Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-08T16:33:36.139Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact evaluation of the effect of an arbitrary mean flow in kinematic dynamo theory

Published online by Cambridge University Press:  20 April 2006

P. Hoyng
P. Hoyng
Affiliation:
Laboratory for Space Research, Beneluxlaan 21, 3527 HS Utrecht, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

An exact derivation is given of the dynamo equation in the presence of an arbitrary incompressible mean flow v0. A mixed representation is used specifying v0 in terms of a Lagrangian displacement, while Eulerian coordinates are employed for the turbulent velocity v superimposed on v0.

When the first-order-smoothing approximation is made (FOSA; valid when the turbulent velocity v has a short correlation time τc) the usual dynamo equation is recovered, except that the turbulent velocity v in the tensors αis and βisk is replaced by $\overline{\boldmath v}$. The bar represents the effect of advection and is expressed solely in terms of the Lagrangian coordinate specifying the mean flow v0. Thus the intuitive idea is confirmed that dynamo action depends only on velocity correlation functions measured at a point comoving with the mean flow. The result admits easy evaluation in actual model situations. This is illustrated with an example tailored to the solar dynamo. A shear in v0 causes a (kinematic) anisotropy in the tensors αis and βisk. This can be a large effect, which comes on top of the intrinsic (dynamical) anisotropy in the velocity correlation functions. Subsequently, the analysis is extended beyond FOSA up to arbitrary order, relevant for long correlation times τc on the basis of the work of Van Kampen (1974). It is shown that the same formalism is also applicable to the problem of turbulent transport of a scalar.

Conditions for applicability of the work are (1) very large magnetic Reynolds number, (2) incompressible flows v and v0, (3) stationary mean flow v0, and (4) correlation time τc [Lt ] period of the dynamo.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 151 , February 1985 , pp. 295 - 309
Copyright
© 1985 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

Andrews, D. G. & McIntyre M. E.1978 J. Fluid Mech. 89, 609.
Braginskii S. I.1965a Soviet Phys. JETP 20, 726.
Braginskii S. I.1965b Soviet Phys. JETP 20, 1462.
Cowling T. G.1981 Ann. Rev. Astron. Astrophys. 19, 115.
Knobloch E.1977 J. Fluid Mech. 83, 129.
Knobloch E.1978 Astrophys. J. 225, 1050.
Krause F.1973 Z. angew. Math. Mech. 53, 475.
Krause F.1976 Z. angew. Math. Mech. 56, 172.
Krause, F. & Rädler K.-H.1971 In Ergebnisse der Plasmaphysik und der Gaselektronik (ed. Rompe, R. & Steenbeck, M.), Bd. II, p. 1.
Krause, F. & Rädler K.-H.1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon.
Moffatt H. K.1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Rädler K.-H.1980 Astron. Nachr. 301, 101.
Roberts P. H.1967 An Introduction to Magnetohydrodynamics: Longmans.
Soward A. M.1972 Phil. Trans. R. Soc. Lond. A 272, 431.
Stix M.1981 Solar Phys. 74, 79.
Van Kampen N. G.1974 Physica 74, 239.
Van Kampen N. G.1976 Phys. Rep. 24, 171.
Van Kampen N. G.1981 Stochastic Processes in Physics and Chemistry. North-Holland.