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

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 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

- Bruno, R.; Carbone, V. The Solar Wind as a Turbulence Laboratory. LRSP
**2013**, 10. [Google Scholar] [CrossRef] [Green Version] - Kolmogorov, A.N. Dissipation of Energy in Locally Isotropic Turbulence. Akad. Nauk SSSR Dokl.
**1941**, 32, 16. [Google Scholar] - Hellinger, P.; Verdini, A.; Landi, S.; Franci, L.; Matteini, L. von Kármán-Howarth Equation for Hall Magnetohydrodynamics: Hybrid Simulations. Astrophys. J. Lett.
**2018**, 857, L19. [Google Scholar] [CrossRef] [Green Version] - Papini, E.; Franci, L.; Landi, S.; Verdini, A.; Matteini, L.; Hellinger, P. Can Hall Magnetohydrodynamics Explain Plasma Turbulence at Sub-ion Scales? Astrophys. J.
**2019**, 870, 52. [Google Scholar] [CrossRef] [Green Version] - Bandyopadhyay, R.; Sorriso-Valvo, L.; Chasapis, A.; Hellinger, P.; Matthaeus, W.H.; Verdini, A.; Landi, S.; Franci, L.; Matteini, L.; Giles, B.L.; et al. In-situ observation of Hall Magnetohydrodynamic Cascade in Space Plasma. Phys. Rev. Lett.
**2020**, 124. [Google Scholar] [CrossRef] - Alexandrova, O.; Carbone, V.; Veltri, P.; Sorriso-Valvo, L. Small-Scale Energy Cascade of the Solar Wind Turbulence. Astrophys. J.
**2008**, 674, 1153–1157. [Google Scholar] [CrossRef] [Green Version] - Howes, G.G.; Cowley, S.C.; Dorland, W.; Hammett, G.W.; Quataert, E.; Schekochihin, A.A. A model of turbulence in magnetized plasmas: Implications for the dissipation range in the solar wind. J. Geophys. Res. Space Phys.
**2008**, 113, A05103. [Google Scholar] [CrossRef] [Green Version] - Schekochihin, A.A.; Cowley, S.C.; Dorland, W.; Hammett, G.W.; Howes, G.G.; Quataert, E.; Tatsuno, T. Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades in Magnetized Weakly Collisional Plasmas. Astrophys. J. Suppl.
**2009**, 182, 310–377. [Google Scholar] [CrossRef] [Green Version] - Sahraoui, F.; Goldstein, M.L.; Belmont, G.; Canu, P.; Rezeau, L. Three Dimensional Anisotropic k Spectra of Turbulence at Subproton Scales in the Solar Wind. Phys. Rev. Lett.
**2010**, 105, 131101. [Google Scholar] [CrossRef] - Boldyrev, S.; Perez, J.C. Spectrum of Kinetic-Alfvén Turbulence. Astrophys. J. Lett.
**2012**, 758, L44. [Google Scholar] [CrossRef] [Green Version] - Wan, M.; Matthaeus, W.H.; Karimabadi, H.; Roytershteyn, V.; Shay, M.; Wu, P.; Daughton, W.; Loring, B.; Chapman, S.C. Intermittent Dissipation at Kinetic Scales in Collisionless Plasma Turbulence. Phys. Rev. Lett.
**2012**, 109, 195001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wu, P.; Perri, S.; Osman, K.; Wan, M.; Matthaeus, W.H.; Shay, M.A.; Goldstein, M.L.; Karimabadi, H.; Chapman, S. Intermittent Heating in Solar Wind and Kinetic Simulations. Astrophys. J. Lett.
**2013**, 763, L30. [Google Scholar] [CrossRef] [Green Version] - Franci, L.; Verdini, A.; Matteini, L.; Landi, S.; Hellinger, P. Solar Wind Turbulence from MHD to Sub-ion Scales: High-resolution Hybrid Simulations. Astrophys. J. Lett.
**2015**, 804, L39. [Google Scholar] [CrossRef] - Franci, L.; Landi, S.; Matteini, L.; Verdini, A.; Hellinger, P. High-resolution Hybrid Simulations of Kinetic Plasma Turbulence at Proton Scales. Astrophys. J.
**2015**, 812, 21. [Google Scholar] [CrossRef] [Green Version] - Sulem, P.L.; Passot, T.; Laveder, D.; Borgogno, D. Influence of the Nonlinearity Parameter on the Solar Wind Sub-ion Magnetic Energy Spectrum: FLR-Landau Fluid Simulations. Astrophys. J.
**2016**, 818, 66. [Google Scholar] [CrossRef] [Green Version] - Franci, L.; Cerri, S.S.; Califano, F.; Landi, S.; Papini, E.; Verdini, A.; Matteini, L.; Jenko, F.; Hellinger, P. Magnetic Reconnection as a Driver for a Sub-ion-scale Cascade in Plasma Turbulence. Astrophys. J. Lett.
