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

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

Abstract

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.

Keywords

Poisson Bracket Cohomology Class Trigonometric Polynomial Compact Riemannian Manifold Galerkin Approximation 
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.

Copyright information

© Russ. Acad. Sciences 1983

Authors and Affiliations

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

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