Negative magnetic eddy diffusivity due to oscillogenic α-effect

Abstract

We study large-scale kinematic dynamo action of steady mirror-antisymmetric flows of incompressible fluid, that involve small spatial scales only, by asymptotic methods of the multiscale stability theory. It turns out that, due to the magnetic α-effect in such flows, the large-scale mean field experiences harmonic oscillations in time on the scale O(εt) without growth or decay. Here ε is the spatial scale ratio and t is the fast time of the order of the flow turnover time. The interaction of the accompanying fluctuating magnetic field with the flow gives rise to an anisotropic magnetic eddy diffusivity, whose dependence on the direction of the large-scale wave vector generically exhibits a singular behaviour, and thus to negative eddy diffusivity for whichever molecular magnetic diffusivity. Consequently, such flows always act as kinematic dynamos on the time scale O(ε 2 t); for the directions at which eddy diffusivity is infinite, the large-scale mean-field growth rate is finite on the scale O(ε 3/2 t). We investigate numerically this dynamo mechanism for two sample flows.

Computation of Kinematic and Magnetic α-Effect and Eddy Diffusivity Tensors by Padé Approximation

“…For η > 0.05, the problem was preconditioned by the operator (−∇ 2 ) −1/2 , readily available in the Fourier space. The resolution of 64 3 Fourier harmonics was used. Energy spectra of the neutral modes s k decay for this flow by at least 9 orders of magnitude for the smallest considered η = 0.035.…”

“…We have tried the algorithm [15] for a set of tol values ranging between 10 −10 and 10 −20 using the MATLAB procedure provided by the authors of [15]. We have computed 65 first coefficients of the power series expansions of the symmetrized α-effect tensor entries ( s A k ) (n) l (which involve only odd powers of 1/η, see section 3.2) up to order η −129 terms with the spatial resolution of 64 3 Fourier harmonics. The MATLAB procedure has been requested to construct the [63/64] Padé Table 2: Order parameter L of the Padé approximants [2L − 1/2L] (ratios of polynomials in 1/η) constructed by the algorithm [15] for six independent entries of the symmetrized α-effect tensor s A. approximants for each entry.…”

“…Four Padé approximants [2L − 1/2L] of the entries of the α-effect tensor have been constructed for each considered L, using the resolution of 64 3 or 512 3 Fourier harmonics, and running our code with the double (real*8) or extended quadruple (real*16) precision of the floating-point number arithmetics. In computers built around Intel and compatible processors, the former is standard, and the latter is not supported by hardware, but is software-emulated; however, many compilers do not require modifying the Fortran source code to use it, all the floating-point data and computations can be readily promoted to the real*16 precision by using the appropriate compiler option such as -r16.…”

“…Mathematically, this does not present any difficulty -the large individual terms in the Taylor expansion around zero are guaranteed to mutually cancel out and yield the final result which is less than the size of the flow periodicity box and the flow velocity are order unity), when it is small, i.e., for large viscosities and diffusivities. When the parameter tends to the critical value for the onset of the small-scale instability (i.e., in the dynamo context, to the value for which generation of smallscale magnetic field starts), the tensors usually exhibit singular behaviour [36,25,3,2] of a simple pole type, bounding from above the radius of convergence of the series. This suggests to try Padé approximants for computing the tensors and the respective large-scale magnetic field / instability growth rates.…”