N Schaeffer

D Jault

P Cardin

M Drouard

On the reflection of Alfvén waves and its implication for Earth's core modeling

Alfvén waves propagate in electrically conducting fluids in the presence of a magnetic field. Their reflection properties depend on the ratio between the kinematic viscosity and the magnetic diffusivity of the fluid, also known as the magnetic Prandtl number P m. In the special case P m = 1, there is no reflection on an insulating, no-slip boundary, and the incoming wave energy is entirely dissipated in the boundary layer.

We investigate the consequences of this remarkable behaviour for the numerical modeling of torsional Alfvén waves (also known as torsional oscillations), which represent a special class of Alfvén waves, in rapidly rotating spherical shells. They consist of geostrophic motions and are thought to exist in the fluid cores of planets with internal magnetic field. In the geophysical limit P m 1, these waves are reflected at the core equator, but they are entirely absorbed for P m = 1. Our numerical calculations show that the reflection coefficient at the equator of these waves remains below 0.2 for P m ≥ 0.3, which is the range of values for which geodynamo numerical models operate. As a result, geodynamo models with no-slip boundary conditions cannot exhibit torsional oscillation normal modes.

I. INTRODUCTION

Hannes Alfvén first showed the theoretical existence, in an inviscid fluid of infinite electrical conductivity, of hydromagnetic waves that couple fluid motion and magnetic field [START_REF] Alfvén | Existence of Electromagnetic-Hydrodynamic waves[END_REF]. The propagation of torsional Alfvén waves in the Earth's fluid core was thereafter predicted by Braginsky [START_REF] Braginsky | Torsional magnetohydrodynamic vibrations in the Earth's core and variations in day length[END_REF]. Such waves arise in rapidly rotating spheres or spherical shells in the presence of a magnetic field. In torsional Alfvén waves, the motions are geostrophic and consist in the rotation ω g (s) of nested cylinders centered on the rotation axis. They thus depend only on the distance s to the rotation axis. The period of the fundamental modes of torsional Alvén waves in the Earth's fluid core was first estimated to be about 60 years. This timescale was inferred from the analysis of the decadal length of day changes since the first half of the 19th century [START_REF] Jordi | Fluctuations in the earth's rotation since 1830 from high-resolution astronomical data[END_REF] and of the geomagnetic secular variation after 1900 [START_REF] Braginsky | Short-period geomagnetic secular variation[END_REF]. With hindsight, these time series were not long enough to show convincingly variations with 60 years periodicity. Torsional waves with much shorter periods have now been extracted from time series of core surface flows for the time interval 1955-1985 [START_REF] Gillet | Fast torsional waves and strong magnetic field within the Earth's core[END_REF]. If this discovery is confirmed, the period of the fundamental modes is of the order of 6 years and, as such, is much shorter than initially calculated.

Several authors have searched for torsional Alfvén waves in geodynamo simulations. Using stress-free boundary conditions, Dumberry & Bloxham [START_REF] Dumberry | Torque balance, Taylor's constraint and torsional oscillations in a numerical model of the geodynamo[END_REF] and Busse & Simitev [START_REF] Busse | Convection in rotating spherical fluid shells and its dynamo states[END_REF] illustrated some parts of the torsional wave mechanism. Dumberry & Bloxham [START_REF] Dumberry | Torque balance, Taylor's constraint and torsional oscillations in a numerical model of the geodynamo[END_REF] found that the whole length of the geostrophic cylinders accelerates azimuthally as if they were rigid. The inertial forces, in their simulation, are however so influential that they dominate the Lorentz forces. Torsional Alfvén waves have finally been detected in a set of numerical simulations of the geodynamo with no-slip boundary conditions, for 0.5 ≤ P m ≤ 10, by Wicht & Christensen [START_REF] Wicht | Torsional oscillations in dynamo simulations[END_REF] (the magnetic Prandtl number P m is the ratio of kinematic viscosity over magnetic diffusivity). In both the geophysical (P m ∼ 10 -5 ) and the numerical studies, there seems to be no reflection of the torsional Alfvén waves upon their arrival at the equator. However, experimental studies in liquid metals have shown resonance effects on Alfvén normal modes [START_REF] Jameson | A demonstration of Alfvén waves part 1. generation of standing waves[END_REF] as well as reflection of wave packets [START_REF] Alboussière | Experimental evidence of Alfvén wave propagation in a gallium alloy[END_REF].

In this paper we elaborate on the remark that reflection of Alfvén waves is controlled not only by the boundary condition, but also by the magnetic Prandtl number of the fluid in which they propagate [see 13, p. 23,24]. In the next section, we discuss the governing equations for one dimensional Alfvén waves and the associated boundary conditions for a solid and electrically insulating wall. We remark that for P m = 1 all the energy of the incident Alfvén wave is dissipated in a boundary layer, resulting in no reflected wave. In the following section, we change geometry to further emphasize our point and briefly present a direct numerical simulation of propagation and reflection of Alfvén wave in a non rotating spherical shell. That introduces the section devoted to the geophysical application, where we investigate torsional Alfvén waves in the Earth's core, modeled as a rapidly rotating spherical shell, calculating the energy loss on reflection at the Equator as a function of P m. Finally, we discuss the implications concerning the ability of geodynamo simulations to produce torsional eigenmodes and waves which are expected in the Earth's core.

II. REFLECTION OF ONE-DIMENSIONAL ALFV ÉN WAVES

We introduce the problem through the example of Alfvén waves, transverse to a uniform magnetic field in an homogeneous and electrically conducting fluid, hitting a solid wall perpendicular to the imposed magnetic field [START_REF] Roberts | An introduction to magnetohydrodynamics[END_REF]. The imposed uniform magnetic field B 0 is along the x-axis, while the induced magnetic field b(x, t) and the velocity field u(x, t) are transverse to this field, along y. Assuming invariance along y and z axes, the problem reduce to a 1-dimensional problem, u and b depending only on x. Projecting the Navier-Stokes equation and the induction equation on the y direction (on which the pressure gradient and the non-linear terms do not contribute), one obtains the following equations:

∂ t u = B 0 µ 0 ρ ∂ x b + ν∂ xx u (1)
∂ t b =B 0 ∂ x u + 1 µ 0 σ ∂ xx b (2)
where µ 0 is the magnetic permeability, ρ is the fluid density, ν the kinematic viscosity, and σ the electrical conductivity.

A. Elsasser variables

Introducing the two Elsasser variables h ± = u ± b/ √ µ 0 ρ, the equation of momentum [START_REF] Alboussière | Experimental evidence of Alfvén wave propagation in a gallium alloy[END_REF] and the equation of magnetic induction (2) can be combined into

∂ t h ± ∓ V A ∂ x h ± - η + ν 2 ∂ xx h ± = ν -η 2 ∂ xx h ∓ (3)
where V A = B 0 / √ µ 0 ρ is the Alfvén wave speed, and η = (µ 0 σ) -1 is the magnetic diffusivity. It is already apparent that when ν = η, the right hand side of the previous equation vanishes, in which case h + and h -are fully decoupled.

One can also show that h -travels in the direction of the imposed magnetic field, while h + travels in the opposite direction.

Introducing a length scale L and the time-scale L/V A , the previous equations take the following non-dimensional form:

∂ t h ± ∓ ∂ x h ± - 1 S ∂ xx h ± = 1 S P m -1 P m + 1 ∂ xx h ∓ (4)
where the Lundquist number S and the magnetic Prandtl number P m are defined as:

S = 2V A L η + ν P m = ν η
The propagation of Alfvén wave requires that the dissipation is small enough for the wave to propagate. This is ensured by S 1. The fact that (P m -1)/(P m + 1) = -(P m -1 -1)/(P m -1 + 1) establishes a fundamental symmetry of these equations: when changing P m into P m -1 , only the sign of the coupling term (right hand side of equations 4) changes.

B. Physical boundary conditions and reflection of Alfvén waves

These equations must be completed by boundary conditions. We assume that the wall is electrically insulating, and that the fluid velocity vanishes at the solid boundary (no-slip boundary condition), which translate to b = 0 and u = 0, leading to h ± = 0.

For P m = 1 the equations for h + and h -are fully decoupled, regardless of the value of S:

∂ t h ± = ±∂ x h ± + 1 S ∂ xx h ± (5)
In addition, for an insulating solid wall, the boundary condition h ± = 0 does not couple h + and h -either. As a result, reflection is not allowed at an insulating boundary when P m = 1, because reflection requires change of traveling direction, and thus transformation of h + into h -and vice versa. The energy carried by the wave has to be dissipated in the boundary layer.

For P m = 1 the equations are coupled: for very small diffusivities (that is large Lundquist number S), the coupling will be effective only in a thin boundary layer. In addition the coupling will be more efficient as P m is further from 1. This gives a mechanism for reflection of Alfvén waves on an insulating boundary when P m = 1. Before giving a numerical illustration, it is instructive to consider the boundary conditions in the two limits P m = 0 and P m = ∞, with S 1 (dissipationless interior). In the limit P m = 0, there is no viscous term and the boundary condition, at the wall x = x 0 , reduces to

b(x 0 , t) = 0 ⇒ h + (x 0 , t) = h -(x 0 , t). (6)
There is perfect reflection. The incident (+) and reflected (-) waves have equal velocities and opposite magnetic fields. This also corresponds to a stress-free boundary condition for the velocity field in combination with an insulating wall (infinitely small vorticity sheet at the wall), leading to perfect reflection regardless of the value of P m used in equation 4. In this case the boundary condition for the velocity field is ∂ x u = 0, which translates into ∂ x (h + + h -) = 0 and h + -h -= 0, effectively coupling h + and h -.

