# Properties of Hall-MHD Turbulence at Sub-Ion Scales: Spectral Transfer Analysis

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Hall-MHD Simulations of Plasma Turbulence: Numerical Setup

#### 2.2. Spectral Transfer Equations

## 3. Results

#### 3.1. General Evolution

#### 3.2. Spectral Properties and Cross-Scales Energy Transfer

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Evolution of the root-mean-square of the current density for the runs listed in Table 1 in code units (

**a**), and in units of the nonlinear time and of the maximum of ${J}_{\mathrm{rms}}$ (

**b**).

**Figure 2.**Colored contours of the amplitude of the total magnetic field fluctuations (

**a**–

**c**) and of the current density (

**d**–

**f**) at the time t = 80 ${\mathsf{\Omega}}_{i}^{-1}$, 45 ${\mathsf{\Omega}}_{i}^{-1}$, and 44 ${\mathsf{\Omega}}_{i}^{-1}$ for Run d16, d08a, and d08b, respectively. To better highlight the differences at small scales between the current structures in the three simulations, panels (

**g**–

**i**) show a zoom of the amplitude of the current density in a particular region.

**Figure 3.**Magnetic and velocity spectra (

**top panels**) and normalized spectral energy transfer (

**bottom panels**) (see Equation (6)) averaged in the interval from 70 to 90 ${\mathsf{\Omega}}_{i}^{-1}$ (∼$0.31{\tau}_{nl}$) for Run d16 (

**left**) and in the interval from 40 to 50 ${\mathsf{\Omega}}_{i}^{-1}$ (∼$0.31{\tau}_{nl}$) for Run d08a (

**right**). Each energy transfer rate term (reported in adimensional units $\left[Q\right]$) is normalized with respect to the total heating rate ${Q}_{0}$ (see Table 1). Vertical dash-dotted and dashed gray lines denote the location of the injection and of the $2/3$-filter cutoff wavenumber, respectively. The dark gray area denotes the range in which the Hall term ${S}_{\mathrm{H},k}$ is bigger than the other spectral energy transfer terms.

**Figure 4.**The same as Figure 3 but for Run d08b, averaged in the interval from 39 to 48 ${\mathsf{\Omega}}_{i}^{-1}$ (∼$0.34{\tau}_{nl}$).

**Table 1.**Physical and numerical parameters of the simulations used in this work (in code adimensional units). Box size ${L}_{0}$, injection scale ${L}_{\mathrm{inj}}$, rms amplitude of the initial fluctuations (${b}_{\mathrm{rms}}={u}_{\mathrm{rms}}$), nonlinear time at the injection scale ${\tau}_{nl}$ (in units of ${\mathsf{\Omega}}_{i}^{-1}$), viscosity $\mu $ and resistivity $\eta $, (magnetic) Reynolds number $\left(\mathrm{Rm}\right)\mathrm{Re}={L}_{\mathrm{inj}}{u}_{\mathrm{rms}}{\rho}_{0}/\mu $ (since $\mu =\eta $ and ${\rho}_{0}=1$), and initial Mach number ${M}_{0}={u}_{\mathrm{rms}}/{c}_{s}$. All simulations have the same grid size (${L}_{\left[xy\right]}=32{d}_{i}$) and resolution ($\Delta \left[xy\right]={d}_{i}/32$). The nonlinear time is defined as ${\tau}_{\mathrm{nl}}={L}_{\mathrm{inj}}/{u}_{\mathrm{rms}}$. ${Q}_{0}$ is the total heating rate (in units of ${\rho}_{0}{c}_{A}^{2}{\mathsf{\Omega}}_{i}$) measured in the time interval of fully developed turbulence considered (see Section 3.2).

Run | ${\mathit{L}}_{0}$ | ${\mathit{L}}_{\mathbf{inj}}$ | ${\mathit{u}}_{\mathbf{rms}}\left({\mathit{b}}_{\mathbf{rms}}\right)$ | ${\mathit{\tau}}_{\mathbf{nl}}$ | $\mathit{\mu},\mathit{\eta}$ | Re (Rm) | ${\mathit{M}}_{0}$ | ${\mathit{Q}}_{0}$ |
---|---|---|---|---|---|---|---|---|

d16 | 32 ${d}_{i}$ | 16 ${d}_{i}$ | 0.25 | 64 | $5\times {10}^{-4}$ | 8000 | ∼$0.19$ | $1.8\times {10}^{-4}$ |

d08a | 32 ${d}_{i}$ | 8 ${d}_{i}$ | 0.25 | 32 | $5\times {10}^{-4}$ | 4000 | ∼$0.19$ | $4.3\times {10}^{-4}$ |

d08b | 32 ${d}_{i}$ | 8 ${d}_{i}$ | 0.30 | $26.6$ | $2.5\times {10}^{-4}$ | 9600 | ∼$0.23$ | $7.2\times {10}^{-4}$ |

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**MDPI and ACS Style**

Papini, E.; Hellinger, P.; Verdini, A.; Landi, S.; Franci, L.; Montagud-Camps, V.; Matteini, L. Properties of Hall-MHD Turbulence at Sub-Ion Scales: Spectral Transfer Analysis. *Atmosphere* **2021**, *12*, 1632.
https://doi.org/10.3390/atmos12121632

**AMA Style**

Papini E, Hellinger P, Verdini A, Landi S, Franci L, Montagud-Camps V, Matteini L. Properties of Hall-MHD Turbulence at Sub-Ion Scales: Spectral Transfer Analysis. *Atmosphere*. 2021; 12(12):1632.
https://doi.org/10.3390/atmos12121632

**Chicago/Turabian Style**

Papini, Emanuele, Petr Hellinger, Andrea Verdini, Simone Landi, Luca Franci, Victor Montagud-Camps, and Lorenzo Matteini. 2021. "Properties of Hall-MHD Turbulence at Sub-Ion Scales: Spectral Transfer Analysis" *Atmosphere* 12, no. 12: 1632.
https://doi.org/10.3390/atmos12121632