doi: 10.3934/dcdsb.2021285
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Phenomenologies of intermittent Hall MHD turbulence

851 S Morgan St, Chicago, IL 60607, USA

*Corresponding author: Mimi Dai

Received  April 2021 Early access December 2021

Fund Project: The author is supported by NSF grants DMS–1815069 and DMS–2009422. She is grateful to the Institute for Advanced Study for its hospitality

We introduce the concept of intermittency dimension for the magnetohydrodynamics (MHD) to quantify the intermittency effect. With dependence on the intermittency dimension, we derive phenomenological laws for intermittent MHD turbulence with and without the Hall effect. In particular, scaling laws of dissipation wavenumber, energy spectra and structure functions are predicted. Moreover, we are able to provide estimates for energy spectra and structure functions which are consistent with the predicted scalings.

Citation: Mimi Dai. Phenomenologies of intermittent Hall MHD turbulence. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021285
References:
[1]

M. AcheritogarayP. DegondA. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magnetohydrodynamic system, Kinet. Relat. Models, 4 (2011), 901-918.  doi: 10.3934/krm.2011.4.901.

[2]

F. AnselmetY. GagneE. J. Hopfinger and R. A. Antonia, High-order velocity structure functions in turbulent shear flow, J. Fluid Mech., 140 (1984), 63-89.  doi: 10.1017/S0022112084000513.

[3]

R. Beekie, T. Buckmaster and V. Vicol, Weak solutions of ideal MHD which do not conserve magnetic helicity, Annals of PDE, 6 (2020), Article number: 1. doi: 10.1007/s40818-020-0076-1.

[4]

A. Beresnyak, The spectral slop and Kolmogorov constant of MHD turbulence, Phys. Rev. Lett., 106 (2011), 075001. 

[5]

A. Beresnyak, Basic properties of magnetohydrodynamic turbulence in the inertial range, Mon. Not. R. Astron. Soc., 422 (2012), 3495. 

[6]

A. Beresnyak, Spectra of strong magnetohydrodynamic turbulence from high-resolution simulations, Astrophys. J., 784 (2014), L20.  doi: 10.1088/2041-8205/784/2/L20.

[7]

A. Beresnyak and A. Lazarian, Polarization intermittency and its influence on MHD turbulence, Astrophys. J., 640 (2006), L175.  doi: 10.1086/503708.

[8]

A. BeresnyakA. Lazarian and F. Cattaneo, Scaling laws and diffuse locality of balanced and imbalanced MHD turbulence, Astrophys. J., 722 (2010), L110. 

[9]

A. Bhattacharjee, Impulsive magnetic reconnection in the Earth's magnetotail and the solar corona, Ann. Rev. Astron. Astrophys., 42 (2004), 365-384.  doi: 10.1146/annurev.astro.42.053102.134039.

[10] D. Biskamp, Magnetic Reconnection in Plasmas, Cambridge University Press, 2000. 
[11] D. Biskamp, Magnetohydrodynamic Turbulence, Cambridge University Press, 2003.  doi: 10.1017/CBO9780511535222.
[12]

S. Boldyrev, On the spectrum of magnetohydrodynamic turbulence, Astrophys. J., 626 (2005), L37. 

[13]

S. Boldyrev, Spectrum of magnetohydrodynamic turbulence, Phys. Rev. Lett., 96 (2006), 115002. 

[14]

S. BoldyrevJ. Mason and F. Cattaneo, Dynamic alignment and exact scaling laws in magnetohydrodynamic turbulence, Astrophys. J., 699 (2009), L39.  doi: 10.1088/0004-637X/699/1/L39.

[15]

R. Bruno and V. Carbone, The solar wind as a tubulence laboratory, Living Rev. Solar Phys., 10 (2013), 2. 

[16]

R. CaflischI. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.  doi: 10.1007/s002200050067.

[17]

D. ChaeP. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Lineaire, 31 (2014), 555-565.  doi: 10.1016/j.anihpc.2013.04.006.

[18]

D. Chae and M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971-3982.  doi: 10.1016/j.jde.2013.07.059.

[19]

D. Chae and S. Weng, Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. I. H. Poincaré-AN, 33 (2016), 1009-1022.  doi: 10.1016/j.anihpc.2015.03.002.

