Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-29T13:13:38.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Fast dynamo action in a steady flow

Published online by Cambridge University Press:  21 April 2006

A. M. Soward
A. M. Soward
Affiliation:
School of Mathematics, The University, Newcastle upon Tyne, NE1 7RU, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The existence of fast dynamos caused by steady motion of an electrically conducting fluid is established by consideration of a two-dimensional spatially periodic flow: the velocity, which is independent of the vertical coordinate z, is finite and continuous everywhere but the vorticity is infinite at the X-type stagnation points. A mean-field model is developed using boundary-layer methods valid in the limit of large magnetic Reynolds number R. The magnetic field is confined to sheets, width of order R−½. The mean magnetic field lies and is uniform on horizontal planes: its direction is independent of time but rotates once about the vertical axis over a short distance 2πl, where l−1 = R½β and β is a vertical stretched wavenumber independent of R. Its alternating direction gives it a rope-like structure within the sheets. An α-effect is calculated for the model, whose strength for a given flow is a function of β and R. Two sources of α-effect are isolated whose relative importance depends critically on the size of β. When the vorticity is finite everywhere and β [Lt ] 1, the dynamo is ‘almost’ fast with growth rates of order (ln R)−1. The maximum growth rate ln (ln R)/ln R occurs when, correct to leading order, β is (ln R)−½. The asymptotic results valid for large R compare excellently with Roberts (1972) modal analysis for finite R.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 180 , July 1987 , pp. 267 - 295
Copyright
© 1987 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

Abramowitz, M. & Stegun, I. A. (eds) 1965 Handbook of Mathematical Functions. Dover.
Anufriyev, A. P. & Fishman, V. M. 1982 Magnetic field structure in the two-dimensional motion of a conducting fluid. Geomag. Aeron. 22, 245–248.Google Scholar
Arnol'D, V. I. & Korkina, E. I.1983 The growth of a magnetic field in a steady incompressible flow. Vest. Mosk. Un. Ta. Ser. 1, Math. Mec. 3, 43–46 (in Russian).Google Scholar
Braginsky, S. I. 1964 Self-excitation of a magnetic field during the motion of a highly conducting fluid. Z. Eksp. teor. Fiz. 47, 1084–1098. (Transl. Sov. Phys., J. Exp. Theor. Phys. 20, (1965) 726–735).Google Scholar
Bullard, E. C. & Gubbins, D. 1977 Generation of magnetic fields by fluid motions of global scale. Geophys. Astrophys. Fluid Dyn. 8, 43–56.Google Scholar
Busse, F. H. 1978 Introduction to the theory of geomagnetism. In Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 361–388. Academic.
Childress, S. 1967 Construction of steady-state hydromagnetic dynamos. 1. Spatially periodic fields. Courant Institute of Mathematical Sciences Rep. MF-53, pp. 1–36.Google Scholar
Childress, S. 1970 New solutions of the kinematic dynamo problem. J. Math. Phys. 11, 3063–3076.Google Scholar
Childress, S. 1979 Alpha-effect in flux ropes and sheets. Phys. Earth Planet. Int. 20, 172–180.Google Scholar
Childress, S. & Soward, A. M. 1984 On the rapid generation of magnetic fields. In Chaos in Astrophysics, Nato Advanced Research Workshop, Palm Coast, Florida, USA.
Dombre, T., Frisch, U., Greene, J. M., Hénon, M., Mehr, A. & Soward, A. M. 1986 Chaotic streamlines and Lagrangian turbulence: the ABC flows. J. Fluid Mech. 167, 353–391.Google Scholar
Galloway, D. J. & Frisch, U. 1984 A numerical investigation of magnetic field generation in a flow with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. 29, 13–18.Google Scholar
Galloway, D. & Frisch, U. 1986 Dynamo action in a family of flows with chaotic streamlines. Geophys. Astrophys. Fluid Dyn. 36, 53–83.Google Scholar
Ghil, M. & Childress, S. 1986 Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics. Applied Mathematical Science Series. Springer.
Krause, F. & Rädler, K.-H. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Akademic-Verlag and Pergamon.
Moffatt, H. K. 1970 Turbulent dynamo action at low magnetic Reynolds number. J. Fluid Mech. 41, 435–452.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Moffatt, H. K. & Proctor, M. R. E. 1985 Topological constraints associated with fast dynamo action. J. Fluid Mech. 154, 493–507.Google Scholar
Parker, E. N. 1979 Cosmical Magnetic Fields. Clarendon.
Roberts, G. O. 1972 Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc. Lond. A 271, 411–454.Google Scholar
Roberts, G. O. 1979 Fast viscous Bénard convection. Geophys. Astrophys. Fluid Dyn. 12, 235–272.Google Scholar
Soward, A. M. & Childress, S. 1986 Analytic theory of dynamos. Adv. Space Res. Proc. COSPAR meeting, Toulouse (to appear).
Vainshtein, S. I. & Zel'dovich, Ya. B.1972 Origin of magnetic fields in astrophysics. Sov. Phys. Usp. 15, 159–172.Google Scholar
Weiss, N. O. 1966 The expulsion of magnetic flux by eddies. Proc. R. Soc. Lond. A 293, 310–328.Google Scholar
Zel'Dovich, Ya. B., Ruzmaikin, A. A., Molchanov, S. A. & Sokolov, D. D.1984 Kinematic dynamo problem in a linear velocity field. J. Fluid Mech. 144, 1–11.Google Scholar
Zel'Dovich, Ya. B., Ruzmaikin, A. A. & Sokolov, D. D.1983 Magnetic Fields in Astrophysics. Gordon & Breach.