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

## Abstract

**:**

## 1. Introduction

## 2. Statement of a Problem

## 3. Mathematical Model

## 4. Numerical Method and Model Parameters

## 5. Study of the Field Generation Conditions in the Linear Approximation

## 6. Results of Simulation and Discussion

## 7. Conclusions

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviation

MHD | Magnetohydrodynamic |

## References

- Vodinchar, G.M. Using modes of free oscillation of a rotating viscous fluid in the large-scale dynamo. Vestn. KRAUNC Fiz. Mat. Nauk.
**2013**, 2, 33–42. (In Russian) [Google Scholar] - Vodinchar, G.M.; Feschenko, L.K. 6-jet kinematic model of geodinamo. Nauchnye Vedom. BelGU Mat. Fiz.
**2014**, 5, 94–102. (In Russian) [Google Scholar] - Feschenko, L.K.; Vodinchar, G.M. Reversals in the large-scale αΩ-dynamo with memory. Nonlinear Process. Geophys.
**2015**, 4, 361–369. [Google Scholar] [CrossRef] - Vodinchar, G.M.; Feshenko, L.K. Reversals in the 6-cells convection driven. Bull. KRASEC Phys. Math. Sci.
**2015**, 4, 41–50. [Google Scholar] - Vodinchar, G.M.; Feschenko, L.K. Model of geodynamo dryven by six-jet convection in the Earth’s core. Magnetohydrodynamics
**2016**, 1, 287–300. [Google Scholar] [CrossRef] - Vodinchar, G.M.; Godomskaya, A.N.; Sheremetyeva, O.V. Reversal of magnetic field in the dynamic system with stochastic αΩ-generators. Vestn. KRAUNC Fiz. Mat. Nauk.
**2017**, 4, 76–82. [Google Scholar] - Sheremetyeva, O.V.; Godomskaya, A.N. Modelling the magnetic field generation modes in the low-mode model of the αΩ-dynamo with varying intensity of the α-effect. Vestn. Yuzhno Ural. Univ. Seriya Mat. Model. I Program.
**2021**, 14, 27–38. (In Russian) [Google Scholar] [CrossRef] - Godomskaya, A.N.; Sheremetyeva, O.V. 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. EPJ Web Conf.
**2021**, 254, 02015. [Google Scholar] - Gledzer, E.B.; Dolzhanskiy, F.V.; Obukhov, A.M. Sistemy Gidrodinamicheskogo Tipa i Ikh Primenenie [Hydrodynamic Type Systems and Their Application]; Nauka: Moscow, Russia, 1981; p. 368. (In Russian) [Google Scholar]
- Kono, M.; Roberts, P.H. Recent geodynamo simulations and observations of the field. Rev. Geophys.
**2002**, 40, B1–B41. [Google Scholar] [CrossRef] - Sokoloff, D.D.; Stepanov, R.A.; Frick, P.G. Dynamos: From an astrophysical model to laboratory experiments. Physics-Uspekhi
**2014**, 3, 313–335. [Google Scholar] [CrossRef] - Riols, A.; Latter, H. Gravitoturbulent dynamos in astrophysical discs. Mon. Notices R. Astronom. Soc.
**2019**, 482, 3989–4008. [Google Scholar] [CrossRef] - Parker, E.N. Hydromagnetic dynamo models. Astrophys. J.
**1955**, 122, 293–314. [Google Scholar] [CrossRef] - Zeldovich, Y.B.; Rusmaikin, A.A.; Sokoloff, D.D. Magnetic Fields in Astrophysics. The Fluid Mechanics of Astrophysics and Geophysics; Gordon and Breach: New York, NY, USA, 1983; p. 382. [Google Scholar]
- Krause, F.; Rädler, K.H. Mean-Filed Magnetohydrodynamics and Dynamo Theory; Pergamon Press: Oxford, UK, 1980; p. 271. [Google Scholar]
- Sokoloff, D.D.; Nefedov, S.N. A small-mode approximation in the stellar dynamo problem. Numer. Methods Program.
**2007**, 2, 195–204. (In Russian) [Google Scholar] - Zhou, T.; Deng, H.-P.; Chen, Y.-X.; Lin, D.N.C. Turbulent Transport of Dust Particles in Protostellar Disks: The Effect of Upstream Diffusion. Astrophys. J.
**2022**, 940, 117. [Google Scholar] [CrossRef] - Yang, L.; Chen, J.; Hu, G. A framework of the finite element solution of the Landau-Lifshitz-Gilbert equation on tetrahedral meshes. J. Comput. Phys.
**2021**, 431, 110142. [Google Scholar] [CrossRef] - Sadovnikov, A.V.; Bublikov, K.V.; Beginin, E.N.; Sheshukova, S.E.; Sharaevskii, Y.P.; Nikitov, S.A. Nonreciprocal propagation of hybrid electromagnetic waves in a layered ferrite-ferroelectric structure with a finite width. JETP Lett.
