Henri-Claude Nataf

Magnetic induction maps in a magnetized spherical Couette flow experiment

Keywords: dynamo, magnetohydrodynamics, omega effect, liquid sodium, DTS magnétohydrodynamique, effet oméga, sodium liquide, DTS

The DT S experiment is a spherical Couette flow experiment with an imposed dipolar magnetic field. Liquid sodium is used as a working fluid. In a series of measurement campaigns, we have obtained data on the mean axisymmetric velocity, the mean induced magnetic field and electric potentials. All these quantities are coupled through the induction equation. In particular, a strong ω-effect is produced by differential rotation within the fluid shell, inducing a significant azimuthal magnetic field. Taking advantage of the simple spherical geometry of the experiment, I expand the azimuthal and meridional fields into Legendre polynomials and derive the expressions that relate all measurements to the radial functions of the velocity field for each harmonic degree. For small magnetic Reynolds numbers Rm the relations are linear, and the azimuthal and meridional equations decouple. Selecting a set of measurements for a given rotation frequency of the inner sphere (Rm ≃ 9.4), I invert simultaneously the velocity and the magnetic data and thus reconstruct both the azimuthal and the meridional fields within the fluid shell. The results demonstrate the good internal consistency of the measurements, and indicate that turbulent non-axisymmetric fluctuations do not contribute significantly to the axisymmetric magnetic induction.

Résumé

Cartographie de l'induction magntique dans un écoulement de Couette sphérique soumis à un champ magnétique. L'expérience DT S consiste en un écoulement de Couette sphérique soumis à un champ magnétique dipolaire. Le fluide utilisé est du sodium liquide. Au cours d'une série de campagnes de mesure, nous avons obtenu des données sur le champ de vitesse moyen axisymétrique, le champ magnétique moyen, et le potentiel électrique. Toutes ces quantités sont couplées à travers l'équation d'induction. En particulier, la rotation différentielle du fluide produit un fort effet ω qui induit un champ magnétique azimutal conséquent. Profitant de la géométrie sphérique de l'expérience, je développe les champs azimutaux et méridionaux en polynômes de Legendre et j'obtiens les expressions qui relient toutes les mesures aux fonctions radiales du champ de vitesse pour chaque degré. Pour de petits nombres de Reynolds magnétiques Rm les relations sont linéaires et les équations azimutale et méridionale sont découplées. Je sélectionne un jeu de mesures pour une vitesse de rotation donnée de la sphère interne (Rm ≃ 9.4) et j'inverse simultanément les données de vitesse et magnétiques, reconstruisant ainsi à la fois les champs azimutaux et méridionaux dans la coquille fluide. Les résultats démontrent la bonne cohérence des mesures et indiquent que les fluctuations turbulentes non-axisymétriques ne contribuent pas de fac ¸on significative à l'induction magnétique axisymétrique.

Introduction

It is now well established that the magnetic field of most planets and stars is generated by the dynamo mechanism [START_REF] Larmor | How could a Rotating Body such as the Sun become a Magnet? Report of the British Association for[END_REF][START_REF] Elsasser | Induction Effects in Terrestrial Magnetism Part I[END_REF][START_REF] Parker | Hydromagnetic dynamo models[END_REF], in which motions within an electrically conducting medium amplify infinitesimally small magnetic field fluctuations up to a level where the Lorentz force that results is large enough to stop their amplification. This is possible for large enough values of the magnetic Reynolds number Rm = UL/η (where U is a typical flow velocity, L a typical length, and η is the magnetic diffusivity of the medium).

After the success of the first fluid dynamo experiments in Riga [START_REF] Gailitis | Magnetic Field Saturation in the Riga Dynamo Experiment[END_REF][START_REF] Gailitis | History and results of the Riga dynamo experiments[END_REF] and Karlsruhe [START_REF] Stieglitz | Experimental demonstration of a homogeneous two-scale dynamo[END_REF][START_REF] Mueller | The Karlsruhe two-scale dynamo experiment[END_REF], it was felt that the next step was to obtain dynamo action in a highly turbulent free-to-adjust flow. Indeed, the flow was very much constrained by walls and pipes in those pioneering liquid sodium experiments, while the flow has much more freedom in natural dynamos, and the amplitude of turbulent fluctuations was less than 10% that of the mean flow, while it can be of order one or more in astrophysical objects.

The challenge was soon addressed by several teams, in Russia [START_REF] Frick | A nonstationary dynamo experiment in a braked torus[END_REF], in the USA [START_REF] Lathrop | Laboratory experiments on the transition to MHD dynamos[END_REF][START_REF] O'connell | On the possibility of an homogeneous MHD dynamo in the laboratory[END_REF], and in Europe [START_REF] Cardin | Towards a rapidly rotating liquid sodium dynamo experiment[END_REF][START_REF] Marié | Water experiments related to the "Von Karman Sodium" dynamo project[END_REF], all using liquid sodium as a working fluid (see recent reviews by [START_REF] Verhille | Laboratory Dynamo Experiments[END_REF] and [START_REF] Lathrop | Magnetic dynamos in the lab[END_REF]). In several of these experiments, the mean flow was well characterized, and was expected to yield a dynamo above a critical magnetic Reynolds number that was achievable. It therefore came as a surprise that none of these experiments produced a dynamo, although a rich variety of dynamo behaviours have been discovered in the VKS experiment in Cadarache, France [START_REF] Berhanu | Magnetic field reversals in an experimental turbulent dynamo[END_REF][START_REF] Monchaux | Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium[END_REF][START_REF] Aumaitre | The VKS experiment: turbulent dynamical dynamos[END_REF] when ferromagnetic disks stir the fluid.

It appears that turbulent fluctuations have a collective contribution to the mean magnetic induction that counteracts that of the mean flow [START_REF] Spence | Observation of a turbulence-induced large scale magnetic field[END_REF]. That small-scale turbulent fluctuations can contribute to a large-scale magnetic field is not surprising. In fact, it is the basis of the α-effect introduced by [START_REF] Steenbeck | A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of Coriolis forces[END_REF], who showed that the interaction of small-scale velocity fluctuations with small-scale magnetic field fluctuations produce an electromotive force proportional to a large-scale magnetic field. The success of the Karlsruhe dynamo experiment relies precisely on this collective effect.

In rotating bodies such as planets and stars, it has long been recognized that the combination of an α-effect with an ω-effect provides an appropriate recipe to produce a large-scale magnetic field [START_REF] Parker | Hydromagnetic dynamo models[END_REF] (see [START_REF] Charbonneau | Dynamo Models of the Solar Cycle[END_REF] and [START_REF] Rieutord | The solar dynamo[END_REF] for reviews). The ω-effect is due to the shear caused by differential rotation in the fluid. It converts poloidal magnetic field into toroidal magnetic field. One then invokes some kind of α-effect, due to non-axisymmetric motions, to convert toroidal field into poloidal field, in order to sustain the dynamo. Many meanfield dynamo models for the Earth or the Sun are based on this mechanism (e.g., [START_REF] Parker | Hydromagnetic dynamo models[END_REF][START_REF] Barenghi | Nonlinear planetary dynamos in a rotating spherical shell. 1. Numerical methods[END_REF][START_REF] Brandenburg | Hydromagnetic alpha-omega-type dynamos with feedback from large-scale motions[END_REF][START_REF] Charbonneau | Dynamo Models of the Solar Cycle[END_REF].

It is therefore of some interest to evaluate these two effects in various situations, in particular to understand when, where and why turbulent fluctuations favor or hinder dynamo action. There has been several efforts in this direction, using either numerical simulations [START_REF] Spence | Fluctuation-driven magnetic fields in the Madison Dynamo Experiment[END_REF][START_REF] Brandenburg | Turbulent transport in hydromagnetic flows[END_REF], or laboratory experiments [START_REF] Pétrélis | Nonlinear magnetic induction by helical motion in a liquid sodium turbulent flow[END_REF][START_REF] Spence | Observation of a turbulence-induced large scale magnetic field[END_REF], by observing the magnetic response of the device when a weak external magnetic test field is applied. The VKS team in particular has obtained clear evidence for these effects (and others) in von Kármán flows, as nicely summarized by [START_REF] Verhille | Laboratory Dynamo Experiments[END_REF]. In a similar geometry, the Madison group has measured a dipolar component of the induced magnetic field, which cannot be explained by the mean flow [START_REF] Spence | Observation of a turbulence-induced large scale magnetic field[END_REF]. More recently, a direct measurement of the α-effect has been obtained on the same set-up [START_REF] Rahbarnia | Direct observation of the turbulent emf and transport of magnetic field in a liquid sodium experiment[END_REF], using a dedicated probe that measures locally the electromotive force. Turbulent fluctuations can also enhance the magnetic diffusivity of the fluid, yielding the so-called β-effect. Impressive measurements of this effect have been obtained by the group in Perm [START_REF] Frick | Direct measurement of effective magnetic diffusivity in turbulent flow of liquid sodium[END_REF][START_REF] Noskov | Turbulent viscosity and turbulent magnetic diffusivity in a decaying spin-down flow of liquid sodium[END_REF] for the transient flow of sodium in a torus abruptly stopped. In these studies, only a global averaged response of the system is obtained, with no information on the spatial distribution of the αand ω-effects.

Several teams have also been able to combine experimental measurements of the induced field with numerical simulations of the flow backed by velocity measurements on a water model of the experiment [START_REF] Spence | Fluctuation-driven magnetic fields in the Madison Dynamo Experiment[END_REF][START_REF] Ravelet | Kinematic alpha Tensors and Dynamo Mechanisms in a von Karman Swirling Flow[END_REF]. The present article presents one effort in the same direction, using the DT S magnetized spherical Couette flow experiment. The DT S experiment is the only experiment so far that combines rotation and a strong applied dipolar magnetic field. It was built to explore the resulting magnetostrophic regime [START_REF] Cardin | Towards a rapidly rotating liquid sodium dynamo experiment[END_REF]. A number of interesting observations have already been presented, relating to the mean flow [START_REF] Nataf | Experimental study of super-rotation in a magnetostrophic spherical Couette flow[END_REF](Nataf et al., , 2008;;[START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] and to the fluctuations [START_REF] Schmitt | Rotating spherical Couette flow in a dipolar magnetic field : Experimental study of magneto-inertial waves[END_REF][START_REF] Schmitt | Magneto-Coriolis waves in a spherical Couette flow experiment[END_REF]. Since the imposed magnetic field governs the dynamics of the system, it is not possible to rely on a water model. However, we already know that turbulent fluctuations are strongly affected by the presence of a large scale magnetic field [START_REF] Nataf | On the peculiar nature of turbulence in planetary dynamos[END_REF][START_REF] Schmitt | Magneto-Coriolis waves in a spherical Couette flow experiment[END_REF][START_REF] Figueroa | Modes and instabilities in magnetized spherical couette flow[END_REF], and it would be most interesting to find out how this translates into the α-, βand ω-effects for this kind of flow. Fortunately, [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] have shown that velocity profiles measured by in situ ultrasound Doppler velocimetry provide excellent constraints of the mean flow in DT S . Furthermore, the simple spherical geometry of this experiment makes it possible to use tools developed in the context of celestial dynamos.

