# Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation

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## Abstract

**:**

## 1. Introduction

## 2. Calculation of Eddy Viscosity

#### 2.1. Eddy Viscosities and Multiscale Techniques

#### 2.2. Eddy Viscosity Expansion in Powers of ${\nu}^{-1}$

#### 2.3. Results of Calculations

## 3. Computation of the Magnetic $\mathit{\alpha}$-Effect Tensor

#### 3.1. The Multiscale Formalism Revealing the Magnetic $\alpha $-Effect

#### 3.2. Padé Approximation

#### 3.3. Numerical Results

#### 3.3.1. Approximation by the Algorithm I

`tol`(i.e., is not smaller than $\mathtt{tol}\parallel ({f}^{\left(1\right)},{f}^{\left(2\right)},\dots ,{f}^{(M+L-1)},{f}^{(M+L)})\parallel $, where $\parallel \xb7\parallel $ is the standard Lebesgue space ${L}_{2}$ norm), decreasing the degrees of the polynomials involved in the Padé approximation, M and L, by the number of the “missing” dimensions. To counter noise due to rounding errors, $\mathtt{tol}={10}^{-14}$ was often used in [35].

- •
- Pseudospectral methods used in the computation of space-periodic solutions ${\mathbf{s}}_{k}$ to the auxiliary problems (14) and their coefficients ${\mathbf{s}}_{k}^{\left(n\right)}$ (22) involve fast Fourier transforms. These algorithms are very efficient. However, they operate by computing various linear combinations of the Fourier coefficients. Typically, at least for moderate molecular diffusivities, the energy spectra of these fields decay fast. In a sum of a large coefficient with a small (in absolute values) one, a significant part of the accuracy of the smaller coefficient is lost.
- •
- Insufficiency of the spatial resolution can result in significant numerical errors. We may note that while increasing the resolution improves solutions, it aggravates the FFT accuracy problems.

#### 3.3.2. Approximation by the Algorithm II

`mprove`in §2.5)”. This is implemented in their

`pade`procedure. We have used it with one alteration: the routine

`mprove`stops when the discrepancy increases; instead, it has been allowed to make up to 1000 improvement iterations, permitting the discrepancy to temporarily grow and storing the minimum-discrepancy solution obtained in the course of these iterations (however, it has often been forced to stop before the allowed number of iterations has been performed, the iterative process blowing up with an overflow).

`−r16`. Higher precision and resolution has been expected to improve the accuracy of the coefficients of the approximants, to augment the orders of the approximants beyond those produced by the algorithm in [35], and to increase the $\eta $ interval, where the growth rate values determined for the approximated $\alpha $-effect tensor are close to the actual growth rates.

## 4. Computation of the Magnetic Eddy Diffusivity Tensor

#### 4.1. The Multiscale Formalism Revealing the Magnetic Eddy Diffusivity

#### 4.2. Padé Approximation

- •
- determine coefficients in the expansion of the neutral magnetic modes ${\mathbf{s}}_{k}$ applying recurrence relations (22);
- •
- in the course of these calculations, determine coefficients in the expansion of $\nabla \xb7{\mathbf{Z}}_{l}$ using (47);
- •
- calculate the coefficients ${\left({\mathfrak{D}}_{mk}^{l}\right)}^{\left(n\right)}$ applying (48).

#### 4.3. Numerical Results

- ${\mathbf{b}}_{\mathbf{n}}={\mathbf{b}}_{-\mathbf{n}}$ are real;
- the numbers ${n}_{1}$ and ${n}_{1}/2+{n}_{2}+{n}_{3}$ are even;
- ${\mathbf{s}}_{1}$ and ${\mathbf{s}}_{2}$ are symmetric in ${x}_{3}$ (i.e., ${b}_{\mathbf{n}}^{1}={b}_{{\mathbf{n}}^{*}}^{1},\phantom{\rule{3.33333pt}{0ex}}{b}_{\mathbf{n}}^{2}={b}_{{\mathbf{n}}^{*}}^{2},\phantom{\rule{3.33333pt}{0ex}}{b}_{\mathbf{n}}^{3}=-{b}_{{\mathbf{n}}^{*}}^{3}$), and ${\mathbf{s}}_{3}$ is antisymmetric in ${x}_{3}$ (i.e., ${b}_{\mathbf{n}}^{1}=-{b}_{{\mathbf{n}}^{*}}^{1},\phantom{\rule{3.33333pt}{0ex}}{b}_{\mathbf{n}}^{2}=-{b}_{{\mathbf{n}}^{*}}^{2},\phantom{\rule{3.33333pt}{0ex}}{b}_{\mathbf{n}}^{3}={b}_{{\mathbf{n}}^{*}}^{3}$), where ${\mathbf{n}}^{*}=({n}_{1},{n}_{2},-{n}_{3})$.

