O Dezellus
email: olivier.dezellus@univ-lyon1.fr

N Eustathopoulos

Fundamental issues of reactive wetting by liquid metals

After a brief presentation of thermodynamics and kinetics of non-reactive wetting, the recent results and theoretical developments concerning the reactive wetting of solids by liquid metals are reviewed. A section is devoted to illustrate and discuss the effect of interfacial reactions in removing the wetting barriers existing on many ceramic and metallic solids.

Introduction

This review concerns the wetting of high temperature solids, mainly ceramics, by liquid metals and alloys. Many metal/ceramic couples are far from equilibrium and the resulting interfacial reactions can strongly modify the chemistry, structure and topography of the contact area .These changes affect both spreading kinetics and the ultimate degree of wetting and those aspects of reactive wetting that can be exploited in practice to control wetting and adhesion are reviewed in this paper.

The paper is organized in three parts. In section 1 the main features of non-reactive wetting of metals on various types of solids are rapidly presented. The main part of the paper (section 2) is on thermodynamics and kinetics of reactive wetting. After briefly describing the basic aspects of the "Reaction Product Control" model, recent results and developments of this model are given in more detail. In a final section the role of interfacial reactions in the

-Non-reactive wetting

Thermodynamics

As a general rule, the interaction energy between a liquid and a solid substrate is quantified by the work of adhesion W a related to the characteristic surface energies σ ij of the solid (S)liquid (L)-vapour (V) system by the Dupré equation: [START_REF] Blake | Wettability, chapter Dynamic Contact Angles and Wetting Kinetics[END_REF] Combining this expression with Young's equation leads to the following fundamental equation of wetting, known as the Young-Dupré equation: [START_REF] Bougiouri | Wetting and reactivity in Ni-Si/C system: experiments versus model predictions[END_REF] This equation shows that the equilibrium contact angle obtained for a given liquid metal on a solid substrate results from the competition of two types of forces: adhesion forces that develop between the liquid and solid phases, expressed by W a (tending to increase the common area thus improving wetting) , and cohesion forces of the liquid taken into account by the surface energy of the liquid acting in the opposite direction (the cohesion energy of the liquid is roughly equal to 2 ). Liquid metals are high cohesion energy liquidsindeed their cohesion energy at melting point represents more than 90% of the cohesion energy of the same metal in the solid state. Thus, according to the Young-Dupré equation, good wetting of a liquid metal on a solid substrate, i.e a close to unity, can be obtained only if the work of adhesion is also high, which is possible if the interfacial bond is strong, i.e. chemical.

Solid metals

Pb/Fe -40° [29,[START_REF] Protsenko | Wetting of Fe-7.5Cr steel by molten Pb and Pb-17Li[END_REF] AgCu/Stainless Steel -10-60° [START_REF] Kozlova | Wetting and brazing of stainless steels by copper-silver eutectic[END_REF] Semi-conductors Sn/Ge -40° [START_REF] Yu | Temperature dependence of spreading kinetics in systems with different types of interaction of contacting phases[END_REF] Ag/SiC -33-65° [START_REF] Rado | Bonding and wetting in non-reactive metal/SiC systems: weak or strong interfaces?[END_REF] Ceramics with a partially metallic character AgCu/Ti 3 SiC 2 -10° [START_REF] Dezellus | in press) Wettability of Ti 3 SiC 2 by Ag-Cu and Ag-Cu-Ti melts[END_REF] θ<<90°

Au/ZrB 2 -25° [START_REF] Voytovych | Reactive vs nonreactive wetting of ZrB 2 by azeotropic Au-Ni[END_REF] Carbon materials Au/C -119-135° [START_REF] Dezellus | The role of van der waals interactions on wetting and adhesion in metal/carbon systems[END_REF] Iono-covalent oxides Cu/Al 2 O 3 , Cu/Si0 2 -120-130° [START_REF] Eustathopoulos | Wettability at High Temperatures[END_REF] Covalent ceramics Au/AlN -134-138° [START_REF] Naidich | Proc. Int. Conf. High Temperature Capillarity[END_REF] θ>>90°

Au/BN -135-150° [START_REF] Naidich | chapter The Wettability of Solids by Liquid Metals[END_REF] For this reason good wetting is the general rule for liquid metals on solid metals whatever the intensity of interaction between liquid and solid, the interfacial bond being metallic, i.e.

strong (see table 1). Liquid metals also wet semi-conductors such as Si, Ge or SiC [START_REF] Yu | Temperature dependence of spreading kinetics in systems with different types of interaction of contacting phases[END_REF][START_REF] Protsenko | Dissolutive wetting of Si by molten Cu[END_REF][START_REF] Rado | Bonding and wetting in non-reactive metal/SiC systems: weak or strong interfaces?[END_REF] because the cohesion of semi-conductors has a metallic character near the free surface.

Finally, liquid metals also wet ceramics such as carbides, nitrides or borides of transition metals because a significant part of the cohesion of these materials is provided by metallic bonds [START_REF] Dezellus | in press) Wettability of Ti 3 SiC 2 by Ag-Cu and Ag-Cu-Ti melts[END_REF][START_REF] Voytovych | Reactive vs nonreactive wetting of ZrB 2 by azeotropic Au-Ni[END_REF]. Solids that are not wetted by non-reactive liquid metals include the different forms of carbon [START_REF] Dezellus | The role of van der waals interactions on wetting and adhesion in metal/carbon systems[END_REF], iono-covalent oxides and covalent ceramics with a high band gap such as BN or AlN [START_REF] Naidich | chapter The Wettability of Solids by Liquid Metals[END_REF][START_REF] Naidich | Proc. Int. Conf. High Temperature Capillarity[END_REF].

It should be emphasized that, even with wettable solids, contact angle values less than 90 ο can be prevented by the presence of wetting barriers on the solid surface. Thus, many of the ceramics with a partially metallic character (such as borides of transition metals), some covalent ceramics (such as SiC and Si 3 N 4 ) and many metallic solids (stainless steels, superalloys, Al containing alloys…) are also easily oxidized and their wettability by nonreactive metals is strongly affected by oxide films that may be very tenacious, especially at low temperatures [START_REF] Dezellus | in press) Wettability of Ti 3 SiC 2 by Ag-Cu and Ag-Cu-Ti melts[END_REF][START_REF] Protsenko | Wetting of Fe-7.5Cr steel by molten Pb and Pb-17Li[END_REF][START_REF] Rado | The role of surface chemistry on spreading kinetics of molten silicides on silicon carbide[END_REF].

Spreading kinetics

In non-reactive systems, the spreading rate is controlled by the viscous flow and described by a power function of the triple line velocity U vs. the instantaneous contact angle θ

U∼θ n (3a)

For a spherical shaped drop with a base radius R, U=dR/dt and equation 3a leads to

R 3n+1 ∼t (3b)
De Gennes has developed an analytical approach in which the spreading rate is considered to be limited by the viscous dissipation in the drop bulk leading to n=3 [START_REF] De Gennes | Wetting: statics and dynamics[END_REF], while Blake's model, which is based on dissipation at atomic level at the triple line, leads to n=2 [START_REF] Blake | Wettability, chapter Dynamic Contact Angles and Wetting Kinetics[END_REF]. Because the viscosity of molten metals is very low, the spreading time (i.e. the time needed for millimetric size droplets to reach capillary equilibrium) is less than 10 -1 s in the case of non-reactive systems with equilibrium contact angle values higher than 20° [START_REF] Yu | Temperature dependence of spreading kinetics in systems with different types of interaction of contacting phases[END_REF][START_REF] Protsenko | Dissolutive wetting of Si by molten Cu[END_REF][START_REF] Saiz | Atomic dynamics and Marangoni films during liquid-metal spreading[END_REF]43]. A significantly longer spreading time was observed in [START_REF] Kozlova | Eustathopoulos (in press) Nonreactive versus dissolutive wetting of Ag-Cu alloys on Cu substrates[END_REF] for a couple with an equilibrium contact angle very close to zero.

-Reactive wetting

In 1996 Landry proposed a model to describe wetting for metal/ceramic systems in which the interfacial reactions lead to the formation of continuous layers of a new compound [START_REF] Landry | Dynamics of wetting in reactive metal/ceramic systems: linear spreading[END_REF].

According to the model ("reaction product control" model), both the final contact angle and spreading kinetics are governed by the interfacial reaction. Note that in 2000, Saiz et al. [START_REF] Saiz | Reactive spreading: adsorption, ridging and compound formation[END_REF] suggested that reactive wetting is caused mainly by adsorption, while spreading kinetics is controlled by the migration of a ridge formed at the solid-liquid-vapour triple line. In 2005 [START_REF] Eustathopoulos | Progress in understanding and modeling reactive wetting of metals on ceramics[END_REF] a detailed analysis of existing experimental data was published that did not confirm the Saiz et al. approach. Since that date no new results or new developments concerning this approach have been published and for this reason the approach of Saiz et al. is not reported in the present review.

2-1 Reactive vs non-reactive wetting

Recently, Bougiouri et al. illustrated the first-order transition, from the point of view of wetting, that is associated with the non-reactive to reactive transition in the case of Ni-Si alloys on vitreous carbon (C v ) substrates [START_REF] Bougiouri | Wetting and reactivity in Ni-Si/C system: experiments versus model predictions[END_REF]: when the silicon content in the liquid alloy is below the minimum silicon content needed for SiC formation (35at% at 1200°C), no SiC is indeed formed at the liquid/solid interface and the system is characterized by a high contact angle value (123°). Conversely, when the Si content exceeds the reactivity limit, a thin continuous SiC layer is formed along the whole interface leading to spreading of the liquid alloy. The continuous layer is revealed for example by the triple line receding during cooling after capillary equilibrium is reached (see Fig. 1). on SiC (circles, [START_REF] Rado | Wetting and bonding of Ni-Si alloys on silicon carbide[END_REF]). T = 1200 ο . The reactivity limit for the C v substrate is shown by the dotted line.

