Kinematic simulation of dynamo action by a hybrid boundary-element/finite-volume method

Abstract

The experimental realization of dynamo excitation as well as theoretical and numerical examinations of the induction equation have shown the relevance of boundary conditions for a self-sustaining dynamo. Within the interior of a field producing domain geometric constraints or varying material properties (e.g. electrical conductivity of the container walls or localized high-permeability material) might also play a role. Combining a grid based finite volume approach with the boundary element method in a hybrid FV-BEM scheme offers the flexibility of a local discretization with a stringent treatment of insulating magnetic boundary conditions in almost arbitrary geometries at comparatively low costs. Kinematic simulations of dynamo action generated by a well known prescribed mean flow demonstrate the reliability of the approach.Future examinations are intended to understand the behavior of the VKS-dynamo experiment where the field producing flow is driven by ferrous propellers and the induction effects of conductivity/permeability inhomogeneities might provide the required conditions for the measured dynamo characteristics.Introduction. Nowadays, there are nearly no doubts that the mechanism which is responsible for the generation of astrophysical or planetary magnetic fields is a dynamo process in which kinetic energy from a suitable flow of a conducting fluid is transfered into magnetic energy. Although the basic idea of this process has already been presented at the beginning of the 20th century [1] only few years ago fluid flow generated dynamo action has been realized in the laboratory [2,3,4].The key parameter that determines the onset of dynamo action is the magnetic Reynolds number Rm = µ 0 σVL where σ denotes the electrical conductivity, µ 0 the vacuum permeability, V a typical velocity magnitude and L the characteristic size of the considered system. Critical values that are necessary to obtain dynamo action in the laboratory are of the order Rm crit ∼ 30...100 which is already technically demanding. Therefore, essential efforts are concentrated on possibilities to reduce this critical value and to increase the actual Rm of the field producing flow.From numerical simulations it is known that the boundary conditions and also boundary layers of stagnant or somehow guided flow could possess supportive as well as obstructive impacts on the onset of dynamo action [5,6]. In the kinematic regime the backreaction of the field on the flow by the Lorentz force can be ignored so that the complexity of the underlying system of equations is significantly reduced because only the induction equation with a prescribed velocity field has to be solved numerically. Nevertheless, analyzing laboratory experiments requires a flexible numerical scheme that is able to consider geometric constraints as well as material properties like conductivity jumps between fluid and container walls or the high-permeability domains brought in by the iron propellers that are used to drive the flow in the VKS experiment, at least in the realizati...

Numerical simulation of the Tayler instability in liquid metals

“…The solution of this equation requires suitable boundary conditions for the magnetic field, which can be implemented either by solving a Laplace equation in the exterior of the fluid [25,26] or by equivalent boundary element methods [27][28][29].…”

Influence of high-permeability discs in an axisymmetric model of the Cadarache dynamo experiment

“…Both approximation methods can account for insulating boundaries and nonuniform permeability and/or conductivity distributions. In the FV/BEM scheme, insulating boundary conditions are treated by solving an integral equation on the boundary, which allows a direct computation of the (unknown) tangential field components by correlating the (known) normal field components on the surface of the computational domain (Iskakov et al 2004, Giesecke et al 2008. In the SFEMaNS code, the magnetic field is computed numerically in a certain domain outside the cylinder and matching conditions at the interfaces with the insulator are enforced weakly by using an interior penalty technique (Guermond et al 2007).…”

Electromagnetic induction in non-uniform domains

“…In the following we only discuss the treatment of the diffusive part of the electric field, E = +η∇ × B/µ r because the advective contributions (∝ −u × B) do not involve the material properties and can be treated separately in the framework of an operator splitting scheme (see e.g. Iskakov et al 2004, Giesecke et al 2008, Ziegler 1999. To obtain the computation directive for the electric field the magnetic field has to be integrated along a (closed path) around E x(,y,z) at the edge of a grid cell (see dotted curve in Fig.…”