A Sozza

F De Lillo

S Musacchio

G Boffetta

Large-scale confinement and small-scale clustering of floating particles in stratified turbulence

We derive a simple model, valid within the Boussinesq approximation, for the dynamics of small buoyant particles in stratified turbulence, in presence of a mean linear density profile. By means of extensive direct numerical simulations, we investigate the statistical distribution of particles as a function of the two dimensionless parameters of the problem. We find that vertical confinement of particles is mainly ruled by the degree of stratification, with a weak dependency on the particle properties. On the contrary, small scale fractal clustering is found to depends on the particles relaxation time and is only slightly dependent on the flow stratification. The implications of our findings for the formation of thin phytoplankton layers are discussed.

Particles of density different from that of the surrounding fluid do not follow the motion of fluid particles, and generate inhomogeneous distributions even in incompressible flows [1]. This phenomenon is crucial in a variety of instances, from cloud formation in the atmosphere, to the dynamics of plankton in the ocean and lakes, to industrial applications [START_REF] Grabowski | Growth of cloud droplets in a turbulent environment[END_REF][START_REF] Post | Modeling the outcome of dropdrop collisions in Diesel sprays[END_REF][START_REF] Seinfeld | Atmospheric Chemistry and Physics of Air Pollution[END_REF]. The formation of inhomogeneous distributions in turbulent flows is also interesting from a theoretical point of view and, in recent years, analytical, numerical and experimental studies led to significant advances in the understanding of this process [START_REF] Squires | Preferential concentration of particles by turbulence[END_REF][START_REF] Balkovsky | Intermittent distribution of inertial particles in turbulent flows[END_REF][START_REF] Bec | Fractal clustering of inertial particles in random flows[END_REF][8][START_REF] Bec | Heavy particle concentration in turbulence at dissipative and inertial scales[END_REF][START_REF] Fouxon | Distribution of Particles and Bubbles in Turbulence at a Small Stokes Number[END_REF]. Most of the studies have considered the case of inertial particles which accumulate in regions of high vorticity (light particles) or high strain (heavy particles) [START_REF] Squires | Preferential concentration of particles by turbulence[END_REF][START_REF] Bec | Fractal clustering of inertial particles in random flows[END_REF][START_REF] Wang | Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence[END_REF], as a consequence of the accelerations induced by the flow. Recent works have studied the interaction between gravity and turbulent accelerations in the dynamics of heavy particles. In particular, it has been shown that turbulence can increase their settling velocity with respect to still fluid, by pushing particles in regions of downward flow [START_REF] Bec | Gravity-driven enhancement of heavy particle clustering in turbulent flow[END_REF][START_REF] Gustavsson | Clustering of particles falling in a turbulent flow[END_REF]. In presence of density fluctuations, gravity also affects the flow itself as in the case of stratified turbulence, which finds many applications in natural flows [START_REF] Lilly | Stratified Turbulence and the Mesoscale Variability of the Atmosphere[END_REF][START_REF] Riley | Fluid motions in the presence of strong stable stratification[END_REF], e.g. ocean dynamics in the presence of the pycnocline resulting from temperature and salinity variations [START_REF] Thorpe | An Introduction to Ocean Turbulence[END_REF][START_REF] Williams | Ocean Dynamics and the Carbon Cycle[END_REF]. Little is known about the distribution of floating particles in stratified turbulence, in spite of its relevance for applications. Recent works have studied the effect of stratification on the clustering of heavy [START_REF] Van Aartrijk | Preferential concentration of heavy particles in stably stratified turbulence[END_REF] and light [START_REF] Van Aartrijk | Vertical dispersion of light inertial particles in stably stratified turbulence: The influence of the Basset force[END_REF] particles, the formation of tangling clustering in stratified turbulence [START_REF] Eidelman | Tangling clustering of inertial particles in stably stratified turbulence[END_REF][START_REF] Elperin | Tangling clustering instability for small particles in temperature stratified turbulence[END_REF] and the effect of a vertical confinement in homogeneous turbulence [START_REF] Pietro | Clustering of vertically constrained passive particles in homogeneous isotropic turbulence[END_REF].

One of the most remarkable examples of confinement of particles in the ocean is the formation of the so-called thin phytoplankton layers (TPL): aggregations of phytoplankton and zooplankton at high concentration with thickness from centimeters to few meters, extending up to several kilometers horizontally and with timescale from hours to days [START_REF] Durham | Thin phytoplankton layers: characteristics, mechanisms, and consequences[END_REF]. Among the possible mechanisms of formation of TPLs, convergence to a depth of neutral buoyancy has been proposed for non-swimming species and aggregates, such as diatoms and marine snow, which are often observed to accumulate in presence of strong stratification [START_REF] Alldredge | Occurrence and mechanisms of formation of a dramatic thin layer of marine snow in a shallow Pacific fjord[END_REF].

In this work we investigate, by means of direct numerical simulations (DNS), the distribution of small buoyant particles transported in a turbulent stratified flow in the regime in which gravity dominates over the fluid acceleration. In this regime we consider a simple model for the particle motion which neglects inertial acceleration and balance the Stokes drag with the buoyancy forces. We study both the large scale vertical distribution and the small scale clustering of particles as a function of the relevant parameters. We find that small scale (fractal) clustering is determined by the particles relaxation time with a weak dependency on the degree of stratification. Conversely, the vertical confinement of particles is found to be mostly controlled by stratification. When rescaled to the typical parameters in the ocean, our results predict a fractal distribution for particles of few millimeter size. We consider a fluid linearly (and stably) stratified in the direction z of gravity g = (0, 0, -g) with a constant mean density gradient dρ/dz = -γ. Within the Boussinesq approximation, the motion for the incompressible velocity field u(x, t) is ruled by

∂u ∂t + u • ∇u = - 1 ρ 0 ∇p + ν∇ 2 u -N 2 θẑ + f (1)
∂θ ∂t + u • ∇θ = u • ẑ + κ∇ 2 θ (2)
together with ∇ • u = 0. The scalar field θ(x, t), which in the above equations has the dimension of a length, represents the deviations of the local density from the linear vertical profile, ρ = ρ 0 -γ(z -θ). ν is the kinematic viscosity, κ the density diffusivity and N = (γg/ρ 0 ) 1/2 is the Brunt-Väisälä frequency. f represents an external mechanical forcing needed to sustain turbulence. In the inviscid, unforced limit (ν = κ = f = 0) equations (1-2) conserve the total energy, sum of kinetic and potential contributions, E = 1 2 |u| 2 + 1 2 N 2 θ 2 where ... denotes the average over the domain.

The velocity v of a small inertial particle transported by the flow u generated by (1) evolves according to [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] dv dt

= β du dt - v -u τ p + (1 -β)g (3)
where β = 3ρ/(ρ + 2ρ p ) is the density ratio (ρ p is the density of the particle of radius a) and

τ p = a 2 /(3νβ) is the viscous (Stokes) time. By introducing the covelocity w = v -u [7] we can rewrite (3) as dw dt = -(1 -β) du dt - w τ p + (1 -β)g . (4)
Without loss of generality, we fix the particle density ρ p = ρ 0 and, consistently with the Boussinesq approximation in (1-2), we neglect the first term in the r.h.s of (4) (as (1 -β) ≃ (ρ 0 -ρ)/ρ 0 is a small parameter). Moreover, having in mind the dynamics of plankton in the ocean, we have that typical turbulent accelerations are much smaller than g. We remark that additional terms (e.g. the Basset history term) are in general present in (3) but they would give a negligible contribution within the above approximation [START_REF] Olivieri | The effect of the Basset history force on particle clustering in homogeneous and isotropic turbulence[END_REF]. Considering now the limit of small particles we obtain at the first order in τ p an explicit expression for the velocity of the floater v whose position x evolves according to

dx dt = v = u - 1 τ (z -θ)ẑ (5)
where we have used the fact that (1 -β)g ≃ 2 3 N 2 (z -θ) and we have introduced the particle relaxation time τ ≡ 3/(2N 2 τ p ) to the isopycnal surface of density ρ 0 . This surface, defined implicitly by the relation z = θ(x, y, z), will be denoted as h(x, y), keeping in mind that in general it can be multivalued. The model ( 5) which, to our knowledge has never been studied before, represents the simplest model for the dynamics of floaters in stratified flows and physically represents the balance between the Stokes drag and the buoyancy forces. A similar model has been developed and studied in [START_REF] Elperin | Tangling clustering instability for small particles in temperature stratified turbulence[END_REF]. Its validity is based on the assumptions of small density fluctuations (i.e. validity of the Boussinesq approximation) and small Stokes time of the particle. More general models, starting from the complete equation (3) can be considered, at the price of increased complexity and number of parameters [START_REF] Van Aartrijk | Vertical dispersion of light inertial particles in stably stratified turbulence: The influence of the Basset force[END_REF].

