Figure C.9: (Color online) Ohmic decay. Decay times againstµeffr (top row) and againstµ0σeff (bottom row) for three disk thicknesses d= 0.6,0.3,0.1 (blue, red, yellow). The solid curves show the results obtained from the hybrid FV/BEM scheme and the dotted curves denote the results from the SFEMaNS scheme.
for Rm6= 0 is weaker compared to the Ohmic decay (green curve in left panel of gure C.13).
An opposite behavior is obtained for a high conducting disk where a reduction of the(m= 1) growth rate is obtained (see right panel of gure C.13).
In both cases the(m= 1)decay rate remains independent ofµr(respectivelyσ) for values exceeding approximately µr ≈ 20 (or µ0σ ≈ 20). The critical magnetic Reynolds number has also been computed for a dierent set-up with the ow restricted to the bulk region : 0≤r≤1.4,−0.9≤z≤0.9 with VTF conditions applied at the boundary of this region. We obtained in Rmc= 39in this case. Note that this pseudo-vacuum set-up under-estimates the threshold by more than 30% when compared to Rmc= 55in the limitµr1. This conrms that a realistic description of the soft iron impellers is crucial to get correct estimates.
The robustness of the results reported above exhibits a rather delicate dependence of the eld behavior on the details of the ow distribution, in particular from the ow in the lid layers. Beside the dynamo killing inuence of the lid ow [137] this is also true for the radial ow in the vicinity of the inner side of the disks. In order to estimate the relative impact of velocity jumps on the two codes, some simulations have been performed by smoothing the radial component of the velocity at the transition between the bulk of the domain and the impeller disk (where ur = 0). The resulting decay rates atRm = 50 (black stars in the left panel of gure C.13 and table C.3) are slightly dierent at µr = 1but the dierence is more signicant for µr= 60.
Figure C.10: (Color online) Decay rates and decay times againstµeffr (left column) and against µ0σeff (right column) for vanishing tangential elds boundary conditions. d= 0.6. The solid (dashed) curves denote the results from the FV/BEM (SFEMaNS) scheme.
Figure C.11: (Color online) Comparison of boundary conditions. Decay times against µeffr (left panel) and against µ0σeff (right panel) for vacuum BC (solid curves) and VTF boundary conditions (dashed curves). d= 0.6. All data results from the SFEMaNS scheme.
communication). This unexplained fact raises the question whether the role of this material
Figure C.12: (Color online) Structure of the prescribed axisymmetric velocity eld. The color coded pattern represents the azimuthal velocity and the arrows show the poloidal velocity eld. The black solid lines represent the shape of the impeller disk.
m=1, Rm=50 µr= 1 µr= 60
FV/BEM SFEMaNS FV/BEM SFEMaNS
M N D −1.218 −1.327 −0.550 −0.655
SM N D −1.51 −1.667 −1.16 −1.291
Table C.3: Decay rate for m = 1 mode for 2 ows MND and a similar ow with slightly modied (smoothed) radial velocity component (SMND).
is only to lower the critical magnetic Reynolds number in the domain of experimental feasi-bility or if the dynamo mechanism is fundamentally dierent when the conducting medium is no longer homogenous. This issue can be addressed numerically in principle . However, to face such problems with heterogenous domains, specic algorithms must be implemented and validated. This was the aim of the present study. Our comparative runs of Ohmic decay problems proved in practice to be extremely useful to optimize both codes and to select some numerical coecients occurring in the algorithms (such as in penalty terms).
The problems which have been successively presented above are standard in MHD, but we were forced to reduce the dimension of the parameter space to congurations more or less related to the Cadarache experiment, where the impellers may be treated as disks in a conducting ow bounded by a cylinder of a given aspect ratio. We have thus considered axisymmetric domains only (see [58] for non-axisymmetric cases), and azimuthal modes of low order (m= 0 and 1).
