Dynamical magnetic relaxation: A nonlinear magnetically driven dynamo

Two spectral regimes of magnetic field amplification in magnetohydrodynamic (MHD) flows can be distinguished by the scale on which fields are amplified relative to the primary forcing scale of the turbulence. For field amplification at or below the forcing scale, the amplification can be called a “small-scale dynamo.” For amplification at and above the forcing scale the process can be called a “large-scale dynamo.” Non - local (in wave number) effects play a key role in both the growth of the small-scale field in non-helical turbulence and the growth of large and smallscale fields in helical turbulence. Mean field dynamo (MFD) theory represents a simple semi-analytic way to get a handle on large-scale field amplification in MHD turbulence. Helicity has long been known to be important for large scale, flux generating, externally forced MFDs. The extent to which such MFDs operate “slow” or “fast” (dependent or independent on magnetic Reynolds number) has been controversial, but there has been recent progress. Simulations of α2 dynamos in a periodic box dynamo and their quenching can now be largely understood within a simplified dynamical non-linear paradigm in which the MFD growth equation is supplemented by the total magnetic helicity evolution equation. For α2 dynamos, the large-scale field growth is directly related to the large-scale magnetic helicity growth. Magnetic helicity conservation then implies that growth of the large-scale magnetic helicity induces growth of small-scale magnetic (and current) helicity of the opposite sign, which eventually suppresses the α effect driving the MFD growth. Although the α2 MFD then becomes slow in the long time limit, substantial large-scale field growth proceeds in a kinematic, “fast” phase before non-linear asymptotic quenching of the “slow” phase applies. Ultimately, the MFD emerges as a process that transfers magnetic helicity between small and large scales. How these concepts apply to more general dynamos with shear, and open boundary dynamos is a topic of ongoing research. Some unresolved issues are identified. Overall, the following summarizes the most recent progress in mean-field dynamo theory: For a closed turbulent flow, the non-linear mean field dynamo, is first fast and kinematic, then slow and dynamic, and magnetic helicity transfer makes it so.KeywordsMagnetic EnergyMagnetic HelicityMagnetic Reynolds NumberDynamo TheoryForce ScaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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 achived for the magnetic component with the same helicity of the flow, in agreement with the Stretch–Twist–Fold mechanism. Vice versa, in the presence of Lorentz feedback on the velocity, we find that the inverse cascade of magnetic helicity is mostly local if magnetic and kinetic helicities have opposite signs, 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.

Context. Obtaining observational constraints on the role of turbulent effects for the solar dynamo is a difficult, yet crucial, task. Without such knowledge, the full picture of the operation mechanism of the solar dynamo cannot be formed. Aims. The magnetic helicity spectrum provides important information about the α effect. Here we demonstrate a formalism in spherical geometry to infer magnetic helicity spectra directly from observations of the magnetic field, taking into account the sign change of magnetic helicity across the Sun’s equator. Methods. Using an angular correlation function of the magnetic field, we develop a method to infer spectra for magnetic energy and helicity. The retrieval of the latter relies on a fundamental definition of helicity in terms of linkage of magnetic flux. We apply the two-scale approach, previously used in Cartesian geometry, to spherical geometry for systems where a sign reversal of helicity is expected across the equator on both small and large scales. Results. We test the method by applying it to an analytical model of a fully helical field, and to magneto-hydrodynamic simulations of a turbulent dynamo. The helicity spectra computed from the vector potential available in the models are in excellent agreement with the spectra computed solely from the magnetic field using our method. In a next test, we use our method to obtain the helicity spectrum from a synoptic magnetic field map corresponding to a Carrington rotation. We observe clear signs of a bihelical spectrum of magnetic helicity, which is in complete accordance to the previously reported spectra in literature from the same map. Conclusions. Our formalism makes it possible to infer magnetic helicity in spherical geometry, without the necessity of computing the magnetic vector potential. It has many applications in solar and stellar observations, but can also be used to analyse global magnetoconvection models of stars and to compare them with observations.

The nonlinear mean-field dynamo due to a shear-current effect in a nonhelical homogeneous turbulence with a mean velocity shear is discussed. The transport of magnetic helicity as a dynamical nonlinearity is taken into account. The shear-current effect is associated with the W × J term in the mean electromotive force, where W is the mean vorticity due to the large-scale shear motions and J is the mean electric current. This effect causes the generation of large-scale magnetic field in a turbulence with large hydrodynamic and magnetic Reynolds numbers. The dynamo action due to the shear-current effect depends on the spatial scaling of the correlation time τ( k ) of the background turbulence, where k is the wave number. For Kolmogorov scaling, τ(k)∝k −2/3, the dynamo instability occurs, while when τ(k) ∝k −2 (small hydrodynamic and magnetic Reynolds numbers) there is no the dynamo action in a sheared nonhelical turbulence. The magnetic helicity flux strongly affects the magnetic field dynamics in the nonlinear stage of the dynamo action. Numerical solutions of the nonlinear mean-field dynamo equations which take into account the shear-current effect, show that if the magnetic helicity flux is not small, the saturated level of the mean magnetic field is of the order of the equipartition field determined by the turbulent kinetic energy. Turbulence with a large-scale velocity shear is a universal feature in astrophysics, and the obtained results can be important for elucidation of origin of the large-scale magnetic fields in astrophysical sheared turbulence.

