An hybrid spectral-finite element method for the 3D MagnetoHydroDynamics equations in heterogeneous domains

C Nore, R Laguerre and A Ribeiro in collaboration with J.-L. Guermond (Texas A&M University, USA) and J. Léorat (*)

(*) Laboratoire Univers et Théories (LUTh) Observatoire de Paris, CNRS, Université Paris Diderot 5 Place Jules Janssen - 92195 Meudon Cedex

Object

The purpose of this paper is to demonstrate the ability of a new Spectral-Finite Element Method (SFEMHD) algorithm to deal with situations described by the MagnetoHydroDynamics (MHD) equations in the presence of conducting and nonconducting regions. In various axisymmetric domains, we have considered ohmic diffusion in conductors or induction in solid rotors submitted to a uniform and constant magnetic field. Comparisons have shown good agreement with analytical or previously published numerical solutions in two and three dimension configurations. We also simulate the kinematic dynamo action in different geometries and currently we are investigating the nonlinear version of the code. More specifically, we present here two cases of kinematic dynamos where the velocity field is imposed and the magnetic field can grow or decay depending on the value of the order parameter, namely the magnetic Reynolds number $R_m \,$.

Overview of the numerical method

In this section we give a brief overview of the numerical method which is used. The problem is modeled by the MHD equations in the eddy current approximation with no retroaction of the magnetic field on the velocity field (kinematic dynamo approximation):

$\mu\partial_t \vec{H} = -\nabla \times \vec{E},$

$\nabla \times \vec{H} = \sigma (\vec{E} + {\vec u} {\times} \mu \vec{H}), \;$

with relevant boundary and initial data, where $\vec{H}$ is the magnetic field, $\vec{E}$ the electric field, $\vec{u}$ an imposed velocity field, $\sigma \,$ the electric conductivity and $\mu \,$ the magnetic permeability. This system is solved in a heterogeneous domain composed of conducting regions of different conductivities ($\sigma_1 > 0, \sigma_2>0,\ldots$) and an insulating region (vacuum, $\sigma_0 = 0 \,$). Since the magnetic field is curl free in vacuum, it can be expressed as the gradient of a scalar potential $\phi \,$, as long as the insulating domain is simply connected. $\vec{E}$ can then be eliminated and the original system can be re-written in term of $\vec{H}$ and $\phi \,$ only (see reference [1]). Enforcing continuity of $\vec{H}$ and $\nabla\phi$ across interfaces is a significant numerical difficulty. In our finite element approximation, continuity is not enforced in the function spaces but is rather weakly imposed using an Interior Penalty Galerkin method (IPG) together with Lagrange elements. This method has been shown to be stable and convergent in [1].

Since the geometry is axisymmetric, the equations are written in cylindrical coordinates and the approximate solution is expanded in Fourier series in the azimuthal direction and nodal Lagrange finite elements in the meridian plane. The time is discretized by means of a semi-implicit Backward Finite Difference method of second order (BDF2). If the boundary of the computational domain is the vacuum boundary, we can impose Robin, Neumann, or Dirichlet boundary conditions on $\phi \,$.

Results

We focus here on the kinematic dynamo problem, in which the velocity field is imposed and the magnetic field is free to evolve. Note that $\vec{H} = 0$ is a solution of the MHD equations which becomes unstable above criticality. The order parameter $R_m= \mu \sigma R U \,$ is the magnetic Reynolds number based on the characteristic velocity $U$ and length $R$. It compares the production term to the dissipative term.

Dynamo in the Perm torus

The configuration we model is inspired from the so-called Perm device [2]. This experiment aims at generating the dynamo effect in a strongly time-dependent helical flow created in a toroidal channel after impulsively stopping the fast rotating container. We consider a torus which is represented on figure 1. The reference length scale is chosen so that the nondimensional mean radius of the torus (the distance between the $z \,$-axis and the center of the cross section) is $R=4 \,$. The nondimensional radius of the circular cross section is $\rho_1=1.6 \,$. The inner part of the torus, $0\le \rho < \rho_0=1.2$, is occupied by conducting fluid of conductivity $\sigma_{fluid} \,$ and is referred to as the fluid channel. The outer part of the torus, $\rho_0< \rho < \rho_1 \,$ is occupied by conducting solid of conductivity $\sigma_{vessel} \,$ . In order to be closer to the real experimental geometry, the conducting domain includes flat equatorial protuberances (see figure 1). The numerical domain has the following discretization characteristics: the conducting domain is discretized using a quasi-uniform grid of meshsize $\delta_x=1/20$ and is embedded in a spherical insulating domain of radius 10.

