Abderaouf Nassim Amalou
email: abderaouf.amalou@irisa.fr

Elisa Fromont

Pr Isabelle

Puaut Pacap

Pr Elisa

Fromont Lacodam

Machine learning for timing estimation

TIMING WORST CASE 1. CONTEXT

The execution time is calculated in specific cases depending on the needed application domain, (e.g, worst case execution for realtime applications, average case for compiler optimization).

TIMING AVERAGE CASE

Machine learning for timing estimation -Instruction == Word.

-Basic block == Sentence.

-Sequence of basic blocks == Paragraph.

RESULTS

PERSPECTIVE

• Transformers [4] are able to handle sequential data, but this is done in parallel, which drastically reduces both training and inference times.

  Machine Learning for timing estimation : • Context-awareness (Cache and Pipline). • Representativity of training data. • Reliability of the estimations. WE-HML [1] (WCET Estimation using an Hybrid Machine-Learning based technique) estimates the WCET for processors that have caches. • Learning phase : Training on Basic Block and WCET. • Estimation phase : ILP for longest path combined with ML model.

Figure 7 Figure 2 Figure 1 Figure 5 Figure 3 Figure 4 Figure 6

 7215346 Figure 7 Absolute Percentage Error on MiBench for CATREEN, ITHEMAL and MLP regressor

improves ITHEMAL to estimate the execution time of a BB in its context.

• ITHEMAL [3] uses 2 stacked LSTMs layers to estimate the execution time for a BB in isolation.

• CATREEN

[2]


W Herreman

M Le Bars

P Le Gal

On the effects of an imposed magnetic field on the elliptical instability in rotating spheroids

The effects of an imposed magnetic field on the development of the elliptical instability in a rotating spheroid filled with a conducting fluid are considered. Theoretical and experimental studies of the spin-over mode, as well as a more general short-wavelength Lagrangian approach, demonstrate that the linear growth rate of the instability and the square amplitude of the induced magnetic field fall down linearly with the square of the imposed magnetic field. Application of the results to the Galilean moon Io confirms the fundamental role played by the elliptical instability at the planetary scale.

Introduction

The elliptical instability is a generic instability of rotating flows with elliptical streamlines. It has for instance been observed in wakes [START_REF] Leweke | Cooperative elliptic instability of a vortex pair[END_REF], [START_REF] Meunier | A merging criterion for two-dimensional co-rotating vortices[END_REF], in elliptically deformed containers, [START_REF] Malkus | An experimental study of global instabilities due to tidal (elliptical) distorsion of a rotating elastic cylinder[END_REF], [START_REF] Eloy | Elliptic and triangular instabilities in rotating cylinders[END_REF], [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF], [START_REF] Lacaze | Elliptical instability of a flow in a rotating shell[END_REF], and more generally in the transition to turbulence of strained vortices [START_REF] Pierrehumbert | Universal short-wave instability of twodimensional eddies in an inviscid fluid[END_REF]. Since its discovery in the mid-1970s, it has received considerable attention, theoretically, experimentally and numerically (see for instance the review by Kerswell [START_REF] Kerswell | Elliptical instability[END_REF]).

Flows with elliptical streamlines arise as a superposition of rotation and a small strain field, and the instability mechanism has been identified as a 1 parametric resonance of pairwise inertial waves coupled by this strain [START_REF] Bayly | Three-dimensional instability of elliptical flow[END_REF], [START_REF] Waleffe | On the three-dimensional instability of strained vortices[END_REF]. In the geophysical context of liquid planetary cores [START_REF] Malkus | Precession of the earth as the cause of geomagnetism[END_REF], [START_REF] Kerswell | Tidal excitation of hydromagnetic waves and their damping in the Earth[END_REF], [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF], the strain comes from the tidal deformations due to gravitational interaction between neighboring celestial bodies. The elliptical instability (also called tidal instability in this context), as well as the closely related precessional instability, may leave traces in the gravitational and magnetic fields of planets [START_REF] Aldridge | Inertial waves identified in Earth's fluid outer core[END_REF], [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF], and may even provide alternative sources to power the geodynamo [START_REF] Malkus | Precession of the earth as the cause of geomagnetism[END_REF], [START_REF] Tilgner | Precession driven dynamos[END_REF]. Even if the hydrodynamic of the elliptical instability is today well known, its planetary consequences are still controversial and necessitate a full understanding of the magnetohydrodynamic (MHD) of the elliptical instability, which remains a mostly open question (e.g. [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF]). Understanding the MHD of the elliptical instability is also important in metallurgic applications, especially regarding its role in the transition from two to three-dimensional MHD-turbulence [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF].

In the present paper, we consider an elliptically deformed rotating sphere filled with a conducting fluid (figure 1) and we study both theoretically and experimentally the effects of an imposed magnetic field parallel to the rotation axis on the development of the elliptical instability. This situation is reminiscent of planetary configurations where a tidally deformed moon with a liquid iron core rotates in the magnetic field of its planet, as for instance the Galilean moon Io in the vicinity of Jupiter. Our purpose is to answer the two following questions. How is the elliptical instability damped by the magnetic field? And what is the amplitude of the magnetic field induced by the elliptical instability?

This article, which completes and extends the previous works of Lacaze et al. [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF] and Thess & Zikanov [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF], is organized as follow. We first focus on the so-called spin-over mode, which corresponds to the simplest mode of the elliptical instability in spheroids, excited at the smallest values of the Reynolds number above threshold. We derive a nonlinear and viscous model of its development under an imposed magnetic field valid for low values of the magnetic Reynolds number, based on the hydrodynamical model of Lacaze et al. [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] and including the magnetic damping term determined by Thess & Zikanov [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF]. These results are validated experimentally using an extended version of the set-up of Lacaze et al. [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF], with stronger imposed magnetic fields. These results are then extended to the large magnetic Reynolds number, large Reynolds number limit relevant to planetary applications, using a short-wavelength Lagrangian theory [START_REF] Lifschitz | Local stability conditions in fluid dynamics[END_REF]. An analytical expression of the growth rate of the elliptical instability is determined and results are finally applied to the case of Io, highlighting the importance of the elliptical instability at the planetary scale.

Set-up and state of the art

We consider the experimental system sketched in figure 1. A spherical cavity with radius R, molded in a deformable silicone block, is filled with a liquid metal, with permeability µ, conductivity σ e , kinematic viscosity ν, and density ρ. It is set in rotation at a constant angular velocity Ω 0 = Ω 0 ẑ around the vertical z-axis, and a homogenous magnetic field B 0 = B 0 ẑ is externally imposed along the same axis with a pair of Helmholtz coils. Following the original idea of Malkus [START_REF] Malkus | An experimental study of global instabilities due to tidal (elliptical) distorsion of a rotating elastic cylinder[END_REF], a pair of fixed and opposed rollers compresses the transverse section of the deformable container, giving it an elliptical crosssection with long axis R √ 1 + along x and short axis R √ 1 -along y, being the eccentricity of the elliptical deformation. Previous experimental studies (e.g. [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF]) have demonstrated that the flow in the volume effectively exhibits elliptical streamlines due to the boundary deformation and approaches the theoretical elliptical base-flow

U b = -Ω 0 1 + 1 - y, Ω 0 1 - 1 + x, 0 . (1)
Together with a homogenous magnetic field along the axis, this flow defines an exact base state of the MHD equations on which perturbations may grow due to the elliptical instability. In our experiments, the magnetic Reynolds number Rm = Ω 0 R 2 /η, where η is the magnetic field diffusion η = (σ e µ) -1 , is small (Rm = O(10 -2 )) and magnetic field diffusion is always dominant over magnetic field advection. In this limit, the scales

[r] = R [t] = Ω -1 0 [u] = Ω 0 R [p] = ρ(Ω 0 R) 2 [b] = Rm B 0 (2)
respectively for space, time, velocity, pressure and magnetic field, are well adapted to non-dimensionalize the perturbation problem. In addition to Rm, the relevant non-dimensional parameters are the Ekmann number E = ν/Ω 0 R 2 , which measures the importance of diffusive effects over inertial terms, and the Elsasser number Λ = σ e B 2 0 /ρ Ω, measuring the ratio of Lorentz force effects over the inertial forces. In our experiments, E = O(10 -4 ) and Λ ≤ O(1) typically.

The hydrodynamic stability of elliptical flow in spheroids was previously studied in [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF], which formalizes an asymptotic theory in terms of inertial wave coupling. In the limit Λ ∼ , Rm → 0, this theory can be extended to include the magnetic field effect perturbatively. We will not go as far, as the combination of the results of [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF], [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF] and [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF], allows us to describe the linear and nonlinear dynamics of the dominant spin-over mode, which is the only mode accessible to purely hydrodynamical experiments using the present device in a spherical geometry with a fixed strain field [START_REF] Bars | Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers[END_REF]. The spinover mode is mainly a solid body rotation around an inclined axis, whose horizontal projection Ω H is aligned with the axis stretched by the strain field, at polar angle in the vicinity of -45 o in the (x, y) plane (see figure 1). A low-dimensional model was derived in [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] in close agreement with the experiments, which describes the nonlinearly purely fluid evolving spin-over mode as a solid body rotation. Even though E is small in the experiment, viscosity plays an essential role. Indeed, it postpones the elliptical instability to a critical eccentricity and allows the nonlinear dynamical system to have stable non-trivial fixed points.

Stays the question whether the spin-over mode remains the most unstable mode in presence of a magnetic field, which seems hard to answer without a more complex global analysis of the elliptical instability. In [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF] (see also section 5), the local growth rates of elliptical instability in an unbounded domain were calculated using Flocquet theory. For the limit of small we are interested in, asymptotic arguments as in [START_REF] Waleffe | On the three-dimensional instability of strained vortices[END_REF] imply a growth rate linear in and a magnetic damping γ M = Λλ 2 /4, where λ is the wave frequency in the rotating frame. Since all elliptically interesting waves have λ 1 [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF], [START_REF] Bars | Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers[END_REF], local theory indicates that the magnetic field damping acts similarly on all couplings, no matter what their spatial structure is. Since in our device the spin-over mode is always the most unstable mode in the hydrodynamical experiments, we expect that it remains the case when a magnetic field is imposed.

Thess and Zikanov [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF] also extended the non-linear, inviscid model of the spin-over dynamics to include the magnetic field effects in the low Rm limit. They found that the magnetic field introduces a Joule damping, which only operates on the rotations with axis transverse to the imposed magnetic field, identically to the Joule damping of solid conductors rotating in a strong magnetic field, commonly used in magnetic brakes. In our experimental set-up, both viscous and magnetic field effects are important. The eccentricity is small 0.1, which means that Λ is at most of order O(10 -1 ) in the experiments where we observe the instability. The important consequence is that the magnetic field in this case is always too small to change the viscous boundary layer into a Hartmann-layer, so that there is no need of a more complex boundary layer analysis. This also implies that the viscous terms in the nonlinear system of [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] may be used here. Finally, there will be no significant contributions to the external and internal magnetic fields due to the boundary layer, which would make the field deviate from the field induced by the non-viscous spin-over mode, calculated in [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF].