Spreading kinetics

The spreading time observed for reactive metal/ceramic systems, where a new compound is formed at the interface during spreading, is in the range 10 1 -10 4 s, i.e. several orders of magnitude higher than for non-reactive systems. As a consequence, the spreading rate in reactive systems is not limited by viscous dissipation but by the rate of interfacial reaction at the triple line [START_REF] Landry | Dynamics of wetting in reactive metal/ceramic systems: linear spreading[END_REF]. In turn, this rate is controlled by the slower of two successive phenomena that take place in the reaction process: diffusive transport of reacting species to or from the triple line and local reaction kinetics at the triple line.

Control by the local reaction at the triple line

In the case of control by the local reaction at the triple line, the rate of reaction and hence the triple line velocity are expected to be constant with time [START_REF] Landry | Dynamics of wetting in reactive metal/ceramic systems: linear spreading[END_REF]. Since the first experimental evidence obtained for unalloyed aluminium on carbon [START_REF] Landry | Dynamics of wetting in reactive metal/ceramic systems: linear spreading[END_REF], constant or nearly constant triple line velocities U have been observed for different systems such as the CuSi/C [START_REF] Dezellus | Chemical reaction-limited spreading: the triple line velocity versus contact angle relation[END_REF] and NiSi/C [START_REF] Bougiouri | Wetting and reactivity in Ni-Si/C system: experiments versus model predictions[END_REF] systems, for CuSi alloys on oxidized SiC [START_REF] Dezellus | Spreading of Cu-Si alloys on oxidized SiC in vacuum: experimental results and modelling[END_REF] ,for pure Si on oxidized Si 3 N 4 [START_REF] Drevet | Wetting and adhesion of Si on Si 3 N 4 and BN substrates[END_REF] and for Ti-containing Ni-based alloys on AlN [START_REF] Koltsov | Influence of Ti on wetting of AlN by Ni-base alloys[END_REF]. An example is given in Fig. 3 for a NiSi alloy on vitreous carbon. After transient spreading, where the contact angle decreases in a few tens of seconds from 160° to 110°, the spreading rate dR/dt continues to decrease but only slightly ("quasilinear spreading") and tends towards a constant value U* when θ tends towards θ N .

The quasilinear spreading proves that the spreading rate is not limited by long-range diffusion of the reactive element (Si) but by the local reaction process at the triple line. At θ=θ N a second wetting stage appears abruptly, accompanied by a net decrease in spreading rate. This stage is discussed later. For the quasilinear stage, a model leading to analytical expressions of contact angle as a function of time, has been developed [START_REF] Dezellus | Chemical reaction-limited spreading: the triple line velocity versus contact angle relation[END_REF][START_REF] Dezellus | Progress in modelling of chemical-reaction limited wetting[END_REF] assuming that, at any time, the instantaneous contact angle θ reflects the local chemistry of the interface at the triple line (referred to hereafter as TL) where the interface is composite, partly of the liquid/initial substrate type and partly of the liquid/reaction product type. Moreover the reaction rate is assumed to be controlled by the dissolution process, i.e., by the process of atom transfer occurring at the substrate/alloy interface.

The following equations describing spreading were derived for the change in contact angle with time t:

cosθ F -cosθ=(cosθ F -cosθ 0 ).exp(-k.t) (4)
And for the dependence of the spreading rate U on the instantaneous contact angle θ:

(

with and V being the volume of the liquid droplet.

The constant k in Eqs. 4 and 5 is proportional to the kinetic constant k d and to the driving force Δµ of the dissolution process. For the NiSi/C system, Δµ is the difference between the chemical potential of carbon in the solid and the liquid and is related to the activity of Si in the alloy a Si by the equation 6:

where a Si I is the minimum value of Si activity for SiC formation by reaction between Si in the alloy and solid carbon (a Si I corresponds to the equilibrium of three phases: initial substrate, reaction product and liquid). According to this expression logarithmic plots of cosθ F -cosθ versus time would be linear with a slope equal to minus k. An example is given in figure 4 for Ni-63 at.% Si alloy on vitreous carbon at different temperatures. The experimental data agree with the linearity predictions in a wide domain of theta. The sharp change in the slope of straight lines reflects the high value of the activation energy of the dissolution process (about 250 kJ.mol -1 in this case) [START_REF] Bougiouri | Wetting and reactivity in Ni-Si/C system: experiments versus model predictions[END_REF]. For dissolution reactions controlled by the atomic process at the interface, the kinetic constant k d is in principle sensitive to the crystallographic structure of the initial substrate. This is confirmed by the strong variation in the triple line velocities observed for Cu-50at.% Si on two different types of carbon: vitreous carbon and pseudo-monocrystalline graphite (Fig. 5). As for the influence of the activity of reactive element on spreading kinetics, according to Eq. 6 this would be weak, in agreement with experimental findings [START_REF] Calderon | Eustathopoulos (in press) Wetting dynamics versus interfacial reactivity of AlSi alloys on carbon[END_REF]. According to Eq. 5, plots of the spreading rate U divided by F(θ) versus the cosine of the instantaneous contact angle would be linear. This was verified by the results of Muolo et al. [START_REF] Muolo | Wetting, spreading and joining in the alumina-zirconia-Inconel 738 system[END_REF] for the CuAg-5at%Ti alloy on three different substrates, two oxidic (an alumina-zirconia ceramic and zirconia stabilized by yttria) and one metallic (the superalloy Inconel 738).

10

Control by local reaction at the triple line: strong vs weak coupling

Fig. 3 shows that the spreading rate changes abruptly at a contact angle noted θ N . As discussed in detail elsewhere [START_REF] Dezellus | Contribution à l'étude des mécanismes de mouillage réactif[END_REF], this reflects the change in the microscopic configuration at TL. At t<t N (θ>θ N ) direct contact between the liquid and the initial substrate results in a high dissolution rate and in a strong coupling between the local chemical reaction and spreading kinetics. The microstructure of the reaction product in this stage is columnar [START_REF] Calderon | Eustathopoulos (in press) Wetting dynamics versus interfacial reactivity of AlSi alloys on carbon[END_REF][START_REF] Dezellus | Chemical reaction-limited spreading: the triple line velocity versus contact angle relation[END_REF] At t>t N the liquid no longer has direct access to the initial substrate surface and a slow reaction takes place by nucleation and growth of reaction product particles occurring on the substrate free surface in front of TL (see Fig. 6 for schematic description of TL configuration). Direct experimental evidence for this mechanism, which leads to formation of an equiaxial microstructure, was given recently in [START_REF] Calderon | Eustathopoulos (in press) Wetting dynamics versus interfacial reactivity of AlSi alloys on carbon[END_REF] (see Fig. 7). Table 2 reports the main features of the two wetting stages. The last column gives the growth rate of the reaction layer on the carbon free surface once the capillary equilibrium was attained (in this case the reaction front advances while TL does not move). The comparison indicates that the rate of the linear spreading stage is much closer to the growth rate of silicon carbide during the decoupled growth than to the TL velocity in the quasilinear stage.

Diffusion-controlled reactive wetting

Diffusion-controlled wetting was modelled by Mortensen et al. in 1997 [23]. According to this model, the rate of isothermal spreading is time-dependent and varies in direct proportion to the instantaneous contact angle θ.: [START_REF] Dezellus | Chemical reaction-limited spreading: the triple line velocity versus contact angle relation[END_REF] where D is the diffusion coefficient in the liquid, n v is the number of moles of reactive solute per unit volume of the reaction product, e is the reaction product thickness at the triple line, C 0 is the nominal (far-field) drop reactive solute concentration, C e is the concentration of reactive solute in equilibrium with the reaction product (such that C = C e at the triple line). F(t) is strictly a function of time but in fact varies so little that it can be taken to be constant, remaining close to 0.04 in usual sessile drop experiments [START_REF] Mortensen | Kinetics of diffusion-limited spreading of sessile drops in reactive wetting[END_REF].

When compared with experimental results, this relation shows good agreement for systems such as CuSn-Ti/C v [START_REF] Dezellus | Diffusion-limited reactive wetting. spreading of Cu-Sn-Ti alloys on vitreous carbon[END_REF], in which the final contact angle is close to zero. However some discrepancies remain in other cases such as Cu-Cr/C v . For this system the intercept of the line giving vs. θ with the θ-axis is not at the origin (θ = 0), but rather at a finite angle close to Fig. 8: Experimental triple line velocity as a function of instantaneous contact angle θ for Cu-1 at.% Cr on vitreous carbon at 1150°C [START_REF] Voitovitch | Diffusionlimited reactive wetting: study of spreading kinetics of Cu-Cr alloys on carbon substrates[END_REF] and spreading rate according to Mortensen's model [START_REF] Mortensen | Kinetics of diffusion-limited spreading of sessile drops in reactive wetting[END_REF] To verify the effect of thickening on the spreading kinetics, numerical calculations of the wetting process by finite element modelling was recently performed in the case of Cu-Cr alloy on C v [START_REF] Hodaj | Diffusion-limited reactive wetting: effect of interfacial reaction behind the advancing triple line[END_REF]. Calculations were performed by arbitrarily varying the value of the diffusion coefficient D C of C in chromium carbide. It was found that for very low values of D C the rate of reaction product growth behind the triple line is negligible. The triple line velocity is then simply proportional to the instantaneous contact angle θ, as predicted by Mortensen's analytical model. When D C increases, the curve giving the triple line velocity U as a function of θ remains linear, retaining the same slope, but is shifted so that its intercept with the θ-axis increases progressively. Finally, the experimental data are relatively well predicted for D C equal to 6.10 -15 m 2 s -1 which is a realistic value for this system.

