Pierre Saulgeot
email: pierre.saulgeot@onera.fr

Vincent Brion
email: vincent.brion@onera.fr

Nicolas Bonne
email: nicolas.bonne@onera.fr

Emmanuel Dormy
email: emmanuel.dormy@ens.fr

Laurent Jacquin
email: laurent.jacquin@onera.fr

Effects of atmospheric stratification and jet position on the properties of early aircraft contrails

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I. INTRODUCTION

Civil aircraft contrails are an increasingly troublesome problem due to their potentially high impact on the aviation radiation budget. In supersaturated regions of the atmosphere, they can degenerate from initial line contrails into high-level cirrus clouds and last several hours. A recent study from [START_REF] Lee | The contribution of global aviation to anthropogenic climate forcing for 2000 to 2018[END_REF] suggests that the impact of contrails on climate change could be larger than that of the release of CO 2 by the worldwide aircraft fleet: the radiative forcing caused by contrails could account for more than half of the radiative forcing of civil aviation while CO 2 one third. However these estimates remain very uncertain, with an uncertainty in the range 30 to 170%.

Contrails form under conditions of pressure, temperature and humidity that are often encountered at cruise altitude. For example, at an altitude of 11 000 m, the standard pressure and temperature are 250 hPa and 217 K respectively, and the homogeneous nucleation threshold of water is about RH i ≈ 170%. Aircraft engine exhaust contains additional water vapor compared to ambient air, typically 1.23 kg per kilogram of fuel burnt (called Emission Index, and hereafter denoted EI). This moist air, initially warm at the jet exit, cools down very rapidly as it mixes with cold ambient air due to turbulent diffusion at the jet interface. During this mixing, air can reach conditions of temperature and moisture that are above the saturation curve of liquid or solid water. A simple model of these effects is implemented in the Schmidt-Appleman criterion [START_REF] Appleman | The formation of exhaust condensation trails by jet aircraft[END_REF]. In this criterion, the exhaust flow follows an isobaric dilution from its thermodynamic state (T en , p v,en ) at the exit of the engines to that of the atmosphere (T atm , p v,atm ), where p v denotes vapor pressure. The trajectory of the exhaust gases in the (T, p v ) plane is then a straight line. In order to predict contrail formation and persistence, this line can be compared to the saturation curves of liquid water (for an appearance criterion) and ice (for a persistence criterion [START_REF] Schumann | On conditions for contrail formation from aircraft exhausts[END_REF]). However, the formation of ice in the aircraft wake cannot be described only by this thermodynamic reasoning. The complete picture requires micro-physical phenomena that are not fully understood at the moment, but, for instance, condensation on soot from kerosene combustion seems to be the dominant factor for current jet engines. At present, sustainable aviation fuels (SAF) as well as hydrogen are being investigated to reduce CO 2 emission. These alternative fuels have different EI and will produce a different mixture of soot and other aerosols that act as ice nucleating particles. More importantly, this model does not account for changes in atmospheric conditions around the jet plume induced by its interaction with the descending aircraft wake.

The interaction between the wake and the exhaust plume is generally divided into five phases [START_REF] Jacquin | On the dynamics of engine jets behind a transport aircraft[END_REF]- [START_REF] Paoli | Contrail modeling and simulation[END_REF]: the jet phase, the deflection phase, the vortex phase, the dissipation phase and the diffusion phase. The aircraft wake is dynamic until the dissipation phase, whereas in the diffusion phase, the wake has lost all its dynamics and the plume evolves under atmospheric effects only. During the jet phase, the dynamics of the jets and the wake are almost independent. The vorticity sheet coming from the different wings of the airplane rolls up in a vortex wake [START_REF] Saffman | Vortex dynamics[END_REF] and the engine jets undergo isobaric dilution and strong turbulent diffusion, owing to the velocity difference at the jet periphery (roughly 200 m s -1 initially). Condensation of water vapor occurs during this phase, if any. Their possible sublimation takes place during the following phases [START_REF] Unterstrasser | Properties of young contrails-a parametrisation based on large-eddy simulations[END_REF]. The deflection phase corresponds to the moment when the axial and longitudinal moments of the plume have the same order of magnitude, where the interaction between the driving jets and the wake can no longer be neglected. This phase occurs in a zone between 1 and 10 wingspans behind the aircraft [START_REF] Jacquin | On the dynamics of engine jets behind a transport aircraft[END_REF], depending on the aircraft type and on the lateral position of the engines. Although it has been the subject of experimental studies [START_REF] Brunet | Numerical/experimental simulation of exhaust jet mixing in wake vortex[END_REF], [START_REF] Jacquin | An experiment on jet-wake vortex interaction[END_REF], this phase is often neglected in numerical studies [START_REF] Paoli | Dynamics and mixing in jet/vortex interactions[END_REF]- [START_REF] Paoli | Effects of jet/vortex interaction on contrail formation in supersaturated conditions[END_REF] because of its complexity and high computational cost. The vortex phase begins when the specific dynamics of the jets becomes negligible [START_REF] Brunet | Numerical/experimental simulation of exhaust jet mixing in wake vortex[END_REF], [START_REF] Garnier | Modelling exhaust plume mixing in the near field of an aircraft[END_REF]. During this phase, the engine plumes are dragged by the wake. An order of magnitude of the thresholds between these different phases is obtained by the ratio R 3 introduced in [START_REF] Jacquin | On the dynamics of engine jets behind a transport aircraft[END_REF]. This ratio compares the axial momentum of the jet to the induction effect by the tip vortices.Then, R 3 ≫ 1 during the jet phase (the vortex induction on the jet is negligible compared to the jet dynamics), R 3 (x jet ) ∼ 1 during the deflection phase (the vortex induction and the jet dynamics have the same order of magnitude) and R 3 ≪ 1 during the vortex phase (the vortex induction on the jet is dominant compared to the jet dynamics). Typically x jet ∈ [START_REF] Lee | The contribution of global aviation to anthropogenic climate forcing for 2000 to 2018[END_REF][START_REF] Saffman | Vortex dynamics[END_REF] depending of the jet position. Beyond this approximate distance, the evolution of the jet is mostly dictated by the effect of entrainment by the vortex flow. At such distances behind the aircraft, the vortex wake emanating from the wing has mostly rolled-up into two trailing vortices that are close to an axisymmetric configuration [START_REF] Jacquin | An experiment on jet-wake vortex interaction[END_REF], [START_REF] Khou | Spatial simulation of contrail formation in near-field of commercial aircraft[END_REF]. The vortices are characterized by their radius r 0 and circulation Γ 0 , the latter being proportional to the ratio of the aircraft weight to the initial vortex spacing b 0 . Their natural motion is a descent at constant speed W 0 caused by mutual induction which can be estimated using a point vortex model W 0 ≃ Γ 0 /2πb 0 . The vortex dipole has a lifetime of a few minutes before the onset of three-dimensional instabilities that lead to its destruction -this is the dissipation phase. According to [START_REF] Spalart | Airplane trailing vortices[END_REF] the maximum lifetime of such a vortex pair is about 5 to 6 characteristic time units, defined as τ 0 = b 0 /W 0 . This corresponds typically to a time of 60 to 360 s or a distance between 15 and 90 km behind the aircraft, depending on its size and weight. Instabilities (e.g., Crow instability [START_REF] Crow | Stability theory for a pair of trailing vortices[END_REF]) can occur and dissipate the vortices [START_REF] Jacquin | On the persistence of trailing vortices[END_REF]. High external turbulence levels can accelerate the start of the diffusion phase [START_REF] Crow | Lifespan of trailing vortices in a turbulent atmosphere[END_REF]. The diffusion phase starts when the wake has lost all its momentum and the plume becomes fully governed by the dynamics of the atmosphere.

The contrail to cirrus transition that may occur at the end of this process is dependent on many parameters. [START_REF] Unterstrasser | Aircraft-type dependency of contrail evolution[END_REF] showed a great dependence of the evolution of contrails on the aircraft type, as well as a good correlation in the total extinction between the cirrus phase and the early phases. Moreover, due to the combined effects of atmospheric stratification on buoyancy forces, the vortex phase plays a particular role in the vertical dispersion of contrails which will then influence their radiative impact in the diffusion phase. Therefore the aim of this study is 1) to describe the influence of the position of the jet on the wake dynamics during the vortex phase in a stratified atmosphere and 2) to evaluate the various scenario of optical impact of the early contrails.

In the present setup, the vortex is located at s = π/4 of the wingspan according to the lifting line theory for an elliptically loaded wing [START_REF] Batchelor | An introduction to fluid dynamics[END_REF], which is a good model of a generic wing, owing to its induced drag optimality. Then, it is the jet position along the wing that needs to be changed to explore the various physics. There are three variants on current aircraft: at the center plane of the aircraft or along the fuselage, at the first third of the wing for twin-engined aircraft, and finally for fourengined aircraft, the first jet is again at about one third and the second at about two thirds along the wing. Our analysis explores these different lateral positions, and other positions, including between the middle of the aircraft (at the fuselage) and at the wing tip.

