# Nonlinear Large-Scale Perturbations of Steady Thermal Convective Dynamo Regimes in a Plane Layer of Electrically Conducting Fluid Rotating about the Vertical Axis

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## Abstract

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## 1. Introduction

## 2. Convective Magnetic Dynamo: Equations and Numerical Methods

#### 2.1. Statement of the Problem

#### 2.2. Numerical Techniques

#### 2.3. Symmetries

## 3. Results of Simulation

#### 3.1. Regimes in the Elongated Boxes with the Periodicities 1x2/2x1

- All of them are quasiperiodic hydrodynamic travelling waves.
- The energy ranges and mean (over time) energies coincide.
- The basic frequencies are identical.
- The ranges of the magnetic Reynolds number estimates Rm${}_{1}^{\prime}$ and Rm${}_{3}^{\prime}$ and their mean (over time) values coincide.
- The ranges of the kinetic helicity and its temporal mean values are the same.
- The growth rates of the dominant magnetic modes are also equal.
- The flows have the symmetry $r{\gamma}_{L/2}^{1}$ in the 1x2 cell and $r{\gamma}_{L/2}^{2}$ in the 2x1 cell.

- Independence of ${x}_{2}$ (the energy of the discrepancy averaged over the periodicity cell, equal to the mean energy of the harmonics associated with wave vectors with a non-zero second component, is below 0.25 in the saturated regime for $t>18$; it shows the tendency to increase towards the ends of the intervals of almost regular helicity behaviour).
- ${s}_{2}$ (for $t>18$, the mean energy of the discrepancy $\mathbf{v}\left(\mathbf{x}\right)-{s}_{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{v}\left(\mathbf{x}\right)$ is below 24.0).
- $r{\gamma}_{L/2}^{1}$ (for $t>18$, the mean energy of the discrepancy $\mathbf{v}\left(\mathbf{x}\right)-r{\gamma}_{L/2}^{1}\phantom{\rule{0.166667em}{0ex}}\mathbf{v}\left(\mathbf{x}\right)$ is below 14.4).
- The flow is approximately 1x1-periodic (for $t>18$, the spatially averaged mean energy of the discrepancy, equal to the mean energy of the harmonics associated with wave vectors, whose first component is odd, is below 6.1).

- We have performed a hydrodynamic run (the magnetic field being set to zero at each step) in a 2x1-periodic fluid box for an initial condition that is the 2x1.1 flow at $t=180$ (close to the local minimum of the magnetic energy), where the ${x}_{2}$-dependence is suppressed by setting to zero all harmonics associated with wave vectors, whose second component is non-zero. This run has converged fast (in roughly 15 time units) to the double-replicated R${}_{1}$, the energy of discrepancy decreasing below ${10}^{-18}$.
- The energy of the discrepancies between the 2x1.1 flow at $t=180$ and the flows of the MHD steady states R${}_{1}$, S${}_{7}^{\mathrm{R}1}$ and S${}_{8}^{\mathrm{R}1}$ is 0.2, 0.2 and 16.1, respectively.

#### 3.2. Regimes in the Elongated Cells with the Periodicities 1x3/3x1

#### 3.3. Regimes in the Elongated Cells with the Periodicities 1x4/4x1

#### 3.4. Regimes in the $({\mathit{M}}_{1},{\mathit{M}}_{2})$-Periodicity Cells for ${\mathit{M}}_{i}\ge 2$

#### 3.5. Videos of the Kinetic and Magnetic Energy Evolution

#### 3.6. Analysis of Integral Parameters of the Regimes

## 4. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Isosurfaces of the kinetic (a) and magnetic (b) energy densities at the levels of 1/2 of the respective maxima of the unperturbed short-scale convective dynamo steady state. One periodicity box is shown. Isolines of the kinetic (c) and magnetic (d) energy densities on the midplane ${x}_{3}=0$, step 1.5 and 0.3, respectively. In (c) and (d), the line width increases with the common along-the-line value of the energy density.

**Figure 2.**Energy spectrum of the flow at $t=220$ in run 2 for ${\mathit{M}}_{1}={\mathit{M}}_{2}=2$; the kinetic energy distribution ${E}_{k}^{\mathbf{v}}$ over shells ${C}_{k}$ is shown for fully populated shells only.

**Figure 3.**Space- and time-averaged kinetic (smaller black dots, left axis) and magnetic (larger blue dots, right axis) energies for the attractors obtained in the runs with the standard ($128{\mathit{M}}_{1}\times 128{\mathit{M}}_{2}\times 97$ harmonics), (a), and alternative ($256\times 256\times 257$ harmonics), (b), resolution. Gray vertical lines: r.m.s. deviations of the space-averaged kinetic energy from their time averages. Thin vertical dotted lines separate the runs into groups of regimes residing in identical periodicity cells.

**Figure 4.**Temporal behaviour of the time-dependent estimates of the length scales (7), ${\ell}_{1}^{\prime}$ (green line) and ${\ell}_{3}^{\prime}$ (blue line), (a), and of the magnetic Reynolds number, Rm${}_{1}^{\prime}$ (green line) and Rm${}_{3}^{\prime}$ (blue line), in run 1x2.2 (b).

**Figure 5.**Temporal behaviour of the kinetic (left vertical axis, black line) and magnetic (right vertical axis, blue line) energies in the double-period runs 1x2.1 (a), 1x2.2 (c), 1x2.3 (e), 2x1.1 (g), 2x1.2 (i), and 2x1.3 (k). Evolution of the magnetic energy versus kinetic energy in runs 1x2.1 (b), 1x2.2 (d), 1x2.3 (f), 2x1.1 (h), 2x1.2 (j), and 2x1.3 (l).

**Figure 6.**Isosurfaces of the kinetic (a) and magnetic (b) energy densities at the level of 1/2 of their respective maxima in run 1x2.3 at time $t=1650.707$. One double-period fluid box is shown.

**Figure 7.**Temporal behaviour of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (left column) and Re ${\widehat{v}}_{222}^{1}$ (right column) in run 2x1.2 in four time intervals: $0\le t\le 320$ (phase T of the initial transient processes, chaotic behaviour and approach to the quasiperiodic metastable state), (a), (b); $800\le t\le 830$ (phase G in the vicinity of the quasiperiodic hydrodynamic metastable state), (c), (d); $1520\le t\le 2020$ (departure from phase G, phase S of the saturated chaotic regime and approach to the chaotic phase D), (e), (f); and $2000\le t\le 2200$ (phase D of magnetic field decay), (g), (h).

