Xu Mingtian, Stefani Frank, Gerbeth Gunter
Forschungszentrum Rossendorf, P.O. Box 510119, D-01314 Dresden, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2004 Nov;70(5 Pt 2):056305. doi: 10.1103/PhysRevE.70.056305. Epub 2004 Nov 16.
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.
均匀发电机效应是宇宙磁场产生的根源。除了极少数例外情况，均匀发电机的数值处理是在微分方程方法的框架内进行的。本文试图促进积分方程在发电机研究中的应用。除了在毕奥 - 萨伐尔定律这一众所周知的图景中阐释发电机作用所具有的教学价值外，积分方程方法还有许多实际优势。第一个优势是其已被证实的数值稳健性和稳定性。第二个或许也是最重要的优势是它适用于任意几何形状的发电机。第三个优势是它与反问题有着紧密联系，这不仅与发电机相关，也与磁流体动力学的技术应用有关。本文给出了积分方程方法对于有限域中具有静止发电机源的随时间变化的运动学发电机的首个通用公式及其应用。时间依赖性仅限于磁场，而发电机作用的速度或相应的平均场源被假定为静止的。对于球对称α²发电机模型，展示了通用公式如何简化为关于极向场和环向场分量的定义标量的两个径向积分方程的耦合系统。还推导了具有一般静止速度场的球发电机的积分方程公式。两个数值例子——α随径向变化的α²发电机模型和布拉德 - 盖尔曼模型——说明了该方法与常用微分方程方法的等效性。该方法的主要优势通过在矩形域中处理α²发电机得到了例证。
