Abstract
We investigate the dynamo effect generated by an incompressible, helicity-free flow drawn from the Kraichnan statistical ensemble. The quantum formalism introduced by Kazantsev [A. P. Kazantsev, Sov. Phys. JETP 26, 1031–1034 (1968)] is shown to yield the growth rate and the spatial structure of the magnetic field. Their dependences on the magnetic Reynolds number and the Prandtl number are analyzed. The growth rate is found to be controlled by the largest between the diffusive and the viscous characteristic times. The same holds for the magnetic field correlation length and the corresponding spatial scales.
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REFERENCES
H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids (Cambridge University Press, Cambridge, 1978).
E. N. Parker, Cosmical Magnetic Fields (Clarendon Press, Oxford, 1979).
R. H. Kraichnan, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids 11:94–953 (1968).
A. P. Kazantsev, Enhancement of a magnetic field by a conducting fluid, Sov. Phys. JETP 26:103–1034 (1968).
F. Krause and K.-H. Rädler, Mean-field Magnetohydrodynamics and Dynamo Theory (Pergamon Press, Frankfurt, 1980).
S. Boldyrev, A Solvable Model for Nonlinear Mean Field Dynamo, astro-ph/0105354 (2001).
A. A. Ruzma?kin and D. D. Sokolov, The magnetic field in mirror-invariant turbulence, Sov. Astron. Lett. 7:38–390 (1981).
V. G. Novikov, A. A. Ruzma?kin, and D. D. Sokolov, Kinematic dynamo in a reflectioninvariant random field, Sov. Phys. JETP 58:52–532 (1983).
A. Gruzinov, S. Cowley, and R. Sudan, Small-scale-field dynamo, Phys. Rev. Lett. 77:434–4344 (1996).
M. Chertkov, G. Falkovich, I. Kolokolov, and M. Vergassola, Small-scale turbulent dynamo, Phys. Rev. Lett. 83:406–4068 (1999).
A. A. Schekochihin, S. A. Boldyrev, and R. M. Kulsrud, Spectra and Growth Rates of Fluctuating Magnetic Fields in the Kinematic Dynamo Theory with Large Magnetic Prandtl Numbers, astro-ph/0103333 (2001); submitted to Astrophys. J. (2001).
M. Vergassola, Anomalous scaling for passively advected magnetic fields, Phys. Rev. E 53:R302–R3024 (1996).
N. V. Antonov, A. Lanotte, and A. Mazzino, Persistence of small-scale anisotropies and anomalous scaling in a model of magnetohydrodynamics turbulence, Phys. Rev. E 61:658–6605 (2000).
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics, Vol. 2, J. Lumley, ed. (MIT Press, Cambridge, MA, 1975).
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, Inc., United States of America, 1953).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1965).
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Vincenzi, D. The Kraichnan–Kazantsev Dynamo. Journal of Statistical Physics 106, 1073–1091 (2002). https://doi.org/10.1023/A:1014089820881
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DOI: https://doi.org/10.1023/A:1014089820881