To investigate the characteristics of large-scale and long-term evolution of magnetic helicity with solar cycles, we use the method of Local Correlation Tracking to estimate the magnetic helicity evolution over solar cycle 23 from 1996 to 2009 using 795 MDI magnetic synoptic charts. The main results are as follows: the hemispheric helicity rule still holds in general, i.e., the large-scale negative (positive) magnetic helicity dominates the northern (southern) hemisphere. However, the large-scale magnetic helicity fluxes show the same sign in both hemispheres around 2001 and 2005. The global, large-scale magnetic helicity flux over the solar disk changes from a negative value at the beginning of solar cycle 23 to a positive value at the end of the cycle, while the net accumulated magnetic helicity is negative in the period between 1996 and 2009.

Electromotive force is an essential quantity in dynamo theory. During a coronal mass ejection (CME), magnetic helicity gets decoupled from the Sun and advected into the heliosphere with the solar wind. Eventually, a heliospheric magnetic transient event might pass by a spacecraft, such as the Helios space observatories. Our aim is to investigate the electromotive force, the kinetic helicity effect (α term), the turbulent diffusion (β term), and the cross-helicity effect (γ term) in the inner heliosphere below 1 au. We set up a one-dimensional model of the solar wind velocity and magnetic field for a hypothetic interplanetary CME. Because turbulent structures within the solar wind evolve much slower than this structure needs to pass by the spacecraft, we use a reduced curl operator to compute the current density and vorticity. We test our CME shock-front model against an observed magnetic transient that passes by the Helios-2 spacecraft. At the peak of the fluctuations in this event we find strongly enhanced α, β, and γ terms, as well as a strong peak in the total electromotive force. Our method allows us to automatically identify magnetic transient events from any in situ spacecraft observations that contain magnetic field and plasma velocity data of the solar wind.

Roles played by the currents in the impulsive phase of a solar flare and in a coronal mass ejection (CME) are reviewed. Solar flares are magnetic explosions: magnetic energy stored in unneutralized currents in coronal loops is released into energetic electrons in the impulsive phase and into mass motion in a CME. The energy release is due to a change in current configuration effectively reducing the net current path. A flare is driven by the electromotive force (EMF) due to the changing magnetic flux. The EMF drives a flare‐associated current whose cross‐field closure is achieved by redirection along field lines to the chromosphere and back. The essential roles that currents play are obscured in the “standard” model and are described incorrectly in circuit models. A semiquantitative treatment of the energy and the EMF is provided by a multicurrent model, in which the currents are constant and the change in the current paths is described by time‐dependent inductances. There is no self‐consistent model that includes the intrinsic time dependence, the EMF, the flare‐associated current, and the internal energy transport during a flare. The current, through magnetic helicity, plays an important role in a CME, with twist converted into writhe allowing the kink instability plus reconnection to lead to a new closed loop and with the current‐current force accelerating the CME through the torus instability.

We consider an unforced, incompressible, turbulent magnetofluid constrained by concentric inner and outer spherical surfaces. We define a model system in which normal components of the velocity, magnetic field, vorticity, and electric current are zero on the boundaries. This choice allows us to find a set of Galerkin expansion functions that are common to both velocity and magnetic field, as well as vorticity and current. The model dynamical system represents magnetohydrodynamic (MHD) turbulence in a spherical domain and is analyzed by the methods similar to those applied to homogeneous MHD turbulence. We find a statistical theory of ideal (i.e. no dissipation) MHD turbulence analogous to that found in the homogeneous case, including the prediction of coherent structure in the form of a large-scale quasistationary magnetic field. This MHD dynamo depends on broken ergodicity, an effect that is enhanced when total magnetic helicity is increased relative to total energy. When dissipation is added and large scales are only weakly damped, quasiequilibrium may occur for long periods of time, so that the ideal theory is still pertinent on a global scale. Over longer periods of time, the selective decay of energy over magnetic helicity further enhances the effects of broken ergodicity. Thus, broken ergodicity is an essential mechanism and relative magnetic helicity is a critical parameter in this model MHD dynamo theory.

