# 3D magnetic field reconstruction in VKS experiment - Iramis

## Transcription

3D magnetic field reconstruction in VKS experiment - Iramis

3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms J. Boisson1 and B. Dubrulle 1 SPEC/IRAMIS/CEA Saclay, and CNRS (URA 2464), 91191 Gif-sur-Yvette Cedex, France Abstract. We present a method for 3D magnetic eld reconstruction based on Galerkin Transforms. We test it over synthetic elds and real solenoidal (velocity) elds, measured in a water experiment. Our study shows that reliable reconstructions are possible provided the probes are suciently sampled and located in shifted congurations. A preliminary application of our method is performed over results obtained in the VKS dynamo experiment [8]. We show that the stationary dynamos obtained with counter-rotating impellers are mainly axisymmetric, with a non- axisymmetric part that decreases with increasing Reynolds numbers. Most of the azimuthal energy is localized near the (iron) impellers, conrming their importance in the dynamo process. PACS numbers: Magnetic elds measurement of, 07.55.Ge; Magnetic elds generation of, 07.55.Db; Galerkin method, 02.70.Dh; Inverse problems, 02.30.Zz Keywords: Dynamo Inversion, Galerkin 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 2 1. Introduction The magnetic eld is an essential ingredient of most natural objects and numerous campaigns of observations or experiments are devoted to its measurement or modeling. A problem often encountered in this process is the sparsity of measurements : only a limited number of measurements, at a limited number of positions are usually available. Specic techniques must then be implemented on the data to reconstruct the magnetic eld outside the measurements points, so as to get information about its geometry, intensity, direction, ... Classical examples are reconstruction of the terrestrial or Solar magnetic eld. In the rst case, data are rather numerous at the surface of the earth and reconstruction is achieved by tting a set of basis functions, usually spherical harmonics or spherical caps, to magnetic data. In the case of the Sun, data are sparser, and the reconstruction can usually only be achieved through physical assumptions. For example, Solanki et al [1] use magnetic data issued from vector spectropolarimetry of the He I 1083 nm multiplet, formed near the coronal base and assume that the He line is formed along tubes of magnetic ux as the active region emerges through the chromosphere into the corona. The He line emission then forms in coronal-like loops. With this assumption a 3D magnetic eld can be constructed that can be compared with extrapolations from deeper layers. Similar problematics can be found in laboratory experiments. For example, electromagnetic velocity probes usually require measuring the distribution of the 3D magnetic eld in the whole measuring volume out of the probe. Clever assumptions about exciting currents and numerical solving of a Laplace equation allow for reconstruction of the 3D magnetic eld from a mere measurement of the distribution of the normal component of the magnetic eld on the surface of the probe [2]. In the present paper, we focus on the case of the von Karman Sodium (VKS) experiment, a laboratory facility designed to studying the turbulent uid dynamo eect, i.e. the spontaneous generation of a magnetic eld from turbulent motion of liquid sodium. In that experiment, the magnetic eld measurements are localized only along a few magnetic sensors, thereby avoiding large disturbances of the measurements onto the ow of liquid sodium. The reconstruction of the full 3D magnetic eld from these sparse measurements therefore implies technical processing of the magnetic data. Here, we describe a solution inspired from geophysical methods, involving tting a set of suitable Galerkin basis functions to magnetic data. 2. Method 2.1. Geometry The setup corresponding to the VKS experiment is sketched in Figure 1 [13]. cylindrical vessel with a xed aspect ratio of h/R = 1.8, where R (h) A denotes the radius (height) of the vessel which is lled with liquid sodium. We consider a cylindrical coordinate system R (r, ϕ, z) such that the axial vessel extension lies along as the length scale, and introducing L = h/2R = 0.9, z. Choosing the variation ranges of the j 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms k 3 octobre 2007 Figure 1. (a) Sketch of the VKS experiment geometry including the probe locations. m, n The z o axis of the cylindrical coordinates is taken as the x axis of the experiment, i.e. the axial extension of the cylinder. The gap in between the inner and the outer cylinder is lled with sodium at rest. The inner container radius R is taken as length scale. (r, ϕ, z) inside the experimental vessel are therefore (0 ≤ r ≤ 1, −π ≤ ϕ ≤ +π, −L ≤ z ≤ +L). Our magnetic eld reconstruction however also extends beyond the vessel boundaries, i.e. for r > 1 thanks to suitable dimensionless cylindrical polar coordinates boundary conditions (see below). The uid is stirred by two independent rotating so-called TM73 impellers [16, 15]. The impellers are tted with curved blades, and can be rotated independently at frequencies F1 and F2 . Magnetic measurements are performed using 3D Hall probes, and recorded with q a National Instruments PXI digitizer. q’ Both a single-point (three components) probe (hereafter label G-probe) and custom-made arrays of 10 (three components) probes (hereafter called SM-array) are used. The array is made from Sentron 2SA-1M Hall sensors located every 28 mm along a line. The dierent possible locations of the probes are sketched in Fig. 1. 