We have rst studied Ohmic decay problems, with disk impellers of various thicknesses to investigate scaling laws and the impact of the spatial resolution. The eects of internal assemblies of high permeability material within the uid container are dierent from those of an enhanced, but homogenous uid permeability because of inner boundary conditions for the magnetic eld (in case of high permeability material), and for the electric eld/current (in case of conductivity jumps). In the free decay problem with thin high permeability disks a selective enhancement of the axisymmetric toroidal eld and the (m= 1)mode is observed whereas the axisymmetric poloidal eld component is preferred in case of high conductive
Figure C.13: Growth rates for the MND ow driven dynamo againstµr(left panel) and against µ0σ(right panel). Solid curves denote data obtained from the FV/BEM scheme, dashed curves denote the results from the SFEMaNS scheme. The green, blue, red, yellow colors denote the casesRm = 0,30,50,70. The black stars in the left panel show the results for the SMND ow atRm = 50 (see text) as reported in Tab. C.3.
disks.
We have also shown that pseudo-vacuum boundary conditions, which are easier to im-plement on the cylinder walls than the jump conditions on the impellers, have only a slight inuence on the decay rates. The impact of the outer container boundaries on the eld be-havior is limited to a shift of the decay/growth rates. This is surprising, insofar as pseudo vacuum boundary conditions resemble the conditions that correspond to an external material with innite permeability. Nevertheless, the presence of high permeability/conductivity disks within the liquid hides the inuence of outer boundary conditions, and the simplifying ap-proach applying vanishing tangential eld conditions at the end caps of the cylinder in order to mimic the eects of the high permeability disks in the VKS experiment ([61]) is not sucient to describe the correct eld behavior. The consideration of impeller disks with (large but nite) permeability remains indispensable in order to describe the inuence of the material properties.
For completeness, we have also considered conductivity domains. From the experimental point of view the utilization of disks with a conductivity that is 100 times larger than the conductivity of liquid sodium remains purely academic. Nevertheless, the simulations show a crucial dierence between heterogeneous permeabilities and conductivities: even if these two quantities may appear in the denition of an eective Reynolds numberRmeff =µ0µeffr σeffU L, they do not play the same role and they select dierent geometries of the dominant decaying mode. It is not only a change of magnetic diusivity that matters.
We have considered kinematic dynamo action, using analytically dened ows in accor-dance with the setting of the VKS mean ow. Since these ows and the variation of µ and σ are axisymmetric, the azimuthal modes are decoupled. An important Fourier mode is the (m= 1) mode which will be excited eventually through dynamo action. We have shown that our codes give comparable growth rates for this mode. We have examined also the growth rate of the (m = 0) magnetic eld in presence of soft iron impellers and the axisymmetric MND ow. Since convergence of results is not achieved in all the cases considered, this comparative
study is still in progress and it has thus not been included in the present paper. We recall that the main surprise of the Cadarache experiment was perhaps the occurrence of the mode (m= 0), which pointed out the possible role of the non-axisymmetric ow uctuations. Non-axisymmetric velocity contributions might be considered in terms of anα-eect as it has been proposed in [119] and [89, 88]. Preliminary examinations applying simple α-distributions are presented in [55] and [58]. However, there is still a lack of knowledge on the details and phys-ical justication of a precise α-distribution which requires a non-linear hydrodynamic code.
The questions related to this empirical fact represent a main issue of the experimental and numerical approaches of the uid dynamo problem and deserve a dedicated study. Our ax-isymmetric model is not intended to explain the main features of the VKS experiment, which are the dominating axisymmetric eld mode and the surprising low critical magnetic Reynolds number of Rm≈32. However, our results give a hint why the (m= 1) mode remains absent in the experiment.
A source term on them= 0mode appears when the ow axisymmetry is broken. Although the relative amplitude of this source cannot be discussed here, we note that the decay time of the(m= 0)toroidal mode becomes the largest when the eective permeability is high enough (see for example gure C.8). It may thus appear as the dominant mode of the dynamo, as it seems to be observed in the VKS experiment. Stated otherwise, the impact of soft-iron impellers on the critical magnetic Reynolds number of the (m= 1)-mode could be rather low (decrease from ∼76to ∼55in the MND case) and could remain unobservable, while it could be strong for the (m= 0) mode (down to 32 in the VKS geometry) when conjugated with a slight departure from axisymmetry of the ow. Numerical evidences for this picture require the consideration of non-axisymmetric velocity contributions, either in terms of vortices as e.g. observed in water experiments by [152] or applying a physically established prole of an α-eect.