## 5. Conclusions

- of Froissart doublets in approximants of tensor entries and their elimination (the approach of [36] may prove useful for monitoring the absence of the doublets);
- of the interval in molecular diffusivity, where the approximation is sufficiently accurate;
- of the realistic orders of a Padé approximant, for which the length of such intervals is close to the maximum.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Padé approximants (vertical axes) ${[L/L]}_{{\nu}_{\mathrm{E}}}\left(\nu \right)$ (top) and ${[L/L]}_{{\nu}_{\mathrm{E}}/\nu}\left({\nu}^{-1}\right)$ (bottom) for $6\le L\le 14$ step 2. Horizontal axes: $\nu $ (top), ${\nu}^{-1}$ (bottom). In the bottom panel, we extend $\nu $ to negative values on purpose to highlight that the Maclaurin expansion of ${\nu}_{\mathrm{E}}\left(\right)open="("\; close=")">{\nu}^{-1}$ is an even function of ${\nu}^{-1}$.

**Figure 2.**Maximum slow-time growth rates ${\gamma}_{\alpha}$ (20) (vertical axis) of large-scale magnetic modes generated by the $\alpha $-effect as a function of the molecular diffusivity $\eta $ (horizontal axis), computed using the $\alpha $-effect tensor, Padé-approximated by the algorithm in [35] for a varying tolerance

`tol`. Thin solid line: the dependence determined by computation of ${\gamma}_{\alpha}$ at individual $\eta $ values (red solid circles) by spectral methods (resolution ${64}^{3}$ Fourier harmonics), thick dashed line: the approximate dependence.

**Figure 3.**Approximate dependencies of the maximum slow-time growth rates ${\gamma}_{\alpha}$ (20) (vertical axis) of large-scale magnetic modes generated by the $\alpha $-effect on molecular diffusivity $\eta $ (horizontal axis). Padé approximants $[2L-1/2L]$ of $\alpha $-effect tensor entries are constructed by the algorithm in [5] for the specified L. Resolution of ${64}^{3}$ (

**c**,

**d**,

**g**,

**h**,

**k**,

**l**,

**o**,

**p**,

**s**,

**t**,

**w**,

**x**) and ${512}^{3}$ (

**a**,

**b**,

**e**,

**f**,

**i**,

**j**,

**m**,

**n**,

**q**,

**r**,

**u**,

**v**,

**y**,

**z**) Fourier harmonics, computations with the double (real*8, left panels except (

**a**)) and quadruple (real*16, right panels and (

**a**)) precision. Thin solid line: the dependence revealed by computation of ${\gamma}_{\alpha}$ at individual $\eta $ values (red solid circles) by spectral methods (resolution ${64}^{3}$ Fourier harmonics), wide dashed line: Padé approximants.

**Figure 5.**Minimum eddy diffusivity (53) (vertical axis) for the sample flow (49), (51) computed using Padé approximants of the quantities ${q}_{i}$ (52) of orders [20/20] (

**a**), [18/18] (green line) and regularized [20/20] (black line) (

**c**), and [24/24] (

**d**). Behavior of the [20/20] Padé approximants of ${q}_{i}$ (vertical axis) near the points, where Froissart doublets are located (

**b**). Horizontal axis: magnetic molecular diffusivity $\eta $. Red dots: minimum eddy diffusivity computed by spectral methods (resolution of ${128}^{3}$ Fourier harmonics) individually for the respective $\eta $ values.

**Figure 7.**The ratios $|{\left({\mathfrak{D}}_{mk}^{l}\right)}^{(2n-1)}/{\left({\mathfrak{D}}_{mk}^{l}\right)}^{(2n+1)}{|}^{1/2}$ (vertical axis) versus $n>0$ (horizontal axis) for five independent entries of the eddy diffusivity tensor for the flow (54), (55) (

**a**). The [25/24] Padé approximants for the five entries ${\mathfrak{D}}_{mk}^{l}$ (vertical axis) versus $\eta $ (horizontal axis) (

**b**) and minimum eddy diffusivity (53) (vertical axis) computed using these approximants for varying $\eta $ (horizontal axis) (

**c**). Red dots: minimum eddy diffusivity computed by spectral methods (resolution of ${128}^{3}$ Fourier harmonics) individually for the respective $\eta $ values. Zoomed-in view of plot (

**c**) for small $\eta $ (black line) and a hyperbolic fit (blue line) through the 20 spectral eddy diffusivity values for 20 $\eta $ points in the interval $0.0045\le \eta \le 0.014$ step 0.0005 (

**d**). The dashed line shows the vertical asymptote of the minimum eddy diffusivity at the onset of a small-scale dynamo in the symmetry subspace, where the neutral mode ${\mathbf{s}}_{3}$ resides.