An important factor leading to this result is atmospheric stratification, which strongly modifies the behavior of the wake. The vertical movement of the wake creates buoyant forces upon it that alter its descent and shape. This evolution depends at first order on the product of the Brunt-Väisälä frequency and wake time scale τ 0 . Typical values range between 0.1 (e.g., fully loaded B737 with weak stratification) and 1.8 (e.g., empty A330 with strong stratification). In a highly stratified atmosphere, the downward motion of the wake is overwhelmed by buoyancy, causing the wake to stagnate at flight altitude. Otherwise, the wake has a downward motion whose velocity and maximum height depend on the level of stratification [START_REF] Sarpkaya | Trailing vortices in homogeneous and density-stratified media[END_REF]. Wake descent creates a density gradient at the edge of the Kelvin oval that causes the formation of secondary vorticity by the baroclinity [START_REF] Scorer | Contrails and aircraft downwash[END_REF]. Once created, this secondary vorticity is drawn towards the top of the oval. This movement leads, by Biot-Savart's law, first to a slow down of the descent of the vortices and then to their convergence [START_REF] Shirgaonkar | Interaction of vortex wakes and buoyant jets: A study of two-dimensional dynamics[END_REF]. The vortices can also transiently move up in this phase. The vortices then accelerate downward because of the implied increase of the induction between the two vortices. At the top of the oval, the secondary vorticity receives an upward motion along the symmetry axis of the flow and rises, generally to flight altitude [START_REF] Spalart | On the motion of laminar wing wakes in a stratified fluid[END_REF], due to buoyancy and the induction of each half of the secondary vorticity onto the other. There, it accumulates and forms a secondary wake which considerably increases the vertical extent of the wake.

From the point of view of ice formation in the exhaust plume, its altitude is important as the phenomenon is related to the value of ambient temperature. Two extreme cases of the jet position can be considered, as a first analysis. The two situations correspond to when the jets are fully entrained in the wake and, conversely, when they are not. In the former case, the jets evolve in an external environment made of variable thermodynamic conditions and condensation is affected. If the ice plume is entrained to a lower altitude, one expects ice sublimation because of the higher temperature. This would be beneficial from the radiative impact point of view. In the latter case, the jets remain at flight altitude and external conditions are stable and as cold as they could be. This configuration, which agrees then with the Schmidt-Appleman criterion, appears worst. Some sensitivity studies [START_REF] Unterstrasser | Large-eddy simulation study of contrail microphysics and geometry during the vortex phase and consequences on contrail-tocirrus transition[END_REF], [START_REF] Unterstrasser | Dimension of aircraft exhaust plumes at cruise conditions: effect of wake vortices[END_REF] already report on such an influence of the initial jet distribution on the plume evolution, especially for extreme values. The sensitivity of the jet entrainment process has also been described experimentally by [START_REF] Jacquin | An experiment on jet-wake vortex interaction[END_REF] within a few wing spans downstream of an idealized aircraft configuration.

The outline of the paper is the following: section II describes the model of the flow and ice, section III analyses the influence of the jet positioning and stratification on the flow and the plume evolutions, and section IV describes the optical features of the ice plume and concludes on their dependence on the two parameters.

II. MODEL

A. Flow

The model consists of a two-dimensional pair of counterrotating vortices and two engines exhaust plumes. This represents a cross section of the aircraft wake during the vortex phase. During this phase, the atmosphere and the wake interact via buoyancy effects that oppose the downward momentum of the vortices and modify the temperature distribution. Buoyant forces result from the vertical non-uniformity of a quantity called potential temperature Θ = T (p 0 /p) R /cp , which corrects the thermodynamic temperature for adiabatic evolution. The atmosphere is stratified when dΘ/dz ̸ = 0. In this relation, R is the perfect gas constant of dry air and c p its heat capacity at constant pressure. The Brunt-Väisälä frequency N = g Θ dΘ dz

gives the frequency of the vertical oscillations of air particles when they are disturbed in such an environment (g is the standard gravity). The wake imposes τ 0 as the reference time scale and the vortex separation b 0 = (π/4) b as the reference distance, with b being the wingspan. In this scaled framework, stratification is effectively measured by the inverse Froude number Fr -1 = N τ 0 . For a single aircraft, τ 0 can vary by almost a factor two as a consequence of the weight decrease In a linearly stratified medium, the Navier-Stokes equations simplify into the Boussinesq approximation, in which the flow is considered as a superposition of a fixed atmospheric state and a perturbation state, which here refers to the wake flow. For a detailed derivation of this approximation and further discussion on the assumptions, see [START_REF] Saulgeot | Effects of atmospheric stratification and jet position on the properties of early aircraft contrails[END_REF].

Soot and water vapor concentrations emitted by the engines are integrated into the model by considering the concentration c of a passive scalar and its transport law. The dimensionless governing equations for the perturbation variables are [START_REF] Chandrasekhar | Hydrodynamic and Hydromagnetic Stability[END_REF]

∇ • u = 0, (1a) du dt = -∇p + θe z + 1 Re ∇ 2 u, (1b)
dθ dt = - 1 Fr 2 u • e z + 1 Pr Re ∇ 2 θ, (1c)
dc dt = 1 Sc Re ∇ 2 c. ( 1d
)
where Re = W 0 b 0 /ν, Pr and Sc are the Reynolds, Prandtl and Schmidt numbers of water vapor respectively, with ν = 3.9 × 10 -5 m 2 s -1 the kinematic viscosity. They are set to Pr = 0.7 [START_REF] List | Smithsonian Meteorological Tables 6th Revised Edition[END_REF], Sc = 1.3 [START_REF] Vallis | A simple system for moist convection: the rainy-bénard model[END_REF] and Re = 10 4 . The diffusivity κ of soot depends on the diameter d of the particles: κ = d -1 × 10 -13 m 2 s -1 [START_REF] Flower | Measurements of the diffusion coefficient for soot particles in flames[END_REF]. The typical size of particles from jet engines is d = 10 -8 m [33] leading to Sc soot = ν/κ = 4. However, on account of the high Reynolds number (i.e., diffusion is negligible compared to advection) and to simplify the setup, we take it equal to that of water vapor: Sc soot = Sc. As discussed earlier, the (turbulent) dynamics of the jet is negligible during the vortex phase. Moreover, trailing vortices have a very low amount of turbulence because the rotational flow is stabilizing according to the Bradshaw-Richardson criterion [START_REF] Cambon | Stability analysis and large-eddy simulation of rotating turbulence with organized eddies[END_REF]. Thus, it is the fluid viscosity that dictates the diffusion of the vortices [START_REF] Govindaraju | Flow in a turbulent trailing vortex[END_REF]. Furthermore, in the vortex jet interaction in a stratified atmosphere, the dominant dynamical mechanisms are the jet entrainment, stirring by the vortices and the baroclinic vorticity production, much more than turbulent or viscous diffusion. This justifies the 2D laminar model used in the present study.

The flow initialization follows the symmetric scheme presented in fig. 1. We introduce two Gaussian vortices of radius r 0 = 0.05 b [START_REF] Gerz | Commercial aircraft wake vortices[END_REF]. The initial distribution of passive scalar and temperature follow a tanh distribution at the location of the jet [START_REF] Paoli | Dynamics and mixing in jet/vortex interactions[END_REF]. The turbulence of the first stage of the jets is taken into account in the radius r jet of the initial conditions of the passive scalar, i.e., it is enlarged compared to the engine exit size by the amount caused by turbulent diffusion. We set r jet ≃ 0.15 b while the jet exhaust half diameter is D(x out )/2 ≃ 0.01 b. Such an initial condition is commonly used in contrail analysis [START_REF] Paoli | Dynamics and mixing in jet/vortex interactions[END_REF], [START_REF] Paoli | Effects of jet/vortex interaction on contrail formation in supersaturated conditions[END_REF]. The initial jet lateral position bjet = b jet /b varies in the range 0 (mid plane) to 1 (wingtip). Their vertical position relative to flight level is fixed at -0.081 b [START_REF] Paoli | Effects of jet/vortex interaction on contrail formation in supersaturated conditions[END_REF]. It is further assumed that stratification has not yet affected the flow at the initial time of the present simulations. A quick calculation allows the estimation of the distance behind the aircraft from which the influence of the buoyant forces becomes significant x = U 2 0 /g, which corresponds to a hundred aircraft wingspans for a typical cruising speed U 0 = 250 m s -1 i.e., within the vortex phase.

The reference equations are solved using the Nek5000 spectral element code in a two-dimensional setup with polynomials of order 12. The size of the numerical domain is between 100 and 200 elements in y and between 100 and 400 elements in z, depending on the stratification level. The size of the computational domain is between [-4, 4] and [-6, 6] in y, and [-4, 4] and [-24, 6] in z. The large number of elements is explained by the fine discretization of the mesh that is required at the mid-plane between the vortices, due to the strong fluid strains that occur at the upper hyperbolic point of the Kelvin oval and the possible close proximity of the vortices. The cost of one simulation is about 100 days of CPU time. The parametric study evaluates 22 b jet values between 0 and 1, and 20 Fr -1 values between 0 and 2 amounting to 440 calculations and 120 years of (sequential) CPU time. Using about 3000 processors, the calculations can be done in two weeks.

