doi: 10.3934/dcds.2020397

The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence

School of Mathematics and Systems Science, Beihang University, Beijing 100191, China

* Corresponding author: Jia Yuan

Received  July 2020 Revised  October 2020 Published  December 2020

Fund Project: The work is supported by NSF grant No.11871087 and No.11771423. The first author is supported by the Academic Excellence Foundation of BUAA for PhD students and China Scholarship Council No.201906020100

In this paper, we consider the Littlewood-Paley $ p $th-order ($ 1\le p<\infty $) moments of the three-dimensional MHD periodic equations, which are defined by the infinite-time and space average of $ L^p $-norm of velocity and magnetic fields involved in the spectral cut-off operator $ \dot\Delta_m $. Our results imply that in some cases, $ k^{-\frac{1}{3}} $ is an upper bound at length scale $ 1/k $. This coincides with the scaling law of many observations on astrophysical systems and simulations in terms of 3D MHD turbulence.

Citation: Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020397
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, 2011. doi: 10.1007%2F978-3-642-16830-7.  Google Scholar

[2]

A. Basu and J. K. Bhattacharjee, Universal properties of three-dimensional magnetohydrodynamic turbulence: do Alfvén waves matter?, J. Stat. Mech., 2005 (2005), P07002. doi: 10.1088/1742-5468/2005/07/P07002.  Google Scholar

[3]

A. BasuA. SainS. K. Dhar and R. Pandit, Multiscaling in models of magnetohydrodynamic turbulence, Phys. Rev. Lett., 81 (1998), 2687-2690.  doi: 10.1103/PhysRevLett.81.2687.  Google Scholar

[4]

D. Biskamp and W-C. Müller, Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence, Phys. Plasmas, 7 (2000), 4889-4900.  doi: 10.1063/1.1322562.  Google Scholar

[5]

M. Cannone, Harmonic analysis tools for solving incompressible Navier-Stokes equations, Handbook of Mathmatical Fluid Dynamics vol 3,161–244, North-Holland, Amsterdam, 2004.  Google Scholar

[6]

Q. ChenC. Miao and Z. Zhang, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271 (2007), 821-838.  doi: 10.1007/s00220-007-0193-7.  Google Scholar

[7]

Q. ChenC. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.  doi: 10.1007/s00220-008-0545-y.  Google Scholar

[8]

Q. ChenC. Miao and Z. Zhang, On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces, Arch. Ration. Mech. Anal., 195 (2010), 561-578.  doi: 10.1007/s00205-008-0213-6.  Google Scholar

[9]

J. ChoE. T. Vishniac and A. Lazarian, Simulations of magnetohydrodynamic turbulence in a strongly magnetized medium, Astrophys. J., 564 (2002), 291-301.  doi: 10.1086/324186.  Google Scholar

[10]

J. Cho and E. T. Vishniac, The anisotropy of magnetohydrodynamic Alfvénic turbulence, Astrophys. J., 539 (2000), 273-282.  doi: 10.1086/309213.  Google Scholar

[11]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Lecture Notes in Math. Vol. 1871, Berlin: Springer, 2006, 1–43. doi: 10.1007%2F11545989_1.  Google Scholar

[12]

P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. Comput. Fluid Dyn., 9 (1997), 183-189.  doi: 10.1007/s001620050039.  Google Scholar

[13]

E. Falgarone and T. Passot, Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics, Springer, 2003. doi: 10.1007%2F3-540-36238-X.  Google Scholar

[14]

Y. GuptaB. J. Rickett and W. A. Coles, Refractive interstellar scintillation of pulsar intensities at 74 MHz, Astrophysical J., 403 (1993), 183-201.  doi: 10.1086/172193.  Google Scholar

[15]

E. Hopf, Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[16]

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

[17]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proceedings of the Royal Society A, 434 (1991), 9-13.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[18]

A. N. Kolmogorov, Dissipation of energy in the locally isotropic turbulence, Proceedings of the Royal Society A, 434 (1991), 15-17.  doi: 10.1098/rspa.1991.0076.  Google Scholar

[19]

R. H. Kraichnan, Lagrangian-history closure approximation for turbulence, Phys. Fluids, 8 (1965), 575-598.  doi: 10.1063/1.1761271.  Google Scholar

[20]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace., Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[21] C. MiaoJ. Wu and Z. Zhang, Littlewood-Paley Theory and Applications to Fluid Dynamics Equations, Monographs on Modern pure mathematics, No. 142, Beijing: Science Press, 2012.   Google Scholar
[22]

W-C. Müller and D. Biskamp, Scaling properties of three-dimensional magnetohydrodynamic turbulence, Phys. Rev. Lett., 84 (2000), 475-478.  doi: 10.1103/PhysRevLett.84.475.  Google Scholar

[23]

F. Otto and F. Ramos, Universal bounds for the Littlewood-Paley first-order moments of the 3D Navier-Stokes equations, Comm. Math. Phys., 300 (2010), 301-315.  doi: 10.1007/s00220-010-1098-4.  Google Scholar

[24]

S. R. Spangler and C. R. Gwinn, Evidence for an inner scale to the density turbulence in the interstellar medium, Astrophys. J., 353 (1990), L29–L32. doi: 10.1086/185700.  Google Scholar

[25]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.  doi: 10.1080/03605300701382530.  Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, 2011. doi: 10.1007%2F978-3-642-16830-7.  Google Scholar

[2]

A. Basu and J. K. Bhattacharjee, Universal properties of three-dimensional magnetohydrodynamic turbulence: do Alfvén waves matter?, J. Stat. Mech., 2005 (2005), P07002. doi: 10.1088/1742-5468/2005/07/P07002.  Google Scholar