The flow velocity is helical with a uniform azimuthal component that we henceforth denote by $U_{a} \,$. The reference velocity $U \,$ is chosen to be $U=U_{a}\rho_0 \,$ so that the magnetic Reynolds number can be rewritten as $R_m = \mu\sigma_{fluid} U_{a} \rho_0 L \,$. The non-dimensional helical flow in the fluid channel is then defined in cylindrical coordinates by $v_r=-\chi \frac{1}{\rho_0} \frac{R z}{\rho_0 r},\quad v_{\theta}= \frac{1}{\rho_0},\quad v_z=\chi \frac{1}{\rho_0} \frac{R (r-R)}{\rho_0 r}$ where $\chi \,$ is a constant hereafter referred to as the poloidal to toroidal velocity ratio. We choose $\chi=1 \,$.

Since the flow is axisymmetric, the magnetic azimuthal modes evolve independently; therefore, the initial magnetic field in the conductor is set to contain all the azimuthal modes that we want to test. After thorough investigations, we have found that $m=3 \,$ is the critical mode corresponding to the lowest critical magnetic Reynolds number. More precisely we have determined $R_m^c(Perm) \simeq 16 \pm 0.5 \,$. We show in figure 2 (a-b) the $H_\theta$-component of the unstable mode at $R_m=17 > R_m^c(Perm) \,$. Observe that the support of this unstable eigenmode is localized close to $\rho\approx \rho_0 \,$, in the region of maximum shear. This eigenmode has a double helix shape and has the same helicity sign as that of the velocity field. It is rotating in a retrograde sense with a period of $T \simeq 4 \,$ (see figure 2 a-b). Our computations are the only ones performed in a torus. For more details, see reference [1].

 Figure 1: Schematic representation of the Perm torus configuration Figure 2a: Perm dynamo at $R_m=17 \,$ at $t=t_1\,$: iso-surfaces of the $H_{\theta} \,$ component of the $m=3 \,$ mode; $H_\theta=25 \%$ of the minimum value in black and $25 \%$ of the maximum value in white Figure 2b: Same as figure 2a at $t=t_1+T/4\,$

Dynamo in the rotating MND flow

We now examine a rotating flow in a cylindrical cavity of unit height-to-diameter aspect ratio. The conducting domain, composed of a finite cylinder (fluid) and an additional external layer (vessel), is discretized using a quasi-uniform grid of meshsize $\delta_x=1/40$. The conducting region is embedded in a spherical insulating domain of radius 10. The fluid flow is modeled by the following so-called MND velocity field [3], used as a convenient simplified model of the actual VKS (von Kármán Sodium) flow, expressed in cylindrical coordinates $(r,\theta,z)\,$:

for $0 \le r \le 1\,$ and $-1 \le z \le 1\,$ :

$v_r=-\frac{\pi}{2}r(1-r)^2(1+2r)\cos(\pi z);\quad v_{\theta}=4 \epsilon r(1-r)\sin(\frac{\pi}{2} z); \quad v_z=(1-r)(1+r-5r^2)\sin(\pi z).$

The parameter $\epsilon\,$ is the toroidal to poloidal flow ratio. We use $\epsilon=0.7259\,$ which yields near optimal dynamo action, as shown in [3]. Since the flow is axisymmetric, the magnetic azimuthal modes evolve independently; as a result, we restrict ourselves to the azimuthal mode 1 which is known to be the most unstable. Hereafter, we use the following definition of the magnetic Reynolds number: $R_m=\mu \sigma_{fluid} r_0 v_{max}\,$, where $r_0\,$ is the radius of the inner cylinder and $v_{max}\,$ the maximum of the velocity. The vessel containing the fluid is a hollow cylinder (see figure 4 a). The nondimensional thickness of the side wall is denoted by $w\,$ and that of the top and bottom lids is denoted by $l\,$. The conductivity ratio $\sigma_{vessel}/\sigma_{fluid}\,$ is denoted by $X\,$.

Inspired by a previous study [5], we study the influence of conducting annuli and top and bottom lids, .

We first consider a vessel having the shape of a cylindrical annulus, i.e. the thickness of the lids $l \,$ is first assumed to be infinitely thin ($l=0 \,$), (see figure 4 b, case S). Figure 3(a) shows $R_m^c(w) \,$ as a function of $w \,$ for $X=1\,$ and $X=5\,$. We observe that $R_m^c\,$ decreases quasi-exponentially with respect to $w\,$ at $X=1\,$ with the typical lengthscale $0.2$, thus confirming results from [4], then $R_m^c\,$ reaches an asymptote for large $w\,$.