B. Ice plume

The humidity in the system comes from the natural humidity in the atmosphere which adds to the humidity released by the aircraft engines. One can then write the decomposition of the partial density of vapor ρ v = ρ v,atm + ρ v,jet as the sum of the contributions of the atmosphere and the jets. The present model follows the approach of [START_REF] Paoli | Dynamics and mixing in jet/vortex interactions[END_REF], [START_REF] Paoli | Effects of jet/vortex interaction on contrail formation in supersaturated conditions[END_REF] and makes several assumptions: (i) surface effects (Kelvin effects) are neglected; (ii) soot act as nuclei for water vapor and there are as many ice particles as there are soot particles; and (iii) phase changes are instantaneous and without feedback to the flow and temperature.

In this framework, the condensation of vapor to ice in the jet plume is calculated using an Eulerian description of the distribution of effective radius r e , mass ρ i and density ρ soot of ice crystals.

Assuming that at equilibrium the vapor pressure is equal to the saturation vapor pressure (i.e., the mass of ice, when present, is such that the vapor pressure is equal to the saturation vapor pressure), the local density of ice (total mass of ice particles in a volume of air) is derived from the perfect gas equation

ρ i = p v -p sat,i R v T 1 pv⩾psat,i = (ρ v -ρ sat,i ) 1 ρv⩾ρsat,i (2)
where we denote by analogy p sat,i (T ) = ρ sat,i (T ) R v T . We denote 1 A the indicator function of the set A. As the expected size of the ice crystals is about 10 to 20 nm [START_REF] Unterstrasser | Numerical simulations of contrail-tocirrus transition -part 1: An extensive parametric study[END_REF], we assume that they can be represented approximately by randomly oriented hexagonal-based cylinders whose size can be characterized by an effective radius r e [START_REF] Ebert | A parameterization of ice cloud optical properties for climate models[END_REF]. For a detailed derivation and some comments on the assumptions, see [START_REF] Saulgeot | Effects of atmospheric stratification and jet position on the properties of early aircraft contrails[END_REF].

While being simple, the present physical model offers the main ingredients for the evaluation of the plume evolution and does so at an acceptable computational cost, hence allowing the present parametric study.

III. FLOW EVOLUTION A. Influence of stratification on the vortex wake

This section is dedicated to the evolution of the vortex flow and effect of the Brunt-Väisälä frequency. Similar simulations were carried out by [START_REF] Scorer | Contrails and aircraft downwash[END_REF]- [START_REF] Spalart | On the motion of laminar wing wakes in a stratified fluid[END_REF], [START_REF] Widnall | The structure and dynamics of vortex filaments[END_REF]. One difficulty is to run the simulation over a long time (t = 8.5 τ 0 ) when the Froude number is close to 1. Indeed in these cases the flow has small spatial scales that require a particularly fine mesh. We provide novel results for these conditions.

The evolution of the wake altitude for various stratification levels is shown in fig. 2 using the vorticity extrema as the vortex locator. Note that there were no hot jets in these simulations. When there is no stratification, the vortex pair descends at constant velocity W 0 , the distance between the vortices varies marginally and their circulation decreases only by diffusion, which is weak. When the stratification is small (Fr -1 ⩽ 0.9), the descent of the wake slows down at around t = 1, then accelerates sharply at around t = 4. For intermediate levels (0.9 ⩽ Fr -1 ⩽ 1.2), the descent of the Fr -1 = 0.4

Fr -1 = 0.6

Fr -1 = 0.9

Fr -1 = 1.18

Fr -1 = 1.2

Fr -1 = 2 z = -t wake stops at t = 2, the wake then rises again somewhat, and then finally descends again at a much higher speed. The comparison between the descent velocity and the distance between the vortices allows one to notice that the increase of the descent speed is synchronous with the increased proximity of the vortices. This can be explained by the fact that the mutual induction of the two vortices is inversely proportional to their separation. Above the threshold value Fr -1 = 1.2, the wake stays at flight altitude. When the atmosphere is stratified, the vorticity distribution separates into two parts: the two patches of vorticity associated with the initial vortices (that is called primary vorticity) and the vorticity generated by buoyancy (that is called secondary vorticity), of opposite sign, which is created at the boundary of the Kelvin oval of the primary vortex pair. The secondary vorticity does not circulate around the vortices but is diverted upwards and rises along the axis of symmetry (fig. 3 (a)). The source term ∂θ ∂y in this equation results from the baroclinic term 3 (c). This can be understood by writing the Helmholtz equation for the axial vorticity d t ω = ∂ y θ + Re -1 ∇ 2 ω. This appearance coincides with the inflection of the descent of the primary vortices. Indeed, the action induced by the secondary vorticity on the primary vortices is a vertical velocity directed upwards. Furthermore, as noted by [START_REF] Spalart | On the motion of laminar wing wakes in a stratified fluid[END_REF], secondary vorticity forms along the streamline that bounds the Kelvin oval, and is therefore stronger above the vortices than below, leading to the contraction of the vortex pair. In fig. 3 (a), one can see that the primary vortices have significantly moved closer together and that the secondary vorticity has escaped from the oval along the axis of symmetry. It then forms a so-called secondary wake at the flight altitude where the width of its horizontal distribution increases.

B. Influence of the hot plume on the vortex wake

The presence of a hot jet can increase the influence of stratification on the motion of the primary vortices [START_REF] Shirgaonkar | Interaction of vortex wakes and buoyant jets: A study of two-dimensional dynamics[END_REF], in particular by increasing their approach and descent speed. Fig. 4 shows the influence of the jet position on the altitude of the primary vortices at t = 5 τ 0 . The presence of a hot jet strengthen the descent of the vortices, but this effect depends on its position. When it is at the level of the plane of symmetry of the aircraft ( bjet = 0), it does not affect the primary vortices. When it is closer to the core of the vortices (0.4 ⩽ bjet ⩽ 1), it has two effects. First, when Fr -1 is close to 1, the vortices descend lower. Moreover, the threshold between the intermediate (slight rise of the vortices and then faster descent) and high (no descent of the vortices) values of stratification is at higher Fr -1 . Between the two regimes (0 ⩽ bjet ⩽ 0.4), the altitude to which the wake descends increases with bjet as does the threshold between the intermediate and high values of Fr -1 . One can notice that the value of the threshold admits a maximum around bjet = 0.33 before decreasing slightly.

C. Jet exhaust evolution

The evolution of the engine plume is governed by the initial rolling around the primary vortices. Fig. 5 shows the time sequence of the field of passive scalar c, which models the plume, for several values of Fr -1 and b jet . The spiral of the winding can be seen in the first column of these figures. Stratification has no effect at this stage since the descent is of low amplitude (fig. 2). The plume then forms a ring around each vortex (second column). As long as the Kelvin oval is large enough to contain the plume, it descends along with the vortices. This is the case for weak stratification levels (first row) or when the plume is initially very close to the vortices (last row). For intermediate levels, most or all of the jet plume escapes from the oval, becomes diverted across the top of the oval boundary and is then driven upwards with the secondary vorticity. This brings part of the plume back to flight altitude (middle row and last row for t = 3.5 τ 0 ). For high levels of stratification, the vortex pair does not descend and the plume maintains its ring shape around the vortices (as for the weak stratification case but without descending).

The oval thus separates two regions of the plane where the flow, and therefore the plume, behave very differently. Everything inside descends with the main vortices, while everything outside extends vertically and rises towards flight altitude with the secondary wake. This behavior depends on stratification and initial jet position, which can be assessed using the mean plume altitude, defined as the average of the altitudes weighted by the amount of passive scalar. The result is shown in fig. 6 (a). Three distinct regions can be identified. When Fr -1 ⩾ 1.3 the plume is always located at flight altitude. When Fr -1 ⩽ 0.5 and bjet ⩾ 0.4 the plume descends by a distance that mostly depends on the Froude number. The lowest altitude is reached for Fr -1 = 0.7 and bjet = 0.8. In the intermediate regime, there is a critical threshold Fr -1 ≃ α ( t -5) bjet + β ( t -5), with α( t) = 0.75 -0.07 t and β( t) = 0.25 -0.07 t, above which the plume exceeds flight altitude and below which it descends by a distance that depends on both parameters. These values are obtained by calculating the same quantities for different time steps and then interpolating linearly. Fig. 6 (b) shows these regimes using the difference between the average altitude of the plume and the altitude of the primary wake. In the first and second cases, the plume and the vortices are at the same altitude. In the intermediate regime, all or part of the plume is driven by the secondary vorticity away from the main wake. In summary stratification and jet to vortex distance are determining factors on the evolution of the altitude and the distribution of the plume. The outcome of the optical properties of the ice plume is explored in the final section.

We have placed in fig. 6 (b) the possible parameter ranges for different twin-engined aircraft. We notice that most of them cross the aforementioned boundary between the intermediate and high levels of stratification, and are mostly in the upper part. It is then understood that the plume is no longer located in the vortices at the end of the vortex phase in many cases. Since the observation of early contrails is the only way to see the flow visually from the ground, this explains why it is so rare to observe Crow instability that affect the primary vortices behind airliners.