Thus, my aim here is to set the stage for further steps that should enable us to map the α-, βand ω-effects for the magnetized spherical Couette flow, using measurements from the DT S experiment. The first step consists in solving the kinematic problem of the large-scale fields, ignoring the contribution of the small-scale fluctuations. For future reference, I write down the equations and the inversion procedure that apply when only the large-scale fields are taken into account. I then apply this procedure to a selected set of actual DT S measurements. The results demonstrate the good internal consistency of the measurements and show that small-scale fluctuations play a negligible role.

The organization of this article is as follows: Section 2 describes the DT S experiment, the measurements and the way they are processed. Section 3 explains why we treat separately the azimuthal part and the meridional part of the fields. The forward problem of predicting the induced field and related quantities from the mean flow is treated in section 4 for the azimuthal fields, and in section 6 for the meridional fields. Section 5 presents the inversion results for the azimuthal fields, and section 7 for the meridional fields. Conclusions and perspectives are discussed in section 8.

b total

Figure 1: Sketch of the DT S set-up. In this northern quadrant of a meridional cross-section, liquid sodium is enclosed between the inner and outer solid shells. The inner shell is made of copper and extends from radius ri (thin line) to r i (thick line). It contains a strong magnet, which produces an axial dipolar magnetic field, as shown by the blue field lines. The fluid is entrained by the rotation of the inner sphere at frequency f around the vertical axis. The outer shell is made of stainless steel and extends from radius r o (thick line) to ro (thin line). It is at rest in the laboratory frame.

Experimental

Experimental set-up

The DT S experiment is a spherical Couette flow experiment with an imposed dipolar magnetic field. Liquid sodium is used as a working fluid. It is sketched in figure 2.1. It is contained between an inner sphere and a concentric outer shell, from radius r * i = 74 mm to r * o = 210 mm (the superscript * refers to the dimensional quantities, and is dropped after adimensionalization by r * o as given in Appendix A). The inner sphere consists of a 15 mm-thick copper shell (dimensionless radius ri to r i ), which encloses a permanent magnet that produces the imposed dipolar magnetic field B d , whose intensity reaches 175 mT at the equator of the inner sphere. The stainless steel outer shell (dimensionless radius r o to ro ) is 5 mm thick. The inner sphere can rotate around the vertical axis (which is the axis of the dipole) at rotation rates f = 2πΩ up to 30 Hz. Although the outer shell can also rotate independently around the vertical axis in DT S , I only consider here the case when the outer sphere is at rest. Additional details can be found in Nataf et al. (2008) and [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF].

Dimensionless numbers

In this article, I concentrate on the magnetic induction equation alone, which is governed by a single dimensionless number: the magnetic Reynolds number Rm, which measures the induction over diffusion ratio. However, the actual flow depends of course on other dimensionless numbers that determine the relative importance of forces in the Navier-Stokes equation. Three independent dimensionless numbers can be constructed, and we choose the magnetic Reynolds number Rm, the Elsasser number Λ and the magnetic Prandtl number Pm, whose expressions are given in table 1, together with their values in the DT S experiment for rotation rates f = 3 Hz and f = 30 Hz.

f Rm Λ Pm Re Ha Lu Hz Ω(r * o ) 2 /η σB 2 0 /(ρΩ) ν/η Ω(r * o ) 2 /ν r * o B 0 / √ ρµ 0 νη r * o B 0 / η 2 ρµ 0 3
9.4 3 10 -2 7.4 10 -6 1.3 10 6 200 0.5 30 94 3 10 -3 1.3 10 7

Table 1: Relevant dimensionless numbers and their values for two rotation rates f of the inner sphere.

The magnetic Prandtl number simply compares the kinetic and magnetic diffusivities, and is a physical property of the fluid used (here liquid sodium). The Elsasser number has been introduced to compare the Lorentz and the Coriolis forces. It looks small in table 1 because we use B 0 , the intensity of the imposed magnetic field at the equator of the outer sphere, to scale the magnetic field, but one should keep in mind that the actual imposed magnetic field is 23 times larger at the equator of the inner sphere.

The three other dimensionless numbers listed in table 1 can be deduced from the previous ones. The Hartmann number Ha indicates that, even at the outer boundary, magnetic forces dominate over viscous forces. The Lundquist number Lu compares the time-scale of Alfvén waves to their life-time. It is needed to convert magnetic energies to the same units as kinetic energies.

Data processing

We want to investigate the interaction of the flow with the magnetic field. We thus want to combine measurements of the velocity field and of the induced magnetic field. [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] have recently shown that it was possible to retrieve the large scales of the mean axisymmetric velocity field in DT S by inverting criss-crossing ultrasound Doppler velocity profiles. The induced magnetic field is measured at several latitudes on the outer sphere. Since the toroidal magnetic field vanishes at the surface, it is crucial to also measure the induced magnetic field inside the fluid. Additional information on the interaction of the flow with the magnetic field is obtained by measuring the electric potential at the surface of the outer sphere, and the torque applied by the fluid on the spinning inner sphere. [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] present these various measurements and display their evolution with the rotation frequency f of the inner sphere up to 30 Hz. Here I concentrate on the case with f = ±3 Hz (Rm ≃ 9.4), for which we have the best velocity data coverage. I give below further indications on the processing of the data I invert.

ultrasound Doppler velocity profiles

Ultrasound Doppler velocimetry provides measurements of the component of the fluid velocity along the shooting line of the ultrasound beam, from the Doppler shift of back-scattered acoustic energy by small heterogeneities in the fluid. In the DT S set-up, ultrasound transducers can be placed in ports at 4 different latitudes (10 • , -20 • and ±40 • ) (see Nataf et al. (2008) and [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] for more details). Purely radial shots provide radial profiles of the radial velocity. Shooting at some angle away from the radial direction, we get profiles that record a combination of meridional and azimuthal velocities. We observe that azimuthal velocities are one order of magnitude larger than the meridional velocities (we record a maximum azimuthal azimuthal meridional Figure 2: Data coverage in the northern quadrant of the (s,z) plane (s: cylindrical radius, z: vertical axis). left: azimuthal fields. The colored curves are the projections in the (s,z) plane of the off-meridian ultrasound beams used to measure the azimuthal velocity of the fluid. The colored dots are the positions where the azimuthal magnetic field is measured. The electric potential is measured in the outer shell at latitudes given by the black squares. right: meridional fields. The straight lines are radial ultrasound shots. The broken straight line is a meridional shot, and the curved lines are the projections in the (s,z) plane of the off-meridian shots. The radial magnetic field is measured at the cyan triangles both inside the fluid and at the surface. The orthoradial magnetic field data are from the magenta triangles.

velocity of 1.8 m/s for f = ±3 Hz, whereas the maximum meridional velocity reaches 0.16 m/s). In a first approximation, the angular velocity in the fluid is thus simply the fluid velocity measured along the ultrasound beam (e.g., [START_REF] Brito | Ultrasonic Doppler velocimetry in liquid gallium[END_REF].

However, we note that the shape of the profiles depends slightly upon the direction of rotation of the inner sphere. This is because, while the sign of the azimuthal velocity is reversed in this operation, the meridional flow keeps the same sign. Therefore, I construct the sum and the difference of profiles shot for two opposite rotation frequencies f andf . I thus obtain refined azimuthal velocity profiles from the difference, and meridional profiles from the sum.

The resulting spatial data coverage in the (s,z) upper quadrant is shown by the colored lines in figure 2.3.1 for the azimuthal velocity (left) and for the meridional velocity (right) (we will see in section 3 that the meridional and azimuthal fields are decoupled under our approximations.)

In this study, I focus on time-averaged properties. For a given rotation frequency f , we record a series of profiles over a time lapse T . For each point of the profiles, the mean velocity is the time-averaged value, and the standard deviation of the mean τ mean is given by τ mean = τ/ T f , where τ is the root mean square deviation, and T f is used as an estimate of the number of statistically independent samples. I retain only one every 4 points along the ray because the original profiles were over-sampled.

magnetic field inside the fluid

The magnetic field is measured by Hall magnetometers placed on a narrow rectangular board inside a cylindrical stainless steel sleeve, which penetrates the fluid radially. The sleeve can be placed at three different latitudes: 10 • , -20 • and ±40 • . The azimuthal field is measured at 6 different radii along the board, while the radial and orthoradial components are obtained at 2 radii. We emplace only one sleeve at a time in order to limit the perturbation of the flow that it produces. As for the Doppler profiles, the collection of data I invert was acquired in different campaigns, with f ranging from 2.7 to 4.0 Hz. All data are scaled by their actual f before being combined. The resulting spatial data coverage in the (s,z) upper quadrant is shown by the colored symbols in figure 2.3.1 for the azimuthal field (left) and for the meridional field (right).

The mean azimuthal field is deduced from a time-average of the records for a given f over a time-lapse of typically 300 turns, and its standard deviation from the root mean square deviations divided by an estimate of the square root of independent samples as described in the previous section. The zero reference is the average value measured when the inner sphere rotates very slowly (in order to average over small deviations away from a perfect dipole, which amount to about 1% of the dipole field).

The induced meridional magnetic field is one order of magnitude smaller than the azimuthal field. For f = ±3 Hz, the maximum measured meridional field is 0.6 mT, to be compared with 5.6 mT for the azimuthal field. The amplitude of the average field is comparable to that of its fluctuations. It can also be contaminated by a projection of the much stronger azimuthal field if the alignment is not perfect. As for the Doppler profiles, I take advantage of the differing symmetries of the azimuthal and meridional fields and correct for misalignment by making the values for the latter coincide when the sign of f is reversed. The misalignments are found to range between 1 • and 6 • . I then deduce the mean meridional field from a time-average of the records for a given f over a time-lapse of typically 300 turns. The zero value remains difficult to establish because the value measured when the inner sphere is at rest varies by up to ±0.2 mT depending on the position in which it stopped, due to the deviations of the imposed field from a perfect dipole. Therefore, I assume that the induced field varies linearly with f for the small values of f that I consider and obtain the value for f = ±3 Hz by rescaling the difference between the average signal at f = ±3 Hz and that at f ≃ 0.7 Hz. The induced meridional magnetic field is thus not very precisely measured in our experiment, which is reflected in its relatively large error bars.

magnetic field at the surface

The three components of the magnetic field are measured at several latitudes from -57 • to 57 • at the surface of the outer sphere. The magnetometers are Giant Magneto Resistive (GMR) devices arranged on an in house designed electronic board (see [START_REF] Schmitt | Magneto-Coriolis waves in a spherical Couette flow experiment[END_REF] for details). The time-averaged azimuthal magnetic field vanishes at the surface of the fluid. Therefore, the surface measurements only constrain the meridional magnetic field. The actual spatial data coverage in the (s,z) upper quadrant is shown by the colored symbols at the surface of the sphere in figure 2.3.1 (right). The maximum induced surface magnetic field we measure at f = 3 Hz is 0.04 mT, which is difficult to measure. In particular, one must take into account the effect of temperature on the signals. We correct for these, using platinum resistance thermometers installed on the GMR board, and calibration curves measured under controlled temperatures and imposed magnetic fields. The high latitude radial probes are saturated by the dipole field and not used in the present analysis. The zero value remains difficult to establish because we cannot use the value measured when the inner sphere is at rest, since it varies by up to 0.03 mT depending on the position in which it stopped, due to the deviations of the imposed field from a perfect dipole. Therefore, I assume that the induced field varies linearly with f for the small values of f that I consider and obtain the value for f = ±3 Hz by rescaling the difference between the average signal at f = ±3 Hz and that at f ≃ 1 Hz. The values obtained from the two hemispheres and for the two signs of rotation are consistent and are combined to yield my final data.