**2017**, 850, L16. [Google Scholar] [CrossRef] [Green Version] - Yang, Y.; Matthaeus, W.H.; Parashar, T.N.; Haggerty, C.C.; Roytershteyn, V.; Daughton, W.; Wan, M.; Shi, Y.; Chen, S. Energy transfer, pressure tensor, and heating of kinetic plasma. Phys. Plasmas
**2017**, 24, 072306. [Google Scholar] [CrossRef] [Green Version] - Ghosh, S.; Siregar, E.; Roberts, D.A.; Goldstein, M.L. Simulation of high-frequency solar wind power spectra using Hall magnetohydrodynamics. J. Geophys. Res.
**1996**, 101, 2493–2504. [Google Scholar] [CrossRef] - Biskamp, D.; Schwarz, E.; Zeiler, A.; Celani, A.; Drake, J.F. Electron magnetohydrodynamic turbulence. Phys. Plasmas
**1999**, 6, 751–758. [Google Scholar] [CrossRef] [Green Version] - Galtier, S.; Buchlin, E. Multiscale Hall-Magnetohydrodynamic Turbulence in the Solar Wind. Astrophys. J.
**2007**, 656, 560–566. [Google Scholar] [CrossRef] - Shaikh, D.; Shukla, P.K. 3D Simulations of Fluctuation Spectra in the Hall-MHD Plasma. Phys. Rev. Lett.
**2009**, 102, 045004. [Google Scholar] [CrossRef] [PubMed] - Roberts, O.W.; Narita, Y.; Escoubet, C.P. Direct Measurement of Anisotropic and Asymmetric Wave Vector Spectrum in Ion-scale Solar Wind Turbulence. Astrophys. J. Lett.
**2017**, 851, L11. [Google Scholar] [CrossRef] - Bandyopadhyay, R.; Chasapis, A.; Chhiber, R.; Parashar, T.N.; Maruca, B.A.; Matthaeus, W.H.; Schwartz, S.J.; Eriksson, S.; Le Contel, O.; Breuillard, H.; et al. Solar Wind Turbulence Studies Using MMS Fast Plasma Investigation Data. Astrophys. J.
**2018**, 866, 81. [Google Scholar] [CrossRef] [Green Version] - Pitňa, A.; Šafránková, J.; Němeček, Z.; Franci, L.; Pi, G. A Novel Method for Estimating the Intrinsic Magnetic Field Spectrum of Kinetic-Range Turbulence. Atmosphere
**2021**, 12, 1547. [Google Scholar] [CrossRef] - Alexandrova, O.; Saur, J.; Lacombe, C.; Mangeney, A.; Mitchell, J.; Schwartz, S.J.; Robert, P. Universality of Solar-Wind Turbulent Spectrum from MHD to Electron Scales. Phys. Rev. Lett.
**2009**, 103, 165003. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, C.H.K.; Boldyrev, S. Nature of Kinetic Scale Turbulence in the Earth’s Magnetosheath. Astrophys. J.
**2017**, 842, 122. [Google Scholar] [CrossRef] - Howes, G.G.; Tenbarge, J.M.; Dorland, W.; Quataert, E.; Schekochihin, A.A.; Numata, R.; Tatsuno, T. Gyrokinetic Simulations of Solar Wind Turbulence from Ion to Electron Scales. Phys. Rev. Lett.
**2011**, 107, 035004. [Google Scholar] [CrossRef] [Green Version] - Franci, L.; Landi, S.; Matteini, L.; Verdini, A.; Hellinger, P. Plasma Beta Dependence of the Ion-scale Spectral Break of Solar Wind Turbulence: High-resolution 2D Hybrid Simulations. Astrophys. J.
**2016**, 833, 91. [Google Scholar] [CrossRef] [Green Version] - Cerri, S.S.; Franci, L.; Califano, F.; Landi, S.; Hellinger, P. Plasma turbulence at ion scales: A comparison between particle in cell and Eulerian hybrid-kinetic approaches. J. Plasma Phys.
**2017**, 83, 705830202. [Google Scholar] [CrossRef] [Green Version] - González, C.A.; Parashar, T.N.; Gomez, D.; Matthaeus, W.H.; Dmitruk, P. Turbulent electromagnetic fields at sub-proton scales: Two-fluid and full-kinetic plasma simulations. Phys. Plasmas
**2019**, 26, 012306. [Google Scholar] [CrossRef] [Green Version] - Loureiro, N.F.; Boldyrev, S. Collisionless Reconnection in Magnetohydrodynamic and Kinetic Turbulence. Astrophys. J.