As a conclusion, the simulation yields the following general result: interfacial reactivity manifests itself by the presence of a finite "dead-angle" that must be subtracted from θ in Mortensen's analytical solution, this equation remaining otherwise essentially valid. The physical meaning of the dead angle is presented on Fig. 9: the flux across the solid angle delimited by the "dead angle" is deviated away from the TL and consumed in thickening of the reaction layer behind the TL.

-Interfacial reactions vs wetting barriers

Most of the non-oxide ceramics are easily oxidized and the oxide films on their surface act as wetting barriers leading to non-wetting by non-reactive liquid metals. Reactivity at liquid/solid interfaces can remove these wetting barriers thus improving wetting.

Symmetrically, reactivity can also lead to the formation of a reaction product less wetted than the initial substrate, i.e to the formation of a wetting barrier.

An interesting example was given by Frage et al. in the Au-TiC system [START_REF] Frage | Wetting of TiC by non-reactive liquid metals[END_REF]. The authors showed that pure Au does not wet TiC the contact angle being about 130° a value that is similar to that obtained for Au on carbon substrates (see Table 1). Because of the strong interaction between Au and Ti slight dissolution of Ti from the substrate into the liquid occurs. However, as the solubility of C in Au is much smaller than that of Ti, graphite precipitates at the interface leading to a contact angle of 130° which is characteristic of Au on graphite (Table 1). A similar case where reactivity leads to the formation of a wetting barrier was already reported by Kalogeropoulou et al. in the Ag/SiC system [START_REF] Kalogeropoulou | Mechanisms of reactive wetting: the wetting to non-wetting case[END_REF] where dissolution of Silicon carbide is a typical example of non-oxide ceramic that is easily oxidized. Indeed even under high vacuum SiC surfaces at temperatures less than 1100°C are usually covered by a silica film that acts as a wetting barrier. However, with alloys containing Si, the removal of silica is possible by the formation of the volatile suboxide SiO by reaction between silicon and silica at the triple line. Dezellus et al. performed a dedicated study with CuSi alloys on silicon carbide substrates which had been deliberately oxidized to obtain oxide films a few tens of nanometers thick [START_REF] Dezellus | Spreading of Cu-Si alloys on oxidized SiC in vacuum: experimental results and modelling[END_REF]. A typical wetting experiment consists of two steps (see Fig. At low temperatures normal metallic solids are covered by thin oxide films, a few nanometers thick, formed rapidly in contact with air at room temperature. Such oxides, that may be stable even in high vacuum or in low partial pressure of oxygen gas mixtures, are not wetted by liquid metals [START_REF] Eustathopoulos | Wettability at High Temperatures[END_REF][START_REF] Kozlova | Wetting and brazing of stainless steels by copper-silver eutectic[END_REF][START_REF] Protsenko | Wetting of Fe-7.5Cr steel by molten Pb and Pb-17Li[END_REF][START_REF] Protsenko | The role of intermetallics in wetting in metallic systems[END_REF]. Wetting can be strongly improved by reactions where intermetallic compounds are formed, leading to the replacement, in situ, of the oxidized surface by a clean surface of an intermetallic compound [START_REF] Protsenko | The role of intermetallics in wetting in metallic systems[END_REF] (see Fig. 12). For this reason, as a general rule, the final degree of wetting in liquid metal/solid metal systems in which intense reactions occur at the interface is much less sensitive to environmental factors than in nonreactive systems.

  Journal of Materials Science, 2010, 45, 4256-4264 2 removal of wetting barriers existing on many metallic and ceramic solids is illustrated with several examples.

Fig. 1 :Fig. 2 :

 12 Fig. 1: SEM image of a triple line region of a Ni-66.8 at% Si alloy on C v observed fromabove. X-ray micro-analysis of area exposed after receding reveals the presence of C and Si only[START_REF] Bougiouri | Wetting and reactivity in Ni-Si/C system: experiments versus model predictions[END_REF]

Fig. 3 :

 3 Fig. 3: Typical change in drop base radius R and contact angle θ with time in the case of reactive spreading kinetics limited by the local process at the triple line. Ni-63 at.%Si alloy on vitreous carbon at 1200°C [2].

Fig. 4 :

 4 Fig.4: Natural logarithm of cosθ F -cosθ versus time t for Ni-63at.% Si alloy/C v system at different temperatures[START_REF] Dezellus | Chemical reaction-limited spreading: the triple line velocity versus contact angle relation[END_REF].

Fig. 5 :

 5 Fig. 5: Change in drop base radius versus time t for Cu-50at.% Si alloy on two different types of carbon: vitreous carbon and pseudo-monocrystalline graphite at 1200°C [5].

Fig. 6 :

 6 Fig. 6: Schematic representation of the configuration at the triple line (a) at θ>θ N , direct contact between the liquid and initial substrate leads to strong coupling between the local chemical reaction and spreading kinetics. (b) at θ<θ N , the liquid no longer has direct access to the initial substrate surface and coupling is weak, (c) when θ P is reached, decoupled growth of reaction product ahead of TL occurs .(according to [5])

Fig. 7 :

 7 Fig. 7: Top view of the reaction layers close to the triple line for Al-25at.%Si / C v .Zone A (observed after removing, by dissolution, the Al-Si alloy) corresponds to SiC formed during weakly coupled spreading (θ P < θ < θ N ), while zone B corresponds to the growth of a film ahead of the TL after capillary equilibrium is reached (θ = θ P ) [3].

40 ο [ 38 ]

 38 as shown in Fig.8. Note that the Mortensen equation was established by assuming that all solute flux is consumed by the reaction at the triple line, thus neglecting any thickening of the reaction layer behind the triple line. However, this assumption is hardly verified in the Cu-Cr/C v system, where chromium carbide interfacial reaction layers are relatively thick (several micrometers) and continue to thicken if the drop is held at temperature, indicating rapid carbide growth kinetics in this system.

Fig. 9 :

 9 Fig. 9: Schematic representation of the physical meaning of the dead angle that must be subtracted from θ in Mortensen's analytical solution (Eq. 7). The flux across the solid angle delimited by the "dead angle" is deviated away from the TL and consumed in thickening of the reaction layer behind the TL.

Fig. 10 :

 10 Fig. 10: Contact angle between stoichiometric TiC and Au alloys at 1150 ο [15]

11): ( 1 )

 1 Firstly, fast spreading is observed towards 100°, the equilibrium contact angle value of Cu-Si alloy on bulk silica. Then, the contact angle remains constant during a time that depends on the initial silica film thickness. During this first step, deoxidation occurs close to the triple line, because in this particular area the distance between the liquid/solid interface (where the dissolution of silica takes place) and the liquid/vapour interface (where SiO evaporation occurs) is the smallest. (2) As soon as the liquid comes into contact with the true SiC surface, spreading occurs and the contact angle decreases toward the equilibrium contact angle on SiC. This second step is characterized by a constant triple line velocity and corresponds to reactive spreading limited by the kinetics of the dissolution reaction of the silica layer at the triple line. Note that when large pieces of SiC are brazed with Me-Si alloys, the deoxidation mechanism of the SiC surface by SiO formation gives the geometrical configuration a crucial role in alloy wetting, the difficulty increasing in the order: sessile drop → "capillary" brazing → "sandwich" brazing[START_REF] Koltsov | Brazing of AlN to Si by a Pr silicide: Physicochemical aspects[END_REF].

Fig. 11 :

 11 Fig. 11: Time-dependent variation in drop base radius R and contact angle θ for Cu-40 at.% Si on the C face of an α-SiC single crystal covered by a silica layer 60-70 nm thick [10].

Fig. 12 :

 12 Fig. 12: Schematic description of the region close to the triple line for an oxidised metal (a) in the case of a non-reactive liquid metal B-solid metal A couple and (b) in the case of formation of a layer of intermetallic compound AB n at the interface [32].

Table 1 :

 1 Wetting by non-reactive liquid metals

	Type of solid	Wetting	Examples

Table 2 :

 2 Wetting of vitreous carbon by Cu-50 at.% Si at T=1200°C. Characteristics of the

	three different regimes [5]
	Regime	θ > θ N	θ P < θ < θ N	θ = θ P
	R(t) law	quasilinear	linear	-
	U (nm/s)	630*	54*	20**
	Activation energy (kJ/mol)	255 ± 30	335 ± 15	-
	Microstructure	columnar	equiaxial	equiaxial
	* At θ=θ N

** Average value, L=30 µm, Δt=1500 s

was improved to take into account the influence on spreading rate of the reaction occurring at the interface behind the triple line. This review does not report on "dissolutive wetting". This is wetting accompanied by dissolution of the solid into the liquid with formation of a macroscopically non-planar solid/liquid interface [14,40]. Dissolutive wetting is a common process in liquid metal /solid metal systems but it is also relevant in some metal/ceramic systems such as Ni / C [14],

Ni/HfB 2 [28] and AuNi/ZrB 2 [39]. Although recent studies performed with model liquid metal/solid metal couples have contributed significantly to improving our knowledge of dissolutive wetting [21,30,40,41,42] more work is needed to obtain a satisfactory description of the thermodynamics and kinetics of this type of wetting.


