# Magnetohydrodynamic Turbulence Mediated by Reconnection

## Abstract

Magnetic field fluctuations in MHD turbulence can be viewed as current sheets that are progressively more anisotropic at smaller scales. As suggested by Loureiro & Boldyrev 2017 and Mallet et al. 2017, below a certain critical thickness $$\lambda_c$$ such current sheets become tearing-unstable. We propose that the tearing instability changes the effective alignment of the magnetic field lines in such a way as to balance the eddy turnover rate at all scales smaller than $$\lambda_c$$. As a result, turbulent fluctuations become progressively less anisotropic at smaller scales, with the alignment angle increasing as $$\theta \sim (\lambda/\lambda_*)^{-4/5+\beta}$$, where $$\lambda_*\sim L_0 S_0^{-3/4}$$ is the resistive dissipation scale. Here $$L_0$$ is the outer scale of the turbulence, $$S_0$$ is the corresponding Lundquist number, and {$$0\leq \beta <4/5$$} is a parameter. The resulting Fourier energy spectrum is $$E(k_\perp)\propto k_\perp^{-11/5+2\beta/3}$$, where $$k_\perp$$ is the wavenumber normal to the local mean magnetic field, and the critical scale is $$\lambda_c\sim S_L^{-(4-5\beta)/(7-{20\beta/3})}$$. The simplest model corresponds to $$\beta=0$$, in which case the predicted scaling formally agrees with one of the solutions obtained in (Mallet et al. 2017) from a discrete hierarchical model of abruptly collapsing current sheets, an approach different and complementary to ours. We also show that the reconnection-mediated interval is non-universal with respect to the dissipation mechanism. Hyper-resistivity of the form $${\tilde \eta}k^{2+2s}$$ leads (in the simplest case of $$\beta=0$$) to the different transition scale $$\lambda_c\sim L_0{\tilde S}_0^{-4/(7+9s)}$$ and the energy spectrum $$E(k_\perp)\propto k_\perp^{-(11+9s)/(5+3s)}$$, where $${\tilde S}_0$$ is the corresponding hyper-resistive Lundquist number.

- Authors:

- Publication Date:

- DOE Contract Number:
- SC0016215

- Research Org.:
- Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Plasma Science and Fusion Center

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES)

- Subject:
- 79 ASTRONOMY AND ASTROPHYSICS

- OSTI Identifier:
- 1881002

- DOI:
- https://doi.org/10.7910/DVN/0JQHWY

### Citation Formats

```
Boldyrev, Stanislav, and Loureiro, Nuno F.
```*Magnetohydrodynamic Turbulence Mediated by Reconnection*. United States: N. p., 2019.
Web. doi:10.7910/DVN/0JQHWY.

```
Boldyrev, Stanislav, & Loureiro, Nuno F.
```*Magnetohydrodynamic Turbulence Mediated by Reconnection*. United States. doi:https://doi.org/10.7910/DVN/0JQHWY

```
Boldyrev, Stanislav, and Loureiro, Nuno F. 2019.
"Magnetohydrodynamic Turbulence Mediated by Reconnection". United States. doi:https://doi.org/10.7910/DVN/0JQHWY. https://www.osti.gov/servlets/purl/1881002. Pub date:Thu Jan 10 00:00:00 EST 2019
```

```
@article{osti_1881002,
```

title = {Magnetohydrodynamic Turbulence Mediated by Reconnection},

author = {Boldyrev, Stanislav and Loureiro, Nuno F.},

abstractNote = {Magnetic field fluctuations in MHD turbulence can be viewed as current sheets that are progressively more anisotropic at smaller scales. As suggested by Loureiro & Boldyrev 2017 and Mallet et al. 2017, below a certain critical thickness $\lambda_c$ such current sheets become tearing-unstable. We propose that the tearing instability changes the effective alignment of the magnetic field lines in such a way as to balance the eddy turnover rate at all scales smaller than $\lambda_c$. As a result, turbulent fluctuations become progressively less anisotropic at smaller scales, with the alignment angle increasing as $\theta \sim (\lambda/\lambda_*)^{-4/5+\beta}$, where $\lambda_*\sim L_0 S_0^{-3/4}$ is the resistive dissipation scale. Here $L_0$ is the outer scale of the turbulence, $S_0$ is the corresponding Lundquist number, and {$0\leq \beta <4/5$} is a parameter. The resulting Fourier energy spectrum is $E(k_\perp)\propto k_\perp^{-11/5+2\beta/3}$, where $k_\perp$ is the wavenumber normal to the local mean magnetic field, and the critical scale is $\lambda_c\sim S_L^{-(4-5\beta)/(7-{20\beta/3})}$. The simplest model corresponds to $\beta=0$, in which case the predicted scaling formally agrees with one of the solutions obtained in (Mallet et al. 2017) from a discrete hierarchical model of abruptly collapsing current sheets, an approach different and complementary to ours. We also show that the reconnection-mediated interval is non-universal with respect to the dissipation mechanism. Hyper-resistivity of the form ${\tilde \eta}k^{2+2s}$ leads (in the simplest case of $\beta=0$) to the different transition scale $\lambda_c\sim L_0{\tilde S}_0^{-4/(7+9s)}$ and the energy spectrum $E(k_\perp)\propto k_\perp^{-(11+9s)/(5+3s)}$, where ${\tilde S}_0$ is the corresponding hyper-resistive Lundquist number.},

doi = {10.7910/DVN/0JQHWY},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2019},

month = {1}

}

Works referencing / citing this record:

##
Magnetohydrodynamic Turbulence Mediated by Reconnection

journal, July 2017

- Boldyrev, Stanislav; Loureiro, Nuno F.
- The Astrophysical Journal, Vol. 844, Issue 2