**Table 1.**The first 39 non-zero coefficients of (6) for the decorated hexagonal flow (DHF, prime decomposition where presented). The first 5 significant figures of these coefficients are given in [31]. $\u201ca\u22d8p\u22d9c\u201d$ denotes a natural number containing p decimal digits between a and b.

n | Coefficient ${\mathit{\nu}}_{\mathbf{E}}^{\left(\mathit{n}\right)}$ (Exact Rational Number) |
---|---|

1 | $\frac{3}{{2}^{2}}$ |

3 | $-\frac{3\times 5\times 11\times 1931\times 80491}{{2}^{9}\times {7}^{4}\times {13}^{2}\times {19}^{2}}$ |

5 | $-\frac{3\times {53}^{2}\times 222967\times 1994517983033813651288306079222192539}{{2}^{19}\times {5}^{2}\times {7}^{9}\times {13}^{6}\times {19}^{7}\times {31}^{3}\times {37}^{2}\times {43}^{3}\times {61}^{2}}$ |

7 | $\frac{{3}^{3}\times 23\times 17401\times 11608063\times 570396658307516795186040829874710499\times 595146062519802577066082838776447096016784218965582671080441286999}{{2}^{25}\times {5}^{2}\times {7}^{17}\times {13}^{10}\times {19}^{11}\times {31}^{5}\times {37}^{7}\times {43}^{5}\times {61}^{6}\times {67}^{3}\times {73}^{3}\times {79}^{3}\times {97}^{3}\times {103}^{2}\times {109}^{2}}$ |

9 | $-\frac{9606359879\phantom{\rule{0.222222em}{0ex}}\u22d8188\u22d9\phantom{\rule{0.222222em}{0ex}}5777697637}{{2}^{33}\times {5}^{2}\times {7}^{24}\times {11}^{2}\times {13}^{14}\times {19}^{15}\times {31}^{7}\times {37}^{11}\times {43}^{7}\times {61}^{10}\times {67}^{5}\times {73}^{5}\times {79}^{5}\times {97}^{5}\times {103}^{3}\times {109}^{6}\times {127}^{3}\times {139}^{3}\times {151}^{3}\times {157}^{2}\times {163}^{2}}$ |

11 | $-\frac{7129561983\phantom{\rule{0.222222em}{0ex}}\u22d8324\u22d9\phantom{\rule{0.222222em}{0ex}}7108258721}{8493879641\phantom{\rule{0.222222em}{0ex}}\u22d8326\u22d9\phantom{\rule{0.222222em}{0ex}}4312960000}$ |

⋮ | ⋮ |

39 | $-\frac{1648936106\phantom{\rule{0.222222em}{0ex}}\u22d89785\u22d9\phantom{\rule{0.222222em}{0ex}}2091564067}{3775138782\phantom{\rule{0.222222em}{0ex}}\u22d89788\u22d9\phantom{\rule{0.222222em}{0ex}}0000000000}$ |

**Table 2.**Order parameter L of the Padé approximants $[2L-1/2L]$ (ratios of polynomials in $1/\eta $) constructed by the algorithm in [35] for six independent entries of the symmetrized $\alpha $-effect tensor ${}^{\mathrm{s}}\mathfrak{A}$.

tol | ${{}^{\mathbf{s}}\mathfrak{A}}_{1}^{1}$ | ${{}^{\mathbf{s}}\mathfrak{A}}_{1}^{2}$ | ${{}^{\mathbf{s}}\mathfrak{A}}_{1}^{3}$ | ${{}^{\mathbf{s}}\mathfrak{A}}_{2}^{2}$ | ${{}^{\mathbf{s}}\mathfrak{A}}_{2}^{3}$ | ${{}^{\mathbf{s}}\mathfrak{A}}_{3}^{3}$ |
---|---|---|---|---|---|---|

${10}^{-10}$ | 5 | 5 | 6 | 5 | 5 | 5 |

${10}^{-12}$ | 6 | 6 | 6 | 6 | 6 | 6 |

${10}^{-14}$ | 7 | 7 | 8 | 7 | 7 | 7 |

${10}^{-16}$ | 8 | 8 | 9 | 8 | 8 | 8 |

${10}^{-18}$ | 9 | 8 | 10 | 10 | 9 | 9 |

${10}^{-20}$ | 10 | 10 | 11 | 10 | 10 | 10 |

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**MDPI and ACS Style**

Gama, S.M.A.; Chertovskih, R.; Zheligovsky, V.
Computation of Kinematic and Magnetic *α*-Effect and Eddy Diffusivity Tensors by Padé Approximation. *Fluids* **2019**, *4*, 110.
https://doi.org/10.3390/fluids4020110

**AMA Style**

Gama SMA, Chertovskih R, Zheligovsky V.
Computation of Kinematic and Magnetic *α*-Effect and Eddy Diffusivity Tensors by Padé Approximation. *Fluids*. 2019; 4(2):110.
https://doi.org/10.3390/fluids4020110

**Chicago/Turabian Style**

Gama, Sílvio M.A., Roman Chertovskih, and Vladislav Zheligovsky.
2019. "Computation of Kinematic and Magnetic *α*-Effect and Eddy Diffusivity Tensors by Padé Approximation" *Fluids* 4, no. 2: 110.
https://doi.org/10.3390/fluids4020110