Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-19T13:51:03.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Magnetic field intermittency and fast dynamo action in random helical flows

Published online by Cambridge University Press:  26 April 2006

Andrew D. Gilbert  and
B. J. Bayly
Andrew D. Gilbert
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver St., Cambridge CB3 9EW, UK
B. J. Bayly
Affiliation:
Mathematics Department, University of Arizona, Tucson, AZ 85721, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The evolution of passive magnetic fields is considered in random flows made up of single helical waves. In the absence of molecular diffusion the growth rates of all moments of a magnetic field are calculated analytically, and it is found that the field becomes increasingly intermittent with time. The evolution of normal modes of the ensemble-averaged field is determined; it is shown that the flows considered give fast dynamo action, and magnetic field modes with either sign of magnetic helicity may grow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 199 - 214
Copyright
© 1992 Cambridge University Press

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

Arnol'd, V. I. & Korkina E. I. 1983 The growth of a magnetic field in a three-dimensional steady incompressible flow. Vest. Mosk. Un. Ta. Ser. 1, Math. Mec. 3, 43–46.Google Scholar
Arnol'd V. I., Zel'dovich Ya, B., Ruzmaikin, A. A. & Sokoloff, D. D. 1981 A magnetic field in a stationary flow with stretching in Riemannian space. Zh. Eksp. Teor. Fiz. 81, 2052–2058 (Transl. Sov. Phys. JETP 54, 1083–1086 (1981)).Google Scholar
Batchelor G. K. 1952 The effect of homogeneous turbulence on material lines and surfaces Proc. R. Soc. Lond. A 313, 349–366.Google Scholar
Bayly, B. J. & Childress S. 1988 Construction of fast dynamos using unsteady flows and maps in three dimensions. Geophys. Astrophys. Fluid Dyn. 44, 211–240.Google Scholar
Bayly, B. J. & Childress S. 1989 Unsteady dynamo effects at large magnetic Reynolds number. Geophys. Astrophys. Fluid Dyn. 49, 23–43.Google Scholar
Childress, S. & Klapper I. 1991 On some transport properties of baker's maps. J. Statist. Phys. 63, 897–914.Google Scholar
Cowling T. G. 1934 The magnetic field of sunspots, Mon. Not. Roy. Astr. Soc. 94, 39–48.Google Scholar
Dittrich P., Molchanov S. A., Sokoloff, D. D. & Ruzmaikin A. A. 1984 Mean magnetic field in renovating random flow. Astron. Nachr. 305, 119–125.Google Scholar
Dombre T., Frisch U., Greene J. M., HeAnon M., Mehr, A. & Soward A. M. 1986 Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353–391.Google Scholar
Drummond, I. T. & Horgan R. R. 1986 Numerical simulation of the -effect and turbulent magnetic diffusion with molecular diffusivity. J. Fluid Mech. 163, 425–438.Google Scholar
Drummond, I. T. & MuUnch W. H. P. 1990 Turbulent stretching of line and surface elements. J. Fluid Mech. 215, 45–49.Google Scholar
Falcioni M., Paladin, G. & Vulpiani A. 1989 Intermittency and multifractality in magnetic dynamos. Europhys. Lett. 10, 201–206.Google Scholar
Finn J. M., Hanson J. D., Kan, I. & Ott E. 1991 Steady fast dynamo flows Phys. Fluids B 3, 1250–1269.Google Scholar
Finn, J. M. & Ott E. 1988 Chaotic flows and fast magnetic dynamos. Phys. Fluids 31, 2992–3011.Google Scholar
Galloway, D. & Frisch U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. 36, 58–83.Google Scholar
Gilbert A. D. 1988 Fast dynamo action in the Ponomarenko dynamo. Geophys. Astrophys. Fluid Dyn. 44, 214–258.Google Scholar
Gilbert, A. D. & Childress S. 1990 Evidence for fast dynamo action in a chaotic web. Phys. Rev. Lett. 65, 2133–2136.Google Scholar