Notice that all these suppositions will be confirmed a-posteriori by the good agreement between the following theory and our experimental results.

Analytical study of the spin-over mode

Combining the results of [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] and [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF], the nonlinear evolution of the spinover mode can be modeled in the laboratory frame of reference as a solid body rotation with angular velocity Ω = Ω 1 (t), Ω 2 (t), Ω 3 (t) , which evolves according to the nonlinear system

Ω1 = - 2 - (1 + Ω 3 ) Ω 2 -(γ so + Λ/4) Ω 1 , (3)
Ω2 = - 2 + (1 + Ω 3 ) Ω 1 -(γ so + Λ/4) Ω 2 , (4)
Ω3 = Ω 1 Ω 2 -γ 3 Ω 3 + ν nl Ω 2 1 + Ω 2 2 . (5)
On the right hand sides, we first recognize the destabilizing terms from the non-viscous system. The damping of the spin-over mode is controlled by the viscous linear boundary layer γ so = 2.62 √ E in the horizontal directions and γ 3 = 2.85 √ E around the vertical axis. Supplementary nonlinear terms arise through the boundary layer, ν nl = 1.42 √ E. All these coefficients are explicitly detailed in Lacaze et al. [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] and find their origin in the classical analysis of Greenspan [START_REF] Greenspan | The Theory of Rotating Fluids[END_REF]. The magnetic field only adds a linear term corresponding to the Joule-damping Λ/4 in the directions perpendicular to the imposed field. The terms due to the viscous frequency detuning are left out from the model as in [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF], since they only introduce negligible differences in the limit of small Ekman numbers we are interested in.

Linearising the system around the trivial fixed point 0, we calculate the linear growth rate of the spin-over mode

σ = √ 4 -2 -2.62 √ E -Λ/4. (6)
In agreement with [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF], the magnetic damping lowers the growth rate of the spin-over mode linearly with Λ, and the system becomes stable above a critical Elsasser number

Λ c = 4( √ 4 -2 -2.62 √ E). (7)
Some time-series for the horizontal projection of the spin-over mode amplitude, Ω H = Ω 2 1 + Ω 2 2 , found by numerical integration of the nonlinear system (3)-( 5), are shown in figure 2(a). After an exponential growth, the flow always goes towards a stable non-zero fixed point which is a stable focus. Before saturation the spin-over mode horizontal amplitude displays a small overshoot which originates from the spiral trajectory around this focus. For increasing magnetic field amplitudes, both the linear growth rate and the saturation amplitude decrease. The non-zero fixed points of (3) can be calculated explicitly. The square of the spin-over mode amplitude at saturation writes Ω 2 H = 4

γ 3 σ -4 ν nl / √ 4 -2 . ( 8
)
Note that Λ only appears in this formula through the growth rate. According to [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF], the field induced by the non-viscous spin-over mode at low Rm is a dipole with axis transverse to the imposed field, in quadrature with the rotation axis of the spin-over mode. On the dipole axis outside the spheroid, the field is purely radial and decays as

b r = |Ω H | 35 1 r 3 . (9)
Combining ( 8) and ( 9), we expect a linear decrease of the square of the induced field amplitude with Λ at fixed and E. The polar angle in the (x, y) plane of the saturated spin-over axis is determined by

φso = ± arctan Ω 2 Ω 1 = ± arctan - 2 - 2 + ( 10
)
so that the vorticity of the saturated spin-over mode is not exactly aligned with the direction of maximum stretching at -45 o of the long axis of the spheroid (e.g. φso = -42.1 o for = 0.10). Lacaze et al. [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] tested experimentally the purely hydrodynamical version of this theory and found good agreement for both linear and nonlinear stages. However, the nonlinear overshoot was not observed and the authors noticed that the experimental saturation amplitudes of the spin-over mode agree better with the maximum theoretical amplitudes attained during the overshoot than with the theoretical saturation amplitudes. We now extend these experimental results by taking the magnetic field into account.

Experiments

Our experimental set-up is an extension of the one presented in [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF] (see figures 1 and 3). The experimental parameters are R = 22.75 mm Ω 0 ∈ [0, 10π] rad s -1 and = 0.10. The imposed field B 0 ranges between 0 and 0.13 T (up to 100 times larger than in the previous set-up [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF]). It is realized with a set of watercooled copper Helmholtz coils, powered by a stabilized DCsupply. The liquid metal we use is Galinstan, a gallium-indium-tin eutectic liquid at room temperature, with ρ = 6440 kg m -3 , ν = 9.5 × 10 -5 m 2 s -1 , σ e = 2.9×10 6 S m -1 and µ = µ 0 = 4π ×10 -7 T A -1 m. According to [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF], the field induced by the non-viscous spin-over mode at low Rm is a dipole with axis transverse to the imposed field, in quadrature with the rotation axis of the spin-over mode. It is measured in the experiment by a radial Hall-probe mounted in the equator plane of the spheroid, facing the compressed direction at a polar angle of 45 o . The probe is 26.5±0.5 mm away from the center of the sphere. The hall probe and the Gaussmeter have a maximum sensibility of s = 300 µT /mV . Since the induced fields are of order O(10 -4 B 0 ), the probes are used at the limit of their sensibility. This implies careful positioning, thorough prefiltering and amplification of the recorded signals. In practice, the electric signal produced by the Hall-probe is put to zero before each experimental run. The recorded signals are prefiltered with a low-pass filter at f c = 2Hz, and amplified by a factor 50. The signals are transferred to the data-acquisition unit on the laboratory computer.

Figure 2(b) shows the experimentally recorded radial components of the induced magnetic fields for different Λ. The shapes and relative positions of the experimental records compare well with the theoretical profiles of figure 2(a), with an exponential growth (see also figure 4) and a slight overshoot preceding a saturation at a constant level. As expected, there is a gradual decrease of growth rates and saturation amplitudes with Λ. Notice again that the amplitude overshoot was not observed in the purely hydrodynamical experiments (see [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF]). At Λ = 0.121 and higher, we continue to observe a nonzero induced magnetic field, but it becomes increasingly difficult to determine a true exponential growth. These fields probably come in our experimental set-up from misalignment between the axis of rotation of the sphere, the axis of the rollers inducing the elliptical deformation, and the axis of the imposed field.

The signals also give us quantitative informations on the growth rates and saturation amplitudes. As shown in figure 5, the growth rate decreases as Λ/4, following the analytical result given by [START_REF] Lacaze | Elliptical instability of a flow in a rotating shell[END_REF]. Also shown are the theoretical dashed curves for = 0.095 and = 0.105 representing the uncertainty in . As can be observed, the experimental data are in complete agreement with the theory within this 5% error range, without any adjustment parameter. Figure 6 shows that the square of the induced field b 2 r behaves as the growth rate, in close agreement with the theory (see formula 8 and 9). Figure 6 also provides an experimental measurement of the critical Elsasser number Λ c 0.096, close to the theoretical value Λ c = 0.103.

Using [START_REF] Bayly | Three-dimensional instability of elliptical flow[END_REF], we systematically translate the magnetic field measurements to spin-over-mode amplitude and show in figure 7 the variations of the ratio ζ 1 of experimental saturation amplitudes to theoretical saturation amplitudes. The ratio ζ 1 significantly decreases with the Elsasser number, where we expected a constant value close to 1. This discrepancy between theoretical and experimental saturation values was already observed in the absence of magnetic field in [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF]. Several explanations can be provided. From a theoretical point of view, all nonlinear viscous corrections as well as possible secondary instabilities are not included in our model but may become important, especially far from the linear instability threshold. And from an experimental point of view, one can notice that measurements at a fixed -45 o angle only take into account a fraction of the spin-over amplitude when φso = -45 o , this effect being also more important far from the linear instability threshold.

Lacaze et al. [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] also remarked that the ratio ζ 2 of experimental saturation amplitudes to the maximal theoretical amplitudes attained during the overshoot remains constant over a rather large range of Ekman number. As shown in figure 7, this remain valid over a large Λ-interval. We expect that this behavior is not a coincidence, but that it could be revealed by a more sophisticated model, beyond the scope of this paper. However, and contrary to the hydrodynamical experiments by [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF] where ζ 2 ∼ 1, theoretical predictions always underestimate the experimental measurements by a factor 1.42 in our case. Possible explanations are error in the positioning of the probe (the field decreases rapidly in r -3 ), but most probably uncertainties in the value of the electrical conductivity σ e of Galinstan (values in the literature typically range between 2.3 and 3.5 × 10 6 S m -1 ). Note also that the elliptical deformation of the spheroid as well as the misalignment between the spin-over axis and the axis of maximum strain are not taken into account in equation ( 9).

From laboratory models to geophysical applications

Magnetic induction by inertial waves is of particular interest in geo-and astrophysical applications. For instance, Kerswell and Malkus [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF] have suggested that Io's magnetic field is induced from Jupiter's magnetic field by tidally driven inertial waves resonance, without dynamo action. However, our previous results derived in the limit of dominant magnetic diffusion (i.e. low Rm) and for the laminar spin-over mode (i.e. at rather large E) cannot apply directly to planetary configurations, corresponding to the limit of small E, large Rm and probably large wavenumbers. As can be seen in the visualizations of figure 8, the flow can then become increasingly complex, especially at small scale, and an extension of our analysis is necessary. Fortunately, a more general expression of the growth rate of the tidal instability, independent of the geometry of the flow, can be derived using the so-called local approach. Our goal here is not to give the exact expression of the growth rate of the various modes explicitly excited in a given planet but to determine an analytical expression able to describe the power dependence of the growth rate on all dimensionless numbers and to determine an order of magnitude of the various prefactors. The local approach is based on the inviscid short-wavelength Lagrangian theory developed in [START_REF] Bayly | Three-dimensional instability of elliptical flow[END_REF] and [START_REF] Craik | Evolution of wavelike disturbances in shear flows -a class of exact solutions of the Navier-Stokes equations[END_REF], then generalized in [START_REF] Friedlander | Instability criteria for steady flows of a perfect fluid[END_REF] and [START_REF] Lifschitz | Local stability conditions in fluid dynamics[END_REF]. There, perturbations are assumed to be sufficiently localized in order to be advected along flow trajectories and are searched as local plane waves of the form (u, p) = u(t), p(t) e ik(t)•x , [START_REF] Malkus | Precession of the earth as the cause of geomagnetism[END_REF] where k(t) is the time-dependent wave vector and x the position vector. This method has been applied to the elliptical instability by Le Dizès [START_REF] Dizès | Three-dimensional instability of a multipolar vortex in a rotating flow[END_REF]. Here, we extend his results by taking into account the induction equation and the Lorentz force in the limit of small Elsasser number (e.g. Λ ≤ O( )) in the presence of an imposed vertical magnetic field B 0 , looking for a perturbed magnetic field under the same wave form

b = b(t)e ik(t)•x . ( 12
)
Details of the analysis are given in the appendix. Notice that in the following, we do not consider the limit of small magnetic Reynolds number anymore; hence, the magnetic field is made dimensionless using the amplitude of the imposed field B 0 rather than Rm B 0 as in the previous sections. MHD equations are solved analytically using a perturbative expansion in eccentricity , supposing that the Lorentz force is of order . In this context, equations for fluid motions at order 0 are similar to the purely hydrodynamical case. Through the Lorentz force, the magnetic field induces a correction in the fluid equation at order 1, hence a correction in the growth rate of the instability.