**2017**, 850, 182. [Google Scholar] [CrossRef] [Green Version] - Mallet, A.; Schekochihin, A.A.; Chandran, B.D.G. Disruption of Alfvénic turbulence by magnetic reconnection in a collisionless plasma. J. Plasma Phys.
**2017**, 83, 905830609. [Google Scholar] [CrossRef] [Green Version] - Parashar, T.N.; Matthaeus, W.H. Propinquity of Current and Vortex Structures: Effects on Collisionless Plasma Heating. Astrophys. J.
**2016**, 832, 57. [Google Scholar] [CrossRef] [Green Version] - Bandyopadhyay, R.; Matthaeus, W.H.; Parashar, T.N.; Yang, Y.; Chasapis, A.; Giles, B.L.; Gershman, D.J.; Pollock, C.J.; Russell, C.T.; Strangeway, R.J.; et al. Statistics of Kinetic Dissipation in the Earth’s Magnetosheath: MMS Observations. Phys. Rev. Lett.
**2020**, 124, 255101. [Google Scholar] [CrossRef] [PubMed] - Hellinger, P.; Papini, E.; Verdini, A.; Landi, S.; Franci, L.; Matteini, L.; Montagud-Camps, V. Spectral transfer and Kármán-Howarth-Monin equations for compressible Hall magnetohydrodynamics. Astrophys. J.
**2021**, 917, 101. [Google Scholar] [CrossRef] - Landi, S.; Del Zanna, L.; Papini, E.; Pucci, F.; Velli, M. Resistive Magnetohydrodynamics Simulations of the Ideal Tearing Mode. Astrophys. J.
**2015**, 806, 131. [Google Scholar] [CrossRef] [Green Version] - Papini, E.; Landi, S.; Zanna, L.D. Fast magnetic reconnection: The ideal tearing instability in classic, Hall, and relativistic plasmas. J. Phys. Conf. Series
**2018**, 1031, 012020. [Google Scholar] [CrossRef] - Papini, E.; Landi, S.; Del Zanna, L. Fast Magnetic Reconnection: Secondary Tearing Instability and Role of the Hall Term. Astrophys. J.
**2019**, 885, 56. [Google Scholar] [CrossRef] - Papini, E.; Franci, L.; Landi, S.; Hellinger, P.; Verdini, A.; Matteini, L. Statistics of Magnetic Reconnection and Turbulence in Hall-MHD and Hybrid-PIC Simulations. Available online: https://www.sif.it/riviste/sif/ncc/econtents/2019/042/01/article/22 (accessed on 2 December 2017).
- Wray, A.A. Minimal Sstorage Time Aadvancement Schemes for Spectral Methods. Available online: https://www.researchgate.net/publication/246830945_Minimal_storage_time-advancement_schemes_for_spectral_methods (accessed on 2 December 2017).
- Orszag, S.A. On the Elimination of Aliasing in Finite-Difference Schemes by Filtering High-Wavenumber Components. J. Atmos. Sci.
**1971**, 28, 1074. [Google Scholar] [CrossRef] [Green Version] - Ghosh, S.; Hossain, M.; Matthaeus, W.H. The application of spectral methods in simulating compressible fluid and magnetofluid turbulence. Comp. Phys. Commun.
**1993**, 74, 18–40. [Google Scholar] [CrossRef] - Schmidt, W.; Grete, P. Kinetic and internal energy transfer in implicit large-eddy simulations of forced compressible turbulence. Phys. Rev. E
**2019**, 100. [Google Scholar] [CrossRef] [Green Version] - Praturi, D.S.; Girimaji, S.S. Effect of pressure-dilatation on energy spectrum evolution in compressible turbulence. Phys. Fluids
**2019**, 31. [Google Scholar] [CrossRef] - Kida, S.; Orszag, S.A. Energy and spectral dynamics in forced compressible turbulence. J. Sci. Comput.
**1990**, 5, 85–125. [Google Scholar] [CrossRef] - Wan, M.; Matthaeus, W.H.; Roytershteyn, V.; Parashar, T.N.; Wu, P.; Karimabadi, H. Intermittency, coherent structures and dissipation in plasma turbulence. Phys. Plasmas
**2016**, 23, 042307. [Google Scholar] [CrossRef] - Yang, Y.; Matthaeus, W.H.; Parashar, T.N.; Wu, P.; Wan, M.; Shi, Y.; Chen, S.; Roytershteyn, V.; Daughton, W. Energy transfer channels and turbulence cascade in Vlasov-Maxwell turbulence. Phys. Rev. E
**2017**, 95, 061201. [Google Scholar] [CrossRef] [Green Version] - Papini, E.; Cicone, A.; Piersanti, M.; Franci, L.; Hellinger, P.; Landi, S.; Verdini, A. Multidimensional Iterative Filtering: A new approach for investigating plasma turbulence in numerical simulations. J. Plasma Phys.