Acknowledgments
Financial support from Deutsche Forschungsgemeinschaft (DFG) in frame of the Collaborative Research Center (SFB) 609 is gratefully acknowledged and from European Commission under contract 028679. The computations using SFEMaNS were carried out on the IBM SP6 com-puter of Institut du Développement et des Ressources en Informatique Scientique (IDRIS) (project # 0254).
Eects of discontinuous magnetic permeability on magnetodynamic problems
J.-L. Guermonda, J. Léoratb, F. Luddensa,c, C. Norec,d,e, A. Ribeiroc
Abstract
A novel approximation technique using Lagrange nite elements is proposed to solve magneto-dynamics problems involving discontinuous magnetic permeability and non-smooth interfaces. The algorithm is validated on benchmark problems and is used for kinematic studies of the Cadarache von Kármán Sodium 2 (VKS2) experimental uid dynamo.
D.1 Introduction
This paper is the third part of a research program whose goal is to develop a solution method for solving the magnetohydrodynamic equations in heterogeneous axisymmetric domains. The computational domain is assumed to be composed of non-conducting and conducting media.
The electromagnetic eld is represented by the pairH−φ, whereHdenotes the magnetic eld in the conducting region and φ denotes the magnetic scalar potential in the non-conducting region. The basic ideas for approximating this class of problems have been introduced in [65]. Lagrange nite elements are used in the median section and variations in the azimuthal direction are approximated with Fourier expansions. The approximation is discontinuous across the interface separating the conducting and the non-conducting domains. This choice allows us to use Lagrange elements. The coupling between the H and φ representations is done by using an Interior Penalty technique [7, 11]. The method has been applied in [65] to the Maxwell equations forced by given velocity elds; this is the so-called kinematic dynamo
aDepartment of Mathematics, Texas A&M University 3368 TAMU, College Station, TX 77843-3368, USA
b Luth, Observatoire de Paris-Meudon, place Janssen, 92195 Meudon, France
cLaboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur, CNRS, BP 133, 91403 Orsay Cedex, France
d Université Paris Sud 11, 91405 Orsay Cedex, France
e Institut Universitaire de France, 103 Bd Saint-Michel, 75005 Paris, France
problem. The solution method has been shown to be stable and convergent. In [66], the method has been generalized to the full magnetohydrodynamic (MHD) problems and has been shown to be capable of solving nontrivial nonlinear dynamo problems. The Navier-Stokes/Maxwell coupling together with details on a parallelization technique for the Fast Fourier Transform (FFT) method are described in [66].
The main restriction of the method introduced in [65, 66] is that the magnetic permeability must be smooth in the conducting region. This is a major impediment since magnetic perme-ability heterogeneousness is suspected to play a key role in the connement of the magnetic eld in some dynamo experiments (we refer in particular to the VKS2 (von Kármán Sodium 2) successful dynamo experiment [110]) and thus signicantly lowers the dynamo threshold, [89]. The second restriction is that our using Lagrange nite elements and penalizing the di-vergence of the magnetic induction inL2 requires all the interfaces to be either smooth or the convexity of the interfaces be oriented towards the non-conducting region. This geometrical restriction is sometimes cumbersome. The objective of the present work is to address the two above issues. We show in the present work that the approximation framework proposed in [65, 66] can be generalized to account for magnetic permeability jumps and possible lack of smoothness of the interfaces where the electric conductivity and the magnetic permeability are discontinuous.
The paper is organized as follows. Notation and basic notions regarding the continuous problem are introduced and discussed in D.2. The nite element approximation is presented in D.3. In addition to accounting for discontinuous magnetic permeability, the main novelty of the method is condensed in the bilinear form D in the weak formulation (D.3.11). The new method is tested numerically on various academic benchmark problems in D.4. The method is shown therein to be robust with respect to geometric singularities and high magnetic permeability contrasts. The method is nally used in D.5 to explore various aspects of the VKS2 experiment. Our numerical results conrm the experimental observation that using soft iron components in the VKS2 experiment signicantly lowers the dynamo threshold.