# Turbulence with Magnetic Helicity That Is Absent on Average

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## Abstract

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## 1. Introduction

## 2. Nonhelical Turbulence and the Hosking Integral

#### 2.1. Nonhelical Inverse Cascading and Scaling Relations

#### 2.2. The Loitsyansky and Saffman Integrals in Hydrodynamics

#### 2.3. The Magnetic Saffman Integral: Comparison with the Hosking Integral

#### 2.4. The Effect of Rotation

## 3. Extensions of the Hosking Idea

#### 3.1. Hall Effect

#### 3.2. Ambipolar Diffusion

#### 3.3. Chiral MHD

## 4. Hosking Integral in Shell Models of Chiral MHD

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) magnetic energy spectra, normalized by ${I}_{\mathrm{H}}^{-1/2}{k}_{0}^{-3/2}$; the dashed dotted line shows the envelope $0.028\phantom{\rule{0.166667em}{0ex}}{(k/{k}_{0})}^{3/2}$ under which the spectrum evolves. The times are $c{k}_{1}t$ = 3, 7, 17, and 58. (

**b**) $qp$ diagram showing as red dots the convergence of $p\left(t\right)$ versus $q\left(t\right)$ toward the Hosking attractor $(q,p)=(4/9,\phantom{\rule{0.166667em}{0ex}}10/9)$. The blue symbols denote the Loitsyansky and Saffman attractors, respectively, and the orange symbol denotes the magnetic helicity attractor.

**Figure 2.**Comparison of $\mathrm{Sp}\left(\mathit{B}\right)$ (

**a**,

**c**) and $\mathrm{Sp}\left(h\right)$ (

**b**,

**d**) for $\alpha =2$ (

**a**,

**b**) and $\alpha =4$ (

**c**,

**d**). The dashed-dotted lines indicate ${k}^{3/2}$ scaling in (

**c**) and ${k}^{2}$ scaling otherwise.

**Figure 3.**Compensated spectra for $\alpha =2$ (

**a**,

**c**,

**e**) and $\alpha =4$ (

**b**,

**d**,

**f**). From (

**a**,

**b**), the horizontal dashed-dotted lines indicate that ${I}_{\mathrm{SM}}\approx 0.23$ and 0.09, respectively, and from (

**c**,

**d**) they indicate that ${I}_{\mathrm{H}}\approx 2\times {10}^{-3}$ and $5\times {10}^{-4}$, respectively. The asymptotic values estimated from (

**e**,

**f**) are discussed in the text and in Table 1.

**Figure 4.**$\mathrm{Sp}\left(\mathit{B}\right)$ (

**a**,

**c**,

**e**,

**g**) and $\mathrm{Sp}\left(h\right)$ (

**b**,

**d**,

**f**,

**h**) for $\Omega /{c}_{\mathrm{s}}{k}_{0}$ from ${10}^{-3}$ (

**a**,

**b**: $\mathrm{Co}=1.2$) to $\Omega /{c}_{\mathrm{s}}{k}_{0}=1$ (

**g**,

**h**: $\mathrm{Co}={10}^{4}$). The times are 220 (black), 1000 (blue), 4600 (orange), and 22,000 (red).

**Figure 5.**$\mathrm{Sp}\left(\mathit{B}\right)$ (

**a**–

**c**) and $\mathrm{Sp}\left(h\right)$ (

**d**–

**f**) for Hall dynamics with $\alpha =2$ (

**a**,

**d**) and $\alpha =4$ (

**b**,

**e**), and for ambipolar diffusion with $\alpha =4$ (

**c**,

**f**). Note the presence of inverse cascading for $\alpha =4$ in panels (

**b**,

**c**), although $\mathrm{Sp}\left(h\right)$ changes at $k/{k}_{0}\ll 1$ in all cases. The straight lines indicate ${k}^{3/2}$ scaling in (

**b**) and ${k}^{2}$ scaling otherwise.

**Figure 6.**(

**a**) Magnetic energy (solid lines) and magnetic helicity spectra (dotted lines), and (

**b**) magnetic helicity variance spectra for a chiral MHD run with balanced chirality and an initial ${k}^{4}$ spectrum for the magnetic field. In (

**a**), positive (negative) magnetic helicities are indicated by small red (blue) dots. The four large dots denote the positions of ${\xi}_{\mathrm{M}}^{-1}$. Their colors are the same as those of the solid lines in (

**b**) and correspond to the times 1500, 5000, 15,000, and 50,000.

**Figure 8.**Evolution of ${E}_{\mathrm{M}}(k,t)$ from shell models of (

**a**) type I and (

**b**) type II. The times are 10 (red), 1 (orange), 0.1 (green), 0.01 (blue), and earlier times are denoted by black lines of different line types. Note the presence of inverse cascading in both cases.

**Table 1.**Summary of nondimensional prefactors in the relations for ${\xi}_{\mathrm{M}}\left(t\right)$, ${\mathcal{E}}_{\mathrm{M}}\left(t\right)$, and ${E}_{\mathrm{M}}(k,t)$. The numbers in parentheses indicate that the slope $\beta $ is incompatible with the value of $\alpha $.

$\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{C}}_{\mathbf{SM}}^{\left(\mathit{\xi}\right)}$ | ${\mathit{C}}_{\mathbf{H}}^{\left(\mathit{\xi}\right)}$ | ${\mathit{C}}_{\mathbf{SM}}^{\left(\mathcal{E}\right)}$ | ${\mathit{C}}_{\mathbf{H}}^{\left(\mathcal{E}\right)}$ | ${\mathit{C}}_{\mathbf{SM}}^{\left(\mathit{E}\right)}$ | ${\mathit{C}}_{\mathbf{H}}^{\left(\mathit{E}\right)}$ |
---|---|---|---|---|---|---|---|

2 | 2 | 0.16 | 0.15 | 4.2 | 3.8 | 0.025 | (0.05) |

4 | 3/2 | 0.15 | 0.13 | 4.0 | 3.5 | (0.02) | 0.037 |

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**MDPI and ACS Style**

Brandenburg, A.; Larsson, G. Turbulence with Magnetic Helicity That Is Absent on Average. *Atmosphere* **2023**, *14*, 932.
https://doi.org/10.3390/atmos14060932

**AMA Style**

Brandenburg A, Larsson G. Turbulence with Magnetic Helicity That Is Absent on Average. *Atmosphere*. 2023; 14(6):932.
https://doi.org/10.3390/atmos14060932

**Chicago/Turabian Style**

Brandenburg, Axel, and Gustav Larsson. 2023. "Turbulence with Magnetic Helicity That Is Absent on Average" *Atmosphere* 14, no. 6: 932.
https://doi.org/10.3390/atmos14060932