**2020**, 86, 871860501. [Google Scholar] [CrossRef] - Uritsky, V.M.; Pouquet, A.; Rosenberg, D.; Mininni, P.D.; Donovan, E.F. Structures in magnetohydrodynamic turbulence: Detection and scaling. Phys. Rev. E
**2010**, 82, 056326. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Miura, H.; Araki, K. Structure transitions induced by the Hall term in homogeneous and isotropic magnetohydrodynamic turbulence. Phys. Plasmas
**2014**, 21, 072313. [Google Scholar] [CrossRef] [Green Version] - Agudelo Rueda, J.A.; Verscharen, D.; Wicks, R.T.; Owen, C.J.; Nicolaou, G.; Walsh, A.P.; Zouganelis, I.; Germaschewski, K.; Vargas Domínguez, S. Three-dimensional magnetic reconnection in particle-in-cell simulations of anisotropic plasma turbulence. J. Plasma Phys.
**2021**, 87, 905870228. [Google Scholar] [CrossRef] - Goldreich, P.; Sridhar, S. Toward a Theory of Interstellar Turbulence. II. Strong Alfvenic Turbulence. Astrophys. J.
**1995**, 438, 763. [Google Scholar] [CrossRef] - Parashar, T.N.; Gary, S.P. Dissipation of Kinetic Alfvénic Turbulence as a Function of Ion and Electron Temperature Ratios. Astrophys. J.
**2019**, 882, 29. [Google Scholar] [CrossRef] - Gary, S.P.; Bandyopadhyay, R.; Qudsi, R.A.; Matthaeus, W.H.; Maruca, B.A.; Parashar, T.N.; Roytershteyn, V. Particle-in-cell Simulations of Decaying Plasma Turbulence: Linear Instabilities versus Nonlinear Processes in 3D and 2.5D Approximations. Astrophys. J.
**2020**, 901, 160. [Google Scholar] [CrossRef] - Oughton, S.; Matthaeus, W.H.; Dmitruk, P. Reduced MHD in Astrophysical Applications: Two-dimensional or Three-dimensional? Astrophys. J.
**2017**, 839, 2. [Google Scholar] [CrossRef] [Green Version] - Bandyopadhyay, R.; Oughton, S.; Wan, M.; Matthaeus, W.H.; Chhiber, R.; Parashar, T.N. Finite Dissipation in Anisotropic Magnetohydrodynamic Turbulence. Phys. Rev. X
**2018**, 8, 041052. [Google Scholar] [CrossRef] [Green Version] - Karimabadi, H.; Roytershteyn, V.; Wan, M.; Matthaeus, W.H.; Daughton, W.; Wu, P.; Shay, M.; Loring, B.; Borovsky, J.; Leonardis, E.; et al. Coherent structures, intermittent turbulence, and dissipation in high-temperature plasmas. Phys. Plasmas
**2013**, 20, 012303. [Google Scholar] [CrossRef] - Franci, L.; Landi, S.; Verdini, A.; Matteini, L.; Hellinger, P. Solar Wind Turbulent Cascade from MHD to Sub-ion Scales: Large-size 3D Hybrid Particle-in-cell Simulations. Astrophys. J.
**2018**, 853, 26. [Google Scholar] [CrossRef] - Franci, L.; Hellinger, P.; Guarrasi, M.; Chen, C.H.K.; Papini, E.; Verdini, A.; Matteini, L.; Landi, S. Three-dimensional simulations of solar wind turbulence with the hybrid code CAMELIA. JPhCS
**2018**, 1031, 012002. [Google Scholar] [CrossRef] - Shay, M.A.; Drake, J.F.; Rogers, B.N.; Denton, R.E. Alfvénic Collisionless Magnetic Reconnection and the Hall Term. J. Geophys. Res.
**2001**, 106, 3759. [Google Scholar] [CrossRef] - Bruno, R.; Trenchi, L.; Telloni, D. Spectral Slope Variation at Proton Scales from Fast to Slow Solar Wind. Astrophys. J. Lett.
**2014**, 793, L15. [Google Scholar] [CrossRef] [Green Version] - Quijia, P.; Fraternale, F.; Stawarz, J.E.; Vásconez, C.L.; Perri, S.; Marino, R.; Yordanova, E.; Sorriso-Valvo, L. Comparing turbulence in a Kelvin-Helmholtz instability region across the terrestrial magnetopause. MNRAS
**2021**, 503, 4815–4827. [Google Scholar] [CrossRef]

**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 | ∼$0.23$ | $7.2\times {10}^{-4}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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