Using the 2D base flow U b given by ( 1), which corresponds to a stationary tidal deformation, we find a non viscous growth rate

σ nv = 9 16 - k 4 Λ 4(Rm 2 + k 4 ) , ( 13
)
where k is a constant equal to the norm of the wave vector k(t) at leading order in (see equation 29 in appendix). The viscous damping rate resulting from the boundary layer can be estimated following [START_REF] Lacaze | Elliptical instability in a rotating spheroid[END_REF], and induces a supplementary correction of the order O(E 1/2 ). Notice that in the limit of small magnetic Reynolds number, one immediately finds the linear Joule damping -Λ/4 determined in section 3. The present result generalizes the validity of this scaling to all possible excited modes of the elliptical instability. Note also that the numerical factor before is different from ( 6), but remains of the same order of magnitude in the relevant limit of small . The previous result can still not be directly applied to the case of Io, where the elliptical deformation is not stationary. Indeed, as explained for instance in [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF], Io is almost synchronized in its revolution around Jupiter, but orbital resonances with Europa and Ganymede force it to follow a slightly elliptical orbit of eccentricity 0.004. As a result, the tidal bulge raised by Jupiter, of magnitude ∼ 6 × 10 -3 , does not rotate exactly at the same velocity as Io's spin, but oscillates back and forth across Io's body with a typical angular velocity 1 -β cos(t), where β ∼ 0.008 is twice the eccentricity of Io's orbit and where time is made dimensionless using Io's spin velocity. In this case, the base flow in Io's core at first order in β writes (see Appendix)

U b = -y + β cos(t)(sin(2t)x -cos(2t)y), (14)
x -β cos(t)(cos(2t)x + sin(2t)y]), 0 , and the growth rate of the elliptic instability writes

σ = 17 64 (β ) 2 - 576 289 Λ 2 Rm 2 k 4 (Rm 2 + 4k 4 ) 2 - 3 4 k 4 Λ Rm 2 + 4k 4 . ( 15
)
Formula ( 15) is closely related to [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF], where the eccentricity in the case of a stationary tidal deformation has been replaced by the product of the tidal bulge times the amplitude of the perturbation β . In particular at small Rm, we once again end up at first order with a Joule damping linear in Λ. As mentioned before, surface viscous effects induce a correction to this formula of order O(E 1/2 ) that could be explicitly determined. This is not done here, since the interest of formula ( 15) is to determine the relevant power law dependence on all dimensionless parameters (i.e. , Rm, Λ, E) as well as the order of magnitude of the various prefactors. In the following, we use for illustration the explicit values shown in [START_REF] Tilgner | Precession driven dynamos[END_REF] as well as the viscous correction 8.8E 1/2 determined in [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF] for the first excited resonance in Io's configuration, but all our conclusions remain valid using prefactors of the same order of magnitude. Formula (15) allows us to compute the order of magnitude of the growth rate of the elliptical instability in Io's core. We take as typical values an imposed magnetic field by Jupiter B 0 = 1850nT , and for Io's core R = 900km, 2π/Ω 0 = 1.77days, and ν = 10 -6 m 2 s -1 , σ e = 4 × 10 5 Sm -1 , ρ = 12000kgm -3 , consistent with a Fe/Fe-S composition. Then, Rm = 1.7 × 10 7 , E = 3.0 × 10 -14 , Λ = 2.8 × 10 -6 and ( 15) implies that none of the elliptical modes is significantly affected by Joule damping. The typical growth rate of the tidal instability in Io is about 0.014years -1 , suggesting rapid largescale variations in its core flows. Supposing that the spin-over mode still has an important component in Io's core (see figure 8 and appendix), its saturation amplitude would be about Ω H = 0.096Ω 0 according to [START_REF] Kerswell | Elliptical instability[END_REF]. The corresponding induced magnetic field would be a dipole aligned with the spinover axis and of typical amplitude b r ∼ sin(0.096) B 0 = 178nT , as derived from [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF] in the relevant limit of large magnetic Reynolds number. We expect this field to fluctuate on rapid times ranging between the rotation period of 1.77 days and the typical time given by the growth rate of the instability of 72 years. The question then remains whether this field is measurable outside the core. Taking as in the Earth a typical mantle conductivity of 0.1 -1S.m -1 and considering the short period signal of about 1 day, we find a skin length of 200 -600km. In the Earth, this low value compared to the typical depth of the mantle means that signals coming from the elliptical instability will be totally filtered. This will not be the case in Io. Hence, continuous field measurements of the ambient field in the vicinity of Io would allow to discriminate between its internal and atmospheric origins, an issue raised since the first punctual measurements provided by the Galileo mission (e.g. [START_REF] Kivelson | Magnetized or unmagnetized: Ambiguity persists following Galileo's encounters with Io in 1999 and[END_REF]).

Conclusion

In this paper, we have studied the effects of an imposed magnetic field on the elliptical instability in spheroids. By combining theoretical elements of previous works [START_REF] Lacaze | Magnetic field induced by elliptical instability in a rotating spheroid[END_REF], [START_REF] Thess | Transition from two-dimensional to three-dimensional magnetohydrodynamic turbulence[END_REF], we have extended the nonlinear system governing the dynamics of the spin-over mode to include simultaneously the magnetic and viscous damping. We have shown theoretically and confirmed experimentally that the linear growth rate of the instability as well as the square amplitude of the induced magnetic dipole fall down linearly with the Elsasser number (i.e. with the square of the imposed magnetic field), with good agreement regarding predicted and measured prefactors. These conclusions have then been extended to all possible resonances of the elliptical instability using a short-wavelength Lagrangian approach. Applied to the specific case of Io in the magnetic field of Jupiter, we conclude that despite the viscous and Joule damping, a tidal instability is more than probable in the Jovian moon's core and induces in the core a relatively important field of about 10% of the ambient value. In addition to the magnetospheric interactions with Jupiter [START_REF] Saur | Plasma interaction of Io with its plasma torus[END_REF], we thus conclude from purely magnetohydrodynamical considerations that the elliptical instability provides a significant and non-stationary contribution to the magnetic field measured in the vicinity of Io, as first suggested by [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF]. Continuous measurements in Io's vicinity should allow to discriminate between internal and external magnetic signatures.

This research was funded by a PHD-scholarschip of the Ministère de la Recherche Française, and by the ANR-IMAGINE (ANR-07BLAN-0182). We are also grateful to H. Bouchiat who kindly provided the Helmholtz coils.

Appendix: WKB analysis of Io's tidal instability

The now classical application of the short-wavelength "WKB" theory to inviscid fluids was developed in [START_REF] Bayly | Three-dimensional instability of elliptical flow[END_REF] and [START_REF] Craik | Evolution of wavelike disturbances in shear flows -a class of exact solutions of the Navier-Stokes equations[END_REF], generalized in [START_REF] Friedlander | Instability criteria for steady flows of a perfect fluid[END_REF] and [START_REF] Lifschitz | Local stability conditions in fluid dynamics[END_REF], and summarized for the elliptical instability in [START_REF] Dizès | Three-dimensional instability of a multipolar vortex in a rotating flow[END_REF]. It consists in looking for a perturbed solution of the full equations of motion under the form of a plane wave along the streamlines of the base flow. In our case, we thus look for a perturbed solution of the non-dimensional system

∇.u = 0, (16)
d t u + (u.∇)u = -∇p + Λ Rm (∇ × b) × b, (17)
∇.b = 0, (18)

d t b + (u.∇)b = (b.∇)u + 1 Rm ∇ 2 b, (19)
under the form

u = U b + u(t)e ik(t)•x , (20) p
= P b + p(t)e ik(t)•x , (21) b
= B 0 + b(t)e ik(t)•x , (22)
along the streamlines of the base flow described by

dx dt = U b , (23)
where U b stands for the two-dimensional base flow, P b for the corresponding pressure field, B 0 = (0, 0, 1) for the (non-dimensional) imposed vertical magnetic field and x for the position vector. The linearised MHD equations then write k.u = 0, (24)

d t u + iu(d t k.x) + i(U b .k)u + (u.∇)U b = -ikp + Λ Rm (ik × b) × B 0 , (25)
k.b = 0, (26)

d t b(t) + ib(d t k.x) + i(U b .k)b = (b.∇)U b + ( B 0 .k)u - k 2 Rm b, (27)
The velocity and induction equations can be decoupled in space and time to give an equation for the wave vector only

d t k.x + U b .k = 0. ( 28
)
Linearized equations are then solved analytically using a perturbative expansion in the small parameter (i.e. the eccentricity in our case), supposing that the Elsasser number is of order 1. In this context, equations for fluid motions at order 0 are similar to the purely hydrodynamical equations, and the Lorentz force only induces a correction at order 1. Technically, the easiest way to solve the MHD equations in our case is to use the vertical velocity u z and the vertical vorticity

W z = ∂ x u y -∂ y u x = i(k x u y -k y u x )
of the perturbed field as unknowns, as well as the vertical component b z of the perturbed magnetic field and the corresponding magnetic vertical vorticity

C z = i(k x b y -k y b x ).
The resolution is then straightforward.

