Dynamo onset as a first-order transition: Lessons from a shell model for magnetohydrodynamics (original) (raw)

Cascade and dynamo action in a shell model of magnetohydrodynamic turbulence

Physical Review E, 1998

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 ''Ϫ5/3'' dominate if the cross helicity vanishes. If the level of cross helicity is high ͑correlated velocity and magnetic field͒ 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 ''Ϫ3/2.'' ͓S1063-651X͑98͒04103-8͔

A shell model for a large-scale turbulent dynamo

Geophysical & Astrophysical Fluid Dynamics, 2013

This article addresses the interesting and important problem of large-scale magnetic field generation in turbulent flows, using a self-consistent dynamo model recently developed. The main idea of this model is to consider the induction equation for the large-scale magnetic field, integrated consistently with the turbulent dynamics at smaller scales described by a magnetohydrodynamic shell model. The questions of dynamo action threshold, magnetic field saturation, magnetic field reversals, nature of the dynamo transition and the changes of small-scale turbulence as a consequence of the dynamo onset are discussed. In particular, the stability curve obtained by the model integration is shown in a very wide range of values of the magnetic Prandtl number not yet accessible by direct numerical simulation but more realistic for natural dynamos. Moreover, from our analysis it is shown that the large-scale dynamo transition displays a hysteretic behaviour and therefore a subcritical nature. The model successfully reproduces magnetic polarity reversals, showing the capability to generate persistence times which are increasing for decreasing magnetic diffusivity. Moreover, when the system reaches a statistically stationary dynamo state, where the large-scale magnetic field can abruptly reverse its polarity (magnetic reversal state) or not, keeping the same polarity (steady state), it shows an unmistakable tendency towards the energy equipartition for the turbulence at small scale.

A shell model for turbulent dynamos

Advances in Plasma Astrophysics, 2011

A self-consistent nonlinear dynamo model is presented. The nonlinear behavior of the plasma at small scale is described by using a MHD shell model for fields fluctuations; this allow us to study the dynamo problem in a large parameter regime which characterizes the dynamo phenomenon in many natural systems and which is beyond the power of supercomputers at today. The model is able to reproduce dynamical situations in which the system can undergo transactions to different dynamo regimes. In one of these the large-scale magnetic field jumps between two states reproducing the magnetic polarity reversals. From the analysis of long time series of reversals we infer results about the statistics of persistence times, revealing the presence of hidden long-time correlations in the chaotic dynamo process.

A SHELL MODEL TURBULENT DYNAMO

The Astrophysical Journal, 2011

Turbulent dynamo phenomena, observed almost everywhere in astrophysical objects and also in the laboratory in the recent VKS2 experiment, are investigated using a shell model technique to describe magnetohydrodynamic turbulence. Detailed numerical simulations at very high Rossby numbers (α 2 dynamo) show that as the magnetic Reynolds number increases, the dynamo action starts working and different regimes are observed. The model, which displays different large-scale coherent behaviors corresponding to different regimes, is able to reproduce the magnetic field reversals observed both in a geomagnetic dynamo and in the VKS2 experiment. While rough quantitative estimates of typical times associated with the reversal phenomenon are consistent with paleomagnetic data, the analysis of the transition from oscillating intermittent through reversal and finally to stationary behavior shows that the nature of the reversals we observe is typical of α 2 dynamos and completely different from VKS2 reversals. Finally, the model shows that coherent behaviors can also be naturally generated inside the many-mode dynamical chaotic model, which reproduces the complexity of fluid turbulence, as described by the shell technique.

A STUDY OF THE DYNAMO TRANSITION IN A SELF-CONSISTENT NONLINEAR DYNAMO MODEL

The Astrophysical Journal, 2011

We develop a nonlinear dynamo model that couples the evolution of a large-scale magnetic field with the turbulent dynamics of a magnetofluid system in the small scale by electromotive force. Because the dynamo effect takes place in astrophysical objects characterized by a range of dynamical parameters (Reynolds numbers, Prandtl number, etc.) which is beyond the current possibilities of direct numerical simulations, we describe the nonlinear behavior of the system at small scales by using a shell model. Under specific conditions of the turbulent state, the field fluctuations at small scales are able to trigger the dynamo instability. The stability curve derived from our simulations allows us to gain some insight not only into the regime of parameters analyzed up to this point but also for very large Prandtl numbers. Moreover, from our analysis, it is shown that the large-scale dynamo transition displays a hysteretic behavior revealing its subcritical nature. The system, undergoing dynamo transition, can reach different dynamo regimes depending on the Reynolds numbers of the magnetic flow. This points out the critical role that turbulence plays in the dynamo phenomenon. Moreover, in this Letter, we show the presence of the natural ordering of dynamo regimes (oscillatory-reversing-steady dynamos) observed in the large-scale magnetic field for increasing magnetic Reynolds numbers. However, the signature of these regimes is also found in the small-scale dynamo by looking at the scaling properties of magnetic fluctuation energy as a function of magnetic Reynolds number.