Although the fluid velocity u is incompressible, the velocity field transporting the floaters is not since ∇ • v = -(1 -∂θ/∂z)/τ which is in general nonzero. This expression represents the rate of contraction of the phase space (here the configuration space) under the dynamics.

When it is negative we expect that trajectories of floaters will collapse on a (dynamical) fractal attractor in the phase space. Examples of the attractor are displayed in Fig. 1 which shows that the large scale confinement in the vertical direction coexists with a small scale clustering with fractal distribution on the isopycnal surface. [START_REF] Brethouwer | Scaling analysis and simulation of strongly stratified turbulent flows[END_REF]. Our simulations are mostly within the turbulent regime, with Re b in the range 2 ≤ Re b ≤ 72. We remark that in stratified turbulence the energy dissipation rate is in general different from the injection rate ε as a fraction of the input is converted into potential energy [START_REF] Sozza | Dimensional transition of energy cascades in stably stratified forced thin fluid layers[END_REF]. In our simulations this difference is small and therefore, following the literature [START_REF] Brethouwer | Scaling analysis and simulation of strongly stratified turbulent flows[END_REF], we use the injection rate to define small scale quantities.

-L/2 0 L/2 0 L/2 L -L/2 0 L/2 0 L/2 L -L/2 0 L/2 0 L/2 L -L/2 0 L/2 0 L/2 L
Together with (1-2), we integrated equation (5) for the particle motion for a set of 10 classes of floaters characterized by different values of τ in the range 0.1 ≤ τ ≤ 10.0. In presenting the results, this time will be made dimensionless with the Kolmogorov time τ D = (ν/ε) 1/2 by introducing the relaxation number R ≡ τ /τ D . We remark that R is proportional to the inverse of the usual Stokes number St ≡ τ p /τ D which is based on the particles Stokes time τ p [START_REF] Bec | Fractal clustering of inertial particles in random flows[END_REF]. Table I reports the most important parameters of the simulations.

In Figure 2 we show vertical sections (at y = 0) of the isopycnal surface h (obtained from the solution of z = θ(x, y, z)) together with positions of the particles on the same sections, for different values of the parameters F r and τ . It is evident that the isopycnal surface h is almost flat for strong stratification and it becomes more bent (and multivalued) as F r increases. Figure 2 shows also the effect of the relaxation time τ on the particles. When R = τ /τ D < 1 the floaters are practically attached to the isopycnal surface, while their positions depart from the surface h by increasing τ . The distribution of the vertical displacement z of particles with respect to their equilibrium position in the absence of turbulence, z = 0, is found to be Gaussian in the wide range of parameters investigated. The standard deviations σ z of these vertical distributions of particles for different values of F r and R are shown in Fig. 3. We obtain a linear scaling of σ z on F r, with a coefficient γ which shows a (weak) dependency on R. For stronger stratification, F r ≤ 0.3, the standard deviation is almost independent on R and the slope converge to γ ≃ 0.84, a feature which can be understood by looking at the plots in Fig. 2.

The effect of the relaxation term in ( 5) is to allow the particle to detach from the level z = θ and to remain "suspended" for a time of order τ before feeling the vertical velocity towards the isopycnal surface. Because in stratified turbulence vertical velocity is suppressed [START_REF] Brethouwer | Scaling analysis and simulation of strongly stratified turbulent flows[END_REF],

there is no mechanism which ejects particles far from the surface. Therefore, as shown by Fig. 2, the vertical region visited by particles (measured by σ z ) reflects the vertical extension of the isopycnal surface (given by its standard deviation σ h ) which, by definition, is independent on τ . The linear dependence of σ h on F r shown in Fig. 3 can be understood within the framework of stratified turbulence, as a manifestation of the presence of the so-called vertical shear layers [START_REF] Billant | Self-similarity of strongly stratified inviscid flows[END_REF] and the associated vertical correlation scale of velocity L v . Physically this scale represents the vertical displacement for converting injected kinetic energy into potential energy and can be estimated dimensionally as L v ≃ U/N (U is a typical large scale velocity) and therefore one obtains L v ∝ F r, i.e. the linear scaling shown in Fig. 3. this equation would simply represent the linear relaxation of particles towards the isopycnal layer z = 0 [START_REF] Pietro | Clustering of vertically constrained passive particles in homogeneous isotropic turbulence[END_REF]. This is not achieved since the term ∂θ/∂z is fluctuating without a definite sign. Figure 4 shows the normalized PDF of the variable ζ for three different values of τ at F r = 1.0. It is evident that the statistics is neither Gaussian nor scale invariant and the PDF develops large tails for small relaxation times [START_REF] Birch | Plankton layer profiles as determined by shearing, sinking, and swimming[END_REF]. These large fluctuations are due to the folding of the isopycnal surface. In correspondence of a fold the stratification is inverted, (1 -∂θ/∂z) changes sign and (z -θ) grows exponentially (assuming that the field θ is quenched) until the particle reaches the nearest branch, above or below, of the isopycnal.

This mechanism, which is enhanced at large F r, produces large fluctuations of the distance between the particles and the isopycnal surface, and causes the development of large tails in the PDF of ζ.

As already discussed, floating particles moving according to [START_REF] Squires | Preferential concentration of particles by turbulence[END_REF] 5) has been derived under the assumption of small τ p /τ D , its validity is guaranteed above a minimum R which depends on F r. In Fig. 5 we plot the values of D 2 corresponding the parameter space which satisfies the above condition.

In the limit R ≫ 1, for which v → u in (5), floaters move as fluid particles in an incompressible velocity and therefore remain uniformly distributed in the volume with D 2 = 3.

We remark that this limit is equivalent to taking τ p → 0 in the general model of inertial particles [START_REF] Post | Modeling the outcome of dropdrop collisions in Diesel sprays[END_REF]. For smaller values of R the fractal dimension decreases and reaches a value in Fig. 1).

The dependence of the fractal dimension on the degree of stratification is shown in the inset of Fig. 5. We observe that, in general, D 2 is virtually independent on F r for F r ≥ 0.5. This is in contrast with the (large scale) vertical confinement shown in Fig. 4 which displays a strong (linear) dependence on F r and a weak dependence on St.

In order to test the applicability of our results to the formation of TPLs we provide a simple example using the typical values observed for diatom-dominated marine snow [START_REF] Alldredge | Occurrence and mechanisms of formation of a dramatic thin layer of marine snow in a shallow Pacific fjord[END_REF].

Let us assume an energy dissipation rate ε ≃ 10 -8 m 2 s -3 (τ D ≃ 10 s) and a Brunt-Väisälä frequency N ≃ 0.1 s -1 . The Froude number corresponding to N ≃ 0.1 s -1 in our simulations is F r ≃ 0.2. At this F r the thickness of the layer, measured by σ z , is almost independent on the size of the particles (see Fig. 3). Conversely, the distribution of the particles within the layer can be very different. Aggregates of size a ≃ 0.5 cm have a relaxation time τ ≃ 18 s which corresponds to R = 1.8, at which we observe clustering at small scales with D 2 < 2 (Fig. 5). Smaller aggregates or single cells of size a = 0.1 cm correspond to R ≃ 45 and, according to our results, would distribute almost homogeneously (with a fractal dimension close to 3) within the thin layer. Our model allows estimating the characteristic time for the formation of TPLs. In particular, in the case of strong stratification in which the isopycnals are almost flat, our model predicts that the floaters converge exponentially toward the isopycnal ρ = ρ p , with a relaxation time τ = 3/(2N 2 τ p ). Further field and laboratory experiments could provide evidence of this stratification-induced patchiness in thin phytoplankton layers.

Beside the applications to thin layers, our results are of general interest as they describe quantitatively and for the first time how the combination of turbulence and stratification generates both large scale confinement and small scale fractal patchiness in a suspension of buoyant particles.

FIG. 1 .

 1 FIG. 1. (Color online). Isopycnal surfaces θ = z corresponding to the density ρ 0 for a run at F r = 0.2. The points represent the (fractal) distribution of bouyant particles with τ = 0.1 in the stationary state.

FIG. 2 .128 5 × 2 Kolmogorov timescale, Re = L 4 / 3 f ε 1 / 3 2 / 3 f

 252431323 FIG. 2. (Color online). Vertical (x, z) sections of isopycnal surfaces h (black points) for F r = 0.3 (upper row) and F r = 1.0 (lower row) together with the positions of bouyant particles (green points) with relaxation numbers R = 1.0 (left column) and R = 60.0 (right column).

