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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

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

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[32]

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

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

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

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[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.   Google Scholar

[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.   Google Scholar

[42]

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

[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.  Google Scholar

[44]

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

[45]

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

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

[48]

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

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[52]

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

[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.  Google Scholar

[54]

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

[55]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[59]

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

[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.  Google Scholar

[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.  Google Scholar

[62]

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

[63]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[7]

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

[8]

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

[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.  Google Scholar

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

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

[13]

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

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[32]

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

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

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

[36]

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

[37]

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

[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. Google Scholar

[39]

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

[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.   Google Scholar

[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.   Google Scholar

[42]

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

[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.  Google Scholar

[44]

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

[45]

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

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

[48]

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

[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.  Google Scholar

[50]

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

[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.  Google Scholar

[52]

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

[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.  Google Scholar

[54]

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

[55]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[59]

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

[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.  Google Scholar

[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.  Google Scholar

[62]

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

[63]

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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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