D. Comparison between the 2D and 3D dynamics

The two-dimensional constraint of the present simulations precludes the capture of the real turbulent dynamics of the secondary wake. Instead, the secondary wake sustains an inverse cascade that favors the formation of larger vortices in the last stages of the flow evolution displayed in the results. The small structures induced by the rising secondary wake and baroclinic torque in reality should induce smaller scale three-dimensional movements. Nevertheless the dynamics of the primary wake -and thus the vertical motion of the vortex dipole -and the entrainment of the plume by one or the other wake -and thus the evolution of its altitude -are intrinsically two-dimensional mechanisms.

IV. OPTICAL PROPERTIES

A. Optical thickness and total extinction

Calculations of the distributions of ice mass per volume ρ i and crystal effective radius r e provide an estimate of the energy absorbed by an ice cloud and thus of its potential effect on the radiative balance. For this we are interested in the relative variation in intensity across the entire transverse extent of the cloud, called total extinction and defined by E = (I 0 -I)/I 0 dy. In this relation, I and I 0 are the transmitted and incident intensities of a vertical ray of light passing through the medium respectively. Their ratio is called transmittance T = I/I 0 and can be calculated from the optical thickness τ = -ln T , which is the integral over . The values of a and b are independent upon wavelength for wavelengths between 250 and 3 500 nm [START_REF] Ebert | A parameterization of ice cloud optical properties for climate models[END_REF]. Therefore the total extinction yields

E = 1 -exp -ρ i a + b r e dz dy. (3)

B. Analysis

The influence of stratification and vortex to jet distance on plume total extinction for RH i = 140% at t = 5 is shown in fig. 7. The possible parameter regions for several aircraft are also indicated. The total extinction calculation requires the transform of the variables into physical units. The reference values used for this are b 0 = 50.9 m (B777) and τ 0 = 27.6 s (see table I).

We find that the total extinction remains low when Fr -1 is small. This corresponds to two possible and distinct situations. The first is when the plume is initially close enough to the vortex. The required proximity to the vortex is all the more strict as stratification is stronger. In this case, the plume descends and this leads to a weak and very local formation of ice crystals. As a result, the optical thickness has a reduced support and reaches small values, which leads to a low total extinction. This regime is illustrated by the cases Fr -1 = 0.2 and bjet set to 0.31 and 0.79 in fig. 8 that displays the spatial distribution of the attenuation coefficient for three stratification rates and three jet-to-vortex spacings.

The second situation is when the distance between the jets and the vortices is large (e.g., for Fr -1 = 0.2 and bjet = 0.16). .

In this case a part of the plume stretches vertically along the symmetry axis above the primary vortices. However the extent of the plume is reduced in the horizontal direction and thus the total extinction remains low. Conversely, total extinction is high when stratification is strong and the engine is initially not too close to the vortices. This region actually breaks into two sub-regimes, depending on the flow dynamics (fig. 6). In the first sub-region a primary and a secondary wake coexist, e.g., Fr -1 = 1 and bjet equal to 0.16 or 0.31. As the plume is not close to the vortex core it rises towards the secondary wake (right panel of fig. 8). This results in a large optical thickness both at the symmetry axis where ice mass is important, and more widely in the horizontal extent of the secondary wake where the low temperature allows crystals to persist. In the second, e.g., Fr -1 = 1.4 and bjet = 0.16 or 0.31, the entire wake is present at flight altitude, but the fairly large distance between the plume and the center of the vortices allows it to extend horizontally, resulting in a widely distributed attenuation coefficient. So does the optical thickness, leading to a high total extinction. Finally, when the stratification is very strong and the plume lies initially close to the vortices, e.g., Fr -1 = 1.4 and bjet = 0.79, the ice remains inside the vortices, as in the weakly stratified case, but this time at flight altitude. Due to lower temperatures at this higher altitude, optical thickness and thus total extinction are slightly higher than in the weakly stratified case.

V. CONCLUSION

A bi-dimensional multi-parametric study on the effects of stratification and engine jet position along the wingspan is carried out to evaluate the potential radiative impact of early contrails during the vortex phase. This phase has the particularity of exhibiting a purely two-dimensional dynamic. It has a particular role in the vertical dispersion of contrails due to the combined effects of stratification and jet-vortex interaction. A model is implemented in a normalized framework to account for the general behavior of the wake flow containing the contrails under these effects. The positions of the jets investigated range from the plane of symmetry of the aircraft to the wing tip. All possible values of stable stratification at flight altitudes are analyzed. Condensation is taken into account to assess the spatial distribution of ice mass and crystal radius and to describe the optical properties of the simulated contrails. This serves as a gauge to examine the radiative impact of the possibly formed cloud as a function of the initial wake and atmospheric properties. It must be denoted that the normalization approach taken here makes the present study a large multidimensional analysis that encompasses the aerodynamic properties of the aircraft and the stratification of the atmosphere. The atmospheric stratification in itself is not enough as the behavior depends on the effective stratification which also accounts for the wake and, indirectly, the aircraft. A return to the physical units is performed for a selection of aircraft to provide real life diagnostics. Important results are found regarding the effects of stratification and jet position along the wing. Three main behaviors are observed. For low levels of stratification, the ice plume descends with the wake vortices, generating a small optical impact. This impact is smaller the closer the jets are to the wingtip vortices. For high levels of stratification and small jet spacing, the ice plume tends to remain at the flight altitude and to expand horizontally, generating a higher optical impact. The boundary between these two regimes is quite clear and lies at a threshold that depends on time, for instance Fr -1 = 0.75 bjet + 0.45 for t = 5 τ 0 . It results from the wake dynamics and is also observable examining the altitude of the plume. In a nutshell, the plume is found at flight altitude (in the secondary wake) for high levels of stratification and small jet spacing, and at a lower altitude (in the primary wake) for lower levels of stratification. At the threshold, the plume is present in both the primary and secondary wake, as well as near the symmetry axis. This generates a larger optical impact than when it remains in the primary wake, but less than when it rises in the secondary wake. Finally, for high values of stratification and jet spacing, the plume remains concentrated in the vortices at flight altitude resulting in low optical impact. A comparison with the possible parameter ranges for current twin-engined aircraft shows that in most cases the ice plume is no longer present in the vortices at the end of the vortex phase. This dynamic is totally realistic, and in particular is not the consequence of the two-dimensional approximation, even if the description of the evolution of the secondary wake at flight altitude is distorted by the turbulence that develops there. A comparison with a 3D simulation confirms that the dynamics of the primary wake -and thus the vertical motion of the vortex dipole -and the entrainment of the plume by one or the other wake -and thus the evolution of its altitude -are intrinsically two-dimensional mechanisms. Importantly, these results indicate the large variability of the contrail appearance depending on the atmospheric conditions but also on the specific aircraft configuration (wing/jet configuration and weight). In general, contrails do not lie with the primary vortices but with the secondary wake, between the flight altitude and the altitude of the vortices, or even at flight altitude. For an observer present at the ground or for automatic observation of contrails (by means of satellites or ground cameras), this sensitivity of the contrail properties must be taken into account to well correlate the relations between contrail, aircraft and atmosphere. Overall this shows that the near wake dynamics has a profound influence on the contrail issue. The results obtained are essentially dependent on the dynamics of the wake and in this sense are applicable to a large number of situations, in particular to different atmospheric or propulsion situations. For current aircraft, the total extinction of contrails, which is a measure of their optical impact, can be low or high. Among other things, these variations are dependent on the weight of the aircraft and stratification of the atmosphere, parameters that cannot be modified. Therefore, the effect of jet spacing is of particular interest as it could be used as a contrail mitigation strategy. From a purely aerodynamic point of view, placing the jet close to the location of the rolled-up vortex ( bjet ≃ 0.8) is on average beneficial to reduce total extinction and is not far off from the most outboard engines of current four-engined aircraft.

Fig. 1 .

 1 Fig. 1. Setup of the initial flow.

Fig. 2 .

 2 Fig. 2. Altitude of the vortices as a function of time for different values of Fr -1 . Distances are normalized by b 0 and time by τ 0 .

1 ρ 2

 12 ∇ρ × ∇p derived from the Boussinesq approximation. The temperature differential at the boundary of the Kelvin oval results from the combined effects of stratification and wake descent. It creates a horizontal potential temperature gradient, as shown in fig. 3 (b). Secondary vorticity of opposite sign is thus generated at the boundary of the Kelvin oval, as shown in fig.

Fig. 3 .

 3 Fig. 3. (a) Vorticity field at t = 4.5 τ 0 for Fr -1 = 1.(b) Vorticity and (c) perturbation of potential temperature fields at t = 2 τ 0 for Fr -1 = 1.

Fig. 4 .

 4 Fig. 4. Altitude modification of the vortex dipole due to hot jets as a function of Fr -1 and bjet .

Fig. 5 .

 5 Fig. 5. Evolution of the passive scalar field over time. From top to bottom: Fr -1 = 0.2, bjet = 0.16 ; Fr -1 = 1, bjet = 0.16 ; Fr -1 = 1, bjet = 0.79.From left to right: t = τ 0 , 2.5 τ 0 , 3 τ 0 , 3.5 τ 0 .

Fig. 6 .Fig. 7 .