FIG. 3 .

 3 FIG. 3. (Color online). Standard deviations σ z of the PDFs of particle vertical position as a function of F r for different values of R at Re = 363. The solid black line represents the standard deviation σ h of the isopycnal surface. Inset: slope γ of the standard deviations obtained from the fit σ z = σ 0 + γF r as a function of R.

FIG. 4 .

 4 FIG. 4. (Color online). Probability density functions of the variable ζ = z -θ computed on particle positions for Re = 454, F r = 1.0 and R = 1.4 (red line with square), R = 7.0 (blue line with circle) and R = 70 (pink line with triangle). The black line is a Gaussian.

  are transported by a compressible velocity field and are therefore expected to relax on a (dynamical) fractal subset of the physical space, as shown in the examples of Fig. 1. In order to characterize this subset, and its dependence on the parameters, we have measured the correlation dimension D 2 of particle distribution, defined as the scaling exponent of the probability of finding two particles at distance less than r: P (|x 1x 2 | < r) ∝ r D 2 as r → 0 [30]. The maximum value D 2 = 3 denotes uniformly distributed particles, while D 2 < 3 indicates fractal patchiness with smaller D 2 corresponding to more clustered distributions and increased probability of finding pairs of particles at close separation. Figure 5 shows D 2 as a function of R for different values of F r. As the model (

D 2 ∼FIG. 5 .

 25 FIG. 5. (Color online). Correlation dimension D 2 versus relaxation time R for different values of stratification parameter F r. The error bars represent the fluctuation of the local scaling exponent in the fitting range of scales (0.1η ≤ r ≤ 10η). Inset: the correlation dimension D 2 plotted as a function of F r for different values of R.

ACKNOWLEDGMENTS

This article is based upon work from COST Action MP1305, supported by COST (European Cooperation in Science and Technology).


G Boffetta

A Mazzino

S Musacchio

Rotating Rayleigh-Taylor turbulence

The turbulent Rayleigh-Taylor system in a rotating reference frame is investigated by direct numerical simulations within the Oberbeck-Boussinesq approximation. On the basis of theoretical arguments, supported by our simulations, we show that the Rossby number decreases in time, and therefore the Coriolis force becomes more important as the system evolves and produces many effects on Rayleigh-Taylor turbulence. We find that rotation reduces the intensity of turbulent velocity fluctuations and therefore the growth rate of the temperature mixing layer. Moreover, in presence of rotation the conversion of potential energy into turbulent kinetic energy is found to be less effective and the efficiency of the heat transfer is reduced. Finally, during the evolution of the mixing layer we observe the development of a cyclone-anticyclone asymmetry.

I. INTRODUCTION

In natural fluid systems, the interface between regions of different density is subject to the well-known Rayleigh-Taylor (RT) instability when the heavier fluid is pushed against the lighter one (see, e.g., [START_REF] Chandrasekhar | Hydrodynamics and Hydromagnetic Stability[END_REF][START_REF] Sharp | An overview of Rayleigh-Taylor instability[END_REF][START_REF] Abarzhi | Review of theoretical modelling approaches of Rayleigh-Taylor instabilities and turbulent mixing[END_REF] for reviews on the subject). In the late stage, this instability evolves in a fully-developed turbulent flow [START_REF] Sharp | An overview of Rayleigh-Taylor instability[END_REF][START_REF] Abarzhi | Review of theoretical modelling approaches of Rayleigh-Taylor instabilities and turbulent mixing[END_REF][START_REF] Boffetta | Incompressible Rayleigh-Taylor Tubulence[END_REF]. Because of the fact that unstably stratified interfaces are ubiquitous in nature, fluid mixing induced by RT instability and turbulence characterizes many systems, over a huge interval of spatial and temporal scales. Among the possible examples, we mention astrophysical systems, in relation to flame acceleration in type Ia supernova [START_REF] Hillebrandt | Type Ia Supernova Explosion Models[END_REF][START_REF] Bell | Direct Numerical Simulations of Type Ia Supernovae Flames. II. The Rayleigh-Taylor Instability[END_REF]; geological systems, where RT instability has been invoked to explain the initiation and evolution of polydiapirs (domes-in-domes) [START_REF] Weinberg | Polydiapirs: multiwavelength gravity structures[END_REF]; atmospheric physics, where RT instability has been invoked as a possible explanation for the formation of the mammatus clouds [START_REF] Agee | Some inferences of eddy viscosity associated with instabilities in the atmosphere[END_REF].

Several experimental [START_REF] Read | Experimental investigation of turbulent mixing by Rayleigh-Taylor instability[END_REF][START_REF] Dalziel | Rayleigh-Taylor instability: experiments with image analysis[END_REF][START_REF] Dimonte | Turbulent Rayleigh-Taylor instability experiments with variable acceleration[END_REF][START_REF] Banerjee | Statistically steady measurements of Rayleigh-Taylor mixing in a gas channel[END_REF][START_REF] Andrews | Small Atwood number Rayleigh-Taylor experiments[END_REF][START_REF] Banerjee | Detailed measurements of a statistically steady Rayleigh-Taylor mixing layer from small to high Atwood numbers[END_REF] and numerical [START_REF] Young | On the miscible Rayleigh-Taylor instability: two and three dimensions[END_REF][START_REF] Dimonte | A comparative study of the turbulent Rayleigh-Taylor instability using high-resolution three-dimensional numerical simulations: the Alpha-Group collaboration[END_REF][START_REF] Cabot | Reynolds number effects on Rayleigh-Taylor instability with possible implications for type Ia supernovae[END_REF][START_REF] Vladimirova | Self-similarity and universality in Rayleigh-Taylor, Boussinesq turbulence[END_REF][START_REF] Matsumoto | Anomalous scaling of three-dimensional Rayleigh-Taylor turbulence[END_REF][START_REF] Biferale | High resolution numerical study of Rayleigh-Taylor turbulence using a thermal lattice Boltzmann scheme[END_REF] works have investigated the evolution of RT instability and turbulence in different physical conditions. In the limit of small density variations, the simplest mathematical description is provided by the Oberbeck-Boussinesq approximation of two incompressible, miscible fluids. Under these conditions, the seminal paper [START_REF] Chertkov | Phenomenology of Rayleigh-Taylor Turbulence[END_REF] has shown that three-dimensional RT turbulence can be described as a timedependent hydrodynamical turbulent flow with a gravity-induced forcing mechanism.

The RT instability and mixing can be significantly affected by the presence of other physical mechanisms, beside gravity, acting on the flow, including compressibility [START_REF] Mitchner | Rayleigh-Taylor Instability for Compressible Fluids[END_REF][START_REF] Scagliarini | Lattice Boltzmann methods for thermal flows: Continuum limit and applications to compressible Rayleigh-Taylor systems[END_REF], viscoelasticity [START_REF] Boffetta | Polymer heat transport enhancement in thermal convection: The case of Rayleigh-Taylor turbulence[END_REF], surface tension [START_REF] Chertkov | Effects of surface tension on immiscible Rayleigh-Taylor turbulence[END_REF] and physical confinement [START_REF] Lawrie | Turbulent diffusion in tall tubes. I. Models for Rayleigh-Taylor instability[END_REF][START_REF] Boffetta | Bolgiano scale in confined Rayleigh-Taylor turbulence[END_REF]. An important instance is the case in which the flow is embedded in a rotating reference frame, which causes the emergence of the Coriolis force in the ruling equations. Previous works have shown that the Coriolis force has a stabilizing effect on the RT instability, both in the linear stage [START_REF] Chandrasekhar | Hydrodynamics and Hydromagnetic Stability[END_REF] and in the weakly nonlinear stage of the perturbation evolution [START_REF] Tao | Nonlinear Rayleigh-Taylor instability of rotating inviscid fluids[END_REF][START_REF] Carnevale | Rotational suppression of Rayleigh-Taylor instability[END_REF][START_REF] Baldwin | The Inhibition of the Rayleigh-Taylor Instability by Rotation[END_REF].

In this study we focus on the effects of rotation on the fully developed RT turbulence. We discuss theoretical arguments based on dimensional scaling analysis which predict that the Rossby number decreases in time during the growth of the mixing layer. The Coriolis force is therefore expected to become dominant as the system evolves. This prediction is confirmed by our numerical simulations, which reveals that the turbulent RT system is strongly affected by rotation. In particular, we find that rotation reduces the efficiency of the heat transfer as well as the conversion of potential energy into turbulent kinetic energy. This causes a reduction of the intensity of turbulent velocity fluctuations and slows down the growth of the mixing layer. Interestingly, rotation allows the velocity fluctuations to extend outside the temperature mixing layer. We also observe the development of a cyclone-anticyclone asymmetry which is weakly dependent on the evolution of the mixing layer.