In the limit P m = ∞, the boundary condition , at the wall x = x 0 , reduces instead to

u(x 0 , t) = 0 ⇒ h + (x 0 , t) = -h -(x 0 , t). (7)
1e-06 1e-05 0,0001 0,001 0,01 0,1 1 10 100 1e+03 Pm, magnetic Prandtl number The incident and reflected waves have opposite velocities and equal magnetic fields. This also corresponds to a no-slip boundary condition for the velocity field in combination with a perfectly conducting wall (infinitely small current sheet at the wall), leading to perfect reflection regardless of the value of P m used in equation 4. In this case the boundary condition for the magnetic field is ∂ x b = 0, which couples h + and h -. Another combination of boundary conditions inhibits reflection for P m = 1: for a stress-free (∂ x u = 0) and perfectly conducting wall (∂ x b = 0), which translates into ∂ x h + = 0 and ∂ x h -= 0, the fields h + and h -are decoupled, as for a no-slip insulating wall. Note finally that a wall with finite conductivity will allow some weak reflection, as illustrated by figure 4h.

C. Numerical simulations

We have performed a numerical simulation in a channel 0 ≤ x ≤ x 0 with a one-dimensional finite difference scheme. The Lundquist number is chosen large enough so that dissipation can be neglected in the interior. The boundary conditions were set to be electrically insulating and no-slip. The grid is refined next to the boundaries, in order to have at least 4 points in each boundary layer, which are Hartmann layers of thickness δ = √ νη/V A (see appendix A).

From the simulation of the traveling wave, we compute the transmission coefficient as the ratio of the velocity amplitude of the reflected and incident waves for different values of P m and S. The results are reported on figure 1.

As expected, there is full dissipation for P m = 1 and energy conservation for P m 1 or P m 1. Furthermore, the reflection coefficient R is independent of S, and exhibits the expected symmetry R(P m -1 ) = -R(P m). The

III. REFLECTION OF A LOCALIZED ALFV ÉN WAVE PACKET ON A SPHERICAL BOUNDARY

The peculiar case where no reflection occurs is not specific to the planar, one-dimensional ideal experiment. Here, we run an axisymmetric simulation in a spherical shell permeated by a non-uniform magnetic field, without global rotation. The imposed magnetic field is the same as in Jault [START_REF] Jault | Axial invariance of rapidly varying diffusionless motions in the Earth's core interior[END_REF], and is represented by the dashed field lines of figure 2. Contrary to the simplest case of the previous section, it is a non-uniform magnetic field, which is not perpendicular to the boundaries. The observed behaviour of Alfvén wave packets hitting the curved boundaries should therefore apply to many systems.

The numerical pseudo-spectral code is the one used in Gillet et al. [START_REF] Gillet | Rationale and geophysical evidence for quasi-geostrophic rapid dynamics within the Earth's outer core[END_REF], but restrained to axisymmetry. It uses the SHTns library [START_REF] Schaeffer | Efficient Spherical Harmonic Transforms aimed at pseudo-spectral numerical simulations[END_REF] for spherical harmonic expansion (Legendre polynomials) in the latitudinal direction, and second order finite differences in radius with many points concentrated near the boundaries. It time-steps both induction and momentum equation in the sphercial shell using a semi-implicit Crank-Nicholson scheme for the diffusive terms, while the coupling and (negligible) non-linear terms are handled by an Adams-Bashforth scheme (second order in time). The number of radial grid points is set to 500 and the maximum degree of Legendre polynomials to 120.

The Alfvén wave packets are generated mechanically by spinning the conducting inner core for a very short duration (compared to the Alfvén propagation time). Since the imposed magnetic field strength is not uniform, the wave front deforms as it propagates along the field lines. When the wave packet hits the outer insulating spherical shell, it does reflect and propagates back towards the inner shell for P m = 0.1 and P m = 10 but there is no reflection for P m = 1. This is illustrated by the snapshots of figure 2.

IV. REFLECTION OF TORSIONAL ALFV ÉN WAVES

Finding evidence of propagation of Torsional Alfvén Waves (TAW) in the Earth's fluid core may open a window on the core interior. Properties of TAW in the Earth's core have thus been thoroughly investigated after the initial study of Braginsky [START_REF] Braginsky | Torsional magnetohydrodynamic vibrations in the Earth's core and variations in day length[END_REF]. They have been recently reviewed by Jault [START_REF] Jault | Electromagnetic and topographic coupling, and LOD variations[END_REF] and Roberts & Aurnou [START_REF] Roberts | On the theory of core-mantle coupling[END_REF].

A. Model of Torsional Alfvén waves

In order to model TAW, magnetic diffusion and viscous dissipation are neglected in the interior of the fluid. The Earth's fluid core is modeled as a spherical shell of inner radius r i , outer radius r o and rotation rate Ω. Rapid rotation introduces an asymmetry between the velocity and magnetic fields and makes the velocity geostrophic, provided that λ ≡ V A /Ωr o 1 [START_REF] Jault | Axial invariance of rapidly varying diffusionless motions in the Earth's core interior[END_REF]. Note that the Lehnert number λ is about 10 -4 in the Earth's core. Geostrophic velocity in a spherical shell consists of the rotation ω g (s) of nested cylinders centered on the rotation axis. It thus depends only on the distance s from the rotation axis. A one-dimensional wave equation for the geostrophic velocity sω g (s) is obtained after elimination of the magnetic field b:

L ∂ 2 ω g (s) ∂t 2 = ∂ ∂s L Ṽ 2 A ∂ω g (s) ∂s (8)
with L = s 3 H(s) and H(s) the half-height of the geostrophic cylinders, and Ṽ 2 A involves only the z-average of the squared s-component of the imposed magnetic field. Braginsky [START_REF] Braginsky | Torsional magnetohydrodynamic vibrations in the Earth's core and variations in day length[END_REF] derived [START_REF] Busse | Parameter dependences of convection-driven dynamos in rotating spherical fluid shells[END_REF] rigorously in the geophysical case for which the viscous Ekman layer is thin compared to the magnetic diffusion layer located at the top and bottom rims of the geostrophic cylinders. This condition amounts to P mλ 1. Then, the velocity remains geostrophic in the magnetic diffusion layer. We have written the equation ( 8) in its simplest form, when the imposed magnetic field is axisymmetric, the mantle is insulating and Ekman friction at the rims of the geostrophic cylinders is neglected. The equation ( 8) needs to be completed by two boundary conditions, which can be derived when either P m 1 or P m 1. Interestingly, the equation ( 8) may be valid in the limit P m 1 but also in the limit P m 1 (provided P mλ 1). In the specific case P m 1, the appropriate boundary condition on the geostrophic velocity at the equator (on the inner edge of the Hartmann boundary layer) can be inferred from the boundary condition on the magnetic field. For an insulating outer sphere, it yields ∂ s ω g = 0 which corresponds to a stress-free boundary, as in the one dimensional wave case with P m → 0. In the case P m 1, the appropriate boundary condition is ω g = 0 as the angular velocity of the outermost geostrophic cylinder is immediately synchronized with the rotation of the solid outer sphere in the course of a spin-up experiment. This is equivalent to a no-slip boundary, as for the one dimensional wave case with P m → ∞.

B. Normal modes

Assuming that ω g varies with time as e ict , the equation ( 8) can be transformed into a normal mode equation:

-c 2 ω g (s) = 1 L ∂ ∂s L Ṽ 2 A ∂ω g (s) ∂s (9)
Transmission and reflection of TAW on the geostrophic cylinder tangent to the inner core set a special problem that we do not address here. As an intermediate step, we simply illustrate our discussion with results for the full sphere case, imposing ∂ s ω g | s=ε = 0, with ε 1 (we have checked the convergence of the numerical results as ε → 0). It is of interest to write the solution of this equation in the case c = 0 and ṼA uniform:

ω g (s) = 1 2 α 1 - √ 1 -s 2 s 2 -log 1 -s 2 + 1 + log(s) + α 2 (10)
A non-zero solution (uniform rotation ω g (s) = α 2 ) exists for the boundary condition ∂ s ω g | s=ro = 0 but not for the condition ω g | s=ro = 0 that applies when P m 1. We are interested in this latter case, despite its lack of geophysical realism, as contrasting the two boundary conditions sheds light on the nature of the constraint ∂ s ω g | s=ro = 0 that has always been used in TAW studies.

In the general case (c = 0, non-uniform ṼA ), it remains easy to calculate numerically a solution of (9) for 0 < s < r o . We have successfully checked our numerical results against the eigenvalues listed in the table C1 of Roberts & Aurnou [START_REF] Roberts | On the theory of core-mantle coupling[END_REF], that have been obtained analytically for ∂ s ω g | s=ro = 0 and ṼA = 1. Then, the first eigenvalues are (0, 5.28, 8.63, 11.87, 15.07, ..), while in the case ṼA = 1 and ω g | s=ro = 0 they are (2.94, 6.35, 9.58, 12.78, 15.95, ..). In the latter case, we recover our previous observation that 0 is not an eigenvalue.

In contrast with an often-made statement [START_REF] Buffett | The core-mantle boundary region[END_REF][START_REF] Jault | Electromagnetic and topographic coupling, and LOD variations[END_REF][START_REF] Roberts | On the theory of core-mantle coupling[END_REF], the study of the case P m 1 shows that it is not required to have ∂ s ω g | s=ro = 0 to obtain solutions with bounded values of ω g for s ≤ r o . On the other hand, the singularity of ∂ s L at s = r o implies a singularity of ∂ s ω g (which is O((1 -s) -1/2 ) as s → 1) . That points to significant viscous dissipation once the viscous term is reintroduced.

When P m is neither very small nor very large, it is not possible to separate the interior region (where (9) applies) and the Hartmann boundary layer.

