Shell Model Of Magnetohydrodynamic Turbulence Research Articles

Small-scale Kazantsev-Kraichnan dynamo in a MHD shell approach

The small-scale magnetic energy generation in a turbulent velocity field is studied by two different approaches. One of them is based on the Kazantsev-Kraichnan model developed for turbulence with short-time velocity correlations, and the other uses the shell model of magnetohydrodynamic turbulence, describing the turbulent energy cascade on a finite number of spectral shells. We have found that the injection of weak magnetic field at the initial moment in both models leads to an exponential growth of magnetic energy and tried to determine whether these effects are of the same or different nature. The investigations have shown that the rates of growths and magnetic energy spectra in two approaches can be very much different, which can be attributed to the contradictions of the model assumptions and unknown correlation time. The discussion of these contradictions allows us to formulate a possible explanation, which is likely related to the fact that the small-scale magnetic field generation is under the influence of some spectral subrange, rather than the entire kinetic spectrum. Varying the correlation time of the velocity field and considering the spectral regions, we have determined the range of kinetic energy spectrum responsible for the small-scale dynamo generation.

JOINT INVERSE CASCADE OF MAGNETIC ENERGY AND MAGNETIC HELICITY IN MHD TURBULENCE

We show that oppositely directed fluxes of energy and magnetic helicity coexist in the inertial range in fully developed magnetohydrodynamic (MHD) turbulence with small-scale sources of magnetic helicity. Using a helical shell model of MHD turbulence, we study the high Reynolds number magnetohydrodynamic turbulence for helicity injection at a scale that is much smaller than the scale of energy injection. In a short range of scales larger than the forcing scale of magnetic helicity, a bottleneck-like effect appears, which results in a local reduction of the spectral slope. The slope changes in a domain with a high level of relative magnetic helicity, which determines that part of the magnetic energy related to the helical modes at a given scale. If the relative helicity approaches unity, the spectral slope tends to $-3/2$. We show that this energy pileup is caused by an inverse cascade of magnetic energy associated with the magnetic helicity. This negative energy flux is the contribution of the pure magnetic-to-magnetic energy transfer, which vanishes in the non-helical limit. In the context of astrophysical dynamos, our results indicate that a large-scale dynamo can be affected by the magnetic helicity generated at small scales. The kinetic helicity, in particular, is not involved in the process at all. An interesting finding is that an inverse cascade of magnetic energy can be provided by a small-scale source of magnetic helicity fluctuations without a mean injection of magnetic helicity.

Cascades and dissipation ratio in rotating magnetohydrodynamic turbulence at low magnetic Prandtl number

A phenomenology of isotropic magnetohydrodynamic (MHD) turbulence subject to both rotation and applied magnetic field is presented. It is assumed that the triple correlation decay time is the shortest between the eddy turn-over time and the ones associated to the rotating frequency and the Alfvén wave period. For Pm=1 it leads to four kinds of piecewise spectra, depending on four parameters: injection rate of energy, magnetic diffusivity, rotation rate, and applied field. With a shell model of MHD turbulence (including rotation and applied magnetic field), spectra for Pm ≤ 1 are presented, together with the ratio between magnetic and viscous dissipations.

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 () or nonlocal () nature of the energy transfers. In both cases, the magnetic energy grows at a small scale and saturates as an inverse "cascade."

Coupling of mean-field equation and shell model of turbulence in the context of galactic dynamo problem

A model of the galactic dynamo has been proposed. It includes the equation of large-scale magnetic field induction written in terms of alpha-omega dynamo approximation and the shell model of magnetohydrodynamic turbulence, which allows evaluation of the turbulent transfer coefficients entering in the mean field equation. Although both models have been studied extensively, their simultaneous usage poses new questions concerning a correct description of the interaction between large-scale and small-scale fields. In this work coupling of the models has been made with the observance of all laws of conservation of the of the ideal magnetic hydrodynamics. As an example, several numerical simulations of the magnetic field dynamics have been carried out and the role of separate mechanisms of the large-scale field -turbulence interaction has been shown.

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.

Cascade and dynamo action in a shell model of magnetohydrodynamic turbulence

A shell model of magnetohydrodynamic turbulence, which allows one to conserve all the integrals of motion in both two and three dimensions, is proposed and studied. We demonstrate that this model reproduces basic facts known in the small-scale turbulent dynamo theory. In particular, we consider a process of redistribution of magnetic helicity generated by the mean-field dynamo, described in the model as magnetic forcing, into a small-scale magnetic field. We argue that the resulting equilibrium magnetic field spectrum strongly depends on the level of magnetic helicity and cross helicity, introduced by the large scales. The spectra with spectral index ``-5/3'' dominate if the cross helicity vanishes. If the level of cross helicity is high (correlated velocity and magnetic field) the spectra depend on the magnetic helicity: the strong magnetic helicity suppresses any cascade providing steep spectra, while the vanishing helicity of turbulent magnetic fields results in the occurrence of Kraichnan-Iroshnikov spectral index ``-3/2.''