2.2. Brief overview A given VKS experiment provides us with a set of measurements of the (three components of the) magnetic eld, at N dierent sparse locations. To reconstruct the magnetic eld in the whole space, we expand it on a set of well chosen (see below) Galerkin functions Gi : B(r, ϕ, z) = M X bi Gi (r, ϕ, z), (1) i=1 where M is the number of modes of the reconstruction. Then, by imposing that the expansion matches the three components of the measured elds at the points, we get a set of bi ). 3N equations with M N measurements unknown (the coecients of the expansion Inverting the systems thus provides the values of the bi , and thus, the reconstruction 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 4 at that resolution. Of course, one cannot hope to reconstruct the eld with a resolution M < 3N to get good results). Besides, possible non uniqueness of solution suggest to use M 3N in order to allow for robustness of the inversion through redundancy. Since N is typically of the order to 1 to 10 (41 at most), this means that only a small number of Galerkin modes will larger than the number of measurements (i.e. be allowed in the reconstruction procedure. To get meaningful results, it is therefore mandatory to choose an optimal Galerkin basis, such that a small number of modes captures the large scale structure of the experimental elds. This imposes that the Galerkin basis satises both the solenoidal condition and the boundary conditions of the magnetic eld. We therefore now describe how to implement these constraints on the choice of the basis, following the recent work of [3, 4]. 2.3. Boundary conditions Through the cylindrical coordinate system, the system is naturally periodic in the direction. ϕ Boundary conditions must however be specied in the vertical and radial directions. 2.3.1. Vertical boundaries z = ±L As in pseudo-spectral simulations [15], the inversion procedure can be greatly simplied if one considers a periodized version of the problem, in which the magnetic eld is assumed periodic in the axial direction. It is however not possible to use as the unit cell the interval not naturally periodic over this cell. z ∈ [−L; L] since the magnetic eld is To avoid discontinuities and Gibbs phenomena (artifact oscillations of the reconstructed magnetic eld near the discontinuity), we have thus chosen to use as a unit cell the whole range the original physical domain. z ∈ [−L; 3L], which is twice Doing so yields magnetic eld components which are continuous everywhere. 2.3.2. Radial boundary r = 1 The boundary conditions at r = 1 are set by the condition that both the magnetic eld components and the derivative continuous at ∂r Br r = 1 -the last condition stems from the solenoidal nature of B -, are i.e. they match continuously the exterior domain solution. Many numerical methods to deal with the r=1 boundary conditions were developped. These schemes are based on boundary elements formalism [24], on integral equation [26] or on spectral nite elements [25]. Here we chose a simpler way to manage with these boundary conditions. In the sequel, we consider three main possibilities: • External insulating: If the medium surrounding the uid is insulating, we get the following condition for the exterior solution: ∇ × B = 0, ∇.B = 0, ⇒ ∆B = 0 (2) In the case of spatially homogeneous magnetic permeability we are considering, the magnetic eld must also be continuous and nite over all space, and vanish for r → ∞. 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms • = ∞) wall at r = 1. In that Bz = Bφ = 0 at r = 1. This Ferromagnetic (µr zero at r > 1, and that 5 case, we assume that the eld is case mimics the situation where the uid is surrounded by a ferromagnetic wall. for • General case. Br and Bφ . In that case, we do not specify the boundary conditions at r=1 2.4. Galerkin basis 2.4.1. General Case Following [3, 4], we rst switch from usual cylindrical components B = (Br , Bϕ , Bz ) to the new components B = (B+ , B− , Bz ) such that B± = (Br ± iBϕ )/2. Due to the periodic conditions in ϕ and z , it is natural to introduce the general decomposition over eigenmodes of the Helmoltz operator as: nmk P π/2L B+ C J (µ r) N M m+1 nmk X + X X nmk B = B− = C J (µ − m−1 nmk r) exp(imϕ + ikz), n=1 m=−M k=−P π/2L Bz Cznmk Jm (µnmk r) (3) n−m−k ∗ nmk = (C±,z C±,z ) to enforce reality of the eld and the numbers µnmk have to nmk be chosen so as to respect boundary conditions. The coecient C±,z are the spectral with coecients of our decomposition and are uniquely associated to our initial eld then write symbolically the value of B in the Galerkin space as GT (B) B. We (for Galerkin Transform) with C+nmk GT (B) = C−nmk . Cznmk For completeness, we introduce a similar decomposition for a scalar eld Ψ= N M X X (4) Ψ as P π/2L X Qnmk Jm (µnmk r) exp(imϕ + ikz), (5) n=1 m=−M k=−P π/2L so that we have GT (Ψ) = Qnmk . 2.4.2. Useful formulae (6) Our decomposition allows for interesting representation of usual derivatives, that are generalization of similar operations in the Fourier space. Indeed, we have for a scalar eld: GT (∆Ψ) = −(µ2nmk + k 2 )Qnmk , − 21 µnmk Qnmk GT (∇Ψ) = 21 µnmk Qnmk . ikQnmk (7) 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 6 In a similar way, we have for a vector eld: GT (∇ · B) = µnmk C+nmk − C−nmk + ikCznmk , i nmk nmk − kC µ C nmk z + 2 GT (∇ × B) = 2i µnmk Cznmk + kC−nmk . −iµnmk C+nmk + C−nmk (8) These simple formulae show that once the eld has been reconstruct, it is easy to perform accurate derivative operations by working in the Galerkin space. Moreover, the formulae can be used to select useful special Galerkin basis for magnetic eld reconstructions. 