Eduard Feireisl

AND Isabelle Gallagher
email: isabelle.gallagher@math.jussieu.fr

Anton Ín

Antonín Novotný
email: novotny@univ-tln.fr

A SINGULAR LIMIT FOR COMPRESSIBLE ROTATING FLUIDS

We consider a singular limit problem for the Navier-Stokes system of a rotating compressible fluid, where the Rossby and Mach numbers tend simultaneously to zero. The limit problem is identified as the 2-D Navier-Stokes system in the "horizontal" variables containing an extra term that accounts for compressibility in the original system.

Introduction

Consider a scaled Navier-Stokes system in the form (1.1)

∂ t ̺ + div x (̺u) = 0, (1.2) ∂ t (̺u) + div x (̺u ⊗ u) + 1 ε (g × ̺u) + 1 ε 2 ∇ x p(̺) = div x S(∇ x u
), with the viscous stress tensor (1.3) S(∇ x u) = µ ∇ x u + ∇ t x u -2 3 div x uI , µ > 0, and g = [0, 0, 1]. Here ̺ = ̺(t, x) ≥ 0 denotes the density and u(t, x) = [u 1 , u 2 , u 3 ](t, x) denotes the velocity of the fluid. Problem (1.1 -1.2) arises in meteorological applications, modeling rotating compressible fluids with the rotation axis determined by g and the Rossby and Mach number proportional to a small parameter ε.

We consider a very simple geometry of the underlying physical space, namely an infinite slab Ω bounded above and below by two parallel planes, (1.4) Ω = R 2 × (0, 1).

The work of E.F. was supported by Grant 201/08/0315 of GA ČR as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503.

The work of I.G is partially supported by the French Ministry of Research grant ANR-08-BLAN-0301-01.

The velocity u satisfies the complete slip boundary conditions, (1.5) u

• n = u 3 | ∂Ω = 0, [Sn] × n| ∂Ω = [S 2,3 , -S 1,3 , 0]| ∂Ω = 0.
For the initial data (1.6) ̺(0, •) = ̺ 0,ε , u(0, •) = u 0,ε , our goal is to study the asymptotic behavior of the corresponding solutions ̺ ε , u ε for ε → 0. We focus on the interplay between the Coriolis force, here proportional to a singular parameter 1/ε, and the acoustic waves created in the low Mach number regime. In particular, we neglect:

• stratification due to the presence of gravitation, here assumed in equilibrium with the centrifugal force; accordingly the action of the centrifugal force is also neglected; • the effect of a boundary layer (Ekman layer), here eliminated by the choice of the complete slip boundary conditions. We consider ill-prepared initial data, specifically,

(1.7)            ̺ 0,ε = ̺ + εr 0,ε , with {r 0,ε } ε>0 bounded in L 2 ∩ L ∞ (Ω), for some positive constant ̺, {u 0,ε } ε>0 bounded in L 2 ∩ L ∞ (Ω; R 3 ).           
Because of the prominent role of the "vertical" direction g in the problem, we introduce the "horizontal" component v h = [v 1 , v 2 , 0] of a vector field v, together with the corresponding differential operators ∇ h , div h , and, notably, curl h , which is represented by the scalar

field curl h [v] = ∂ x 1 v 2 -∂ x 2 v 1 . Let ̺ ε , u ε be a solution of problem (1.1-1.6). Introducing a new quan- tity r ε = ̺ ε -̺ ε which satisfies ∂ t r ε + 1 ε div x (̺u) + div x (r ε u) = 0,
we easily check that if

r ε → r, u ε → U in some sense,
then, at least formally, the limits satisfy a diagnostic equation

(1.8) g × U + p ′ (̺) ̺ ∇ x r = 0, which in turn implies that (1.9) r = r(x 1 , x 2 ), U = [U h , 0], U h = U h (x 1 , x 2 ).
Moreover, as we will see below, div x U = div h U h = 0, and denoting ∇ ⊥ h r the vector (∂ x 2 r, -∂ x 1 r),

(1.10) ∂ t ∆ h r - 1 p ′ (̺) r + ∇ ⊥ h r • ∇ h (∆ h r) = µ ̺ ∆ 2 h r.
Thus r may be interpreted as a stream function associated to the vector field U h , therefore (1.10) can be viewed as a 2D Navier-Stokes system describing the motion of an incompressible fluid in the horizontal plane R 2 , supplemented with an extra term (1/p ′ (̺))∂ t r.

The main goal of the present paper is to provide a rigorous justification of the target system (1.10) in the framework of weak solutions to the primitive equations (1.1), (1.2). In Section 2, we introduce the weak solutions to both systems, recall their basic properties, and state our main result. In Section 3, we derive the necessary uniform bounds on the family of solutions {̺ ε , u ε } ε>0 , and pass formally to the limit when ε → 0. In Section 4, the associated wave equation describing propagation of the acoustic waves in the low Mach number regime is introduced. Using the celebrated RAGE theorem, we show that the acoustic energy tends to zero, at least locally in space. The proof of the main result is completed in Section 5.

Preliminaries

To begin, we point out that system (1.1 -1.3), endowed with the boundary conditions (1.5) can be recast as a purely periodic problem with respect to the vertical coordinate x 3 provided ̺, u 1 , u 2 were extended as even functions in the x 3 -variable defined on

Ω = R 2 × T 1 , T 1 ≡ [-1, 1]| {-1,1} ,
while u 3 is extended to be odd in x 3 on the same set. A similar convention is adopted for the initial data.

2.1. Weak solutions. We shall say that functions ̺, u represent a weak solution to problem (1.1 -1.6) in (0, T ) × Ω if:

• ̺ ≥ 0, ̺ ∈ L ∞ (0, T ; L γ (Ω)) for a certain γ > 3/2, u ∈ L 2 (0, T ; W 1,2 (Ω; R 3 )); • equation of continuity (1.1) is satisfied in the sense of renormal- ized solutions, namely (2.1) T 0 Ω (̺+ b(̺))∂ t ϕ + (̺+ b(̺))u• ∇ x ϕ + (b(̺) -b ′ (̺)̺)div x uϕ dx dt = - Ω ̺ 0,ε + b(̺ 0,ε ) ϕ(0, •) dx for any b ∈ C ∞ [0, ∞), b ′ ∈ C ∞ c [0, ∞), and any test function ϕ ∈ C ∞ c ([0, T ) × Ω); • p = p(̺) ∈ L 1 ((0, T )×Ω), momentum equation (1.2) is replaced by a family of integral identities (2.2) T 0 Ω ̺u • ∂ t ϕ + ̺(u ⊗ u) : ∇ x ϕ + 1 ε (g × ̺u) • ϕ + 1 ε 2 p(̺)div x ϕ dx dt = T 0 Ω S(∇ x u) : ∇ x ϕ dx dt - Ω ̺ 0,ε u 0,ε • ϕ(0, •) dx for any ϕ ∈ C ∞ c ([0, T ) × Ω; R 3 ); • the energy inequality (2.3) Ω 1 2 ̺|u| 2 + 1 ε 2 E(̺, ̺) (τ, •) dx + τ 0 Ω S(∇ x u) : ∇ x u dx dt ≤ Ω 1 2 ̺ 0,ε |u 0,ε | 2 + 1 ε 2 E(̺ 0,ε , ̺) dx
holds for a.a. τ ∈ (0, T ), where

E(̺, ̺) = H(̺) -H ′ (̺)(̺ -̺) -H(̺), with
H(̺) = ̺ ̺ 1 p(z) z 2 dz.
Note that, by virtue of hypothesis (1.7), the quantity on the righthand side of (2.3) is bounded uniformly for ε → 0.

Existence of global-in-time weak solutions to problem (1.1 -1.6) can be established by the method developed by P.-L. Lions [START_REF] Lions | Mathematical topics in fluid dynamics[END_REF], with the necessary modifications introduced in [START_REF] Feireisl | On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids[END_REF] in order to accommodate a larger class of physically relevant pressure-density state equations, specifically, (2.4)

p ∈ C 1 [0, ∞), p(0) = 0, p ′ (̺) > 0 for ̺ > 0, lim ̺→∞ p ′ (̺) ̺ γ-1 = p ∞ > 0, for a certain γ > 3/2.

Main result.

The main result of the present paper can be stated as follows.

Theorem 2.1. Assume that the pressure p satisfies (2.4).

Let {̺ ε , u ε } ε>0 be a family of weak solutions to problem (1.1 -1.6) in (0, T ) × Ω, where Ω is specified through (1.4), with the initial data satisfying (1.7), where

r 0,ε → r 0 weakly in L 2 (Ω), u 0,ε → U 0 weakly in L 2 (Ω; R 3 ).
Then after taking a subsequence, the following results hold

r ε ≡ ̺ ε -̺ ε → r weakly-(*) in L ∞ (0, T ; L 2 (Ω) + L γ (Ω)), u ε → U weakly in L 2 (0, T ; W 1,2 (Ω; R 3 )), and u ε → U strongly in L 2 loc ((0, T ) × Ω; R 3
), where r and U satisfy (1.8), div x U = 0, and, moreover, the stream function r solves equation (1.10) in the sense of distributions, supplemented with the initial datum

(2.5) r(0, •) = r, where r ∈ W 1,2 (R 2 ) is the unique solution of -∆ h r + 1 p ′ (̺) r = ̺ 1 0 curl h U 0,h dx 3 + 1 0 r 0 dx 3 .
If, in addition, curl h U 0,h ∈ L 2 (Ω), then the solution r of (1.10) is uniquely determined by (2.5) and the convergence holds for the whole sequence of solutions.

