Modes of magnetic field generation in the low-mode $\alpha\Omega$-dynamo model with dynamic regulation of the $\alpha$-effect by the field energy

In this paper, we use a low-mode $\alpha$Ω-dynamo model to simulate the modes of magnetic field generation with insignificant changes in the velocity field of a viscous fluid. Within the framework of this model, an additive correction is introduced into the magnetohydrodynamic system to control the intensity of the $\alpha$-effect in the form of a function Z(t) from the field energy. As the kernel J(t) of the function Z(t) is chosen the function that determines damped oscillations with the different values of the damping coefficient and a constant damping frequency taken equal to one. The study of the magnetic field behavior is carried out on a large time scales, therefore, for numerical calculations, a rescaled and dimensionless MHD-system is used, where the time of the magnetic field dissipation (104 years) is accepted as the unit of time. The main parameters of the system are the Reynolds number and the amplitude of the $\alpha$-effect, which contains information about the large-scale and turbulent generators, respectively. According to the results of numerical simulation, an increase in the values of the damping coefficient is characterized an increase in the inhibition effect of the process Z(t) on the $\alpha$-effect and decrease of the magnetic field divergence region on the plane of the main parameters.