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Critical magnetic Prandtl number for small-scale dynamo.

Alexander A Schekochihin1, Steven C Cowley, Jason L Maron

  • 1Plasma Physics Group, Imperial College, Blackett Laboratory, Prince Consort Road, London SW7 2BW, United Kingdom. as629@damtp.cam.ac.uk

Physical Review Letters
|March 6, 2004
PubMed
Summary

The critical magnetic Reynolds number for small-scale dynamos depends on the Reynolds number. A low magnetic Prandtl number can shut down dynamos, even when the magnetic Reynolds number is sufficient.

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Area of Science:

  • Astrophysics
  • Geophysics
  • Plasma Physics

Background:

  • Small-scale dynamos are crucial for generating magnetic fields in various astrophysical and geophysical contexts.
  • Understanding the conditions for dynamo operation is essential for explaining cosmic magnetic fields.

Purpose of the Study:

  • To investigate the dependence of the critical magnetic Reynolds number (Rm(c)) for nonhelical small-scale dynamos on the Reynolds number (Re).
  • To determine the influence of the magnetic Prandtl number (Pr(m)) on dynamo suppression.
  • To explore the existence of finite critical values for Pr(m) and Rm(c) in the limit of infinite Re.

Main Methods:

  • Numerical simulations were employed to explore the parameter space of the small-scale dynamo.
  • The study systematically varied the magnetic Reynolds number (Rm) and the Reynolds number (Re).

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  • The magnetic Prandtl number (Pr(m) = Rm/Re) was analyzed in relation to dynamo onset and suppression.

    • Main Results:

    • The critical magnetic Reynolds number (Rm(c)) for nonhelical small-scale dynamos is shown to be dependent on the Reynolds number (Re).
    • A critical magnetic Prandtl number (Pr(m,c) < 1) was identified, below which dynamo action is suppressed, even for sufficiently large Rm.
    • Two scenarios for the limit of Re → ∞ were proposed: a finite Pr(m,c) or Pr(m,c) → 0 with Rm(c) approaching a large constant.
    • If Pr(m,c) is finite, dynamo sustainability requires magnetic field existence at sub-flow scales, effectively becoming a large-Pr(m) dynamo.
    • A lower bound of Rm(c) ≈ 220 was established for Pr(m) ≤ 1/8, exceeding values in planets and liquid-metal experiments.

    Conclusions:

    • The onset and sustainability of small-scale dynamos are sensitive to the interplay between Re and Pr(m).
    • Dynamo suppression at low Pr(m) suggests that certain astrophysical and geophysical environments may not sustain small-scale dynamos.
    • The findings provide crucial insights into the fundamental physics of magnetic field generation and have implications for planetary magnetism and laboratory experiments.