The remaining part of the paper is devoted to the proof of Theorem 2.1. The crucial point of the proof is, of course, the strong (a.a. pointwise) convergence of the velocity field that enables us to carry out the limit in the convective term. Here, the desired pointwise convergence will follow from the celebrated RAGE theorem, together with the fact that the wave propagator in the associated acoustic equation commutes with the Fourier transform in both the horizontal variables (x 1 , x 2 ) and the vertical variable x 3 .

Related results.

This work is a contribution to a general research direction consisting in studying singular limits in PDEs arising in fluid mechanics. Without giving an extensive bibliography, one should refer for the first works in this line to Klainerman and Majda [START_REF] Klainerman | Singular limits of quasilinear hyperbolic systems with large parameters, and the incompressible limit of compressible fluids[END_REF] and Ukai [START_REF] Ukai | The incompressible limit and the initial layer of the compressible Euler equation[END_REF] for the incompressible limit (actually [START_REF] Ukai | The incompressible limit and the initial layer of the compressible Euler equation[END_REF] is probably the first work in which dispersive estimates were established in order to prove strong convergence in the whole space), followed by Desjardins et al. [START_REF] Desjardins | Low Mach number limit of compressible flows in the whole space[END_REF] and [START_REF] Desjardins | Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions[END_REF]. In the context of rotating fluids one should mention the important work of Babin, Mahalov and Nicolaenko [START_REF] Babin | Global regularity of 3D rotating Navier-Stokes equations for resonant domains[END_REF], as well as the book [START_REF] Chemin | Basics of Mathematical Geophysics[END_REF] and references therein; one also refers to [START_REF] Gallagher | On the influence of the Earth's rotation on geophysical flows[END_REF] for a survey. Few studies combine both rotation and compressible effects. We refer to Bresch, Desjardins and Gérard-Varet [START_REF] Bresch | Rotating fluids in a cylinder[END_REF] for an analysis in a cylinder, where the well prepared case is studied precisely; the ill prepared case is also addressed but only a conditional result is proved.

Uniform bounds

We start reviewing rather standard uniform bounds that follow directly from the energy inequality (2.3). To this end, it is convenient to introduce a decomposition

h = [h] ess + [h] res , where [h] ess = ψ(̺ ε )h, ψ ∈ C ∞ c (0, ∞), 0 ≤ ψ ≤ 1, ψ ≡ 1 in
a neighborhood of ̺ for any function h defined on (0, T ) × Ω. It is understood that the essential part [h] ess is the crucial quantity that determines the asymptotic behavior of the system while the residual component [h] res "disappears" in the limit ε → 0.

As already pointed out, our choice of the initial data (1.7) guarantees that the right-hand side of energy inequality (2.3) remains bounded for ε → 0. After a straightforward manipulation, we deduce the following estimates:

(3.1) { √ ̺ ε u ε } ε>0 bounded in L ∞ (0, T ; L 2 (Ω; R 3 )), (3.2) {[r ε ] ess } ε>0 bounded in L ∞ (0, T ; L 2 (Ω)), (3.3) ess sup t∈(0,T ) [̺ ε ] res γ L γ (Ω) ≤ ε 2 c, (3.4) ess sup t∈(0,T ) [1] res L 1 (Ω) ≤ ε 2 , and (3.5)
∇ x u ε + ∇ t x u ε - 2 3 div x u ε I ε>0 bounded in L 2 ((0, T ) × Ω; R 3×3 ).
In addition, it is easy to observe that (3.2), (3.3) yield

(3.6) ̺ ε → ̺ in L ∞ (0, T ; L γ + L 2 (Ω)),
which, together with (3.1), (3.6) and the standard Korn inequality, gives rise to

(3.7) {u ε } ε>0 bounded in L 2 (0, T ; W 1,2 (Ω; R 3 )).
In accordance with (3.2), (3.3), we may assume that

(3.8) [r ε ] ess → r weakly-(*) in L ∞ (0, T ; L 2 (Ω)),
and, taking (3.4) into account, (3.9) [r ε ] res → 0 in L ∞ (0, T ; L q (Ω)) for any 1 ≤ q < min{γ, 2}.

Moreover, by virtue of (3.7),

(3.10)

u ε → U weakly in L 2 (0, T ; W 1,2 (Ω; R 3 )),
passing to suitable subsequences as the case may be.

Letting ε → 0 in the weak formulation of the continuity equation (2.1), with b ≡ 0, we obtain (3.11) div x U = 0 a.a. in (0, T ) × Ω.

Finally, multiplying momentum balance (2.2) by ε, we recover (1.8)

(3.12) ̺ -U 2 U 1 = p ′ (̺)∇ h r, ∂ 3 r = 0,
in particular, r = r(x 1 , x 2 ) is independent of the vertical variable, and

(3.13) U h = U h (x 1 , x 2 ), div h U h = 0,
which, together with (3.11), implies U 3 is independent of x 3 . However, as U satisfies the complete-slip boundary conditions (1.5) on ∂Ω, we may infer that (3.14) U 3 ≡ 0.

Propagation of acoustic waves

Assume from now on, to simplify notation, that p ′ (̺) = 1. System (2.1), (2.2) can be written in the form (4.1)

ε∂ t r ε + div x V ε = 0, (4.2) ε∂ t V ε + (g × V ε + ∇ x r ε ) = εf ε ,
where we have set

r ε = ̺ ε -̺ ε , V ε = ̺ ε u ε , and
f ε = div x S(∇ x u ε )-div x (̺ ε u ε ⊗u ε )- 1 ε 2 ∇ x p(̺ ε )-p ′ (̺)(̺ ε -̺)-p(̺)
. More precisely, system (4.1), (4.2) should be understood in the weak sense:

(4.3) T 0 Ω εr ε ∂ t ϕ + V ε • ∇ x ϕ dx dt = -ε Ω r 0,ε ϕ(0, •) dx for any ϕ ∈ C ∞ c ([0, T ) × Ω), (4.4) T 0 Ω εV ε •∂ t ϕ-(g ×V ε )•ϕ+r ε div x ϕ dx dt = -ε T 0 < f ε , ϕ > dt -ε Ω ̺ 0,ε u 0,ε • ϕ(0, •) dx, for any test function ϕ ∈ C ∞ c ([0, T ) × Ω; R 3 ), ϕ • n| ∂Ω = 0, where -< f ε , ϕ >= Ω S(∇ x u ε ) : ∇ x ϕ -(̺ ε u ε ⊗ u ε ) : ∇ x ϕ - 1 ε 2 p(̺ ε ) -p ′ (̺)(̺ ε -̺) -p(̺) div x ϕ dx. It follows from the uniform bounds established in (3.1 -3.7) that (4.5) < f ε , ϕ >= Ω F 1 ε : ∇ x ϕ + F 2 ε : ∇ x ϕ dx, with (4.6) {F 1 ε } ε>0 bounded in L ∞ (0, T ; L 1 (Ω; R 3×3 )), (4.7) {F 2 ε } ε>0 bounded in L 2 (0, T ; L 2 (Ω; R 3×3 )). 4.1.
Point spectrum of the acoustic propagator. Consider an operator B defined, formally, in

L 2 (Ω) × L 2 (Ω; R 3 ), B r V ≡ div x V g × V + ∇ x r .
As a matter of fact, it is more convenient to work in the frequency space, meaning, we associate to a function v its Fourier transform ṽ ṽ = ṽ(ξ h , k),

ξ h ≡ (ξ 1 , ξ 2 ) ∈ R 2 , k ∈ Z, where ṽ(ξ h , k) = 1 0 R 2 exp -i(ξ h • x h ) v(x h , x 3 ) dx h exp(-ikx 3 ) dx 3 .
We investigate the point spectrum of B, meaning, we look for solutions of the eigenvalue problem

(4.8) div x V = λr, ∇ x r + g × V = λV,
or, in the Fourier variables, i

2 j=1 ξ j Ṽj + k Ṽ3 -λr = 0, i[ξ 1 , ξ 2 , k]r -[ Ṽ2 , -Ṽ1 , 0] -λ Ṽ = 0.
After a bit tedious but straightforward manipulation, we obtain (4.9)

λ 2 = -µ, µ = 1 + |ξ| 2 + k 2 ± (1 + |ξ| 2 + k 2 ) 2 -4k 2 2 ;
whence the only eigenvalue is λ = 0, for which k = 0, and consequently, the space of eigenvectors coincides with the null-space of B,

(4.10) Ker(B) = [r, V] r = r(x 1 , x 2 ), V = [V 1 (x 1 , x 2 ), V 2 (x 1 , x 2 ), V 3 (x 1 , x 2 )], div h V h = 0, ∇ h r = [V 2 , -V 1 ] . 4
H = H c ⊕ cl H span{w ∈ H | w an eigenvector of A} . Then 1 τ τ 0 exp(-itA)CP c exp(itA) dt L(H) → 0 for τ → ∞.
In addition to the hypotheses of Theorem 4.1, suppose that C is non-negative and self-adjoint in H. Thus we may write

1 T T 0 exp -i t ε A C exp i t ε A P c X, Y H dt ≤ h(ε) X H Y H ,
where h(ε) → 0 as ε → 0. Taking Y = P c X we deduce

(4.11) 1 T T 0 √ C exp i t ε A P c X 2 H dt ≤ h(ε) X 2 H .
Similarly, for X ∈ L 2 (0, T ; H), we have

1 T 2 √ CP c t 0 exp i t -s ε A X(s) ds 2 L 2 (0,T ;H) (4.12) ≤ 1 T T 0 T 0 √ C exp i t -s ε A P c X(s) 2 H dt ds ≤ h(ε) T 0 exp -i s ε A X(s) 2 H ds = h(ε) T 0 X(s) 2 ds.
4.3. Application of RAGE theorem. For a fixed M > 0, we introduce a Hilbert space

H = H M ≡ {[r, V] | r(ξ h , k) = 0, Ṽ(ξ h , k) = 0 whenever |ξ h |+|k| > M}. Let P M : L 2 (Ω) × L 2 (Ω; R 3 ) → H M denote the associated orthogonal projection onto H M .
Our goal is to apply RAGE theorem to the operators

A = iB, C[v] = P M [χv], χ ∈ C ∞ c (Ω), 0 ≤ χ ≤ 1, considered on the Hilbert space H M .
Going back to system (4.3), (4.4), we obtain that

(4.13) ε d dt r ε,M V ε,M + B r ε,M V ε,M = ε 0 f ε,M , where [r ε,M , V ε,M ] = P M [r ε , V ε ], and 0 f ε,M ∈ H * M ≈ H M , 0 f ε,M , s w H M = - Ω F 1 ε : ∇ x w + F 2 ε : ∇ x w dx whenever (s, w) ∈ H M . Since w W m,∞ ∩W m,2 (Ω;R 3 ) ≤ c(m) w W m+2,2 (Ω;R 3 ) ≤ cM m+2 w L 2 (Ω;R 3 ) ,
we may use the uniform bounds (4.6), (4.7) in order to conclude that

0 f ε,M L 2 (0,T ;H M ) ≤ c(M)
uniformly for ε → 0.