electric potentials at the surface

Electric currents are produced through the interaction of the fluid flow with the magnetic field. Through Ohm's law, the electric field carries some information about these currents, together with the induction term U × B. The time-averaged potentials do not vary with azimuth, and thus relate only to the azimuthal flow. I therefore include in my azimuthal inversion the differences between the electric potential measured at 5 different latitudes on the outer shell, as indicated by the squares in figure 2.3.1 (left). The time-averaged values and their standard deviations are deduced from 640s-long records. The maximum electric potential difference between electrodes 10 • apart in latitude is 0.84 mV for f = ±3 Hz.

torque on the inner sphere

We will see that the magnetic torque applied by the fluid on the inner sphere is directly related to the strength of the induced azimuthal magnetic field at its surface. We expect the magnetic torque to dominate over the viscous torque on the inner sphere (whereas the opposite holds on the outer sphere). I therefore include this data as a further constraint in the azimuthal inversion. The torque is recorded for all runs, as given by the electronic drive of the motor. We substract from the reading the mechanical torque estimated for the given rotation rate from runs without liquid sodium (see [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] for further details). The torque is equal to -1.2 ± 0.1 Nm at f = 3 Hz, which translates into -0.29 ± 0.03 in our dimensionless units (see Appendix A).

Field decomposition

The problem that I consider here is purely kinematic: we do not solve for the dynamics but we want to relate the velocity field to the magnetic field it interacts with, using the induction equation:

∂B ∂t = ∇ × (U × B) + η∆B, ( 1
)
where η is the uniform magnetic diffusivity of the fluid.

Both the velocity field U and the magnetic field B are divergence-free and can therefore be decomposed into their poloidal and toroidal components. I use spherical coordinates (r, θ, ϕ). Since I only consider axisymmetric mean fields, which do not depend upon the azimuth ϕ, the decomposition can be further simplified into the following expressions (e.g., Roberts (2007)):

U = u P + U ϕ e ϕ = ∇ × ue ϕ + U ϕ e ϕ , (2)
where u is the potential of the meridional flow and U ϕ is the azimuthal velocity field. The magnetic field is given by:

B = B P + b ϕ e ϕ = B d + b P + b ϕ e ϕ = ∇ × (A d + a) e ϕ + b ϕ e ϕ , (3)
where A d is the potential of the imposed dipolar field B d , a the potential of the induced meridional magnetic field b P , and b ϕ the induced azimuthal magnetic field.

All fields are made dimensionless, using Ω -1 for a time scale and r * o for a length scale. The imposed dipolar magnetic field is scaled by its intensity B o at the equator of the outer sphere. The expression of B d is thus:

B d = ∇ × A d e ϕ = 1 r 3 (2 cos θ e r + sin θ e θ ) .
(4)

The induced magnetic fields b P and b ϕ are scaled by RmB o , where Rm = Ω(r * o ) 2 /η is the magnetic Reynolds number, since we expect the induced field to vary almost linearly with Rm and with B o in the moderate magnetic Reynolds number regime that we consider.

Led by our experimental measurements, I further consider the following approximations: U ϕ >> u P and B d >> b ϕ >> b P . In other words, the flow is predominantly azimuthal, and it induces a magnetic field that is also mostly azimuthal, and which remains small compared to the imposed dipolar field.

Under all these conditions, and assuming steady state, the induction equation decomposes into two independent scalar equations (e.g., [START_REF] Fearn | Convection, stability and the dynamo[END_REF][START_REF] Barenghi | Nonlinear planetary dynamos in a rotating spherical shell. 1. Numerical methods[END_REF]: The induction equation for the azimuthal field b ϕ yields:

sB d • ∇ω + ∆ 1 b ϕ = 0, (5)
with ∆ 1 = ∇ 2 -1/s 2 , and where we introduced the angular fluid velocity ω = U ϕ /s, with s = r sin θ the cylindrical radius. The induction equation of the meridional magnetic potential a becomes:

u p s • ∇(sA d ) -∆ 1 a = 0. (6)
In classical mean-field dynamo theory, one requires mechanisms to convert large-scale poloidal magnetic field into large-scale toroidal magnetic field, and vice versa. For rapidly rotating bodies, it is believed that azimuthal shear is an efficient way of achieving the former. It corresponds to the induction term of equation 5 and is therefore called the 'ω-effect'. Our equation 6 for the meridional (poloidal) magnetic field only converts poloidal magnetic field (the imposed dipole) into poloidal magnetic field. One usually resorts to the contribution of small-scale fluctuations to produce an 'α-effect', which converts toroidal magnetic field into poloidal magnetic field. The objective of our study is to see how much can be explained by the large-scale flow alone, in order to measure the need for an α-effect in our experiment.

Azimuthal fields: forward problem

In this section, I want to express all the observables as functions of the azimuthal velocity field U ϕ , which I will invert for. U ϕ projects onto the associated Legendre functions P 1 l of order 1 as:

U ϕ (r, θ) = odd l U l (r)P 1 l (cos θ), (7)
where only odd degrees l are considered because U ϕ is symmetric with respect to the equator.

induction equation

Similarly, b ϕ is written as:

b ϕ (r, θ) = even n b n (r)P 1 n (cos θ), (8)
where the sum is over even degrees n since we anticipate that b ϕ is equatorially antisymmetric (I use n to denote even degrees and l for odd degrees). With this decomposition, the diffusion term of the induction equation ( 5) becomes:

∆ 1 b ϕ = even n 1 r d 2 (rb n ) dr 2 - n(n + 1) r 2 b n P 1 n (cos θ), (9)
while the induction term is expressed as:

s B d • ∇ U ϕ s = 1 r 4 odd l U l (r) d dθ sin θ P 1 l (cos θ) +2 r dU l (r) dr -2U l (r) cos θ P 1 l (cos θ) . (10)
In order to keep all θ dependence in the associated Legendre polynomials only, we make use of the following relationships:

d dθ sin θ P 1 l = l(l + 1) 2l + 1 P 1 l+1 -P 1 l-1 (11) cos θ P 1 l = 1 2l + 1 l P 1 l+1 + (l + 1) P 1 l-1 . (12)
Injecting into equation ( 10), the induction term becomes:

s B d • ∇ U ϕ s = 1 r 4 odd l 1 2l + 1 (l -3)U l + 2r dU l dr l P 1 l+1 + -(l + 4)U l + 2r dU l dr (l + 1) P 1 l-1 , (13)
which confirms that only even degrees n contribute to the azimuthal magnetic field b ϕ . Equating equation 9 and the opposite of equation 13, we obtain for each even degree n a linear relationship between b n (r) and U n±1 (r). Appendix B indicates the truncation degree to be considered for the various fields when l max is the truncation degree of the expansion of the azimuthal velocity field U ϕ in equation 7.

electric field and potential

In order to compare with the differences in electric potential measured at the surface of the sphere, we need to relate the electric potential V to the azimuthal velocity U ϕ . The E θ component of the electric field E = -∇V can be decomposed as:

E θ (r, θ) = even n e n (r)P 1 n (cos θ), (14)
which we integrate to get:

V(r, θ) = V(r) - even n r e n (r)P 0 n (cos θ). (15)
We compute the radial function V(r) by integrating E r along the z-axis. The E θ and E r components of the electric field are derived from Ohm's law:

E = j -U × B, (16)
which yields:

E θ (r, θ) = - ∂ r∂r rb ϕ - 2 r 3 odd l U l (r) l + 1 2l + 1 P 1 l-1 + l 2l + 1 P 1 l+1 . ( 17
)
We get the e n (r) coefficients of equation 14 by identification, using the expansion of b ϕ in b n , which were related to U l in the previous paragraph.

boundary conditions

We make use of the continuity of the magnetic field and of the tangential components of the electric field at the interface between the fluid and the solid shells. At these interfaces, thin velocity boundary layers form in the fluid, which we will not be able to resolve from our Doppler profiles. I therefore authorize azimuthal velocity discontinuities on the walls. Appendix C details how we deal with these discontinuities, as well as those in the electrical conductivity between the three shells. Note that electric currents are present in the solid shells and that the radial derivative of b ϕ is discontinuous. In the end, we obtain a relation between the magnetic field and its radial derivative (on the fluid side) at the interface r = r i given by:

d b n dr r i = -P n b n (r i ) -Q n , (18)
and similarly for r = r o . The expressions of P n and Q n are given by C.15 for r = r i .

magnetic torque

The magnetic torque is the torque due to the Lorentz force acting on the copper shell. We can compute it by integrating:

Γ M = 4π r i ri dr π/2 0 dθ r 3 sin 2 θ F ϕ (r, θ), (19)
where F ϕ is the azimuthal component of the Lorentz force per unit volume given by:

F ϕ (r, θ) = ( j × B) ϕ = 1 r 4 ∂ ∂θ (sin θ b ϕ ) + 2 cos θ ∂ ∂r (rb ϕ ) . ( 20
)
Since we could project b ϕ over the P 1 l (cos θ), and because of the recursive relations given in equations 12, the azimuthal component of the Lorentz force also projects onto the P 1 l (cos θ):

F ϕ (r, θ) = odd l f l (r)P 1 l (cos θ). (21)
Developing the expansion for b ϕ , we obtain:

f l (r) = 1 r 4 l + 2 2l + 3 (1 -l)b l+1 + 2r db l+1 dr + l -1 2l -1 (l + 2)b l-1 + 2r db l-1 dr . ( 22
)
Getting back to the expression of the magnetic torque (equation 19), we note that because of the orthogonality of the P m l (cos θ), the only degree of the expansion of the Lorentz force that contributes is l = 1, which only implies the degree 2 component of b ϕ . It is then straightforward to see that the magnetic torque is directly proportional to the value of b 2 at the surface of the copper shell, and does not depend on its thickness. We simply get:

Γ M = - 16π 5 b 2 (r i ). ( 23
)
Note that, more generally, the integral of the magnetic torque over the volume of a conductive shell reduces to a surface integral of the Maxwell tensor expressed as B r b ϕ .

Azimuthal inversion

inversion set-up

We now have all the expressions that relate our various measurements to the radial functions U l (r) of the different degrees l of the expansion of the azimuthal velocity field. In order to carry out the inversion, I define a regular radial grid of Nr = 137 points between r i = 74/210 and r o = 1, on which the U l functions are discretized. The model vector M consists of Nr (l max + 1)/2 unknowns.