2.4.3. Useful basis General Galerkin transform of an arbitrary vector eld requires 3 independent coecient for each (nmk). In the case of a solenoidal eld, we must nmk − C−nmk + ikCznmk = 0 for any (nmk), so that only two independent have µnmk C+ coecients remain. In the sequel, we describe even simpler cases, with only one independent coecient, that are appropriate for elds and boundary conditions usually met in our von Karman experiment. 2.4.4. Beltrami case In this case, the Galerkin transform writes GT (B) = Dnmk Dnmk are p 2 ± µnmk + k 2 . This where the 1 2 1 2 (λ± − k) (λ± + k) , −iµnmk complex numbers such that ∗ Dnmk = Dn−m−k (9) and λ± = corresponds to a decomposition into Beltrami waves, polarization given by the sign of with λ. To guarantee some interesting orthogonality th conditions, we further take µnmk as the n root of Jm . The corresponding eld is solenoidal, and such that Bz = 0 at r = 1. 2.5. Marié Normand Daviaud (MND) eld case There are two cases depending on the axial wave number. case k 6= 0 where the c+ Km+1 /Jm+1 (µnmk ) GT (B) = Dnmk c− Km−1 /Jm−1 (µnmk ) , Km /Jm (µnmk ) Dnmk µnmk ∗ Dnmk = Dj−m−k , Km expressed through √ Km = Km (|k|), c± = i|k|/k 2, and the are complex numbers such that the modied Bessel function of order parameter (10) is determined as the n m th as root of the equation: 2|k| Km (|k|) Km−1 (|k|) Km+1 (|k|) = − µ Jm (µ) Jm−1 (µ) Jm+1 (µ) (11) 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms For each k, there is an innite set of real positive values µnmk , 7 which we truncate to N, the number of radial modes in the Galerkin expansions (3). Marié et al. [3] determined the roots of (11) for m=1 and discrete values of the axial wave number (kπ/2L). The roots of (11) for higher values of m, up to m ≤ 5, were computed later by Leprovost k the successive roots of (11) take increasingly large values as n increases (µ(n−1)mk < µnmk ). As an example, for m = 1 and k = 1 their values range between µ111 = 2.217 and µn11 = 32.22 for n = 20. [7]. For a given value of case k = 0 In that case, there are two dierent families of modes. The rst family has shape GT (B) = Dnm0 where µnm0 th is the n root of 1 µ /Jm+1 (µnm0 ) 2m nm0 1 µ /Jm−1 (µnm0 ) 2m nm0 0 , (12) Jm−1 . The second family has shape: 0 GT (B) = Dnm0 0 , 1/Jm (µ̄nm0 associated with a dierent set of eigenvalues µ̄nm0 which are the zeros of This Galerkin basis is solenoidal and matches at ∆B = 0. (13) r = 1 Jm . with a eld such that This case is therefore suitable to describe a medium surrounded by an insulating uid. 2.6. Ferro case For this case, we choose: −µnmk (k − γ) GT (B) = Dnmk µnmk (k − γ) , µ2nmk (14) ∗ where the Dnmk are complex numbers such that Dnmk = Dn−m−k and with th the n root of Jm and γ is solution of the dispersion equation: γ=k Jm+1 (µnmk ) + Jm−1 (µnmk ) . Jm+1 (µnmk ) − Jm−1 (µnmk ) In this case, the eld is solenoidal, and the boundary at r = 1. Bz = Bφ = 0 at r = 1, µnmk being (15) i.e. is perpendicular at This case therefore describes the case when the uid is enclosed inside a ferromagnetic boundary. 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 8 2.7. Inverse problem In the three cases we considered above, the nal Galerkin expansion for the magnetic eld at a given resolution coecients‡. To (N, M, P ) compute Galerkin expansion of B these requires 2N (M + 1)(2P + 2) coecients, one matches the measured eld writes Bmes , the independent real condition that the at dierent probes location (rmes , ϕmes , zmes ): B(rmes , ϕmes , zmes ) = Bmes . (16) This procedure can be recast formally into the equation: Aij Xj = Yi , where Xj , j = 1....2N (M + 1)(2P + 2) (17) is the vector formed with the real part and the imaginary part of the unknown complex coecients Dimk , Yi , i = 1....3Nmes is the vector formed by each of the component of the measured magnetic eld at each probe location, and locations. Aij is the matrix formed by the Galerkin basis evaluated at the probe Inverting formally (17), one then obtains the unknown coecient of the Galerkin expansion through: Xi = A−1 ij Yj , where A−1 ij is the pseudo-inverse of Aij . (18) Indeed, to ensure robustness of the procedure, we choose in the sequel the resolution such that 2N (M +1)(2P +2) ≤ 3Nmes (see discussion above). The problem is therefore overdetermined so that only pseudo-inverse is dened (in the sense that there is an innity of solutions for (18)). We have tested two Matlab procedures for the pseudo-inverse resulting in two sets of solutions that are both solutions in the least square sense (i.e. minimizing the norm of norm of of Xi (/ Xi Aij Xi − Yj ): one minimizing the (pinverse function) and one minimizing the number of non zero component function). In addition, we have also implemented a standard regularization procedure based on Tikhonov method using Matlab codes developed by Prof. Per Christian Hansen [5] and provided in http://www2.imm.dtu.dk/ pch/Regutools/. All three procedures give similar results for smooth elds (for example time-averaged elds). The regularization method allows elimination of spurious noise in the case of rough elds (such as instantaneous elds). We have therefore privileged this method in the sequel. To test the method and its sensitivity to the dierent parameters, we use two types of elds: i) synthetic elds, constructed from the Galerkin basis; ii) real elds, issued from Stereoscopic Particle Image Velocimetry performed on a turbulent von Karman water ow, in a half scale cylindrical vessel with respect to the VKS experiment [17]. ‡ (M + 1) takes into account Dnmk , the factor 2P + 2 takes The factor 2 comes from the complex nature of the coecients, the factor the reality of the eld, thereby imposing a symmetry conditions on the into account the existence of two families of modes at k = 0. 