Saturation of a Ponomarenko Type Fluid Dynamo

Abstract

The kinematic dynamo problem is rather well understood in the case of laminar flows [1]. Several simple but clever examples have been found in the past [2, 3, 4, 5] and more realistic geometries can be easily studied numerically [6]. However, most flows of liquid metal are fully turbulent before reaching the dynamo threshold: indeed, the magnetic Prandtl number, Pm = μ₀σν, where μ₀ is the magnetic permeability of vacuum, σ is the electric conductivity and ν is the kinematic viscosity, is smaller than 10^-5 for all liquid metals. Since the dynamo action requires a large enough magnetic Reynolds number, Pm = μ₀σLU, where U is the fluid characteristic velocity and L is the characteristic scale, one expects to observe the dynamo effect when the kinetic Reynolds number, Re = UL/ν, is larger that 10⁶. The kinematic dynamo problem with a turbulent flow is much more difficult to solve. A theoretical approach exists only when the magnetic neutral modes grow at large scale. It has been shown that the role of turbulent fluctuations may be twofold: on one hand, they decrease the effective electrical conductivity and thus inhibits dynamo action by increasing Joule dissipation. On the other hand, they may generate a large scale magnetic field through the“alpha effect” or higher order similar effects [7]. Consequently, it is not known whether turbulent fluctuations inhibits or help dynamo action. More precisely, for a given configuration of the moving solid boundaries generating the flow, the behavior of the critical magnetic Reynolds number Rm _c for the dynamo threshold, as a function of the flow Reynolds number Re (respectively Pm) in the limit of large Re (respectively small Pm), is not known.