Reynolds numbers studied numerically. It appears that one fourth of the dynamics are identical, regardless of the Reynolds number and even when the spot eventually vanishes (see the brightest lines corresponding to Re = 320). During this first phase, the spot growth rate is roughly independent of Re and so are the mechanisms responsible for the spot spreading that can possibly involve non-normal growth.

Spreading in experiments

Figure 8 illustrates the time evolution of the four quantities defined in table 1 for 6 values of the Reynolds number spanning the range of Re studied here. We do not present the 11 available Re values for the sake of clarity but the limiting cases detailed below take them into account.

In figure 8(a), σ z sf is seen to monotonically increase with time, while saturating at the end of the growth, resulting in S-shaped curves whose typical amplitude is 0.13 U . When varying the Reynolds number, the overall shape of these curves does not change, but their starting point is higher and occurs at earlier time for larger Re so that they can be deduced one from another by simple translations. The spot growth is faster at larger Re and, with the type of perturbation used here, it also starts earlier, even in dimensionless units. This is probably due to faster development of the perturbation around the bead in the first stages of the experiment (see section 2.2).

In figure 8(b), σ z adv , the velocity of the vortices is seen to linearly increase with time. The rate of this increase grows with the Reynolds number so that, at the end of the growth process, vortices travel much faster at larger Re while in the first steps of the growth dynamics, all vortices go roughly at the same speed (0.05 U within our measurement accuracy), except at the very large Re values. Note that the time span on which the velocity of the vortices is displayed is different from the displayed time span for the spot front position. This is because vortices are not always found in the space-time diagram at all times, either because they are not present, or because the visualization quality does not allow us to accurately track them. As σ z loc and σ z loc /σ z sf are derived from σ z adv , the same remark obviously applies for them too.

Due to the different dependencies for σ z sf and σ z adv on Re, their difference σ z loc has more complicated dynamics that strongly depends on Re. The local contribution to the total growth is most of the time between 10% and 30% at any Re. This shows that most of the spot growth has to be attributed to the global mechanism as already proven from the time-average study by [START_REF] Couliou | Large scale flows in transitional plane couette flow: a key ingredient of the spot growth mechanism[END_REF].

Spreading in numerics

Figure 9 gathers the time evolution of the three growth rates defined above and of their ratio for different Reynolds numbers in the numerical simulations. As a preliminary remark, we can note that all these figures differ from the experimental ones. Precise differences will be explained in this section. Note that for Re = 320, spots grow during a first phase but eventually vanish. For Re = 340, spots arise in most cases but their sizes stay moderate and fluctuate.

Contrary to what was observed in the experiment, the behaviour of σ z sf strongly depends on Re. For Re > 360, its maximum is reached at about 125 h/U . Below this value, σ z sf monotonically decreases with time and the decreasing rate is stronger with lower Reynolds number. Note that for Re = 320, the spreading rate ends up negative since the considered spot actually vanishes (not presented). Regardless of the Reynolds number, the total growth rate σ z sf is of the order of 0.12 U at short times and takes values between 0.05 U and 0.15 U . This order of magnitude is similar to the experimental one.

The quantity σ z adv also displays a dynamics that is more complicated than in the experimental case and that is Re dependent. It is roughly constant at low Re, slowly increasing at intermediate Re values but presents a maximum for the larger Re. This maximum is once again reached around 125 h/U . Regarding the order of magnitudes, they increase with Re from 0.05 U to 0.1 U . Once again, this order of magnitude is similar to the experimental one, although slightly smaller for larger Re.

The quantity σ z loc has time and Re dependencies very similar to that of σ z sf , except that the maxima are reached at earlier times. As in the experimental case, its order of magnitude is around 0.05 U .

The trend of σ z loc /σ z sf are almost identical regardless of the Reynolds number. They increase with time from values around 50% to 60% in the early stage of the growth before decreasing down to values around 20%. The order of magnitude is clearly smaller than in the experimental case.

Coming back to the underlying mechanisms, these numerical results show that the global growth induced by large-scale flows dominates the growth rate at long times while local growth is much more active at the beginning of the growth process, leading to a balance between the two mechanisms. It is interesting to note that, when comparing the order of magnitudes seen in the experiment and in the simulation, they are very similar for the local growth (σ z loc ) but clearly larger in the experiment for the global growth (σ z adv ). This will be further discussed in section 4.