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 3 2 1 5 Figure 9 4 2. Projection of synthetic elds at dierent resolution meridional plane passing through the symmetry axis. component Bφ (N, M, P ) in a The out of plan (toroidal) is coded with color and the in-plane (poloidal) component (Br , Bz is coded with arrows (this representation is used in the whole article). Top left: N=1, M=0, P=0; Top right : N=1, M=1,P =1; bottom left : N=2, M=1, P=1; bottom right : N=3, M=2, P=1. We represent the 5 possible probes locations in the top left gure. 3. Tests on synthetic elds 3.1. Description of synthetic elds The synthetic elds are constructed from Eq. (3) by xing a resolution then choosing the 2N(M+1)(2P+2) independent (N, M, P ) and Dnmk so that the energy is concentrated in the lowest modes and decreases with increasing mode number- the idea being to i mimic a "realistic" experimental eld. In practice, we set Dnmk = (−1) /i where i = |j| + |m| + |k|. Example resolution (N, M, P ). of such synthetic elds in provided in Fig. 2, for dierent The complexity of the synthetic eld topology increases with the resolution. The synthetic measurements are extracted from these elds by computing the value of the eld at some selected location. In the sequel, we test various reconstruction by using "measurements" performed at the location given by VKS magnetic probes. There are three probes in the vertical plane (probes 1 2 3), and three again in the central plane (probes 1 4 5), at the location of the velocity mixing layer in the exact contra-rotation 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 10 regime. The location of these probes is reported in Fig. 2 for illustration and clarity. 3.2. Diagnostic To quantify the quality of the reconstruction, we dene scalar quantities inspired from classical residue denition as follow. We rst divide our available measures in two j distinct parts: i) a large set (X = {rmeas }), including about 95 percent of the measures, j that is kept to reconstruct the eld; ii) a second, smaller set (Y = {rtest }), that is used to compute "residue-tests". From the rst set X, we dene a classical residue characterizing the quality of the reconstruction from the measures in set X as: v u N 3 j j i uX X (B i (rmeas ) − Breconst (rmeas ))2 real r=t j i (Breal (rmeas ))2 j i with i = r, φ, z coordinates and j = 1, N (19) the number of the measure in X. This residue quanties the ability of the reconstruction to t the observed magnetic eld with the set of measures X. Y , we dene another residue based on the measures not involved in the magnetic eld reconstruction. This residue is named Residue test and is dened j j like in Eq. (19) by substituting rmeas with rtest . Finally, we can dene a third residue based on energy as: ∆E = (Esynth − Ereconst )/Esynth . We refer to it as the Energy residue. From the second set 3.3. Inuence of mode numbers We rst explore the inuence of the mode number on the reconstruction. Clearly, we cannot get more reliable information than present in the eld to be reconstructed. Therefore, the number of modes used in the reconstruction Nr cannot be larger than 3Nmes . Also, in order to get stable results, it is interesting that Nr = (1 − fr )3Nmes where 0 < fr < 1 quanties the the total number of measures to use some redundancy, so percentage of redundancy of the data. In order to get a hint about typical admissible of values for fr , we use a given (N, M, P ) synthetic eld with varying Nr = 2N (M + 1)(2P + 2) modes and extract 3Nmes = 120 synthetic measures at the VKS sensor positions (10 sensors measuring three components per probe on the 2 - 3 - 4 - 5 probes represented in gure 2). then rst reconstruct the eld from the synthetic measure with the same set We (N, M, P ) of modes, the necessary test which have driven the elaboration of the reconstruction method. We are not representing the results here, however in all the cases, the residue −15 are very low (r ∼ 10 ) i. e. the reconstruction with the same set of modes than the initial eld always reproduces well the eld at the measurement points. residue is fairly low until Nr ∼ 50 (fr = 0.6) The energy where the reconstructed energy starts to 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 11 1 6 10 0.9 4 0.8 10 Energy Residue 0.7 Residue 0.6 0.5 0.4 0.3 2 10 0 10 −2 10 −4 10 0.2 −6 10 0.1 0 0 10 20 30 40 50 Modes Number 60 70 0 80 10 20 30 40 Modes Number 50 60 70 Figure 3. Evolution of residue (bottom left) and of the energy residue (bottom right) with the magnetic eld modes number. The (4, 2, 2) initial eld with 144 modes is the in the top left gure, the ltered version is represented in the top middle gure, the reconstruction (3, 1, 1) with 48 modes is plotted on the top right gure. diverge form the synthetic eld energy. Therefore, the degree of redundancy is necessary to reconstruct the eld in this case must be at least The previous situation is however quite 60 per cent. ideal. In real experiments, the magnetic