On Kinematic Generation of the Magnetic Modes of Bloch Type

Zheligovsky, Vladislav; Chertovskih, Roman

doi:10.1134/s1069351320010152

Cited by 4 publications

(3 citation statements)

References 27 publications

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“…In agreement with [Zheligovsky and Chertovskih, 2020], our numerical results demonstrate that, at least in the limited interval of the molecular diffusivity parameters values for which we have performed computations of the Bloch eigenmodes, the three linear stability problems under consideration: the kinematic dynamo problem, the hydrodynamic and MHD stability problems -show a remarkably similar behavior, irrespective of the energy spectrum of the steady flow and magnetic field (V, B) subjected to perturbation. For generic steady flows and MHD states, the Bloch modes, that have the maximum over q growth rates, feature spatial scale separation that enhances on increasing the diffusivity ν and/or η.…”

Section: Discussionsupporting

confidence: 83%

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

Chertovskih,

Zheligovsky

2023

Russian Journal of Earth Sciences

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We consider Bloch eigenmodes of three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic ﬂows and MHD states comprised of randomly generated Fourier coeﬃcients and having energy spectra of three types: exponentially decaying, Kolmogorov with a cut oﬀ, or involving a small number of harmonics (“big eddies”). A Bloch mode is a product of a ﬁeld of the same periodicity as the perturbed state and a planar harmonic wave, exp(iq · x). Such a mode is characterized by the ratio of spatial scales which, for simplicity, we identify with the length |q| < 1 of the Bloch wave vector q. Computations have revealed that the Bloch modes, whose growth rates are maximum over q, feature the scale ratio that decreases on increasing the nondimensionalized molecular diﬀusivity and/or viscosity from 0.03 to 0.3, and the scale separation is high (i.e., |q| is small) only for large molecular diﬀusivities. Largely this conclusion holds for all the three stability problems and all the three energy spectra types under consideration. Thus, in a natural MHD system not aﬀected by strong diﬀusion, a given scale range gives rise to perturbations involving only moderately larger spatial scales (i.e., |q| only moderately small), and the MHD evolution consists of a cascade of processes, each generating a slightly larger spatial scale; ﬂows or magnetic ﬁelds characterized by a high scale separation are not produced. This cascade is unlikely to be amenable to a linear description. Consequently, our results question the allegedly high role of the α-eﬀect and eddy diﬀusivity that are based on spatial scale separation, as the primary instability or magnetic ﬁeld generating mechanisms in astrophysical applications. The Braginskii magnetic α-eﬀect in a weakly non-axisymmetric ﬂow, often used for explanation of the solar and geodynamo, is advantageous not being upset by a similar deﬁciency.

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Section: Discussionsupporting

confidence: 83%

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

Chertovskih,

Zheligovsky

2023

Russian Journal of Earth Sciences

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“…In other words, under the stated conditions, half-integer wave vectors are stationary points of the growth rates γ(q). We consider the two cases following [Zheligovsky and Chertovskih, 2020], where this was shown in the context of the kinematic dynamo problem. The existence of the derivatives is assumed.…”

Section: Stationary Half-integer Wave Vectorsmentioning

confidence: 99%

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries

Chertovskih¹,

Zheligovsky

2023

Russian Journal of Earth Sciences

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We consider Bloch eigenmodes in three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states. A Bloch mode is a product of a field of the same periodicity, as the state subjected to perturbation, and a planar harmonic wave, exp(iqx). The complex exponential cancels out from the equations of the respective eigenvalue problem, and the wave vector q remains in the equations as a numeric parameter. The resultant problem has a significant advantage from the numerical viewpoint: while the Bloch mode involves two independent spatial scales, its growth rate can be computed in the periodicity box of the perturbed state. The three-dimensional space, where q resides, splits into a number of regions, inside which the growth rate is a smooth function of q. In preparation for a numerical study of the dominant (i.e., the largest over q) growth rates, we have derived expressions for the gradient of the growth rate in q and proven that, for parity-invariant flows and MHD steady states or when the respective eigenvalue of the stability operator is real, half-integer q (whose all components are integer or half-integer) are stationary points of the growth rate. In prior works it was established by asymptotic methods that high spatial scale separation (small q) gives rise to the phenomena of the α-effect or, for parity-invariant steady states, of the eddy diffusivity. We review these findings tailoring them to the prospective numerical applications.

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“…The model considers one layer of differential rotation. The source of large-scale differential rotation is turbulence [27]. When a largescale toroidal field appears, Coriolis drift leads to the appearance of a large-scale poloidal field, which also gives large-scale convection.…”

Section: Mathematical Modelmentioning

confidence: 99%

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

Sheremetyeva

2023

Mathematics

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The article considers a large-scale model of an αΩ-dynamo in the low-mode approximation. The intensity of the α-effect is regulated by a process that depends on the energy of the magnetic field and has hereditarity properties (finite “memory”). The regulation process is included in the MHD-system in the form of an additive correction. The action character of the process is defined by the alternating kernel with variable parameters: the damping frequency and the damping coefficient. The Reynolds number and the α-effect measure are the control parameters of the system. Information about the action of a large-scale generator is contained in the Reynolds number, and that about the action of a turbulent one is contained in the measure of the α-effect. The stability of the solution of the MHD-system is studied depending on the values of the control parameters and the parameters of the alternating kernel. Based on the results of numerical simulation of the dynamical regimes, limitations are determined for the values of the model parameters at which the regimes are reproduced against the background of small oscillations of the viscous liquid velocity field. The results of the study of the stability of solutions and numerical simulations of the dynamical regimes are represented on the phase plane of the control parameters. The paper investigates the question of changing the pattern on the phase plane depending on the values of the damping coefficient, the damping frequency, and the waiting time. A comparison is made with the results obtained earlier, when the α-effect intensity is a constant or is regulated by a process with an exponential kernel and the same values of the damping coefficient.

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On Kinematic Generation of the Magnetic Modes of Bloch Type

Cited by 4 publications

References 27 publications

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. I. Mathematical preliminaries

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

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