The study of the 2D base flow U b given by [START_REF] Leweke | Cooperative elliptic instability of a vortex pair[END_REF], which corresponds to a stationary tidal deformation, closely follows the results already presented by Le Dizès [START_REF] Dizès | Three-dimensional instability of a multipolar vortex in a rotating flow[END_REF]. From (28), one immediately finds the wave vector

k(t) = k sin(a) √ A cos(χt), sin(a) √ A sin(χt), cos(a) , ( 29
)
where k is a constant, A = (1 + )/(1 -) is the ellipticity, χ = √ 1 -2 , and a is the angle between the flow rotation axis and the wavevector. Equations for fluid motions at order 0 give the expression of u at order 0 with a temporal frequency f f = ±2 cos(a),

whereas the linearised induction equation immediately gives the expression of b at order 0. According to [START_REF] Dizès | Three-dimensional instability of a multipolar vortex in a rotating flow[END_REF], an elliptical instability is possible if the forcing terms due to the elliptical deformation oscillate at the same frequency as the inertial wave, which means in our case f = 1. Then, at order 1 in , inertial waves resonate implying an exponential growth rate of the elliptical instability [START_REF] Dizès | Three-dimensional instability of a multipolar vortex in a rotating flow[END_REF]

σ nv = 9 16 - k 4 Λ 4(Rm 2 + k 4 ) . ( 31
)
This result is not directly applicable to the case of Io, where the elliptical deformation is not stationary. Indeed, Io is almost synchronized with Jupiter. It means that, in an isolated Jupiter-Io system, Io's spinning and orbital periods would be exactly equal: Io would always present the same face to Jupiter, and the tidal bulge would rotate at exactly the same frequency as Io. However, well-known orbital resonances with the other Galilean satellites maintain a 0.004 eccentricity in Io's orbit. The equality of orbital and spinning velocities is only true on average: in reality the orbital angular velocityhence the tidal bulge angular velocity -varies periodically with the orbital radius around this mean. Focusing on the first harmonic of this oscillation, the orbital angular velocity in the absolute frame of reference writes 1 -β cos(t), where β ∼ 0.008 is twice the eccentricity of Io's orbit and where time is made dimensionless using Io's (constant) spin velocity. As demonstrated by Kerswell and Malkus [START_REF] Kerswell | Tidal instability as the source for Io's magnetic signature[END_REF], the fluid's laminar response in Io's core driven by this tidally distorted mantle motion corresponds in the bulge frame to the simple elliptical flow

U b B.F. = -(1 + )β cos(t)Y, ( 1
-)β cos(t)X, 0 , (32)
which is an exact nonlinear solution to the incompressible Navier-Stokes equations of motion for any finite viscosity in the spheroid of equation X 2 /(1+ ) + Y 2 /(1 -) + Z 2 = 1. A simple change of frame then gives the base flow in the absolute frame of reference at first order in β

U b = -y + β cos(t)(sin(2t)x -cos(2t)y), (33)
x -β cos(t)(cos(2t)x + sin(2t)y]), 0 .

For a given initial position (R, 0), streamlines are described by

x(t) = R cos(t) + β R 2 (1 -cos(2t)), (34)
y(t) = R sin(t) - β R 2 sin(2t), (35)
(note that the results of the WKB theory do not depend on the chosen initial position along a closed trajectory). The solution to the wave vector equation along this streamline then writes

k(t) = k sin(a) cos(t + φ) + β /2(cos(2t -φ) -cos(φ)), (36)
sin(a) sin(t) + β /2(sin(2t -φ) + sin(φ)), cos(a) , Note that the resonance condition f = 1/2 corresponds to the resonance condition for the closely related precession instability, where the spin-over mode is also known to be excited [START_REF] Malkus | Precession of the earth as the cause of geomagnetism[END_REF]. This validates the evaluation of the spin-over induction performed at the end of section 5.

H A liquid metal in a deformable spheroidal cavity rotates at Ω 0 . A strong magnetic field B 0 is imposed along the rotation axis. Fixed rollers induce an elliptical deformation of the streamlines. Also shown here is the horizontal projection of the spin-over mode, corresponding to a transverse solid body rotation Ω H in the stretched direction (dashed arrow), which tilts the rotation axis of the fluid to Ω. In the limit of low magnetic Reynolds number, Ω H induces in the compressed direction a dipolar magnetic field b r ∼ Ω H , which is measured by a Hall probe. Ekmann number E = 10 -5 and increasing values of (purely hydrodynamical experiment). As suggested by [START_REF] Lacaze | Elliptical instability of a flow in a rotating shell[END_REF], the relevant parameter to describe the dynamics of the elliptical instability is α = E 1/2 / . Decreasing α from 0.11 to 0.053, the flow becomes more and more complex, especially at small scale, but the spin-over mode remains present at large scale. The same behavior is expected to remain valid at the planetary scale, for instance in Io's core where α ∼ 0.0036.

Figure 1 :

 1 Figure 1: Sketch of the set-up, side and top-view (see also Lacaze et al.A liquid metal in a deformable spheroidal cavity rotates at Ω 0 . A strong magnetic field B 0 is imposed along the rotation axis. Fixed rollers induce an elliptical deformation of the streamlines. Also shown here is the horizontal projection of the spin-over mode, corresponding to a transverse solid body rotation Ω H in the stretched direction (dashed arrow), which tilts the rotation axis of the fluid to Ω. In the limit of low magnetic Reynolds number, Ω H induces in the compressed direction a dipolar magnetic field b r ∼ Ω H , which is measured by a Hall probe.

Figure 2 :

 2 Figure 2: (a) Theoretical nonlinear temporal evolution of the horizontal projection of the spin-over mode amplitude Ω H = (Ω 2 1 + Ω 2 ) 1/2 , for various values of the Elsasser number Λ. Ekmann number and eccentricity are fixed, E = 8.53 × 10 -5 , = 0.10. Calculations started from the initial state Ω 1 = 10 -3 , Ω 2 = -10 -3 , Ω 3 = 0, which is the linearly unstable spin-over mode with small amplitude. The arrow on the right side, indicate the saturation level of the slowly growing spin-over mode horizontal amplitude at Λ = 0.095. The critical Elsasser number is Λ c = 0.103 for this parameter set. (b) Typical recorded magnetic field-signals for varying Elsasser number. Ekmann number and eccentricity are fixed E = 8.53 × 10 -5 , = 0.10 ± 0.005 . The experiments agree with the theoretical profiles of figure 2(a).

Figure 3 :

 3 Figure 3: Picture of the experimental set-up. Large watercooled Helmholtz coils provide a homogenous magnetic field up to B 0 = 0.1 T . Induced fields are measured with a radial Hall probe.

Figure 4 :

 4 Figure 4: Logarithm of the induced magnetic field signal (full line) at E = 8.53 × 10 -5 , = 0.100 ± 0.005, Λ = 0.01. The slope of the linear fit (dashed line) provides the initial linear growth rate.

Figure 5 : 2 ΛFigure 6 :

 526 Figure5: Linear growth rates σ as a function of Elsasser number Λ. E = 8.5 × 10 -5 , = 0.100 ± 0.005 . The experimental measurements (•) are in good agreement with the theoretical values for = 0.10 (soft line). Also shown are the theoretical (dashed) curves for = 0.095 and = 0.105, representing the uncertainty in ; note however that there is no adjusting parameter in the comparison between theory and experiment. Growth was no longer exponential beyond Λ ≥ 0.121.

2 Figure 7 :

 27 Figure 7: Ratios of the experimentally observed saturation amplitude, to the theoretical saturation amplitudes ζ 1 (•) and to the maximum amplitude attained during the overshoot ζ 2 (+), as a function of the Elsasser number Λ. E = 8.5 × 10 -5 , = 0.10.

Figure 8 :

 8 Figure 8: Kalliroscope visualization of the elliptical instability for a fixedEkmann number E = 10 -5 and increasing values of (purely hydrodynamical experiment). As suggested by[START_REF] Lacaze | Elliptical instability of a flow in a rotating shell[END_REF], the relevant parameter to describe the dynamics of the elliptical instability is α = E 1/2 / . Decreasing α from 0.11 to 0.053, the flow becomes more and more complex, especially at small scale, but the spin-over mode remains present at large scale. The same behavior is expected to remain valid at the planetary scale, for instance in Io's core where α ∼ 0.0036.

Jupiter. The planet, satellites and magnetosphere. Edited by F. Bagenal, T. E. Dowling, W. B. McKinnon. Cambridge planetary science, Vol. 1, Cambridge, UK: Cambridge University Press, ISBN 0-521-81808-7, 537-560.

where k, a and φ are constant. At order zero in β , the system can be reduced to a single equation for u z

whose solution writes

where c 1 , c 2 are constant and f is the frequency determined as a function of the wave vector, i.e. f = 2 cos(a). At order 1, we allow a long term variation of the solution at order zero, i.e.

where σ is the growth rate of the instability. The system then reduces to the same type of equation as (37), with a forcing term directly coming from the first order terms in the base flow. According to (33), terms of type cos(t) sin(2t) and cos(t) cos(2t) arise here, and the equation at order 1 schematically reads

, e -it/2 , e 3it/2 , e -3it/2 ). (40)

Solvability conditions then imply

2π/f 0 F (e it/2 , e -it/2 , e 3it/2 , e -3it/2 )e if t = 0, (41)

which validate the whole asymptotic approach by ensuring that forcing terms are not secular. This system with unknown c 1 , c


Sandrine Prat
email: sandrine.prat@mnhn.fr

Emergence of the genus Homo: from concept to taxonomy Emergence du genre Homo : du concept à la taxonomie

Keywords: Homo habilis, Homo rudolfensis, taxonomie, définition Homo habilis, Homo rudolfensis, taxonomy, definition

The main goal of this paper is to present an overview of hypotheses concerning early Homo specimens and to discuss the definition of the genus Homo in the light of recent discoveries. For some authors, all the specimens attributed to early Homo belong to one unique species. For others, this group (Homo habilis sensu lato) is heterogeneous and could be splitted into two groups: Homo habilis and Homo rudolfensis. Some researchers have also proposed to put the species habilis and rudolfensis into the genera Australopithecus or Kenyanthropus. Therefore, two scenarios concerning first humans seem to emerge. An emergence of the genus Homo, as early as 2.8 Ma, with Homo sp. specimens and the species Homo habilis and Homo rudolfensis, another at 1.9 Ma with Homo ergaster. According to the recent archaeological and palaeoanthropological discoveries, these criteria often considered to be crucial for the definition of the genus Homo, as the cranial capacity, the human-like manipulative abilities, the habitual erect posture and bipedal gait, the language ability and the capacity to make tools are now obsolete.

Résumé

Le but de cet article est de présenter une vue d'ensemble des hypothèses concernant les premiers représentants du genre Homo et de discuter de la définition notre genre à la lumière des découvertes récentes. Pour certains auteurs, tous les spécimens attribués aux premiers représentants du genre Homo appartiennent à une unique espèce. Pour d'autres auteurs, ce groupe (Homo habilis sensu lato) est hétérogène et pourrait être séparé en deux groupes : Homo habilis et Homo rudolfensis. Certains chercheurs ont également proposé de placer les espèces habilis et rudolfensis dans les genres Australopithecus ou Kenyanthropus. Par conséquent, deux scénarios pour l'émergence du genre Homo semblent se dessiner. Une émergence du genre Homo, dès 2,8 Ma, avec les spécimens Homo sp. et les espèces Homo habilis et Homo rudolfensis, ou à 1,9 Ma avec Homo ergaster. D'après les récentes découvertes archéologiques et paléoanthropologiques, les critères souvent considérés comme cruciaux pour la définition du genre Homo, comme la capacité crânienne, les capacités de manipulation de type humain, la posture érigée et la démarche bipède, la capacité de langage et la capacité de fabriquer des outils sont désormais caduques.

