On magnetic boundary conditions for non-spectral dynamo simulations

Iskakov, Alexey B.; Dormy, Emmanuel

doi:10.1080/03091920500337145

Cited by 18 publications

(20 citation statements)

References 16 publications

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“…Equation (1) is time stepped, applying a finite volume method where a constraint transport scheme ensures the exact treatment of the solenoidal property of B (if the initial field is divergence free). Insulating boundary conditions are treated with a modified boundary integral equation approach which yields the tangential field components on the boundary from the normal field components on the whole surface of the computational domain [39,40].…”

Section: Numerical Modelmentioning

confidence: 99%

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

2012

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We present numerical simulations of the kinematic induction equation in order to examine the dynamo efficiency of an axisymmetric von Kármán-like flow subject to time-dependent nonaxisymmetric velocity perturbations. The numerical model is based on the setup of the French von Kármán-sodium dynamo (VKS) and on the flow measurements from a water experiment conducted at the University of Navarra in Pamplona, Spain. The principal experimental observations that are modeled in our simulations are nonaxisymmetric vortexlike structures which perform an azimuthal drift motion in the equatorial plane. Our simulations show that the interactions of these periodic flow perturbations with the fundamental drift of the magnetic eigenmode (including the special case of nondrifting fields) essentially determine the temporal behavior of the dynamo state. We find two distinct regimes of dynamo action that depend on the (prescribed) drift frequency of an (m = 2) vortexlike flow perturbation. For comparatively slowly drifting vortices we observe a narrow window with enhanced growth rates and a drift of the magnetic eigenmode that is synchronized with the perturbation drift. The resonance-like enhancement of the growth rates takes place when the vortex drift frequency roughly equals the drift frequency of the magnetic eigenmode in the unperturbed system. Outside of this small window, the field generation is hampered compared to the unperturbed case, and the field amplitude of the magnetic eigenmode is modulated with approximately twice the vortex drift frequency. The abrupt transition between the resonant regime and the modulated regime is identified as a spectral exceptional point where eigenvalues (growth rates and frequencies) and eigenfunctions of two previously independent modes collapse. In the actual configuration the drift frequencies of the velocity perturbations that are observed in the water experiment are much larger than the fundamental drift frequency of the magnetic eigenmode that is obtained from our numerical simulations. Hence, we conclude that the fulfillment of the resonance condition might be unlikely in present day dynamo experiments. However, a possibility to increase the dynamo efficiency in the VKS experiment might be realized by an application of holes or fingers on the outer boundary in the equatorial plane. These mechanical distortions provoke an anchorage of the vortices at fixed positions thus allowing an adjustment of the temporal behavior of the nonaxisymmetric flow perturbations.

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Section: Numerical Modelmentioning

confidence: 99%

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

2012

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“…Using this formulation [24], a stationary kinematic dynamo problem was solved in a cylindrical geometry. In [25,26], a finite volume method is used to discretize the solution in the interior, which is matched to that in the exterior vacuum via a boundary element method. An integral equation formulation was applied to the entire domain in [27], and [28] uses finite elements with a penalty method to apply boundary conditions.…”

Section: Towards An Mhd Solvermentioning

confidence: 99%

Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

Boronski

Tuckerman

2007

Journal of Computational Physics

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The Navier-Stokes equations and magnetohydrodynamics equations are written in terms of poloidal and toroidal potentials in a finite cylinder. This formulation insures that the velocity and magnetic fields are divergence-free by construction, but leads to systems of partial differential equations of higher order, whose boundary conditions are coupled. The influence matrix technique is used to transform these systems into decoupled parabolic and elliptic problems. The magnetic field in the induction equation is matched to that in an exterior vacuum by means of the Dirichlet-to-Neumann mapping, thus eliminating the need to discretize the exterior. The influence matrix is scaled in order to attain an acceptable condition number. Motivation and Governing EquationsThe requirement that velocity and magnetic fields be solenoidal, i.e. divergence-free, represents one of the most challenging difficulties in hydrodynamics and in magnetohydrodynamics [1,2,3,4,5,6,7]. For the velocity field, this condition is the fundamental approximation used in incompressible fluid dynamics. For the magnetic field, this condition is the statement of the non-existence of magnetic monopoles.Two main approaches exist for imposing this requirement. The first is to use three field components and to project three-dimensional fields onto a divergence-free field. In an incompressible fluid, the pressure serves to counterbalance the nonlinear term which is the source of the divergence in the Navier-Stokes equations; the pressure also plays this role numerically. The divergence of the Navier-Stokes equations is taken, leading to a Poisson problem for the pressure. However, the boundary conditions on the

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“…The search for a self-sustaining magnetohydrodynamic dynamo has taken on great momentum in recent years, as researchers have sought to produce dynamos in the laboratory [1][2][3][4][5] and in simulations [6][7][8][9][10][11][12][13][14][15][16]. One of the fundamental problems in numerical magnetohydrodynamics is the formulation of boundary conditions.…”

Section: Motivationmentioning

confidence: 99%

Spectral method for matching exterior and interior elliptic problems

Boronski

2007

Journal of Computational Physics

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A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green's function with a surface density; this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum

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On magnetic boundary conditions for non-spectral dynamo simulations

Cited by 18 publications

References 16 publications

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

Impact of time-dependent nonaxisymmetric velocity perturbations on dynamo action of von Kármán-like flows

Poloidal–toroidal decomposition in a finite cylinder. I: Influence matrices for the magnetohydrodynamic equations

Spectral method for matching exterior and interior elliptic problems

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