 67 Fig. 6. (a) Mean altitude of the plume and (b) difference between the mean altitude of the plume and that of the vortices at t = 5 τ 0 as a function of Fr -1 and bjet . The vertical lines represent the of possible parameters for a selection of twin-engined aircraft (from left to right): CRJ, B737, B777, A330, A320, B787. The purple dashed line on the right panel corresponds to the orange line on the left panel.

Fig. 8 .

 8 Fig. 8. Attenuation coefficient µ at t = 5 τ 0 for RH i = 130%. From left to right bjet = 0.16, 0.31 and 0.79. From top to bottom Fr -1 = 0.2, 1 and 1.4

TABLE I VORTEX

 I WAKE AND STRATIFICATION PARAMETERS FOR A COMMON AIRCRAFT. THE RANGE OF Fr -1 IS FOR N BETWEEN 0.01 AND 0.03 S -1 . b (m): WINGSPAN, b 0 (m): VORTEX SEPARATION, Γ 0 (m 2 s -1 ): VORTEX CIRCULATION, W 0 (m s -1 ): VORTEX DESCEND SPEED, τ 0 (s): VORTEX TIME SCALE, b jet (%b): JET SEPARATION.

		CRJ-200	A320	B737	A330	B777	A380
	b	21.2	35.8	35.8	64.0	64.8	79.7
	b 0	16.7	28.1	28.1	50.3	50.9	62.6
	Γ 0,min	93.3	161	161	266	326	430
	Γ 0,max	162	296	298	506	683	893
	W 0,min	0.89	0.91	0.91	0.84	1.02	1.09
	W 0,max	1.55	1.67	1.68	1.60	2.14	2.27
	τ 0,min	10.7	16.8	16.7	31.4	23.8	27.6
	τ 0,max	18.7	30.8	30.9	59.6	49.9	57.3
	Fr -1 min Fr -1 max b jet /b	0.107 0.560 20	0.168 0.167 0.313 0.238 0.923 0.927 1.79 1.50 32 28 33 31	0.276 1.72 40 a 67 b
	a For the inboard engine;

b

For the outboard engine during flight time. τ 0 is also a function of the wingspan. The natural Brunt-Väisälä frequency of the atmosphere at flight altitude lies generally between 0.01 and 0.03 s -1 . As a consequence, the possible values for Fr -1 range from 0 to 2. The Froude number is set to be uniform in the present setup. In reality, it varies temporally and spatially but with sufficiently slow and large scales so that a constant approximation is locally adequate. Table I displays wake and stratification data for a selection of aircraft.


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Leila Amgoud
email: amgoud@irit.fr

Souhila Kaci
email: kaci@cril.univ-artois.fr

On the generation of bipolar goals in argumentation-based negotiation

Keywords:

The notion of agent's goals is crucial in negotiation dialogues. In fact, during a negotiation, each agent tries to make and to accept the offers which satisfy its own goals. Works on negotiation suppose that an agent has a set of fixed goals to pursue. However, it is not shown how these goals are computed and chosen by the agent. Moreover, these works handle one kind of goals: the ones that an agent wants to achieve. Recent studies on psychology claim that goals are bipolar and there are at least two kinds of goals: the positive goals representing what the agent wants to achieve and the negative goals representing what the agent rejects. In this paper, we present an argumentation-based framework which generates the goals of an agent. The framework returns three categories of goals: the positive goals, the negative ones and finally the goals in abeyance.

Introduction

In most agent applications, the autonomous components need to interact with one another because of the inherent interdependencies which exist between them, and negotiation is the predominant mechanism for achieving this by means of an exchange of offers. The purpose of negotiation is to make a deal and each agent aims to maximize its profit. In fact, an agent makes and accepts only offers that satisfy its goals. Works in multi-agents negotiation can be roughly divided into two categories. The first one has mainly focused on the numerical computation of trade-offs in terms of utilities, and the search for concessions which still preserve the possibility of reaching preferred states of affairs e.g. [START_REF] Luo | A fuzzy constraint based model for bilateral, multi-issue negotiations in semi-competitive environments[END_REF][START_REF] Wong | Possibilistic reasoning for intelligent payment agents[END_REF]. These works suppose that each agent has a set of fixed goals that it should pursue. Recently, a second line of research [START_REF] Amgoud | Arguments, dialogue, and negotiation[END_REF][START_REF] Kraus | Reaching agreements through argumentation: a logical model and implementation[END_REF][START_REF] Sierra | A framework for argumentation-based negotiation[END_REF] has focused on the necessity of supporting offers by arguments during a negotiation. Indeed, an offer supported by a good argument has a better chance to be accepted by an agent and may lead an agent to revise its goals. However, in these works, it is not clear how the goals are handled and updated if necessary. In sum, in all these approaches, it is not shown how the goals are computed and chosen by the agent and how they can be revised. Moreover, in all the above approaches, only one kind of goals is considered: the ones that an agent wants to achieve. However, in [START_REF] Benferhat | Bipolar representation and fusion of preferences in the possibilistic logic framework[END_REF] the authors argued that when an agent expresses its goals, it usually does that in a bipolar way. On one hand, it expresses what it really wants, what it considers as really satisfactory. These are positive goals. They will represent the goals which will be pursued by the agent and each offer satisfying these goals is rewarded. On the other hand, it expresses what it definitely rejects, what it considers as unacceptable. These are negative goals. They represent the goals which will not be pursued by the agent. This category of goals is very important in a negotiation since each offer satisfying a negative goal will automatically be rejected by the agent. Indeed, reasoning on both what an agent likes and what it rejects, enriches the negotiation process since an offer can be evaluated w.r.t. both kinds of goals. For example an agent may consider an offer better than another if both falsify all its positive goal but the first one does not satisfy any negative goal (i.e., it is not rejected) while the second one satisfies at least one negative goal (i.e., it is rejected). Beware that positive goals do not just mirror what is not rejected since a goal which is not rejected is not necessarily pursued. This category of goals which are neither negative nor positive are said to be in abeyance. Note however that positive and negative goals are related by a coherence condition which says that what is pursued should be among what is not rejected. This distinction between positive and negative goals is supported by recent studies in cognitive psychology which have shown that these two types of goals are independent and processed separately in the mind [START_REF] Cacioppo | The affect system: Architecture and operating characteristics[END_REF][START_REF] Cacioppo | Beyond bipolar conceptualizations and measures: The case of attitudes and evaluative space[END_REF][START_REF] Rolls | Precis of "brain and emotion[END_REF][START_REF] Borod | The neuropsychology of emotion[END_REF]. This paper focuses on the computation of goals. It particularly answers the following questions: what is a goal? what is its origin? what is its nature? what are the different kinds of arguments supporting it? and how it is computed or obtained? We present an argumentation-based framework which returns the three categories of goals, namely positive goals, negative goals and goals in abeyance.

The paper is organized as follows: section 2 studies the nature of a goal. Section 3 introduces two different types of arguments supporting a goal: explanatory arguments and instrumental arguments. Section 4 presents an argumentation framework which evaluates the explanatory arguments and section 5 presents another framework which evaluates the instrumental arguments. Section 6 computes the positive and the negative goals of an agent. Section 7 shows through an example how the positive and negative goals of an agent may change in a negotiation dialogue.

The nature of a goal

In this section, we will discuss the nature of a goal according to three different criteria: subjectivity which has been already discussed in [START_REF] Rahwan | Towards interest-based negotiation[END_REF], bipolarity and finally the origin.

The subjective nature

As it has already been mentioned in [START_REF] Rahwan | Towards interest-based negotiation[END_REF] goals are considered as motivational attitudes of an agent and they are therefore by nature intrinsic to an agent. Indeed, one cannot say, as for beliefs, that a given goal of an agent is correct or incorrect. But, we can attempt to establish that a goal seems unachievable, not useful or unsupported. Let's take the following dialogues between two agents: Example 1. P: I would like to fly to Algiers with Algerian Airlines because it is not expensive. C: But flying with Algeria Airlines means changing your flight arrangements is not flexible. P: I know that.

In this case even if the argument given by C seems acceptable and in some sense defeats the argument supporting the goal to fly with Algerian Airlines, P maintains its goal.

Example 2. P: I would like to fly to Algiers with Algerian Airlines because it is not expensive. C: Actually flying with Algeria Airlines can be quite expensive because it is the holiday season. P: I didn't know that.

In this case, if the agent P does not find another company which is cheaper than Algerian Airlines, then we can also imagine that the agent keeps its goal.

A bipolar nature

As we said in the introduction, an original representation of goals called bipolar goals [START_REF] Benferhat | Bipolar representation and fusion of preferences in the possibilistic logic framework[END_REF] has been proposed. Indeed, we distinguish two independent types of goals: positive goals describing the goals pursued by the agent and negative goals describing the goals rejected, not pursued by the agent. The goals which are neither negative nor positive are called goals in abeyance. In example 1, we can imagine that the flexibility arrangements of a flight is not very important for the agent. Goals are matter of degrees. Thus an agent expresses its goals by means of two different bases. A base representing what is more or less rejected by the agent and another base representing what is more or less satisfactory for him. To illustrate the idea of bipolar goals, let us consider the following example (introduced in [START_REF] Kaci | Bipolar goals. A possibilistic logic characterization of preferred choices[END_REF]) of an agent who goes to an agency in order to buy a house or an apartment. It gives the following positive and negative goals to the seller.