Introduction

For nearly 70 years, Homo erectus (formerly Pithecanthropus erectus), whose first remains were discovered at Java (Indonesia) in 1890 by Eugène Dubois, were considered to be the oldest specimens of the genus Homo. In 1964, Louis Leakey, Phillip Tobias and John Napier, by discovering and naming the species Homo habilis, from individuals dating back to 1.75 million years (Ma), changed the picture of our family tree [START_REF] Leakey | A new species of the genus Homo from Olduvai Gorge[END_REF]. Our oldest ancestors are now considered as african and are dated nowadays to 2.8 Ma with the discovery of the hemi-mandible from Ledi Geraru site [START_REF] Villmoare | Early Homo at 2.8 Ma from Ledi-Geraru, Afar, Ethiopia[END_REF]. The debate that ensued, and which is still ongoing, concerns the number of species within the early taxa of the genus Homo and therefore the definition of our genus. The purpose of this paper is to present an overview of problematics concerning the early Homo specimens and to discuss about the definition of the genus Homo in the light of recent discoveries. Three species currently seem to be subject of consensus: habilis, rudolfensis and ergaster. However, two major debates concern these fossil groups. One relates to early Homo, the other to the definition of Homo erectus and on the possible inclusion of african and caucasian individuals in this species, which is not the focus of this paper. These two debates refer to a recurrent discussion in palaeoanthropology concerning the existence of one or several species within fossil hominin groups. Two schools of thought diverge on this point. The first, called "splitting" leads to multiply the number of taxa discovered. The second, called "lumpers", reduces the number of fossil species to a minimum. It has to be noticed that the definition of a fossil species is highly subjective since, due to the nature of the object studied: the fossil, it cannot follow the notion of species defined by [START_REF] Mayr | Speciation phenomena in birds[END_REF][START_REF] Mayr | The species, category, taxon and population[END_REF] as 'species are groups of interfertile natural populations that are reproductively isolated from other similar groups' (Mayr, 1987: 306).

Some historical milestones

As early as 1871, Darwin already mentioned in "the Descent of Man" [START_REF] Darwin | The descent of man and selection in relation to sex[END_REF] that human origin must have been in Africa. It was with the prospect to discover the oldest human remains in Africa that Louis and Mary Leakey have undertaken research in the Olduvai Gorge in Tanzania. These fossils were allocated to the genus Homo and named Homo habilis in 1964 [START_REF] Leakey | A new species of the genus Homo from Olduvai Gorge[END_REF] after a lot of reconstructions and comparisons. It was the subject of much criticism. These concerned in particular the nature of the anatomical characters used and the characteristics of the metric variability that made it possible to distinguish the species Homo habilis from both Australopithecus and Homo erectus, leading some authors (e.g. [START_REF] Holloway | Cranial capacity of the hominine from Olduvai Bed I[END_REF][START_REF] Robinson | Homo habilis" and the Australopithecines[END_REF][START_REF] Brace | Tooth measurements and the rejection of the taxon "Homo habilis[END_REF] to consider Homo habilis as an artificial cluster of Australopithecus and Homo erectus specimens. Despite these discussions, the concept of the existence of the species Homo habilis was definitively validated by the scientific community in 1968 following the discovery of the OH 24 skull in Olduvai. The original description of Homo habilis emphasized the similarity of its features to Homo sapiens and the increase in brain volume compared to australopithecines. By integrating the species habilis into the genus Homo, it became necessary to revise the diagnosis of this genus established by [START_REF] Gros | The fossil evidence for human evolution[END_REF] as a genus of the family Hominidae, distinguished by a high cranial capacity with an average value of over 1100 cm 3 but with a range of variation from about 900 to 2000 cm 3 . The discoverers of the species Homo habilis therefore proposed this new definition: "... the cranial capacity is very variable but is, on the average, larger than the range of capacities of members of the genus Australopithecus, although the lower part of the range of capacities in the genus Homo overlaps with the upper part of the range in Australopithecus; the capacity is (on the average) large relative to body-size and ranges from about 600 c.c. in earlier forms to more than 1600 c.c...." (Leakey et al., 1964, p. 7). In 1972, Richard and Meave Leakey found nearly three hundred cranial fragments of a single individual on the surface, which, when completely reconstructed, became the skull KNM-ER 1470. Its chronological dating was established at 2.42 Ma at the time of discovery [START_REF] Fitch | 40 Ar/3 9 Ar dating of the KBS tuff in Koobi Fora Formation, East Rudolf, Kenya[END_REF] and later re-evaluated at 1.85 Ma [START_REF] Feibel | Stratigraphic context of fossil hominids from the Omo Group deposits : Northern Turkana Basin, Kenya and Ethiopia[END_REF]) and now to 2.05 +/-0.034 Ma [START_REF] Joordens | Improved age control on early Homo fossils from the upper Burgi Member at Koobi Fora, Kenya[END_REF]. Its specific attribution was (and still is) subject of debate. Richard Leakey initially placed this specimen in the taxon Homo sp. (Leakey, 1976a), but later considered that it belongs to Homo habilis (Leakey, 1976b). After a re-examination of this specimen, Alexeev proposed in 1978 a new species: Homo rudolfensis. The diagnosis and description of this species was published in Russian [START_REF] Alexeev | Paleoantropologia Zemnogo shara i formirosanie chelovetceskikh ras Paleolit (The Palaeonthropology of the World and the forming of the Human races. Paleolithic)[END_REF] and later in English [START_REF] Alexeev | The origin of Human race[END_REF], based on the Kenyan specimen KNM-ER 1470. In this paper, the author mentions neither the type locality, nor the distribution, nor the hypodigm of this taxon, and puts forward only few distinctive anatomical characters. This species was not recognized by the scientific community until the early 1990s, partly due to the Bernard Wood's work [START_REF] Wood | Origin and evolution of the genus Homo[END_REF]. The systematic validity of this species, with respect to the rules of international nomenclature, was and remains the subject of controversy [START_REF] Kennedy | Is "Homo rudolfensis" a valid species?[END_REF][START_REF] Wood | Homo rudolfensis" Alexeev, 1986-fact or phantom[END_REF].

Since the first discoveries in the 1960s, more than 200 remains, mainly cranial and dental elements, have been attributed to early Homo (habilis, rudolfensis and Homo sp. older than 2 Ma). These remains are mostly located at Hadar and Omo Valley in Ethiopia, West and East Turkana and Chemeron in Kenya, Olduvai in Tanzania, Uraha in Malawi and Drimolen, Sterkfontein and Swartkrans in South Africa (Figure 1). From a chronological point of view, the early Homo individuals range from 2.8 Ma to 1.65 Ma.

INSERT FIGURE 1 : Map

3. Current issues. One or more species? Homo or Australopithecus?

Since the mid-1980s, the development of statistical methods (particularly multivariate) and cladistics has contributed to questioning the certainties concerning the unity of Homo habilis sensu lato and its generic allocation. The debate has indeed refocused on the problem of intra-specific variability within this taxon. Are we dealing with one or two palaeontological species? Two schools of thought illustrating the different interpretations were put forward. For some, the variability is consistent with intra-specific variability, i.e. all these specimens belong to the same group (1 species). Conversely, for others, two groups can be determined from these specimens that are both metrically and morphologically distinct (2 species).

3.1. One or two species? 3.1.1. One species: Homo habilis sensu lato For some authors (e.g. [START_REF] Howell | Hominidae[END_REF][START_REF] Blumenschine | Late Pliocene Homo and hominid land use from western Olduvai gorge,Tanzania[END_REF][START_REF] Tobias | Position et rôle des Australopithécinés dans la phylogénèse humaine, avec étude particulière d'Homo habilis et des théories controversées avancées à propos des premiers Hominidés fossiles de Hadar et de Laetolil[END_REF][START_REF] Tobias | The brain of Homo habilis : a new level of organization in cerebral evolution[END_REF][START_REF] Tobias | Olduvai gorge Vol 4a et 4b. Homo habilis : skulls, endocasts and teeth[END_REF][START_REF] Tobias | Encore Olduvai[END_REF], all the specimens attributed to early Homo belong to one unique species. The metric and morphological variability between these specimens would be compatible with an intra-specific variability. Homo habilis is thus considered as a polymorphic species. The specimens grouped in a single species Homo habilis sensu lato show a reduction of the masticatory apparatus, with an insertion of the masticatory muscles less important compared to australopithecines. The facial and alveolar regions are less prognathic, the anterior pillar of the face as well as the frontal trigone (observed in Paranthropus) and the maxillary sulcus are absent. The mandible is parabolic in shape, the mandibular body is less high and robust, and the chin prominence is slightly developed in some individuals. The temporo-mandibular articulation is close to the third molar, which is related to the low projection of the face. The nuchal region is relatively smaller than in australopithecines. The foramen magnum is located more anteriorly. Concerning the dentition, the upper anterior teeth (incisors, canines) are not aligned and large, unlike those of specimens attributed to Paranthropus. The upper molars are small. The upper third molar is the smallest while the lower third molar is sometimes larger than the second. The lower molars are characterized by an elongation in length (mesiodistal diameter) and a narrowing in width (vestibulo-lingual diameter). Other distinctions are essentially related to brain growth and concern the development of the cranial vault (especially the development of the parietal and upper temporal regions). Homo habilis is characterized by a cranial capacity between 509 and 824 cm3 (Holloway, 2004;[START_REF] Spoor | Reconstructed Homo habilis type OH 7 suggests deep-rooted species diversity in early Homo[END_REF], which is on average greater than that of australopithecines. The differences are important between cranial specimens allocated to early Homo and Paranthropus. However, they are less marked with the A. africanus and A. afarensis individuals. Some remains have been attributed either to Homo habilis or to Australopithecus africanus or another hominin taxa, depending on the authors. This is notably the case for the South African specimen Stw 53 (e.g. [START_REF] Kuman | Stratigraphy, artefact industries and hominid associations for Sterkfontein, Member 5[END_REF][START_REF] Curnoe | Description, new reconstruction, comparative anatomy, and classification of the Sterkfontein Stw 53 cranium, with discussions about the taxonomy of other southern African early Homo remains[END_REF][START_REF] Curnoe | A review of early Homo in southern Africa focusing on cranial, mandibular and dental remains, with the description of a new species (Homo gautengensis sp. nov.)[END_REF][START_REF] Zanolli | Dental data challenge the ubiquitous presence of Homo in the Cradle of Humankind[END_REF].