**Figure 8.**Temporal power spectra of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (left column) and Re ${\widehat{v}}_{222}^{1}$ (right column) in run 2x1.2 in the time intervals $285\le t\le 1555$ (phase G), (a), (b); $1580\le t\le 1940$ (phase S), (c), (d); and $2000\le t\le 2200$ (phase D), (e), (f).

**Figure 9.**Projection of the Poincaré map of the hyperplane Re ${\widehat{v}}_{222}^{3}=0$ on the plane (Re ${\widehat{v}}_{111}^{1}$,Re ${\widehat{v}}_{111}^{2}$) in the time intervals $300\le t\le 1500$ (phase G), (a); $1600\le t\le 2015$ (phase S and its small neighbourhood), (b); and $2000\le t\le 2200$ (phase D), (c), in run 2x1.2.

**Figure 10.**Temporal behaviour of the kinetic (left vertical axis, black line) and magnetic (right vertical axis, blue line) energies in the double-period run 2x1.2 for $1540\le t\le 1960$, phase S (a). Evolution of the magnetic energy versus kinetic energy in this run in the same time interval (b).

**Figure 11.**Temporal behaviour of the mean kinetic helicity (green line, right vertical axis) together with the kinetic energy (black line, left vertical axis) for reference in runs 1x2.2 (a), 2x1.2 (b) and 2x1.1 (c).

**Figure 12.**Temporal behaviour of the Fourier coefficients Re ${\widehat{b}}_{111}^{1}$ (a) and Im ${\widehat{b}}_{111}^{1}$ (b) in run 2x1.1.

**Figure 13.**Temporal behaviour of the kinetic (left vertical axis, black line) and magnetic (right vertical axis, blue line) energies in triple-period runs (left column), and the projection of the trajectories on the plane of magnetic and kinetic energies (right column). Runs 3x1.1, (a), (b), and 3x1.2, (c), (d), with the standard space resolution ($128\phantom{\rule{-0.166667em}{0ex}}\times 384\phantom{\rule{-0.166667em}{0ex}}\times 97$ harmonics); and run 3x1.2, (e), (f), with the alternative resolution ($256\phantom{\rule{-0.166667em}{0ex}}\times 256\phantom{\rule{-0.166667em}{0ex}}\times 257$ harmonics).

**Figure 14.**Isosurfaces of the kinetic, (a), (c) and magnetic (b), (d) energy densities in run 3x1.2 (the standard resolution) at times $t=380.671$ (a), (b) and $t=566.671$ (c), (d), at the levels of 15% (d), 30% (b) and 50% (a), (c) of the respective maxima of the densities. One triple-period fluid box is shown.

**Figure 15.**Isosurfaces of the flow components: ${v}^{1}=0$ (a), (d); ${v}^{2}=0$ (b), (e); ${v}^{3}=0$ (c), (f) in run 3x1.2 (the standard resolution) at times $t=380.671$ (a)–(c) and $t=566.671$ (d)–(f). Two adjacent triple-period fluid boxes are shown. However, by the boundary conditions (2.1), ${v}^{3}=0$ on the horizontal boundaries, they are not shown as isosurfaces in (c), (f), because that would block the view of the isosurfaces inside the fluid volume.

**Figure 16.**Temporal behaviour of the magnetic Reynolds number estimates Rm${}_{1}^{\prime}$ (green line) and Rm${}_{3}^{\prime}$ (blue line), (a), and of the kinetic helicity (green line, right vertical axis) together with the kinetic energy (black line, left vertical axis) for reference, (b), in run 3x1.2.

**Figure 17.**Temporal behaviour of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (a) and Re ${\widehat{v}}_{222}^{1}$ (b) for $0\le t\le 600$ in run 3x1.2.

**Figure 18.**Temporal power spectra of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (left column) and Re ${\widehat{v}}_{222}^{1}$ (right column) in the interval $50\le t\le 533$ in run 3x1.2 (a), (b) and in the interval $75\le t\le 975$ in run 1x3.3 (c), (d).

**Figure 19.**Projection of the Poincaré map of the hyperplane Re ${\widehat{v}}_{222}^{3}=0$ on the plane (Re ${\widehat{v}}_{111}^{1}$, Re ${\widehat{v}}_{111}^{2}$) for the amagnetic attractors in runs 1x3.1 (constructed over the time interval $100\le t\le 487$) (a) and 3x1.3 (over $300\le t\le 550$) (b).

**Figure 20.**Temporal behaviour of the mean kinetic (left vertical axis, black line) and magnetic (right vertical axis, blue line) energies (left column), and projection of the trajectories on the plane of magnetic and kinetic energies (right column) in the quadruple-period runs 4x1.2 for the standard resolution, (a), (b); 4x1.1 for the standard resolution, (c), (d); and 4x1.1 for the alternative resolution (e), (f).

**Figure 21.**Temporal behaviour of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (a) and Re ${\widehat{v}}_{222}^{1}$ (b) in run 4x1.1.

**Figure 22.**Temporal behaviour of the magnetic Reynolds number estimates, Rm${}_{1}^{\prime}$ (green line) and Rm${}_{3}^{\prime}$ (blue line), (a); and of the kinetic helicity (green line, right vertical axis) together with the kinetic energy (black line, left vertical axis) for reference, (b), in run 4x1.1.

**Figure 23.**Temporal power spectra of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (left column) and Re ${\widehat{v}}_{222}^{1}$ (right column) in the standard-resolution run 4x1.1 in the time intervals $15\le t\le 120$, (a), (b); $132\le t\le 212$, (c), (d); and $223\le t\le 242$, (e), (f).

**Figure 24.**Isosurfaces of the kinetic (a) and magnetic (b) energy densities at the levels of 50% and 35% of their respective maxima in run 4x1.1 at time $t=222.626$. One quadruple-period fluid box is shown.

**Figure 25.**Temporal behaviour of the kinetic (left vertical axis, black line) and magnetic (right vertical axis, blue line) energies in the double-period runs 4x4.2 (a) and 2x4.2 (b).

**Figure 26.**Temporal behaviour of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (a) and Re ${\widehat{v}}_{222}^{1}$ (b) in run 2x4.2.

**Figure 27.**Temporal power spectra of the Fourier coefficients Re ${\widehat{v}}_{111}^{1}$ (a) and Re ${\widehat{v}}_{222}^{1}$ (b) in the time interval $40\le t\le 381.21$ in run 2x4.2.

**Figure 28.**Projection of the Poincaré map of the hyperplane Re ${\widehat{v}}_{222}^{3}=0$ on the plane (Re ${\widehat{v}}_{111}^{1}$, Re ${\widehat{v}}_{111}^{2}$) constructed over the time interval $40\le t\le 381.21$ for run 2x4.2.