Writing solutions to (4.13) by means of Duhamel's formula we get (4.14)

r ε,M V ε,M = exp(iA t ε ) r ε,M (0) V ε,M (0) + t 0 exp i t -s ε A 0 f ε,M ds;
whence a direct application of (4.11), (4.12), recalling that the only point spectrum is reduced to 0, yields (4.15)

Q ⊥ r ε,M V ε,M → 0 in L 2 ((0, T ) × K; R 4 )) as ε → 0,
for any compact K ⊂ Ω and any fixed M, where we have denoted

Q : L 2 (Ω) × L 2 (Ω; R 3 ) → Ker(B)
the orthogonal projection onto the null space of B. Indeed observe that

√ CQ ⊥ r ε,M V ε,M 2
H M = CQ ⊥ r ε,M V ε,M , Q ⊥ r ε,M V ε,M H M = Ω χ Q ⊥ r ε,M V ε,M 2 dx,
where we have used the fact that P M and Q commute. Finally, a direct inspection of (4.14) yields

(4.16) Q r ε,M V ε,M → r M ̺U M in L 2 ((0, T ) × K; R 4 )) as ε → 0,
where r and U are the asymptotic limits identified through (3.8 -3.12).

4.4.

Strong convergence of the velocity fields. Relations (4.15), (4.16), together with (3.7 -3.10), may be used to obtain the desired conclusion (4.17)

u ε → U in L 2 ((0, T ) × K; R 3 ) for any compact K ⊂ Ω.
Indeed, by virtue of (3.8), (3.9), (4.15), (4.16), we obtain

P M [u ε ] → P M [U] in L 2 ((0, T ) × K; R 3 )
for any fixed M, which, together with (3.10) and compactness of the embedding W 1,2 (K) ֒→ L 2 (K), yields (4.17).

5. The limit system 5.1. Identifying the limit system. With the convergence established in (3.8 -3.10), and (4.17), it is not difficult to pass to the limit in the weak formulation (2.1), (2.2). To this end, we take

ϕ ≡ [∇ ⊥ h ψ, 0], ψ ∈ C ∞ c ([0, T ) × Ω) as a test function in momentum equation (2.2) to obtain (5.1) T 0 Ω ̺ ε u ε • ∂ t ϕ + ̺ ε u ε ⊗ u ε : ∇ x ϕ - 1 ε ̺ ε [u ε ] h • ∇ x ψ dx dt = - Ω ̺ 0,ε u 0,ε • ϕ(0, •) dx + T 0 Ω S(∇ x u ε ) : ∇ x ϕ dx dt.
Moreover, (4.3) yields (5.2)

T 0 Ω r ε ∂ t ψ + 1 ε ̺ ε [u ε ] h • ∇ x ψ dx = - Ω r 0,ε ψ(0, •) dx. Ý
Letting ε → 0 in (5.1), (5.2) we may infer that

T 0 Ω ̺U h • ∂ t ∇ ⊥ h ψ + ̺[U h ⊗ U h ] : ∇ x (∇ ⊥ h ψ) + r∂ t ψ dx = - Ω ̺U 0,h • ∇ ⊥ h ψ(0, •) + r 0 ψ(0, •) dx + T 0 Ω µ∇ h U h : ∇(∇ ⊥ h
ψ) dx dt. Moreover, as the limit functions are independent of x 3 , we get,

(5.3) T 0 R 2 ̺U h • ∂ t ∇ ⊥ h ψ + ̺[U h ⊗ U h ] : ∇ h (∇ ⊥ h ψ) + r∂ t ψ dx h dt = - R 2 ̺ 1 0 U 0,h dx 3 • ∇ ⊥ h ψ(0, •) + 1 0 r 0 dx 3 ψ(0, •) dx h + T 0 R 2 µ∇ h U h : ∇ h (∇ ⊥ h ψ) dx h dt for all ψ ∈ C ∞ c ([0, T ) × R 2 )
. Finally, by virtue of (1.8), U h = ∇ ⊥ h r, and (5.3) coincides with a weak formulation of (1.10), (2.5). We have completed the proof of the convergence result, up to a subsequence, of Theorem 2.1. 5.2. Uniqueness for the limit system. In this final section we shall prove that the limit system has a unique solution provided the initial data are more regular. In order to do so we shall simply write an energy-type estimate on the difference of two solutions, called r 1 and r 2 , associated with two initial data r1 and r2 . This will provide a stability estimate, whose immediate consequence will be a uniqueness result. Notice that the diagnostic equation (1.8) implies that r should be taken in W 1,2 (R 2 ).

The limit system writes

∂ t (∆ h r -r) + ∇ ⊥ h r • ∇ h (∆ h r) = µ ̺ ∆ 2 h r
recalling that for simplicity we have chosen p ′ (̺) = 1. Multiplying (formally) this equation by ∆ h r and integrating over R 2 yields d dt

∆ h r 2 L 2 + ∇ h r 2 L 2 + µ ̺ ∇ h ∆ h r 2 L 2 = 0,
whence the estimate

∆ h r(t) 2 L 2 + ∇ h r(t) 2 L 2 + 2µ ̺ t 0 ∇ h ∆ h r(t ′ ) 2 L 2 dt ′ = ∆ h r 2 L 2 + ∇ h r 2 L 2 .
Now suppose r 1 and r 2 are two solutions as described above, and define δ := r 1 -r 2 . Then of course δ satisfies

∂ t (∆ h δ -δ) + ∇ ⊥ h δ • ∇ h (∆ h r 2 ) + ∇ ⊥ h r 1 • ∇ h (∆ h δ) = µ ̺ ∆ 2 h δ
with initial data δ 0 = r1 -r2 . Writing a similar energy estimate to the one above yields formally

d dt ∆ h δ 2 L 2 + ∇ h δ 2 L 2 + 2µ ̺ ∇ h ∆ h δ 2 L 2 = - R 2 ∇ ⊥ h δ • ∇ h (∆ h r 1 )∆ h δ dx.
Then we simply write, by Hölder's inequality followed by Gagliardo-Nirenberg's inequality

R 2 ∇ ⊥ h δ • ∇ h (∆ h r 1 )∆ h δ dx ≤ ∇ ⊥ h δ L 4 ∇ h ∆ h r 1 L 2 ∆ h δ L 4 ≤ C ∇ h δ 1 2 L 2 ∆ h δ 1 2 L 2 ∇ h ∆ h r 1 L 2 ∆ h δ 1 2 L 2 ∇ h ∆ h δ 1 2 L 2 . This implies that R 2 ∇ ⊥ h δ • ∇ h (∆ h r 1 )∆ h δ dx ≤ µ ̺ ∇ h ∆ h δ 2 L 2 + ∇ h δ 2 L 2 +C ̺ µ ∆ h δ 2 L 2 ∇ h ∆ h r 1 2 L 2 .
Finally Gronwall's inequality allows to obtain

∆ h δ(t) 2 L 2 + ∇ h δ(t) 2 L 2 + µ ̺ t 0 ∇ h ∆ h δ(t ′ ) 2 L 2 dt ′ ≤ ∆ h δ 0 2 L 2 + ∇ h δ 0 2 L 2 exp C ̺ µ t 0 ∇ h ∆ h r 1 (t ′ ) 2 L 2 dt ′ + Ct .
This allows to conclude to stability, hence uniqueness for the limit system (leaving the usual regularization procedure to make the above arguments rigorous to the reader) provided the initial datum enjoys the extra regularity stated in Theorem 2.1.

  .2. RAGE theorem. Our goal is to show that the component of the field [r ε , V ε ], orthogonal to the null space Ker(B) decays to zero on any compact subset of Ω. To this end, we use the celebrated RAGE theorem in the following form (seeCycon et al. [4, Theorem 5.8]):

	Theorem

4.1. Let H be a Hilbert space, A : D(A) ⊂ H → H a selfadjoint operator, C : H → H a compact operator, and P c the orthogonal projection onto H c , where

EDUARD FEIREISL, ISABELLE GALLAGHER, AND ANTON ÍN NOVOTN Ý

EDUARD FEIREISL, ISABELLE GALLAGHER, AND ANTON ÍN NOVOTN Ý

EDUARD FEIREISL, ISABELLE GALLAGHER, AND ANTON ÍN NOVOTN Ý


Philippe Herbaux
email: philippe.herbaux@univ-lille2.fr

Keywords: mutualisation, semiology, sustainable development, anticipation. TITRE : ANTICIPATION ET TERRITOIRE mutualisation, sémiologie, développement durable, anticipation

In 2002, the author led the regional plan of territorial intelligence Nord-Pas de Calais in

Introduction

If, according to Gaston Berger, prospective is 'an analysis tool of potential futures', its complex use is left to the discretion of scientists.