The paper is organized as follow. In Sec. II we introduce the rotating RT system under the Oberbeck-Boussinesq framework and we present mean field (scaling) arguments through which the role of rotation can be inferred. Numerical results from high-resolution DNS of the rotating RT system are presented and discussed in Sec. III concerning the effects of rotation on i) the mixing-layer growth, ii) the heat transfer, iii) the energy spectra, and, finally, iv) the symmetry breaking between cyclones and anticyclones. Final discussions and perspectives are drawn in Sec. IV

II. EQUATION OF MOTION AND THEORETICAL CONSIDERATIONS

We consider miscible Rayleigh-Taylor turbulence generated at the interface of two layers of fluid with small temperature (density) difference. The initial (at t = 0) interface is the plane z = 0, perpendicular to gravity g = (0, 0, -g) and to the uniform rotation Ω = (0, 0, Ω). The dynamics is ruled by the Boussinesq equations for an incompressible (u(x, 0) = 0) velocity field u(x, t) = (u, v, w) coupled with a temperature field T (x, t)

∂u ∂t + u • ∇u + 2Ω × u = -∇p + ν∇ 2 u -βgT (1)
∂T ∂t + u • ∇T = κ∇ 2 T (2)
The reference temperature is set to T = 0, ν and κ are the kinematic viscosity and diffusivity respectively and β is the thermal expansion coefficient. The initial conditions are u(x, 0) = 0, T (x, 0) = -(θ 0 /2)sgn(z) where θ 0 is the temperature jump at the interface at z = 0, which fixes the Atwood number A = βθ 0 /2.

As turbulence develops from the instability at the interface, turbulent kinetic energy in the mixing layer E = (1/2) |u| 2 is produced at the expense of potential energy P = -βg zT (brackets represent integral over the physical domain of volume L x × L y × L z ). The energy balance is written from (1-2) as

- dP dt = βg wT = dE dt + ε ν + βg κθ 0 2L z (3)
where ε ν = ν (∇u) 2 is the viscous energy dissipation and the last term is negligible in the limit of small diffusivity. We define the efficiency Σ of turbulent generation as the ratio of the kinetic energy production over potential energy consumption, Σ = -dE/dt dP/dt . We anticipate that, although the Coriolis force 2Ω × u does not enter directly in the energy balance [START_REF] Abarzhi | Review of theoretical modelling approaches of Rayleigh-Taylor instabilities and turbulent mixing[END_REF] it does affect the efficiency of the process which, for Ω > 0 is reduced with respect to the value Σ = 0.5 observed for Ω = 0 [START_REF] Boffetta | Statistics of mixing in three-dimensional Rayleigh-Taylor turbulence at low Atwood number and Prandtl number one[END_REF].

The development of the RT instability produces a mixing zone of width h(t) which grows in time. Dimensional arguments derived in absence of rotation [START_REF] Boffetta | Incompressible Rayleigh-Taylor Tubulence[END_REF] predicts that in the turbulent regime the mixing layer grows as h(t) ≃ Agt 2 and the typical velocity fluctuations inside the mixing layer grow as u ≃ Agt (here u indicates an arbitrary component of the velocity).

Assuming that the dimensional scalings hold also in the rotating case, one obtains the prediction that the dimensionless Rossby number Ro = w rms /(2Ωh), which measures the relative strength of the inertial forces to the Coriolis forces, should decrease in time as Ro ≃ 1/(Ωt). In particular, in the case of a weak rotation rate Ω even if the effect of rotation is negligible at the initial times (ensuring the validity of the dimensional scalings), it becomes more important and eventually competes with the inertial and buoyancy forces for t 1/Ω. For large Ω, this time can be sufficiently small for the rotation to affect already the linear (or quasi-linear) phase of the instability [START_REF] Carnevale | Rotational suppression of Rayleigh-Taylor instability[END_REF][START_REF] Baldwin | The Inhibition of the Rayleigh-Taylor Instability by Rotation[END_REF].

We performed direct numerical simulations (DNS) of the system of equations (1-2) in a periodic domain of size L x × L y × L z with L y = L x and L z = 4L x by means of a fully parallel pseudo-spectral code at resolution 512 × 512 × 2048 at different values of rotation Ω. Time evolution is ruled by a second-order Runge-Kutta scheme with explicit integration of the linear terms. For all runs we have Ag = 0.25, θ 0 = 1 and P r = ν/κ = 1. Viscosity is sufficiently large to resolve small scales (k max η ≥ 1.3 at the end of the simulations) and the results are made dimensionless with the box size L x , the characteristic time τ = (L x /Ag) 1/2 and the characteristic velocity U = L x /τ . Rayleigh-Taylor instability is seeded by adding to the initial density field a 10% of white noise in a thin layer around the unstable interface at z = 0. The same realization of the random perturbation is used for all the simulations at different rotation. The simulation is stopped before the mixing layer reaches the dimension of the vertical scale L z . We remark that the average quantities in (3) are defined by the integral over the space and therefore their time dependence has an additional t 2 factor due to the growth of the support of the integral given by the mixing layer width h(t).

III. NUMERICAL RESULTS

One effect of rotation is to reduce the growth of the mixing layer, a feature which is evident at a qualitative level from Fig. 1. More quantitatively, this reduction is due to a suppression of the vertical velocity fluctuations, measured by w rms and shown in Fig. 2. After the instability, for t > τ , vertical velocity fluctuations in the non-rotating case Ω = 0 are observed to grow linearly in time, in agreement with the dimensional prediction w rms ≃ Agt.

By increasing the rotation velocity, the growth of w rms is strongly depleted and for Ωτ = 12.5 vertical velocities almost saturates at a constant value. The inset of Fig. 2 shows the ratio w rms /u rms of the vertical to the horizontal velocity fluctuations which is found to be weakly dependent on Ω. Therefore also horizontal velocities are reduced and the degree of large-scale anisotropy in the velocity field is not strongly affected by rotation.

One consequence of the reduction of velocity fluctuations is the slowing down of the growth of the mixing layer in presence of rotation. Different definitions of the mixing layer width h have been proposed, based on either threshold values or integral quantities [START_REF] Dalziel | Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability[END_REF]. In the inset of Fig. 3 we plot the time evolution of h defined in terms of the threshold value z = h/2 at which the temperature profile T (z) reaches a fraction r of the maximum value, i.e. T (±h/2) = ∓rθ 0 [START_REF] Dalziel | Self-similarity and internal structure of turbulence induced by Rayleigh-Taylor instability[END_REF][START_REF] Boffetta | Nonlinear diffusion model for Rayleigh-Taylor mixing[END_REF] (the overline indicates average over the horizontal planes L x × L y ). We observe that rotation indeed induces a strong suppression of the growth of

h(t).

A remarkable result shown in Fig. 3 is that the ratio w rms /h is almost independent on Ω in the late turbulent phase. The natural interpretation is that, since the mixing layer is produced by the velocity fluctuations, h is essentially given by the integral in time of w rms and their ratio becomes Ω-independent.

In Figure 4 we plot the temporal evolution of the Rossby number defined as Ro = w rms /(2Ωh). Even though in the strongly rotating case the w rms and h might deviate from the scaling expected for Ω = 0, we find that the Rossby number, which is proportional to their ratio, decreases with time close to the dimensional prediction Ro ≃ 1/(Ωt).

Since vertical velocity fluctuations are responsible for the conversion of potential energy into kinetic energy, the potential energy loss ∆P (t) = P (0) -P (t) is reduced for Ω > 0, as shown in Fig. 5. We find that also the efficiency of the process is reduced by rotation: For Ω = 0 we have Σ ≃ 0.5, i.e. about one half of potential energy is converted into large-scale kinetic energy while the other half feeds the turbulent cascade and is eventually dissipated by viscosity [START_REF] Boffetta | Statistics of mixing in three-dimensional Rayleigh-Taylor turbulence at low Atwood number and Prandtl number one[END_REF]. For Ω > 0 the efficiency is reduced and we find Σ ≃ 0.3 for the case Ωτ = 12.5, as shown in Fig. 5, i.e. in this case more potential energy is transferred to small scales and dissipated.

A. Integral lengths

Rotation produces different effects on the temperature and on the velocity fields in RT convection. To investigate this issue, we introduce a set of integral lengths based on the vertical profile f (u,w,T ) (z) of velocity or temperature variance, defined as

f (u) (z) = u 2 (z), 0
f (w) (z) = w 2 (z), f (T ) (z) = T 2 (z) -T (z) 2 : L (u,w,T ) = f (u,w,T ) (z)z 2 dz f (u,w,T ) (z)dz 1/2 (4)
where the integral are defined over the whole domain [0, L z ]. The advantage of this definition with respect to the usual h(t), is that it provides information on the extension of the mixing layer both in terms of (horizontal/vertical) velocity and temperature fluctuations.

