Fully developed MHD turbulence near critical magnetic Reynolds number

Abstract

Liquid-sodium-cooled breeder reactors may soon be operating at magnetic Reynolds numbers R^M where magnetic fields can be self-excited by a dynamo mechanism (as first suggested by Bevir 1973). Such flows have kinetic Reynolds numbers R^V of the order of 10⁷ and are therefore highly turbulent.This leads us to investigate the behaviour of MHD turbulence with high R^V and low magnetic Prandtl numbers. We use the eddy-damped quasi-normal Markovian closure applied to the MHD equations. For simplicity we restrict ourselves to homogeneous and isotropic turbulence, but we do include helicity.We obtain a critical magnetic Reynolds number R^M_c of the order of a few tens (non-helical case) above which magnetic energy is present. R^M_c is practically independent of R^V (in the range 40 to 10⁶). R^M_c can be considerably decreased by the presence of helicity: when the overall size of the flow L is much larger than the integral scale l₀, R^M_c can drop below unity as suggested by an α-effect argument. When L ≈ l₀ the drop can still be substantial (factor of 6) when helicity is a maximum. We examine how the turbulence is modified when R^M crosses R^M_c: presence of magnetic energy, decreased kinetic energy, steepening of kinetic-energy spectrum, etc.We make no attempt to obtain quantitative estimates for a breeder reactor, but discuss some of the possible consequences of exceeding R^M_c, such as decreased turbulent heat transport. More precise information may be obtained from numerical simulations and experiments (including some in the subcritical regime).