Example 3. The agent does not want a house or an apartment with a small surface and which is expensive. This negative goal is encoded as ( ¢¡ ¤£ Note that none of positive goals is rejected. Also the goal to have a medium and cheap apartment without garden is neither a negative goal nor a positive one. This is a goal in abeyance.

The origins of the goals

Agent's goals come generally from two different sources:

-from beliefs that justify their existence. So, the agent believes that the world is in a state that warrants the existence of its goals. These goals are called the initial ones or also conditional goals. They are conditional because they depend on the beliefs. -an agent can adopt a goal because it allows him to achieve an initial goal. These are called sub-goals or adopted goals.

Example 4 (Trip to Central Africa).

Let's consider an agent who wants to go to Central Africa because there is a conference there. The goal ¢¡ E£ is derived from the belief £ ! ¥¤ 4© 0 © ¢ ¡ © . The agent believes that to go to Central Africa, it should get tickets (¦ ) and to be vaccinated ( 1£ §¡ ). To get tickets, the agent can either pass to an agency (£ ( ) or ask a friend of him to get them ( ¤ 0 ). Similarly, to be vaccinated, the agent has the choice between going to a doctor (2 0 ) or going to the hospital ( ! ). Thus, ¦ , £ §¡ , £ ( , ¤ 0 , 2 0 and "! become sub-goals of that agent.

Arguing about goals

As mentioned above, there are two kinds of goals: the initial/conditional goals and the adopted ones called also sub-goals. These goals are justified or supported by two different kinds of arguments: explanatory arguments and instrumental arguments. Before presenting formally these two types of arguments, we will start by presenting the logical language which will be used throughout this paper. In what follows, ¨will denote a propositional language. © denotes classical inference and denotes logical equivalence.

Definition 1 (Conditional Rules).

A conditional rule is an expression of the form

§ ¢ § ! #" $
where is the name of the rule and each &% and are literals of ¨. The conjunction at the left of the arrow is the antecedent and the literal at the right is its consequent.

A conditional rule expresses the fact that if ' are true then the agent will have the goal

. Similarly, we will define the planning rules.

Definition 2 (Planning Rules).

A planning rule is an expression of the form

( ) § 0 ¢ § ) #1 2 )
where ( is the name of the rule and each ) 3% and ) are literals of ¨. The conjunction at the left of the arrow is the antecedent and the literal at the right is its consequent. Such a formulae means that the agent believes that if he realizes

) , ,
) then he will be able to achieve

) .

Remark 1. Note that the implications used to define both conditional rules and planning rules are not the material implication.

In what follows, we suppose that the agent's beliefs are more or less certain and that its conditional goals or planning rules may not have equal priority.

Definition 3. An agent is equipped with three bases ¡ £¢ , ¢ ¥¤ , ¢ §¦ ©¨s uch that:

-¢ = % £ % ¥ ! #" with %
is a propositional formulae of the language ¨and £ % its certainty degree. This base contains the basic beliefs of the agent.

-¢ $¤ = &% (' % ) 0 ¡ "
where &% is a conditional rule and ' % represents the priority degree of the consequent of

1% . -¢ §¦ = 2 ( §3 ¡ 3 54 ! ¥ 6"
where ( #3 is a planning rule and ¡ 3

represents the priority degree of this rule.

In what follows, we suppose that £ %

, ' % and ¡ % belong to the interval 7 98 . Moreover, we shall denote by ¢ 1@ , ¢ £@ ¤ and ¢ &@ ¦ the corresponding sets when the weights are ignored i.e.

-¢ @ = A % % B £ % C ¥D ¢ § E #" -¢ @ ¤ = % % F G' % H ID ¢ ¤ P E ¡ " -¢ £@ ¦ = ( #3 ( #3 ¡ 3 ¥D ¢ §¦ 4 E ¥ 6"
Once the language is introduced, we are now able to define formally the potential initial goals and the sub-goals.

Definition 4 (Initial goal -Sub-goal).

Let an agent be equipped with ¡ £¢ , ¢ ¤ , ¢ ¦ ¨.

-Q SR = s.t T § § ' " D ¢ @ ¤ "
is the set of potential initial goals of the agent.

-U # ' 9V

is the set of potential sub-goals of the agent: A literal

) SW D U # ' XV iff there exists a rule ) § ) #W § ) #1 2 ) D ¢ @ ¦ with ) D Q SR or ) D U # ' 9V . In that case, ) #W is a sub-goal of ) .
Remark 2. Note that in the above definition, we speak about potential initial goals of the agent. The reason is that we are not sure that the antecedents of the corresponding rules are true. Consequently, if a potential goal is not adopted by the agent (the antecedents are not true), then it is not useful for the agent to realize its plan.

Example 5 (Trip to Central Africa).

Let us consider an agent who has the two following goals:

1. To go on a journey to central Africa if there is a Conference there. ( ¡ £ ) 2. To finish a publication before going on a journey. ( ¤ 1 )

Thus, ¢ $¤ = ¡ E! ¥¤ " ¢¡ E£ 7 " ¤ 1 7 ¡ " . In addition to the goals, the agent is supposed to have beliefs on the way of achieving a given goal (we suppose that all the rules have equal priority):

¢ @ ¦ ¢ £ £ £ £ £ £ ¤ £ £ £ £ £ £ ¥ ¦ § £ §¡ 1 2 ¢¡ E£ ¦ 1 2 ¤ 1 £ ( 1 2 ¦ ¤ 0 1 2 ¦ "! 1 2 £ ¢¡ 2 0 1 2 £ ¢¡ and ¢ = ¦ 2 & £ ( , 0.8), ( ¦ 2 & 2 0 , 0.8), (¡ E! ¥¤ , 0.8), (¡ E£ , 0.4), (¡ E£ 2 & ¡ E! ¥¤ "
. with: conf = "a conference", can = "to be canceled", t = "to get the tickets", vac = "to be vaccinated", w = "to work", ag = "to go to the agency", fr = "to have a friend who may bring the tickets", hop = "to go to the hospital", dr = "to go to a doctor".

In this example, Q SR = jca, fp "

and U # ' 9V = t, vac, ag, fr, dr, hop, w "

.

Explanatory arguments

Explanatory arguments are used to explain / to give a reason of adopting a given goal. They are also used to give reasons for and against beliefs. In this section, we will propose an argumentation system which constructs explanatory arguments from the different bases of an agent and which evaluates them.

Basic definitions Definition 5 (Explanatory argument).

An explanatory argument is a pair ¡ H, h ¨such that:

- § ©¨¢ @ ¢ @ ¤ .

- § © .

- § is consistent and minimal (for set inclusion).

denotes the set of all arguments such that D Q SR . In other terms, it gathers all the arguments supporting initial goals. gathers all the arguments supporting beliefs (i.e D Q SR ). Finally, = .

Definition 6 (Sub-argument).

Let ¡ H, h ¨ ¡ § W W ¨D . ¡ § ¨is a sub-argument of ¡ § W W ¨iff § ©¨ § W . Example 6 (Trip to Central Africa). The arguments ¡ ¡ ! ¥¤ , ¡ E! ¥¤ " ¢¡ E£ " , ¢¡ E£ ¨and ¡ " ¤ 1 " , ¤ 1 ¨D ! . However, the argument ¡ ¡ E£ , ¡ E£ 2 & ¡ E! ¥¤ " , & ¡ ! ¥¤ ¨D . Remark 3.

The strength of explanatory arguments

As mentioned before, each of the three bases ¡ £¢ , ¢ I¤ , ¢ §¦ ¨i s pervaded with certainty or priority. From the certainty degrees, we define the certainty level of an argument.

Definition 7 (Certainty level of an explanatory argument)

.

Let = ¡ § , ¨D . The certainty level of ¡ § , ¨, denoted by ¥ © © $¥ ¡ = ¡ $£ % ¢ ) % §D § ¤£ ¢ @ and ) % £ % SD ¢ " . If § ¥£ ¢ @ = ¦ then ¥ © © $¥ ¡ = 1.
Remark 4. Note that the priority degree of a given conditional goal is not taken into account in the definition of the strength of its supporting argument. In fact, the intuition behind a conditional goal is that: "the agent will adopt the goal, with its associated priority degree, if the condition is satisfied". So even if the goal is very desired by the agent, if the conditions are not satisfied, then that goal will not be pursued.

The certainty level of the arguments makes it possible to compare different arguments as follows:

Definition 8 (Preference relation). Let

and ¡ be two arguments in

. is preferred to ¡ , denoted by ¨ § ¡ , iff ¥ © © $¥ ¡ ¨¥ ¦© © $¥ © ¡ .
Example 7 (Trip to Central Africa).

In the above example, the certainty level of the argument ¡ ¡ E! ¥¤

, ¡ ! ¥¤ " ¢¡ E£ "

, ¢¡ E£ ïs 0.8. Whereas, the certainty level of the argument

¡ ¡ E£ , ¡ E£ 2 & ¡ E! ¥¤ " , & ¡ E! ¥¤ ïs 0.4. The certainty level of the argument ¡ " ¤ 1 " , ¤ 1 ¨is 1. Thus, ¡ " ¤ 1 " , ¤ 1 § ¡ ¡ E! ¥¤ , ¡ ! ¥¤ " ¢¡ E£ " , ¢¡ E£ ¨ § ¡ ¡ E£ , ¡ E£ 2 & ¡ E! ¥¤ " , & ¡ E! ¥¤ ¨.

