Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T16:05:58.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Path integrals for mean-field equations in nonlinear dynamos

Part of: 50 years of Mean Field Electrodynamics

Published online by Cambridge University Press:  14 June 2018

Dmitry Sokoloff  and
Nobumitsu Yokoi
Dmitry Sokoloff*
Affiliation:
Department of Physics, Moscow State University, Moscow, 119991, Russia IZMIRAN, Kaluzhskoe shosse, Troitsk, Moscow, 108840, Russia
Nobumitsu Yokoi
Affiliation:
Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Mean-field dynamo equations are addressed with the aid of the path integral method. The evolution of magnetic field is treated as a three-dimensional Wiener random process, and the mean magnetic-field equations are obtained with the Wiener integrals taken over all the trajectories of the fluid particles. The form of the equations is just the same as the conventional mean-field equations, but here the equations are derived with the velocity field realisation affected by the force exerted by the magnetic field. In this sense, we derive nonlinear dynamo equations.

Keywords

astrophysical plasmasspace plasma physics
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 3 , June 2018 , 735840307
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brandenburg, A., Schober, J. & Rogachevskii, I. 2017 The contribution of kinetic helicity to turbulent magnetic diffusivity. Astron. Nachr. 338, 790–793.Google Scholar
Brandenburg, A., Rädler, K.-H., Rheinhardt, M. & Subramanian, K. 2008 Magnetic quenching of $\unicode[STIX]{x1D6FC}$ and diffusivity tensors in helical turbulence. Astrophys. J. Lett. 687, L49.Google Scholar
Courvoisier, A., Hughes, D. W. & Proctor, M. R. E. 2010 Self-consistent mean-field magnetohydrodynamics. Proc. R. Soc. Lond. 466, 583–601.Google Scholar
Gantmacher, F. R. 1959 Applications of The Theory of Matrices. Interscience.Google Scholar
Ito, K. 1946 On stochastic integral equation. Proc. Japan. Acad. 1, 32–35.Google Scholar
Kleeorin, N. & Rogachevskii, I. 1994 Nonlinear theory of magnetic fluctuations in random flow: the Hall effect. Phys. Rev. E 50, 493–501.Google Scholar
Kraichnan, R.-H. 1965 Lagrangian-history closure approximation for turbulence. Phys. Fluids 8 (1965), 575–598.CrossRefGoogle Scholar
Krause, F. & Rädler, K.-H. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon Press.Google Scholar
Molchanov, S. A., Ruzmaikin, A. A. & Sokoloff, D. D. 1983 Equation of dynamo in random velocity field with short correlation time. Magnetohydrodynamics 19, 402–407.Google Scholar
Parker, E. N. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293–314.Google Scholar
Peng, S. 1991 Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics Stochastic Rep. 37 (1991), 61–74.Google Scholar
Plunian, F., Stepanov, R. & Frick, P. 2013 Shell models of magnetohydrodynamic turbulence. Phys. Rep. 523, 1–60.Google Scholar
Rheinhardt, M. & Brandenburg, A. 2010 Test-field method for mean-field coefficients with MHD background. Astron. Astrophys. 520, A28.Google Scholar
Shiryaev, A. N. 1999 Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific.Google Scholar
Tomin, D. & Sokoloff, D. 2010 Dynamo in fluctuating ABC flow. Geophys. Astrophys. Fluid Dyn. 104, 183–188.Google Scholar
Yokoi, N. 2013 Cross helicity and related dynamo. Geophys. Astrophys. Fluid Dyn. 107, 114–184.Google Scholar
Zeldovich, Ya. B., Ruzmaikin, A. A. & Sokoloff, D. D. 1990 The Almighty Chance. World Scientific.CrossRefGoogle Scholar