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The amplification of a weak magnetic field by turbulent motion of a fluid of large conductivity

Saffman, P. G. (1964) The amplification of a weak magnetic field by turbulent motion of a fluid of large conductivity. Journal of Fluid Mechanics, 18 (3). pp. 449-465. ISSN 0022-1120. https://resolver.caltech.edu/CaltechAUTHORS:20181119-161053393

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Abstract

This paper considers the question of how large the magnetic energy can be in stationary homogeneous turbulence at large Reynolds number of an incompressible conducting fluid for which the magnetic diffusivity λ is much less than the kinematic viscosity ν. An approximate equation, that takes account of the effect of the Lorentz forces on the turbulence, is proposed for the spectrum function of the magnetic energy for wave-numbers lying in the equilibrium range. This equation is used to determine the magnetic spectrum function and the level of magnetic energy for the case when a statistically steady magnetic field is maintained by a relatively small input of magnetic energy, by a weak applied field say, on the scale of the energy-containing eddies; it being supposed as suggested by earlier work that the magnetic energy would eventually die away in the absence of external electromotive forces. The results are complicated, there being essentially four different régimes depending in a fairly involved way on the relative values of λ/ν, the turbulent Reynolds number, and the ratio of the energy of the applied field to the energy of the turbulence. The main conclusions about the amplification factor are shown diagrammatically in figure 4. The wavenumber at which the magnetic spectrum has its maximum tends to decrease, as the applied field is increased, from the conduction cut-off wave-number (ε | νλ^2)^(1/2) to values lying in the inertial subrange, much less than (ε | ν^3)^(1/4). The case of a turbulent dynamo is also examined, and it is concluded that, if it exists, the equilibrium magnetic energy would be given by Batchelor's criterion of equipartition of energy between the magnetic field and the small, energy-dissipating eddies. The magnetic energy in the dynamo is found to lie mainly in Fourier components of wave-number about (ε | νλ^2)^(1/2).


Item Type:Article
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https://doi.org/10.1017/s0022112064000337DOIArticle
Additional Information:© 1964 Cambridge University Press. (Received 6 June 1963) This paper was written during the tenure of a Resident Research Fellowship at the Jet Propulsion Laboratory, California Institute of Technology.
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Issue or Number:3
Record Number:CaltechAUTHORS:20181119-161053393
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20181119-161053393
Official Citation:Saffman, P. (1964). The amplification of a weak magnetic field by turbulent motion of a fluid of large conductivity. Journal of Fluid Mechanics, 18(3), 449-465. doi:10.1017/S0022112064000337
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:91055
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:20 Nov 2018 16:06
Last Modified:03 Oct 2019 20:31

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