[3]

A. BasuA. SainS. K. Dhar and R. Pandit, Multiscaling in models of magnetohydrodynamic turbulence, Phys. Rev. Lett., 81 (1998), 2687-2690.  doi: 10.1103/PhysRevLett.81.2687.  Google Scholar

[4]

D. Biskamp and W-C. Müller, Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence, Phys. Plasmas, 7 (2000), 4889-4900.  doi: 10.1063/1.1322562.  Google Scholar

[5]

M. Cannone, Harmonic analysis tools for solving incompressible Navier-Stokes equations, Handbook of Mathmatical Fluid Dynamics vol 3,161–244, North-Holland, Amsterdam, 2004.  Google Scholar

[6]

Q. ChenC. Miao and Z. Zhang, A new Bernstein's inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271 (2007), 821-838.  doi: 10.1007/s00220-007-0193-7.  Google Scholar

[7]

Q. ChenC. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations, Comm. Math. Phys., 284 (2008), 919-930.  doi: 10.1007/s00220-008-0545-y.  Google Scholar

[8]

Q. ChenC. Miao and Z. Zhang, On the well-posedness of the ideal MHD equations in the Triebel-Lizorkin spaces, Arch. Ration. Mech. Anal., 195 (2010), 561-578.  doi: 10.1007/s00205-008-0213-6.  Google Scholar

[9]

J. ChoE. T. Vishniac and A. Lazarian, Simulations of magnetohydrodynamic turbulence in a strongly magnetized medium, Astrophys. J., 564 (2002), 291-301.  doi: 10.1086/324186.  Google Scholar

[10]

J. Cho and E. T. Vishniac, The anisotropy of magnetohydrodynamic Alfvénic turbulence, Astrophys. J., 539 (2000), 273-282.  doi: 10.1086/309213.  Google Scholar

[11]

P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in Mathematical Foundation of Turbulent Viscous Flows, Lecture Notes in Math. Vol. 1871, Berlin: Springer, 2006, 1–43. doi: 10.1007%2F11545989_1.  Google Scholar

[12]

P. Constantin, The Littlewood-Paley spectrum in two-dimensional turbulence, Theor. Comput. Fluid Dyn., 9 (1997), 183-189.  doi: 10.1007/s001620050039.  Google Scholar

[13]

E. Falgarone and T. Passot, Turbulence and Magnetic Fields in Astrophysics, Lecture Notes in Physics, Springer, 2003. doi: 10.1007%2F3-540-36238-X.  Google Scholar

[14]

Y. GuptaB. J. Rickett and W. A. Coles, Refractive interstellar scintillation of pulsar intensities at 74 MHz, Astrophysical J., 403 (1993), 183-201.  doi: 10.1086/172193.  Google Scholar

[15]

E. Hopf, Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr., 4 (1951), 213-231.  doi: 10.1002/mana.3210040121.  Google Scholar

[16]

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

[17]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proceedings of the Royal Society A, 434 (1991), 9-13.  doi: 10.1098/rspa.1991.0075.  Google Scholar

[18]

A. N. Kolmogorov, Dissipation of energy in the locally isotropic turbulence, Proceedings of the Royal Society A, 434 (1991), 15-17.  doi: 10.1098/rspa.1991.0076.  Google Scholar

[19]

R. H. Kraichnan, Lagrangian-history closure approximation for turbulence, Phys. Fluids, 8 (1965), 575-598.  doi: 10.1063/1.1761271.  Google Scholar

[20]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace., Acta Math., 63 (1934), 193-248.  doi: 10.1007/BF02547354.  Google Scholar

[21] C. MiaoJ. Wu and Z. Zhang, Littlewood-Paley Theory and Applications to Fluid Dynamics Equations, Monographs on Modern pure mathematics, No. 142, Beijing: Science Press, 2012.   Google Scholar
[22]

W-C. Müller and D. Biskamp, Scaling properties of three-dimensional magnetohydrodynamic turbulence, Phys. Rev. Lett., 84 (2000), 475-478.  doi: 10.1103/PhysRevLett.84.475.  Google Scholar

[23]

F. Otto and F. Ramos, Universal bounds for the Littlewood-Paley first-order moments of the 3D Navier-Stokes equations, Comm. Math. Phys., 300 (2010), 301-315.  doi: 10.1007/s00220-010-1098-4.  Google Scholar

[24]

S. R. Spangler and C. R. Gwinn, Evidence for an inner scale to the density turbulence in the interstellar medium, Astrophys. J., 353 (1990), L29–L32. doi: 10.1086/185700.  Google Scholar

[25]

J. Wu, Regularity criteria for the generalized MHD equations, Comm. Partial Differential Equations, 33 (2008), 285-306.  doi: 10.1080/03605300701382530.  Google Scholar

[1]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[2]

Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020366

[3]

Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85

[4]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351

[5]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

[6]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605

[7]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[8]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[9]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[10]

Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028

[11]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093

[12]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[13]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[14]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015

[15]

Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143

[16]

Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif, Tewfik Sari. Mathematical analysis of a three-tiered food-web in the chemostat. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020369

[17]

Shanding Xu, Longjiang Qu, Xiwang Cao. Three classes of partitioned difference families and their optimal constant composition codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020120

[18]

Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng. A construction of $ p $-ary linear codes with two or three weights. Advances in Mathematics of Communications, 2021, 15 (1) : 9-22. doi: 10.3934/amc.2020039

[19]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[20]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

2019 Impact Factor: 1.338

Article outline

[Back to Top]