HYSTERESIS BETWEEN DISTINCT MODES OF TURBULENT DYNAMOS

The Astrophysical Journal, 2015

ABSTRACT Nonlinear mean-field models for the solar dynamo show long-term variability, which may be relevant to different states of activity inferred from long-term radiocarbon data. This paper is aimed to probe the dynamo hysteresis predicted by the recent mean-field models of Kitchatinov & Olemskoy (2010) with direct numerical simulations. We perform three-dimensional simulations of large-scale dynamos in a shearing box with helically forced turbulence. As initial condition, we either take a weak random magnetic field or we start from a snapshot of an earlier simulation. Two quasi-stable states are found to coexist in a certain range of parameters close to onset of the large-scale dynamo. The simulations converge to one of these states in dependence on the initial conditions. When either the fractional helicity or the magnetic Prandtl number is increased between successive runs above the critical value for onset of the dynamo, the field strength jumps to a finite value. However, when the fractional helicity or the magnetic Prandtl number is then decreased again, the field strength stays at a similar value (strong field branch) even below the original onset. We also observe intermittent decaying phases away from the strong field branch close to the point where large-scale dynamo action is just possible. The dynamo hysteresis seen previously in mean-field models is thus reproduced by 3D simulations. Its possible relation to distinct modes of solar activity such as grand minima is discussed.

The Onset of a Small-Scale Turbulent Dynamo at Low Magnetic Prandtl Numbers

The Astrophysical Journal, 2005

We study numerically the dependence of the critical magnetic Reynolds number for the turbulent small-Rm c scale dynamo on the hydrodynamic Reynolds number. The turbulence is statistically homogeneous, isotropic, Re and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number , which Pm p Rm/Re ! 1 is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are as (a small-scale dynamo exists at low) or Rm r const Re r ϱ Pm Rm /Re p c c as (no small-scale dynamo exists at low). Results obtained in two independent sets of Pm r const Re r ϱ Pm c simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, grows with with no sign of approaching a Rm Re c constant limit. We reach the maximum values of for. By comparing simulations with Rm ∼ 500 Re ∼ 3000 c Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that is not sensitive to the particular form of the viscous cutoff. This work represents a significant Rm c extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve. Rm (Re) c

Dynamo transition in low-dimensional models

Physical Review E, 2008

Two low-dimensional magnetohydrodynamic models containing three velocity and three magnetic modes are described. One of them (nonhelical model) has zero kinetic and current helicity, while the other model (helical) has nonzero kinetic and current helicity. The velocity modes are forced in both these models. These low-dimensional models exhibit a dynamo transition at a critical forcing amplitude that depends on the Prandtl number. In the nonhelical model, dynamo exists only for magnetic Prandtl number beyond 1, while the helical model exhibits dynamo for all magnetic Prandtl number. Although the model is far from reproducing all the possible features of dynamo mechanisms, its simplicity allows a very detailed study and the observed dynamo transition is shown to bear similarities with recent numerical and experimental results. PACS numbers: 91.25.Cw, 47.65.Md, 05.45.Ac

Dynamo Effect in the Kraichnan Magnetohydrodynamic Turbulence

Journal of Statistical Physics, 2007

The existence of a dynamo effect in a simplified magnetohydrodynamic model of turbulence is considered when the magnetic Prandtl number approaches zero or infinity. The magnetic field is interacting with an incompressible Kraichnan-Kazantsev model velocity field which incorporates also a viscous cutoff scale. An approximate system of equations in the different scaling ranges can be formulated and solved, so that the solution tends to the exact one when the viscous and magnetic-diffusive cutoffs approach zero. In this approximation we are able to determine analytically the conditions for the existence of a dynamo effect and give an estimate of the dynamo growth rate. Among other things we show that in the large magnetic Prandtl number case the dynamo effect is always present. Our analytical estimates are in good agreement with previous numerical studies of the Kraichnan-Kazantsev dynamo by Vincenzi (J. Stat. Phys. 106:1073–1091, 2002).

Magnetic reversals in a modified shell model for magnetohydrodynamics turbulence

Physical Review E, 2010

The aim of the paper is the study of dynamo action using a simple nonlinear model in the framework of magnetohydrodynamic turbulence. The nonlinear behavior of the system is described by using a shell model for velocity field and magnetic field fluctuations, modified for the magnetic field at the largest scale by a term describing a supercritical pitchfork bifurcation. Turbulent fluctuations generate a dynamical situation where the large-scale magnetic field jumps between two states which represent the opposite polarities of the magnetic field. Despite its simplicity, the model has the capability to describe a long time series of reversals from which we infer results about the statistics of persistence times and scaling laws of cancellations between opposite polarities for different magnetic diffusivity coefficients. These properties of the model are compared with real paleomagnetic data, thus revealing the origin of long-range correlations in the process.