Growth rate degeneracies in kinematic dynamos

Abstract

We consider the classical problem of kinematic dynamo action in simple steady flows. Due to the adjointness of the induction operator, we show that the growth rate of the dynamo will be exactly the same for two types of magnetic boundary conditions: the magnetic field can be normal (infinite magnetic permeability, also called pseudo-vacuum) or tangent (perfect electrical conductor) to the boundaries of the domain. These boundary conditions correspond to well-defined physical limits often used in numerical models and relevant to laboratory experiments. The only constraint is for the velocity field u to be reversible, meaning there exists a transformation changing u into -u. We illustrate this surprising property using S2T2 type of flows in spherical geometry inspired by Dudley and James (1989). Using both types of boundary conditions, it is shown that the growth rates of the dynamos are identical, although the corresponding magnetic eigenmodes are drastically different.

5 pages, 3 figures

References in corpus (2)

• Generation of magnetic field by dynamo action in a turbulent flow of liquid sodium
• Kinematic dynamo action in square and hexagonal patterns

Cited by in corpus (8)

• Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers
• Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: single mode solutions
• Kinematic dynamo action in square and hexagonal patterns
• Enhanced dynamo growth in nonhomogeneous conducting fluids
• Fast and Furious dynamo action in the anisotropic dynamo
• Kinematic dynamos in triaxial ellipsoids
• Large $\mathrm{Pm}$ small-scale kinematic dynamo in protoneutron stars
• Effects of radial conductivity variation on the Ponomarenko dynamo