With materials of anisotropic electrical conductivity, it is possible to generate a dynamo with a simple velocity field, of the type precluded by Cowling's theorems with isotropic materials. Following a previous study by Ruderman and Ruzmaikin [M. S. Ruderman and A. A. Ruzmaikin, Magnetic field generation in an anisotropically conducting fluid, Geophys. Astrophys. Fluid Dyn. 28, 77 (1984)], who considered the dynamo effect induced by a uniform shear flow, we determine the conditions for the dynamo threshold when a solid plate is sliding over another one, both with anisotropic electrical conductivity. We obtain numerical solutions for a general class of anisotropy and obtain the conditions for the lowest magnetic Reynolds number, using a collocation Chebyshev method. In a particular geometry of anisotropy and wave number, we also derive an analytical solution, where the eigenvectors are just combinations of four exponential functions. An explicit analytical expression is obtained for the critical magnetic Reynolds number. Above the critical magnetic Reynolds number, we have also derived an analytical expression for the growth rate showing that this is a “very fast” dynamo, extrapolating on the “slow” and “fast” terminology introduced by Vainshtein and Zeldovich [S. I. Vainshtein and Y. B. Zeldovich, Reviews of topical problems: Origin of magnetic fields in astrophysics (turbulent “dynamo” mechanisms), Sov. Phys. Usp. 15, 159 (1972)].

中文翻译：

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各向异性导电板中的发电机作用

使用具有各向异性导电性的材料，可以生成具有简单速度场的发电机，其类型是各向同性材料的考林定理所排除的类型。根据Ruderman和Ruzmaikin的先前研究[MS Ruderman和AA Ruzmaikin，各向异性导电流体Geophys 中的磁场产生。天体。流体动力学 28，77（1984）]，他们考虑了均匀剪切流引起的发电机效应，我们确定了当固体板在另一块均具有各向异性电导率的滑行时，发电机阈值的条件。我们使用搭配Chebyshev方法获得了一般各向异性的数值解，并获得了最低雷诺数的条件。在各向异性和波数的特定几何形状中，我们还导出了解析解，其中特征向量只是四个指数函数的组合。对于临界磁雷诺数获得了明确的解析表达式。在临界磁雷诺数以上，我们还得出了增长率的解析表达式，表明这是一个“非常快”的发电机，Sov。物理 Usp。 15，159（1972）]。