**Figure 29.**Time evolution of the quantities ${D}_{q}^{\mathbf{v}}$ (a), ${D}_{q,s}^{\mathbf{v}}$ (c), (e), ${D}_{q}^{\mathbf{b}}$ (b) and ${D}_{q,s}^{\mathbf{b}}$ (d), (f) for $s=2$ and q varying in the range $2\le q\le 10$ (c), (d), and for $q=2$ and s in the range $1\le s\le 10$ (e), (f) in run 4x1.1.

**Table 1.**Minimum energy spectrum decay (orders of magnitude) in the simulated regimes of spatial periods ${\mathit{M}}_{1}L$ and ${\mathit{M}}_{2}L$ in ${x}_{1}$ and ${x}_{2}$, respectively. In each cell of the table, three lines show the decays in the runs for the three different initial conditions that are labelled by the suffices 1, 2 and 3 in the names of the runs (see the beginning of the present section). In each line, commas separate three groups of numbers showing the energy spectra decays of the flow, $\mathbf{v}$, magnetic field, $\mathbf{b}$, and the difference, $\theta $, between the temperature and the linear temperature profile for the fluid at equilibrium in the respective run. A single number in a group, or the first number in a group of two numbers refers to the value in the run with the standard resolution of $128{\mathit{M}}_{1}\times 128{\mathit{M}}_{2}\times 97$ harmonics. The second number (after the slash) in a group of two numbers refers to the decay in the alternative resolution run involving $256\times 256\times 257$ harmonics, when such runs have been performed.

${\mathit{M}}_{1},{\mathit{M}}_{2}$ | 1 | 2 | 3 | 4 |

1 | 15.8,9.6,21.8 | 21.4,8.0,24.6 | 7.8/11.7,4.3/7.2,12.7/17.6 | |

15.2,12.2,21.7 | 9.9/13.4,6.3/10.5,15.9/18.8 | 7.2/11.6,4.3/6.3,12.0/16.8 | ||

17.1,9.6,23.4 | 11.1/13.6,6.0/13.0,16.0/19.6 | 9.5/11.0,4.2/6.8,13.6/16.2 | ||

2 | 16.2,11.4,21.8 | 36.0,14.1,38.0 | 25.3,9.0,27.3 | 18.8/12.2,6.5/7.7,20.8/17.1 |

15.0,8.5,20.5 | 37.8,15.0,38.1 | 24.6,7.9,26.3 | 18.3/11.2,6.5/7.6,20.0/15.7 | |

25.0,9.0,27.2 | 36.9,14.6,38.0 | 24.9,8.5,26.5 | 18.6/10.9,5.9/7.2,20.2/15.7 | |

3 | 9.8/14.9,5.7/12.1,14.7/20.9 | 25.1,8.2,26.6 | 24.7,8.8,27.1 | 18.4/11.8,6.4/7.2,20.9/17.17 |

9.4/14.1,5.6/11.4,14.7/20.1 | 25.0,8.7,26.9 | 19.7,8.4,25.3 | 18.2/10.2,6.1/7.6,20.3/15.6 | |

10.0/13.9,5.8/11.0,15.1/19.2 | 25.2,8.8,27.2 | 18.9,8.3,24.6 | 18.1/10.4,6.1/7.3,20.6/15.2 | |

4 | 7.4/11.6,4.0/6.9,13.1/16.7 | 18.4/12.6,5.9/7.3,20.7/17.5 | 18.0/23.8,6.3/7.5,20.4/26.2 | 14.1/24.4,6.1/7.9,19.5/26.5 |

7.3/10.8,4.2/7.3,13.2/16.3 | 18.8/11.4,6.0/7.5,20.5/16.6 | 18.6/11.0,6.0/6.9,20.3/14.6 | 18.1/10.8,5.6/6.8,20.5/15.5 | |

7.2/11.3,4.1/7.2,12.9/15.7 | 18.5/10.6,6.3/7.5,20.6/16.1 | 18.6/10.8,6.2/7.6,20.9/15.7 | 10.7/10.2,5.7/7.2,16.1/15.4 |

**Table 2.**Attractors (A, column 2) and metastable states (MS) in simulations in the $({\mathit{M}}_{1},{\mathit{M}}_{2})$-periodicity boxes. Columns 3–6: kinetic, magnetic, heat (top to bottom lines in each cell of the table) initial, minimum, average and maximum energies, respectively, in the saturated regimes (A) or when the trajectory approaches the metastable states (MS). Column 7: time intervals when the trajectory is close to the attractor (the right end is then the time at which the run was terminated) or the metastable state. Column 8 (regime type): C—chaotic; HD—hydrodynamic; MHD—magnetohydrodynamic; Q—quasiperiodic; R1—steady rolls parallel to the ${x}_{2}$ axis; S—steady; S2—steady rolls parallel to the ${x}_{2}$ axis; and TW—travelling wave. Column 9: the symmetry pair for the flow and magnetic field.

Run | A/MS | ${\mathit{E}}_{\mathrm{in}}$ | ${\mathit{E}}_{min}$ | ${\mathit{E}}_{\mathrm{av}}$ | ${\mathit{E}}_{max}$ | Time | Type | Symmetries |

1x2.1 | A | 135.92 | 109.88 | 133.34 | 156.64 | [75:1016.90] | MHD | |

4 | 4.22$\times {10}^{-6}$ | 8.64 | 20.11 | |||||

0.0476 | 0.0392 | 0.0452 | 0.05 | |||||

MS | 138.66 | 143.55 | 148.48 | [85:152], | HD | $r{\gamma}_{L/2}^{1},$ | ||

1.54$\times {10}^{-5}$ | 0.012 | 0.13 | [655:663], | Q | $r{\gamma}_{L/2}^{1}$ | |||

0.0441 | 0.0451 | 0.0464 | [805:812] | TW | ||||

MS | 130.80 | 130.87 | 130.93 | [48:50.5] | HD | |||

2.09$\times {10}^{-5}$ | 1.63$\times {10}^{-4}$ | 5.57$\times {10}^{-4}$ | S | |||||

0.0446 | 0.0446 | 0.0460 | ||||||

1x2.2 | A | 100 | 112.36 | 134.09 | 154.22 | [20:300.08] | MHD | |

25 | 0.0350 | 7.76 | 19.27 | |||||

0.01 | 0.0413 | 0.0451 | 0.0493 | |||||

MS | 138.04 | 143.32 | 148.73 | [24:38], | HD | $r{\gamma}_{L/2}^{1},$ | ||

0.0602 | 0.21 | 0.55 | [120:128], | Q | $r{\gamma}_{L/2}^{1}$ | |||

0.0441 | 0.0451 | 0.0477 | [170:190] | TW | ||||

1x2.3 | A | 100 | 34.38 | 133.34 | 158.94 | [40:1691.82] | MHD | |

400 | 4.42$\times {10}^{-74}$ | 0.66 | 19.65 | |||||

0.01 | 0.0115 | 0.0426 | 0.0506 | |||||

MS | 138.84 | 143.56 | 148.48 | [325:1560] | HD | $r{\gamma}_{L/2}^{1},$ | ||

4.60$\times {10}^{-71}$ | 5.44$\times {10}^{-5}$ | 1.23 | Q | $r{\gamma}_{L/2}^{1}$ | ||||

