Effects of Magnetic and Kinetic Helicities on the Growth of Magnetic Fields in Laminar and Turbulent Flows by Helical Fourier Decomposition

Abstract

We present a numerical and analytical study of incompressible homogeneous conducting fluids using a helical Fourier representation. We analytically study both small-and large-scale dynamo properties, as well as the inverse cascade of magnetic helicity, in the most general minimal subset of interacting velocity and magnetic fields on a closed Fourier triad. We mainly focus on the dependency of magnetic field growth as a function of the distribution of kinetic and magnetic helicities among the three interacting wavenumbers. By combining direct numerical simulations of the full magnetohydrodynamics equations with the helical Fourier decomposition we numerically confirm that in the kinematic dynamo regime the system develops a large-scale magnetic helicity with opposite sign compared to the small-scale kinetic helicity, a sort of triad-by-triad α-effect in Fourier space. Concerning the small-scale perturbations, we predict theoretically and confirm numerically that the largest instability is achieved for the magnetic component with the same helicity of the flow, in agreement with the Stretch-Twist-Fold mechanism. Vice versa, in presence of a Lorentz feedback on the velocity, we find that the inverse cascade of magnetic helicity is mostly local if magnetic and kinetic helicities have opposite sign, while it is more nonlocal and more intense if they have the same sign, as predicted by the analytical approach. Our analytical and numerical results further demonstrate the potential of the helical Fourier decomposition to elucidate the entangled dynamics of magnetic and kinetic helicities both in fully developed turbulence and in laminar flows.

Fully resolved array of simulations investigating the influence of the magnetic Prandtl number on magnetohydrodynamic turbulence

“…Analyses of triad interactions and shell-to-shell energy transfers show that energy is transferred from the velocity field at the forcing scale to the magnetic and velocity fields at all scales in a way that depends on the separation between the giving and receiving scales and the energy contained in the involved scales, amongst other things [29,[36][37][38][39][40]. Therefore it is reasonable to expect a consistent scaling of ε K /ε M with Pm that is not affected by whether Pm < 1 or Pm > 1, as we see in Fig.…”

Helicity Dynamics, Inverse, and Bidirectional Cascades in Fluid and Magnetohydrodynamic Turbulence: A Brief Review

“…This result further led to the derivation of global regularity in these truncated systems (Biferale & Titi, ). Such a helical decomposition is rather common; it has been used in so‐called shell models (Lessines et al, ) or in the context of the dynamo problem (Linkmann et al, ).…”

“…The figure here is a simple illustration of the phenomenon. Detailed triadic interactions of such an inverse cascade are studied in Linkmann et al () and Linkmann and Dallas () using a classical helical decomposition, with both analytical and numerical tools at resolutions of 512 3 points. These authors show in particular that the relative signs of kinetic and magnetic helicity play a direct role in the emergence of large‐scale dynamo fields, with growth rates that are determined in the ideal (nondissipative) case and thus independent of the magnitude of the magnetic Reynolds number, with no subsequent so‐called α quenching (see also Pouquet et al, ).…”

Triad interactions and the bidirectional turbulent cascade of magnetic helicity

“…(11) consisting of the coupling among velocity (left) and magnetic-velocity (right) triadic interactions as in Ref. [21]. The magnetic-velocity interactions correspond to several terms in Eqs.…”