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Abstract
The statistical theory of inhomogeneous turbulence is applied to develop a system of model equations for magnetohydrodynamic (MHD) turbulence. The statistical descriptors of MHD turbulence are taken to be the turbulent MHD energy, its dissipation rate, the turbulent cross helicity (velocity-magnetic field correlation), turbulent MHD residual energy (difference between the kinetic and magnetic energies), and turbulent residual helicity (difference between the kinetic and current helicities). Evolution equations for these statistical quantities are coupled to the mean-field dynamics. The model is applied to two MHD-plasma phenomena: turbulence evolution with prescribed mean velocity and magnetic fields in the solar wind, and mean flow generation in the presence of a mean magnetic field and cross helicity in tokamak plasmas. These applications support the validity of the turbulence model. In the presence of a mean magnetic field, turbulence dynamics should be subject to combined effects of nonlinearity and Alfvén waves; consequences for the dissipation rate of MHD residual energy are discussed.
Acknowledgments
One of the authors (Nobumitsu Yokoi) would like to thank the organizing and executive committees of the fifth international symposium on Turbulence and Shear Flow Phenomena (TSFP-5) for providing him with a good opportunity to present some parts of this work at the well-organized symposium. His thanks are also due to the guest editors for TSFP-5 focus issue of the Journal of Turbulence (JoT) for inviting him to submit his work as a paper to the issue. The authors are grateful to anonymous referees for their attentive comments improving the presentation of the manuscript. The authors would like to dedicate this work to the memory of Robert Harry Kraichnan (RHK) who died on February 26, 2008. Apart from his giant steps in closure theories of turbulence, by introducing the response function formalism into turbulence research, RHK also paved the way up for later research on the foundations of turbulence modeling.