0.0441 | 0.0451 | 0.0464 | TW | |||||

MS | 34.38 | 82.10 | 143.21 | [40:293] | HD | |||

5.97$\times {10}^{-74}$ | 4.06$\times {10}^{-7}$ | 8.07$\times {10}^{-5}$ | C | |||||

0.0115 | 0.0291 | 0.0478 | ||||||

2x1.1 | A | 135.92 | 97.18 | 137.37 | 150.22 | [40:341.81] | MHD | ${\gamma}_{L/2}^{2},$ |

3.76 | 0.91 | 6.98 | 23.93 | C | $q{\gamma}_{L/2}^{2}$ | |||

0.0476 | 0.0381 | 0.0475 | 0.0501 | |||||

MS | 150.14 | 150.19 | 150.22 | [62.8:63.25] | HD | |||

0.96 | 1.19 | 1.45 | [182.4:182.7] | S2 | ||||

0.0498 | 0.0499 | 0.0500 | [298.8:299.1] | R1 | ||||

2x1.2 | A | 100 | 34.03 | 133.89 | 159.26 | [40:2218.50] | MHD | |

25 | 1.61$\times {10}^{-73}$ | 2.01 | 19.70 | |||||

0.01 | 0.0109 | 0.0432 | 0.0505 | |||||

MS | 138.85 | 143.56 | 148.48 | [300:1500], | HD | $r{\gamma}_{L/2}^{2},$ | ||

3.99$\times {10}^{-70}$ | 5.83$\times {10}^{-6}$ | 1.32 | [1661:1678], | Q | $r{\gamma}_{L/2}^{2}$ | |||

0.0441 | 0.0451 | 0.464 | [1723:1740] | TW | ||||

MS | 34.03 | 83.34 | 133.71 | [40:270], | HD | |||

3.14$\times {10}^{-73}$ | 5.28$\times {10}^{-15}$ | 1.12$\times {10}^{-12}$ | [2000:2218] | C | ||||

0.0109 | 0.0294 | 0.0445 | ||||||

2x1.3 | A | 1 | 33.85 | 133.05 | 158.90 | [40:3505.05] | MHD | |

400 | 1.71$\times {10}^{-158}$ | 0.15 | 18.76 | |||||

0.01 | 0.0110 | 0.0424 | 0.0505 | |||||

MS | 138.85 | 143.56 | 148.48 | [670:3400] | HD | $r{\gamma}_{L/2}^{2},$ | ||

1.66$\times {10}^{-154}$ | 3.04$\times {10}^{-6}$ | 1.18 | Q | $r{\gamma}_{L/2}^{2}$ | ||||

0.0441 | 0.0451 | 0.0464 | TW | |||||

MS | 33.85 | 82.71 | 153.46 | [40:633.2] | HD | |||

1.83$\times {10}^{-158}$ | 7.26$\times {10}^{-12}$ | 5.66$\times {10}^{-9}$ | C | |||||

0.0110 | 0.0292 | 0.0504 | ||||||

1x3.1 | A | 135.92 | 132.64 | 142.87 | 156.49 | [25:483.79] | HD | $r{\gamma}_{L/2}^{1},$ |

4 | 1.34$\times {10}^{-26}$ | 6.34$\times {10}^{-5}$ | 5.03$\times {10}^{-3}$ | Q | $r{\gamma}_{L/2}^{1}$ | |||

0.0476 | 0.0435 | 0.0453 | 0.0476 | TW | ||||

1x3.2 | A | 100 | 132.58 | 142.87 | 156.50 | [100:558.16] | HD | $r{\gamma}_{L/2}^{1},$ |

25 | 7.25$\times {10}^{-26}$ | 5.26$\times {10}^{-4}$ | 0.0512 | Q | $r{\gamma}_{L/2}^{1}$ | |||

0.01 | 0.0435 | 0.0453 | 0.0477 | TW | ||||

1x3.3 | A | 1 | 132.64 | 142.87 | 156.46 | [85:1992.79] | HD | $r{\gamma}_{L/2}^{1},$ |

400 | 3.78$\times {10}^{-103}$ | 1.86$\times {10}^{-8}$ | 6.68$\times {10}^{-6}$ | Q | $r{\gamma}_{L/2}^{1}$ | |||

0.01 | 0.0435 | 0.0453 | 0.0476 | TW | ||||

3x1.1 | A | 135.92 | 132.59 | 142.85 | 156.49 | [1275:1348.48] | HD | $r{\gamma}_{L/2}^{2},$ |

3.76 | 3.06$\times {10}^{-6}$ | 2.71$\times {10}^{-3}$ | 0.0432 | Q | $r{\gamma}_{L/2}^{2}$ | |||

0.0476 | 0.0435 | 0.0453 | 0.0477 | TW | ||||

3x1.2 | A | 100 | 132.64 | 142.80 | 156.46 | [560:597.69] | HD | $r{\gamma}_{L/2}^{2},$ |

25 | 5.42$\times {10}^{-6}$ | 2.25$\times {10}^{-4}$ | 1.76$\times {10}^{-3}$ | Q | $r{\gamma}_{L/2}^{2}$ | |||

0.01 | 0.0435 | 0.0453 | 0.0476 | TW | ||||

MS | 139.71 | 139.73 | 139.85 | [8:16] | HD | ${\gamma}_{L/2}^{2},$ | ||

3.60$\times {10}^{-5}$ | 1.38$\times {10}^{-4}$ | 5.35$\times {10}^{-4}$ | S2 | ${\gamma}_{L/2}^{2}$ | ||||

0.0468 | 0.0468 | 0.0468 | ||||||

3x1.3 | A | 100 | 132.51 | 142.87 | 156.63 | [160:561.06] | HD | $r{\gamma}_{L/2}^{2},$ |

400 | 2.54$\times {10}^{-22}$ | 1.79$\times {10}^{-3}$ | 0.0991 | Q | $r{\gamma}_{L/2}^{2}$ | |||

0.01 | 0.0435 | 0.0453 | 0.0477 | TW | ||||

MS | 139.71 | 139.72 | 139.74 | [10:15.5] | HD | |||

5.84$\times {10}^{-5}$ | 2.84$\times {10}^{-4}$ | 1.27$\times {10}^{-3}$ | S2 | |||||