Large-scale flows

As explained in section 2, large-scale flows and spot growth are not measured experimentally during the same experiment. Thus a straightforward temporal comparison of the associated dynamics is not possible. We have chosen to time-shift the large-scale flow quantities so that time t = 0 corresponds to early stage of the growth presented above, at around one fourth of the overall process.

In section 2.6, we have introduced two quantities associated with large-scale flows:

A LSF z and M LSF z . The quantity A LSF z
gives a global measure of the large-scale flow amplitude while the quantity M LSF z is a local direct estimation of the order of magnitude of the advection of edge vortices by large-scale flows (see section 1). Figures 10(a) and 10(c) illustrate the evolution of these global and local measurements in our experiment on time scales relevant to the growth process as explained above. Both have relatively similar time dependencies. At low Re, they are fluctuating around a low constant value during the whole growth process. When Re > 330, they increase from the beginning of the growth process and then saturate or slowly decay (actually, at later non-represented times, they all decay). For higher values of Re, the increase is faster and the decay starts sooner. Similar observations can be made for the numerical data presented in figure 10(b) and 10(d). The maximum of max(U LSF z

) is reached at about t 120 -150 h/U when the spot front velocity was found to be maximal too.

Discussion

Comparison to former work

The spot front velocity as a function of time for different Re is represented in figure 8. Contrary to some of the previous works that found the spot front velocity to be independent of time, particularly those by [START_REF] Tillmark | On the Spreading Mechanisms of a Turbulent Spot in Plane Couette Flow[END_REF] and by [START_REF] Dauchot | Finite amplitude perturbation and spots growth mechanism in plane Couette flow[END_REF], these plots show a time dependent spot front velocity, even if our experimental setup is very similar to the ones cited. In fact, these researchers have followed spot growth over shorter times and have performed linear fits of the measured spot fronts equivalent to that presented here in figure 6(a). Restricting the spot growth presented here to their first part, performing the same linear fitting seems reasonable since the non-linear evolution clearly appears only when considering the full temporal signal.

On average, the spot front velocity σ z sf is found both in our experiment and in our simulations to be of the same order of magnitude than the one presented in Dauchot & Daviaud (1995) even though slightly larger, particularly in the experimental case. The dependence of σ z sf on Re is nevertheless similar in all cases as illustrated with the three parallel curves presented in figure 2.

The velocities of edge vortices measured here as a function of both time and Re have an order of magnitude similar to those found by [START_REF] Dauchot | Finite amplitude perturbation and spots growth mechanism in plane Couette flow[END_REF] and by [START_REF] Tillmark | On the Spreading Mechanisms of a Turbulent Spot in Plane Couette Flow[END_REF]. Unfortunately, the dependencies of these velocities with time or Reynolds numbers were not provided, so that the comparisons can not be pushed further. As we are, to our knowledge, the first to investigate the time evolution of the spot development in plane Couette flow and the link to with large-scale flow development, we cannot compare this part of our study to former contributions. The next paragraph is dedicated to a discussion of these aspects.

Experiments vs numerics

As already mentioned in the last section, there are differences between the numerical and experimental velocity evolution presented here. When the numerical domain size is modified, we have seen in section 2 that the growth of turbulent spots is affected. When the spot fronts reach 0.87 L z which corresponds to a distance to the domain boundary of 0.13 L z , the fronts slow down. We have thus restricted our analysis to regions far enough from the domain boundaries in order to limit the influence of this finite-size effect. We have done so for both numerical and experimental data as already explained. Nevertheless, boundary conditions are very different in both cases as sketched in figure 11. In simulations, they are periodic while in the experiment, they consist of walls where the spanwise velocity vanishes. The distance between these walls and the belt edges is about L z /7. A buffer area thus exists between the shear zone and these walls. As a consequence, numerical and experimental fronts are analysed in physical zones of similar sizes but enclosed within very different boundary conditions. The main consequence of these differences is that large-scale flows associated with a numerical spot are tempered by their periodic images at times when, in the experiment, the buffer zone still allows these structures to develop. We indeed observe experimentally that the largescale recirculation is actually present in the buffer zone as sketched in figure 11. Largescale flows and thus the spot fronts are slowed down earlier in the numerical simulations than in the experiment. In the experiment, this slowing down should also be observed, but at later times. Since our analysis is restricted to the same physical space in both cases, and since the growth velocities have similar magnitudes, the numerical analysis spans a larger part of the whole dynamics than the experimental one. Indeed, we observe in figure 9(a) and 9(b) that σ z sf and σ z adv decrease in the numerical case when they are still increasing in the experimental plots of figure 8(a) and 8(b). Suppressing the buffer area in the experiment should lead to a more similar behaviour.