eld has a (presumably fairly large) unknown number of modes and robust reconstruction attempts use a lower number of modes than in the experimental eld. To test this eect, we choose an synthetic eld with a fairly large number of modes ((4, 2, 2) resulting in with Nr = 144 modes) and reconstruct it with dierent sets of modes, Nr < 96. Results are presented in gure 3. We see that the residue decreases fairly monotonously along with the energy residue until Nr ≈ 20(fr = 0.8). Above this, the energy residue starts diverging, while the residue continues to decrease until Nr = 40 (fr = 0.67), where it saturates towards a value of the order of 0.5 (with big oscillations). Note that the oscillations are mainly due to reconstructions that have either M = 2 or P = 2. These modes are dicult to capture in our measurement set-up, since we have a maximum of three values in both φ and z directions (see gure 2). Thus, introducing a number of modes superior to one in these directions can lead to systematic error coming φ coordinates. Keeping in mind this restriction, and focusing on reconstruction with M ≤ 1, P ≤ 1, we get a "best" reconstruction with (3, 1, 1) modes conguration, corresponding to Nr = 48 and δE ∼ 0.03 shown in gure 3. One sees that the reconstruction captures the "large- from the low number of measures points along the z and scale" structure of the initial eld. To quantify further this, we may also compare the 12 Residue Energy Residue 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms Sensors Number Sensors Number Figure 4. Evolution of residue (right gure) and energy residue (left gure) of the reconstruction with varying sensors number Ns . Ns /4 measurements have 32 modes. There are probes. Here, both the initial and the reconstructed eld per reconstruction with a ltered version of the initial eld, computed using a sgolay lter with a length scale of the order of a tenth of the cylinder radius. One sees that the reconstruction captures even ner details than the coarse version. 3.4. Inuence of sensors numbers In this section, we explore the dependence of the reconstruction quality with the number of sensors (varying Nmes number of measurements points). For this, we consider a (2, 1, 1) synthetic eld and reconstruct it with a varying number of measurements 3Nmes . In order to minimize the inuence of probes location, we always consider the same number of sensor Ns /4 on each of the four probes (probes 2, 3, 4 and 5 reported in Ns . The total number of measurements is then 3Nmes = 3Ns . In the sequel, we vary Ns /4 from 1 to 4. The synthetic elds is reconstructed with a (2, 1, 1) set of modes, with Nr = 32. The inuence of Ns over the reconstruction is given in gure 4. The residue is −15 almost constant, maintaining very low values (r ∼ 10 ). The energy residue ∆E is more sensitive to Ns and takes very large values for Ns ≤ 12, i.e. a total measurement number less than 36 . This corresponds precisely to a range of values for which Nr = 32 > 3(1 − 0.67)Nmes , conrming the importance of keeping at least 60 per cent of redundancy. Above the value Ns = 12, the energy residue becomes quite low −5 (∼ 10 ) and the reconstruction quality is excellent. gure 2). The number of measurements is then increased by increasing 3.5. Inuence of probe location We have worked so far assuming that the measurements are performed at the present VKS probe location. (see gure 2). It is however interesting to check the inuence of the probe location onto the reconstruction, so as to potentially optimize experimental measurements. The question is then how to optimize the probes location to get maximal quality of magnetic eld reconstruction with minimal impact on the experimental 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 2 1 5 Shifted (3,5) Central (1,5) Axial (3,2) Residus Energy res. 3 Residus Energy res. 3 5 13 20 10 10 10 −5 10 Energy Residue Shifted (3,5) Residue Axial (3,2) Central (1,5) −10 10 0 10 −10 10 Shifted (3,5) Axial (3,2) Central (1,5) −20 10 −15 10 −30 10 0 10 20 30 40 Modes Number 50 60 70 0 10 20 30 40 Modes Number 50 60 70 Figure 5. Top gure: Example of a synthetic eld with indication of the three probes conguration (axial, central, shifted ). Bottom: evolution of residue (left) and energy residue (right) with number of initial eld modes (equal to the number of reconstructed modes) depending on the three sets of probes location axial, central and shifted. sodium ow. Such impact is decreased with decreasing the number of probes and with a probe conguration respecting as much as possible the symmetries of the ow. We thus consider now measurements performed with only two probes. Given the present experimental conguration, three dierent sets of symmetric probe locations can then be constructed. They are reported in gure 5. The three conguration are: i) the axial conguration with two probes in the same vertical plane symmetric with respect to the meridional plane; ii) the conguration with two probes in the meridional plane of the cylinder (z angles diering by dierent φ angles. π; iii) the shifted conguration = 0) with two probes at dierent central at two z and We consider 10 sensors on each probes, and reconstruct the elds with the same increasing number of modes for both the synthetic eld and the reconstructed eld. The residue behavior for the three congurations is given in gure 5. Residue in the central conguration are large and decreases with the mode number, while it is small or close to zero for axial and shifted congurations. The