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Bochner's theorem and mean-field electrodynamics

Published online by Cambridge University Press:  26 February 2010

F. Krause  and
P. H. Roberts
F. Krause
Affiliation:
Zentralinstitut für Astrophysik, Potsdam, der Akademie der Wissenschaften der DDR.
P. H. Roberts
Affiliation:
School of Mathematics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, England.
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Extract

Central to the study of electromagnetic induction in turbulently moving conductors is the evolution of the ensemble averaged field, and so the subject of “mean field electrodynamics” has been born. It has provided a particularly fruitful approach to the dynamo problem, that is the study of the self-excitation of magnetic field by a moving fluid. It has been shown by Steenbeck, Krause and Rädler, in studies dating from 1966, that regeneration of field is efficient when the motions are sufficiently vigorous and lack mirror-symmetry. Using a commonly accepted approximation of the subject (the neglect of third order cumulants) which is valid when the correlation length and/or the correlation time of the turbulence are sufficiently small, and paying proper regard to a theorem due to Bochner, it is shown here that mirror-symmetric homogeneous turbulence cannot be regenerative.

Type
Research Article
Information
Mathematika , Volume 20 , Issue 1 , June 1973 , pp. 24 - 33
Copyright
Copyright © University College London 1973

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References

Bochner, S., Math. Ann. 108 (1933), 378.CrossRefGoogle Scholar
Bullard, E. C., in The Nature of the Solid Earth, ed. Robertson, E. C. (New York, McGraw-Hill, 1973).Google Scholar
Childress, S., Reports AFOSR-67-0124 and 0976 (New York University; Courant Institute, 1967).Google Scholar
Krause, F., Habilitationsschrift (University of Jena, 1967).Google Scholar
Krause, F. and Rädler, K.-H., in Handbuch der Plasmaphysik und Gaselektronik, ed. Rompe, R. and Steenbeck, M. (Berlin; Acad. Verlag., 1970).Google Scholar
Krause, F. and Roberts, P. H., Astrophys. J., 181 (1973), 977.CrossRefGoogle Scholar
Lerche, I., Astrophysics and Space Science, 19 (1973), 189.CrossRefGoogle Scholar
Moffatt, K., J. Fluid Mech., 35, (1968), 117.CrossRefGoogle Scholar
Moffatt, K., J. Fluid Mech., 41 (1969), 435.CrossRefGoogle Scholar
Radler, K.-H., Dissertation (University of Jena, 1966).Google Scholar
Radler, K.-H., Z. Naturforsch., 23a (1968), 1841.Google Scholar
Roberts, P. H., in Lectures in Applied Mathematics, 14. ed. Reid, W. H. (Providence: American Math. Soc, 1971).Google Scholar
Roberts, P. H. and Stix, M., Technical note NCAR/TN/1A-60 (Boulder; National Center for Atmospheric Research, 1971).Google Scholar
Steenbeck, M. and Krause, F., Astronom. Nach., 291 (1969), 49.CrossRefGoogle Scholar
Steenbeck, M. and Krause, F., Krause, F. and Radler, K.-H., Z. Naturforsch., 21a (1966), 369.Google Scholar
Sweet, P. A., Mon. Not. Roy. Astro. Soc, 110 (1950), 69.Google Scholar
Vainshtein, S. I., Soviet Phys. JETP, 34 (1972), 327.Google Scholar
Vainshtein, S. I. and Zel'dovich, Ya. B., Soviet Phys. Usp., 15 (1972), 159.CrossRefGoogle Scholar
Weiss, N. O., Q. J. Roy. Astro. Soc, 12 (1971), 432.Google Scholar