Two-Mode Hereditary \(\alpha ^2\omega \) -Dynamo Model

The generation of planets and stars magnetic fields is usualy described using the hydromagnetic dynamo mechanism. Within the framework of the mean field theory, two types of generation arise—due to large-scale differential rotation of a celestial body ( \(\omega \) -effect) and due to small-scale pulsations of velocity and magnetic field ( \(\alpha \) -effect). A two-mode hereditary model is described, in which the generation of a toroidal mode from a poloidal one is carried out due to the \(\omega \) -effect and/or the \(\alpha \) -effect, and the generation of a poloidal mode from a toroidal one is ensured only by the \(\alpha \) -effect. This mechanism is known as \(\alpha ^2\omega \) -dynamo. The model implements hereditary quenching of the \(\alpha \) -effect by a quadratic form of the field components, which describes the quenching by energy and helicity. The quenching functional includes a fairly general kernel. The model equations are an integro-differential system of equations for the mode amplitudes. A theorem on the existence and uniqueness of a solution to the Cauchy problem for such a system of equations is proven. It is proven that if the kernel of the quenching functional is a solution to a homogeneous linear differential equation with constant coefficients, then the integral terms can be eliminated. Two difference schemes were developed for numerical simulation. The first scheme is a combination of the 2nd order implicit Runge-Kutta method for the differential part and the trapezoidal method for the integral part. The second is a combination of the Adams scheme of the predictor-corrector type and the Simpson method. A numerical study of the order of accuracy of schemes combined according to Runge’s rule showed that both of them are of first order. It is shown that in the case of an exponential kernel of the suppression functional and one special type of quadratic form, the model reduces to the classical Lorenz system. The known nature of the dynamics of the Lorentz system for various parameters made it possible to verify the circuit and code. In a series of computational experiments, various dynamic regimes that arise in the system by varying the main control parameters the dynamo number and the kernel time scale were studied. Using the calculation of maximum Lyapunov exponents for the case of nuclei with exponential asymptotics, maps of dynamic regimes in the plane of these parameters were constructed, highlighting areas of asymptotically stationary regimes, quasiperiodic regimes and chaotic regimes. It turned out that the regions of chaos and regular regimes alternate in a complex way on the plane.