energy residue for the axial and central conguration increases signicantly with the mode number. Thus the reconstruction in these geometry is not reliable with only two probes. In contrast, the shifted conguration provides fairly reasonable results. We focus on this conguration 14 Residue Residue 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms Energy res. Noise Energy res. Noise Noise Noise Figure 6. Evolution of residue and energy residue with the noise intensity. On the left panel, the reconstructed and the initial eld have the same set of modes; On the right panel the reconstructed has a lower number of modes than the initial eld. from now on. 3.6. Inuence of noise So far, we have performed reconstruction using "perfect" measures, matching exactly the synthetic eld. In real experiments, various noise sources pollute the measurements, and it is important to test the sensitivity of the reconstruction to measurement noise. We therefore extract articial measurements from a synthetic a shifted conguration Anoise = Rnoise /Rmeas (4, 2, 2) eld (144 modes) in and superimpose to them a white noise of increasing amplitude before reconstructing the eld. We consider two cases : i) a reconstruction with the same number of modes as the synthetic eld, and varying noise amplitude. Results are provided on the left panel of gure 6. ii) a reconstruction using an under sampled number of modes (Nr = 48 for (3, 1, 1) experimental conditions- and varying noise amplitude. -to remain close to actual Results are provided on the right panel of gure 6. In all cases, we observe a decrease of the reconstruction quality with increasing noise. The impact is larger for the under-sampled case but grows less rapidly. The energy residue however stays close from the initial energy. Our conclusion is that the energy measurements from reconstruction techniques are reliable up to 20 percent even in the presence of noise. From our series of tests, we obtained two main conditions for reliable reconstruction: shifted probe conguration and number of reconstruction mode smaller than number of measurements. Furthermore, we observed that residue by itself is not a reliable test of the reconstruction quality, since we can get very good residue and poor energy residue. Since in experimental elds, we dot not have access to energy of the eld (i.e. we cannot compute energy residue), we shall use residue test to discriminate the reconstruction quality. 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 15 1 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.8 0.6 0.4 r 0.2 0 −0.2 −0.4 0 0 −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −1 −0.8 −0.6 −0.4 −0.2 0 z 0.2 0.4 0.6 −1 −0.8 0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −1 −0.8 0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Figure 7. On the left gure, a real Von Karman water velocity eld (F=16Hz); on the middle gure, the ltered version; and on the right gure, the reconstructed eld. The out of plan (toroidal) component Vφ is coded with color (red are positive values while blue are negative values) and the in-plane (poloidal) component (Vr , Vz ) is coded with arrows. The 3 probes of 10 sensors used to reconstruct the eld are represented in grey on the left gure. 4. Test on a real eld In this section we just focus on testing the reconstruction method over of an experimental velocity eld measured through a Stereoscopic Particle Image Velocimetry (SPIV) on a Von Karman water ow in counter-rotation at a frequency of 16 Hz [23]. This velocity eld corresponds to the mean sodium ow involved in the VKS experiment if one neglect the magnetic eld retroaction e.g. this experimental eld is used in cinematic dynamo study. The SPIV provides the three velocity components in a meridional plane passing through the axis of rotation. The time average velocity eld is shown in Fig. 7. It is obviously a fairly large scale axisymmetric eld, with four recirculating cells. To test the reconstruction, we extract 30 articial measurements from three ctive probes located at the locations of the VKS experiment, shown in Fig. 7. eld using the inversion procedure described in section 2. We then reconstruct this However in order to take into account the only hydrodynamic ow structure of water, we thus use the Galerkin basis based on Beltrami waves (such that rot(V ) = λV , and Vz = 0 at basis presented in section 2.4.4 respects the inertial nature of the ow. r = 1). This Indeed from the minimization of the energy at xed angular momentum and helicity ref. [22] shows that the ow vorticity and the velocity are aligned everywhere and the mean ow is a Beltrami state. Therefore the reconstructed eld is incompressible (div V =0) and presents boundary layers. Since the ow is axisymmetric, we consider only cases with of the reconstruction is provided in Fig. 7, for is 0.26 while the energy residue is 0.25. (5, 0, 1) M = 0. The result modes. The associated residue Comparing with the full experimental eld, we see that the main topology of the velocity eld is well captured: the cell limits are close to reality and their amplitudes near the centre are well reproduced. However, the positions of the maxima of each cells is not well reproduced and the velocity amplitude of in-plane components is overestimated near the disks. To check whether this is an eect of "under-resolution", we also made a comparison with a ltered-version of the 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 16 experimental eld, using a sgolay lter, so as to keep only largest scale. One sees that indeed the position of the maxima of each cell is closer to that of the ltered eld. Note that we reconstructed this topology from an axial conguration of probes (the only one available in this situation) which is not as reliable as a shifted conguration, as proved in Section 3.5. From this test over an experimental eld, we conclude that the reconstruction is able to extract general behavior (energy and global topology) of a given eld from a restrain number of measures. Smaller scale details like the exact maxima positions, or the eld amplitudes near the boundaries, require a much larger number of measurements, that are for the moment unavailable in the VKS experiment. 