The evolution of L (u,w,T ) is shown in Fig. 6 for two values of Ω. In the absence of rotation the three integral lengths are approximately the same, indicating that temperature and velocity fluctuations are tightly associated within the mixing layer. On the contrary, for Ωτ = 12.5 we observe that while L (T ) is reduced (which is already clear from Fig. 1), both L (u) and L (w) increase, approximately by the same amount. This is a manifestation of the Taylor-Proudman phenomenon of bidimensionalization of the velocity field in presence of rotation [START_REF] Tritton | Physical fluid dynamics[END_REF] which, becoming almost independent on z, extends outside the mixing layer region defined by temperature fluctuations.

To investigate more in details this issue, in Fig. 7 we plot the temperature profile T and the vertical velocity profile w rms = w 2 1/2 at a late time. It is evident that while in the case Ω = 0 the two profiles extend over comparable regions, for the case Ωτ = 12.5 the vertical extension of temperature fluctuations is reduced and at the same time the vertical extension of velocity fluctuations is increased. Summarizing, our findings show that in presence of rotation there is not a unique scale characterizing the mixing layer as velocity and temperature fluctuations define different vertical scales.

B. Heat transfer

The efficiency of the transfer of heat in turbulent convection is quantified by the Nusselt number N u ≡ wT h/(κθ 0 ) (ratio of the convective to conductive heat transfer) while turbulence intensity is provided by the Reynolds number Re ≡ |u| rms h/ν. These numbers depends on the dimensionless temperature difference quantified by the Rayleigh number Ra ≡ βgθ 0 h 3 /(νκ). High resolution direct numerical simulations have shown that in Rayleigh-Taylor turbulence N u and Re displays the so-called ultimate state regime according to which N u ≃ Ra 1/2 P r 1/2 and Re ≃ Ra 1/2 P r -1/2 [START_REF] Celani | Rayleigh-Taylor Turbulence in Two Dimensions[END_REF][START_REF] Boffetta | Kolmogorov scaling and intermittency in Rayleigh-Taylor turbulence[END_REF][START_REF] Boffetta | The ultimate state of thermal convection in RayleighTaylor turbulence[END_REF] both in two and three dimensions.

We find that rotation reduces the value of N u both as a function of time (mainly as a consequence of the reduction of h(t)) and as a function of Ra, i.e. at fixed h as shown in Fig. 8. This latter result means that also the correlation between w and T is reduced by rotation. We find that also Re at given Ra decreases in presence of rotation, mainly as a consequence of the suppression of the velocity fluctuations. Moreover we find that, at larger values of Ω, the ultimate state scaling is violated as both N u and Re show a dependence on

Ra slower than Ra 1/2 .
We remark that our results on the suppression of the heat transport in rotating RT tur- and contributes to the heat flux [START_REF] Kunnen | Heat flux intensification by vortical flow localization in rotating convection[END_REF]. At large values of rotations (and for large Ra) this mechanisms is less efficient and a reduction of N u is observed. Because of the absence of boundary layers in RT convection, it is not contradictory that here we do not observe an increase of N u for any value of Ω and Ra.

Beside the differences between RT and RB turbulence due to the boundary conditions, other phenomena occurring in the bulk of rotating convection display interesting similarities between the two systems. In particular, studies of rotating RB has found an increase of the internal mean gradient with decreasing Ro [START_REF] Julien | Rapidly rotating turbulent rayleigh-bénard convection[END_REF][START_REF] Hart | On the thermal offset in turbulent rotating convection[END_REF][START_REF] Liu | Local temperature measurements in turbulent rotating rayleigh-bénard convection[END_REF] and a corresponding decrease in horizontal versus vertical fluctuations of temperature, which indicates that rotation favors the horizontal transport and suppresses the vertical mixing.

C. Energy spectra

In order to investigate the degree of anisotropy at different scales, we computed at given times during the evolution of the system the energy spectra of the horizontal and vertical components of the velocity, indicated respectively as E x (k) and E z (k). The spectra has been computed from the Fourier transform of the velocity fields in the whole periodic domain, and are plotted as a function of the wavenumber k = (k

2 x + k 2 y + k 2 z ) 1/2
. In Fig. 9 we compare the horizontal and vertical spectra computed at the same time t = 6τ with and without rotation. We see that for Ω = 0 the two spectra are almost identical at high-wavenumbers, indicating the recovering of isotropy in the flow at small scales. [START_REF] Boffetta | Statistics of mixing in three-dimensional Rayleigh-Taylor turbulence at low Atwood number and Prandtl number one[END_REF].

The recovery of small-scale isotropy does not occur in presence of rotation. For Ωτ = 12.5 we find that vertical velocity fluctuations dominates over horizontal velocity fluctuations at all scales. These results indicate that the ratio E z (k)/E x (k) is changed by rotation mostly at small scales, while at large scales rotation reduces both the spectra keeping their ratio almost constant. This is consistent with the weak dependence on Ω of the ratio w rms /u rms shown in Fig. 2. On the other hand, rotation changes significantly the degree of anisotropy of the velocity gradient. Indeed we measure that the ratio (∂ z w) rms /(∂ x u) rms changes from a value close to one for Ω = 0 [START_REF] Boffetta | Statistics of mixing in three-dimensional Rayleigh-Taylor turbulence at low Atwood number and Prandtl number one[END_REF] to about 0.65 for Ωτ = 12.5.

As rotation slows down the development of turbulence, it is interesting to compare spectra with and without rotation at two different times at which the peaks of the vertical energy spectrum are the same. The comparison of the two spectra is shown in Fig. 9 where we compare the case Ωτ = 0 at t = 2.8τ with the case Ωτ = 12.5 at t = 6.0τ . Vertical spectra are almost identical (within statistical fluctuations) while for horizontal spectra the effect of rotation is to reduce small-scale fluctuation in favor of large-scale components.

D. Cyclonic-anticyclonic asymmetry

A remarkable feature of turbulent rotating flows is the predominance of cyclones, i.e., vortices co-rotating with the flow, over anticyclones. The cyclone-anticyclone asymmetry has been observed in the atmosphere [START_REF] Hakim | Observed cycloneanticyclone tropopause vortex asymmetries[END_REF], and has been studied both in experiments and numerics (for a review see, e.g., [START_REF] Godeferd | Structure and dynamics of rotating turbulence: a review of recent experimental and numerical results[END_REF]).

Previous studies [START_REF] Bartello | Coherent structures in rotating three-dimensional turbulence[END_REF][START_REF] Deusebio | Dimensional transition in rotating turbulence[END_REF][START_REF] Gallet | Scale-dependent cyclone-anticyclone asymmetry in a forced rotating turbulence experiment[END_REF][START_REF] Naso | Cyclone-anticyclone asymmetry and alignment statistics in homogeneous rotating turbulence[END_REF] have investigated the dependence of the asymmetry on the Rossby number and the aspect ratio of the flow. The asymmetry is found to be maximum for Ro ≃ O(1), and vanishes when the flow is confined in a quasi two-dimensional geometry. These results suggest that the mechanisms which produce the asymmetry require the interaction between rotation and turbulent vortex stretching.

Here we address the question whether or not the cyclonic-anticyclonic asymmetry develops also in the rotating Rayleigh-Taylor turbulence. To address this issue we compute the probability density function (PDFs) of the components of the vorticity field within the mixing layer, where the vorticity is significantly different from zero.

In Figure 10 we compare the PDFs of the vertical component ω z of the vorticity with and without rotation. The PDFs are computed in the mixing layer of width h at times at which h is the same in the two cases. We find that in the non-rotating case the PDF is symmetric, while in the rotating case it is asymmetric with a positive skewness, which indicates the prevalence of cyclonic vortices, as shown in Fig. 10. A similar result has been observed also for the rotating Rayleigh-Bénard convection [START_REF] Julien | Rapidly rotating turbulent rayleigh-bénard convection[END_REF].

A quantitative measure of this asymmetry is provided by the skewness of the vertical vorticity:

S ω = ω 3 z ω 2 z 3/2 (5)
where the average is here computed within the mixing layer.

The time evolution of S ω is shown in Fig. 11 for two cases with and without rotation.

While for Ω = 0 we have a skewness consistent with zero, for Ωτ = 12.5 we observe, after the development of the turbulent phase, a value S ω ≃ 0.15 almost independent on time.