We can conclude the discussion of normal modes by noting that the solutions for the two cases P m 1 and P m 1 differ in a significant way at the equator. In both cases, solutions with bounded values of ω g in the interval The Lundquist number is always large (S > 5000 for P m ≥ 0.01 and S > 600 otherwise). For reference, the black curve is the planar Afvén wave reflection coefficient (1 -√ P m)/( √ P m + 1), and the red line marks the reflection coefficient for a stress-free boundary with P m = 1 (corresponding to a no-slip boundary with P m → 0) [0, r o ] and satisfying the appropriate boundary conditions are obtained. However, reintroducing dissipation modifies the eigensolutions in the vicinity of the equator and the eigenvalues in the second case only.

C. Numerical experiments

In order to determine the reflection coefficient of TAW at the equator of the outer shell, we use a set-up that resembles the Earth's core. The code is the same as the one described in section III, but this time with imposed global rotation. The total number of radial points is typically 1200 and the maximum degree of Legendre polynomials is set to 360.

For reflection to occur, there must be a non-zero imposed magnetic field B s at the equator. Hence we set the simplest potential quadrupolar field (generated from outside the sphere): B s = B 0 s, B z = 2B 0 z and B φ = 0. This ensures a local traveling speed V A (s) = B s (s)/ √ µ 0 ρ that is large near the reflection point (s = 1). The Lehnert number is small and always set to λ = V A /(Ωr o ) = 5 × 10 -4 , so that λP m is also small. The initial velocity field is along the azimutal direction φ and depends only on the cylindrical radius s: u φ (s) = sω g (s) = u 0 s exp(-(s -s 0 ) 2 / 2 ) with s 0 = 0.675. We used two different width = 0.02 and = 0.063. This initial velocity field splits into a torsional Alfvén wave packet propagating inwards that we do not consider here, and another traveling outwards that we carefully follow and we focus on the reflection of this wave packet at the equator of the outer shell (s = 1). The Lundquist number S based on the size of the spherical shell ranges from 6 × 10 2 to 8 × 10 4 and the Ekman number E = ν/Ωr 2 o and magnetic Ekman number Em = η/Ωr 2 o are both always very low and range from 5 × 10 -10 to 5 × 10 -7 over a wide range of magnetic Prandtl number: from P m = 10 -3 to P m = 10 2 .

We measure the extremum of the velocity field in the wave packet before and after the reflection, a i and a r respectively, at a fixed radius (s = 0.925 for = 0.02 and s = 0.75 for = 0.063), from which we compute the corresponding reflection coefficient R = a r /a i , reported in figure 3 for an insulating outer shell. We found no significant dependence with the Lundquist number S or the width of the initial pulse (R varies by less than 0.03).

As expected from the discussion of Alfvén waves equations, the combination P m = 1, no-slip boundary condition and insulating wall corresponds to a special case whereby no reflection at all occurs at the equator (see also fig. 4g).

However, there are differences with the planar case. First, the reflection coefficient is not symmetric with respect to P m = 1, as expected from our discussion of torsional eigenmodes in spherical geometry in the previous section. For large P m there is high dissipation and very little reflection compared to low P m. Second, the reflection coefficient is not as large.

Space-time diagrams of the reflection of the wave at the equator are presented in figure 4 for a few representative cases. The highest reflection coefficient occurs for the stress-free insulating case at P m = 1: from R = 0.86 at S = 1000 to R = 0.88 at S = 1.5 × 10 4 . In this case (fig. 4ab) one can also see the amplification of the velocity field very near the boundary, as the magnetic field must vanish, doing so by producing the reflected wave, just as in the planar case. This is not a boundary layer, but simply the superposition of the incident and reflected wave (see also appendix A). The Hartmann boundary layer is too small to be seen on these plots, but we checked that its size and relative amplitude for velocity and magnetic fields do match the analytic theory developed in appendix A.

For P m = 0.1, the reflected wave carries only 16% of the energy, the remaining being dissipated in the boundary layer. The magnetic field changes sign at the reflection, while the velocity keeps the same sign (fig. 4cd). For P m = 10, the reflected energy drops to 3% and the small reflected velocity field has opposite sign, while the magnetic field (barely visible on figure 4) keeps the same sign (fig. 4ef). During its propagation, the incoming wave is also much more damped than for P m = 0.1, even in the case where S or E have comparable values. This is due to strong dissipation at the top and bottom boundaries, which increases as the wave propagates toward the equator (visible in figure 4e) for P m > 1. This may not be unrelated to the previously discussed singularity for normal modes in the case P m > 1. A consequence of this large dissipation, is the difficulty to clearly identify the reflected wave, and to properly define a reflection coefficient. The values reported in figure 3 are thus not very precise for P m > 1.

It may also be worth noting that changing the magnetic boundary from insulating to a thin conducting shell allows weak reflection for P m = 1 and no-slip velocity (fig. 4h), in agreement with the analysis of the governing equations (section II B).

D. Energy dissipation and normal modes

We want to emphasize that when no reflection occurs, the energy of the wave is dissipated very quickly. However, for liquid metals (P m 1), only a small amount of the wave energy is absorbed in the event of a reflection, but many successive reflections can lead to significant dissipation. Using the theoretical reflection coefficient, we can estimate the time-scale of dissipation of an Alfvén wave due to its reflections at the boundaries. In the case of an Alfvén wave turbulence (many wave packets) in a spherical shell of radius L with homogeneous mean energy e, permeated by a magnetic field of rms intensity B 0 , any wave packet will reach the outer insulating boundary once (on average) in the time interval L/V A . When it reflects on the boundary, it loses the fraction 1 -R 2 (P m) of its energy, where R(P m) is the reflection coefficient (in amplitude). We can then estimate the dissipation rate of energy e due to this process:

∂ t e ∼ R 2 (P m) -1 B 0 L √ µ 0 ρ e (11)
Hence, the time-scale of dissipation at the boundaries

τ s = L V A 1 1 -R 2 (P m) (12)
which is inversely proportional to the strength of the magnetic field, and depends on the diffusivities only through P m.

We can compare this to the dissipation of Alfvén waves of length scale in the bulk of the fluid: τ v = 2 2 /(η + ν). It appears that the length scale where surface and bulk dissipation are comparable is such that

L/ = √ S 1 -R 2 (13)
Replacing R by its theoretical expression A11 and assuming P m 1, we find for liquid metals

L/ = 2 √ S P m 1/4 (14)
Hence for the Earth's core with S ∼ 10 4 , and P m ∼ 10 -5 , the dissipation of Alfvén waves is dominated by the partial absorption at the boundaries for length scales larger than L/10. For numerical simulations of the geodynamo with S ∼ 10 3 ans R ∼ 0.2, we have L/ ∼ 30. These time-scales are also relevant for torsional normal modes. In one dimension, normal modes are a superposition of waves propagating in opposite directions. Hence, if the dissipation of waves is dominated by their reflection, so will it be for the normal modes. From the previous estimation of L/ in the Earth's core, we expect the dissipation of large wave-length torsional Alfvén waves (the ones that can be observed) to be dominated by the effect of reflection. Furthermore, in order to detect a normal mode, its dissipation time must be much larger than its period T = 2πL(cV A ) -1 . The pulsation c of the first torsional normal modes are given in section IV B in Alfvén frequency units, and their dissipation time can be estimated by τ s for the large-scale normal modes. We define a quality factor for torsional normal modes by

Q = τ s T = c 2π 1 1 -R 2 . ( 15
)
Presence of normal modes requires Q 1. In the Earth's core, we find Q E 13 c and for numerical simulations of the geodynamo we find Q sim 0.16 c. Considering the largest modes (with c 5), the torsional oscillations should therefore be present in the Earth's core, but are completely absent even from the best current geodynamo simulations.

V. DISCUSSION: IMPLICATION FOR NUMERICAL GEODYNAMO MODELS AND THE EARTH-CORE

We showed that numerical simulations conducted for P m ∼ 1 cannot adequately reproduce the boundary conditions for torsional Alfvén waves in the Earth's core (where P m 1). The small reflection coefficient observed for TAW (figure 3) means that it is hard to observe TAW reflection at the equator in numerical simulations of the geodynamo which currently operate with 0.1 < P m < 10 [e.g. [START_REF] Sakuraba | Generation of a strong magnetic field using uniform heat flux at the surface of the core[END_REF][START_REF] Takahashi | Scale variability in convection-driven MHD dynamos at low ekman number[END_REF], where the waves are moreover mixed with thermal convection.

As for possible torsional eigenmodes, it is almost impossible to observe them with such low reflection coefficients. Unfortunately, that severely limits the ability of geodynamo simulations to exhibit torsional oscillation normal modes, because normal modes require a large reflection coefficient to be observable: their period (of order L/V A ) must be much larger than the energy dissipation time τ s (see expression 12). A few studies have tried to pin down torsional eigenmodes [START_REF] Dumberry | Torque balance, Taylor's constraint and torsional oscillations in a numerical model of the geodynamo[END_REF][START_REF] Sakuraba | Torsional oscillations in a numerical geodynamo operating in a regime of low ekman and magnetic prandtl numbers[END_REF][START_REF] Wicht | Torsional oscillations in dynamo simulations[END_REF] but even though they report waves propagating with the appropriate speed, they report neither reflection of these waves, nor eigenmodes.

Another issue for geodynamo models with very low diffusivities, is that the part of the energy carried by Alfvén waves (regular or torsional) is dissipated very quickly (on an Alfvén wave time-scale), so that an Alfvén wave turbulence would be damped much faster, and the turbulent state may be far from what we would expect in the Earth's core.