**2015**, 102, 142–147. [Google Scholar] [CrossRef] - Odintsov, S.A.; Beginin, E.N.; Sheshukova, S.E.; Sadovnikov, A.V. Reconfigurable lateral spin-wave transport in a ring magnonic microwaveguide. JETP Lett.
**2019**, 110, 430–435. [Google Scholar] [CrossRef] - Steenbeck, M.; Krause, F. Zur Dynamotheorie stellarer und planetarer Magnetfelder. I. Berechnunug sonnenähnlicher Wechselfeldgeneratoren. Astron. Nachr.
**1969**, 291, 49–84. [Google Scholar] [CrossRef] - Merril, R.T.; McElhinny, M.W.; McFadden, P.L. The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle; Academic Press: London, UK, 1996; p. 531. [Google Scholar]
- Glatzmaier, G.; Roberts, P. A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature
**1995**, 377, 203–209. [Google Scholar] [CrossRef] - Kuang, W.; Bloxman, J. An Earth-like numerical dynamo model. Nature
**1997**, 389, 371–374. [Google Scholar] [CrossRef] - Sheremetyeva, O.V. Modes of magnetic field generation in the low-mode αΩ-dynamo model with dynamic regulation of the α-effect by the field energy. Vestn. KRAUNC Fiz. Mat. Nauk.
**2021**, 4, 92–103. (In Russian) [Google Scholar] - Sheremetyeva, O.V. Dynamics of generation modes changes in magnetic field depending on the oscillation frequency of the α-effect suppression process by field energy in the αΩ-dynamo model. Vestn. KRAUNC Fiz. Mat. Nauk.
**2022**, 4, 107–119. (In Russian) [Google Scholar] - Zheligovsky, V.A.; Chertovskih, R.A. On kinematic generation of the magnetic modes of bloch type. Izvestiya Phys. Solid Earth
**2020**, 56, 103–116. [Google Scholar] [CrossRef] - Rozenknop, L.M.; Reznikov, E.L. On the free oscillations of a rotating viscous in the outer Earth core. Vychislitelnaya Seismol. Pryamye Zadachi Mat. Fiz
**1998**, 30, 121–132. [Google Scholar] - Vodinchar, G.M. Database ≪Parameters of Eigenmodes of Free Oscillations of MHD Fields in the Earth’s Core≫. Certificate of State Registration No. 2019620054. 2019. Available online: http://new.fips.ru/registers-doc-view/fips_servlet?DB=DB&DocNumber=2019620054&TypeFile=html (accessed on 11 May 2023). (In Russian).
- Vodinchar, G.M.; Feshenko, L.K. Library of Programs for the Research of ≪Low-Mode Geodynamo Model≫: ≪LowModedGeodinamoModel≫, Certificate of State Registration No. 50201100092. 2011. (In Russian)
- Vodinchar, G. Using symbolic calculations to calculate the eigenmodes of the free damping of a geomagnetic field. E3S Web Conf.
**2018**, 62, 02018. [Google Scholar] [CrossRef] - Godomskaya, A.N.; Sheremetyeva, O.V. A Program for the Research of Reversals in Magnetohydrodynamic Type Systems. Certificate of State Registration No. 2016613520. 2016. Available online: http://new.fips.ru/registers-doc-view/fips_servlet?DB=EVM&DocNumber=2016614121&TypeFile=html (accessed on 11 May 2023). (In Russian).
- Elsholts, L.E. Differential Equations and Calculus of Variations; Nauka: Moscow, Russia, 1965; p. 424. (In Russian) [Google Scholar]
- Kurosh, A.G. Course of Higher Algebra; Nauka: Moscow, Russia, 1968; p. 431. (In Russian) [Google Scholar]
- Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems: A method for computing all of them. Part I: Theory. Meccanica
**1980**, 15, 9–20. [Google Scholar] [CrossRef] - Kuznetsov, S.P. Dynamic Chaos and Hyperbolic Attractors: From Mathematics to Physics; Institute of Computer Science Izhevsk: Izhevsk, Russia, 2013; p. 488. (In Russian) [Google Scholar]

**Figure 1.**Distribution of solutions based on the Lyapunov stability criterion on the phase plane of control parameters $R{e}_{m}$ and ${R}_{\alpha}$ (on a double logarithmic scale) for: (

**a**) a system (11) with a constant velocity field and without the $\alpha $-quenching, (

**b**–

**f**) a system (10) includes $\alpha $-quenching defined by a process $Z\left(t\right)$ with a kernel $J\left(t\right)$. Regions for stable solution: red—field decay without oscillations, green—field decay with oscillations. Regions for unstable solution: white—field generation without oscillations, gray—field generation with oscillations.