The weak time-dependence of the skewness is somehow surprising, because the cyclonicanticyclonic asymmetry is known to depend on the Rossby number [START_REF] Deusebio | Dimensional transition in rotating turbulence[END_REF], which in this case decreases in time as Ro ≃ 1/Ωt. One possible explanation is that is our simulations the time dependence of Ro is too weak (see Fig. 3) to appreciate a significant variation of the asymmetry. We conclude by remarking that the other two components of the vorticity are symmetric also in presence of rotation.

IV. CONCLUSIONS

In this paper we have studied the effect of rotation along the vertical axis on the development and the mixing properties of Rayleigh-Taylor turbulence in the Boussinesq approximation of incompressible flow. We have found that rotation reduces the intensity of turbulent velocity fluctuations, a generalization of the known stabilizing effect on the linear phase [START_REF] Chandrasekhar | Hydrodynamics and Hydromagnetic Stability[END_REF][START_REF] Carnevale | Rotational suppression of Rayleigh-Taylor instability[END_REF]. As a consequence, the temporal growth of the mixing layer is observed to be reduced with respect the case without rotation. Rotation also reduces the efficiency of the conversion of the available potential energy into large-scale kinetic energy, and relatively more energy is dissipated at small scales. The analysis of the kinetic energy spectra revealed that rotation induces a strong anisotropy at small scales, where vertical velocity fluctuations dominate over horizontal ones.

In spite of the reduction of the intensity of velocity fluctuations, we found that rotation allows them to extends in a broader region, a manifestation of the Taylor-Proudman theorem on the bidimensionalization of the flow, while the scale associated to temperature fluctuations is reduced. These results imply that in presence of rotation the mixing layer has a complex structure which cannot be characterized by a single scale. The presence of different characteristic scales for velocity and temperature has important dynamical consequences.

Indeed, the analysis of our numerical simulations show that the heat transfer, quantified by the Nusselt number, is reduced by rotation not only as a function of time, but also at a given Rayleigh number (i.e. width of the mixing layer). This means that rotation reduces the correlation wT between the temperature and the vertical velocity.

Finally, we have analyzed the cyclone-anticyclone asymmetry induced by rotation. We have found that, as in other turbulent systems, rotation induces an asymmetry in the PDF of the vertical vorticity which displays a a positive (with respect to the background rotation)

skewness. The skewness is found to be approximatively constant during the development of the turbulent mixing layer.

In conclusion, we found that rotation induces a rich and complex phenomenology in Rayleigh-Taylor turbulence. As RT turbulence is a prototypical example of bulk turbulent convection, not driven by boundary layers, we expect similar effects in other convective flows dominated by bulk properties. Further numerical and experimental work in this direction would be extremely interesting.

FIG. 1 .

 1 FIG. 1. Snapshots of the temperature field at time t = 5.2τ for two simulations with Ωτ = 0 (left) and Ωτ = 10 (right) starting from the same initial conditions. Yellow (blue) represents hot (cold), light (heavy) fluid.

Figure 1

 1 Figure 1 shows two snapshots of the temperature field for two simulations at Ωτ = 0 and Ωτ = 10. Rotation produces clear qualitative effects on the temperature field: Thermal plumes are more coherent and elongated in the vertical direction while their vertical velocity is reduced.

FIG. 2 .

 2 FIG. 2. Vertical velocity fluctuations w rms in the mixing layer as a as a function of time for three different rotations, Ωτ = 0 (red squares), Ωτ = 6.25 (pink circles) and Ωτ = 12.5 (blue triangles). Inset: ratio of vertical to horizontal velocity fluctuations as a function of time, for the same values of rotation.

FIG. 3 .

 3 FIG.3. Ratio of the vertical velocity fluctuations w rms to the width of the mixing layer h as a as a function of time for three different rotations, Ωτ = 0 (red squares), Ωτ = 6.25 (pink circles) and Ωτ = 12.5 (blue triangles). Inset: time evolution of the mixing layer width h defined in terms of a threshold rθ 0 in the temperature profile, with r = 0.9.

FIG. 4 . 25 (

 425 FIG. 4. Rossby number Ro = w rms /(2Ωh) as a function of time for the two rotations Ωτ = 6.25 (pink circles) and Ωτ = 12.5 (blue triangles). Inset: the same quantities in log-log plot. The black line represents t -1 scaling.

FIG. 5 .

 5 FIG. 5. Efficiency of kinetic energy production Σ = -(dE/dt)/(dP/dt) as a function of time for three different rotations, Ωτ = 0 (red squares), Ωτ = 6.25 (pink circles) and Ωτ = 12.5 (blue triangles). Inset: potential energy loss -∆P (t) = P (0)-P (t) for the same three values of rotations (from top to bottom).

FIG. 6 .

 6 FIG.6. Integral lengths based on the standard deviation of the fluctuation profile of temperature L (T ) (red squares), horizontal velocity L (u) (pink circles) and vertical velocity L (w) (blue triangles) without rotation (empty symbols) and for Ωτ = 12.5 (filled symbols).

1 -FIG. 7 .

 17 FIG. 7. Vertical profiles of mean temperature T (z) (upper plot) and of vertical velocity w rms (z) = w 2 (z) 1/2 for the simulation with Ωτ = 0 (red lines) and Ωτ = 12.5 (blue lines) at time t = 6τ .

FIG. 8 .

 8 FIG. 8. Reynolds number (filled blue symbols) and Nusselt number (open red symbols) as a function of Rayleigh number for the simulations with Ωτ = 0 (squares), Ωτ = 6.25 (circles) and Ωτ = 12.5 (triangles). The black lines represent the ultimate state behavior Ra 1/2 .

FIG. 9 . 5 (FIG. 10 .FIG. 11 . 3 z / ω 2 z 3 / 2

 9510113232 FIG.9. Upper plot: Vertical E z (k) (red) and horizontal E x (k) (blue) kinetic energy spectra computed at the time t = 6.0τ both in the absence of rotation (continuous lines) and with Ωτ = 12.5 (dashed lines). Lower plot: Vertical and horizontal kinetic energy spectra computed at time t = 2.8τ for Ωτ = 0 and at time t = 6.0τ for Ωτ = 12.5 at which the peaks of the vertical spectra

ACKNOWLEDGMENTS

We acknowledge support from the European COST Action No. MP1305 Flowing Matter.

AM thanks the financial support from the PRIN 2012 project n. D38C13000610001 funded by the Italian Ministry of Education. AM is also grateful for the financial support for the computational infrastructure from the Italian flagship project RITMARE. HPC center CINECA is gratefully acknowledged for computing resources.


Marianne Allanic
email: marianne.allanic@utc.fr

Thierry Brial

Alexandre Durupt

Marc Joliot

Philippe Boutinaud

Benoit Eynard

Towards an enhancement of relationships browsing in mature PLM systems

Keywords: Product Lifecycle Management (PLM), Heterogeneous Data, Information Visualisation, Graph Theory, Bio-Medical Imaging (BMI)

Product Lifecycle Management (PLM) domain is at a key point in its development: its concepts and technologies are mature. PLM systems not only manage documents but information associated to a product along its lifecycle, such as Bills-Of-Material (BOM) or requirements at different levels of granularity. All the dependencies between concepts lead to complex relationships from which it is not easy to get a coherent overview. The purpose of the paper is to know whether PLM systems are able to deal relationships complexity. Two case studies -one form manufacturing industry and the other one from a new application domain of PLM, Bio-Medical Imaging -are developed in the paper. They show that hierarchical browsing of existing PLM systems is not suitable to manage relationships complexity and must evolve to graph browsing.

Introduction

Product Lifecycle Management (PLM) began to emerge in the late 1980s as an integrated approach for the design management of products developed by automotive and aerospace industries [START_REF] Konstantinov | Emerging standards for design management systems[END_REF]. The complexity of products in these domains, as well as the increasing competition caused by global markets, posed the need for a better product management system [START_REF] Ming | Technology solutions for collaborative product lifecycle management-status review and future trend[END_REF]. To stay competitive by reducing product lifecycle duration, the main concern was to provide the right information in the right context at the right time among the global flow of information and throughout the lifecycle of the product [START_REF] Ameri | Product Lifecycle Management : Closing the Knowledge Loops[END_REF].