Changing the boundary condition to stress-free simulates the case P m = 0 with a high reflection coefficient (R = 0.88), but still lower than the planar case. Even though this may still be problematic to observe eigenmodes, numerical models that use stress-free boundaries [e.g. [START_REF] Busse | Parameter dependences of convection-driven dynamos in rotating spherical fluid shells[END_REF][START_REF] Dumberry | Torque balance, Taylor's constraint and torsional oscillations in a numerical model of the geodynamo[END_REF][START_REF] Kuang | Numerical modeling of magnetohydrodynamic convection in a rapidly rotating spherical shell: Weak and strong field dynamo action[END_REF][START_REF] Sreenivasan | Helicity generation and subcritical behaviour in rapidly rotating dynamos[END_REF] are intrinsically much more suited for the study of torsional normal modes. Quasi-geostrophic dynamo models that can compute dynamo models at very low magnetic Prandtl numbers [P m < 10 -2 in 24], could also provide an interesting tool to study torsional oscillations.

In the case of the Earth's core, a recent study [START_REF] Gillet | Fast torsional waves and strong magnetic field within the Earth's core[END_REF] found no clear evidence for reflection at the equator, although this has yet to be confirmed. One might want to invoke turbulent viscosity (see the contrasted views of Deleplace & Cardin [START_REF] Deleplace | Viscomagnetic torque at the core mantle boundary[END_REF] and Buffett & Christensen [START_REF] Buffett | Magnetic and viscous coupling at the core-mantle boundary: inferences from observations of the Earth's nutations[END_REF] in a different context) to explain this fact, leading to an effective P m close to 1 and inhibiting reflection of torsional Alfvén waves. This would make numerical models more relevant, but is rather speculative. A solid conductive layer at the top of the core can also have a damping effect on the propagation and reflection of torsional waves, and we plan to investigate these matters in a forthcoming study. apparent only in the velocity field component (eq. A9), whereas in the limit P m → ∞, we have β → 0 and α k2 β → -2, so that the boundary layer is apparent only in the magnetic field component (eq. A10).

Finally, we remark that if one sets ν = 0 or η = 0 from the beginning in equations A1 and A2, the solution corresponding to the boundary layer does not exist anymore.

FIG. 1 .

 1 FIG. 1. Reflection coefficient for a one-dimensional Alfvén wave packet hitting an insulating boundary with normal incidence, as a function of P m and for different magnetic Lundquist numbers Lu = VaL/η. The theoretical value for plane waves R(P m) = (1 -√ P m)/(1 + √ P m) fits the numerical simulation results perfectly.

FIG. 2 .

 2 FIG. 2. Snapshot of the azimuthal velocity component of Alfvén waves propagating in a non-rotating spherical shell. The dashed-lines are the imposed magnetic field lines. From left to right: (a) the incoming waves traveling from the inner shell to the outer shell along magnetic field lines; (b) case P m = 0.1, S = 1800 showing reflection with the same sign; (c) case P m = 1, S = 1000 with total absorption at the wall; (d) case P m = 10, S = 1800 showing reflection with opposite sign.

FIG. 3 .

 3 FIG.3. Reflection coefficient for a torsional Alfvén wave for insulating and no-slip boundary conditions, as a function of P m. The Lundquist number is always large (S > 5000 for P m ≥ 0.01 and S > 600 otherwise). For reference, the black curve is the planar Afvén wave reflection coefficient (1 -√ P m)/( √ P m + 1), and the red line marks the reflection coefficient for a stress-free boundary with P m = 1 (corresponding to a no-slip boundary with P m → 0)

FIG. 4 .

 4 FIG. 4. Space-time diagrams of the reflection of a TAW for S 10 4 and = 0.02 recorded in the equatorial plane, near the equator. Top row : stress-free boundary with P m = 1 (R = 0.88), (a) the azimuthal angular velocity u φ /s and (b) the azimuthal magnetic field b φ (changing sign). Second row : No-slip boundary with P m = 0.1 (R = 0.40), (c) the azimuthal angular velocity u φ /s and (d) the azimuthal magnetic field b φ (changing sign). Third row : No-slip boundary with P m = 10 (R = -0.17), (e) the azimuthal angular velocity u φ /s (changing sign) and (f) the azimuthal magnetic field b φ . Bottom row : azimuthal angular velocity u φ /s for no-slip boundary with P m = 1 showing no reflection (R = 0) for insulating boundary (g), and little reflection when the insulator is replaced by a solid conductive layer (h).

ACKNOWLEDGMENTS

The numerical simulations were run at the Service Commun de Calcul Intensif de l'Observatoire de Grenoble (SCCI). We want to thank Mathieu Dumberry and an anonymous reviewer for their help in improving this paper, and Henri-Claude Nataf for useful comments.

Appendix A: Analytic Alfvén wave solutions in one dimension 1. Plane wave solutions Following Jameson [13, p. 15-18], we look for plane wave solutions of equations 1 and 2, substituting u = U e i(ωt+kx) and b = √ µ 0 ρ Be i(ωt+kx) :

which we can combine into

for which the exact solutions are:

where is the reciprocal Lundquist number based on the frequency:

In the regime where Alfvén waves do propagate, we have 1 and also ω √ νη/V 2 A 1 so we can approximate the square root by its first order Taylor expansion, which leads to two solutions k 2 1 and k 2 2 :

The solutions k = ±k 1 = ±ω/V A (1 -i ), correspond to the propagation in both directions of an Alfvén wave at the speed V A and with attenuation on a length scale V A /( ω). The solutions k = ±k 2 ±i/δ correspond to a Hartmann boundary layer of thickness δ ≡ √ νη/V A .

Finally, from equation A1 and A2 we know that U and B are related for each k by:

and for the solutions k = ±k 1 and k = ±k 2 , it reduces to

This means that for the travelling wave solution, U and B have always the same amplitude and the same phase when propagating in the direction opposite to the imposed magnetic field, or opposite phase when propagating in the same direction. For the boundary layers, in the limit P m 1 they involve the velocity field alone, whereas for P m 1 they involve only the magnetic field.

Reflection coefficient at an insulating wall

In order to derive the reflection coefficient, we consider an insulating wall at x = 0 with an incoming Afvén wave from the x > 0 region (k = +k 1 ), giving rise to a reflected wave (k = -k 1 ). The boundary conditions are matched by a boundary layer (k = +k 2 ) localized near x = 0 (the solution k = -k 2 is growing exponentially for x > 0 and has to be rejected for this problem). The solution to this problem reads

where we have taken into account the fact that α -k1 = -α k1 (see eq. A8). The boundary conditions u = 0 and b = 0 at x = 0 lead to:

from which we find the amplitude β of the velocity boundary layer contribution, and the reflection coefficient R of the amplitude of the velocity component:

We are left to evaluate α k2 /α k1 using equations A8, which gives α k2 /α k1 = ν/η at leading order in , and thus

which is independent of ω and V A . In the case P m = 1, we then have R = 0 and β = -1 which means that no reflection occurs and that the amplitude of the incoming wave is canceled by the boundary layer alone.

It may be worth emphasizing that, although the boundary layer has the same thickness δ in the velocity and the magnetic field components, in the limit P m → 0, we have β → -2 and α k2 β → 0, so that the boundary layer is


Marie Ferru
email: marie.ferru@univ-poitiers.fr

Partners connection process and geography of innovation: new insights from a comparative inter-organizational partnerships analysis

Keywords: Collaborations, Embeddedness, Spatial patterns, Social ties, Coordination JEL O33, O31, O12

This paper attempts to give new explanations of the spatial patterns of collaborations focusing on the partners' connection process. Taking into account actors are embedded in an historical and socio-economical space, we first consider organizations can construct a new collaboration or renew an old one when they decide to build a research project with a partner. Then, going back to the moment when they initially connect with their partners through the analysis of the genesis of collaborations, we assume that they can turn to their interpersonal ties or to coordination resources. This process of connection may influence the spatial dimensions of collaborations regarding notably the literature linking the use of social ties and the spatial proximity effects observed in innovation activities. We test empirically these theoretical propositions through the collection of individual data about more than 200 histories of inter-company (IC) and scienceindustry (SI) partnerships. The qualitative and quantitative treatments of these data reveal the way partners connect each other and the spatial patterns of collaborations are significantly dependant to the nature of the partnerships. A strong regularity is nevertheless highlighted: for both partnerships (IC and SI ones), actors renewed prior collaborations in 57% of the total of studied partnerships.

INTRODUCTION

Given the strong polarization of innovation activities [START_REF] Puga | The rise and fall of regional inequalities[END_REF]Caniëls, 1997, etc.), an abundant literature has developed, particularly over the last twenty years, around the question of the geography of innovation. This issue is given to the foreground with the generalization of territorialized policies that promote innovation. To go a step further through this discussion, we propose to explore the spatial dimensions of the research collaborations, the major part of the innovation projects being nowadays built up with partners (Godin and Gingras, 1999;[START_REF] Wagner | Network structure, self-organization, and the growth of international collaboration in science[END_REF][START_REF] Hagedoorn | Inter-firm R&D partnering in pharmaceutical biotechnology since 1975: trends, patterns, and networks[END_REF]Ferru, 2009). Authors have generally focused exclusively on one stage of the collaboration process: the achievement of joint projects. They consider that the exchange of tacit knowledge between partners during this step is crucial and required face to face interactions [START_REF] Feldman | The Geography of Innovation[END_REF][START_REF] Foray | Distribution et expansion de la base de connaissances scientifiques et technologiques[END_REF]etc.). Thus, regarding the dominant thesis, partnerships are more likely to be built up locally (ie. between partners located in a same city, county or region). In this paper, we assume that the prior stages -the formation of the collaboration and notably the connection process of partners -also imply coordination problems that are necessarily prior to the exchange of knowledge and could also affect the spatial patterns of partnerships. The choice of the partner and its location depend indeed of this moment of collaborations formation and could thus explain the spatial proximity effects observed in the innovation activities.