The data vector D consists of a set of 5 azimuthal velocity profiles (with about 150 independent point measurements each) measured by ultrasound Doppler velocimetry, 16 azimuthal magnetic field measurements inside the fluid from 3 different latitudes, 4 electric potential differences at the surface, and the value of the torque on the inner sphere. The spatial data coverage is shown in figure 2.3.1.

matrix formulation

Radial derivatives are computed by finite difference. The diffusion operator in equation 9 is expressed in matrix form, taking into account the conditions at the boundaries expressed by equation 18. Under these conditions, the inverse problem is linear: all measured quantities are linearly related to the model vector, which consists of the U l (r k ) values. Note that the contribution of the spinning inner sphere is treated separately, since there is no need to invert for this known (imposed) velocity.

The best fitting model M is obtained with the classical generalized least square inverse method [START_REF] Tarantola | Generalized non-linear inverse problems solved using the least-squares criterion[END_REF]:

M = C pp G T C dd + GC pp G T -1 D, ( 24
)
where G is the matrix from which the predicted data are computed by D f it = GM. C dd is the covariance matrix of the data, and C pp the a priori covariance matrix of the model parameters, which I describe below.

covariance matrices

The covariance matrix of the data is taken diagonal. The diagonal terms are the square of the standard deviations presented in the data processing section 2.3. I use the a priori covariance matrix of the model to control the smoothness of the model in the radial direction, while the smoothness in the latitudinal direction is controlled by the truncation degree l max of the Legendre expansion. Within an l-block of the a priori covariance matrix of the parameters, the elements are taken as:

C pp (r a , r b ) = τ 0 l 2 exp - r a -r b δ 2 , ( 25
)
where δ is a smoothing parameter and τ 0 /l is a damping parameter.

In order to limit the velocity jump at the inner and outer spheres, where the Doppler profiles cannot resolve the boundary layers, we add fake profiles with zero velocity and a large error bar (τ = 0.1).

inversion results

I present the results of a simultaneous inversion of the velocity profiles, the induced magnetic field inside the fluid, the electric potential differences at the surface, and the torque on the inner sphere. The data coverage is shown in figure 2.3.1. The Legendre expansion of the azimuthal velocity is carried up to l max = 7 (l odd), and the parameters of the a priori covariance matrix in expression 25 are chosen as δ = 0.2 and τ 0 = 0.01.

radial profiles of the modes

The radial profiles of the Legendre modes are shown in figure 3 for the coefficients of the azimuthal velocity U l , the azimuthal magnetic field b n and the θ-component of the electric field e n . The l = 1 mode dominates the velocity field. The magnetic field modes show that the induction at the inner sphere boundary is strong, but that the shear inside the fluid produces large dipole (n = 2) and octupole (n = 4) field as well. The electric field mode is almost entirely n = 2 and does not drop to zero at the outer surface (r = ro ).

maps of the fields

Figure 4 shows contour maps (in the northern meridional quadrant) of the angular velocity of the fluid ω, the azimuthal magnetic field b ϕ , and the electric potential V. All quantities are dimensionless, as given in Appendix A.

The map of angular velocity illustrates some important properties of the flow already discussed by [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF]. Note the region near the equator of the inner sphere where the angular velocity of the fluid gets larger (1 < ω < 1.3) than the angular velocity of the inner sphere. In that region, the fluid obeys Ferraro's law of isorotation [START_REF] Ferraro | The non-uniform rotation of the sun and its magnetic field[END_REF]: the angular velocity is constant along magnetic field lines (white dashed lines). Further away from the inner sphere, the mean flow is geostrophic: the velocity does not vary along the rotation axis (vertical axis). Also note that the magnetic coupling between the inner copper shell and the fluid is very efficient and entrains the fluid at large angular velocities throughout the fluid volume. There is a region of marked weak velocity at the pole of the inner sphere, for which we have no explanation yet.

The map of the induced azimuthal field illustrates that the field is mainly produced in two distinct regions: at the surface of the inner sphere, with alternating polarities, and in the large geostrophic shear zone. Note that the amplitudes range from -0.08 to 0.02 in the RmB 0 units we chose for b ϕ and that Rm ≃ 9.4 for f = 3 Hz.

The electric potential map appears dominated by the geometry of the imposed dipole field.

fits to the data

Let's now see how well the reconstructed azimuthal velocity field fits the observations under the ω-effect approximation. Figure 5 compares the predictions of our model with the velocity measurements along the ultrasound Doppler profiles, the azimuthal magnetic field inside the fluid, and the electric potential differences at the surface. In all cases, we compute the normalized misfit defined as:

mis f it = 1 N N d pred -d obs τ obs 2 , ( 26
)
where d obs are the observed data with their error τ obs , and d pred the model prediction. The ultrasound Doppler profiles are well fit by our simple velocity model. The normalized misfit is 1.9. The fact that it is larger than one indicates that representation errors are present. Some of our assumptions (such as symmetry with respect to the equator) are partially violated. We observe for example that slightly different velocities are measured at points where Doppler profiles intersect in the (s,z) plane.

The predicted induced azimuthal magnetic field under the ω-effect approximation (i.e., taking into account the induction from the mean flow alone) falls in the right range, but deviates strongly from the observations at all latitudes for r < 0.6. The normalized misfit is 8.

With a normalized misfit of 3.2, the differences in electric potential at the surface are not perfectly fit, but display the proper trend.

The predicted magnetic torque is -0.27, slightly smaller than the observed value of -0.29 ± 0.03. I find it quite remarkable that the simple velocity model I invert for provides a reasonable fit to observations as diverse as induced magnetic field, electric potential and torque, when plugged into the mean flow induction equation. By playing with the a priori covariance matrix, it is possible to improve the fit to the observations, at the expense of added complexity in the velocity field. However, the velocity field is very strongly constrained by our direct measurements of azimuthal velocity along criss-crossing profiles inside the fluid, and I have not been able to find models that could significantly improve the fit of the observed induced magnetic field b ϕ inside the fluid.

Meridional fields: forward problem

We now consider the poloidal (meridional) part of the velocity field. According to equation 2, the poloidal velocity field is expressed as:

u p = ∇ × ue ϕ . ( 27
)
The direction of rotation of the meridional flow changes sign at the equator, which implies that the potential u projects on associated Legendre polynomials with even degree n:

u(r, θ) = even n u n (r)P 1 n (cos θ). (28)
We derive the expressions of the u r and u θ components, which we use to relate to the velocity profiles measured by ultrasound Doppler velocimetry:

u r (r, θ) = even n u n (r) r 1 sin θ d dθ sin θ P 1 n (cos θ) (29) u θ (r, θ) = - even n u n (r) r + du n (r) dr P 1 n (cos θ). (30)

induction equation

Our next step is to derive a linear relation between the potential a of the magnetic field and the potential u of the flow from the induction equation 6, which I recall here:

u p s • ∇(sA d ) -∆ 1 a = 0. ( 31
)
The potential a projects onto the associated Legendre polynomials:

a(r, θ) = odd l a l (r)P 1 l (cos θ). ( 32
)
where the sum is over odd degrees l since we anticipate that a is symmetric with respect to the equator. With this decomposition, the diffusion term of the induction equation 31 becomes:

∆ 1 a = odd l 1 r d 2 (ra l ) dr 2 - l(l + 1) r 2 a l P 1 l (cos θ), (33)
while the induction term becomes:

u p s • ∇(sA d ) = 1 r 3 (-u r sin θ + 2u θ cos θ) , (34)
since the potential A d of the dipolar field is (sin θ)/r 2 . Injecting the expressions for u r and u θ (equations 30), one gets the following expression:

u p s • ∇(sA d ) = - 1 r 4 even n u n d dθ sin θ P 1 n (cos θ) +2 u n + r du n dr cos θ P 1 n (cos θ) . ( 35
)
Injecting the recursive relations 12 into the previous equation, we finally obtain:

u p s • ∇(sA d ) = - 1 r 4 even n 1 2n + 1 (3 + n)u n + 2r du n dr n P 1 n+1 + (2 -n)u n + 2r du n dr (n + 1) P 1 n-1 , . (36)
which confirms that only odd degrees l contribute to the potential a of the meridional magnetic field b P . Equating equation 33 and equation 36, we obtain for each odd degree l a linear relationship between a l (r) and u l±1 (r). Appendix B recalls the truncation degree to be considered for the various fields when n max is the truncation degree of the expansion of the meridional velocity field potential u in equation 28.

boundary conditions

We impose non-penetration of the fluid at the interface with the solid shells. However, as for the azimuthal velocity, I allow for a discontinuity of the tangential velocity u θ since we cannot resolve the very thin boundary layers. The continuity of the magnetic field implies that both a and its radial derivative are continuous. Only azimuthal currents contribute to the meridional magnetic field. Thus there is no electric current in the solid shells since both the azimuthal electric field and the U × B contribution to Ohm's law (equation 16) are null. It means that the meridional field matches a potential field on both sides of the fluid shell, yielding a simple relation between a and its derivative:

da l dr r i = l a l (r i ) ( 37
)
da l dr r o = -(l + 1) a l (r o ). ( 38
)

Meridional inversion

inversion set-up

We now have all the expressions that relate our various measurements to the radial functions u n (r) of the different degrees n of the expansion of the meridional velocity potential. In order to carry out the inversion, I use the same regular radial grid of Nr = 137 points between r i = 74/210 and r o = 1, on which the u n functions are discretized. The model vector M consists of Nr n max /2 unknowns.

The data vector D consists of a set of 8 Doppler profiles (radial profiles of the radial velocity at 3 latitudes, one meridional profile, and meridional velocities retrieved from 4 off-meridional profiles by summing azimuthal profiles for opposite rotation direction), measurements of b r and b θ inside the fluid at 3 latitudes, and measurements of b r and b θ at the surface from the equator up to 57 • in latitude. The spatial data coverage is shown in figure 2.3.1.

matrix formulation

We follow the same approach as for the azimuthal inversion. Radial derivatives are computed by finite difference. The diffusion operator in equation 33 is expressed in matrix form, taking into account the conditions at the boundaries expressed by equation 38. Under these conditions, the inverse problem is linear: all measured quantities are linearly related to the model vector, which consists of the u n (r k ) values. The best fitting model M is obtained using the classical generalized least square inverse given by equation 24.

C dd is the covariance matrix of the data, and C pp the a priori covariance matrix of the model parameters, which I describe below.

covariance matrices

The covariance matrix of the data is taken diagonal. The diagonal terms are the square of the standard deviations presented in the data processing section 2.3. The a priori covariance matrix of the model parameters has the same expression 25 as for the azimuthal inversion. In order to impose non-penetration of the fluid at the inner and outer shells, I force u n (r i ) = u n (r o ) = 0 for all degrees n.

inversion results

I present the results of a simultaneous inversion of the velocity profiles and the induced magnetic field inside the fluid and at the surface. The data coverage is shown in figure 2.3.1. The Legendre expansion of the meridional velocity potential is carried up to n max = 8 (n even), and the parameters of the a priori covariance matrix in expression 25 are chosen as δ = 0.15 and τ 0 = 0.001.

radial profiles of the modes

The radial profiles of the Legendre modes are shown in figure 6 for the coefficients of the meridional potentials of velocity (u n ) and magnetic field (a l ) . All quantities are dimensionless, as given in Appendix A. Note that the velocity modes show more structure in radius than the azimuthal velocity modes. Note that the induced dipole field (l = 1) dominates near the inner sphere, but vanishes at the surface as it should [START_REF] Spence | Observation of a turbulence-induced large scale magnetic field[END_REF].

maps of the fields

I show the streamlines of the meridional flow, and the field lines of the induced poloidal magnetic field in figure 7.