If territory needs to enlighten its 'possbilities', it also benefits from preserving its patrimonial, economic or nonmaterial capital. The convergence principle does not spare competition from Nation-States which cannot translate into Regions nowadays and, eventually, within soil zones -industrial, touristic routes, theme parks etc. Thus, information and its treatment are at the centre of anticipation forces for risks and outbreaks in a collective posture termed 'territorial intelligence '. From Latin inter (between) and ligere (link with), the word 'intelligence' does not deny its Indo-European etymological root, leg (gather together) which de facto provides us with its meaning. Applied locally, it thus encourages the demand the link has for creating or developping within territorial information mechanisms.

For the past few years, in information transmission processes and in order to get rid of the frame of the useroriented paradigm [START_REF] Polity | L'évolution des paradigmes dans le domaine de la recherche d'information[END_REF], it has become important that the very nature of information is not seen like a data -'what is given' -only, which the user spots in a document or a corpus. Information can also be apprehended like an interpretation process; it then turns into an individual construction within cognitive appropriation, which, in the case of territorial intelligence, gradually shifts into collective construction. Isn't it a translation of another paradigm? actor-oriented paradigm like?

In such a case, how is the process to be analysed? For each individual in the group, one may undertake to understand the nature of the transformation of those signs or of spotted clues and mechanisms which can turn it into a conjecture. The latter, subjected to appropriation and validation by the community, would constitute the pillars of a collective construction.

Another plausible hypothesis is that information mutualisation activity achieved with the project's partners, takes informational mechanisms into account. In this type of paradigm and in order to act, the social actor needs to interact with the group, backed by the use of information systems to that end.

Nevertheless, both hypotheses constitute the two focals of a same landscape. That of understanding individual mechanisms of translating information to forces of the mediological sort participating in their transmissions.

We offer to comfront these propositions with the contents of an explorative work led during the first half of 2008, within two local associations with a national audience. These actions combine the tools for scientific research with collecting data realised by the 'ordinary individual'1 . The first one looks at air surveillance in the regions of France and more particularly inside controversial zones within those regions. On top of the existing technological process, it makes sure a number of inhabitants -called 'noses' -participate in the process of collecting and identifying unpleasant smells within a definite environment.

The second one is about all volunteers -identified on a national collaborative platform -observing plants, trees, birds and insects phenology. To study phenology means to study seasonal rhythms of plant or animal life according to the weather. The results of the collection of such data participate in a large survey on climatic changes led by various groups -Tela botanica2 , Crea3 -and the CNRS and INRA's participation.

methodology

presentation

This work was realised between the second quarter of 2007 and the end of the first half of 2008. Collecting those data required the involvement of individuals and it also benefited from complimentary inputs on websites, blogs and fora dealing with this issue. Two observations made locally spotted the founding elements of those actions and accompanying means. A set of measures taken by the participating members, their geographical origins and their selected favourite topics was enforced. Two interviews with semi-directive dialogues took place with pilots for such actions, on the basis of an interview guide elaborated on the object of the research. A phone survey using the method of questionnaire was also led on nine participants chosen at random.

theorETiCAL AND methodologiCAL ASPECTs

epistemological tie

They are set into a rather classical triangulation 'space -actors -time' in an evolution perspective of useroriented paradigm towards a territorial actor-oriented paradigm.

Through his notion of 'public space', Habermas provides the first term; not in his 1962 postulate, now drained to its foundations 4 , but in an updated declention of this 'public pace' starring the virtual and new implicit cooperations. [START_REF] Wolton | espace public, un concept à retravailler[END_REF] had given a new perspective backing up his articulation with political space and [START_REF] Charaudeau | Les discours 1 politique, les masques du pouvoir[END_REF] had specified its discursive evolutions noticed in its new uses.

In his proposal for dynamic representations, Lévy (1997) presents a second pressure point by means of his collective intelligence protoparadigm now noticed by Noubel (2007) in the migration of collective intelligences towards one collective intelligence. That is to say the conjunction of knowledge distributed within multiple collectivities and in that environment, the ordering of new technopolitical steps for training, apprenticeship, assessment of knowledge and skills. These proposals are to be coupled with the practice of the grouped putting into speech process as described by [START_REF] Wenger | Knowledge Management Takes Community Spirit[END_REF], viz. knowing the actors' capacity to mutualise and capitalise their knowledge.

methodology

Presentation This study was realised between the second quarter of 2007 and the end of the first half of 2008. The data collection benefited from individual interviews as well as from further input that websites, blogs and fora linked with these topics constitute. Two observations managed to single out the founding elements for these actions and the means accompanying them. A set of measures related to the audience involved, of their geographical origin and the chosen favourite themes was implemented. Two interviews with semi directive dialogues were led with these actions pilots on the basis of an interview guide woven on the object of the research. A phone survey using the questionnaire method was realised on nine participants randomly selected.

FROM sign TO decision

Signs and time-space

Signs get particular attention from De Saussure (1922) in the field of linguistics, particularly in societal situations and he says: 'semiology studies the life of signs within social life'. Barthes had illustrated a few dimensions of the sign that constitutes information. As soon as he returned from Japan, his work, L'empire des signes (Empire of Signs, 1970), was about to give a new impulse to semiology and fertilise the previously ploughed fields of research anew.

The sign is only a sign as long as there is an agent, i.e. someone interpreting it and considered an informationbearer, De Saussure says. Signs are sometimes mistaken for the signified according to the structural translation which can be made in linguistics; as it happens, in that field, it constitutes one of the main contributions Peirce's pragmatic semiotics. Most often, the interpretation of signs is personal and varied, except when it is the attribute of a shared culture (religious, masonic, technical, philosophical signs etc.). As an example and in a more prosaic manner, the symbols expressing the recommendations of the highway code are the signs of a culture shared by drivers and road users.

Here, the signifying sign and the matching decision kin are a prerequisite for entry in that community (the highway code examination).

If information is necessary to the decision peculiar to anticipation, one may wonder if the decision-maker must satisfy him-/herself waiting for validated, formalised and public information in order to make his/her decision; in such a case, it will provide him/her with precise information but in a fashion dramatically too late. Therefore, action at 'news', and even sign, level is necessary in order to hope to participate in the elaboration of that information. Time and the construction of information are correlated. If information formally manifests itself, the price to pay is tragic time for maturation for the judicious decision most of the time (see Graf. 1).

Graf. 1: construction of information

Making signs stand out is a constant glance on the insignificant, but what is the insignificant? Insignificant: that which does not mean anything says the dictionary (source: Robert 2001). Senses are not resorted to by any signifier. Yet, the object, be it sign or symbol, may acquire meaning; it then becomes the indication of an individual representation. The insignificant has relative value in regards to the individual charge it is given, says Barthes. In terms of perceived risk, the uninformed inhabitant of the territory turns into a passive observer of the insignificant. A territorial intelligence system can thus make the actor of the local evolve from a function of receptacle to that of recipient. Thus, the interpretation of a sign's possibilities starts as closely as possible to reception. Given the importance of the population concerned, visual angles and individual experiences a reduced sign treatment exists such as shared resolution programmes confided to thousands of inviduals computers throughout the world can be. The territory endows itself with a mechanism of sustainable anticipation thanks to a continuous mutualisation system of the insignificant as information between inhabitants. This is an evolution of the culture of organisations.

From the technocratic concept to the collaborative concept

In its administrative dimension, the territory is most often subjected to a political decision forced to proposed orientations by numerous partners (elected representatives, communities, social agents, media or even groups with various interests). Conjecture -and therefore opinion -construction modalities yet rely on a collection of multiple data from which a few main pressure points can be spotted:

-territorial databases -statstic treatement -GIS (geographic information system) -media production -computerised management system -Etc. All of that information abound or refute hypotheses which provide with one or more conjectures necessary for the decision (see fig. 1).

Fig 1: Emergence of the territorial decision

Most of the time, the latter then establishes itself on the priority conjecture affected by surrounding constraints pressures caused by personalities pressure groups. Can it be said that the decision then made is the best answer to conservation, if not extension, of territorial heritage? Authorisations to build in areas subjected to floods, which used to be given by some elected officials, show the opposite. If the technocratic construction of conjectures is being subjected to further input made up of a continuous collection of signs and indicators, we introduce a crossing made up of the individual input of the 'ordinary individual''s observations. Thus, the hypothesis is shaped by the conjunction of participation between formal information and signs, between established data and indicators (see fig. 2), thus establishing a sort of diachronic 'grammar' of the upcoming event. This cross-over characteristic may constitute further interference, i.e. a piling up of each piece of information but turns into the consultant of all the data offered by the afore-mentioned pressure points being put into perspective. Our statement is illustrated by reporting about two budding achievements attaching itself to develop the notion of anticipation and proactivity, resorting to the 'ordinary individual'. Both these devices have authority to association and to national audience and they combine tools for scientific research with a vast data collection realised by what I would term the 'ordinary individual'.