0.0468 | 0.0468 | 0.0468 | ||||||

1x4.1 | A | 135.92 | 56.81 | 123.87 | 183.43 | [50:202.18] | MHD | |

4 | 1.92$\times {10}^{-5}$ | 1.15 | 14.76 | C | ||||

0.0476 | 0.0180 | 0.0359 | 0.0487 | |||||

MS | 131.67 | 142.41 | 151.88 | [115:121], | HD | |||

1.92$\times {10}^{-4}$ | 8.71$\times {10}^{-4}$ | 3.95$\times {10}^{-3}$ | [9:15.5] | Q | ||||

0.0439 | 0.0460 | 0.0478 | ||||||

1x4.2 | A | 100 | 53.62 | 128.35 | 215.63 | [50:200.89] | MHD | |

25 | 3.88$\times {10}^{-4}$ | 3.21 | 17.27 | C | ||||

0.01 | 0.0183 | 0.0389 | 0.05 | |||||

1x4.3 | A | 1 | 43.52 | 122.10 | 200.92 | [50:211.14] | MHD | |

400 | 3.01$\times {10}^{-7}$ | 0.27 | 2.92 | C | ||||

0.01 | 0.0136 | 0.0330 | 0.0485 | |||||

4x1.1 | A | 135.92 | 49.93 | 128.33 | 203.82 | [15:381.79] | MHD | |

4 | 1.77$\times {10}^{-6}$ | 2.15 | 16.71 | C | ||||

0.0476 | 0.0168 | 0.0391 | 0.0487 | |||||

MS | 131.45 | 142.83 | 152.27 | [223:242] | HD | |||

1.92$\times {10}^{-4}$ | 4.22$\times {10}^{-4}$ | 8.51$\times {10}^{-4}$ | Q | |||||

0.0444 | 0.0462 | 0.0476 | TW | |||||

4x1.2 | A | 100 | 51.86 | 124.30 | 205.75 | [50:310.23] | MHD | |

25 | 3.60$\times {10}^{-6}$ | 1.65 | 16.04 | C | ||||

0.01 | 0.0172 | 0.0358 | 0.0499 | |||||

MS | 146.06 | 147.19 | 147.87 | [246:247.75] | HD | |||

7.85$\times {10}^{-3}$ | 0.0799 | 0.30 | S2 | |||||

0.0484 | 0.0486 | 0.0487 | ||||||

4x1.3 | A | 100 | 51.20 | 126.69 | 187.53 | [25:286.47] | MHD | |

900 | 9.13$\times {10}^{-6}$ | 2.33 | 15.10 | C | ||||

0.01 | 0.0179 | 0.0388 | 0.0487 | |||||

MS | 130.19 | 143.60 | 151.34 | [46:51.5] | HD | |||

8.92$\times {10}^{-4}$ | 1.78$\times {10}^{-3}$ | 3.03$\times {10}^{-3}$ | Q | |||||

0.0441 | 0.0466 | 0.0483 | ||||||

MS | 122.37 | 123.00 | 123.76 | [12.5:15] | HD | ${\gamma}_{L/2}^{2},$ | ||

0.47 | 0.89 | 1.50 | S2 | ${\gamma}_{L/2}^{2}$ | ||||

0.0419 | 0.0419 | 0.0419 | ||||||

2x2.1 | A | 135.92 | 47.29 | 79.2 | 131.63 | [40:1001.34] | HD | |

4 | 9.49$\times {10}^{-244}$ | 2.01$\times {10}^{-11}$ | 6.90$\times {10}^{-9}$ | C | ||||

0.0476 | 0.0160 | 0.0267 | 0.0393 | |||||

2x2.2 | A | 100 | 48.34 | 78.59 | 118.35 | [40:331.19] | HD | |

25 | 1.77$\times {10}^{-75}$ | 3.59$\times {10}^{-12}$ | 4.69$\times {10}^{-10}$ | C | ||||

0.01 | 0.0166 | 0.0265 | 0.0373 | |||||

2x2.3 | A | 1 | 46.40 | 78.83 | 135.50 | [40:409.44] | HD | |

1225 | 4.30$\times {10}^{-96}$ | 1.18$\times {10}^{-11}$ | 1.98$\times {10}^{-9}$ | C | ||||

0.01 | 0.0171 | 0.0265 | 0.0399 | |||||

2x3.1 | A | 135.92 | 55.55 | 77.88 | 135.61 | [40:204.50] | HD | |

4 | 1.06$\times {10}^{-35}$ | 3.82$\times {10}^{-8}$ | 1.60$\times {10}^{-6}$ | C | ||||

0.0476 | 0.0176 | 0.0258 | 0.0358 | |||||

2x3.2 | A | 100 | 53.37 | 80.47 | 159.21 | [40:540.45] | HD | |

25 | 8.70$\times {10}^{-109}$ | 1.47$\times {10}^{-11}$ | 7.92$\times {10}^{-9}$ | C | ||||

0.01 | 0.0177 | 0.0263 | 0.0413 | |||||

2x3.3 | A | 1 | 47.40 | 79.94 | 149.96 | [40:417.59] | HD | |

400 | 8.98$\times {10}^{-76}$ | 1.51$\times {10}^{-11}$ | 2.21$\times {10}^{-9}$ | C | ||||

0.01 | 0.0161 | 0.0263 | 0.04 | |||||

3x2.1 | A | 135.92 | 55.47 | 80.48 | 136.12 | [40:318.22] | HD | |

4 | 1.56$\times {10}^{-48}$ | 4.52$\times {10}^{-9}$ | 1.16$\times {10}^{-6}$ | C | ||||

0.0476 | 0.0174 | 0.0264 | 0.0395 | |||||

3x2.2 | A | 100 | 53.78 | 80.50 | 149.84 | [40:364.87] | HD | |

25 | 1.79$\times {10}^{-66}$ | 1.12$\times {10}^{-9}$ | 2.41$\times {10}^{-7}$ | C | ||||

0.01 | 0.0188 | 0.0263 | 0.0394 | |||||

3x2.3 | A | 100 | 54.77 | 78.96 | 146.56 | [40:363.77] | HD | |

400 | 1.61$\times {10}^{-65}$ | 1.10$\times {10}^{-7}$ | 2.59$\times {10}^{-5}$ | C | ||||

0.01 | 0.0183 | 0.0261 | 0.0382 | |||||

2x4.1 | A | 135.92 | 56.35 | 83.93 | 165.97 | [40:244.38] | HD | |

4 | 1.25$\times {10}^{-27}$ | 1.77$\times {10}^{-6}$ | 2.77$\times {10}^{-4}$ | C | ||||