Gilbert A. D., Frisch, U. & Pouquet A. 1988 Helicity is unnecessary for alpha effect dynamos, but it helps. Geophys. Astrophys. Fluid Dyn. 42, 151–161.Google Scholar
Hoyng P. 1987a Turbulent transport of magnetic fields: I. A simple mechanical model. Astron. Astrophys. 171, 348–356.Google Scholar
Hoyng P. 1987b Turbulent transport of magnetic fields: II. The roCle of fluctuations in kinematic theory. Astron. Astrophys. 171, 357–367.Google Scholar
Kazantsev A. P. 1967 Enhancement of a magnetic field by a conducting fluid. Zh. Eksp. Teor. Fiz. 53, 1806–1813. [Transl. Sov. Phys. JETP 26, 1031–1034 (1968)].Google Scholar
Klapper I. 1992 On fast dynamo action in chaotic helical cells. J. Fluid Mech. 239, 359–381.Google Scholar
Knobloch E. 1977 The diffusion of scalar and vector fields by homogeneous stationary turbulence. J. Fluid Mech. 83, 129–140.Google Scholar
Kraichnan R. H. 1974 Convection of a passive scalar by a quasi-uniform random straining field. J. Fluid Mech. 64, 737–762.Google Scholar
Kraichnan R. H. 1976a Diffusion of weak magnetic fields by isotropic turbulence. J. Fluid Mech. 75, 657–676.Google Scholar
Kraichnan R. H. 1976b Diffusion of passive-scalar and magnetic fields by helical turbulence. J. Fluid Mech. 77, 753–768.Google Scholar
Malik N. A. 1990 Distortion of material surfaces in steady and unsteady flows. In Proc. IUTAM Symp. on Topological Fluid Mechanics, Cambridge, U.K. August 1989 (ed. A. B. Tsinober & H. K. Moffatt), pp. 617–627. Cambridge University Press.
Moffatt H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.Google Scholar
Moffatt H. K. 1983 Transport effects associated with turbulence with particular attention to the influence of helicity. Rep. Prog. Phys. 46, 621–664.Google Scholar
Molchanov S. A., Ruzmaikin, D. D. & Sokoloff D. D. 1984 A dynamo theorem. Geophys. Astrophys. Fluid Dyn. 30, 242–259.Google Scholar
Novikov V. G., Ruzmaikin, A. A. & Sokoloff D. D. 1983 Kinematic dynamo in a reflection-invariant random field. Zh. Eksp. Teor. Fiz. 85, 909–918. [Transl. Sov. Phys. JETP, 58, 527–532 (1983)].Google Scholar
Otani N. J. 1988 Computer simulation of fast kinematic dynamos EOS. Trans. Am. Geophys. Union, 69, No. 44, Abstract No. SH51–15, p. 1366.Google Scholar
Ott, E. & Antonsen T. M. 1989 Fractal measures of passively convected vector fields and scalar gradients in chaotic fluid flows. Phys. Rev. 39A, 3660–3671 (1989).Google Scholar
Roberts P. H. 1967 An Introduction to Magnetohydrodynamics. Elsevier.
Soward A. M. 1987 Fast dynamo action in a steady flow. J. Fluid Mech. 180, 267–295.Google Scholar
Steenbeck, M. & Krause F. 1969 On the dynamo theory of stellar and planetary magnetic fields I., A.C. dynamos of solar type. Astron. Nachr. 291, 49–84.Google Scholar
Thompson M. 1990 Kinematic dynamo in random flows. Mat. Aplic. Comput. 9, 213–245.Google Scholar
Vainshtein S. I. 1981 Theory of small-scale magnetic fields. Zh. Eksp. Teor. Fiz. 83, 161–175 (1981). [Transl. Sov. Phys. JETP, 56, 86–94 (1982)].Google Scholar
Vainshtein, S. I. & Zel'dovich Ya, B. 1972 Origin of magnetic fields in astrophysics. Sov. Phys. Usp. 15, 159–172.Google Scholar
Zel'dovich Ya. B. 1957 The magnetic field in the two-dimensional motion of a conducting turbulent fluid. Sov. Phys. JETP 4, 460–462.Google Scholar
Zel'dovich Ya. B., Molchanov S. A., Ruzmaikin, A. A. & Sokolov D. D. 1988 Intermittency, diffusion and generation in a nonsteady random medium. Sov. Sci. Rev. C. Math. Phys. 7, 1–110.Google Scholar
Zel'dovich Ya. B., Ruzmaikin A. A., Molchanov, S. A. & Sokoloff D. D. 1984 Kinematic dynamo problem in a linear velocity field. J. Fluid Mech. 144, 1–11.Google Scholar