Abstract
We derive a hierarchy of evolution equations for multi-point probability density functions in magneto-hydrodynamic (MHD) turbulence. We discuss the relation to the moment hierarchy in MHD turbulence formulated by Chandrasekhar (S. Chandrasekhar, Proc. R. Soc. Lond. A 1951, 204, 435–449) and derive a functional equation for a joint characteristic functional, which can be considered as the analogon to the Hopf functional in hydrodynamic turbulence. Furthermore, we develop a closure method for the evolution equation of the single-point magnetic field probability density function, which is based on a joint Gaussian assumption for unclosed terms. It is explicitly shown that this closure, together with the assumptions of homogeneity and isotropy, leads to vanishing nonlinear terms. We discuss the implications of this finding for magnetic field generation and give a brief outlook on an axisymmetric theory, which includes a mean magnetic field.
Highlights
The investigation of magneto-hydrodynamic (MHD) turbulence by statistical methods has a longstanding tradition, which can be traced back to the works of Chandrasekhar [1] and Batchelor [2], as well as to the subdivision of mean field electrodynamics put forth by Steenbeck, Krause, and Rädler [3](for further references, see [4,5])
Recent developments in statistical magnetohydrodynamics include an analytical treatment of weak MHD turbulence [6], a phenomenological description for the energy spectrum based on the dynamical alignment of velocity and magnetic field fluctuations [7,8], and the derivation of relations between longitudinal and transverse structure functions [9] similar to the ones of hydrodynamic turbulence [10,11,12]
We presented a comprehensive statistical description of MHD turbulence
Summary
The investigation of magneto-hydrodynamic (MHD) turbulence by statistical methods has a longstanding tradition, which can be traced back to the works of Chandrasekhar [1] and Batchelor [2], as well as to the subdivision of mean field electrodynamics put forth by Steenbeck, Krause, and Rädler [3]. Recent developments in statistical magnetohydrodynamics include an analytical treatment of weak MHD turbulence [6], a phenomenological description for the energy spectrum based on the dynamical alignment of velocity and magnetic field fluctuations [7,8], and the derivation of relations between longitudinal and transverse structure functions [9] similar to the ones of hydrodynamic turbulence [10,11,12]. Similar to the case of hydrodynamic turbulence, such a statistical description of MHD turbulence is complicated by the occurrence of anomalous statistics of velocity and magnetic field fluctuations at small scales, which is commonly referred to as intermittency [18]. Due to pronounced deviations from Gaussianity and additional nonlocalities in the multi-point hierarchy of MHD turbulence, finding appropriate methods or assumptions to close the hierarchy at a certain stage [22,23] might prove to be even more difficult than in ordinary turbulence. We will proceed to discuss a possible closure method on the basis of a joint Gaussian assumption for velocity and magnetic field statistics
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.