5. Application to the VKS experiment In this section we nally apply the reconstruction method to the magnetic eld in a particular VKS experiment. We consider a set up with two iron impellers. is surrounded by sodium at rest in an outer cylinder (radius 604 mm) delineated by a copper cylinder. in [20]. Rc = 289 The ow mm, length This conguration is discussed at length In exact counter-rotation, it leads to a statistically stationary dynamo for F1 = F2 > 16Hz . Measurement are made through three probes: one probe with 10 sensors, located at position 5 in gure 2 (SM1), one probe with 8 sensors at the position 2 (SM2) and a Gaussmeter (1 sensor) in position 3 (G). Due to failure of some experimental probe, we can consider only a restricted number of reliable measures for reconstruction. Here, we focus on 7 exact contra-rotation regimes at frequencies (F1 = F2 = 12, 17, 18, 19, 20, 21, 22Hz ) corresponding to a magnetic Reynolds number Rm = 0.6µ0 σ(F1 + F2 )/2πRC2 at temperature 120o C Rm = 24, 34, 36, 38, 40, 42, 44. The last six regimes in this study corresponds to dynamo regime. For each regime, measurements are made during 60 seconds at frequency of 2000 Hz. In order to extract a general large-scale behavior, we average in time the measurements at each frequency. According to results of previous section, our measurement number prevents reconstruction of the magnetic eld with more than 54 modes. We then reconstruct the eld with a non-axisymmetric set of modes : (2, 1, 1) corresponding to 32 modes (the maximum number of modes we can reconstruct the magnetic eld with). We compute the residue and the residue test taken on one of the sensors of SM1 (as dened in section 3.2). These residue for dierent Rm are shown in Fig. 8. The (2, 1, 1) set of modes gives a residue ∼ 0.2 and a residue test ∼ 0.15 for the dynamo regimes. This set of mode provides a reasonable t of the measurement, as seen in Fig. 8. Its corresponding total energy as a function of The total energy growth is roughly quadratic in Rm is given in Fig. 8. Rm. A rst striking result of our reconstruction is that the time averaged dynamo eld is not purely axisymmetric: m = 1 mode is needed to t the measurements. is probably due to the presence of the 3 This probes that break the axisymmetry of the experiment,while the presence of the dominant m=0 already pointed in [13] is the 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 0.25 SM1 reconstruction SM1 SM2 reconstruction SM2 10 Residue Residue test 17 Br 5 0 0.2 0.4 0.6 r 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 0.8 1 1.2 0.2 20 10 Bphi Residue and Residue test −5 0 0 −10 0 0.15 r Bz 20 10 0 0.1 32 34 36 38 40 42 0 44 0.2 0.4 0.6 r Rm 5000 8 4500 7 4000 6 5 3000 Em0/Em1 Total Energy 3500 2500 2000 4 3 1500 2 1000 1 500 0 32 34 36 38 40 42 0 32 44 34 36 Rm Figure 8. 38 40 42 44 Rm Top left: reconstruction residue (in blue) and residue test (in red) for dierent magnetic Reynolds number in a VKS2 experiment; Top right: comparison between reconstructed eld (lines) and real probes measures (crosses); one sees that the reconstruction captures well the behavior. Bottom left: total energy of the reconstruction as a function of magnetic Reynolds number; Bottom right: ratio of the energy of m=0 modes on the energy of m=1 modes as a function of the magnetic Reynolds number. We also add the error bars extracted from the residue. subject of many interpretations, [27] [28] suggest a simple α -ω dynamo model where the helical nature of the velocity is ejected by the impellers or [29] where the soft iron impellers favour the growth of m=0 mode against the m=1 mode. To quantify this m = 0 mode to the energy of Rm: the dynamo becomes more result, we show in Fig. 8, the ratio of the energy of the the m=1 mode. We see that this ratio increases with m = 1 does not disappear and still contains largest Rm: for Rm = 34, Em=0 /Em=1 = 1.8; and more axisymmetric. However the mode a signicant amount of energy at the Rm = 44, Em=0 /Em=1 = 5.4. The corresponding dynamo eld structure is given in Fig. 9 at Rm = 34 close to the dynamo onset. It is mainly an axial dipole, with concentration of azimuthal energy near the iron propellers. Taking a closer look at this phenomenon through iso-density of energy surface at 25 % of the maximum energy (Fig. 9), we observe that the energy distribution is asymmetric, with more energy close to the left disk than close to the right disk. At the present stage, we do not know whether this is an artifact of the reconstruction due to the lack of measures or if this reects an actual breaking of the 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 18 40 1 30 0.8 0.6 20 0.4 10 r 0.2 0 0 −0.2 −10 −0.4 −0.6 −20 −0.8 −30 −1 −0.6 −0.4 −0.2 0 z 0.2 0.4 0.6 −40 Figure 9. Left: Total reconstructed magnetic eld in a VKS