[20]

D. Chae and J. Wolf, On partial regularity for the 3D non-stationary Hall magnetohydrodynamics equations on the plane, Comm. Math. Phys., 354 (2017), 213-230.  doi: 10.1007/s00220-017-2908-8.

[21]

B. D. G. ChandranA. A. Schekochihin and A. Mallet, Intermittency and alignment in strong RMHD turbulence, Astrophys. J., 807 (2015), 39. 

[22]

A. Cheskidov and M. Dai, Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 429-446.  doi: 10.1017/prm.2018.33.

[23]

A. Cheskidov and M. Dai, Regularity criteria for the 3D Navier-Stokes and MHD equations, Proceedings of the Edinburgh Mathematical Society, 2020.

[24]

A. CheskidovM. Dai and L. Kavlie, Determining modes for the 3D Navier-Stokes equations, Phys. D, 374/375 (2018), 1-9.  doi: 10.1016/j.physd.2017.11.014.

[25]

A. Cheskidov and R. Shvydkoy, Euler equations and turbulence: Analytical approach to intermittency, SIAM J. Math. Anal., 46 (2014), 353-374.  doi: 10.1137/120876447.

[26]

L. ComissoM. LingamY. M. Huang and A. Bhattacharjee, General theory of the plasmoid instability, Phys. Plasmas, 23 (2016), 100702.  doi: 10.1063/1.4964481.

[27]

M. Dai, Non-uniqueness of Leray-Hopf weak solutions of the 3D Hall-MHD system, SIAM J. Math. Anal., 53 (2021), 5979-6016.  doi: 10.1137/20M1359420.

[28] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511626333.
[29] P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers, Oxford University Press, 2004. 
[30]

M. DobrowolnyA. Mangeney and P. Veltri, Fully developed anisotropic hydromagnetic turbulence in interplanetary space, Phys. Rev. Lett., 45 (1980), 144-147.  doi: 10.1103/PhysRevLett.45.144.

[31]

E. Dumas and F. Sueur, On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-magnetohydrodynamic equations, Comm. Math. Phys., 330 (2014), 1179-1225.  doi: 10.1007/s00220-014-1924-1.

[32]

W. M. Elsässer, The hydromagnetic equations, Phys. Rev., 79 (1950), 183. 

[33] U. Frisch., Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995. 
[34] S. Galtier, Introduction to Modern Magnetohydrodynamics, Cambridge University Press, London, 2016.  doi: 10.1017/CBO9781316665961.
[35]

P. Goldreich and S. Sridhar, Toward a theory of interstellar turbulence. Ⅱ: Strong Alfvénic turbulence, Astrophys. J., 438 (1995), 763-775. 

[36]

P. Goldreich and S. Sridhar, Magnetohydrodynamic turbulence revisited, Astrophys. J., 485 (1997), 680.  doi: 10.1086/304442.

[37]

P. S. Iroshnikov, Turbulence of a conducting fluid in a strong magnetic field, Sov. Astron., 7 (1964), 566-571. 

[38]

I. Jeong and S. Oh, On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity Ⅰ: Illposedness near degenerate stationary solutions, arXiv: 1902.02025.

[39]

D. Khoshnevisan, Analysis of Stochastic Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 2014. doi: 10.1090/cbms/119.

[40]

A. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynold's numbers, C. R. (Doklady) Acad. Sci. URSS (N.S.), 30 (1941), 301-305. 

[41]

A. Kolmogorov, On the decay of isotropic turbulence in an incompressible viscous fluid, C. R. (Doklady) Acad. Sci. URSS (N.S.), 31 (1941), 538-540. 

[42]

A. Kolmogorov, Dissipation of energy in locally isotropic turbulence, C. R. (Doklady) Acad. Sci. URSS (N.S.), 32 (1941), 16-18. 

[43]

A. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech., 13 (1962), 82-85.  doi: 10.1017/S0022112062000518.