We concentrate here on the connection process between organizations-partners that is relatively underexplored as [START_REF] Thune | University-industry collaboration: the network embeddedness approach[END_REF] underlined: "research focusing on process of forming, developing and coordinating university-industry collaborations has been fairly absent" (p.159). Even less authors link the formation process of partnerships with the spatial dimensions of collaborations: only few authors interestingly show the importance of "star scientists" relationships in the genesis of research collaborations built up at the local level [START_REF] Zucker | Intellectual human capital and the birth of U.S. biotechnology enterprises[END_REF][START_REF] Almeida | Localization of knowledge and the mobility of engineers In regional networks[END_REF][START_REF] Singh | Collaborative networks as determinants of knowledge diffusion patterns[END_REF][START_REF] Breschi | Mobility of skilled workers and co-invention networks: an anatomy of localized knowledge flows[END_REF], assuming that the spatial proximity of the innovation activities results from interpersonal relationships. These scholars mainly focus on the role of interpersonal ties in the formation of local partnerships and leave aside a great range of the collaborations: those created without going through social relationships and those built up at the non local level, however major nowadays (Hagedoorn, 2002;[START_REF] Giuri | Everything you always wanted to know about inventors but never asked : evidence from the PatVal-EU survey[END_REF]etc.). In addition, these researches confront to an empirical problem: « most of existing contribution on this matter lacks data at the individual level » [START_REF] Giuri | Inventors and the geographical breadth of knowledge spillovers[END_REF]. Thus, « a growing number of studies use patent information to apply social networks analysis (...) assuming relations between inventors who jointly worked on patents (Cantner et Graf, 2006). Social ties are identified through their results (co-patent, publications) but « this usual procedure is not appropriate » (ibidem). The goal of our paper is to deepen and enrich these researches at the theoretical and empirical levels 1) by analyzing widely and precisely the methods of linkage between two organizations-partners, 2) by exploring how this process of connection can participate to the local dimensions of collaborations and affect more generally the spatial patterns of partnerships.

To fulfill these intentions, we first propose a theoretical analysis of the process of partners' connection and its spatial impact on the geography of collaborations. Taking into account that actors 1 are embedded in a historical and socio-economic context, we first consider that organizations can decide to construct a new collaboration or renew a previous one. Coming back to the moment when they initially connect with their partners (ie. the genesis of new and renewed collaborations), we then assume that they can turn to their interpersonal relationships or to more classic forms of mediation -we called here resources of coordination -such as the Internet, transfer centers, conferences, etc. Our theoretical analysis has to explain how the use of these two different connection modalities can affect the spatial dimensions of collaborations and how it contributes (or not) to the existence of spatial proximity effects. We will be able to test if the recourse to one particular modality of connection favors local (or non local) partnerships.

We need to create an empirical methodology consistent with this theoretical grid that can help us to observe the formation of collaborations and its impact on the geography of partnerships. Hence, individual data was collected through semi-structured interviews with R&D managers directly involved in research partnerships. This method provides a vast amount of information about more than 200 pairs of collaborations: the history of the collaboration (new vs. renewed), the modalities used to connect a partner (interpersonal relationships vs. coordination resources), the spatial dimensions of the collaboration studied (local vs. non local) and the scientific domain or economic activity of the partners. Among the cases studied, 130 deal with science-industry (SI) collaborations and 93 with inter-company (IC) collaborations. A comparison of these two types of interorganizational partnerships is thus possible. This has been rarely built up until now, except [START_REF] Ponds | The geographical and institutional proximity of research collaboration[END_REF] that observe that spatial proximity is more important for SI collaborations than IC ones. We will be able to confirm (or not) these results and more generally explore the formation and the geography of inter-organizational collaborations in two different contexts. From a methodological standpoint, our analysis of collaboration stories combines qualitative and quantitative methods to answer several questions such as: Do the organizations renew collaborations in the major cases? What is the respective weight of interpersonal relationships and coordination resources in the connection process of two organizations-partners? Does the use of interpersonal ties lead necessarily to spatial proximity effects? More generally, does one particular modality of connection foster the local collaborations or the non-local ones? And finally, do these results differ according to the type of partnerships?

The paper is organized as follows: the second section presents our theoretical grid and deduces propositions concerning the link between the modalities of partners' linkages and the geography of collaborations. The empirical work is introduced in a third section with a description of the method used to collect qualitative data about the genesis of collaborations. The fourth section gives the results of our empirical study. We finally conclude and discuss our main results in a last section.

II. THEORY AND PROPOSITIONS

Collaborations are increasingly crucial in the innovation process of companies (Godin and Gingras, 1999;[START_REF] Wagner | Network structure, self-organization, and the growth of international collaboration in science[END_REF][START_REF] Hagedoorn | Inter-firm R&D partnering in pharmaceutical biotechnology since 1975: trends, patterns, and networks[END_REF]. Hence, studies have been developed around this theme. They have mainly focused on the achievement of the partnership, considering the knowledge exchange during this stage as a crucial step and facilitated by the spatial proximity between partners [START_REF] Feldman | The Geography of Innovation[END_REF]Foray, 1995, etc.). This thesis gives an explanation to the spatial agglomeration of innovation activities observed for some years and to the success of clusters such as the Silicon Valley. Some authors in the nineties, mainly from United States, like Powell on biotechnologies in California [START_REF] Powell | Competitive cooperation In biotechnology : learning through networks ?[END_REF][START_REF] Powell | Networks and Economic Life[END_REF] or Almeida and Kogut (1997) or [START_REF] Zucker | Intellectual human capital and the birth of U.S. biotechnology enterprises[END_REF] observed the important role of interpersonal ties of star-scientists in the establishment of the local collaborations studied. More recently, [START_REF] Breschi | Tracing the links between science and technology: An exploratory analysis of scientists' and inventors' networks[END_REF] or [START_REF] Breschi | Mobility of skilled workers and co-invention networks: an anatomy of localized knowledge flows[END_REF] confirm the important role of author-inventors in the networking between academic and innovation circles. They conclude that the local dimension of collaborations result partially from the use of interpersonal relationships tied locally and they give an alternative explanation to the polarization of innovation activities. These scholars underlined implicitly the importance of the connection process between the partners. Nevertheless, they focus exclusively on interpersonal ties and they consider them as the only way of connecting. Moreover, these analyses are focused on local territories. Thus, they failed to address a great range of collaborations cases, those created without going through social ties and those built up at the nonlocal level.

Our work is in line with these latest studies but seeks to complete them through an enlarged analysis of the connection process of partners and thanks to the integration of the non-local partnerships. For this purpose, we propose, first, to "decrypt" the connection process of the organizations-partners by taking into account the embeddedness of actors and, then, we deduce how this formation of partnerships affects the spatial dimensions of collaborations. We detail this theoretical grid of analysis sum up in the next figure. Scholars dealing with the formation of research collaborations are fewer even though this stage constitutes a key stage in the collaboration process by determining the choice of the partner and hence the partner localization. Some scholars shed light on the reasons and motives of these partnerships; others studied the partner choice [START_REF] Gulati | Network location and learning: the influence of network resources and firm capabilities on alliance formation[END_REF][START_REF] Ahuja | The duality of collaboration: inducements and opportunities in the formation of Interfirm Linkages[END_REF][START_REF] Tether | Who co-operates for innovation, and why. An empirical analysis[END_REF][START_REF] Miotti | Co-operative R&D: why and with whom?: An integrated framework of analysis[END_REF][START_REF] Giuliani | What drives the formation of `valuable' university-industry linkages? Insights from the wine industry[END_REF]etc.) and analyse the formation of collaborations [START_REF] Kreiner | Informal Collaboration in R&D. The Formation of Networks Across Organization[END_REF] but nobody has given a precise understanding of the genesis of the interorganizational collaborations and really explain the connection process of the partners.

Regarding the literature about inter-organizational collaborations, the concept of social ties is increasingly used to show their embeddedness in interpersonal relationships and their advantages in terms of information access, reliability and trust. The embeddedness concept, however relevant, is nevertheless limited to the social context of actors and "it is crucial to bring history and complex social structures back into the analysis of socioeconomic phenomena such as inter-firm partnerships"

Actors renew prior collaborations

Genesis of the collaboration

Actors rely on coordination resources

Social embeddedness

Economic embeddedness [START_REF] Hagedoorn | Understanding the cross-level embeddedness of inter-firm partnership formation[END_REF]. We propose to take into account that actors are embedded in a social, economic and historic context (through the terms of historical, economical and social embeddedness) to better understand the connection process between research partners.

