Dynamo instability and feedback in a stochastically driven system

  • H. K. Moffatt
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 12)


In §1, a method of treatment of the equation
where aijk(
,t) is a random tensor field of known statistical properties is reviewed, with particular reference (i) to the magnetohydrodynamic turbulent dynamo problem, and (ii) to the problem of diffusion of a passive scalar field by turbulent motion. The result of Steenbeck, Krause and Radler (1966), that in the former context dynamo action can occur (i.e. the ensemble average of
can grow without limit) provided the statistics of the turbulent field lack reflexional symmetry, is discussed within the framework of the above general equation.

In §2 and 3, the feedback mechanism in the magnetohydrodynamic context is considered. It is supposed that a velocity field lacking reflexional symmetry is generated in an electrically conducting fluid by a random body force of known statistical properties.Conditions are then conducive to the growth of large scale magnetic field perturbations. The growth is limited by the fact that the growing Lorentz force progressively modifies the statistical structure of the velocity field, until ultimately a statistical equilibrium is achieved. It is shown that in this equilibrium the magnetic energy density may exceed the kinetic energy density by a factor O(L/ℓ) ≫ 1, where L is the scale of the magnetic field, and ℓ the scale of the turbulence.


Velocity Field Fourier Component Eddy Diffusivity Kinetic Energy Density Reflexional Symmetry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Moffatt, H.K. 1970a,b. J. Fluid Mech. 41, 435 and 44, 705Google Scholar
  2. Moffatt, H.K. 1971. (submitted for publication).Google Scholar
  3. Parker, E.N. 1955. Astrophys, J. 122, 2930Google Scholar
  4. Roberts, P.H. 1971. Lectures in applied mathematics (ed. W.H. Reid). Providence: American Mathematical Society.Google Scholar
  5. Saffman, P.G. 1960. J. Fluid Mech. 8, 273.Google Scholar
  6. Steenbeck, M., Krause, F. and Radler, K.-H., 1966. Z. Naturforsch. 21a, 369.Google Scholar
  7. Taylor, G.I. 1921. Proc. Lond. Math. Soc. 20, 196.Google Scholar
  8. Weiss, N.O. 1971. Quart. J. Roy. Astr. Soc. (to appear).Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • H. K. Moffatt
    • 1
  1. 1.Dept. of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeEngland

Personalised recommendations