# Kinematic mean-field αα-dynamo of magnetohydrodynamics

Dynamics of an electrically conducting medium (plasma or liquid metal) can produce a magnetic field. Electric currents and fields that are generated by the moving fluid, amplify the existing magnetic field and maintain it against resistive decay. The process of conversion of mechanical energy into magnetic energy is called dynamo. The magnetohydrodynamics (MHD) dynamo is the main mechanism that creates magnetic fields of stars and planets.

The motion of a conducting fluid in a magnetic field is described by the MHD equations that basically couple the Navier-Stokes equations of the viscous fluid to the induction equation for the magnetic field. This set of nonlinear partial differential equations with boundary conditions requires as a rule computationally expensive numerical simulations. Nevertheless, the dynamo action can be understood already by means of simplified models of the kinematic dynamo theory based on the assumption that the velocity field of the fluid is prescribed and not affected by the magnetic field. In the kinematic regime, the time evolution of the magnetic field is reduced to the linear induction equation.

In 1966, in order to further simplify the model it was proposed to split the magnetic field and the flow into mean and fluctuating components, respectively. The fluctuating magnetic field is generated by the interaction of the fluctuating fluid velocity with the mean magnetic field. Then, the averaged induction equation contains the term corresponding to the mean electromotive force due to fluctuations. For its occurrence, known as $\alpha$-effect, the small scale (convective or turbulent) motions of the fluid must on average be non-mirror-symmetric, e.g. helical.