Nonlinear dynamo action in hydrodynamic instabilities driven by shear

Abstract

The present article considers dynamo action in flows driven by Ekman and Taylor–Couette instabilities. The system comprises a rotating, horizontal plane layer of electrically conducting fluid. The top of the layer is at rest, while the base has a given velocity that drives a net shear across the layer. The fluid flow in the system is parameterised by a Reynolds number Re, the square root of the Taylor number τ, and the angle ϑ between the rotation vector and the vertical. When the rotation vector has a vertical component, ϑ ≠ π/2, then in the limit of large τ, the shear takes the form of an thin Ekman layer localised at the bottom boundary. This may be subject to a hydrodynamic Ekman instability which for moderate Re generates steady cat's eyes in a co-moving frame. The flow gives dynamo action, amplifying magnetic fields in the form of sheets concentrated on hyperbolic points and their unstable manifolds. At greater Re, the cat's eyes become three-dimensional, and time-dependent amplifying fields with a more irregular structure. In nonlinear regimes the magnetic field and fluid flow show a relaxation oscillation for moderate Re, and for higher values, oscillatory fluid flows sustain tubes of magnetic field. When the rotation vector is entirely horizontal, ϑ =π /2, the shear driven is uniform across the layer. This can become subject to a primary Taylor–Couette instability driving a cellular flow, and with a further increase of Re, there is a secondary bifurcation to steady, wavy rolls. This flow supports a dynamo and its properties are studied in kinematic and nonlinear regimes.

Keywords:

• Dynamo
• Ekman instability
• Taylor–Couette instability
• Astrophysical magnetic fields

Acknowledgements

The authors are grateful to the Leverhulme Trust who support the work of PZ through a research grant (F00/144/H). They thank Dr Yannick Ponty for use of some of his two-dimensional kinematic codes and stability codes, and for valuable discussions, and also benefitted from many useful discussions with professors Chris Jones, Andrew Soward and Keke Zhang, and comments from the referees. The computations were undertaken on the cluster `saturn' at Exeter, supported by a PPARC grant (PPA/E/S/2000/00673).

Notes

¹ There is an error in the discussion of the last paragraph of paper II: there the quantity q should be replaced by Pm.