Anisotropic nonperturbative zero modes for passively advected magnetic fields

Alessandra Lanotte; Andrea Mazzino

doi:10.1103/PhysRevE.60.R3483

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Anisotropic nonperturbative zero modes for passively advected magnetic fields

Alessandra Lanotte¹ and Andrea Mazzino^2,3

¹CNRS, Observatoire de la Côte d’Azur, Boîte Postale 4229, 06304 Nice Cedex 4, France
²INFM-Dipartimento di Fisica, Università di Genova, I-16142 Genova, Italy
³The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

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Phys. Rev. E 60, R3483(R) – Published 1 October, 1999

DOI: https://doi.org/10.1103/PhysRevE.60.R3483

Abstract

An analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d-dimensional kinematic magnetohydrodynamics problem in the presence of a mean magnetic field. The velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent $ξ .$ Inertial-range anisotropic contributions to the scaling exponents, $ζ_{j},$ of second-order magnetic correlations are associated with zero modes and have been calculated nonperturbatively. For $d = 3,$ the limit $ξ \mapsto 0$ yields $ζ_{j} = j - 2 + ξ {(2 j}^{3} {+ j}^{2} - 5 j - 4) / [{2 (4 j}^{2} - 1)],$ where j $(j >~ 2)$ is the order in the Legendre polynomial decomposition applied to correlation functions. Conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made.

References (13)

A. N. Kolmogorov, C. R. (Dokl.) Acad. Sci. URSS 30, 301 (1941).
N. V. Antonov, e-print archive chao-dyn/9808011.
S. G. Saddoughi and S. V. Veeravalli, J. Fluid Mech. 268, 333 (1994).
V. Borue and S. A. Orszag, J. Fluid Mech. 306, 293 (1994).
I. Arad, B. Dhruva, S. Kurien, V. S. L’vov, I. Procaccia, and K. R. Sreenivasan, Phys. Rev. Lett. 81, 5330 (1998).
I. Arad, L. Biferale, I. Mazzitelli, and I. Procaccia, Phys. Rev. Lett. 82, 5040 (1999).
Ya.B. Zeldovich, A.A. Ruzmaikin, and D.D. Sokoloff, Magnetic fields in Astrophysics (Gordon and Breach, New York, 1983).
U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, England, 1995).
M. Vergassola, Phys. Rev. E 53, R3021 (1996).
L. Ts. Adzhemyan and N. V. Antonov, Phys. Rev. E 58, 7381 (1998).
I. S. Gradshtejn and I.M. Ryzhik, Table of Integrals Series and Products (Academic Press, New York, 1965).
We are particularly grateful to N.V. Antonov, I. Arad, L. Biferale, and I. Procaccia to have found an error in a previous expression for $ζ_{j} .$
A. P. Kazantzev, Zh. Eksp. Teor. Fiz. 53, 1806 (1967) [Sov. Phys. JETP 26, 1031 (1968)].

[c1] A. N. Kolmogorov, C. R. (Dokl.) Acad. Sci. URSS 30, 301 (1941).

[c2] N. V. Antonov, e-print archive chao-dyn/9808011.

[c3] S. G. Saddoughi and S. V. Veeravalli, J. Fluid Mech. 268, 333 (1994).

[c4] V. Borue and S. A. Orszag, J. Fluid Mech. 306, 293 (1994).

[c5] I. Arad, B. Dhruva, S. Kurien, V. S. L’vov, I. Procaccia, and K. R. Sreenivasan, Phys. Rev. Lett. 81, 5330 (1998).

[c6] I. Arad, L. Biferale, I. Mazzitelli, and I. Procaccia, Phys. Rev. Lett. 82, 5040 (1999).

[c7] Ya.B. Zeldovich, A.A. Ruzmaikin, and D.D. Sokoloff, Magnetic fields in Astrophysics (Gordon and Breach, New York, 1983).

[c8] U. Frisch, Turbulence. The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, England, 1995).

[c9] M. Vergassola, Phys. Rev. E 53, R3021 (1996).

[c10] L. Ts. Adzhemyan and N. V. Antonov, Phys. Rev. E 58, 7381 (1998).

[c11] I. S. Gradshtejn and I.M. Ryzhik, Table of Integrals Series and Products (Academic Press, New York, 1965).

[c12] We are particularly grateful to N.V. Antonov, I. Arad, L. Biferale, and I. Procaccia to have found an error in a previous expression for $ζ_{j} .$

[c13] A. P. Kazantzev, Zh. Eksp. Teor. Fiz. 53, 1806 (1967) [Sov. Phys. JETP 26, 1031 (1968)].

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Anisotropic nonperturbative zero modes for passively advected magnetic fields

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