We now set the thickness of the annulus to $w=0.5\,$, and we analyze the effect of top and bottom lids of thickness $l\,$. This case has been studied in [5] with no conductivity jump, i.e. $X=1\,$, the main result being that $R_m^c\,$ dramatically increases with respect to $l\,$. We display in figure 3(b) $R_m^c(w=0.5,l)\,$ as a function of $l\,$ for $X=1\,$ and $X=5\,$. We observe that $R_m^c\,$ increases with $l\,$ and that this effect is again amplified by the conductivity jump, the variation being the largest in the range $0 \le l \le 0.2\,$.

 Figure 3 a: Critical magnetic Reynolds number $R_m^c(w) \,$ as a function of conducting annulus width $w \,$ for $X=1\,$ and $X=5\,$ Figure 3 b: Critical magnetic Reynolds number $R_m^c \,$ as a function of conducting top and bottom lid height $l \,$ for $X=1\,$ and $X=5\,$ with a fixed wall thickness $w = 0.5\,$

To better comment on the behavior of $R_m^c\,$ with respect to $w\,$ and $l\,$, we now compare the spatial distribution of dissipation and currents in three typical cases: (i) case N: No envelope; (ii) case S (Side): an annulus of thickness $w=0.5\,$ with $X=5\,$ and no lids $l=0\,$; (iii) case SL (Side-Lids): an annulus of thickness $w=0.5\,$ and two lids of thickness $l=0.8\,$ with $X=5\,$. The results are reported in figure 4 (a-c).

For the cases N and S, most of the dissipation is concentrated in the fluid, close to the cylinder axis, whereas in the SL case, new dissipation patterns occur within the lids. The figure shows some characteristic lines of the current vector field. The spatial distribution of the lines close to the axis is pretty similar in the three cases, but it significantly differs in regions close to the envelope. Going from the case N to the case S, the currents formerly constrained in the flow volume are deformed and expand to fill the annulus on large scales, suggesting that the dissipation decreases as indeed observed. Going from the case S to the case SL, additional current loops at small scales appear within the top and bottom lids leading to more Ohmic dissipation. For more details, see reference [6].

 Figure 4 a (Case N): Isosurface of $50\%$ of the total dissipation and current lines for $X=5$ and $R_m=65 > R_m^c=63.5$ Figure 4 b (Case S): $X=5, \; w=0.5, \; l=0$ and $R_m=42 > R_m^c=41$ Figure 4 c (Case SL): $X=5, \; w=0.5, \; l=0.8$ and $R_m=65 > R_m^c=63$

Discussion and conclusion

We have shown the ability of our SFEMANS (Spectral Finite Element Maxwell And Navier-Stokes) code to simulate kinematic dynamo action in various geometries with different physical properties. Our results confirm that the impact of an envelope is crucial for the design of experimental fluid dynamos and this effect is enhanced by the difference of vessel and fluid conductivities. In the near future, we plan to optimize the geometry of the envelope of the VKS (von Kármán Sodium) set-up by using the experimental mean flow [4] and the actual sizes and conductivities of the materials that are used (sodium, copper, stainless steel). We are also studying the nonlinear dynamo action in a Taylor-Couette flow between two finite cylinders.

References

[1] J.-L. Guermond, R. Laguerre, J. Léorat and C. Nore, A Discontinuous Galerkin Method for the MHD equations in heterogeneous domains, J. Comp. Physics, 221, 349-369 (2007).

[2] W. Dobler, P. Frick and R. Stepanov, Screw dynamo in a time-dependent pipe flow, Phys. Rev. E, 67, 056309 (2003).

[3] L. Marié, C. Normand and F. Daviaud, Galerkin analysis of kinematic dynamos in the von Kármán geometry, Phys. Fluids, 18, 017102 (2006).

[4] F. Ravelet, A. Chiffaudel, F. Daviaud, J. Léorat, Towards an experimental von Kármán dynamo: numerical studies for an optimized design, Phys. Fluids, 17, 117104 (2005).

[5] F. Stefani, M. Xu, G. Gerbeth, F. Ravelet, A. Chiffaudel, F. Daviaud and J. Léorat, Ambivalent effects of added layers on steady kinematic dynamos in cylindrical geometry: application to the VKS experiment, Eur. J. Fluid Mech./B, 25, 894–908 (2006).

[6] R. Laguerre, C. Nore, J. Léorat and J.-L. Guermond, Induction effects of conductivity jumps in the envelope of a kinematic dynamo flow, C. R. Mécanique, 349-369 (2006).