Rodion Stepanov

A non-local shell model of hydrodynamic and magnetohydrodynamic turbulence

We derive a new shell model of magnetohydrodynamic (MHD) turbulence in which the energy transfers are not necessarily local. Like the original MHD equations, the model conserves the total energy, magnetic helicity, cross-helicity and volume in phase space (Liouvilles theorem) apart from the effects of external forcing, viscous dissipation and magnetic diffusion. The model of hydrodynamic (HD) turbulence is derived from the MHD model setting the magnetic field to zero. In that case the conserved quantities are the kinetic energy and the kinetic helicity. In addition to a statistically stationary state with a Kolmogorov spectrum, the HD model exhibits multiscaling. The anomalous scaling exponents are found to depend on a free parameter α that measures the non-locality degree of the model. In freely decaying turbulence, the infra-red spectrum also depends on α. Comparison with theory suggests using α = −5/2. In MHD turbulence, we investigate the fully developed turbulent dynamo for a wide range of magnetic Prandtl numbers in both kinematic and dynamic cases. Both local and non-local energy transfers are clearly identified.

Fully developed turbulent dynamo at low magnetic Prandtl numbers

We investigate the dynamo problem in the limit of small magnetic Prandtl number (Pm) using a shell model of magnetohydrodynamic turbulence. The model is designed to satisfy conservation laws of total energy, cross helicity and magnetic helicity in the limit of inviscid fluid and null magnetic diffusivity. The forcing is chosen to have a constant injection rate of energy and no injection of kinetic helicity nor cross helicity. We find that the value of the critical magnetic Reynolds number (Rm) saturates in the limit of small Pm. Above the dynamo threshold we study the saturated regime versus Rm and Pm. In the case of equipartition, we find Kolmogorov spectra for both kinetic and magnetic energies except for wave numbers just below the resistive scale. Finally the ratio of both dissipation scales (viscous to resistive) evolves as Pm −3/4 for Pm < 1.

FARADAY SIGNATURE OF MAGNETIC HELICITY FROM REDUCED DEPOLARIZATION

Using one-dimensional models, we show that a helical magnetic field with an appropriate sign of helicity can compensate the Faraday depolarization resulting from the superposition of Faraday-rotated polarization planes from a spatially extended source. For radio emission from a helical magnetic field, the polarization as a function of the square of the wavelength becomes asymmetric with respect to zero. Mathematically speaking, the resulting emission occurs then either at observable or at unobservable (imaginary) wavelengths. We demonstrate that rotation measure (RM) synthesis allows for the reconstruction of the underlying Faraday dispersion function in the former case, but not in the latter. The presence of positive magnetic helicity can thus be detected by observing positive RM in highly polarized regions in the sky and negative RM in weakly polarized regions. Conversely, negative magnetic helicity can be detected by observing negative RM in highly polarized regions and positive RM in weakly polarized regions. The simultaneous presence of two magnetic constituents with opposite signs of helicity is shown to possess signatures that can be quantified through polarization peaks at specific wavelengths and the gradient of the phase of the Faraday dispersion function. Similar polarization peaks can tentatively also be identified for the bi-helical magnetic fields that are generated self-consistently by a dynamo from helically forced turbulence, even though the magnetic energy spectrum is then continuous. Finally, we discuss the possibility of detecting magnetic fields with helical and non-helical properties in external galaxies using the Square Kilometre Array.

Dissipation scales of kinetic helicities in turbulence

A systematic study of the influence of the viscous effect on both the spectra and the nonlinear fluxes of conserved as well as nonconserved quantities in Navier–Stokes turbulence is proposed. This analysis is used to estimate the helicity dissipation scale which is shown to coincide with the energy dissipation scale. However, it is shown using the decomposition of helicity into eigenmodes of the curl operator that viscous effects have to be taken into account for wave vectors smaller than the Kolmogorov wave number in the evolution of these eigencomponents of the helicity.

PHENOMENOLOGY OF TURBULENT DYNAMO GROWTH AND SATURATION

With a nonlocal shell model of magnetohydrodynamic turbulence we investigate numerically the turbulent dynamo action for low and high magnetic Prandtl numbers (Prm). The results obtained in the kinematic regime and along the way to dynamo saturation are understood in terms of a phenomenological approach based on the local (

Direct Numerical Simulation of Helical Magnetohydrodynamic Turbulence with TARANG Code

m{P}ml 1

A non local shell model for MHD turbulence

-->) or nonlocal (

Influence of low Pm on the turbulent MHD dynamo

m{P}mg 1

Investigation of dynamo action in the small Pm limit using a MHD shell model

-->) nature of the energy transfers. In both cases, the magnetic energy grows at a small scale and saturates as an inverse cascade.