Pointwise vanishing velocity helicity of a flow does not preclude magnetic field generation

Abstract

Pointwise zero velocity helicity density is shown not to prevent steady flows from acting as kinematic dynamos. We present numerical evidences that such flows can generate both small-scale magnetic fields as well as, by the magnetic α-effect or negative eddy diffusivity mechanisms, largescale ones. The flows are constructed as curls of analytically defined space-periodic steady solenoidal flows, whose vorticity helicity (i.e., kinetic helicity) density is everywhere zero.

Chaotic transients and hysteresis in an $α^{2}$ dynamo model

“…Next, we set f 0 = 0.07, ν = η = 0.005 and vary σ as a control parameter. Using σ as the control parameter is a natural choice, since it is known that the presence of kinetic helicity in the flow can be favorable for magnetic field generation [25], although it is not strictly needed for either small-or large-scale dynamo to operate (see [26,27] and references therein).…”

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).…”

“…Consequently, for large L the Toeplitz matrix of size L × (L + 1) in the l.h.s. of (24) becomes numerically degenerate, i.e., its rank effectively falls below L. To construct an approximant under such adverse numerical conditions, it was proposed in [15] to compute the singular value decomposition of this matrix and to regard its effective rank as equal to the number of singular values whose absolute value exceeds the given relative tolerance tol (i.e., is not smaller than tol (f (1) , f (2) , ..., f (M +L−1) , f (M +L) ) , where · is the standard Lebesgue space L 2 norm), decreasing the degrees of the polynomials involved in the Padé approximation, M and L, by the number of the "missing" dimensions. To counter noise due to rounding errors, tol = 10 −14 was often used in [15].…”

“…We have checked that for η > η cr = 0.00420516 small-scale modes from the subspace, where s 1 and s 2 are located, are not generated, but at η = η cr the small-scale generation starts in the subspace, where s 3 resides. It is known [36,25,2] that the point of the onset of the small-scale generation is typically associated with a singularity of the α-effect or eddy diffusivity tensors; this is the case for the flow under consideration. We observe in Fig.…”

“…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.…”