0.0476 | 0.0171 | 0.0263 | 0.0382 | |||||

2x4.2 | A | 100 | 54.33 | 82.05 | 159.80 | [40:381.79] | HD | |

25 | 1.94$\times {10}^{-60}$ | 9.28$\times {10}^{-7}$ | 1.62$\times {10}^{-4}$ | C | ||||

0.01 | 0.0196 | 0.0262 | 0.0375 | |||||

2x4.3 | A | 100 | 57.54 | 80.22 | 152.08 | [40:326.87] | HD | |

400 | 6.23$\times {10}^{-49}$ | 1.55$\times {10}^{-7}$ | 1.46$\times {10}^{-5}$ | C | ||||

0.01 | 0.0188 | 0.0259 | 0.0348 | |||||

4x2.1 | A | 135.92 | 56.54 | 79.66 | 153.39 | [40:290.67] | HD | |

4 | 6.00$\times {10}^{-36}$ | 1.60$\times {10}^{-5}$ | 8.01$\times {10}^{-4}$ | C | ||||

0.0476 | 0.0194 | 0.0261 | 0.0355 | |||||

4x2.2 | A | 225 | 55.31 | 80.67 | 146.53 | [40:366.17] | HD | |

25 | 1.49$\times {10}^{-51}$ | 7.81$\times {10}^{-6}$ | 1.50$\times {10}^{-3}$ | C | ||||

0.01 | 0.0194 | 0.0260 | 0.0349 | |||||

4x2.3 | A | 1 | 52.71 | 81.42 | 164.80 | [40:391.69] | HD | |

100 | 1.88$\times {10}^{-58}$ | 6.09$\times {10}^{-7}$ | 7.67$\times {10}^{-5}$ | C | ||||

0.01 | 0.0161 | 0.0260 | 0.0379 | |||||

3x3.1 | A | 135.92 | 59.34 | 80.38 | 133.18 | [40:201.10] | HD | |

4 | 4.48$\times {10}^{-27}$ | 6.42$\times {10}^{-7}$ | 5.76$\times {10}^{-5}$ | C | ||||

0.0476 | 0.0195 | 0.0262 | 0.0354 | |||||

3x3.2 | A | 100 | 55.04 | 79.20 | 108.71 | [40:321.50] | HD | |

25 | 9.71$\times {10}^{-49}$ | 4.57$\times {10}^{-6}$ | 4.75$\times {10}^{-4}$ | C | ||||

0.01 | 0.0186 | 0.0259 | 0.0327 | |||||

3x3.3 | A | 100 | 56.12 | 78.41 | 106.72 | [40:312.34] | HD | |

400 | 3.92$\times {10}^{-40}$ | 2.25$\times {10}^{-5}$ | 2.38$\times {10}^{-3}$ | C | ||||

0.01 | 0.0182 | 0.0259 | 0.0334 | |||||

3x4.1 | A | 135.92 | 60.18 | 78.74 | 120.72 | [40:200.17] | HD | |

4 | 6.08$\times {10}^{-30}$ | 3.19$\times {10}^{-6}$ | 3.62$\times {10}^{-4}$ | C | ||||

0.0476 | 0.0197 | 0.0256 | 0.0319 | |||||

3x4.2 | A | 225 | 63.75 | 79.86 | 121.48 | [40:295.39] | HD | |

100 | 1.11$\times {10}^{-40}$ | 1.20$\times {10}^{-6}$ | 8.99$\times {10}^{-5}$ | C | ||||

0.01 | 0.0203 | 0.0259 | 0.0323 | |||||

3x4.3 | A | 1 | 62.18 | 79.26 | 117.60 | [40:285.44] | HD | |

900 | 5.22$\times {10}^{-38}$ | 7.20$\times {10}^{-6}$ | 5.29$\times {10}^{-4}$ | C | ||||

0.01 | 0.0206 | 0.0258 | 0.0327 | |||||

4x3.1 | A | 135.92 | 60.75 | 79.99 | 112.65 | [40:200.01] | HD | |

4 | 6.26$\times {10}^{-25}$ | 1.07$\times {10}^{-5}$ | 5.83$\times {10}^{-4}$ | C | ||||

0.0476 | 0.0190 | 0.0259 | 0.0338 | |||||

4x3.2 | A | 225 | 59.81 | 79.15 | 110.28 | [40:337.28] | HD | |

100 | 2.34$\times {10}^{-44}$ | 7.68$\times {10}^{-6}$ | 5.46$\times {10}^{-4}$ | C | ||||

0.01 | 0.0202 | 0.0258 | 0.0321 | |||||

4x3.3 | A | 100 | 53.78 | 78.80 | 109.92 | [40:296.86] | HD | |

400 | 1.44$\times {10}^{-33}$ | 6.20$\times {10}^{-7}$ | 5.71$\times {10}^{-5}$ | C | ||||

0.01 | 0.0183 | 0.0257 | 0.0324 | |||||

4x4.1 | A | 135.92 | 65.55 | 79.53 | 106.56 | [40:200.51] | HD | |

4 | 5.33$\times {10}^{-23}$ | 1.03$\times {10}^{-6}$ | 7.88$\times {10}^{-5}$ | C | ||||

0.0476 | 0.0212 | 0.0257 | 0.0313 | |||||

4x4.2 | A | 225 | 61.65 | 79.41 | 105.68 | [40:528.55] | HD | |

100 | 8.18$\times {10}^{-54}$ | 3.14$\times {10}^{-6}$ | 5.88$\times {10}^{-4}$ | C | ||||

0.01 | 0.0201 | 0.0258 | 0.0333 | |||||

4x4.3 | A | 100 | 62.51 | 80.20 | 113.30 | [40:342.84] | HD | |

400 | 1.16$\times {10}^{-38}$ | 2.58$\times {10}^{-5}$ | 2.61$\times {10}^{-3}$ | C | ||||

0.01 | 0.0210 | 0.0259 | 0.0319 |

**Table 3.**Magnetic Reynolds number estimates in the runs for ${\mathit{M}}_{i}\le 4$ (in the order of increasing Rm${}_{4}$). Column G: “M” indicates that magnetic field generation is sustained in the perturbed regime; Interval: the time interval, over which the magnetic Reynolds number estimates are computed; ${\ell}_{n}$: estimates (7) of the internal spatial scale of the perturbed regime; Rm${}_{n}$: the magnetic Reynolds number estimate based on ${\ell}_{n}$; $\left|\mathbf{v}\right|$: the characteristic flow velocity in the time interval under consideration.