PLM can be defined as a "product centric -lifecycle-oriented business model, [...] in which product data are shared among actors, processes and organizations in the different stages of the product lifecycle" [START_REF] Terzi | Product lifecycle management from its history to its new role[END_REF]. PLM integrates modelling, engineering, manufacturing and project management software into one collaborative platform [START_REF] Ming | Technology solutions for collaborative product lifecycle management-status review and future trend[END_REF]. When PLM concepts matured from Product Data Management (PDM), manufacturing companies evolved from managing documents to the management of enterprise concepts. Indeed, users at different stages along the product lifecycle need to access to different information, and PLM systems manage product data at many levels of granularity. Some of them are Bill Of Material (BOM) which describes assemblies and parts that constitute the product, engineering BOM (eBOM) which decomposes the product as it is designed, and manufacturing BOM (mBOM) as it is built. The relationships that link the product information levels are of major importance to be able to manage in a consistent way all the phases of the product lifecycle and to enable collaboration between the different teams.

PDM systems were initially designed to manage Computer Aided Design (CAD) files, thus PLM systems were developed mostly around this type of data in the context of manufacturing industry. However, more and more works address other domains, such as Computer Aided Engineering (CAE), Mechatronics, Architecture Engineering and Construction (AEC), Services and Bio-Medical Imaging (BMI). These domains are multidisciplinary and handle heterogeneous data linked by complex relationships inside which it is not easy to browse. The specific needs coming from new application domains of PLM have been overlooked so far, but the whole PLM community could benefit from them.

Because the complexity of data relationships is growing in manufacturing industry and new application domains of PLM, it is important to study relationships management in PLM systems so as to improve their understanding and analysis. Current data visualisation and relationships browsing in PLM systems are presented in the paper. Then two case studies are developed. They were chosen to show two current challenges of applications of PLM. The first case study deals with the migration from one PLM system to another inside the manufacturing company ACME1 . The second case study focuses on specific needs of a new application domain of PLM: neuroimaging. A discussion about the conclusions from the two case studies ends the paper.

Visualisation and browsing in mature PLM systems

PLM data models

PLM is a mature technology known to increase productivity, maximise product value and reduce cost in organisations [17]. Object-oriented approach was shown to be adequate to model and integrate product, process and resource data through UML diagrams [START_REF] Eynard | UML based specifications PDM product structure and workflow[END_REF]. The Core Product Model (CPM), which defined form, function and behaviour of a product, has been extended [START_REF] Fenves | CPM2: a core model for product data[END_REF] and enables the design of product information-modelling framework to support the full range of PLM requirements [START_REF] Sudarsan | A product information modeling framework for product lifecycle management[END_REF]. Data organisation models in PLM systems are now well-tried.

Three phases define PLM: the design and manufacturing phase is the Beginningof-life (BOL), the distribution and use phase is the Middle-of-life (MOL), and the retired phase is the End-of-life (EOL) [START_REF] Kiritsis | Research issues on product lifecycle management and information tracking using smart embedded systems[END_REF]. Only the BOL phase -with few exceptions -is managed in PLM systems, as most of the research work focuses on this phase. However, many teams need to collaborate and interact during the BOL phase. So they require to access to different levels of information that have to be properly connected, for instance requirement specifications, design or manufacturing.

Product modelling in PLM systems

Important information about a product is stored in product representation or modelling. In manufacturing companies, product modelling is mostly composed of Computer-Aided Design (CAD) models, but these last are heavy, application dependent and focus on one information level. Some lightweight product modelling, on which Ding et al. [START_REF] Ding | Product representation in lightweight formats for product lifecycle management (PLM)[END_REF] made a survey, support users at different stages of the product lifecycle in rapidly browsing, retrieving and manipulating product information across platforms. They constitute interesting exchange formats. Examples of CAD lightweight modelling available in PLM systems are U3D, JT format and XML-based formats (X3D, 3D XML and PLMXML). However these product modelling only address one degree of information complexity for a product. Some works address the design of a multi viewpoints framework [START_REF] Davies | Multiple viewpoint design modelling through semantic markup[END_REF][START_REF] Demoly | Product relationships management enabler for concurrent engineering and product lifecycle management[END_REF] with semantics and relationships management, but they do not indicate how to visually read and understand the relationships complexity.

Interface and browsing in current PLM systems

The interfaces of PLM systems have not been studied by the PLM research community up to now, although they are widely and daily used. It is nothing to say that PLM software interfaces are overall not ergonomic nor intuitive. Companies spend too much time and money in training their employees on information systems, PLM not being an exception. The main critics on the interface are the number of sub-windows and the profusion of menus and icons. Product data instances are presented under the shape of a list and only one level of product information can be displayed at a time. Besides, the interface vocabulary is too manufacturing-industry oriented, which limits the extensibility of PLM systems to other domains.

Rich client installed as a standalone application on local device is widespread among the PLM systems. However, web-based clients are little by little becoming a unavoidable evolution of PLM systems in manufacturing industry in order to access data everywhere and on new mediums such as tablets. The main purpose of web-based clients is to enable easy browsing and retrieval of data, but their interface is not suitable enough. Some interesting PLM web clients have been developed recently -for instance web ARAS system or new Teamcenter 10 active workspace client -, but they still propose an overloaded interface.

Therefore the study of two case studies is proposed to decide whether:

1. The interfaces of PLM systems both comply with the requirements of manufacturing industry and of a new application domain of PLM, such as Bio-Medical Imaging, 2. The PLM research community should address more work on PLM interfaces.

Case study 1: Migration of ACME PLM system

Manufacturing companies are now mostly equipped with PLM systems, and because of information management strategies or costs, they decide to change their PLM system more and more frequently. Therefore, new issues come out such as data consistency in source PLM systems and data import in target PLM systems. In this section, the migration of ACME PLM system is developed to highlight the need for relationships management and visualisation.

Context and data to migrate

In 2011, ACME decided to migrate its Windchill PLM system to Teamcenter PLM system. The CAD software Pro/E remained the same after the migration, so there was no conversion operation. A PLM system handles two distinct types of data: metadata which are stored in a relational database, and data files which are stored in the vault, a securised file system accessible only through PLM. Relationships between data files and metadata are described in the PLM system. In the case of ACME migration, data files are drawings and 3D CAD assemblies.

An Extract Transform Load (ETL) type of implementation was set up for the migration. Indeed, Windchill and Teamcenter do not have the same concepts and data models, so an intermediate migration platform is needed to transform the data. The migration platform is composed of a temporary data files storage and a Graph DataBase (GDB), Neo4J. PLMXQuery tool [START_REF] Sriti | Plmxquery: Towards a standard PLM querying approach[END_REF] is used to populate the migration platform, which constitutes the Extract phase of ETL. Through XQuery language, the metadata are converted into PLMXML language [START_REF] Sdk | Plm xml schema functional description v6[END_REF] and the data files are represented with their dependencies in a ASCII instruction file. Finally, during the load phase of ETL, the target Teamcenter platform is populated thanks to PLMXML Import Tool for the metadata and IPEM tool for the data files.

A GDB was chosen to store temporarily the metadata, because a PLM system can be seen as a set of unique objects having a set of attributes (key or value) and being linked together by typed relationships. A graph corresponds as well to this definition, as it is composed of nodes and edges having attributes. Moreover, it is easy to populate a GDB through standard formats. A screenshot of an item extracted from Windchill and displayed in Neo4J is shown in figure 1.

Issues during the migration

CAD files are by nature tricky data: when they are opened by appropriate CAD software, some CAD files require other dependent CAD files to be opened. This

The IFIP WG5.1 11th

Yokohama, Japan, Fig. 1. Screenshot in its revisions, drawing, means that these dependency files, but must also PLM system to open CAD files are downloaded taken in account during be migrated after in the right order.

Many errors occurred objects: obsolete instances, templates and drawings, during the migration planned initially, because called Kdeep -was to locate when the issues. This algorithm algorithm had passed due to inconsistency hand.

The main lesson referencing of the relationships The ACME compan system. Obviously, problems between the migration turned 11th International Conference -PLM14 Japan, 7th-9th July 2014 5 in Neo4J of an item extracted from Windchill with relationships wing, relatives (family items) and BOM components.

dependency relationships are described somewhere also be described in the PLM systems. When a user open a CAD file locally, the target CAD file and all downloaded from the vault. The relationships dependency during the migration, as data files with dependencies the dependent files. Thus, the imports have to be occurred during the migration, due to wrong referencing obsolete instances, missing references between CAD files and drawings, dependency cycles. These issues were a real migration process which lasted for one year instead of the because the initial data had to be "cleaned". An algorithm was designed to analyse referencing errors on the GDB. the errors occurred in time, and to solve automatically algorithm was applied on the Neo4J database. Once passed through the whole database, there were still some inconsistency of complex relationships, and they had to be lesson from the migration is the unexpected number relationships between the levels of granularity in PLM company was not aware that these issues were existing it is difficult to analyse dependencies and identify ween objects in a PLM system. Besides, the Neo4J GDB turned out to be useful to do the job. Indeed, algorithms relationships to somewhere in the CAD user asks the all dependant dependency must be dependencies can only be scheduled

referencing between and between real burden few months algorithm -GDB. It is able automatically these

Once the Kdeep still some errors be solved by number of wrong PLM systems. existing in its PLM referencing GDB used for algorithms can

The IFIP WG5.1 11th International Conference -PLM14

Yokohama, Japan, 7th-9th July 2014 6 be easily computed on graphs to retrieve statistics and apply display filters for a better understanding of the relationships between graph nodes. Some of the errors could not have been solved out without a visual representation of the dependencies.