Temporal dynamics

Considering numerical or experimental data, it is clear that at any time, the local and the global growth mechanisms are at work since σ z adv and σ z loc are non vanishing. Their relative contributions depend mostly on time. The quantities σ z adv and M z LSF have very similar temporal dynamics: in the experimental case, they are both increasing functions of time and they take values of the same order of magnitude in the range 0.05-0.17 U . In the numerical case they exhibit maxima realized at the same time, around t = 125 h/U . These strong similarities strengthen our conviction that the large-scale flows are responsible for the advection of the edge vortices and of the turbulent spot. From a more quantitative point of view, it is not easy to find a relevant measure of the large-scale flow intensity to be compared to the advection rate σ z adv . We have used the maximum, which is local and by nature an upper bound. A more refined estimation could be done by an integral measure (over x) at a given distance upstream the front and over a given streamwise length. This would imply the introduction of two free parameters and could lead to questionable results. Note nevertheless that the large-scale flow maximum is found to strictly remain at a distance 2 -5 h upstream of the spot front at any time and for any Re (figure not shown).

Growth mechanisms

Both in our previous work [START_REF] Couliou | Spreading of turbulence in plane couette flow[END_REF] and in the present work we have reported the existence of a global growth mechanism due to advection of the spot by the large-scale flows induced by the laminar-turbulent coexistence. This mechanism is associated with a growth rate σ z adv that takes non-zero values at any Re and at any time. This implies that large-scale flows are present throughout the growth process as already shown by direct PIV measurements of these flows by [START_REF] Couliou | Large scale flows in transitional plane couette flow: a key ingredient of the spot growth mechanism[END_REF]. As a consequence, the laminar profile outside of the turbulent spot differs from the linear Couette profile at any time and at any Re. A classical hypothesis to explain the growth of turbulent spots is, as detailed in the introduction, a local destabilization of the laminar profile at the spot spanwise tips. With our notation, the corresponding growth rate is σ z loc . At Re = 335, σ z loc equals zero at any time but large-scale flows are present. The configuration is such that the laminar profile is modified but does not trigger local growth. At Re = 346, the local growth is active as σ z loc is non-zero. We can thus infer the existence of a critical Re value between 335 and 346 below which the laminar profile modification induced by the large-scale flow is not strong enough to trigger the instability. From the numerical simulations, the modified laminar flow is entirely known and can be extracted to perform linear stability analysis as was done by [START_REF] Henningson | The wave structure of turbulent spots in plane poiseuille flow[END_REF] for the plane Poiseuille case.

As presented in the introduction, [START_REF] Duguet | Stochastic and deterministic motion of a laminar-turbulent front in a spanwisely extended Couette flow[END_REF] suggest a local stochastic growth of the turbulent spot. This suggestion comes from observations in narrow domains where associated large scale flows do not develop. This stochastic growth is associated with a growth rate σ z stoch , whose order of magnitude is around 0.01 U in their narrow domains. This is one order of magnitude smaller than the two growth rates introduced in the present work. Actually, [START_REF] Duguet | Stochastic and deterministic motion of a laminar-turbulent front in a spanwisely extended Couette flow[END_REF] observe numerous streak-retreat events at any Re while, in our numerical simulations, these events are very rare (5 streak-retreats over the 35 simulations performed). If this stochastic growth exists in large domains, it can not explain the measured local growth rate that, as shown here, contributes to about 50% of the total growth rate.

Conclusions and perspectives

Spot growth in the transitional regime of plane Couette flow has previously been studied numerically and experimentally by several researchers, revealing the existence of waves at the spot spanwise tips and pointing out the possible role played by largescale flows surrounding the developing spot. The actual mechanisms at work to achieve the observed growth when Re is large enough are still unclear although most of the former studies suggest a destabilization taking place at the spot spanwise tips. This suggestion is based on results time-averaged over the whole growth process. In our recently published work [START_REF] Couliou | Spreading of turbulence in plane couette flow[END_REF], we have revealed the occurrence of streak nucleation within the turbulent phase pointing to a new global growth mechanism involving large-scale flows. In the present work, we have presented a temporal version of this study showing that the two suggested mechanisms are actually both present throughout the growth dynamics. At all times they contribute significantly to the total growth. Temporal comparison of the global growth rate σ z adv and of measures of the large-scale flow intensities strengthen our interpretation that the large-scale flows are responsible for the advection of edge vortices and thus for the global spot growth rate. The role of nonlinear advection in the growth of turbulent spots was also noted from a recent model of pipe and channel flows proposed by [START_REF] Barkley | The rise of fully turbulent flow[END_REF]. From visualisations of both our experiments and simulations, we clearly see that large-scale flows change in shape and intensity during the reorganisation occurring when the spot is large enough to form the laminar-turbulent stripe pattern expected at long times. We suggest they also do play an important role in this reorganisation that we have planned to study.

