Magnetic field generation by pointwise zero-helicity three-dimensional steady flow of an incompressible electrically conducting fluid

Abstract

We introduce six families of three-dimensional space-periodic steady solenoidal flows, whose kinetic helicity density is zero at any point. Four families are analytically defined. Flows in four families have zero helicity spectrum. Sample flows from five families are used to demonstrate numerically that neither zero kinetic helicity density nor zero helicity spectrum prohibit generation of large-scale magnetic field by the two most prominent dynamo mechanisms: the magnetic α-effect and negative eddy diffusivity. Our computations also attest that such flows often generate small-scale field for sufficiently small magnetic molecular diffusivity. These findings indicate that kinetic helicity and helicity spectrum are not the quantities controlling the dynamo properties of a flow regardless of whether scale separation is present or not.

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

“…For numerical experimentation we have applied two types of flows: the so-called cosine flows introduced in [25], and their curls considered in [2]. They are of interest in that the latter have a pointwise zero vorticity (kinematic) helicity, and the former have a pointwise zero velocity helicity, and nevertheless they are capable of both small-and large-scale magnetic field generation (see ibid).…”

“…While the entries of the α-effect tensor, A k l , are smooth functions of η, the graph of γ α (17) has cusps at the points η, where α 1 < α 2 = 0 < α 3 [25] (see, e.g., two such cusps in Fig. 2).…”

“…is the so-called tensor of magnetic eddy diffusivity correction. Again assuming that the mean field b 0 is a Fourier harmonics (15), we find [25]…”

“…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.…”

“…In fact, in this class of MHD systems the inverse energy cascade is important, energy proliferating from small scales towards large ones; the source of energy for the developing large-scale magnetic, hydrodynamic or combined MHD perturbation is the forcing applied to maintain the perturbed (also sometimes called basic) fluid flow.Evaluation of the α-effect and eddy diffusivity tensors involves solving the so-called auxiliary problems, which are linear problems for the respective elliptic operators of linearization. In the large-scale dynamo problem, computing the magnetic α-effect tensor requires considering three such problems; the number increases to 12, when the tensor of eddy diffusivity is sought (unless auxiliary problems for the adjoint operator come into play, decreasing the number of auxiliary problems to be treated to just 6, see, e.g., [1,25]). Interesting results (e.g., instability to largescale perturbation or dynamos) are typically obtained for relatively small molecular viscosity or magnetic diffusivity.…”