[44]

R. H. Kraichnan, Inertial-range spectrum of hydromagnetic turbulence, Phys. Fluids, 8 (1965), 1385-1387. 

[45]

R. H. Kraichnan, Inertial ranges in two dimensional turbulence, Phys. Fluids, 10 (1967), 1417.  doi: 10.1063/1.1762301.

[46] L. D. Landau and E. M. Lifschitz, Fluid Mechanics, Pergamon Press, Oxford, 2nd Edition, 1987. 
[47]

A. Mallet and A. A. Schekochihin, A statistical model of three-dimensional anisotropy and intermittency in strong Alfvénic turbulence, Mon. Not. R. Astron. Soc., 466 (2017), 3918-3927.  doi: 10.1093/mnras/stw3251.

[48]

J. Maron and P. Goldreich, Simulations of incompressible magnetohydrodynamic turbulence, Astrophys. J., 554 (2001), 1175.  doi: 10.1086/321413.

[49]

J. MasonF. Cattaneo and S. Boldyrev, Dynamic alignment in driven magnetohydrodynamic turbulence, Phys. Rev. Lett., 97 (2006), 255002.  doi: 10.1103/PhysRevLett.97.255002.

[50]

J. MasonF. Cattaneo and S. Boldyrev., Numerical simulations of the spectrum in magnetohydrodynamic turbulence, Phys. Rev. E., 77 (2008), 036403. 

[51]

C. M. Meneveau and K. R. Sreenivasan, The multifractal nature of turbulent energy dissipation, J. Fluid Mech., 224 (1991), 429-484.  doi: 10.1017/S0022112091001830.

[52]

W. C. MüllerD. Biskamp and R. Grappin, Statistical anisotropy of magnetohydrodynamic turbulence, J. Fluid Mech., 224 (1991), 429-484. 

[53]

S. V. Nazarenko and A. A. Schekochihin, Critical balance in magnetohydrodynamic, rotating and stratified turbulence: Towards a universal scaling conjecture, J. Fluid Mech., 677 (2011), 134-153.  doi: 10.1017/S002211201100067X.

[54]

A. M. Obukhov, Some specific features of atmosphere turbulence, J. Fluid Mech., 13 (1962), 77-81.  doi: 10.1017/S0022112062000506.

[55]

L. Onsager, Statistical hydrodynamics, Nuovo Cimento (Supplemento, (Convegno Internazionale di Meccanica Statistica)), 6 (1949), 279-287.  doi: 10.1007/BF02780991.

[56]

J. C. PerezJ. MasonS. Boldyrev and F. Cattaneo, On the energy spectrum of strong magnetohydrodynamic turbulence, Phys. Rev. X, 2 (2012), 041005.  doi: 10.1103/PhysRevX.2.041005.

[57]

J. C. Perez, J. Mason, S. Boldyrev and F. Cattaneo, Comment on the numerical measurements of the magnetohydrodynamic turbulence spectrum by A. Beresnyak, arXiv: 1409.8106.

[58]

H. PolitanoA. Pouquet and P. L. Sulem, Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence, Phys. Plasmas, 2 (1995), 2931-2939.  doi: 10.1063/1.871473.

[59]

A. PouquetU. Frisch and M. Meneguzzi, Growth of correlations in magnetohydrodynamic turbulence, Phys. Rev. A, 33 (1986), 4266. 

[60]

A. A. Schekochihin and S. C. Cowley, Turbulence and magnetic fields in astrophysical plasmas, Magnetohydrodynamcis, 80 (2007), 85-115.  doi: 10.1007/978-1-4020-4833-3_6.

[61]

J. Squire and A. Bhattacharjee, Generation of large-scale magnetic fields by small-scale dynamo in shear flows, Phys. Rev. Lett., 115 (2015), 175003.  doi: 10.1103/PhysRevLett.115.175003.

[62]

J. B. Taylor, Relaxation of toroidal plasma and generation of reverse magnetic fields, Phys. Rev. Lett., 33 (1974), 1139. 

[63]

J. B. Taylor, Relaxation and magnetic reconnection in plasmas, Reviews of Modern Physics, 58 (1986), 741.  doi: 10.1103/RevModPhys.58.741.

show all references

References:
[1]

M. AcheritogarayP. DegondA. Frouvelle and J.-G. Liu, Kinetic formulation and global existence for the Hall-Magnetohydrodynamic system, Kinet. Relat. Models, 4 (2011), 901-918.  doi: 10.3934/krm.2011.4.901.

