The Integral Equation Approach To Hydromagnetic Dynamos In A Spherical Region

Not only the Earth and the Sun; it is probably safe to state that a magnetic field is a normal accompaniment of any cosmic body that is both fluid (wholly or in part) and rotating. But how does the cosmic body generate and maintain a magnetic field? In the past understanding the origin of the Earth’s magnetic field has captured the attention of many renowned scientists. Albert Einstein considered it as one of the major unsolved problems in physics. The hydromagnetic dynamo effect is thought to be the best explanation for the origin of cosmic magnetic fields, including the fields of planets, stars, and galaxies. So it is of scientific significance to study the hydromagnetic dynamo effect. In fast breeder reactors and magnetic confinement fusion reactors, liquid metal is used to be coolant. In these reactors, the magnetic Reynolds number can reach20. Thus the liquid metal flow may cause a dynamo effect and affect the safety of the reactors. Therefore it is of great application value to study the hydromagnetic dynamo.In1999, magnetic field self-excitation was observed in the large scale liquid sodium facilities in Riga and Karlsruhe. Self-excitation was also achieved in the French VKS experiment. All these succeeded experiments were done in cylindrical vessels. Interestingly, none of the self-excited dynamo experiments in spherical vessels has been successful. So it is an urgent need to study the self-excitation in spherical containers. In order to make the dynamo experiment successful, it is essential to numerically simulate the self-excited dynamo experiment. The integral equation approach for simulating the self-excited dynamo actions has not only been examined by the three successful experiments, but also played a key role for the success of the VKS experiment. Therefore we attempt to use the integral equation approach to study the hydromagnetic dynamo driven by the Beltrami flows in a spherical container. The main work is summarized as follows:Firstly, we briefly review the development of the magnetohydrodynamics (MHD) and introduce the hydromagnetic dynamo theory. Then the governing equation in the form of integral equations for the kinematic dynamos is given under the spherical coordinate, and the extended trapezoidal rule is employed to discretize the obtained integral equations. Finally, we investigate the variations of the growth rates of the m=0and m=1modes of the hydromagnetic dynamos driven by the Beltrami flows with respect to the magnetic Reynolds number. It is found that the Beltrami flows can induce the self-excitation, the generated magnetic field is the m=1mode and the smallest critical magnetic Reynolds number for this kind of flows is about39.69.