Historical embeddedness: the renewed partnerships. By focusing on the moment when the organizations build up their research project and their collaboration, we can easily consider that they do not search necessarily a new partner, they are able to turn to a former partner and renew a previous partnership, revealing the importance of the historical embeddedness of actors. The existence of renewed collaborations has been already underlined by [START_REF] Gulati | Social structure and alliance formation patterns: a longitudinal analysis[END_REF] in a longitudinal analysis of partnerships: "prior alliances play an important role in shaping future alliance formation" (p.620). "The specifics of the partnering history of companies jointly affect their future partnerships formation" (Hagedoorn, 2006, p.674). [START_REF] Thune | University-industry collaboration: the network embeddedness approach[END_REF] recognized empirically that "direct experience can lead to the development of trust between the parties and a reputation for being trustable and reliable which as seen as vital resources for formation of further collaborative relationships" (p.156). The information that the previous alliances provide counterbalances the risks associated to alliances. In addition, through prior collaborations the partners learn about each other's needs, capabilities, know-how and might manage easily the relationships [START_REF] Thune | University-industry collaboration: the network embeddedness approach[END_REF]. Mutual knowledge gained through past interactions is the basis of "learning by co-operating", which allows these partners to be relatively more effective than others when working together because everyone knows the expectations, capabilities and methods of the other [START_REF] Brousseau | La gouvernance des processus de coopération[END_REF]. In sum, the renewal of previous collaborations brings benefits in terms of trust [START_REF] Granovetter | Economic Action and Social Structure: the Problem of Embeddedness[END_REF], transaction costs [START_REF] Williamson | The economic institutions of capitalism, The free press[END_REF] and learning [START_REF] Dosi | Les frontières des entreprises: vers une théorie de la cohérence de la grande entreprise[END_REF]. Hence, given the costs of finding new partners, one can expect the company to first consider organizations with which it has shared a common partnerships experience. This result already showed by [START_REF] Thune | University-industry collaboration: the network embeddedness approach[END_REF] -"the large majority of the University-Industry collaborations investigated have been initiated and formed through the use of already established contacts"-and corroborated by other data sources [START_REF] Schartinger | Knowledge interactions between universities and industry in Austria: sectoral patterns and determinants[END_REF]. Based on these scholars, we formulate a first proposition:

(P1) Organizations renew partnerships in the major cases whatever the nature of the collaboration.

Such repetitive behavior results from the establishment of routines. This highlights the inertial behavior of organizations and their dependence on the past trajectory of their learning [START_REF] Maskell | Localized learning and industrial competitiveness[END_REF], as well as the existence of « lock-in » [START_REF] Arthur | Competing technologies, increasing returns and lock-In by historical events[END_REF] or over-embeddedness effects [START_REF] Uzzi | The sources and consequences of embeddedness for the economic performance of organizations: the network effect[END_REF].

Economic and social embeddedness: interpersonal ties vs. resources of coordination.

What is even more underexplored and crucial in the formation of collaborations is the genesis of partnerships, the very first stage when partners initially connect with each other. We have to identify the key element allowing the organizations to build the partnership with this partner for the first time.

We can consider first, in the light of the approach developed by Granovetter, that organizations are embedded in interpersonal ties. In a famous article published in 1985, Granovetter defended the thesis that economic activities depend on interpersonal relationships in which the actors are involved. He called this dependence embeddedness. Thus, relying on a study by [START_REF] Eccles | The quasi firm in the construction industry[END_REF], Granovetter shows that relations among companies (prime manufacturers and subcontractors in the construction field) are underlain by interpersonal relationships. By integrating the social embeddedness of actors in our analysis, we want to take into account the borders porosity of organizations from the individual exchanges (Grossetti and Bès, 2003): "Behind the formal ties [ie.

the collaboration] we find informal relations that give them life, support them and frame their development" (Powell and Smith-Doerr, 1994, p.384). We thus need to analyze the interorganizational collaborations at the individual level and we approach the social embeddedness by measuring the place of direct and indirect interpersonal ties in the formation of pairs of collaborations. Thus, we do not search to reconstruct the whole social network structure of the organization involved in the partnership and we use a restrictive definition of social ties compared to authors interested in social network2 since we consider dyadic inter-personal ties only. Authors that underlined interpersonal exchanges in the formation of partnerships neglect more classic form of coordination [START_REF] Bouty | Interpersonal and Interaction Influences on Informal Resources Exchanges Between R&D Reasearchers Across Organizational Boundaries[END_REF][START_REF] Hansen | The Search-Transfer Problem: The Rôle of Weak Ties in Sharing Knowledge across Organization Subunits[END_REF].

Organizations are indeed not only embedded in a social context but also in an economic one. Hence, when they are not connected by interpersonal relationships, they can use material or cognitive resources that serve to mediate between the players and facilitate the bringing together of partners. The very first contact between organizations can result from a meeting in exhibitions, conferences or from public information through the Internet, through public resources dedicated to making links (for instance in France, the centres de ressources technologiques (CRT) and the competitiveness clusters). [START_REF] Eom | Determinants of industry-academy linkages and their impact on firm performance: The case of Korea as a latecomer in knowledge industrialization[END_REF] have notably already underlined the effect of technology transfer centres. These various resources of coordination correspond to a continuum ranging between the two extreme forms of market and hierarchy (see list 1). A connection of partners through these resources has to be differentiated from a connection through interpersonal ties. When using a coordination resource such as reputation, there is not a real interpersonal relationship that linked the two organizations but an impersonal structure that mediate them. And even, when an individual actor is involved, through a conference or an exhibition for instance, this latter behaves under the auspices of the organization and the connection does not depend on one particular individual. Relying on these resources facilitates the partners' connection while reducing various transaction costs. Like interpersonal ties, they limit the uncertainty in the selection process of a partner and the risk of opportunism of this latter [START_REF] Williamson | The economic institutions of capitalism, The free press[END_REF] since they give a framework to the economic action, set rules of interaction and are sources of information considered reliable.

Finally, we consider that the very first connection between the two organizations is achieved by a key modality: it could be a chain of interpersonal relationships or a coordination resource. This analysis of the genesis of inter-organizational collaborations involves the mixing and matching of the individual and organizational levels to identify the key modality of linkages between partners.

Regarding the theoretical literature, collaborations through coordination resources and notably those that are close to the market can be less used than interpersonal ties since « new knowledge creation goes along with uncertainty and opportunism, markets often cannot offer this because it would involve too high transaction costs » (Boschma, 2005, p. 65). We thus formulate a second proposition:

(P2) Partners are more likely to turn to interpersonal relationships than coordination resources.

By integrating the socio-economic and historic context in which actors are embedded, we have reported the existence of renewed collaborations, on the one hand, and of two modalities of connection, on the other, in the formation process of collaborations. We need to determine now the spatial effects resulting from this process.

2.2.

How does the connection process affect the spatial patterns of partnerships?

We search to know if one particular modality of connection fosters the local collaborations or the non-local ones. As explained before some authors linked the spatial proximity effects observed in the innovation process to the use of interpersonal ties [START_REF] Powell | Competitive cooperation In biotechnology : learning through networks ?[END_REF]Almeida and Kogut, 1997;[START_REF] Breschi | Tracing the links between science and technology: An exploratory analysis of scientists' and inventors' networks[END_REF]etc.). This thesis is reinforced by the scholars that show the local dimension of interpersonal ties [START_REF] Fischer | To dwell among friends[END_REF]. Relationships are assumed to be tied more easily in the neighbourhood: "the greater the distance, the less contact and support" (Mok et al., 2007, p.434). Some empirical studies - [START_REF] Wellman | Are personal communities local? A Dumptarian Reconsideration[END_REF] on a sample of residents of Toronto, [START_REF] Fischer | To dwell among friends[END_REF] on the population of San Francisco and [START_REF] Grossetti | Are French networks different?[END_REF] on Toulouse -confirm that personal ties include a large share of local relationships. This is also the case in studies directly inspired by the new economic sociology [START_REF] Saxenian | Regional advantage: culture and competition in Silicon Valley and Route 128[END_REF], [START_REF] Ferrary | The role of venture capital firms in Silicon Valley's complex innovation network[END_REF], etc.). Economists and geographers observe that strong and weak ties exist between agents in a limited area, describing this situation as "local buzz" [START_REF] Bathelt | Clusters and knowledge: local buzz, global pipelines and the process of knowledge creation[END_REF][START_REF] Thune | Proximity and interactive learning in University-firm relationships[END_REF]. We thus formulate a third proposition:

(P3) The use of Interpersonal ties leads to the formation of local partnerships.

On the contrary, coordination resources that may be similar to institutions are assumed to have national characteristics [START_REF] Lundvall | National systems of innovation: Towards a theory of innovation and interactive learning[END_REF][START_REF] Amable | Les systèmes d'innovation à l'ère de la globalisation[END_REF][START_REF] Bathelt | Clusters and knowledge: local buzz, global pipelines and the process of knowledge creation[END_REF] although they seem to grow more and more at a sub-national level (decentralization), as well as on a supranational (eg. European) scale. Considering the previous studies, we propose the following proposition:

(P4) The use of coordination resources fosters national partnerships.

Finally, we can consider the renewal of prior partnerships determines at least a part of the dynamic of the spatial patterns of collaborations: it participates to its spatial concentration (if the players renew mainly local collaborations), its spatial dispersion over time (if the players renew essentially non-local collaborations) or to its spatial inertia (if the players renew their collaborations at the same scale). With regard to the existing literature that assumes the higher efficiency of spatial agglomeration of innovators (see [START_REF] Krugman | Increasing returns and economic geography[END_REF] for instance), we can deduce that partners are more likely to renew local partnerships than non-local ones. Hence, we formulate a fifth proposition:

(P5) The renewal of prior partnerships favors the spatial agglomeration over time.

III. DATA, METHOD AND FIRST DESCRIPTIVE STATISTICS

We need to test the five theoretical propositions by exploring in detail how collaborative projects emerged. With this perspective, a set of qualitative data is collected through interviews involving the reconstruction of histories of collaborations. The codification of this information enables the building of a qualitative database sufficiently consistent for statistics tests.