Conflicts between explanatory arguments

An explanatory argument can be defeated either on one of its beliefs or one of its conditional goals. For example, the argument supporting the goal of going to Central Africa because there is a conference can be defeated by another argument stating that the conference has actually been canceled. This kind of defeat is modeled by the relation of "undercut" defined as follows:

Definition 9 (Undercut relation).

Let ¡ § ¨, ¡ § W W ¨D . ¡ § ¨undercuts ¡ § W W ¨iff T W W D § W £ ¢ such that & W W .
A conditional goal can also be defeated. For instance, the argument of going to Central Africa can be defeated by an argument stating that there is no money and if there is no money then the agent cannot go to Central Africa ¡ ! ! © "

, ! ! © " 2 & ¡ £ "
, & ¢¡ E£ ¨. This kind of defeat is modeled by the following relation of "rebut".

Definition 10 (Rebut relation)

.

Let ¡ § ¨, ¡ § W W ¨D . ¡ § ¨rebuts ¡ § W W ¨iff W D Q V and & W .
The two relations of conflicts are brought together in a unique relation called attack:

Definition 11 (Attack relation).

Let ¡ § ¨and ¡ § W W ¨D . ¡ § ¨attacks ¡ § W W ¨iff: -¡ § ¨undercuts ¡ § W W ¨and not( ¡ § W W ¨ § ¡ § ¨) or -¡ § ¨rebuts ¡ § W W ¨and not( ¡ § W W ¨ § ¡ § ¨) or -¡ § ¨rebuts a sub-argument of ¡ § W W ¨and not( ¡ § W W ¨ § ¡ § ¨).

The acceptability of explanatory arguments

We can now define the argumentation system we will use to evaluate our arguments: Definition 12 (Argumentation system).

An argumentation system (AS) is a pair ¢¡ ¤£ ¥£ §¦ © such that is the set of all explanatory arguments built from ¢ ¢ ¤ . This system will return three categories of explanatory arguments:

-The class S of acceptable explanatory arguments. Goals supported by such arguments are really justified and they may be the "positive goals" that an agent will pursue, if they are achievable. -The class R of rejected arguments. An argument is rejected if it is attacked by an acceptable one. Goals supported only by such arguments will be rejected by the agent even if they can be achieved. They will represent the negative goals of the agent. -The class C of arguments in abeyance. Such arguments are neither acceptable nor

rejected. £ = U .
In what follows, we will define the class of acceptable arguments. For that purpose, we will start by presenting the notion of defence introduced in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and -person games[END_REF].

Definition 13 (Defence).

Let , be two arguments of and U ¨ . U defends iff for every argument where attacks , there is some argument in U which attacks .

Henceforth, the set £ ! #" %$ ¥& will gather all non-attacked arguments. We can show that the set ' of acceptable arguments of the argumentation system %¡ ¤£ (£ )¦ 0¢ is the least fixpoint of a function 1 :

' ¨ 1 #' § ¥D ¥2 ¢ § 43 !5 76 98 A@ B8 AC D6 98 E6 GF IH ' " ¢ Proposition 1. Let ¢¡ ¤£ ¥£ )¦ 0¢ be an an argumentation system. The set of its accept- able arguments is:

' P Q1 % SR UT ©¦ £ ! #" %$ ¥& WV P X1 % SR £ ! Y" %$ §& 8

Example 8 (Trip to Central Africa).

In this example, the argument ¡ " ¤ 1 " , ¤ 1 ¨is not attacked then it is acceptable.

The argument ¡ ¡ E! ¥¤

, ¡ E! ¥¤ " ¢¡ E£ "

, ¢¡ E£ ¨is preferred to its unique undercutting argument

¡ ¡ £ , ¡ E£ 2 & ¡ E! ¥¤ " , & ¡ E!
¥¤ ¨. Then it is not attacked and consequently it is also acceptable.

Let be a set of arguments. The function U # ¡ = § % such that ¡ § % , % ¨D . In other terms, the function U # returns the union of all the supports of arguments of . We can show the following result: Proposition 2. Let ¢¡ ¤£ ¥£ §¦ © D be an argumentation framework and ' its set of acceptable arguments. Then U # B' is consistent.

Property 1. Let D . If
is acceptable then each sub-argument of is also acceptable.

Instrumental arguments

An agent may have another kind of goals. These last are not derived from the current beliefs of the agent, but from the plans to achieve the initial goals. In fact, they are justified by the fact that they will contribute to the achievement of initial goals. They are thus considered as sub-goals of the initial goals. In [START_REF] Amgoud | A formal framework for handling conflicting desires[END_REF], an argumentation framework which handles conflicting goals has been developed. This framework takes as input a set of initial goals, a belief base and a base of planning rules and returns the goals which can be achieved together, as well as the appropriate plans (i.e. the sub-goals). In this section, we will present an extended version of that framework which takes into account the priorities of the goals.

Basic definitions

An agent may have one or several ways to achieve a given goal. We bring the two notions together in a new notion of partial plan.

Definition 14 (Partial plan).

A partial plan is a pair £ = ¡ § ¨such that:

-is an initial goal or a sub-goal.

- § = ) ) "
if there exists a rule

) § § ) 1 2 D ¢ @ ¦ , § = ¦ otherwise.
The function R ! 3£ ¥ £ = returns the initial goal or sub-goal of a given partial plan "£ " and the function ( ¥ £ £ = § returns the support of the partial plan. ¢ will gather all the partial plans that can be built from ¡ Q V , ¢ , ¢ ¦ ©¨.

Note that a goal may have several partial plans.

Remark 5. Let £ = ¡ § ¨be a partial plan. Each element of the support § is a sub- goal of .

Definition 15.

A partial plan £ = ¡ § ¨is elementary iff § = ¦ . Remark 6. If there exists an elementary partial plan for a goal , it means that the agent knows how to achieve directly .

A partial plan shows the actions that should be performed in order to achieve the corresponding goal (or sub-goal). However, the elements of the support of a given partial plan are considered as sub-goals that must be achieved at their turn by another partial plan. The whole way to achieve a given goal is called in [START_REF] Amgoud | A formal framework for handling conflicting desires[END_REF] a complete plan. A complete plan for a given goal ( is an AND tree. Its nodes are partial plans and its arcs represent the sub-goal relationship. The root of the tree is a partial plan for the goal ( . It is an AND tree because all the sub-goals of ( must be considered. When for the same goal, there are several partial plans to carry it out, only one is considered in a tree.

Definition 16 (Instrumental argument).

An instrumental argument is a pair ¡ G, g ¨such that ( D Q V and R is a finite tree such that:

-The root of the tree is a partial plan ¡ H, g ¨.

-A node ¡ )

, ,

) " W ¨has

exactly children ¡ § W ) ¨, , ¡ § W , )
ẅhere

¡ § W % ,
) % ¨is a partial plan for ) %

. -The leaves of the tree are elementary partial plans.

The function ! 32 © 2 R returns the set of all the partial plans of the tree R . W will denote the set of all the instrumental arguments that can be built from the bases ¡ Q V , ¢ , ¢ §¦ ©¨.

The strength of instrumental arguments

As mentioned before, the base ¢ ¤ is pervaded with priority. From the priority degrees, we define the weight of an instrumental argument.

Definition 17 (Weight of an instrumental argument).

Let = ¡ R , ( ¨D W . The weight of ¡ R , ( ¨is © ( ¦ 9 © = ¡ ' % " such that % G' % $D ¢ $¤ and % = ) § 0 ¢ § ) " ( .
In other words, the weight of an instrumental argument is exactly the degree of priority/importance of the corresponding goal. The weights make it possible to compare different arguments as follows:

Definition = ¡ ( ¡¤ ¤ 1 ¨is 0.8. Hence, ¡ ( ¡¤ ¤ 1 ¨ § ¡ ( ¡ £ ¨, ¡ ( ¡ ¢¡ E£ ¨, ¡ ( ¡ ¢¡ E£ änd ¡ ( £¢ ¢¡ E£ ¨.

Conflicts between instrumental arguments

In [START_REF] Amgoud | A formal framework for handling conflicting desires[END_REF], it has been shown that partial plans may be conflicting for several reasons.These different kinds of conflicts are brought together in a unique relation of conflict defined as follows:

Definition 19 (Conflict).

Let £ and £ ¡ be two partial plans of ¢ . £ conflicts with £ ¡ iff:

R ! 3£ ¥ £ , R ! 3£ ¥ ¦£ ¡ " ( ¥ £ ¦£ ( ¥ £ £ ¡ ¢ &@ ¢ &@ ¦ © ¢¡ . Example 11.
In example 5, £ ¤£ = ¡ ag "

, t ¨conflicts with £ ¡ = ¡ ¦ "

, fp ¨. Indeed, Plan(£ ¤£ ) ¢ @ © ) & ¦ " and Plan(£ ¡ ) = ¦ "

. More generally, a set of partial plans may be conflicting.