Two species: Homo habilis sensu stricto and Homo rudolfensis

For other authors, this group (Homo habilis sensu lato) is heterogeneous and could be separated into two groups: Homo habilis and Homo rudolfensis (e.g. [START_REF] Alexeev | Paleoantropologia Zemnogo shara i formirosanie chelovetceskikh ras Paleolit (The Palaeonthropology of the World and the forming of the Human races. Paleolithic)[END_REF][START_REF] Alexeev | The origin of Human race[END_REF][START_REF] Chamberlain | Early hominid phylogeny[END_REF][START_REF] Donnelly | How different are KNM-ER 1470 and KNM-ER 1813? A multivariate comparaison using randomization methods[END_REF][START_REF] Ferguson | A new species of the genus Homo (Primates : Hominidae) from the Plio/Pleistocene of Koobi Fora, Kenya[END_REF]Lieberman and Pilbeam, 1988;Lieberman et al., 1996;[START_REF] Prat | Problème taxinomique des premiers représentants du genre Homo. Etudes crâniennes des individus d'Olduvai et de Koobi Fora[END_REF][START_REF] Prat | Origine et taxinomie des premiers représentants du genre Homo[END_REF][START_REF] Prat | Les premiers représentants du genre Homo, en quête d'une identité. Apports de l'étude morphologique et de l'analyse cladistique[END_REF][START_REF] Rightmire | Variation among early Homo crania from Olduvai Gorge and the Koobi Fora region[END_REF][START_REF] Stringer | The credibility of Homo habilis[END_REF][START_REF] Walker | The hominids of East Turkana[END_REF][START_REF] Ward | Koobi Fora Project Vol 4 : Hominid Remains From Koobi Fora[END_REF][START_REF] Wood | Origin and evolution of the genus Homo[END_REF]. Different parameters were used: morphological characters, variation in endocranial volume, degree and pattern of variation in craniofacial measurements and cranial angles. This view is followed by the majority of researchers, although there is no consensus, concerning the specimens allocated to each taxon.

The table presented here (Table 1) is based on a literature review of the work of different researchers [START_REF] Chamberlain | A taxonomic review and phylogenetics analysis of Homo habilis[END_REF]Lieberman et al., 1996;[START_REF] Prat | Problème taxinomique des premiers représentants du genre Homo. Etudes crâniennes des individus d'Olduvai et de Koobi Fora[END_REF][START_REF] Prat | Origine et taxinomie des premiers représentants du genre Homo[END_REF][START_REF] Rightmire | Variation among early Homo crania from Olduvai Gorge and the Koobi Fora region[END_REF][START_REF] Skelton | Phylogenetic analysis of early hominids[END_REF][START_REF] Skelton | Evolutionary relationships among early hominids[END_REF][START_REF] Strait | A reappraisal of early hominid phylogeny[END_REF][START_REF] Tobias | Olduvai gorge Vol 4a et 4b. Homo habilis : skulls, endocasts and teeth[END_REF][START_REF] Ward | Koobi Fora Project Vol 4 : Hominid Remains From Koobi Fora[END_REF]. This table therefore offer a synthetic view, reflecting the greatest consensus on anatomical distinctions. It should be noted, however, that there is some divergence with respect to the individuals that constitute each taxon. Some trait assignments are related to the developmental age of the individual or are highly dimorphic in extant great apes (see for discussion and data [START_REF] Prat | Origine et taxinomie des premiers représentants du genre Homo[END_REF]).

INSERER Table 1

Thirty-two morphological traits allow to distinguish these two species habilis and rudolfensis (Table 1). The maxillofacial complex plays a major role in this differentiation (e.g. [START_REF] Prat | Problème taxinomique des premiers représentants du genre Homo. Etudes crâniennes des individus d'Olduvai et de Koobi Fora[END_REF][START_REF] Prat | Les premiers représentants du genre Homo, en quête d'une identité. Apports de l'étude morphologique et de l'analyse cladistique[END_REF][START_REF] Rightmire | Variation among early Homo crania from Olduvai Gorge and the Koobi Fora region[END_REF]. They are discriminated by the different proportions of the upper and middle face (identical width in Homo habilis), the alveolar prognathism (smaller in Homo rudolfensis) and the size of the zygomatic bone (smaller in Homo habilis). Among the latter, a zygomatico-maxillary fossa can be observed in some specimens. The orientation of the zygomatic process of the maxillary bone is vertical, when the skull is positioned in the Frankfurt reference plane. Distinctions can also be seen in the orbital region and superior nasal bones. The shape of the orbits is more square in Homo rudolfensis specimens (notably KNM-ER 1470) and the superior orbital margin is ovoid. The horizontal distance between the tempo-mandibular joint and the upper M2/M3 is small in Homo habilis, which is related to the low projection of the face for these individuals. Anatomical differences can also be observed between the mandibular remains attributed to early Homo, but it should be noticed that only two main mandibles KNM-ER 1802 and UR 501 are related to the species Homo rudolfensis. The two species differ mainly in the shape of the chin region, which is sub-rectangular in Homo rudolfensis specimens. In the latter, the alveolar planum is large. Concerning the upper dentition, the anterior teeth (incisors, canines) are smaller in Homo habilis. The occlusal surface of the first molar is small. The premolars generally have two roots, unlike Homo rudolfensis, where three roots are observed. In the lower dentition, the anterior dentition is also smaller in Homo habilis specimens. The premolar of these individuals generally has only one root, unlike Homo rudolfensis, where the first premolar can either have two roots or a bifid root and the second premolar has two roots. The third molar has a larger occlusal surface than the second molar in Homo rudolfensis fossils. The occlusal surface of the first molar is larger than in Homo habilis.

Do the species habilis and rudolfensis belong to the genus Homo?

For several years, the criteria used by Leakey and co-authors in 1964 to redefine the genus Homo have been considered insufficient and/or inappropriate by many researchers. Traditionally, there are two ways of grouping several species within a single genus: 1) a genus groups together species sharing a common ancestor that occupy different ecological areas which are different from those of species belonging to other genera. A group of species can be either monophyletic (i.e. it includes the common ancestor and all its descendants) or paraphyletic (i.e. it includes a common ancestor and some of its descendants); 2) a genus is a monophyletic group according to cladistic classification methods.

Wood and Collard have proposed in 1999 (Wood andCollard, 1999a, Wood andCollard, 1999b) to modify the definition of the genus by taking into account both conditions: species will be grouped in a genus if and only if they are part of the same monophyletic group and share the same adaptative strategies. The application of this definition to hominin fossils therefore leads them to suggest that a fossil species can be included in the genus Homo if it can be shown that: 1-it is closer, based on cladistic analysis, to Homo sapiens than to Australopithecus; 2-it shares more adaptive strategies with Homo sapiens than with australopithecines; -body mass and body proportions closer to Homo sapiens than to australopithecines; -post-cranial skeleton exhibiting a morphology consistent with terrestrial bipedalism and a weak climbing capacity; -dentition and masticatory system closer in relative size to Homo sapiens than to australopithecines; -period of growth and development close to that of Homo sapiens.

In order to test the monophyly of the Homo genus, these authors had taken into account the six main cladistic studies carried out on hominins in the previous fifteen years. They have showed that the species habilis and rudolfensis do not form a monophyletic group with the later species allocated to the genus Homo (from ergaster to sapiens). Then, they have assessed the adaptative coherence of species within the genus Homo by examining the criteria described above. For all these criteria, the species Homo habilis and Homo rudolfensis are closer to australopithecines than to Homo sapiens.

In 2001, Meave Leakey and his team [START_REF] Leakey | New hominin genus from eastern Africa shows diverse middle Pliocene lineages[END_REF] created a new genus Kenyanthropus and a new species Kenyanthropus platyops. The holotype skull KNM-WT 40000 (found in the western part of Lake Turkana and dated at 3.5 Ma) shows some morphological similarities with the Koobi Fora specimen KNM-ER 1470, notably in the shape and orientation of the naso-alveolar clivus, and the size and orientation of the zygomatic region. Based on these similarities, these authors have proposed that "the transfer to this species to Australopithecus has been recommended, but Kenyanthropus may be a more appropriate genus" (Leakey et al., 2001 p.439).

However, other studies based on the same cladistic approach have shown that the genus Homo is monophyletic even if the specimens of the species habilis and rudolfensis are included into this genus and therefore these species did not belong to the genus Australopithecus and Kenyanthropus, using either as terminal taxa, the specimen [START_REF] Prat | Les premiers représentants du genre Homo, en quête d'une identité. Apports de l'étude morphologique et de l'analyse cladistique[END_REF] or the species (e.g., [START_REF] Dembo | Bayesian analysis of a morphological supermatrix sheds light on controversial fossil hominin relationships[END_REF][START_REF] Dembo | The evolutionary relationships and age of Homo naledi: An assessment using dated Bayesian phylogenetic methods[END_REF][START_REF] Strait | Inferring hominoid and early hominid phylogeny using craniodental characters: the role of fossil taxa[END_REF].

Moreover, the study undertaken by Caparros and Prat [START_REF] Caparros | A Phylogenetic Networks perspective on reticulate human evolution[END_REF] using phylogenetic approach combining Maxium Parsimony and Phylogenetic Networks methods [START_REF] Caparros | Protocol combining tree-based Maximum Parsimony and web-like Phylogenetic Networks analyses to investigate reticulate human evolution[END_REF] show that these two species belong to Homo sensu lato, that the side-edge H. habilis, H. floresiensis represents probable phylogenetic divergence that might be construed as evolutionary dead-ends, and H. rudolfensis is part of a clear reticulate radiative structure called by the authors as Homo hybridization subnetwork.

Therefore, two scenarios of the first humans seem to emerge (Figure 2). An emergence of the genus Homo, as early as 2.8 Ma, with Homo sp. specimens and the species Homo habilis and Homo rudolfensis, or at 1.9 Ma with Homo ergaster.

INSERER FIGURE 2. scenarios

Can we define the genus Homo?

The inclusion of the new species Homo habilis into the genus Homo had required a revision of the diagnosis of the genus Homo. Three criteria appeared to be highly important in the definition of Leakey and his collaborators [START_REF] Leakey | A new species of the genus Homo from Olduvai Gorge[END_REF] : endocranial volume, power and precision grip, habitual erect posture and bipedal gait.