The meridional circulation is poleward beneath the outer boundary, where the lines are squeezed, indicating large orthoradial velocities. Our solution is very similar to that of [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] who inverted the velocity data alone, using a sine decomposition in radius.

Not surprisingly, the induced magnetic field is largest near the inner sphere, where the imposed dipole is strongest. The field lines are clockwise around the equator. Note that the induced field remains very small compared to the imposed dipole field, in agreement with the approximation used in our forward problem (see section 3 ). Indeed, I have plotted the field lines of the total field in figure 2.1 (dashed lines), but they hardly depart from those of the dipole.

fits to the data

Let's now see how well the reconstructed meridional velocity field fits the observations. Figure 8 compares the predictions of our model with the measured radial velocity profiles and the other meridional velocity profiles, all obtained by Doppler acoustics. The fit looks rather good, with an overall normalized misfit of 1.2. The more complex radial structure, as compared to the azimuthal case, seems required by the measured profiles.

Figure 9 compares the observations to the predicted induced magnetic field, under the approximation that it all results from induction by the mean meridional flow alone, The b r and b θ components inside the fluid are given in the first two panels. The model predicts reasonable amplitudes, which are in the range ±0.01 in the RmB 0 units we chose for the induced field. Unfortunately, our data coverage is rather sparse, and does not extend deep into the fluid. The overall normalized misift is 2.4. The trends for b r are compatible with the data, while there is a clear disagreement for b θ .

The right panel of figure 9 shows the predicted magnetic field at the surface of the outer shell. It fits rather nicely the data we measure, with a normalized misfit of 1.1.

As in the azimuthal case, I find it rather remarkable that the velocity model, which is strongly constrained by our Doppler profiles, predicts rather well our magnetic observations. It turns out that the magnetic field at the surface is quite sensitive to subtle changes in the velocity field. In the case I present, I over-emphasized the importance of the surface measurements to obtain a good fit. The slight modifications it brought to the velocity potential did not degrade the fit to the Doppler profiles. In contrast, I was not able to provide a better fit to the b θ field measured inside the fluid without introducing velocity features in contradiction with the velocity measurements. Remember however that all meridional observables are at least one order of magnitude less than their azimuthal counterpart, and that some biases cannot be totally excluded.

Discussion

In this article, I have exposed the formalism with which one can model the induced azimuthal magnetic field from the azimuthal velocity field (the ω-effect), as well as other related measurable quantities (electric potentials and magnetic torque), taking into account realistic conductivity and velocity boundary conditions (section 4). I have written the inverse problem and inverted simultaneously ultrasound Doppler velocity profiles and magnetic observations from the DT S experiment, in order to recover the mean azimuthal flow (section 5). I have carried out a similar procedure for the meridional flow and the magnetic field it induces (sections 6 and 7).

The results are rather remarkable, in that I obtain a full coherent solution for the mean velocity and magnetic fields inside the fluid (figures 4 and 7), which can be used to evaluate in a quantitative fashion the distribution of energies and dissipation, the force balance, and so on. In a sense, we have performed a numerical simulation of a magnetohydrodynamic flow at high Reynolds number (Re ≃ 10 6 ), using the experiment as a Navier-Stokes solver, and working out the induction equation numerically.

As an illustration, I report in table 2 the magnetic and kinetic energies of the azimuthal and meridional flows. They were obtained by integration of our solutions over the fluid shell (with the addition of the inner and outer solid shells for the magnetic field). All energies are scaled by ρΩ 2 (r * o ) 5 , yielding a value of order 1 for the kinetic energy of the azimuthal flow. Note that kinetic energies E az K and e mer K are larger their magnetic counterparts e az M and e mer M . The magnetic energy of the imposed dipole field, whose energy, given by E dipole M = 51/Rm 2 , is in fact independent of the spin rate of the inner sphere. For the spin rate f = 3 Hz considered in this article (Rm ≃ 9.4), the energy of the azimuthal flow E az K is comparable to that of the imposed magnetic field. Table 2: Energies of the imposed dipolar field, the azimuthal magnetic field and the meridional magnetic field, and of the azimuthal and meridional flows, all in units of ρΩ 2 (r * o ) 5 = ρη 2 r * o Rm 2 . The table also lists estimates of the energies of the magnetic and velocity fluctuations ẽM and ẽK deduced from [START_REF] Figueroa | Modes and instabilities in magnetized spherical couette flow[END_REF] in the same units.

The mean flow and magnetic field in the DT S experiment have been examined by [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF], who show that at first order both the azimuthal and meridional velocities and the induced azimuthal magnetic field increase linearly with the rotation frequency f of the inner sphere (or equivalently with the magnetic Reynolds number Rm) for f up to 30 Hz. The induced meridional magnetic field evolves in a more complex fashion, but I will assume here that it remains linear in Rm as well. As a consequence, the fields we have inverted for at f = 3 Hz give a rough first order image of the fields at higher rotation frequency.

Mean-field theory indicates that correlated fluctuations of the velocity and magnetic fields can contribute to the axisymmetric mean fields we are dealing with. Fluctuations in the DT Sexperiment are described in [START_REF] Schmitt | Rotating spherical Couette flow in a dipolar magnetic field : Experimental study of magneto-inertial waves[END_REF][START_REF] Schmitt | Magneto-Coriolis waves in a spherical Couette flow experiment[END_REF]; [START_REF] Figueroa | Modes and instabilities in magnetized spherical couette flow[END_REF]. [START_REF] Figueroa | Modes and instabilities in magnetized spherical couette flow[END_REF] show that velocity and magnetic fluctuations also increase linearly with Rm at first order, and give an estimate for the energy of the fluctuations, which is listed in table 2 for comparison.

I will use my inversion results and these various estimates to evaluate the contribution of the terms that have been left out of the induction equation under my simplifying assumptions. The complete induction equation for the axisymmetric steady state, which replaces equations 5 and 6, can be written as:

         ∆ 1 b ϕ = -sB d • ∇ω + Rm -sb p • ∇ω + su p • ∇ b ϕ s -∇ × ũ × b ϕ ∆ 1 a = u p s • ∇(sA d ) + Rm u p s • ∇(sa) -ũ × b ϕ . (39)
All symbols have the same meaning and adimensionalization as in the rest of this article, yielding the Rm-dependence of the additional terms. Let us now check that these additional terms are negligible indeed at the rotation rate we considered ( f = 3 Hz, which yields Rm = 9.4), and evaluate their influence at the highest rotation rate we achieve ( f = 30 Hz, yielding Rm = 94). Before doing so, we examine the various additional terms inside the braces of equation 39. 22 Compared with equation 5 used for the inversion, the equation for b ϕ contains three additional terms. The first one corresponds to the ω-effect applied to the induced meridional magnetic field. The second one is the advection of the azimuthal field by the meridional flow. The third one describes the mean-field axisymmetric induction caused by the correlated non-axisymmetric fluctuations of velocity ũ and magnetic field b. We can compute the first two coupling terms by combining the results of our azimuthal and meridional inversions. The products are computed in the physical space, and projected onto the P 1 n Legendre polynomials (see Appendix B for the parity and bounds of the projection). Solving the induction equation for these two coupling terms, I get the map of the azimuthal magnetic field b ϕ they produce (figure 10). Its amplitude amounts to about 10 -4 Rm, yielding a contribution of less than 1% to b ϕ for Rm = 9.4 (compare with figure 4).

Only one coupling induction term appears in the equation for the meridional magnetic potential a. It corresponds to the advection of the induced meridional field by the meridional flow. I compute it and solve the induction equation to get the field lines of its contribution to the meridional magnetic field, shown in figure 10. Its amplitude is about 10 -6 Rm, which modifies the inverted meridional magnetic field of figure 7 by about 1% for Rm = 9.4.

I now turn to the last term of both equations, which is the most interesting one because it describes the contribution of non-axisymmetric fluctuations to the mean axisymmetric magnetic field. This effect is supposed to play a major role in the generation of large-scale magnetic fields by dynamo action. There exists a vast literature on the axisymmetric mean-field dynamo equations, which have proven very useful in deciphering the magnetic behaviour of the Sun and stars. The excellent review by [START_REF] Charbonneau | Dynamo Models of the Solar Cycle[END_REF] recalls that the electromotive force E = ũ × b term is often expressed as a truncated series expansion of the large-scale magnetic field: E = α : B + β : ∇ × B , where α and β are two tensors, which depend upon the turbulent characteristics of the flow. The α-effect is crucial for transferring energy from the azimuthal magnetic field to the meridional field, while the β-effect can be seen as enhancing the magnetic diffusivity. Although some theories can provide expressions for these two effects in idealized situations such as homogeneous isotropic turbulence [START_REF] Steenbeck | A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion, under the influence of Coriolis forces[END_REF] or magnetohydrodynamic turbulence [START_REF] Pouquet | Strong MHD helical turbulence and nonlinear dynamo effect[END_REF], or phase fluctuations of simple flows [START_REF] Pétrélis | Inhibition of the dynamo effect by phase fluctuations[END_REF], it remains a challenge to estimate their contribution in actual experiments and in celestial objects. Recent experiments have shed some light on this issue:

The torus experiment in Perm consists in spinning at rotation rates up to 45 Hz a torus filled with liquid sodium. The torus is then stopped abruptly, generating a highly turbulent decaying flow. Using this device, [START_REF] Frick | Direct measurement of effective magnetic diffusivity in turbulent flow of liquid sodium[END_REF] have measured the electromagnetic response of the torus channel to an alternating magnetic field, thereby measuring the effective global magnetic diffusivity. They have observed an enhanced diffusivity during the turbulent decay, which they attribute to a β-effect. The magnetic diffusivity was found to be up to 30% larger than the molecular diffusivity, for the highest magnetic Reynolds number Rm ≃ 30. Local velocities were measured in the same experiments, enabling [START_REF] Noskov | Turbulent viscosity and turbulent magnetic diffusivity in a decaying spin-down flow of liquid sodium[END_REF] to compare turbulent magnetic diffusivity with turbulent viscosity, and to propose a scaling for turbulent diffusivity as: β ∼ Rm k rms , where Rm rms = u rms r o /η is a magnetic Reynolds number based on the amplitude of the velocity fluctuations u rms , and r o is the channel radius. The exponent k is about 2 for Rm rms < 1, decreasing to about 1.3 above that. A maximum value of Rm rms ≃ 1.5 was achieved in this experiment.