Run | G | Interval | ${\mathit{\ell}}_{1}$ | Rm${}_{1}$ | ${\mathit{\ell}}_{2}$ | Rm${}_{2}$ | ${\mathit{\ell}}_{3}$ | Rm${}_{3}$ | ${\mathit{\ell}}_{4}$ | Rm${}_{4}$ | $\left|\mathbf{v}\right|$ |

4x4.2 | 51–528.55 | 0.223 | 22.413 | 0.223 | 22.416 | 0.206 | 20.722 | 0.206 | 20.720 | 12.58 | |

4x2.1 | 232.40–290.65 | 0.219 | 21.769 | 0.219 | 21.770 | 0.241 | 23.988 | 0.241 | 23.983 | 12.43 | |

3x3.3 | 220.46–312.34 | 0.220 | 21.843 | 0.220 | 21.842 | 0.241 | 24.011 | 0.241 | 24.004 | 12.43 | |

2x2.3 | 310.15–409.44 | 0.219 | 21.879 | 0.220 | 21.885 | 0.242 | 24.115 | 0.242 | 24.103 | 12.46 | |

2x3.1 | 100–204.50 | 0.219 | 21.920 | 0.219 | 21.923 | 0.241 | 24.113 | 0.241 | 24.108 | 12.50 | |

4x3.3 | 223.17–296.86 | 0.221 | 22.050 | 0.221 | 22.047 | 0.242 | 24.176 | 0.242 | 24.167 | 12.48 | |

2x3.2 | 505.52–540.45 | 0.220 | 22.092 | 0.220 | 22.101 | 0.242 | 24.280 | 0.242 | 24.278 | 12.53 | |

3x2.3 | 251.39–363.77 | 0.221 | 22.183 | 0.221 | 22.190 | 0.243 | 24.365 | 0.242 | 24.358 | 12.55 | |

3x4.1 | 42.18–200.17 | 0.222 | 22.273 | 0.222 | 22.277 | 0.243 | 24.377 | 0.243 | 24.369 | 12.55 | |

3x4.3 | 219.42–295.39 | 0.221 | 22.281 | 0.221 | 22.283 | 0.243 | 24.400 | 0.242 | 24.393 | 12.57 | |

4x3.2 | 290.74–337.38 | 0.222 | 22.337 | 0.222 | 22.332 | 0.243 | 24.414 | 0.243 | 24.406 | 12.56 | |

4x4.3 | 201–336 | 0.223 | 22.385 | 0.223 | 22.385 | 0.243 | 24.439 | 0.243 | 24.436 | 12.56 | |

4x3.1 | 42–200.02 | 0.223 | 22.396 | 0.223 | 22.397 | 0.243 | 24.481 | 0.243 | 24.480 | 12.57 | |

3x4.2 | 246.87–285.44 | 0.223 | 22.454 | 0.223 | 22.446 | 0.243 | 24.497 | 0.243 | 24.486 | 12.57 | |

3x3.2 | 278.28–321 | 0.224 | 22.467 | 0.223 | 22.459 | 0.244 | 24.533 | 0.244 | 24.518 | 12.56 | |

2x3.3 | 309.16–417.59 | 0.221 | 22.382 | 0.221 | 22.415 | 0.243 | 24.554 | 0.243 | 24.560 | 12.65 | |

3x3.1 | 42.91–201.10 | 0.222 | 22.468 | 0.222 | 22.473 | 0.243 | 24.574 | 0.243 | 24.568 | 12.62 | |

2x2.2 | 200–331.19 | 0.221 | 22.333 | 0.221 | 22.356 | 0.245 | 24.698 | 0.244 | 24.680 | 12.62 | |

4x4.1 | 100–200.51 | 0.223 | 22.619 | 0.224 | 22.623 | 0.244 | 24.689 | 0.244 | 24.685 | 12.65 | |

2x2.1 | 719.70–800.70 | 0.221 | 22.484 | 0.221 | 22.502 | 0.243 | 24.788 | 0.243 | 24.779 | 12.73 | |

2x4.2 | 317.09–381 | 0.223 | 22.741 | 0.224 | 22.784 | 0.243 | 24.795 | 0.243 | 24.810 | 12.73 | |

2x4.3 | 223.93–326 | 0.224 | 22.806 | 0.224 | 22.816 | 0.244 | 24.850 | 0.244 | 24.846 | 12.72 | |

3x2.1 | 231.77–318 | 0.223 | 22.723 | 0.223 | 22.728 | 0.244 | 24.863 | 0.244 | 24.857 | 12.72 | |

4x2.3 | 268.69–366.17 | 0.224 | 22.855 | 0.224 | 22.860 | 0.244 | 24.902 | 0.244 | 24.895 | 12.73 | |

2x4.1 | 102–207.69 | 0.225 | 22.959 | 0.226 | 22.973 | 0.245 | 24.954 | 0.245 | 24.961 | 12.73 | |

3x2.2 | 304.32–364 | 0.223 | 22.978 | 0.224 | 23.022 | 0.244 | 25.093 | 0.244 | 25.104 | 12.86 | |

4x2.2 | 334.41–391.69 | 0.225 | 23.106 | 0.226 | 23.135 | 0.245 | 25.106 | 0.245 | 25.113 | 12.81 | |

3x1.1 | 100–249.39 | 0.227 | 27.379 | 0.227 | 27.279 | 0.247 | 29.761 | 0.246 | 29.608 | 15.04 | |

2x1.2 | M | 1946.90–1980.90 | 0.227 | 26.638 | 0.228 | 26.815 | 0.254 | 29.868 | 0.255 | 29.949 | 14.68 |

3x1.2 | 285.85–377.67 | 0.227 | 28.423 | 0.226 | 28.342 | 0.245 | 30.675 | 0.244 | 30.537 | 15.65 | |

1x4.3 | M | 130–211.14 | 0.242 | 29.726 | 0.243 | 29.786 | 0.254 | 31.235 | 0.254 | 31.163 | 15.34 |

4x1.3 | M | 108.81–241.99 | 0.233 | 29.797 | 0.233 | 29.727 | 0.248 | 31.701 | 0.247 | 31.589 | 15.96 |

4x1.2 | M | 200–310.23 | 0.238 | 30.258 | 0.238 | 30.170 | 0.253 | 32.107 | 0.252 | 31.986 | 15.85 |

4x1.1 | M | 168–330 | 0.238 | 29.814 | 0.238 | 29.770 | 0.256 | 32.060 | 0.256 | 32.015 | 15.65 |

1x4.1 | M | 140–192 | 0.242 | 30.969 | 0.241 | 30.894 | 0.252 | 32.300 | 0.251 | 32.165 | 16.00 |