We study the electrodynamics of the chiral medium with electric and magnetic charges using the effective Maxwell–Chern–Simons theory extended to include the magnetic current. The exchange of helicity between the chiral medium and the magnetic field, known as the inverse cascade, is controlled by the chiral anomaly equation. In the presence of the magnetic current, the magnetic helicity is dissipated, so that the inverse cascade stops when the magnetic helicity vanishes while the chiral conductivity reaches a non-vanishing stationary value satisfying σχ2<4σeσm, where σe, σm and σχ are the electric, magnetic and chiral conductivities respectively. We argue that this state is superconducting and exhibits the Meissner effect for both electric and magnetic fields. Moreover, this state is stable with respect to small magnetic helicity fluctuations; the magnetic helicity becomes unstable only when the inequality mentioned above is violated.

Recent theoretical and observational developments in magnetic helicity studies

It is demonstrated that the two-point correlation function of the magnetic helicity in the case of zero mean magnetic field has anomalous scalings for both compressible and incompressible turbulent helical fluid flow. The magnetic helicity in the limit of very high electrical conductivity is conserved. This implies that the two-point correlation function of the conserved property does not necessarily have normal scaling. The reason for the anomalous scalings of the magnetic helicity correlation function is that the magnetic field in the equation for the two-point correlation function of the magnetic helicity plays a role of a pumping with anomalous scalings. It is shown also that when magnetic fluctuations with zero mean magnetic field are generated the magnetic helicity is very small even if the hydrodynamic helicity is large.

Three closely related stumbling blocks of solar mean field dynamo theory are discussed: how dominant are the small scale fields, how is the alpha effect quenched, and whether magnetic and current helicity fluxes alleviate the quenching? It is shown that even at the largest currently available resolution there is no clear evidence of power law scaling of the magnetic and kinetic energy spectra in turbulence. However, using subgrid scalemodeling, some indications of asymptotic equipartition can be found. The frequently used first order smoothing approach to calculate the alpha effect and other transport coefficients is contrasted with the superior minimal tau approximation. The possibility of catastrophic alpha quenching is discussed as a result of magnetic helicity conservation. Magnetic and current helicity fluxes are shown to alleviate catastrophic quenching in the presence of shear. Evidence for strong large scale dynamo action, even in the absence of helicity in the forcing, is presented. (© 2005 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

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.''

Context. There are indications that the magnetic field evolution in galaxies is influenced by tidal interactions and mergers between galaxies. Aims. We carried out a parameter study of interacting disk galaxies with impact parameters ranging from central collisions to weakly interacting scenarios. The orientations of the disks were also varied. In particular, we investigated how magnetic field amplification depends on these parameters. Methods. We used magnetohydrodynamics for gas disks in combination with live dark-matter halos in adaptive mesh refinement simulations. The disks were initialized using a setup for isolated disks in hydrostatic equilibrium. Since we focused on the impact of tidal forces on magnetic field evolution, adiabatic physics was applied. Small-scale filtering of the velocity and magnetic field allowed us to estimate the turbulent electromotive force (EMF) and kinetic helicity. Results. Time series of the average magnetic field in central and outer disk regions show pronounced peaks during close encounters and mergers. This agrees with observed magnetic fields at different interaction stages. The central field strength exceeds 10 μG (corresponding to an amplification factor of 2–3) for small impact parameters. As the disks are increasingly disrupted and turbulence is produced by tidal forces, the small-scale EMF reaches a significant fraction of the total EMF. The small-scale kinetic helicity is initially antisymmetric across the disk plane. Though its evolution is sensitive to both the impact parameter and inclinations of the rotation axes with respect to the relative motion of the disks, antisymmetry is generally broken through interactions and the merger remnant loses most of the initial helicity. The EMF and the magnetic field also decay rapidly after coalescence. Conclusions. The strong amplification during close encounters of the interacting galaxies is mostly driven by helical flows and a mean-field dynamo. The small-scale dynamo contributes significantly in post-interaction phases. However, the amplification of the magnetic field cannot be sustained.

A statistical analysis of magnetic flux ropes, identified by large-amplitude, smooth rotations of the magnetic field vector and a low level of both proton density and temperature, has been performed by computing the invariants of the ideal magnetohydrodynamic (MHD) equations, namely the magnetic helicity, the cross-helicity, and the total energy, via magnetic field and plasma fluctuations in the interplanetary medium. A technique based on the wavelet spectrograms of the MHD invariants allows the localization and characterization of those structures in both scales and time: it has been observed that flux ropes show, as expected, high magnetic helicity states (|σ m | [0.6: 1]), but extremely variable cross-helicity states (|σ c | [0: 0.8]), which, however, are not independent of the magnetic helicity content of the flux rope itself. The two normalized MHD invariants observed within the flux ropes tend indeed to distribute, neither trivially nor automatically, along the curve, thus suggesting that some constraint should exist between the magnetic and cross-helicity content of the structures. The analysis carried out has further showed that the flux rope properties are totally independent of their time duration and that they are detected either as a sort of interface between different portions of solar wind or as isolated structures embedded in the same stream.