experiment at just above dynamo onset. The out of plan (toroidal) component color and the in-plane (poloidal) component (Br , Bz Bφ Rm = 34, is coded with is coded with arrows; Right: Corresponding stream lines and energy density at 25% of the maximum 40 1 30 0.8 0.6 20 0.4 10 r 0.2 0 0 −0.2 −10 −0.4 −0.6 −20 −0.8 −30 −1 −0.6 −0.4 −0.2 0 z 0.2 0.4 0.6 −40 m = 0 of the reconstructed magnetic eld in a VKS experiment Bφ is the in-plane (poloidal) component (Br , Bz is coded with arrows; Figure 10. Left: Mode at Rm = 34, just above dynamo onset. The out of plan (toroidal) component coded with color and Right: Corresponding stream lines and energy density at 25% of the maximum. axial symmetry across the z=0 plane, generated by the dynamo. More measurements would be needed. To try and determine the nature of the two modes, we plot them separately in Fig. 10 for the m=0 m = 1 mode. The m = 0 mode is clearly The m = 1 mode has a meridional dipole mode and in Fig. 11 for the an axial dipole, as already discussed in [13]. component, localized near the propeller which concentrates the largest energy. It does not however not resemble the neutral mode predicted by kinematic dynamos based on the mean velocity ow [3], conrming that the present dynamo is mainly generated by velocity uctuations [21]. To quantify further how the repartition between the m = 0 and m = 1 mode proceeds during dynamo instability, we apply a reconstruction procedure during the 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms 19 40 1 30 0.8 0.6 20 0.4 10 r 0.2 0 0 −0.2 −10 −0.4 −0.6 −20 −0.8 −30 −1 −0.6 −0.4 −0.2 0 z 0.2 0.4 0.6 −40 m = 1 of the reconstructed magnetic eld in a VKS experiment Bφ is the in-plane (poloidal) component (Br , Bz is coded with arrows; Figure 11. Left: Mode at Rm = 34, just above dynamo onset. The out of plan (toroidal) component coded with color and Right: Corresponding stream lines and energy density at 25% of the maximum. 14 1 Em0/(Em0+Em1) Em1/(Em0+Em1) Normalised Energy 0.9 12 Em0/(Em0+Em1) Em1/(Em0+Em1) Normalised Energy 0.8 0.7 10 0.6 0.5 0.4 8 0.3 0.2 6 0.1 0 0 10 20 30 40 50 time (s) 60 70 80 90 100 4 2 0 0 10 20 30 Figure 12. Total Energy, 40 50 time (s) m=0 and 60 m=1 70 80 90 100 modes relative energies as a function of time during the initial growth of the magnetic eld (Rm evolution at early time, to capture the crossover behavior. = 36). Insert: zoom of the 3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms growth of the magnetic eld. Rm = 32, 20 For this, we consider dynamical measurements at during a growth stage, and apply a moving average over 0.1 second upon all measurements. We then reconstruct in time the magnetic eld in order to get the mode topology during the growth. The result is given in Fig. 12 where the normalised magnetic energy measured in the experiment is compared with the relative energy ratios Em=0 /(Em=0 +Em=1 ) and Em=1 /(Em=0 +Em=1 ) obtained through the reconstruction. In this measurement, the experiment is tuned at a value of Rm larger than critical at t = 0. The magnetic eld then increases in time due to the dynamo instability. One sees that at the very early stage of the growth, the mode crossover with the mode Then, after about 30 m = 0 m=1 dominates, with an occasional corresponding to a localized magnetic energy burst. seconds, the magnetic eld growth accelerates, corresponding to a crossover between the m = 1 and m = 0 mode, that predominates after this stage. Note further that the magnetic eld growth proceeds in roughly three steps: one between t=0 and t = 30 where the magnetic eld is small, with only one energy burst m = 0 and m = 1 mode). A second stage, corresponding to the crossover between the m = 0 and m = 1 mode, with larger uctuations, but still moderate values of the magnetic eld. A third stage, from t = 60, where the energy explodes and the dynamo settles, with a predominance of the m = 0 (corresponding to the rst crossover in between the mode. This interesting step-by-step dynamo onset suggests the the dynamo mechanism is non-trivial, and may involve the action of the m = 1 as a catalyst. Further analysis with more measurements will be necessary to clarify this point. 6. Discussion We have presented a method of reconstruction for 3D magnetic eld reconstruction based on Galerkin Transforms and tested it over synthetic elds and real solenoidal (velocity) elds, measured in a water experiment. Our study shows that reliable reconstructions are possible provided the probes are suciently sampled and located in shifted congurations. The present sets of available measurements in VKS experiments prevent reconstruction of modes higher than m = 1. In that respect, it would be interesting to develop azimuthal belt probes around the cylinder in order to extract m > 1 modes without perturbing the ow with intrusive sensors. To illustrate the possibility of our method, we have applied it to an old VKS experimental conguration. Due to the fairly low available number of measurements, we could only obtain rough results regarding the dynamo geometry, conrming previous results. In future VKS dynamo campaigns, as many as four probes with 10 sensors each will be available. We have therefore good hope that reconstruction in such a case will be more reliable, and that we can draw interesting results concerning the magnetic eld geometry during dynamical events (reversals, periodic cycles, extinction, bursts,... improving our understanding of turbulent dynamos. 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