A non-linear, 3-D spherical α2 dynamo using a finite element method

Chan, Kit H.; Zhang, Keke; Zou, Jun; Schubert, G.

doi:10.1016/s0031-9201(01)00276-x

Cited by 38 publications

(40 citation statements)

References 24 publications

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“…The first attempt at using finite element methods for numerical simulations of spherical dynamos was made in [11]. The current work presents the first mathematical theory and numerical analysis for mean-field spherical dynamos, and it has made contributions in the following aspects: (1) The well-posedness of the mean-field dynamo system is rigorously demonstrated; the dynamo system is characterized in terms of a saddle-point type formulation which can be conveniently approximated by finite element methods.…”

Section: Discussionmentioning

confidence: 99%

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Spherical Interface Dynamos: Mathematical Theory, Finite Element Approximation, and Application

Chan¹,

Zhang²,

Zou³

2006

SIAM J. Numer. Anal.

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Abstract. Stellar magnetic activities such as the 11-year sunspot cycle are the manifestation of magnetohydrodynamic dynamo processes taking place in the deep interiors of stars. This paper is concerned with the mathematical theory and finite element approximation of mean-field spherical dynamos and their astrophysical application. We first investigate the existence, uniqueness, and stability of the dynamo system governed by a set of nonlinear PDEs with discontinuous physical coefficients in spherical geometry, and characterize the system by a saddle-point type variational form. Then we propose a fully discrete finite element approximation to the dynamo system and study its convergence and stability. For the astrophysical application, we perform some fully threedimensional numerical simulations of a solar interface dynamo using the proposed algorithm, which successfully generates the equatorially propagating dynamo wave with a period of about 11 years similar to that of the Sun.

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Section: Discussionmentioning

confidence: 99%

“…simulations was made in [11] and proved to be very promising. The current work presents the first mathematical theory and numerical analysis for mean-field spherical dynamos and their application to astrophysical and planetary problems.…”

mentioning

confidence: 99%

Spherical Interface Dynamos: Mathematical Theory, Finite Element Approximation, and Application

Chan¹,

Zhang²,

Zou³

2006

SIAM J. Numer. Anal.

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“…Thus if the numerical algorithm creates divergence, there is no mechanism for eliminating it and it may accumulate [6]. For this reason, magnetohydrodynamic codes sometimes include a fictitious term analogous to the hydrodynamic pressure, which must be treated numerically [7].…”

Section: Motivation and Governing Equationsmentioning

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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“…However, we cannot apply such boundary conditions here because the FEM is based on local operations. Following Chan et al (2001b), the finite element mesh is extended outside the fluid shell. The basic equations for the vector potential in the insulator are…”

Section: Basic Equations For the Magnetic Fieldmentioning

confidence: 99%

Treatment of the magnetic field for geodynamo simulations using the finite element method

Matsui

Okuda

2014

Earth Planet Sp

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We propose a scheme for calculating the magnetic field in a spherical shell, based on Earth's outer core, using the finite element method (FEM). The two most difficult problems for magnetohydrodynamics (MHD) simulations in a rotating spherical shell with FEM are solving the magnetic field outside the fluid shell, and connecting the magnetic field in the fluid shell to the exterior potential field at the boundary. To solve these problems, we extend the finite element mesh beyond the fluid shell and compute the vector potential of the magnetic field. To verify the present scheme, we consider three test case. First, we compare the FEM model with an analytical solution of Laplace's equation outside the fluid. Second, we evaluate free decay of a dipole field and compare the results with a spectral solution. Finally, compare the results of a simple kinematic dynamo problem with a spectral solution. The results suggest that the accuracy of the dipole field depends on the radius of the simulation domain, and that this error becomes sufficiently small if the radius of the outer region is approximately 6 times larger than the radius of the fluid shell.

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A non-linear, 3-D spherical α2 dynamo using a finite element method

Cited by 38 publications

References 24 publications

Spherical Interface Dynamos: Mathematical Theory, Finite Element Approximation, and Application

Spherical Interface Dynamos: Mathematical Theory, Finite Element Approximation, and Application

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

Treatment of the magnetic field for geodynamo simulations using the finite element method

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