[2]

F. AnselmetY. GagneE. J. Hopfinger and R. A. Antonia, High-order velocity structure functions in turbulent shear flow, J. Fluid Mech., 140 (1984), 63-89.  doi: 10.1017/S0022112084000513.

[3]

R. Beekie, T. Buckmaster and V. Vicol, Weak solutions of ideal MHD which do not conserve magnetic helicity, Annals of PDE, 6 (2020), Article number: 1. doi: 10.1007/s40818-020-0076-1.

[4]

A. Beresnyak, The spectral slop and Kolmogorov constant of MHD turbulence, Phys. Rev. Lett., 106 (2011), 075001. 

[5]

A. Beresnyak, Basic properties of magnetohydrodynamic turbulence in the inertial range, Mon. Not. R. Astron. Soc., 422 (2012), 3495. 

[6]

A. Beresnyak, Spectra of strong magnetohydrodynamic turbulence from high-resolution simulations, Astrophys. J., 784 (2014), L20.  doi: 10.1088/2041-8205/784/2/L20.

[7]

A. Beresnyak and A. Lazarian, Polarization intermittency and its influence on MHD turbulence, Astrophys. J., 640 (2006), L175.  doi: 10.1086/503708.

[8]

A. BeresnyakA. Lazarian and F. Cattaneo, Scaling laws and diffuse locality of balanced and imbalanced MHD turbulence, Astrophys. J., 722 (2010), L110. 

[9]

A. Bhattacharjee, Impulsive magnetic reconnection in the Earth's magnetotail and the solar corona, Ann. Rev. Astron. Astrophys., 42 (2004), 365-384.  doi: 10.1146/annurev.astro.42.053102.134039.

[10] D. Biskamp, Magnetic Reconnection in Plasmas, Cambridge University Press, 2000. 
[11] D. Biskamp, Magnetohydrodynamic Turbulence, Cambridge University Press, 2003.  doi: 10.1017/CBO9780511535222.
[12]

S. Boldyrev, On the spectrum of magnetohydrodynamic turbulence, Astrophys. J., 626 (2005), L37. 

[13]

S. Boldyrev, Spectrum of magnetohydrodynamic turbulence, Phys. Rev. Lett., 96 (2006), 115002. 

[14]

S. BoldyrevJ. Mason and F. Cattaneo, Dynamic alignment and exact scaling laws in magnetohydrodynamic turbulence, Astrophys. J., 699 (2009), L39.  doi: 10.1088/0004-637X/699/1/L39.

[15]

R. Bruno and V. Carbone, The solar wind as a tubulence laboratory, Living Rev. Solar Phys., 10 (2013), 2. 

[16]

R. CaflischI. Klapper and G. Steele, Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD, Comm. Math. Phys., 184 (1997), 443-455.  doi: 10.1007/s002200050067.

[17]

D. ChaeP. Degond and J.-G. Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Lineaire, 31 (2014), 555-565.  doi: 10.1016/j.anihpc.2013.04.006.

[18]

D. Chae and M. Schonbek, On the temporal decay for the Hall-magnetohydrodynamic equations, J. Differential Equations, 255 (2013), 3971-3982.  doi: 10.1016/j.jde.2013.07.059.

[19]

D. Chae and S. Weng, Singularity formation for the incompressible Hall-MHD equations without resistivity, Ann. I. H. Poincaré-AN, 33 (2016), 1009-1022.  doi: 10.1016/j.anihpc.2015.03.002.

[20]

D. Chae and J. Wolf, On partial regularity for the 3D non-stationary Hall magnetohydrodynamics equations on the plane, Comm. Math. Phys., 354 (2017), 213-230.  doi: 10.1007/s00220-017-2908-8.

[21]

B. D. G. ChandranA. A. Schekochihin and A. Mallet, Intermittency and alignment in strong RMHD turbulence, Astrophys. J., 807 (2015), 39. 

[22]

A. Cheskidov and M. Dai, Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 429-446.  doi: 10.1017/prm.2018.33.

[23]

A. Cheskidov and M. Dai, Regularity criteria for the 3D Navier-Stokes and MHD equations, Proceedings of the Edinburgh Mathematical Society, 2020.