COLLECT OF QUALITATIVE DATA: RECONSTRUCTING COLLABORATION STORIES

Empirical data is qualitative and is collected through semi-structured interviews. More specifically, we selected researchers or R&D managers who held scientific responsibility for research contracts achieved with a partner and we asked them to tell us about these projects. It is crucial to have a sufficiently sound method for obtaining relevant information about the formation of partnerships and revealing the modalities of connection between partners. To identify interpersonal ties, the relational chains method is used. Famous examples of this method utilization are the survey by Milgram on "small worlds" (1967) and that by Granovetter on access to employment (1974). In this method, it is not a matter of analyzing static structures but rather of recourse to interpersonal relations in order to access resources. Interpersonal relationships actually mobilized to connect a partner are taken into account (and not the whole social network of the organization studied). Applied to the case of collaborations, this method consists of reconstructing the histories of various couples of collaborations. The interviews each lasted about 90 minutes and were recorded and transcribed after each interview. More than 95% of them were conducted face to face between 2007 and 2008. These interviews give precise information we could not have with statistics or even questionnaires. We manage to avoid as much as possible the potential bias linked to this method (problems of incomplete memories of interviewees, or false re-constructed memories that are not reflecting the real process, etc.) thanks to complementary sources of information (documents in local media or local organizations) and cross-referenced interviews (conducted by phone in the major cases and lasting an average of 40 to 45 minutes). In addition, to be as neutral as possible and not influence the interviewed person, we let her/him explained in detail the research project from the beginning to the end. To obtain precisions about the linking of partners, questions can be raised: "With whom did you collaborate to implement this project?", "Is it the first innovation project with this partner?, "How did you find this partner?", etc3 . We are thus able to determine if the partners build a new partnership for the research project studied or renew an old one. We can also identify the genesis of the collaborations (new and renewed ones) and determine if organizations have collaborated for the first time through interpersonal relationships (or coordination resources. We also obtain information about the spatial dimension of the collaboration studied and the scientific domain/economic activity of the two partners.

In the interviews, the unit of analysis is the collaboration that corresponds to any research projects built up with one other organization, with or without success. Data collection involves formal collaboration contracts (either under way or completed) between a laboratory and a firm or between two companies. Innovation partnerships between firms, by definition, consist of bringing together partners from the same milieu -as opposed to SI collaborations.

The construction of the overall sample relies on two databases: one covers all of the information collected by the research development unit of Poitiers and relating to the contracts signed by Poitiers University academics from 2004 to 2007 (about 300 with companies). The other, a public directory, brings together all of the information about companies located in the Châtellerault area. Finally, we have been able to reconstruct 203 collaborations stories. 115 concern contracts made by the laboratories of the University of Poitiers -a relatively large and multidisciplinary university -with a company. 88 collaboration stories deal with contracts made by companies located in Châtelleraulta French average-sized area historically industrial and specializing in sectors of medium-low technology -with another company. Unlike previous works which were generally focused on large urban high-tech agglomerations, considered as the main driver of innovation, the two sites studied here are located in Poitou-Charentes, covering areas of average size, adding another perspective to the existing literature.

CODING AND TREATMENT OF QUALITATIVE DATA

Key information relating to the formation process is extracted and coded (according to theoretical typology previously presented) to build a database. All the information is turned into a dichotomous or a polytomous variable.

Data about the collaborations stories indicate the history of the collaborations (RENEWED = 1 if the organization has renewed an old collaboration, 0 if the organization built a new partnership), the modalities of connection (TIES = 1 if the organization initially relied on interpersonal ties to connect with its partner, 0 if the organization initially had recourse to a coordination resource), the nature of the collaboration (IC_COLLAB= 1 if the collaboration is achieved by a Châtelleraudais firm with another company, 0 if the collaboration is established by a laboratory of the University of Poitiers with a firm). We also include measures of the interviewed partner specialization: the scientific domain of the laboratories of Poitiers (Engineering Sciences (), Chemical Sciences () Physical Sciences (), Biological Sciences, Humanities and Social Sciences, Information Technology ()) and the sectoral activity of the Chatelleraudais companies (Chemical industry (), Advices and assistance (), Household equipments (), Electric and electronic equipments (), Mechanical equipments (), Food industry (), Automotive industry (), Metallurgy ()) involved in the collaborations studies.

Our analysis of the collected data combines qualitative and quantitative methods. A qualitative study gives, first, precisions about the formation of collaborations and determines the role of past and the weight of each modality of connection. By cross-tabulating the two modalities of connection (interpersonal ties vs. coordination resources) with the economic/scientific specialization of the interviewed partners, the nature and the spatial dimensions of the partnerships, we give descriptive statistics. Independence tests and notably khi² tests are done to give more rigorous testing of these links. We can verify whether the characteristics of the actors involved in the collaborations and the nature of the partnership influence recourse to one particular modality of connection. Thus, we can deduce if the way that partners connect is empirically stable. Moreover, we test the dependence between each modality of connection and the spatial dimensions of the collaborations. Hence, we verify whether the interpersonal tie is significantly associated with the local collaborations and the coordination resources with the national ones (P3 and P4). We also check if the renewal of previous partnerships leads to spatial agglomeration effects (P5).

SAMPLE AND FIRST DESCRIPTIVE STATISTICS

The table below presents the data. We give first descriptive statistics about our sample and we notably describe the scientific / economic specialization of one of the partners as well as the spatial patterns of the studied partnerships. Specialization of the partners involved. Among the academic partners, a great part is specialized in the engineering or chemical domains (in 57% and 28% of the interviewed laboratories respectively). Among the industrial partners, they are generally large-sized firms (72% of the interviewed companies have more than 250 salaries) and belong to a group (67% of the companies) and to the construction of automotives, other vehicles or R&D (12%, 19% and 11% of the companies respectively).

In relation to the IC collaborations, the partners located in the Châtellerault area are mainly specialized in the mechanical and metallurgy industries, since these two sectors constitute more than 40% of the total. The over-representation of these industries is representative of the sectoral specialization of the studied area.

Localization of partners and spatial patterns of partnerships. In the collaborations studied, at least one of the partners is located in Poitou-Charentes. This region has a great endowment in terms of R&D thanks to the University of Poitiers 4 . Nevertheless, the region is quite small (regarding different size indicators) and is the 15th out of the 22 regions in France. It gathers a fewer number of companies (mostly SME's specialized in low-medium technology) and the private R&D expenses are lower compared to other regions. Poitou-Charentes present an ambiguous situation: it is the fifth most important region in France regarding the public research expenses and the weight of public researchers whereas it is the 18th in terms of private researchers.

The partners of the company or the laboratory located in Poitou-Charentes are located within or outside the partner's region. They are not concentrated in one particular area revealing the multiscalar character of inter-organizational partnerships. The extra-regional collaborations, however, appear largely dominant, in particular those established at national level, since more than 60% is established at this level (see table above). The local dimension is quite low: less than one fourth of the partners are located within Poitou-Charentes. The international level is even lower, a foreign partner being involved in only 14% of the studied collaborations. We can nevertheless observe specificities according to the nature of the partnerships: for IC ones, the local dimension appears more important (almost 40% of the partnerships are made within the region) whereas for SI collaborations, we note the hypertrophy of Ile-de-France (more than half of all French contracts being established with a firm located in the capital region). These results are close to those obtained in previously conducted studies (see notably [START_REF] Giuri | Everything you always wanted to know about inventors but never asked : evidence from the PatVal-EU survey[END_REF]) but the number of contracts with the Ile-de-France is however relatively higher and the local collaborations are fewer (notably for SI collaborations) in our empirical study. This difference between our results and those from previous studies may be explained, at least in part, by the structural characteristics of the area under investigation and, in particular, by its economic specialization and its relatively low industrial density.

As said previously, the studied area is mainly composed of small unit of production without R&D center. The industrial specialization of these companies is moreover hardly compatible with the scientific expertise of the laboratories of the University of Poitiers. These structural effects play a structuring role in the spatial patterns of the collaborations have already shown [START_REF] Ferru | Formation process and geography of science-industry partnerships: the case of the University of Poitiers[END_REF][START_REF] Bouba-Olga | Exploring spatial features of science-industry partnerships: a study on French data[END_REF])

IV. RESULTS

The following paragraphs detail the main results that emerge from the qualitative and statistical treatments of the collected data. We first give precisions about the formation of partnerships before analyzing the impact of each modality of connection on the spatial patterns of collaborations.

EMBEDDEDNESS OF ACTORS AND MODALITIES OF CONNECTION WITH PARTNERS

Taking into account the historical and socio-economic context in which actors are embedded when they connect with a partner, we analyze the 203 collaboration stories and test the two first propositions introduced in our theoretical grid.

First, in 57% of the studied collaborations, project leaders have renewed past partnerships. Hence, as predicted (P1), in most cases, actors have renewed their previous collaborations to achieve their collaborative project, in both partnerships the SI and IC ones (57% for each). A great number of partnerships exist for a long time since project leaders have a preference for a former partner: "renewed collaborations are really easy, we know each other" commented a researcher. The interviews confirm the renewal of partnerships gives advantages in terms of trust and learning, reducing transaction costs. They also interestingly show that the research of the two partners is defined according to their respective needs in some "historical" partnerships. These historical partnerships are considered to be necessary for the organizations with a high turn-over of scientific leaders because "the laboratory constitutes the memory of the research of the firm". This result confirms that history matters and it emphasizes the inertial behavior of the players concerned and the path dependant innovation trajectory. Secondly, the analysis of the genesis of collaborations reveals that partners use their interpersonal ties in 33% to connect for the first time with their partner and their coordination resources in 67%. Contrary to P2, interpersonal ties are not the major modality of connection. Interpersonal ties are even less used in our study than in previous investigations about the genesis of SI collaborations. Based on Granovetter's methodology (1973), Grossetti and Bès (2003) latter observed that interpersonal ties are used in 44% to find their partner.

Taking into account the nature of the partnership, we add some precisions: the recourse to one modality of connection differs whether the partnership is IC or SI. Social ties are even less mobilised in IC collaborations, the use of interpersonal ties is the exception and the linking occurs more often through a coordination resources with about 85% of project leaders relying on such a procedure to construct the partnership initially. These modalities and notably the conference meetings are essential. Industry becomes familiar with research undertaken by laboratories on specific themes.

