A nonlinear vacillating dynamo induced by an electrically heterogeneous mantle

Abstract

Abstract.This paper reports the first spherical numerical dynamo based on a three-dimensional finite element method. We investigate a nonlinear dynamo in a turbulent electrically conducting fluid spherical shell of constant electric conductivity surrounded by an electrically heterogeneous mantle. Magnetic fields in the form of a threedimensional azimuthally traveling dynamo wave are generated by a prescribed time-dependent a in the fluid shell. In the inner sphere, we assume that there is a solid electrical conductor with the same conductivity as that of the fluid shell. Equilibration of the generated magnetic fields is achieved by the nonlinear process of a-quenching. We show for the first time that finite element methods can be effectively and efficiently employed to simulate three-dimensional dynamos in spherical systems. We also show that an electrically heterogeneous mantle can modulate the core dynamo, leading to a vacillating dynamo whose amplitude depends upon the relative phases between the generated magnetic field and the heterogeneous mantle.

A finite-volume solution method for thermal convection and dynamo problems in spherical shells

“…Unfortunately the implementation of insulating or finite conducting boundary conditions in a local method is not straightforward, but it is possible—as demonstrated by Chan et al (2001a) or Matsui & Okuda (2002). Also a fit of spherical harmonics (as in a spectral method) or the application of a boundary element method are possible and would avoid the gridding of the exterior domain necessary in the former two methods.…”

“…Since local methods, like finite‐element or finite‐volume approaches, are much more easily adapted to parallel architectures, there is currently an increased interest in applying these methods to dynamo simulations. Whereas the older work of Kageyama & Sato (1997) uses finite differences, a variety of approaches is used in the recent and ongoing work of Chan et al (2001a), Matsui & Okuda (2002) (both finite elements), Hejda & Reshetnyak (2003) (finite volume) and Fournier et al (2004) (spectral elements). However, most of the mentioned work is far from completed, either only a part of the magnetohydrodynamic dynamo problem has been solved or a parallel approach has not yet been implemented.…”

Some new additive Runge–Kutta methods and their applications

“…with appropriate initial and boundary conditions [2,3], where B is the magnetic field of the dynamo system, (x) is the magnetic Reynolds number, and f(, x, t; u, B) is a nonlinear vector-valued function of flow u and magnetic field B, which determines the key dynamo system so that the magnetic field can sustain in the corresponding physical system. Now, as an example, we apply the additive method RK.2.A.2 to the temporal semi-discretization of this system and obtain the following scheme advancing from time t n to t n+1 :…”

An iterative scheme for a geodynamo system

“…This model is a coupled nonlinear system involving velocity field u, total kinematic pressure P and magnetic field B. Similar models have recently been widely studied; see, for example, [3,4]. They use the finite-element method to simulate 3D dynamos in spherical systems.…”