[24]

A. CheskidovM. Dai and L. Kavlie, Determining modes for the 3D Navier-Stokes equations, Phys. D, 374/375 (2018), 1-9.  doi: 10.1016/j.physd.2017.11.014.

[25]

A. Cheskidov and R. Shvydkoy, Euler equations and turbulence: Analytical approach to intermittency, SIAM J. Math. Anal., 46 (2014), 353-374.  doi: 10.1137/120876447.

[26]

L. ComissoM. LingamY. M. Huang and A. Bhattacharjee, General theory of the plasmoid instability, Phys. Plasmas, 23 (2016), 100702.  doi: 10.1063/1.4964481.

[27]

M. Dai, Non-uniqueness of Leray-Hopf weak solutions of the 3D Hall-MHD system, SIAM J. Math. Anal., 53 (2021), 5979-6016.  doi: 10.1137/20M1359420.

[28] P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511626333.
[29] P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers, Oxford University Press, 2004. 
[30]

M. DobrowolnyA. Mangeney and P. Veltri, Fully developed anisotropic hydromagnetic turbulence in interplanetary space, Phys. Rev. Lett., 45 (1980), 144-147.  doi: 10.1103/PhysRevLett.45.144.

[31]

E. Dumas and F. Sueur, On the weak solutions to the Maxwell-Landau-Lifshitz equations and to the Hall-magnetohydrodynamic equations, Comm. Math. Phys., 330 (2014), 1179-1225.  doi: 10.1007/s00220-014-1924-1.

[32]

W. M. Elsässer, The hydromagnetic equations, Phys. Rev., 79 (1950), 183. 

[33] U. Frisch., Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, 1995. 
[34] S. Galtier, Introduction to Modern Magnetohydrodynamics, Cambridge University Press, London, 2016.  doi: 10.1017/CBO9781316665961.
[35]

P. Goldreich and S. Sridhar, Toward a theory of interstellar turbulence. Ⅱ: Strong Alfvénic turbulence, Astrophys. J., 438 (1995), 763-775. 

[36]

P. Goldreich and S. Sridhar, Magnetohydrodynamic turbulence revisited, Astrophys. J., 485 (1997), 680.  doi: 10.1086/304442.

[37]

P. S. Iroshnikov, Turbulence of a conducting fluid in a strong magnetic field, Sov. Astron., 7 (1964), 566-571. 

[38]

I. Jeong and S. Oh, On the Cauchy problem for the Hall and electron magnetohydrodynamic equations without resistivity Ⅰ: Illposedness near degenerate stationary solutions, arXiv: 1902.02025.

[39]

D. Khoshnevisan, Analysis of Stochastic Partial Differential Equations, CBMS Regional Conference Series in Mathematics, 2014. doi: 10.1090/cbms/119.

[40]

A. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynold's numbers, C. R. (Doklady) Acad. Sci. URSS (N.S.), 30 (1941), 301-305. 

[41]

A. Kolmogorov, On the decay of isotropic turbulence in an incompressible viscous fluid, C. R. (Doklady) Acad. Sci. URSS (N.S.), 31 (1941), 538-540. 

[42]

A. Kolmogorov, Dissipation of energy in locally isotropic turbulence, C. R. (Doklady) Acad. Sci. URSS (N.S.), 32 (1941), 16-18. 

[43]

A. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech., 13 (1962), 82-85.  doi: 10.1017/S0022112062000518.

[44]

R. H. Kraichnan, Inertial-range spectrum of hydromagnetic turbulence, Phys. Fluids, 8 (1965), 1385-1387. 

[45]

R. H. Kraichnan, Inertial ranges in two dimensional turbulence, Phys. Fluids, 10 (1967), 1417.  doi: 10.1063/1.1762301.

[46] L. D. Landau and E. M. Lifschitz, Fluid Mechanics, Pergamon Press, Oxford, 2nd Edition, 1987. 
[47]

A. Mallet and A. A. Schekochihin, A statistical model of three-dimensional anisotropy and intermittency in strong Alfvénic turbulence, Mon. Not. R. Astron. Soc., 466 (2017), 3918-3927.  doi: 10.1093/mnras/stw3251.

