The modes of magnetic field generation in a low-mode model of αΩ-dynamo with α-generator varying intensity regulated by a function with an alternating kernel

Abstract

The low-mode model αΩ-dynamo is used in this paper to simulate the modes of magnetic field generation with insignificant changes in the velocity field of a viscous fluid. In the framework of those model the α-effect intensity is regulated by the process that is included in the magnetohydrodynamic system (MHD-system) as an additive correction as a functional Z(t) depended on the magnetic field energy. Function that determines damped oscillations with variable damping frequency and constant damping coefficient, taken equal to one, is selected as kernel J(t) of functional Z(t). The research of the behavior of the magnetic field is carried out on large time scales, therefore, a rescaled and dimensionless MHD-system with the unit of time iquel the time of the magnetic field dissipation (104 years) for numerical calculations is used. The control parameters of the system are the Reynolds number and the amplitude of the α-effect, that include information about the large-scale and turbulent generators, respectively. Numerical simulation of the magnetic field generation modes was carried out for the values of the damping coefficient b = 1 and frequency a = 0.1, 0.5, 1, 5, 10. According to the results of numerical simulation, an increase in the values of the damping frequency, when the damping coefficient is equal to one, is characterized by a decrease in the inhibitory effect of the process Z(t) on the α-effect and an increase in the region of divergence of the magnetic field on the phase plane of the control parameters. In a comparative analysis with the results of the authors’ work, where the change of the α-effect intensity was determined by the function Z(t) with an exponential kernel and the same value of the damping coefficient, the following differences were noted: an increase in oscillations in both a magnetic and a velocity fields, the appearance of a chaotic regime of magnetic field generation at the value of the damping frequency equal to one, and also insignificant narrowing of the region of α-effect suppression at values of the damping frequency increasing to one.

Magnetic Field Dynamical Regimes in a Large-Scale Low-Mode αΩ-Dynamo Model with Hereditary α-Quenching by Field Energy

“…This article generalizes research in one of the directions of the cycle of papers of the IKIR FEB RAS on the study of a low-mode model of Earth's dynamo [1][2][3][4][5][6][7][8]. Despite the progress made in geomagnetism theory, the problem of the origin of the Earth's magnetic field and that of space objects has not been fully resolved.…”

“…Therefore, the scalar parametrization can be given by a function α(r, θ) = α 0 a(r) cos θ = α 0 α(r, θ), where the coefficient α 0 is positive and determines the highest intensity of the α-effect, α(r, θ) is a dimensionless quantity that satisfies the condition max|α(r, θ)| = 1 and the radial component a(r) = r. In accordance with the considered dynamo model, it is assumed that the action of the Archimedean (buoyancy) force with a mass density f out determines the differential rotation of the averaged flow v [14,15,21]. Thus, the MHD-equations, including the equations of the Navier-Stokes and the magnetic field induction B, taking into account the turbulent α-effect, the continuity condition of the velocity field v, the solenoidality condition of the magnetic field B and boundary conditions of adhesion, take the form [8]:…”

“…Taking into account the α-quenching in the form of the functional (2), the MHDsystem (1) can be written as follows [8]…”

“…A detailed description of the rescaling and the dimensionlessness of MHD-system (3) is carried out in the article [7]; therefore, within the framework of this paper, we use the already transformed system of MHD-equations with the α-quenching [8]…”

“…In the deterministic dynamic model of αΩ-dynamo, a process with a finite "memory" is used as an additive regulator of the α-effect intensity (the quenching process of the α-effect, i.e., α-quenching). The process is defined by a functional, the kernel of which is an alternating function J(t) = e −bt cos at with a coefficient b and a frequency a of the damped oscillations [8,25,26]. It should be noted that a rescaled and dimensionless MHD-system is used in the paper.…”