Abstract

Magnetic field generation by the motion of an incompressible conductive fluid in a bounded axisymmetric cavity embedded in a dielectric is studied as a model for the Earth's dynamo. The case is considered where the velocity of the conductive fluid experiences spatial fluctuations of finite amplitude along the azimuthal direction, while the amplitude of the mean velocity field and the magnetic diffusion are infinitely small. The ratio of the amplitudes of the fluctuating and mean component of the velocity is chosen to increase as the square root of the inner spatial scale of the velocity. A complete asymptotic expansion of magnetic induction operator eigenvalues and associated eigenvectors is constructed, providing a solution to this problem. It is shown that the $\alpha$-effect takes place in the system under consideration, with the magnitude of the coefficient of generation controlled by the helicity of the velocity vector potential. In the particular case of axial symmetry the magnetic field is shown to obey Braginsky's equations of generation.