1x4.2 | M | 120–200 | 0.237 | 31.140 | 0.236 | 31.098 | 0.259 | 34.047 | 0.257 | 33.888 | 16.45 |

1x2.1 | M | 185–245 | 0.238 | 30.339 | 0.238 | 30.333 | 0.267 | 34.018 | 0.267 | 34.003 | 15.92 |

1x2.3 | M | 1606–1691 | 0.238 | 30.534 | 0.238 | 30.526 | 0.267 | 34.206 | 0.267 | 34.181 | 16.01 |

2x1.3 | M | 3460–3505.05 | 0.239 | 30.794 | 0.238 | 30.787 | 0.266 | 34.407 | 0.266 | 34.383 | 16.13 |

1x2.2 | M | 50–300 | 0.239 | 31.193 | 0.239 | 31.188 | 0.266 | 34.715 | 0.266 | 34.684 | 16.32 |

2x1.1 | M | 152–277.81 | 0.248 | 33.000 | 0.248 | 32.987 | 0.261 | 34.777 | 0.261 | 34.739 | 16.64 |

1x3.3 | 50–354.35 | 0.234 | 31.566 | 0.234 | 31.568 | 0.263 | 35.499 | 0.263 | 35.483 | 16.87 | |

1x3.1 | 25–487.12 | 0.234 | 31.668 | 0.234 | 31.659 | 0.263 | 35.589 | 0.263 | 35.566 | 16.90 | |

1x3.2 | 83–558.16 | 0.234 | 31.669 | 0.234 | 31.660 | 0.263 | 35.590 | 0.263 | 35.567 | 16.90 | |

3x1.3 | 330.20–365.60 | 0.234 | 31.676 | 0.234 | 31.667 | 0.263 | 35.597 | 0.263 | 35.574 | 16.90 |

**Table 4.**Magnetic Reynolds number estimates in various phases of the runs in the double-period periodicity cells. Interval: the time interval, for which the magnetic Reynolds number estimates are computed; ${\ell}_{n}$: estimates (7) of the internal spatial scale of the perturbed regime; Rm${}_{n}$: the magnetic Reynolds number estimate based on ${\ell}_{n}$; $\left|\mathbf{v}\right|$: the characteristic flow velocity in the time interval under consideration.

Run | Phase | Interval | ${\mathit{\ell}}_{1}$ | Rm${}_{1}$ | ${\mathit{\ell}}_{2}$ | Rm${}_{2}$ | ${\mathit{\ell}}_{3}$ | Rm${}_{3}$ | ${\mathit{\ell}}_{4}$ | Rm${}_{4}$ | $\left|\mathbf{v}\right|$ |

1x2.1 | T | 10–65 | 0.215 | 22.690 | 0.214 | 22.615 | 0.249 | 26.325 | 0.249 | 26.265 | 13.20 |

G | 75–170 | 0.240 | 32.566 | 0.240 | 32.553 | 0.263 | 35.653 | 0.263 | 35.638 | 16.93 | |

S | 185–250 | 0.238 | 30.425 | 0.238 | 30.418 | 0.267 | 34.099 | 0.267 | 34.083 | 15.96 | |

1x2.2 | T | 0–13.8 | 0.220 | 24.205 | 0.220 | 24.172 | 0.245 | 26.928 | 0.244 | 26.765 | 13.72 |

G | 20–40 | 0.240 | 32.557 | 0.240 | 32.543 | 0.263 | 35.665 | 0.263 | 35.648 | 16.93 | |

S | 50–115 | 0.238 | 30.439 | 0.238 | 30.432 | 0.267 | 34.165 | 0.267 | 34.151 | 15.98 | |

1x2.3 | D | 10–293 | 0.214 | 22.367 | 0.214 | 22.350 | 0.246 | 25.699 | 0.246 | 25.627 | 13.03 |

G | 300–1597 | 0.240 | 32.584 | 0.240 | 32.571 | 0.263 | 35.667 | 0.263 | 35.653 | 16.94 | |

S | 1606–1691.82 | 0.238 | 30.534 | 0.238 | 30.526 | 0.267 | 34.206 | 0.267 | 34.181 | 16.01 | |

2x1.1 | T | 0–8 | 0.244 | 32.150 | 0.244 | 32.151 | 0.260 | 34.363 | 0.260 | 34.362 | 16.50 |

S | 12–277.81 | 0.248 | 33.010 | 0.248 | 32.993 | 0.261 | 34.789 | 0.261 | 34.737 | 16.65 | |

2x1.2 | D | 40–270 | 0.216 | 22.533 | 0.214 | 22.374 | 0.246 | 25.743 | 0.245 | 25.585 | 13.06 |

G | 285–1555 | 0.241 | 32.658 | 0.240 | 32.621 | 0.263 | 35.727 | 0.263 | 35.699 | 16.95 | |

S | 1570–1962 | 0.239 | 31.254 | 0.239 | 31.170 | 0.266 | 34.814 | 0.266 | 34.743 | 16.33 | |

2x1.3 | D | 40–633.2 | 0.215 | 22.265 | 0.214 | 22.161 | 0.246 | 25.559 | 0.245 | 25.448 | 12.97 |

G | 646–3445 | 0.240 | 32.611 | 0.240 | 32.579 | 0.263 | 35.686 | 0.263 | 35.662 | 16.95 | |

S | 3460–3505.05 | 0.239 | 30.794 | 0.238 | 30.787 | 0.266 | 34.407 | 0.266 | 34.383 | 16.13 |

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**MDPI and ACS Style**

Jeyabalan, S.R.; Chertovskih, R.; Gama, S.; Zheligovsky, V.
Nonlinear Large-Scale Perturbations of Steady Thermal Convective Dynamo Regimes in a Plane Layer of Electrically Conducting Fluid Rotating about the Vertical Axis. *Mathematics* **2022**, *10*, 2957.
https://doi.org/10.3390/math10162957

**AMA Style**

Jeyabalan SR, Chertovskih R, Gama S, Zheligovsky V.
Nonlinear Large-Scale Perturbations of Steady Thermal Convective Dynamo Regimes in a Plane Layer of Electrically Conducting Fluid Rotating about the Vertical Axis. *Mathematics*. 2022; 10(16):2957.
https://doi.org/10.3390/math10162957

**Chicago/Turabian Style**

Jeyabalan, Simon Ranjith, Roman Chertovskih, Sílvio Gama, and Vladislav Zheligovsky.
2022. "Nonlinear Large-Scale Perturbations of Steady Thermal Convective Dynamo Regimes in a Plane Layer of Electrically Conducting Fluid Rotating about the Vertical Axis" *Mathematics* 10, no. 16: 2957.
https://doi.org/10.3390/math10162957