Case study 2: Management of GIN research studies

A growing number of domains outside manufacturing industry are implementing PLM systems. Most of these domains handle heterogeneous data with complex relationships, which require to browse, retrieve and visualise information efficiently. Browsing needs of neuroimaging domain are developed in this section.

A PLM database in neuroimaging domain

Neuroimaging domain is multidisciplinary "by its very nature" [START_REF] Van Horn | The functional magnetic resonance imag-ing data center (fmridc): the challenges and rewards of large-scale databasing of neuroimaging studies[END_REF]: the study of brain require an active interaction between many specialities -physics, medicine, mathematics and engineering among others. Magnetic Resonance Imaging (MRI) is one of the most promising imaging technique to study brain complexity. Structural MRI examines brain anatomy, whereas functional MRI analyses what happens while a subject is performing a given task. A Bio-Medical Imaging (BMI) research study can be represented by four stages that constitute a cycle [START_REF] Allanic | Application of plm for bio-medical imaging in neuroscience[END_REF]: study specifications (stage 1), raw data (stage 2), derived data (stage 3) and published results (stage 4). Between stages, it is crucial to keep all the information to be able to understand the context of computing and the history of data, which is a requirement to reproduce derived data result or to reuse them. What a piece of data is, when, where and how it was produced, why and for whom it was performed is called provenance [START_REF] Simmhan | A survey of data provenance in escience[END_REF]. Due to costs, trends to huge cohorts of subjects and growing complexity of analyses, neuroimaging researchers must collaborate and share data between disciplines and laboratories [START_REF] Yarkoni | Cognitive neuroscience 2.0: building a cumulative science of human brain function[END_REF], which are similar issues than the one of manufacturing industry.

PLM was proposed and shown to be relevant to manage efficiently neuroimaging heterogeneous data and to enable the conditions of sharing and reuse of data [START_REF] Allanic | Towards a data model for plm application in bio-medical imaging[END_REF]. Since 2010, the GIN2 has been using the GIN first Brain Imaging Laterality (BIL&GIN1) dataset, which is composed of 300 subjects -balanced by gender and handedness -and which was acquired between 2009 and 2011 [START_REF] Petit | BIL&GIN: a database for the study of hemispheric specialization[END_REF]. In 2013 a PLM system -Teamcenter 9 -was installed at the GIN, populated with the BIL&GIN1 dataset.

Limits of PLM systems browsing

The figure 2 shows the two main ways of browsing information in Teamcenter -the concepts are similar in other PLM systems -: hierarchy view (descending) and impact analysis view (ascending). It is easy to see where an object has been Fig. 2. Screenshots shows that acquisition processing sequence top objects PCR_ROIgraph hierarchy.

used on close levels, of relationships. However, be able to navigate requires data exploration allows no overall view The feedback of the capacities of the ever, they were very rich client display): vironment. Besides, What they are looking with few choices to ment (browsing, con they express the will end, the GIN researchers 11th International Conference -PLM14 Japan, 7th-9th July 2014 7 Screenshots of Teamcenter rich client browsing interfaces. Impact analysis acquisition ACQima_ epi repos is referenced in a raw data branch sequence branches(a) Similar information in shown in hierarchy OIgraph_ 1 (b) and GINT1 Humant0444 (c) through els, as well as referencing sequences, with a limited However, users have to switch from one view to vigate freely among objects relationships. Indeed, research exploration to build new hypotheses, but the PLM view of the complexity.

of the GIN researchers on the PLM system was good the system to manage neuroimaging data organisation. ery critical about the interface of the software (basic y): there are too many menus, icons and sub-windows Besides, the actions of searching for information is not looking for in terms of ergonomics is an over-simplified to make for each options, as well as a relationships full consistency checking, visual information retrieval. In will to handle the system almost without any training. researchers are reluctant to use the PLM database, even analysis view branch and two chy view from through descending limited number another to research work PLM system concerning organisation. How-Teamcenter windows in the enimmediate. simplified interface full manage-In addition, training. In the even if they recognise that it is of great benefits for storing and sharing data and associated provenance.

Discussion

The two case studies developed in the paper show some limits of the interfaces of current PLM systems. This section discusses the limits divided in three categories: query, analyse and browsing. Leads for future work are proposed at the end of each subsection.

Information access: query

In neuroimaging there is a trend for cross-domain analyses: imaging results are correlated with demographical, clinical, psychological and genetics data. Therefore, numerous joins between concepts are required for one single query, which make the search process complex. Indeed, to design customized queries, users have to know the exact data model organisation. From user's point of view, querying is textual browsing: the search fields that are filled in make the query engine browsing among data relationships. Without efficient retrieval capacities, a database is just a storage room. Apart from the limitation of users' daily data search, the lack of effective querying in PLM systems has an impact on the understanding of data provenance and therefore data reuse. PLM data models should be transparent to the users and a proposal is to design a graphical query builder, with which users would be query-autonomous. Another future work would be semantic enrichment of PLM systems, based on ontology, and which handles the management of relationships between objects in a flexible way [START_REF] Assouroko | Semantic relationship knowledge management and reuse in collaborative product development[END_REF]. This would improve performances. Besides, managing PLM concept as a graph would facilitate search processes as well as query performances, thanks to graph theory algorithms.

Information visualisation: reading and analyse

The way the information is displayed inside the space of the PLM systems windows is meaningless. Only one information level is displayed at a time, under the shape of a list. As a result, it is difficult to read and understand two information levels, especially because they do not require the same display format. PLM systems interfaces are not ergonomic nor intuitive, and companies spend significant time and money in training. If in the manufacturing industry world companies have to use a PLM system to stay competitive, other PLM application domains would be reluctant to use PLM systems because of their current interfaces, which is emphasised by the case study 2 of the paper. So it is of importance that the PLM community addresses the limits of PLM interfaces. Moreover, reducing users' training time on information systems would be obviously a competitive advantage for companies.

The IFIP WG5.1 11th International Conference -PLM14

Yokohama, Japan, 7th-9th July 2014 9 Some future work should address new ways of information display in data management systems, particularly focusing on multi viewpoints display. On figure 1, different levels of information are displayed in the Neo4J graph. The positive aspect is that it is easy to access to all the relationships between data and concepts, but the positive aspect is that all the edges look the same, and nothing distinguishes one level of information from the other, for instance CAD files view and product view. So multi viewpoints developments should take this remark into account.

Information access and visualisation: browsing

The two case studies developed in the paper show that graphical relationships browsers in current PLM systems are not satisfactory. Ascending and descending data relationships hierarchy can only be browsed in two different windows, which prevent from consulting two product information levels at a time or from checking relationships consistency between information levels throughout product lifecycle. As shown in the case study 1, there exists no way to analyse references consistency in a PLM system, which implies a loss of information that could be important for the competitiveness of a company.

Therefore, a major concern in the upcoming works is to visualise data relationships by graphs in one single sub-window, in order to improve browsing and visualisation of every component of data provenance in PLM systems. Further researches should be conducted regarding the application of graph theory to the analysis of relationships in PLM systems.

Conclusion

Current PLM systems present limits, notably relationships browsing and analysis, as well as meaningful information visualisation. PLM community would gain to address the design of a simplified graphical user interface that presents multi viewpoints product information. Leads of future work include semantic enrichment and application of graph theory. It is time to make PLM systems evolve towards a more adequate relationships browsing and visualisation.

ACME is a French company designing and manufacturing thermal systems. The name is modified, as the authorisation of the company to use their name in the paper is under approval.

Groupe d'Imagerie Neurofonctionnelle -Neurofunctional Imaging Research Group

Yokohama, Japan, 7th-9th July 201411

Acknowledgments

The authors wish to thank in particular the Association Nationale de la Recherche et de la Technologie (ANRT) for its financial support to their work (CIFRE 2012/0420).

The work presented in the paper is conducted within an ANR (Agence Nationale de la Recherche) founded project (n o ANR-13-CORD-0007) for thematic axis n o 2 of the Contint 2013 Call for Proposal: from content to knowledge and big data.

The IFIP WG5.1 11th International Conference -PLM14

Yokohama, Japan, 7th-9th July 201410