Evolution of a magnetic field under the action of transfer and diffusion

  • Vladimir I. Arnold
Part of the Vladimir I. Arnold - Collected Works book series (ARNOLD, volume 2)


The equation of a (divergence-free) magnetic field H in the flow of an incompressible conducting fluid with velocity field v and coefficient of diffusion μ has the form

= {v,H} + μΔH,

where Δ = −rot rot and {·,·} is the Poisson bracket. The fields H and v are assumed to be 2π-periodic in (x, y, z). The flow

v = (cos y + sin z, cos z + sin x, cos x + sin y)

exponentially stretches the fluid particles (the increment is of order 0.15, cf. [1]). The calculation of the eigenvalue of the operator A = μ −1{v,·} + Δ with the largest real part was carried out by Korkin [2] for μ −1 ≤ 18 (about 20,000 harmonics were taken into account in the Galerkin approximation). The growth of H was discovered for 9 ≤ μ −1 ≤ 17, with the largest increment for μ −1 ≈ 12 being of order 0.01.


Poisson Bracket Cohomology Class Trigonometric Polynomial Compact Riemannian Manifold Galerkin Approximation 
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Copyright information

© Russ. Acad. Sciences 1983

Authors and Affiliations

  • Vladimir I. Arnold
    • 1
  1. 1.HeidelbergGermany

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