The Madison liquid sodium experiment has two counter-rotating impellers facing each other in a sphere filled with 520 liters of liquid sodium. With a maximum rotation rate of 30 Hz, magnetic Reynolds numbers up to 160 are achieved. [START_REF] Spence | Observation of a turbulence-induced large scale magnetic field[END_REF] have reported a turbulentinduced large-scale magnetic field in this set-up. Fluctuations induce some α-effect, which decreases the induction produced by the mean large-scale flow, thereby preventing dynamo onset. More recently, [START_REF] Rahbarnia | Direct observation of the turbulent emf and transport of magnetic field in a liquid sodium experiment[END_REF] have installed a probe that measures the velocity and magnetic field vectors simultaneously at a single position. This gives access to components aligned with the mean magnetic field (akin to an α-effect) and to those aligned with the mean electric current (akin to a β-effect). [START_REF] Rahbarnia | Direct observation of the turbulent emf and transport of magnetic field in a liquid sodium experiment[END_REF] find that the latter dominates in the current setup that includes an equatorial baffle, which reduces the amplitude of the largest-scale turbulent eddies. For their highest magnetic Reynolds number Rm tip = 160, they measure an excess magnetic diffusivity of about β ≃ 30%. As [START_REF] Noskov | Turbulent viscosity and turbulent magnetic diffusivity in a decaying spin-down flow of liquid sodium[END_REF] they introduce a fluctuation-based magnetic Reynolds number: Rm * rms = u rms ℓ/η, which differs because the typical dimension ℓ is the correlation length of the fluctuations, which they find to be at least one order of magnitude smaller than the integral scale. The maximum value they reach is Rm * rms ≃ 1.2, corresponding to Rm rms ≃ 12. The relation β ≃ Rm * rms looks compatible with their data. Getting back to DT S and to equations 39, I note that, unlike the coupling terms, the induction due to non-axisymmetric fluctuations cannot be computed from my inversion results. In fact, we have no access to the ũ × b term. However, we can estimate a reasonable upper bound from the product ũ b . In the example shown by [START_REF] Rahbarnia | Direct observation of the turbulent emf and transport of magnetic field in a liquid sodium experiment[END_REF], the average electromotive force amounts to about 60% of this product. Figure 10 displays a tentative map of ũ b in the DT S experiment. Unlike the other maps in this article, it does not result from a rigorous inversion. I have assumed a decomposition on P 0 n (with even n) of the average velocity and magnetic fluctuations. I thus constructed a map of the velocity fluctuations from the inversion of the root-mean square fluctuations measured along the same Doppler profiles as for the meanflow inversions. Similarly, a map of the magnetic fluctuations was built by inversion of the rms fluctuations of the magnetic field in the sleeve at the three different latitudes. The map shown in figure 10 is the product of these two maps.

The lack of directional and time-correlation information precludes using this map to solve for the induced magnetic field directly, but we can compare the amplitude of the estimate of this induction term with that of the other additional terms discussed above. The amplitude of ũ b reaches about 10 -4 Rm, while the amplitude of the additional coupling terms is about 2 10 -2 Rm in the b ϕ equation and 5 10 -4 Rm in the equation for a. Therefore, non-axisymmetric fluctuations can contribute at best to a few percents of the total induction at our highest magnetic Reynolds number (Rm ≃ 100), and should always remain smaller than the terms that couple the meridional and azimuthal flows and magnetic fields.

It may look surprising that fluctuations have so little effect in our case, while we reach magnetic Reynolds number larger than [START_REF] Noskov | Turbulent viscosity and turbulent magnetic diffusivity in a decaying spin-down flow of liquid sodium[END_REF]. We know that velocity fluctuations are hampered by the presence of the strong imposed magnetic field [START_REF] Figueroa | Modes and instabilities in magnetized spherical couette flow[END_REF]. Nevertheless, using our estimate of ẽK in table 2, we get Rm rms ≃ 4.5 10 -2 Rm, yielding a value Rm rms ≃ 4 at our highest rotation rate. This is at least twice larger than the maximum value of [START_REF] Noskov | Turbulent viscosity and turbulent magnetic diffusivity in a decaying spin-down flow of liquid sodium[END_REF], but comparable to the lowest value reported by [START_REF] Rahbarnia | Direct observation of the turbulent emf and transport of magnetic field in a liquid sodium experiment[END_REF] (Rm rms ≃ 3.4 for Rm tip = 60), for which the β-effect is only 2%.

However, one should keep in mind that I have focused on the inversion of data obtained at low magnetic Reynolds number (Rm = 9.4), and assumed that the results can be extrapolated to our maximum Rm because [START_REF] Brito | Zonal shear and super-rotation in a magnetized spherical Couette-flow experiment[END_REF] observe that velocity and induced magnetic field are proportional to Rm at first order. But they also observe some important differences when Rm increases and causes the relative strength of the imposed dipolar field to weaken: the zone of super-rotation shrinks and the intensity of the induced field increases more rapidly with Rm. We will have to investigate the role of these modifications. It remains that I was not able to perfectly fit all data at Rm = 9.4. In particular, my model does not predict high enough values for b ϕ , while it overestimates the electric potential differences measured at the surface. It could due to the fact that the experiment deviates somewhat from the assumptions used: the imposed magnetic field is not perfectly dipolar, the inner sphere is held by shafts, the equatorial symmetry could be slightly violated, etc. [START_REF] Nataf | On the peculiar nature of turbulence in planetary dynamos[END_REF] note that fluctuations get even smaller when both the Coriolis and the Lorentz forces are dominant, as evidenced from measurements in the DT S experiment when the outer sphere spins. It would be interesting to investigate the role of fluctuations when these constraints are still present but at a much larger magnetic Reynolds number, such as can be achieved in the 3m-diameter BigSister experiment of Dan Lathrop and his group at the University of Maryland. acknowledgments I thank D. Jault for his guidance throughout this study. This manuscript was greatly improved by appreciated comments from an anonymous referee and from the Editor. I thank N. Plihon for useful comments. I am very grateful to all past and present members of the geodynamo team for their active participation in the DT S adventure. I gratefully acknowledge the support of CNRS and Université de Grenoble through the collaborative program "Turbulence, Magnetohydrodynamics and Dynamo". where Na σ and Cu σ are the electrical conductivities of liquid sodium and of the copper inner shell, respectively. The azimuthal velocity of the inner sphere is Cu U ϕ (r i , θ) = r i sin θ = -r i P 1 1 (cos θ). We now evaluate the derivative term of equations C.8 in the solid part, using expression C. where δ 1l is the Kronecker symbol. By identification of the P 1 l of identical degree, we deduce a relationship between the magnetic field and its derivative at the interfaces for each even degree 2 ≤ n ≤ l max + 1: (C.13) with:

d b n dr r i = -P n b n (r i ) -Q n ,
P n = 1 r i - Na σ Cu σ C n (C.14) Q n = 2 r 3 i n + 2 2n + 3 U n+1 (r i ) + n -1 2n -1 (U n-1 (r i ) + r i δ n2 ) . (C.15)
This condition is similar (albeit more complex) to the one encountered for the magnetic potential induced by the meridional circulation. The boundary condition at the interface between the fluid and the outer shell is exactly the same, replacing r i by r o and ri by ro , Cu σ by S S σ (the electric conductivity of the stainless steel outer shell), and omitting the δ n2 term in Q n since the outer sphere is at rest.

Appendix C.2. electric field

Using the expression C.7 of the θ component of the electric field in the solid shell, and injecting the decomposition C.2 of the b n functions, we obtain an analytical expression for the e n functions in the inner copper shell:

Cu e n (r) = Na σ Cu σ (n + 1)r -(n+1) i r n-1 + n rn i r -(n+2) b n (r i ) D n + 2 3r 2 δ n2 . (C.16)
A similar relation is easily derived for the outer stainless steel shell.

Figure 3 :

 3 Figure 3: Azimuthal inversion: radial profiles of the modes of different harmonic degree. From left to right: radial functions of the azimuthal velocity U l (r), the azimuthal magnetic field b n (r) and the orthoradial electric field e n (r). All fields are dimensionless as given in Appendix A. The radius axis extends from ri to ro . horizontal dotted lines indicate the fluid/solid interfaces at r i and r o .

Figure 4 :

 4 Figure 4: Contour maps of the reconstructed angular velocity of the fluid (left), induced azimuthal magnetic field (center), and electric potential (right), from the inversion of DT S measurements. The amplitudes are given by the color bars. All fields are dimensionless as given in Appendix A.

Figure 5 :

 5 Figure5: Azimuthal inversion: fits to the data. These plots compare the measured data, with their error bars, to the predictions of our inversion model for the azimuthal fields. We use the same colors and symbols as in figure 2.3.1 (left), which gives the spatial distribution of the measurements. All data are dimensionless as given in Appendix A. Left: angular velocity from Doppler velocimetry, as a function of the distance d along the ray of the ultrasound beam. Center: azimuthal magnetic field measured inside the fluid at different radii for 3 latitudes. Right: differences in electric potential between electrodes 10 • degrees apart at the surface of the outer shell.

Figure 6 :

 6 Figure 6: Meridional inversion: radial profiles of the modes of different harmonic degrees. From left to right: radial functions of the meridional velocity potential u n (r), and of the meridional magnetic potential a l (r). All fields are dimensionless as given in Appendix A. The radius axis extends from ri to ro . horizontal dotted lines indicate the fluid/solid interfaces at r i and r o .

Figure 7 :

 7 Figure7: Stream lines of the meridional flow (left) and field lines of the induced meridional magnetic field (right) reconstructed from the inversion of DT S measurements. The flow is poleward beneath the outer shell. The values of the stream function range from -2.0 10 -3 to 0.2 10 -3 . Those for the magnetic field from -1.0 10 -3 to 0.2 10 -3 . The blue solid lines indicate anti-clock-wise circulation, and the pink dotted lines clok-wise circulation. The zero line is magenta.

Figure 8 :Figure 9 :

 89 Figure8: Meridional inversion: fits to the velocity data. These plots compare the measured data, with their error bars, to the predictions of our inversion model for the meridional velocity field. We use the same colors as in figure 2.3.1 (right), which gives the spatial distribution of the measurements. All data are dimensionless as given in Appendix A. Left: radial velocity from the radial Doppler velocimetry profiles at three latitudes, as a function of radius. Right: meridional velocity projected on the ultrasound beam as a function of the distance d along the beam for the non-radial profiles.

  2 1.9 10 -4 2.3 10 -5 0.45 4 10 -4 ∼ 2 10 -6 ∼ 4 10 -3

Figure 10 :

 10 Figure 10: (left and center) maps of the contributions of the additional induction terms from equations 39 to the azimuthal magnetic field b ϕ and to the meridional field b p . The values of the magnetic stream function for b p range from -1.2 10 -6 to 0.5 10 -6 . (right) tentative map of the product of average fluctuations in velocity and magnetic field ũ b .