[48]

J. Maron and P. Goldreich, Simulations of incompressible magnetohydrodynamic turbulence, Astrophys. J., 554 (2001), 1175.  doi: 10.1086/321413.

[49]

J. MasonF. Cattaneo and S. Boldyrev, Dynamic alignment in driven magnetohydrodynamic turbulence, Phys. Rev. Lett., 97 (2006), 255002.  doi: 10.1103/PhysRevLett.97.255002.

[50]

J. MasonF. Cattaneo and S. Boldyrev., Numerical simulations of the spectrum in magnetohydrodynamic turbulence, Phys. Rev. E., 77 (2008), 036403. 

[51]

C. M. Meneveau and K. R. Sreenivasan, The multifractal nature of turbulent energy dissipation, J. Fluid Mech., 224 (1991), 429-484.  doi: 10.1017/S0022112091001830.

[52]

W. C. MüllerD. Biskamp and R. Grappin, Statistical anisotropy of magnetohydrodynamic turbulence, J. Fluid Mech., 224 (1991), 429-484. 

[53]

S. V. Nazarenko and A. A. Schekochihin, Critical balance in magnetohydrodynamic, rotating and stratified turbulence: Towards a universal scaling conjecture, J. Fluid Mech., 677 (2011), 134-153.  doi: 10.1017/S002211201100067X.

[54]

A. M. Obukhov, Some specific features of atmosphere turbulence, J. Fluid Mech., 13 (1962), 77-81.  doi: 10.1017/S0022112062000506.

[55]

L. Onsager, Statistical hydrodynamics, Nuovo Cimento (Supplemento, (Convegno Internazionale di Meccanica Statistica)), 6 (1949), 279-287.  doi: 10.1007/BF02780991.

[56]

J. C. PerezJ. MasonS. Boldyrev and F. Cattaneo, On the energy spectrum of strong magnetohydrodynamic turbulence, Phys. Rev. X, 2 (2012), 041005.  doi: 10.1103/PhysRevX.2.041005.

[57]

J. C. Perez, J. Mason, S. Boldyrev and F. Cattaneo, Comment on the numerical measurements of the magnetohydrodynamic turbulence spectrum by A. Beresnyak, arXiv: 1409.8106.

[58]

H. PolitanoA. Pouquet and P. L. Sulem, Current and vorticity dynamics in three-dimensional magnetohydrodynamic turbulence, Phys. Plasmas, 2 (1995), 2931-2939.  doi: 10.1063/1.871473.

[59]

A. PouquetU. Frisch and M. Meneguzzi, Growth of correlations in magnetohydrodynamic turbulence, Phys. Rev. A, 33 (1986), 4266. 

[60]

A. A. Schekochihin and S. C. Cowley, Turbulence and magnetic fields in astrophysical plasmas, Magnetohydrodynamcis, 80 (2007), 85-115.  doi: 10.1007/978-1-4020-4833-3_6.

[61]

J. Squire and A. Bhattacharjee, Generation of large-scale magnetic fields by small-scale dynamo in shear flows, Phys. Rev. Lett., 115 (2015), 175003.  doi: 10.1103/PhysRevLett.115.175003.

[62]

J. B. Taylor, Relaxation of toroidal plasma and generation of reverse magnetic fields, Phys. Rev. Lett., 33 (1974), 1139. 

[63]

J. B. Taylor, Relaxation and magnetic reconnection in plasmas, Reviews of Modern Physics, 58 (1986), 741.  doi: 10.1103/RevModPhys.58.741.

Figure 1.  Structure function exponent as a function of p for EMHD with different intermittency level
Figure 2.  Second (red) and third (blue) order structure functions for homogeneous isotropic self-similar EMHD turbulence
Figure 3.  Second (red) and third (blue) order structure functions for extremely anisotropic EMHD turbulence
Figure 4.  Magnetic energy spectra of Hall MHD when $ \delta_u = \delta_b = 3 $ (blue lines) and when $ \delta_u = \delta_b = 0 $ (red lines)
Figure 5.  Negative exponent $\gamma$ of perpendicular energy spectrum with dependence on intermittency dimension $ \delta_{\perp} $
Figure 6.  Energy spectra of perpendicular cascade under different assumptions
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