Regarding the spot growth in the streamwise direction, if the same kind of mechanisms act in this direction, due to their geometry, large-scale flows should be acting against the growth in this case. We are currently investigating these aspects from the data presented here. As plane Couette flow shares most of its features with the other extended shear flows, including the existence of large-scale flows around turbulent patches, it is worth performing the same analysis in plane Poiseuille and Taylor-Couette flows to see if the same conclusions can be formulated.

Figure 1 .

 1 Figure 1. Snapshots of a spot growing around a bead during a step experiment (sudden increase at t = 0 of Re from 0 to 403). Times are given in unit of h/U . First phase last till t 400 h/U when the reorganization process starts. See text for details. The corresponding space-time diagram can be found in figure 5.

Figure 3 .

 3 Figure 3. Left: general sketch of the plane Couette flow set-up. Right: snapshot obtained from our visualisation system during the growth of the turbulent spot. The PIV window is highlighted by the red frame. The orange dot corresponds to the position of the bead.

Figure 4 .

 4 Figure 4. Side and top sketch of the perturbation set-up.

Figure 5 .

 5 Figure 5. Left: Ux/U space-time diagram for a DNS at Re = 380. Nucleations of streak inside ( ) and outside (•) of the spot are labelled. Only half the spot is presented in the spanwise direction. Blue: negative velocities, red: positive velocities, white: zero velocity. Right: Space-time diagram from the experimental visualisation corresponding to the growing spot presented in figure 1. Final Re = 403 is reached after 10 h/U and, according to Tillmark & Alfredsson (1992), the linear Couette profile is established after about 0.25 Reh/U corresponding to 100 h/U in this case. The white vertical band from 450 h/U to 480 h/U corresponds to the tape which closes the plastic belt.

Figure 6 .

 6 Figure 6. Methodology figures: (a) front position as dots and corresponding smoothing spline as black line from an experiment at Re = 365. (b) same in a numerical simulation at Re = 380. (c) Front velocities obtained by derivating the cubic spline of figure (a). (d) Front velocities obtained by differentiating the cubic spline of figure (b).

Figure 7 .

 7 Figure 7. Methodology figures: (a) Front position for three numerical simulations performed at Re = 380 for three different domain size (Lx * Lz = 360 h * 160 h, Lx * Lz = 180 h * 80 h and Lx * Lz = 90 h * 40 h). Lightest lines correspond to smaller boxes. (b) Superposition of all fronts obtained at various Re by numerical simulations for a domain size of Lx * Lz = 180 h * 80 h . From brightest to darkest, Re increases from 320 to 420 by a step of 20.

Figure 8 .

 8 Figure 8. Total spot growth rate σ z sf (a), edge vortices advection velocity / advective growth rate σ z adv (b), local growth rate σ z loc (c) and ratio σ z loc /σ z sf (d) as a function of time for various Re. Experimental data. Color online.

Figure 9 .

 9 Figure 9. Total spot growth rate σ z sf , edge vortices advection velocity / advective growth rate σ z adv , local growth rate σ z loc and ratio σ z loc /σ z sf as a function of time for various Re. Numerical data.

  Figure 10. A LSF z (top line) and M LSF z (bottom line) as a function of time for different Re in the experiment (left) and in the DNS (right). See text for precise definitions.

Figure 11 .

 11 Figure 11. Schematic representation of the periodic boundary conditions in numerical simulation (a) and of water at rest and wall boundary conditions in the experiment (b).

Table 1 .

 1 Table gathering the different growth rates used in the text, their definition and the way they are measured.

	Quantity	Definition	Measurement
	σ sf	Total growth rate of the spot	From laminar-turbulent front velocity
	σ adv	Growth rate fraction due to advection	From the velocity of edge vortices
	σ loc	Remaining growth rate fraction	as σ sf -σ adv

† Email address for correspondence: monchaux@ensta.fr

Acknowledgments

This project was supported by ANR Jeunes chercheurs/Jeunes chercheuses QAN-COUET.