-

  written as:s b n (r) = c + n r n + c - n r -(n+1) , (C.2)where the s left superscript refers to the solid shell. Since electric currents cannot circulate outside the shells, b ϕ = 0 at r = ri and r = ro . Adding the continuity of b ϕ across the boundaries, we obtain the following relations between the + and -coefficients of equation (C.2): the θ component of the electric field yields the final relation. On the two sides of the solid/fluid interface, we have: l left superscript refers to the liquid. The continuity of E θ thus yields the following relation between the jumps in velocity and conductivity, and that in the derivative of the magnetic field at r i (with our adimensionalization given in Appendix A): U ϕ (r i , θ) -Cu U ϕ (r i , θ) B r (r i , θ), (C.8)

  .9 in C.8 and expressing the radial derivative of rb ϕ in the fluid, we get:θ) = -U ϕ (r i , θ)r i sin θ B r (r i , θ). (C.11)Making use of the Legendre projection (17) for the u × B term, we finally obtain:

Email address: Henri-Claude.Nataf@ujf-grenoble.fr (Henri-Claude Nataf) Preprint submitted to Comptes Rendus Physique December 10, 2012

Appendix C. Dealing with discontinuities in velocity and conductivity

In this appendix, I give the analytic expressions of the magnetic and electric fields in the solid spherical shells that contain the liquid sodium. I take into account the different electric conductivities and the contribution of a discontinuity of the tangential velocity at the interface between the solid and the fluid. I take advantage of the fact that we only consider the steady-state solution.

Appendix C.1. magnetic field

In the solid inner and outer shells, equation ( 5) reduces to ∆ 1 b ϕ = 0, which implies that the b n (r) functions of the decomposition (8) satisfy:


D Loheas

Croissant

C Barral

N Touze-Foltz

E Loheas

D Croissant

Comparative study of hydraulic behaviour of Geosynthetic Clay Liners exhumed from a landfill cover and from a dam after several years in service

Keywords: Eurogeo 5, 16-19/09/2012, Valence, ESP

Comparative study of hydraulic behaviour of Geosynthetic Clay Liners exhumed from a landfill cover and from a dam after several years in service

INTRODUCTION

Geosynthetic Clay Liners (GCLs) are sealing elements which contain bentonite encapsulated between geotextile components. They have been mostly employed to replace clay liners in landfill cover systems over the past years [START_REF] Egloffstein | Natural bentonites-Influence of the ion exchange and partial desiccation on permeability and selfhealing capacity of bentonites used in GCLs[END_REF][START_REF] Bouazza | Geosynthetic clay liners[END_REF]. A significant number of studies have been published on GCLs behavior based on field data and have underlined several parameters of influence on GCL field performance, as cation exchange and desiccation hydration cycles, for the case of this specific application of GCLs (Touze-Foltz et al. 2010a).

In comparison to landfill applications during the last 20 years, the acceptance of GCLs as sealing systems in hydraulic engineering applications increased more slowly [START_REF] Heibaum | Geosynthetic clay liners (GBR-C) for hydraulic structures. 3 rd International Symposium on Geosythetic Clay Liners[END_REF][START_REF] Werth | 20 years experience with GCLs in dams and dykes. 3 rd International Symposium on Geosythetic Clay Liners[END_REF]. Different requirements have to be taken into account for the lining elements for ponds (with permanent hydraulic loads), dams (with permanent or temporary hydraulic loads) or dykes (with wave pressure loads).

The objective of this paper is to present the results obtained on GCL samples exhumed from a landfill cover and from a dam after 8 and 3 years of service respectively.

The small dams under study used as barrier protection against flooding are only subjected to hydraulic loads during short periods when water levels are high following heavy rainfalls; most of the time the dams under study are "dry" and the GCL may be subjected to cation exchange and stressed by desiccation as in landfill covers.

The general features of the landfill cover, of the dam and of the GCL samples at the time of sampling are presented. Recommendations made by [START_REF] Zanzinger | Clay geosyhtetic barriers performance in landfill covers[END_REF] as to how GCLs should be exhumed, transported and tested were followed in order to obtain reliable and complete results. GCLs from landfill cover and from the dam were tested for saturated hydraulic conductivity, water content, swell index and composition of the exchange complex. Samples of the overlying and underlying soils from each field site were also tested for water content, cations in the pore water and carbonate content.

Data from these tests were evaluated in conjunction with data reported by others recent studies conducted on virgin GCL samples in laboratory with Ca dilute solutions and conducted on GCL samples exhumed from landfill covers.

EXHUMATION OF GCLS AND COVER SOILS

GCLs were exhumed from a landfill final cover and from a small dam. The landfill site is located in the east of France (continental climate) and receives 730 mm of precipitation annually. GCLs were overlain by a protective geotextile and a 500 mm vegetated surface layer containing some gravels, tile pieces and roots (Fig. 1). The dam is located in the west of France (oceanic climate) where the precipitation is comprised between 700 and 1200 mm/year. This dam is used to prevent flooding during heavy rainfalls and since its construction it has never been subjected to high hydraulic loads (Fig. 2). The cover profile consists of a 200 mm vegetated surface layer with flint stone pieces and roots. A description of both sites is summarized in Table 1. All of the stitch bonded GCLs were originally composed of natural powdered sodium bentonite sandwiched between two woven geotextiles. At the landfill site, samples L-1 and L-2 were exhumed from two adjacent areas whereas in the dam, the two GCLs D-1 and D-2 were installed one on top of the other. In fact the first GCL was installed during a rainfall event and had swelled without confining stress; a second GCL was thus placed above the first one a few days later to address this installation problem. For both exhumations, the cover soil was carefully removed to a depth near the GCL using a digging machine. The remaining soil was then removed by hand to prevent damage to the GCL. Rectangular GCL samples (0.3×0.6 m) were cut using a razor knife (Fig. 3), transferred onto rigid plates and hermetically sealed in plastic bags to prevent the loss of moisture during transport and storage. All GCL samples were penetrated by fine roots but no cracks in the bentonite were apparent. Samples of cover soils and subgrades were also collected for the determination of water content and soluble cation content.

Hydraulic conductivity

Hydraulic conductivity tests were conducted on GCL specimens in rigid-wall permeameters following the procedures in standard test NF P84-705 (AFNOR 2008). The testing device was previously presented in [START_REF] Guyonnet | Geosynthetic clay liner interaction with leachate: correlation between permeability, microstructure, and surface chemistry[END_REF]. A 10 -3 M NaCl solution was used as the permeant liquid; this solution is specified in NF P84-705 as the testing solution and is believed to yield conservative hydraulic conductivities. An average vertical confining stress of 10 kPa was selected to represent the in situ condition in a capping system. Tests are composed of an initial swelling phase and a percolation phase. During the swelling phase in the case of the specimens tested here, no swell occurred at all. Consequently, the percolation phase started when 90% of equilibrium adsorption had occurred. Several hydraulic gradients were applied to specimens, ranging between 1 and 18, corresponding to hydraulic heads depending on the observed hydraulic conductivity of the specimens. When the obtained flow rate was too large, only small hydraulic heads could be applied, in order to ensure that the flow rate could be measured along time considering the capabilities of the measuring devices (Fig. 5).

To check that sidewall leakage and preferential flow were not occurring, a blue dye (Brilliant Blue) was added to the influent liquid of all exhumed GCLs at the end of testing. The blue dye spread in a homogeneous way through specimens; no sidewall leakage or preferential flow was evident in any test (Fig. 4).

A synthesis of the hydraulic conductivity values and water contents of bentonite in the GCLs before starting the permeability test and at the end of the permeability test are presented in Table 2. Very high values of hydraulic conductivities were obtained for the GCLs exhumed from the landfill cover (>10 -7 m/s) (Fig. 5). Lower hydraulic conductivities were obtained for the GCLs exhumed from the dam (~10 - 10 m/s). Those very different results will be subsequently discussed.

CEC and soluble cations

Cation exchange capacity (CEC) and the mole fraction of cations present in bentonite were determined following the procedures in standard test NF X 31-130 (AFNOR 1999). The CEC values were determined by extraction using the cobaltihexamine chloride method and chemical analysis of the extracts was conducted using Inductively Coupled Plasma Atomic Emission Spectroscopy (ICP-AES).

CEC and exchangeable Na, K, Ca and Mg are summarized in Table 2. Cation exchange has occurred nearly completely in the bentonites; Na originally contained in the bentonites was replaced by Ca and Mg originated from overlying soils and from dissolution of carbonates within the GCLs (Guyonnet et al. 2009). Those results are consistent with the low swell indices.

Surprisingly the quantity of Na cations of the bentonite contained in the D-2 GCL sample (GCL sample located below the D-1 GCL sample) is greater than the one measured in the other GCL samples. This tends to indicate that the GCL located above has protected the GCL located below by limiting the water percolation from the overlying soil; the cation exchange was thus slowed down.

Cover soils

Samples were collected from soils above and below GCLs. Water content, CEC and exchangeable Na, K, Ca and Mg were measured for each soil. Methods used to determine CEC and exchangeable cations were the same as those presented for the GCL samples.

Properties of the soils are summarized in Table 3. Ca is the dominant cation in the exchange complex of the landfill cover layer and subgrade. Ca and Na cations are equally abundant in the soil cover of the dam whereas Na appears as the dominant exchangeable cation present in the underlying soil of the dam. This result is consistent with the greater quantity of Na cations contained in the D-2 GCL bentonite sample.

DISCUSSION OF GCL PROPERTIES

Comparison with tests performed in laboratory with calcium chloride solutions

Many studies showed that the hydraulic conductivity of GCLs is sensitive to the concentration of the permeant solution and the cation valence [START_REF] Lin | Effect of Wet-Dry Cycling on Swelling and Hydraulic Conductivity of GCLs[END_REF], Jo et al. 2001[START_REF] Vasko | Hydraulic conductivity of partially prehydrated geosynthetic clay liners permeated with aqueous calcium chloride solutions[END_REF], Lee et al. 2005, Jo et al. 2005).

Results of tests performed on virgin non prehydrated GCLs in the laboratory with calcium chloride solutions are summarized in Figure 6 [START_REF] Gleason | Calcium and sodium bentonite for hydraulic containment applications[END_REF][START_REF] Norotte | Evolution of GCL hydraulic performance during contact with landfill leachate[END_REF][START_REF] Katsumi | Hydraulic conductivity of nonprehydrated geosynthetic clay liners permeated with inorganic solutions and waste leachates[END_REF], Katsumi et al. 2008a, Katsumi et al. 2008b).

The highest hydraulic conductivities were obtained with concentrated CaCl 2 solutions (>0.2 M). Nevertheless many of these studies have shown that GCLs maintain low hydraulic conductivity in the range of 10 -11 to 10 -9 m/s when permeated with lowest calcium concentration solutions which are close to the frequent calcium concentration in the pore water of soils (range between 0.001 M and 0.004 M) [START_REF] Egloffstein | Natural bentonites-Influence of the ion exchange and partial desiccation on permeability and selfhealing capacity of bentonites used in GCLs[END_REF]). The CaCl 2 maximal concentrations existing in the real landfill and dam overlying and underlying soils range between 0.0005 M and 0.0037 M (Table 3). For the same range of concentrations GCLs which were ion-exchanged in the laboratory by CaCl 2 solutions and exhumed GCLs are compared. The comparison shows that the low hydraulic conductivity values obtained in D-1 and D-2 GCL samples are in good agreement with the laboratory results.