The issue about a minimum cranial capacity associated with the genus Homo was raised by Keith as early as 1948 [START_REF] Keith | A new theory of human evolution[END_REF]. This limit was, according to him, 750 cm3. Leakey and co-authors later have lowered this threshold to 600 cm3. This notion of "cerebral rubicon" remains problematic today because there is an overlap between the cranial capacities of Australopithecus and Homo : A. africanus (428-515 cm 3 ), Homo habilis (509-824 cm 3 ), Homo floresiensis [426 cm 3 [START_REF] Kubo | Brain size of Homo floresiensis and its evolutionary implications[END_REF] dated to 100-60 ka [START_REF] Sutikna | Revised stratigraphy and chronology for Homo floresiensis at Liang Bua in Indonesia[END_REF]], or Homo naledi [465-560 cm 3 [START_REF] Garvin | Body size, brain size, and sexual dimorphism in Homo naledi from the Dinaledi Chamber[END_REF] dated to 335-236 Ka; [START_REF] Dirks | The age of Homo naledi and associated sediments in the Rising Star Cave[END_REF]]. This has no real meaning in terms of cognitive capacities as it has been shown that there is no direct and causal influence between total brain volume and overall cognitive performance (e.g. [START_REF] Schoenemann | Brain size does not predict general cognitive ability within families[END_REF][START_REF] Louail | Refining the ecological brain: Strong relation between the ventromedial prefrontal cortex and feeding ecology in five primate species[END_REF]. Furthermore, it is problematic because the meaning of cranial capacity when related to body mass is questionable [START_REF] Martin | Relative brain size and basal metabolic rate in terrestrial vertebrates[END_REF]. Moreover, the human-like cortical reorganization does not appear to be specific to the genus Homo (see [START_REF] Beaudet | Evolution of the modern human brain[END_REF] for a review of hominin brain reorganization). It is already present 3-4 Ma ago in Australopithecus africanus specimens (occipital (visual cortex) and prefrontal (inferior frontal gyrus) reorganizations, e.g. Holloway et al. (2004) and [START_REF] Beaudet | The endocranial shape of Australopithecus africanus: surface analysis of the endocasts of Sts 5 and Sts 60[END_REF]), Paranthropus robustus (occipital reorganization, prefrontal reorganization(?) [START_REF] Holloway | New australopithecine endocast, SK 1585, from Swartkrans, South Africa[END_REF]) and debated in Australopithecus afarensis (occipital organization (e.g. [START_REF] Gunz | Australopithecus afarensis endocasts suggest ape-like brain organization and prolonged brain growth[END_REF][START_REF] Holloway | Endocast morphology of Hadar hominid AL 162-28[END_REF]). Furthermore, it is also observed in more recent Australopithecus species as Australopithecus sediba (1.9 Ma) with the reorganization of the prefrontal cortex [START_REF] Carlson | The endocast of MH1, Australopithecus sediba[END_REF]. Moreover, according to Ponce de León and collaborators, the brain of the early specimens from Africa and Georgia retained a great ape-like organization of the frontal lobe, and by contrast, specimens younger than 1.5 Ma in Africa and Asia exhibited a human-like brain organization (Ponce de [START_REF] Ponce De León | The primitive brain of early Homo[END_REF].

The second criterion advanced concerns the human-like manipulative abilities which involves the last phalanges of the thumb and index finger. In a broader sense, these manual abilities are forceful precision grips and precision handling sensu Marzke [START_REF] Marzke | Tool making, hand morphology and fossil hominins[END_REF] or pad-to-pad precision grasping sensu [START_REF] Napier | The prehensile movements of the human hand[END_REF] and [START_REF] Almécija | On manual proportions and pad-to-pad precision grasping in Australopithecus afarensis[END_REF], which are linked to the evolution of increasingly complex tool technologies within the hominin clade [START_REF] Marzke | Tool making, hand morphology and fossil hominins[END_REF]. However, this anatomical ability to use and even make lithic tools is not exclusive to the genus Homo. Indeed, these anatomical and potential behavioral abilities seem also present in two hominin genera different from Homo. As a matter of fact, morphological evidence (even if they are limited) suggests the potential for human-like precision grip in Australopithecus (afarensis, africanus, sediba) and Paranthropus (robustus and boisei). Indeed, A. afarensis is mainly characterized by human-like hand functions illustrated for example by substantial mobility of carpometacarpal joints, the overall morphology of the distal pollical phalanx [START_REF] Ward | New postcranial fossils of Australopithecus afarensis from Hadar, Ethiopia (1990-2007)[END_REF] and modern human-like manual proportions with a relatively long thumb and short medial digits (e.g. Alba et al., 2013;Almécija andAlba, 2014, Drapeau et al., 2005;[START_REF] Ward | New postcranial fossils of Australopithecus afarensis from Hadar, Ethiopia (1990-2007)[END_REF]. According to Marzke [START_REF] Marzke | Joint functions and grips of the Australopithecus afarensis hand, with special reference to the region of the capitate[END_REF][START_REF] Marzke | Tool making, hand morphology and fossil hominins[END_REF][START_REF] Marzke | Tool making, hand morphology and fossil hominins[END_REF], A. afarensis individuals were able to use pad-to-side as well as threejaw chuck precision grips and precision handling, all three manipulative abilities which are involved in stone tools behaviors and considered unique to the human hand compared to extant hominids. However, Rolian and Gordon [START_REF] Rolian | Reassessing manual proportions in Australopithecus afarensis[END_REF] have proposed that the manual proportions of A. afarensis are intermediate between gorillas and humans, and have suggested that A. afarensis did not possess the efficient precision grip typical of modern humans. Concerning Australopithecus africanus, based on the specimens classically put into Member 4 deposit from Sterkfontein site (recently dated to 3.4 Ma; [START_REF] Granger | Cosmogenic nuclide dating of Australopithecus at Sterkfontein, South Africa[END_REF], it has been shown that the individuals of this species was able to use a human-like pad-to-pad precision grasp [START_REF] Green | Metacarpal proportions in Australopithecus africanus[END_REF], pad-to-side and handling grips [START_REF] Marzke | Tool making, hand morphology and fossil hominins[END_REF][START_REF] Ricklan | Functional anatomy of the hand of Australopithecus africanus[END_REF], but with not the full range of human-like manipulative behaviors seen in later hominins [START_REF] Green | Metacarpal proportions in Australopithecus africanus[END_REF]. More recently, based on the internal structure of metacarpals, Skinner and co-authors [START_REF] Skinner | Human evolution. Human-like hand use in Australopithecus africanus[END_REF] confirm that A. africanus would have had, during precision and power grips, the capacity of forceful human-like opposition of the thumb and fingers. Furthermore, according to Kivell and co-authors [START_REF] Kibunjia | Australopithecus sediba hand demonstrates mosaic evolution of locomotor and manipulative abilities[END_REF], individuals belonging to a more recent Australopithecus species: A. sebiba (Malapa site, South Africa, 1.977 Ma) possessed forceful precision gripping and hand proportions that show enhanced dexterity, confirmed by other recent studies using trabecular bone analyses and musculoskeletal modeling [START_REF] Bardo | The impact of hand proportions on tool grip abilities in humans, great apes and fossil hominins: A biomechanical analysis using musculoskeletal simulation[END_REF][START_REF] Dunmore | The position of Australopithecus sediba within fossil hominin hand use diversity[END_REF]. Concerning Paranthropus genus, the anatomy of the hand of specimens belonging to P. boisei (e.g. specimens from East Turkana, Kenya), highlights that this species exhibit manual dexterity required to use or make stone tools, but did not possess the robust thumb observed in Homo erectus, which are linked to the intensification of forceful precision grips and tool use [START_REF] Richmond | The upper limb of Paranthropus boisei from Ileret, Kenya[END_REF]. In southern Africa, the remains (phalanges and metacarpals) discovered mainly from Member 1 and allocated to P. robustus [START_REF] Susman | Hand of Paranthropus robustus from Member 1, Swartkrans: fossil evidence for tool behavior[END_REF][START_REF] Susman | Fossil evidence for early hominid tool use[END_REF][START_REF] Susman | Hand function and tool behavior in early hominids[END_REF], made it possible to highlight that these hominins possessed, in particular, the opposing muscle of the thumb and the fifth finger. These muscles allow respectively a human-like precision grasping by stabilizing the opposing the thumb and the fifth finger to the other fingers which are considered as adaptations for tool-using and tool-making [START_REF] Susman | Fossil evidence for early hominid tool use[END_REF][START_REF] Susman | Hand function and tool behavior in early hominids[END_REF]. However, according to Moyà-Solà and co-authors (which consider that OH 7 hand bones belong to Paranthropus boisei), Paranthropus individuals exhibit more robust hand bone than Homo and lack the disproportionately hypertrophied first metacarpal characteristic of the genus Homo [START_REF] Moyà-Solà | Taxonomic attribution of the Olduvai hominid 7 manual remains and the functional interpretation of hand morphology in robust australopithecines[END_REF]. For these authors, the Paranthropus pattern probably reflects an adaptation to manual activities other than toolmaking more related to manipulative feeding behaviours like specialized food gathering and processing, such as it can be observed in extant hominids [START_REF] Gérard | Manipulative repertoire of bonobos (Pan paniscus) in spontaneous feeding situation[END_REF][START_REF] Marzke | Grips and hand movements of chimpanzees during feeding in Mahale Mountains National Park, Tanzania[END_REF][START_REF] Neufuss | Manual skills for food processing by mountain gorillas in Bwindi Impenetrable National Park, Uganda[END_REF].

The third argument relating to bipedal gaits is also questioned. From an evolutionary perspective, bipedalism is now perceived as a complex evolutionary process and not as a one-off event in human evolution (e.g. [START_REF] Crompton | Locomotion and posture from the common hominoid ancestor to fully modern hominins, with special reference to the last common panin/hominin ancestor[END_REF][START_REF] Niemitz | The evolution of the upright posture and gait--a review and a new synthesis[END_REF][START_REF] Preuschoft | Mechanisms for the acquisition of habitual bipedality: are there biomechanical reasons for the acquisition of upright bipedal posture?[END_REF]. Recent analyses in functional anatomy and biomechanics and the locomotor record of extant primates and fossil hominins illustrate several forms of bipedalism (i.e. arboreal locomotion with some adaptation to bipedal walking, effective bipedal walking with some arboreal grasping adaptations, long distance walking and running (e.g. [START_REF] Holowka | Rethinking the evolution of the human foot: insights from experimental research[END_REF]) which can occur in genera other than Homo genus. As for example occasional bipedal walking can represent a part of the positional repertoire of nonhuman primates (in Catarrhines e.g. [START_REF] Berillon | Bipedal versus Quadrupedal hind limb and foot kinematics in a captive sample of Papio anubis: Setup and preliminary results[END_REF][START_REF] Carvalho | Chimpanzee carrying behaviour and the origins of human bipedality[END_REF][START_REF] Druelle | Bipedalism in non-human primates: A comparative review of behavioural and experimental explorations on catarrhines[END_REF]. Furthermore, during the course of hominin evolution, first bipedal gait are considered to be present in the oldest hominins taxa, with evidence of an upright terrestrial bipedalism with substantial non-stereotypical arboreal behaviors. We can mention Salehanthropus tchadensis [START_REF] Daver | Postcranial evidence of late Miocene hominin bipedalism in Chad[END_REF][START_REF] Zollikofer | Virtual cranial reconstruction of Sahelanthropus tchadensis[END_REF], Orrorin tugenensis [START_REF] Senut | First hominid from the Miocene (Lukeino Formation, Kenya)[END_REF], Ardipithecus ramidus [START_REF] Lovejoy | The pelvis and femur of Ardipithecus ramidus: the emergence of upright walking[END_REF], Ardipithecus kaddaba [START_REF] Haile-Selassie | Hominidae in Ardipithecus kadabba[END_REF], and in Australopithecus species for example A. anamensis (e.g. [START_REF] Ward | Koobi Fora Project Vol 4 : Hominid Remains From Koobi Fora[END_REF][START_REF] Ward | New fossils of Australopithecus anamensis from Kanapoi, West Turkana, Kenya (2012-2015)[END_REF], A. afarensis (e.g. [START_REF] Ward | New postcranial fossils of Australopithecus afarensis from Hadar, Ethiopia (1990-2007)[END_REF]; with a debate if the initial hypodigm is based on a single species developing upright bipedalism associated or not with arboreal behaviors (see review in [START_REF] Ward | Interpreting the posture and locomotion of Australopithecus afarensis: where do we stand?[END_REF] and [START_REF] Chevalier | Australopithecus afarensis: bipédie stricte ou associée à une composante arboricole? Critiques et révision du matériel fémoral[END_REF]) and in Paranthropus species [START_REF] Domínguez-Rodrigo | First partial skeleton of a 1.34-million-year-old Paranthropus boisei from Bed II, Olduvai Gorge, Tanzania[END_REF], with different bipedal patterns. Nowadays, only long-distance walking and running are considered to be specific to humans (e.g. [START_REF] Bramble | Endurance running and the evolution of Homo[END_REF][START_REF] Harcourt-Smith | The Origins of Bipedal Locomotion[END_REF][START_REF] Holowka | Rethinking the evolution of the human foot: insights from experimental research[END_REF].