One scientific leader of an automotive company explained he went to a conference in relation to an item that the firm was interested in and met researchers who presented their research. The scientist responsible was impressed by the results of one researcher of a laboratory of Poitiers. As a result, the project leader of the company discussed with this researcher to know whether the laboratory of Poitiers would be interested in working with his company in relation to the research he presented.

The interpersonal ties and notably the teaching ties are over-represented in SI collaborations (relatively to the intercompany partnerships). The relationship with former doctoral students is a recognized as a source of partnerships these last years [START_REF] Fleming | The evolution of inventor networks in the Silicon Valley and Boston regions[END_REF][START_REF] Thune | Proximity and interactive learning in University-firm relationships[END_REF][START_REF] Bozeman | Editor's introduction: building and deploying scientific and technical human capital[END_REF]. It facilitates the reconciliation of two worlds relatively distinct and separate, as is the case for science and industry, through their knowledge of these two fields. In our interviews, one researcher states that "They are a key means", while another adds that "They make the contact easier with the manufacturer, because they are familiar with the laboratory's know-how".

One researcher explains that one of his collaboration with a foreign firm would not have been possible without this relation associated with education: before this former doctoral student was recruited into the company, the laboratory was already interested in the firm's know-how and wanted to collaborate with it; it had tried to contact it via the internet to propose collaborative projects, but the firm had never followed up.

The khi² test confirms the significant (p=0,000) dependence between the modality of linkage and the nature of the partnership (see table 2). The statistic Cramer's V that measures the strength of association or dependency between the variables is around 0.32 indicating the high degree of dependency5 . The cross-tabulation of the modalities of connections with the other characteristics of the partners and the khi² tests associated reveal that the recourse to one modality of connection does not depend significantly of the scientific or sectoral specialization.

SPATIAL PATTERNS OF PARTNERSHIPS

The table 3 gives some results about the modality of connection used and the spatial patterns of the collaboration. First, we cannot conclude that interpersonal relationships are necessarily linked to the local dimension contrary to P3 and the literature that highlights the positive effects of the "local buzz". Ties can emerge and can increasingly be kept at a distance through ICT as assumed [START_REF] Lorentzen | The spatial dimension of innovation. Embedding proximity in socio-economic space[END_REF]. Thus, the use of interpersonal relationships does not necessarily lead to local collaborations. The structural characteristics of the Poitou-Charentes region (where one of the partners is located) can explained a part of this result: its low industrial and scientific density limits the opportunities for local ties and notably those related to education since the recruitment of former doctoral students in the region is really low.

Secondly, it appears that the use of renewed partnerships is not linked to the local dimension and thus the renewal of partnerships does not favor the spatial agglomeration of collaborations over time. This result contradicts P5 and the existing scholars that underline the higher efficiency of colocation of innovators. dependence of actors on the past collaborations leads mostly to the strengthening of the existing geography of partnerships over time.

Thirdly, we observe that the recourse to the coordination resources to connect a partner is generally to the non-local collaborations, confirming P4. The interviews also revealed that the public coordination resources created to facilitate the linking of regional partners (such as CRT in France) are rarely used.

These results differ again according to the nature of the partnership. The cross-tabulations and the Khi² tests summarized in the table 5 reveal the significant relation of dependence between the nature of the partnerships and its spatial patterns. The IC collaborations are over-represented within the region and it seems that collaborations with a partner that belongs to the same milieu (industry) favour the co-location of partners contrary to Ponds et al. Our result could be explained by the low level of skill/education of workers that gives to these latter a lower spatial mobility and a lower capacity of absorption. It thus leads to the construction of collaborations at a more local level. The partner searched in the IC collaborations is also less specific and could be found in various places and hence more easily in the region. The values of the Cramer's V (between 0.17 and 0.32) indicate also the important degree of dependency. NB: X² corresponds to the sum of the differences between the theoretical values and the observed values, and p corresponds to the likelihood of an associated error.

V. DISCUSSION AND CONCLUSION

This paper has searched to give new explanations of the spatial dimensions of collaborations focusing on the formation process of partnerships. Taking into account that actors are embedded in an historical and socio-economical space, we first consider that when organizations decide to build a research project with a partner, they can construct a new collaboration or renew an old one. Then, going back to the very first moment when partners connect to each other -through the genesis of new and renewed collaborations -we assume that they can turn to their interpersonal ties or to coordination resources. We believe that the way that organizations form their partnership can influence the spatial patterns of the partnership created. We have thus proposed a theoretical analysis of this connection process between partners and its spatial impact. We have then verified the validity of this theoretical grid through the collection of individual data from more than 200 accounts of SI and IC partnerships.

The qualitative and quantitative treatments of these data have revealed several interesting results. First of all, in a large majority of collaborations, organizations have renewed prior collaborations whatever the form of the partnership (SI or IC). Secondly, interpersonal ties are not the major modality to access a partner for the very first time. They are even less used in IC collaborations, revealing differences in the connection process according to the nature of the partnership. Our data also qualifies the dominant literature that associates interpersonal ties to the spatial proximity effects: interpersonal ties are underrepresented when partners are co-located. Finally, dependence tests highlight differences in the spatial patterns of partnerships according to the nature of the collaborations: those built up between companies are deployed more frequently at a local level than is the case for SI collaborations, which may merely reflect the importance of the partnership nature.

Scope and limitations of our findings. Our results cannot be generalized and are not widelyapplicable. They are context sensitive: the structural characteristics of the studied area, Poitou-Charentes (at least one partner belongs to this region), have to be taken into account. The low local dimension observed in our empirical study can be explained by the relatively lower density of this region, a factor which greatly reduces the possibilities for local relationships and collaborations [START_REF] Ferru | Formation process and geography of science-industry partnerships: the case of the University of Poitiers[END_REF][START_REF] Bouba-Olga | Exploring spatial features of science-industry partnerships: a study on French data[END_REF]. The structural specificities of Poitou-Charentes could explain our findings contradict the previous ones. The investigations by Grossetti and Bès (2003) about the genesis of SI partnerships in five French agglomerations with the same methodology gave different results and confirmed the spatial proximity effects generally associated to interpersonal ties. As explained in the previous section, according to the industrial and scientific density of the territory studied, opportunities for partners to tie local relationships (related to education notably) differ.

Implications. By shedding light on the importance of renewed partnerships and on the weakness of local collaborations, our results challenge the clusters policies that promote local collaborations and want to foster them. Organizations are more likely to renew a previous partnership regardless where the partner is located. Our study reveals also the diversity of situations according to the nature of the partnerships. This result is important as it sheds light on the necessity to qualify what the literature show when it tends to generalize its results. It also sets new challenges for policy makers who have to integrate this complexity.

Figure 1 :

 1 Figure 1 : Theoretical grid of analysis

  associated with the vector of connection? ! Yes: Co-location of partners ! Local collaborations ! No: The connection process crosses territories ! Non local collaborations

Actors turn to interpersonal ties

	2.1
	How do
	partners
	connect
	each other?
	2.2
	How does it
	affect the
	spatial
	patterns of
	partnerships?

Table 1 : Data and first descriptive statistics Variable Procedures Number (out of 203)

 1

	Type of partnership	IC collaborations SI collaborations	88 115
		Engineering sciences	66
		Chemical sciences	32
		Physical sciences	8
		Biological sciences, humanities and social sciences, information technology	8
	Scientific / sectoral specialization of partner	Chemical industry Advices and assistance Household equipments	7 11 12
		Electric and electronic equipments	9
		Mechanical equipments	21
		Food industry	7
		Automotive industry	6
		Metallurgy	15
	History of collaborations	Renewed partnerships New partnerships	115 88
	Initial modalities of	Interpersonal ties	20
	Connection	Coordination resources	68
	Type of inter-personal ties	Non professional ties Professional ties Teaching ties	11 34 21
		Total number of observations=203

Table 2 : Dependence between the modalities of connection and the nature of partnerships

 2

		IC	SI	Total	Results of khi² tests
	Coordination resources	75 (85%)	62 (54%)	137 (67%)	X²=20.87
	Interpersonal ties	13 (1%)	53 (46%)	66 (33%)	p=0.000
	Total	88	115	203	Dependence
	NB: X² corresponds to the sum of the differences between the theoretical values and the observed values, and p
	corresponds to the likelihood of an associated error.

Table 3 : Formation and spatial patterns of partnerships Interpersonal ties Coordination ressources Renewed collaborations

 3 The collaboration is qualified as local when the two partners are located in the same region.

	Local	a	1	16	30
	Non local	19	52	85
	a

Table 5 : Nature of the partnership and its spatial patterns SI collaborations IC collaborations Results of Khi² test

 5

	Local	12%	25%	X²=5,6287
				p=0,017
	Non local	88%	75%	Dependence

By actors, we mean organizations or individuals since the connection process between partners combines and mixes these two levels of analysis (cf. infra).

The term of « interpersonal ties » is privileged to the "social network" one to reduce ambiguity. Our analysis is nevertheless in the line of[START_REF] Granovetter | Economic Action and Social Structure: the Problem of Embeddedness[END_REF] and[START_REF] Wellman | Are personal communities local? A Dumptarian Reconsideration[END_REF] studies since they have been also interested in dyadic ties.

We never use explicitly during the interview the terms of "interpersonal relationship" or "coordination resource" to avoid any biases. We search to understand precisely how the first collaboration happened and reinterpret thereafter according to our theoretical grid.

The University of Poitiers is one of the older in France and in Europe: it gathers 24 000 students or so and 934 researchers.

The value of the contingency coefficient confirms this result.