**Figure 2.**The magnetic field dynamical regimes on the plane of the control parameters ${R}_{\alpha}$ and $R{e}_{m}$: the white region is the infinitely increasing magnetic field, the red is the damped, the green is the steady, the blue is the steady-state, the yellow is the vacillation, the grey is the chaotic regime. The $\alpha $-quenching is defined by a process $Z\left(t\right)$ (9) with a kernel $J\left(t\right)={e}^{-5t}cos10t$.

**Figure 4.**The dependence of the waiting time ${t}_{0}$ [26] on the damped oscillations frequency a at a constant damping coefficient $b=0.5$.

s | j | ${P}_{sj}$ |
---|---|---|

2 | 1 | −0.1929 |

1 | 2 | 0.1929 |

3 | 2 | −0.3648 |

11 | 2 | 0.402 |

13 | 2 | 0.2993 |

2 | 3 | 0.3648 |

2 | 11 | −0.402 |

2 | 13 | −0.2993 |

s | ${\mathit{\lambda}}_{\mathit{s}}$ | ${\mathit{\alpha}}_{\mathit{s}}$ |
---|---|---|

1 | 28.1592 | −0.9353 |

2 | 86.5734 | $1.7497\times {10}^{-6}$ |

3 | 50.9094 | −0.33353 |

11 | 98.8577 | 0.1048 |

13 | 125.9759 | 0.0556 |

s | i | j | ${L}_{sij}$ |
---|---|---|---|

2 | 1 | 1 | −0.1697 |

2 | 1 | 3 | 0.1459 |

1 | 2 | 1 | 0.4873 |

3 | 2 | 1 | −0.7552 |

11 | 2 | 1 | 1.2321 |

13 | 2 | 1 | 0.1692 |

2 | 2 | 2 | 0.2108 |

1 | 2 | 3 | −0.5719 |

3 | 2 | 3 | −0.4535 |

11 | 2 | 3 | −1.0872 |

13 | 2 | 3 | −0.0769 |

2 | 3 | 1 | 0.6108 |

2 | 3 | 3 | 0.3764 |

i | s | j | ${W}_{isj}$ |
---|---|---|---|

2 | 1 | 1 | −0.4873 |

2 | 1 | 3 | 0.5719 |

1 | 2 | 1 | 0.1697 |

3 | 2 | 1 | −0.6108 |

2 | 2 | 2 | −0.2108 |

1 | 2 | 3 | −0.1459 |

3 | 2 | 3 | −0.3764 |

2 | 3 | 1 | 0.7552 |

2 | 3 | 3 | 0.4535 |

2 | 11 | 1 | −1.2321 |

2 | 11 | 3 | 1.0872 |

2 | 13 | 1 | −0.1692 |

2 | 13 | 3 | 0.0769 |

i | j | ${\mathit{W}}_{ij}^{\mathit{\alpha}}$ |
---|---|---|

2 | 1 | 0.7883 |

3 | 1 | 1.6361 |

1 | 2 | 1.0728 |

1 | 3 | 1.2005 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sheremetyeva, O.
Magnetic Field Dynamical Regimes in a Large-Scale Low-Mode *α*Ω-Dynamo Model with Hereditary *α*-Quenching by Field Energy. *Mathematics* **2023**, *11*, 2297.
https://doi.org/10.3390/math11102297

**AMA Style**

Sheremetyeva O.
Magnetic Field Dynamical Regimes in a Large-Scale Low-Mode *α*Ω-Dynamo Model with Hereditary *α*-Quenching by Field Energy. *Mathematics*. 2023; 11(10):2297.
https://doi.org/10.3390/math11102297

**Chicago/Turabian Style**

Sheremetyeva, Olga.
2023. "Magnetic Field Dynamical Regimes in a Large-Scale Low-Mode *α*Ω-Dynamo Model with Hereditary *α*-Quenching by Field Energy" *Mathematics* 11, no. 10: 2297.
https://doi.org/10.3390/math11102297