The aptitude for language, the manufacture of lithic tools and the notion of social behaviour (hunting, etc.), although they are not included in the diagnosis of the genus Homo, are often considered to be characteristic of our genus and would constitute key stages in the hominization process. Nowadays, there is no consensus on the definition of language. Nevertheless, the notion of articulation or more precisely of double articulation of language has been put forward as a characteristic of human language, but it does not refer to physiological and/or anatomical characteristics relating to a phonatory capacity. Indeed, a language is said to be doubly articulated, i.e. constructed of minimal units at two levels. The second articulation is that of sound units which can be assembled to form units of first articulation. The first articulation which corresponds to the units of meaning which allow an infinite number of statements to be composed. Furthermore, this linguistic notion seems to be obsolete, as there are non-oral natural languages (for example, sign language). The language ability of a fossil species is therefore difficult to demonstrate on the basis of fossil remains. Indeed, as pointed out by some researchers this ability is difficult to deduce from endocranial casts (e.g. [START_REF] Gannon | Asymmetry of chimpanzee planum temporale: humanlike pattern of Wernicke's brain language area homolog[END_REF][START_REF] Holloway | Cerebral brain endocast pattern of Australopithecus afarensis hominid[END_REF] because very often the areas considered to be involved in the function of language, such as Broca's and Wernicke's areas, are not easily observable on endocranial casts or their functional implications and direct relationship with the origin of language seem unclear, i.e. size, shape and asymmetries of the third frontal convolution (e.g. [START_REF] Tobias | The brain of Homo habilis : a new level of organization in cerebral evolution[END_REF] and [START_REF] Balzeau | Variations in size, shape and asymmetries of the third frontal convolution in hominids: paleoneurological implications for hominin evolution and the origin of language[END_REF]; see also review in [START_REF] Beaudet | The Emergence of Language in the Hominin Lineage: Perspectives from Fossil Endocasts[END_REF][START_REF] Bruner | Language, Paleoneurology, and the Fronto-Parietal System[END_REF]). Furthermore, the human-like structural brain asymmetries are observed in several nonlinguistic primates such as chimpanzees or baboons which can be related to the lateralization of gestural communicative system (see review in [START_REF] Becker | Structural Brain Asymmetries for Language: A Comparative Approach across Primates[END_REF]. Morevover, the present of these neuroanatomical landmarks clearly states that they are not human-or language specific [START_REF] Becker | Structural Brain Asymmetries for Language: A Comparative Approach across Primates[END_REF]. Therefore, endocranial morphology does not imply de facto linguistic ability. Considering the fossil record, according to Holloway in 2017 [START_REF] Holloway | Brain Fossils: Endocasts[END_REF], Australopithecines lack a human-like Broca's cap region and Australopithecines and early Homo individuals could have used vocal signals as a way of communication [START_REF] Aboitiz | A Brain for Speech. Evolutionary Continuity in Primate and Human Auditory-Vocal Processing[END_REF] Finally, tools and the ability to make tools take a special place in the literature in the definition of the genus Homo. The possible relationship between the Oldowan lithic industry and the Olduvai H. habilis specimens is however not mentioned in the diagnosis of [START_REF] Leakey | A new species of the genus Homo from Olduvai Gorge[END_REF] but in the following paragraph entitled "Cultural Association", as follows: "When the skull of Australopithecus (Zinjanthropus) boisei was found a living floor at F.L.K. 1, no remains of any other type of hominid were known from the early part of the Olduvai sequence. It seemed reasonable, therefore, to assume that this skull represented the makers of the Oldowan culture. The subsequent discovery of remains of Homo habilis in association with the Oldowan culture at three other sites has considerably altered the position. While it is possible that Zinjanthropus and Homo habilis both made stone tools, it is probable that the latter was the more advanced tool maker and that the Zinjanthropus skull represents an intruder (or a victim) on a Homo habilis living site" (Leakey et al., 1964, p.9). The attribution of the lithic remains in the Olduvai site to Homo habilis rather than Zinjanthropus boisei (Paranthropus boisei) thus appears to be more an ideological choice than a scientific interpretation. The oldest known oldowan tools found in situ, which are dated at 2.6 Ma in Ethiopia [START_REF] Semaw | 2.5-million-year-old stone tools from Gona, Ethiopia[END_REF][START_REF] Braun | Earliest known Oldowan artifacts at >2.58 Ma from Ledi-Geraru, Ethiopia, highlight early technological diversity[END_REF] and 2.34 Ma in Kenya [START_REF] Kibunjia | Australopithecus sediba hand demonstrates mosaic evolution of locomotor and manipulative abilities[END_REF][START_REF] Kibunjia | Pliocene archaeological occurrences in the Lake Turkana basin[END_REF][START_REF] Roche | Early hominid stone tool production and technical skill 2.34 Myr ago, in West Turkana, Kenya[END_REF], are contemporaneous in eastern Africa with both early Homo, Paranthropus (P. aethiopicus) and Australopithecus (A. garhi, [START_REF] Semaw | 2.6-Million-year-old stone tools and associated bones from OGS-6 and OGS-7, Gona, Afar, Ethiopia[END_REF]. Furthermore, some stone and bone artefacts have been discovered in association with individuals allocated to P. robustus which supports the hypothesis that P. robustus probably used and/or made bone tool [e.g. Swartkrans [START_REF] Brain | Swartkrans. A Cave's Chronicle of Early Man[END_REF][START_REF] Backwell | Evidence of termite foraging by Swartkrans early hominids[END_REF][START_REF] Backwell | Additional evidence on the early hominid bone tools from Swartkrans with reference to spatial distribution of lithic and organic artefacts[END_REF]; Drimolen [START_REF] Backwell | Early hominid bone tools from Drimolen, South Africa[END_REF]; Coopers'D [START_REF] Hanon | New evidence of bone tool use by Early Pleistocene hominins from Cooper's D, Bloubank Valley, South Africa[END_REF]]. The attribution of artefacts to Homo habilis rather than Paranthropus appears to be once again to be more ideological than scientific. In addition, the recent discovery in Kenya of stone-tools dated to 3.3 Ma (LOM3 site) [START_REF] Harmand | 3.3-million-year-old stone tools from Lomekwi 3, West Turkana, Kenya[END_REF], i.e. 700 000 years older than earliest Oldowan technology and 500,000 years before the oldest fossils attributed to the genus Homo (Homo sp. remains at Ledi Geraru [START_REF] Villmoare | Early Homo at 2.8 Ma from Ledi-Geraru, Afar, Ethiopia[END_REF]). They have been found in the same chronological and geographical frameworks than specimens allocated to Kenyanthropus platyops and are also contemporaneous with individuals attributed to Australopithecus afarensis and Australopithecus deyiremeda. These findings have reopened the debate on the association between lithic industry and the genus Homo [START_REF] Harmand | 3.3-million-year-old stone tools from Lomekwi 3, West Turkana, Kenya[END_REF][START_REF] Lewis | An earlier origin for stone tool making: implications for cognitive evolution and the transition to Homo[END_REF]. The stone artefacts from Lomekwi 3 are larger in width, length and thickness than those coming from Oldowan sites. Two mode of knapping have been recognized (passive hammer and bipolar). The knappers have combined battering activities and core reduction, and may have used the lithic artefacts variously: as anvils, core in order to produce flakes and/or as pounding tools. These findings put into the light that species that occurred far long before the emergence of the genus Homo present important manipulative and cognitive ability.

Moreover, the unique and specific attribution to the genus Homo of a capacity to make lithic tools is therefore no longer valid for those periods.

Conclusion

In fine, the increase in the fossil record has shown the extent of both the morphological and cultural complexity of hominins and evolutionary scenarios that led to the appearance the genus Homo. According to the recent archaeological and palaeoanthropological discoveries, the criteria often considered to be crucial for the definition of the genus Homo, as the cranial capacity, the human-like manipulative abilities, the habitual erect posture and bipedal gait, the language ability and the capacity to make tools are not anymore valid. Trying to characterize the genus Homo factually through for example the notion of hominization leads to a certain arbitrariness inherent in any definition and conceptually questionable: is humans are capable of defining themselves, and it is possible to be both the object of study and the one who studies him? In any case, as demonstrated in previous paper [START_REF] Caparros | A Phylogenetic Networks perspective on reticulate human evolution[END_REF], hominin biological evolution reveals a reticulate nature. The incompleteness of the fossil record, the complexity of genomic transfers within this evolution and the persistence of this taxon in time and of their behavioural cultural evolutions question the definitive conceptual limit of the species in Palaeoanthropology, where species can be seen more as conceptual objects that can fuel debates on evolutionary scenarios.

Figure 1 .

 1 Figure 1. Location of the main sites where early Homo remains have been discovered Figure 1. Localisation des principaux sites où des restes attribués aux premiers représentants du genre Homo ont été mis au jour

Figure 2 .

 2 Figure 2. Scenarios relatifs à l'émergence du genre Homo

Table 1 .

 1 Morphological comparisons between Homo habilis sensu stricto and Homo rudolfensis. * : trait assignment related to the developmental age of the individual, # high polymorphism in extant great apes).Tableau 1 : Comparaisons morphologiques entre Homo habilis sensu stricto et Homo rudolfensis : * : état de caractère lié à l'âge developmental de l'individu, # fortement dimorphique chez les grands singes actuels.

	Homo habilis sensu stricto	Homo rudolfensis

Acknowledgment

I would like to thanks the reviewers, Pr. Henry de Lumley and the editors of l'Anthropologie for their invitation to participate to this special issue, Ameline Bardo and Philippe Arab for their comments. This work is supported by the French Agence national de la recherché the ANR HOMTECH (ANR-17-CE27-0005).