The former looks at air surveillance and its unpleasant drifis by a group of local volunteers. The second device, a partner of GIEC (Intergovernmental Group on Climate Evolution) is of interest for the observation by all citizens, thanks to a national collaborative platform for plant, tree, bird and insect phenology5 .

observations

Atmo Picardie

In France, Atmo Picardie is a structure in charge of the surveillance of air quality for the Picardie region, encompassing the départements (or counties) of Somme (Amiens), Aisne (Chauny, St Quentin) and Oise (Creil). This structure gives out an indicator of air quality for the towns concerned. Three codes of coulours, ranging from green, for a satisfying air quality, to orange, for mediocre air quality, and red for a bad air quality, present a indicating scale from 1 to 10 broadcasted daily to the public. Apart from ozone, hydrogen dioxide concentration and polluting particles concentration (< 10 micrometres) are particularly sought. The drift in the rates leads the Préfet (or chief of police) to take conservatory measures upon still or mobile sources (reducing traffic, actions upon emission origins, public information etc.).

To this day, despite many existing captors, there is no measuring device capable of following certain smelly molecules in the environment, considered unpleasant by residents of the emission zones (e.g. industrial zones).

Please notice that the notion of toxic risk is limited since polluting and dangerous products forbidden by law are already spotted by electronic captors; nevertheless, the permanence of undesirable smells may be considered a threat to indivudal well-being. Following Atmo Picardie's initiative, a network of volunteer captor citizens, whom journalists nicknamed 'northern noses', set itself up as an olfactive watchdog system. According to our data collection, the presence of a few company officials among the fifty or so registered volunteers is evidence of the interest they feel as citizens and their care to act in an anticipative wasy within their framework.

During the last quarter of 2007 and the first of 2008, a biweekly training aimed at learning and spotting fifty or so isolated smells and their multiple combinations. The typology of this 'field of odours'® is established in a way specific to the environment in order to watch and to potential pollution sources. It is implemented under a referent laboratory control6 . Thus, five poles of odours have been defined in a virtual space limited by five families of molecules (amines 7 , pinene, citral8 , pyrazine and DMDS9 ) As soon as they start training, each volunteer takes daily data collection in specific field of odours within n urban or suburban zone. This does not prevent from occasionally multiply notes in moves outside the collection zones. The results expected through that watch by 'the ordinary individual' should allow to establish a cartography of polluting emissions and their evolution over time, thanks to a GIS10 . The transmission of information usually happens under several modes: meetings, text messages, fax and phone. An interactive platform, accessible on the net and over the phone (voice mail) is currently being studied. In this example, the 'data collection' phase only is sollicited. Information being put into perspective depends on the institution which has the last say on a return to volunteers' conclusions.

Access to maps and pollution evolutions make up a sort of virtual appointment of individual investment, according to some participants. Nevertheless, it can be observed that so far this device limits the notion of coaction (collection, mutualisation, treatment, transmission). Involve citizens in data interpretation constitutes a second step according to the action pilot. Yet, the difficulty of the device seems to lie in the durability of the volunteers' action. The construction of a mutualised management of knowledge seems slowed down by the 'turn over' noticed by participants (30 % over 6 months); the main reasons evoked for these abandonments are the lack of availability related to the training follow-up mainly 11 ; the difficulty of the training is put forward as the second reason. Moreover, that community of practices usually dislikes being subjected into a system of formal functioning, excessively organised in favour of a liberty process otherwise of lesser constraint. The observation of the construction communities of knowledge by Wenger had headed towards a dynamic process of the group, those 'hesitations' he considers necessary and constitute compulsory evolutions of that construction to him. Among others, he reduced these steps down to seven progressive phases [START_REF] Wenger | Knowledge Management Takes Community Spirit[END_REF]. We think that fitting this experiment into a quadrennial programme would allow to progressively set up the ' hard core' of volunteers and to establish a regional network fully using the resources of a TIS (territorial information system). Reactive, proactive and anticipative logics then could fully express themselves according to the following triptych (fig 3) : When asked about their motivations for joining this network, some contributors mainly mention their care for curiosity, alterity and pride to participate in this programme with a globaphobe essence. Attendance is a constant put forward by participants over a quarter of whom also take part in other data-collection networks. Curiosity and attendance themes may be put together with the sought after return on investment issue. It seems that frequent reports given to participants, whose conclusive content is submitted to the IPCC, participate in an effective recognition of their on-site collection. That would consitute some sort of symbolical remuneration for Levy (1997a, p. 154) says that typology of public 'expresses the aspiration to constructing a social link not founded on institutional or power relationships but in association centred round common interests'. Beyond this collectively completed data collection, we observe a cohabitation and crossing of various expertises:

5.2.
-individual and collective, -beginner and asserted, -profane and scientific It is not a question of mistaking genres but, postulating, rather of the aptitude to consider the other, 'the ordinary individual', like a potential actor in a local or extended process. In the case of the Seasonal Observatory and whereas the territorial field of observation is pushed to national limits, we may state that this mutualisation of information participate in a process of territorial intelligence. Lévy (ibid) says there is no 'transcending knowledge tank' and knowledge is nothing more than what people know. Among the few volunteers who were asked, a great interest for knowledge capitalisation offered by the Seasonal Observatory programme is observed. It appears that this possibility to access knowledge offers actors a baggage or some sort of 'seed' apt to favour new fields of curiosity 14 and in the present case, more particularly in the sustainable development (observation of the transformation of organic waste, of selected waste habits, of supposed and verified performances of the habitat declared in 'high environmental quality'). In this collective operation, together with Noubel (ibid), we notice that there is a migration of collective intelligences towards a collective intelligence; not in the fusion of individual intelligences but in the recognition of singularities a few attributes of which can be distinguished within the Seasonal Obervatory operation a) autonomy of individuals, creators of meaning b) emancipation of a subjective space freed from state and economic constraints c) decentralisation of knowledge and powers d) aggregation of small exchange units

The logic of sign mutualisation which presides over the functioning of the Seasonal Observatory system exists in other countries too: 'blossom' operation in New Brunswick, 'apfelblütenland' (apple tree blossom) operation in Germany and more recently in 2008 in Sweden, 'Svenska fenologinätverket' operation (Swedish phenological research). Yet, this rationalisation of information mutualisation usually is correletad to internet usage; but the equipment rate in France and in Europe, still fluctuating still, puts even more limits to a balance representativity. The extension of public equipments and emerging years and generations will gradually erase this technological entry pass to turn sign mutualisation into one of the tools of territorial anticipation.

Conclusion

Beyond technical experiments led by experts and specialists, the territory develops new approaches of knowledge management. Among other things, they are based on the multiplication of information collection and its display to 'ordinary individual'. If the programme's frame and protocols naturally belong to the scientific domain, mixing scientists and territory's inhabitants to data interpretation is a way to develop. Participating in mutualised data collection makes inhabitants go from a passive state to that of judge voter, actor of local governance. Citizens ought to be further associated to collected signs being put into perspective and thus give them back their part in the 'local orchestra'. This collaborative concept, a tool for governance, lifts citizens Von Reibnitz says from an ignoring certainty to a thought uncertainty.

Fig 2 :

 2 Fig 2: Collaborative emergence of the territorial decision

Fig

  Fig 3: Action states facing rupture5.3. Seasonal observatoryThe field of environment is not left out. The CNRS (National Centre for Scientific Research), the INRA (National Institute for Agronomical Research) and a few associative structures implemented the ODS 12 (Seasonal Observatory). This operation occurs throughout the year and offers anyone in France to take part in a scientific research programme. These works are based on the alarm climatologists pulled about ten years ago concerning climate change on global scale. The analyses produced contribute to feed the IPCC (intergovernmental panel on climate change)13 . Each volunteer's contribution may be slightly time-consuming and forces him/her to a regular and precise committment. By joining the electronic platform, the citizen commits to observe and report on-line all of these observations of the evolution of wildlife and flora over the seasons. The method is ancient; as early as the mid-19 th century, a sort of phenological observation norm in each county, piloted by what was then called the Ministry of Public Instruction. In the present instance of the Seasonal Observatory, a simple protocol established by scientists allows everyone to collect an ensemble of signs which will be analysed and put into perspective by laboratories. From North to South and East to West, the quest for evolution modifying signs is done by hundreds of volunteers in real time (fig.4). The observatory then exchange with different researchers who can then refine the field of research.

  Through our surveys, it appears that this form of cyberculture catches an audience yearning to get out of the conventional work group frame.

	provided committments.
		Event
	Anticipativ	Proactif	Reactive
	e state	state	state
	Mutualisation	Collection	'Fireman'
	'ordinary	Experts	effect
	individual' +	+
	experts	technology
		.

12 ODS: http://www.obs-saisons.fr as of 05/06/08 13 GIEC in French (Groupe Inter-gouvernemental d'Experts sur le Changement climatique): http://www.ipcc.ch/

The phrase 'ordinary individual' means any individual randomly chosen from within a population.

A network of Francophone botanists grouped together within the Tela Botanica organisation. www.tela-botanica.org

Centre de recherche sur les écosystèmes d'altitude (Centre for Research on altitude ecosystems, non-profit organisation) http://www.crea.hautesavoie.net

i.e. in the 'tidal bore place' of bourgeois expression meeting civil society expression

The study of phenology is the study seasonal rhythms in the life of a plant or an animal according to climate.

IAP SENTIC research department in Evreux, specialised in olfactive expertise (environment, food industry, perfumery etc.)

among which the rotten egg odour hydrogen sulfide

among which limonen with a citrus smell (citral and pinene family)

Dimethyl sulphide with a carrion smell (fertiliser).

GIS: geographic information system which can be seen by cartography most of the time multilayered.

Each participant lives less than 10 minutes away from the training centre; this constitutes one of the selection criteria for selecting volunteers.