Definition 20. Let U ¨¢ . U is conflicting iff ¥ £ §¦ ©¨ AR ! 3£ ¥ ¦£ " ( ¥ £ £ B ¢ @ ¢ @ ¤ © ¡ .
Since partial plans may be conflicting, two instrumental arguments may be conflicting too.

Definition 21 (Defeat).

Let ¡ R ( ¨, ¡ R ¡ ( ¡ ¨D W . ¡ R ( ¨defeats ¡ R ¡ ( ¡ ¨iff T 6£ D ! 32 © 2 R and T 6£ ¡ D '! 32 © 2 R ¡ such that £ conflicts with £ ¡ and not( ¡ R ¡ ( ¡ ¨ § ¡ R ( ¨).
More generally we are interested in conflict-free sets of instrumental arguments.

Definition 22 (Conflict-free).

Let U = ¡ R % ( % ¨: i = 1, . . . , n " ¨ W . S is conflict-free iff

V ¥ ¡ £¢ ¤ ¦¥ ¦ ©¨V ¥ £ ¦ ¨ § © ¢ ( ( ¥ £ £ AR ! 3£ ¥ ¦£ " 8 ¢ @ ¢ @ ¦ ¡! © ¡ 18 .
If U = ¡ R ( ¨" then we say that the argument ¡ R ( ¨is conflict-free.

Proposition 3. Let U ¨ W

. If U is conflict-free then " and " ¡ in S such that defeats ¡ .

The following example shows that we can find an instrumental argument which is not conflict-free even if it does not defeat itself.

Example 12. X is an agent equipped with the following bases:

Q V = $2 " , ¢ @ = ' W § ¡ W 2 & £ " and ¢ @ ¦ = ¢ £ £ ¤ £ £ ¥ £ W 1 2 £ ' W 1 2 ' ¡ W 1 2 ¡ £ § ' § ¡ 1 2 2
There is a unique instrumental argument for 2 whose set of nodes is conflicting.

Obviously a goal which has no conflict-free instrumental argument will be called unachievable. This means it is impossible to carry out such a goal.

Acceptability of instrumental arguments

From the preceding definitions, we can now present the formal system for handling instrumental arguments.

Definition 23. Let's consider a triple ¡ £¢ , ¢ ¤ , ¢ ¦ ¨. The pair ¡ W , # © ¤ 4© £ ¢¦ ¨will be called a system for handling instrumental arguments.

As for explanatory arguments, this system will return three categories of instrumental arguments:

-The acceptable instrumental arguments. These arguments represent the good plans to achieve their corresponding goals. -The class R' of rejected instrumental arguments. This class gathers the arguments which are not conflict-free and those defeated by acceptable arguments. Goals supported only by such arguments are unachievable. -The class C' of arguments in abeyance. Such arguments are neither acceptable nor rejected.

Unlike the previous system, we may have here several acceptable sets. Each of them will correspond to a set of goals which can be achieved together.

Definition 24. Let ¡ W , # © ¤ 4©
£ ¦ ¨be a system and U ¨ W

. S is an acceptable set of arguments iff:

-S is conflict-free.

-S is maximal (for set inclusion). Let U , , U be the different sets of acceptable arguments. Example 13. In example 5, there are four instrumental arguments ( , ¡ , ,

¢

) for the goal "going on a journey to Central Africa" and exactly one argument ¤ for the goal "finishing the paper". Moreover, ¤ defeats , ¡ and . We have exactly two acceptable sets of arguments:

-U = ¡ ¢ " , -U ¡ = ¢ ¤ "
The purpose of an agent is to achieve a maximal subset of Q SR . Consequently, among the sets of acceptable arguments, we will choose the ones which achieve maximal sets of desires (for set inclusion). In the above example, we will choose the set U ¡ .

Computing bipolar goals

Once we have defined the two frameworks which evaluate the different arguments supporting goals, we are now able to define among the potential initial goals and sub-goals the positive goals of the agent, the negative ones and finally the goals in abeyance. - § ( SD , § ( D ¢¡ , or -R ( SD W , R ( $D ¢¡ W Indeed, a goal is negative if it is not justified or if it is unachievable.

Definition 27 (Goals in abeyance). Let

( D Q SR . (
is in abeyance iff it is neither positive nor negative.

Example 14 (Trip to Central Africa).

In this example, the two initial potential goals of the agent ( ¢¡ E£ , ¤ 1 ) become positive goals of the agent. Moreover, the sub-goals (¦ , 1£ §¡ , ¤ 0 , "! , ¦ ) are also positive goals.

However, the following sub-goals are in abeyance: (2 0 , £ ( ).

Handling bipolar goals

Let's consider an agent ( who has the following belief base: ¢ & £ ( © ¤£ 7 ( # 7 ¥ " . Suppose that this agent has two potential goals: to be a president and/or to be a profes-

sor. ¢ $¤ 2 ¦£ ( © ¦£ 7 " 0 © ¢ 2 © ¢ ¦ 9 ( # " 0 3! ¤ 4© $ ¢! $0 "
. To be a president, the agent knows that he should have more than 40 years old however he has less than 40 years. According to its beliefs, there is no argument in favor of this goal. Consequently, the agent will keep this goal in abeyance.

Concerning the goal of becoming a professor, it has the following explanatory argument:

¡ ( # ( # " 0 $! ¤ 4© $ ! $0 " 0 3! ¤ 4©
$ ! $0 ¨. This argument is not attacked at all, thus it is acceptable. In this example, there are no instrumental arguments. So, the goal 0 $! ¤ 4© $ ! $0 is a positive one. Note that at this stage, the agent has one positive goal, one goal in abeyance an no negative goals. Suppose that one year later, the agent has more than 40 years old then ¢ is updated as follows: ¢ 2 ¦£ ( © ¤£ 7 9 ( # 7 ¥ " . Using this new base, we can find an argument in favor of the goal ¨.

The certainty level of , ¢ and ¤ is 0.5 and the certainty level of ¡ and is 1. Thus, ¡ , § , ¢ and ¤ . We can check easily that ¢ rebuts ¡ but since ¡ is preferred to ¢ then ¡ is not attacked and consequently it is acceptable. The argument rebuts and defeats . Moreover, is preferred to its unique undercutting argument ¤ . Then is also acceptable. Consequently, is rejected. The goal of being president is supported by an acceptable argument ¡ then this goal is positive. However, the goal of being a professor is supported only by the rejected argument then it is a negative goal.

Conclusion

In most negotiation literature, each negotiating agent is supposed to have a set of fixed and predefined goals. It is not clear where do these goals come from and how an agent selects them. Argumentation-based negotiation makes an advance by supposing that the goals are not fixed during a negotiation and may change. However, even in these works the goals are predefined and it is not clear how they are changed. In fact, since there is no work on how goals are computed, it seems difficult to model the way in which they are updated.

The aim of this paper is twofold. First, it presents the goals in a bipolar way. In fact, an agent has positive goals that it will pursue and also negative goals that it does not want to achieve. This second category of goals is very important in negotiation since the offers that satisfy such goals will be rejected by the agent. The second aim of this paper is to present a formal framework which computes the goals of an agent. We have shown through an example how an agent may change its goals during a negotiation. The principle of goals generation proposed in this paper is close to the one proposed in [START_REF] Thomason | Desires and defaults: A framework for planning with inferred goals[END_REF] in the context of planning, where an argument-based generation of goals is implicitly used. However in this latter, the author only generates the positive goals (called wants), those that the agent will pursue, from a set of initial conditional goals called wishes. Moreover, the set of beliefs on which initial conditional goals are based is flat (i.e., all the beliefs are equally certain) then there is no evaluation of arguments when they are conflicting.

In this paper, we have shown that a goal may be supported by two different kinds of arguments: the exaplanatory arguments and the instrumental ones. We have then presented two different systems for handling each category of arguments. An extension of our work will be to handle the two kinds of arguments in a unique framework. We are actually working in this direction. We are also planning to investigate more deeply the handling of bipolar goals in a negotiation dialogue.

Example 9 .Fig. 1 .

 91 figure 1 for the trees ( % ).

Example 10 .

 10 18 (Preference relation). Let and ¡ be two arguments in W . is preferred to ¡ , denoted by § ¡ The weight of the four instrumental arguments = ¡ ( ¢¡ E£ ¨, , = ¡ ( ¢¡ E£ ¨and ¢ = ¡ ( ¢ ¢¡ E£ ¨is 0.6 whereas the weight of the argument ¤

  Note that the implication used in conditional rules is not material and it has

	no contrapositive. So, for example the set Consequently, ¡ , 2 & #" , ( " " "	,	2 & #"	,	( " " "	does not infer	& )( .

, & )( ¨is not an explanatory argument.

Definition 25 (Positive goals).

	Let -T § ( SD ( D Q SR . ( -T R ( SD W is a positive goal iff: such that § ( $D and T PU % such that R ( $D U % ' , and

. This means that a goal is positive if it is justified and it is achievable. Note that the sub-goals of a positive initial goal are also considered as positive.

Definition 26 (Negative goals). Let

( D Q SR . ( is a negative goal iff:

Acknowledgments

This work was supported by the Commission of the European Communities under contract IST-2004-002307, ASPIC project "Argumentation Service Platform with Integrated Components".