Mingtian Xu

The integral equation method for a steady kinematic dynamo problem

With only a few exceptions, the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method. However, the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics. The paper is intended to facilitate the use of integral equation solvers in dynamo theory. In concrete, the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter α. For the case of spherical geometry, three examples of the function α(r) with steady and oscillatory solutions are considered. A convergence rate proportional to the inverse squared of the number of grid points is achieved. Based on this method, a convergence accelerating strategy is developed and the convergence rate is improved remarkably. Typically, quite accurate results can be obtained with a few tens of grid points. In order to demonstrate its suitability for the treatment of dynamos in other than spherical domains, the method is also applied to x2 dynamos in rectangular boxes. The magnetic fields and the electric potentials for the first eigenvalues are visualized.

Integral equation approach to time-dependent kinematic dynamos in finite domains

The homogeneous dynamo effect is at the root of cosmic magnetic field generation. With only a very few exceptions, the numerical treatment of homogeneous dynamos is carried out in the framework of the differential equation approach. The present paper tries to facilitate the use of integral equations in dynamo research. Apart from the pedagogical value to illustrate dynamo action within the well-known picture of the Biot-Savart law, the integral equation approach has a number of practical advantages. The first advantage is its proven numerical robustness and stability. The second and perhaps most important advantage is its applicability to dynamos in arbitrary geometries. The third advantage is its intimate connection to inverse problems relevant not only for dynamos but also for technical applications of magnetohydrodynamics. The paper provides the first general formulation and application of the integral equation approach to time-dependent kinematic dynamos, with stationary dynamo sources, in finite domains. The time dependence is restricted to the magnetic field, whereas the velocity or corresponding mean-field sources of dynamo action are supposed to be stationary. For the spherically symmetric alpha2 dynamo model it is shown how the general formulation is reduced to a coupled system of two radial integral equations for the defining scalars of the poloidal and toroidal field components. The integral equation formulation for spherical dynamos with general stationary velocity fields is also derived. Two numerical examples--the alpha2 dynamo model with radially varying alpha and the Bullard-Gellman model--illustrate the equivalence of the approach with the usual differential equation method. The main advantage of the method is exemplified by the treatment of an alpha2 dynamo in rectangular domains.

Oscillation or rotation: a comparison of two simple reversal models

The asymmetric shape of reversals of the Earths magnetic field indicates a possible connection with relaxation oscillations as they were early discussed by van der Pol. A simple mean-field dynamo model with a spherically symmetric u2009αu2009coefficient is analysed with view on this similarity, and a comparison of the time series and the phase space trajectories with those of paleomagnetic measurements is carried out. For highly supercritical dynamos a very good agreement with the data is achieved. Deviations of numerical reversal sequences from Poisson statistics are analysed and compared with paleomagnetic data. The role of the inner core is discussed in a spectral theoretical context and arguments and numerical evidence is compiled that the growth of the inner core might be important for the long term changes of the reversal rate and the occurrence of superchrons.The asymmetric shape of reversals of the Earths magnetic field indicates a possible connection with relaxation oscillations as they were early discussed by van der Pol. A simple mean-field dynamo model with a spherically symmetric α coefficient is analysed with view on this similarity, and a comparison of the time series and the phase space trajectories with those of paleomagnetic measurements is carried out. For highly supercritical dynamos a very good agreement with the data is achieved. Deviations of numerical reversal sequences from Poisson statistics are analysed and compared with paleomagnetic data. The role of the inner core is discussed in a spectral theoretical context and arguments and numerical evidence is compiled that the growth of the inner core might be important for the long term changes of the reversal rate and the occurrence of superchrons.

Cylindrical anisotropic α 2 dynamos

We explore the influence of geometry variations on the structure and the time-dependence of the magnetic field that is induced by kinematic α 2 dynamos in a finite cylinder. The dynamo action is due to an anisotropic α effect which can be derived from an underlying columnar flow. The investigated geometry variations concern, in particular, the aspect ratio of height to radius of the cylinder, and the thickness of the annular space to which the columnar flow is restricted. Motivated by the quest for laboratory dynamos which exhibit Earth-like features, we start with modifications of the Karlsruhe dynamo facility. Its dynamo action is reasonably described by an α 2 mechanism with anisotropic α tensor. We find a critical aspect ratio below which the dominant magnetic field structure changes from an equatorial dipole to an axial dipole. Similar results are found for α 2 dynamos working in an annular space when a radial dependence of α is assumed. Finally, we study the effect of varying aspect ratios of dynamos with an α tensor depending both on radial and axial coordinates. In this case only dominant equatorial dipoles are found and most of the solutions are oscillatory, contrary to all previous cases where the resulting fields are steady.

An integral equation approach to two-dimensional incompressible resistive magnetohydrodynamics

In the present work, an integral equation approach is developed to solve two-dimensional incompressible resistive magnetohydrodynamic equations. This approach is examined by simulating the magnetic reconnection driven by the Orszag–Tang vortex and the doubly periodic coalescence instability. The results show that when the viscosity and magnetic resistivity of the plasma are reduced, the current sheet forming in the magnetic reconnection driven by the Orszag–Tang vortex becomes thinner. In comparison with the spectral method, the integral equation approach has much better accuracy and convergence.

Integral equations in MHD: theory and application

The induction equation of magnetohydrodynamics (MHD) is mathematically equivalent to a system of integral equations for the magnetic field in the bulk of the fluid and for the electric potential at its boundary. We summarize the recent developments concerning the numerical implementation of this scheme and its applications to various forward and inverse problems in dynamo theory and applied MHD.

Forward and inverse problems in fundamental and applied magnetohydrodynamics

This minireview summarizes the recent efforts to solve forward and inverse problems as they occur in different branches of fundamental and applied magnetohydrodynamics. For the forward problem, the main focus is on the numerical treatment of induction processes, including self-excitation of magnetic fields in non-spherical domains and/or under the influence of non-homogeneous material parameters. As an important application of the developed numerical schemes, the functioning of the von-Kármán-sodium (VKS) dynamo experiment is shown to depend crucially on the presence of soft-iron impellers. As for the inverse problem, the main focus is on the mathematical background and some initial practical applications of contactless inductive flow tomography (CIFT), in which flow induced magnetic field perturbations are utilized to reconstruct the velocity field. The promises of CIFT for flow field monitoring in the continuous casting of steel are substantiated by results obtained at a test rig with a low-melting liquid metal. While CIFT is presently restricted to flows with low magnetic Reynolds numbers, some selected problems from non-linear inverse dynamo theory, with possible applications to geo- and astrophysics, are also discussed.

Inverse problems in magnetohydrodynamics: theoretical and experimental aspects

We consider inverse problems related to velocity reconstruction in electrically conducting fluids from externally measured magnetic fields. The underlying theory is presented in the framework of the integral equation approach to homogeneous dynamos in finite domains, which can be cast into a linear inverse problem in case that the magnetic Reynolds number of the flow is not too large. Some mathematical problems of the inversion, including the uniqueness problem in the sphere and a paradigmatic isospectrality problem for mean-field dynamos, are touched upon. For practical purposes, the inversion is carried out with the help of Tikhonov regularization using a quadratic functional of the velocity as penalty function. We present results of an experiment in which the three-dimensional (3D) velocity field of a propeller driven flow in a liquid metal is reconstructed by a contactless inductive measuring technique.