The difference between results obtained for GCL samples L-1 and L-2 exhumed from the landfill (k>10 -7 m/s) and the GCL samples D-1 and D-2 exhumed from the dam (k<10 -10 m/s) can not be explained only by the cation exchange occurred in the bentonites but by a combination of various criteria: cation exchange and desiccation-hydration cycles experienced by the GCLs. Indeed the GCL placed in the landfill cover had a greater life service (8 years) which lead to a greater exposure to climatic condition compared to the GCL installed in the dam which was only subjected to three dry seasons.

Comparison with other recent field studies

Only a few recent field studies report on the behavior of GCLs in covers in a detailed way as the one described in this paper. [START_REF] Benson | Hydraulic performance of geosynthetic clay liners in a landfill final cover[END_REF] studied needle-punched GCL samples exhumed from the cover of a landfill located in Wisconsin after 2 and 4 years of service. GCLs initially contained Na bentonite and were covered by 0.76 m of a vegetated surface layer. The hydraulic conductivities of the GCLs were 4.2×10 -10 and 9.4×10 -8 m/s for the GCLs exhumed after 2 years service and 1.4×10 -8 and 8.1×10 -7 m/s for the GCLs exhumed after 4 years of service. Replacement of the native Na cations in the exchange complex by Ca and Mg combined with dehydration of the bentonite appeared to be the key factors causing the large increase in hydraulic conductivity. The overlying and underlying soils appear to be the source of Ca and Mg cations that exchanged with Na. [START_REF] Meer | Hydraulic conductivity of geosynthetic clay liners exhumed from landfill final covers[END_REF] detailed the investigation of needlepunched GCLs exhumed from three landfills in Wisconsin and in Georgia after 4.6, 4.1 and 5.6 years of service. All GCLs contained Na bentonite and were covered by 0.75 m and 0.80 m of vegetated surface layer depending on the site. Most of the Na initially in the bentonite had been replaced by Ca and Mg in the exhumed GCLs and the bentonites had swell indices typical of Ca bentonite (~10 mL/2g). Hydraulic conductivities of the exhumed GCLs varied over a wide range (5.2×10 -11 to 1.3×10 -6 m/s) and were strongly related to the gravimetric water content at the time of sampling: GCL samples with a gravimetric water content less than 80% had high hydraulic conductivities (10 -8 to 10 -6 m/s)(Fig. 7) whereas GCLs with gravimetric water contents greater than 100% had lower hydraulic conductivity (10 -11 to 10 -9 m/s). According to Meer and Benson the abrupt change in hydraulic conductivity shown in Figure 7 suggests that GCL desiccation to lower water content has a more dramatic effect on hydraulic conductivity.

Hydraulic conductivities exhibited no relationship with mole fraction of exchangeable Na because the Na mole fraction was low on all of the exhumed GCLs but the data could be segregated into two groups regardless of the Na mole fraction: GCLs with higher hydraulic conductivity (>10 -7 m/s) and GCLs with lower hydraulic conductivity (<10 -9 m/s) (Fig. 8). The lower hydraulic conductivities were similar to those obtained from hydraulic conductivity tests performed on GCLs in the laboratory with dilute calcium chloride solutions (10 mM) by [START_REF] Egloffstein | Natural bentonites-Influence of the ion exchange and partial desiccation on permeability and selfhealing capacity of bentonites used in GCLs[END_REF] and Jo (2005); whereas the very high hydraulic conductivities obtained were caused by other factors in conjunction with ion exchange. 2010b) presented the results obtained on needlepunched GCL samples exhumed from a cover of a landfill site in France after 3 and 6 years of service. GCLs contained calcium activated bentonite. The thickness of the cover soil was found to be approximately 0.2 and 0.5 m respectively and contained a large amount of calcium carbonate. The bentonite in the GCL showed a full exchange of sodium to calcium; the GCL samples did not exhibit any swelling capacity and the hydraulic conductivity measured ranged between 1.07×10 -6 and 6.91×10 -6 m/s. The GCL was no longer performing its lining function in relation with significant root intrusion in the GCL, desiccation-hydration cycles and cation exchange. Except for some points, hydraulic conductivity values as a function of the gravimetric water content of the bentonite at the time of sampling for all exhumed GCL samples are in good agreement with the tendencies observed by [START_REF] Meer | Hydraulic conductivity of geosynthetic clay liners exhumed from landfill final covers[END_REF] (Fig. 7); low hydraulic conductivity is obtained when the water content of GCL exceeds 80%. GCL sample D-1 and 6 (Touze-Foltz et al. 2010) have very low gravimetric water contents at the time of sampling due to the very low cover soil thickness (0.2 m for both sites) but higher hydraulic conductivities obtained for the GCL samples 6 (Touze-Foltz et al. 2010) could be explained by the long service life of the GCLs (6 years) which led to a greater exposure to desiccation-hydration cycles. A large difference in hydraulic conductivities between the GCL samples 1 [START_REF] Benson | Hydraulic performance of geosynthetic clay liners in a landfill final cover[END_REF]) is also noticed in Fig. 7. Indeed, one GCL sample had a very low hydraulic conductivity (k = 4.2×10 -10 m/s) whereas the second sample had a hydraulic conductivity close to 10 -7 m/s; but no reason for this difference could be determined from data collected by [START_REF] Benson | Hydraulic performance of geosynthetic clay liners in a landfill final cover[END_REF].

As described by [START_REF] Meer | Hydraulic conductivity of geosynthetic clay liners exhumed from landfill final covers[END_REF] even if there is no apparent relationship between hydraulic conductivity and Na mole fraction, the data collected from the various studies can be segregated into two groups regardless of the Na mole fraction (Fig. 8), except for two data points. Even if all the GCL samples exhumed from the landfill cover and the dam exhibit a low content in sodium ions (Na mole fraction range from 0.15 to 0.35), the hydraulic conductivities obtained for the GCL samples D-1 and D-2 exhumed from the dam match with the low conductivity value group whereas the hydraulic conductivities obtained for the GCL samples exhumed from the landfill L-1 and L-2 belong to the high value group.

The difference between the hydraulic conductivities of the GCL samples D-1, D-2 and L-1, L-2 can be linked to the very different service lives of the GCLs (3 years for D-1, D-2 and 8 years for L-1, L-2); the full exchange of sodium ions and an insufficient cover soil thickness which could not prevent the GCL from desiccation combined with a long time exposure to climatic conditions led to the formation of irreversible cracks in the bentonite.

CONCLUSION

GCL samples were exhumed from a landfill cover and from a dam after 8 and 3 years of service and tested for water content, swell index, saturated hydraulic conductivity and exchangeable cations. The GCLs were located under a shallow soil cover with a thickness in the range 0.2 to 0.5 m; samples of the overlying and underlying soils were also collected and tested for water content and exchangeable cations.

The bentonite in the GCL showed an exchange of sodium to calcium evidenced by swell indices and quantification of the concentration in various cations in the bentonite. The difference between hydraulic conductivity results obtained for GCL samples exhumed from the landfill (k>10 -7 m/s) and the dam (k<10 -10 m/s) cannot be explained only by the cation exchange occurred in the bentonites but by a combination of cation exchange and desiccation-hydration cycles experienced by the GCLs.

Indeed, the comparison of the exhumed GCLs and GCLs which were ion-exchanged by CaCl 2 solutions showed that GCLs maintain low hydraulic conductivity in the (range of 10 -11 to 10 -9 m/s) when permeated with lowest calcium concentration solutions (<0.004 M) which are the maximal calcium concentration found in the pore water of overlying and underlying soils collected in the landfill and in the dam.

The results from this study are consistent with previous recent results from the literature which indicate that cation exchange combined with hydration-desiccation cycles can adversely affect GCL in its function of hydraulic barrier.

ACKNOWLEDGMENTS

The urban area community of Le Havre (CODAH) permitted sampling and testing of the GCLs exhumed from the dam. Assistance provided by the CODAH is gratefully acknowledged.

  Swell index Swell index tests were conducted according to methods described in standard test XP P84-703 (AFNOR, 2002) which is almost similar to ASTM D 5890 (ASTM, 2011). The swell indices of the exhumed GCLs are consistent with typical values obtained for calcium bentonites (<10 mL/2g) (Egloffstein 2001).

Figure 1 .

 1 Figure 1. Landfill cover layer.

Figure 2 .

 2 Figure 2. Dam to prevent flooding.

Figure 3 .

 3 Figure 3. Exhumation of GCL samples.

  in : Eurogeo 5, 16-19/09/2012, Valence, ESP

Figure 4 .Figure 5 .

 45 Figure 4. GCL specimen after test with the blue dye.

Figure 6 .

 6 Figure 6. Hydraulic conductivity of GCLs percolated with CaCl 2 solutions.

Figure 7 .

 7 Figure 7. Hydraulic conductivity of exhumed GCLs versus gravimetric water content at the time of exhumation.

Figure 8 .

 8 Figure 8. Hydraulic conductivity of exhumed GCLs versus Na mole fraction in the exchange complex of the bentonite.

Table 1 .

 1 Description of lining systems at field sites.

		Site
		Landfill cover Dam
	Installation date	2003		March 2008
	Sampling date	August 2011	May 2011
	Service life (year)	8		3.17
	Cover soil thickness (m)	0.5		0.2
	Sample name	L-1	L-2	D-1	D-2
	Sample location			Above Below

Table 2 .

 2 Summary of properties of exhumed GCLs.

				in : Eurogeo 5, 16-19/09/2012, Valence, ESP
	Initial	Post test	Thickness	Bentonite		Hydraulic	Post test
	water content	water content	under 10 kPa	mass per unit area	Swell index	conducti-vity a	water content CEC	Exchange Cations (cmol/kg)
	Site (%)	(%)	(mm)	(kg/m 2 )	(mL/2g) (m/s)	(%)	(meq/100g) Na K	Ca Mg
	L-1 55	122	5.0	3.6	< 10	3.2×10 -7	122	84.3	8.3 0.9 37.9 2.4
	L-2 49	98	6.2	4.1	< 10	2.4×10 -6	97	94.2	8.5 1.2 41.9 3.3
	D-1 30	74	6.9	5.3	< 10	7.7×10 -11 74	85.2	7.5 1.0 29.3 8.3
	D-2 83	148	11.8	5.7	11.0	1.0×10 -10 148	92.6	19.6 2.3 25.9 8.1
	a Mean value obtained for the maximum hydraulic gradient

Table 3 .

 3 Summary of properties of soils.

								a
			Initial water
			content	CEC	Exchange Cations (cmol/kg)
	Site		(%)	(meq/100g) Na	K	Ca	Mg
		Surface layer 19	28.0	6.54	0.60	19.79	1.15
	L-1	Subgrade	21	32.7	5.43	0.60	16.79	0.72
		Surface layer 18	38.0	5.31	0.00	22.36	0.64
	L-2	Subgrade	18	31.4	8.49	0.51	13.98	0.67
		Surface layer 5	24.3	5.87	0.31	5.63	0.29
	D	Subgrade	17	17.5	5.72	0.33	3.13	0.29

in : Eurogeo 5, 16-19/09/2012, Valence, ESP