Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-19T13:11:15.158Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamo properties of the turbulent velocity field of a saturated dynamo

Published online by Cambridge University Press:  12 February 2009

FAUSTO CATTANEO  and
STEVEN M. TOBIAS
FAUSTO CATTANEO
Affiliation:
Department of Astronomy and Astrophysics and The Computation Institute, University of Chicago, Chicago, IL 60637, USA
STEVEN M. TOBIAS*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: smt@amsta.leeds.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In order better to understand how dynamo systems saturate, we study the kinematic dynamo properties of velocity fields that arise from nonlinearly saturated dynamos. The technique is implemented by solving concurrently, in addition to the momentum equation, two induction equations, one for the actual magnetic field and one for an independent passive vector field. We apply this technique to two illustrative examples: convectively driven turbulence and turbulence represented by a shell model. In all cases we find that the velocity remains an efficient kinematic dynamo even after nonlinear saturation occurs. We discuss the implications to the process of dynamo saturation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 621 , 25 February 2009 , pp. 205 - 214
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Boldyrev, S. & Cattaneo, F. 2004 Magnetic-field generation in Kolmogorov turbulence. Phys. Rev. Lett. 92 (14), 144501.CrossRefGoogle ScholarPubMed
Brummell, N. H., Cattaneo, F. & Tobias, S. M. 2001 Linear and nonlinear dynamo properties of time-dependent ABC flows. Fluid Dyn. Res. 28, 237–265.CrossRefGoogle Scholar
Cattaneo, F., Emonet, T. & Weiss, N. 2003 On the interaction between convection and magnetic fields. Astrophys. J. 588, 1183–1198.CrossRefGoogle Scholar
Cattaneo, F. & Hughes, D. W. 2006 Dynamo action in a rotating convective layer. J. Fluid Mech. 553, 401–418.CrossRefGoogle Scholar
Cattaneo, F. & Hughes, D. W. 2008 Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers. Mon. Not. Roy. Ast. Soc. submitted.Google Scholar
Cattaneo, F., Hughes, D. W. & Kim, E. 1996 Suppression of chaos in a simplified nonlinear dynamo model. Phys. Rev. Lett. 76, 2057–2060.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Frick, P. 1983 Hierarchical models of two-dimensional turbulence. Magnitnaia Gidrodinarnika 19, 60–66.Google Scholar
Frick, P. & Sokoloff, D. 1998 Cascade and dynamo action in a shell model of magnetohydrodynamic turbulence. Phys. Rev. emphE 57, 4155–4164.Google Scholar
Gledzer, E. B. 1973 System of hydrodynamic type admitting two quadratic integrals of motion. Sov. Phys. Dok. 18, 216–217.Google Scholar
Kazantsev, A. P. 1968 Enhancement of a Magnetic Field by a Conducting Fluid. Sov. J. Exp. Theoret. Phys. 26, 1031–1034.Google Scholar
Klapper, I. & Young, L. S. 1995 Rigorous bounds on the fast dynamo growth-rate involving topological entropy. Comm. Math. Phys. 175, 623–646.CrossRefGoogle Scholar
Kraichnan, R. H. & Nagarajan, S. 1967 Growth of turbulent magnetic fields. Phys. Fluids 10, 859–870.CrossRefGoogle Scholar
Plunian, F. & Stepanov, R. 2007 A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence. New J. Phys. 9, 296–319.CrossRefGoogle Scholar
Schekochihin, A. A., Cowley, S. C., Maron, J. L. & McWilliams, J. C. 2004 Critical magnetic Prandtl number for small-scale dynamo. Phys. Rev. Lett. 92 (5), 054502.CrossRefGoogle ScholarPubMed
Stellmach, S. & Hansen, U. 2004 Cartesian convection driven dynamos at low Ekman number. Phys. Rev. E 70 (5), 056312.Google ScholarPubMed
Vainshtein, S. I. & Cattaneo, F. 1992 Nonlinear restrictions on dynamo action. Astrophys. J. 393, 165–171.CrossRefGoogle Scholar
Vainshtein, S. I. & Kichatinov, L. L. 1986 The dynamics of magnetic fields in a highly conducting turbulent medium and the generalized Kolmogorov–Fokker–Planck equations. J. Fluid Mech. 168, 73–87.CrossRefGoogle Scholar
Vishik, M. M. 1989 Magnetic field generation by the motion of a highly conducting fluid. Geophys. Astrophys. Fluid Dyn. 48, 151–167.CrossRefGoogle Scholar
Yamada, M. & Ohkitani, K. 1987 Lyapunov spectrum of a chaotic model of three-dimensional turbulence. J. Phys. Soc. Jpn 56, 4210–4213.CrossRefGoogle Scholar