Inverse Cascade Of Magnetic Helicity Research Articles

From incoherent field to coherent reconnection

Context. Magnetic reconnection in the quiet Sun is a phenomenon that is consistently observed, however, its conditions of occurrence are not as well known as for more energetic events. It has recently become feasible to address this issue with 3D numerical simulations of realistically stratified and convection-driven reconnection. Aims. We aim to illustrate ways by which quiet Sun fields may contribute to solar atmospheric heating via magnetic reconnection that is driven by convective motion. We also aim to compare our complex stratified model to earlier idealized coronal models in terms of reconnection drivers and topological conditions. Methods. We analyzed a simulation of the quiet Sun in which a complex coronal magnetic field is self-consistently driven by the underlying convection. We employed a selection of Lagrangian markers to trace the spatiotemporal behavior of specific magnetic features that are relevant to magnetic reconnection and atmospheric heating. Results. A relatively large-scale reconnection-driven heating event occurs in the simulated corona, in a flattened X-shaped feature characterized by a weak field and high current. It is reminiscent of a hyperbolic flux tube (HFT), which is located at the interface between multiple flux systems. One of these is a smooth overlying horizontal field and the two most relevant others are located below the HFT. They consist of an arcade and a horizontal flux rope which eventually reconnect with the overlying field, raising coronal plasma temperatures up to 1.47 MK. Conclusions. We have identified a reconnection-driven coronal heating event in a quiet Sun simulation. We find that our results are in good phenomenological agreement with idealized coronal flare models, which demonstrates that the same general physical concepts are valid. However, we also find that the reconnecting flux rope and arcade are neither formed by any obvious coherent flux emergence, nor by any ordered photospheric motion or flux cancellation. Instead, they seem to develop merely from the self-consistent convective driving of pre-existing tangled field lines. This gradual and smooth ordering suggests an inverse cascade of magnetic helicity via smaller reconnection events, located at or above slowly-moving photospheric flux concentrations. We suggest that this case is representative of many heating events that may be ubiquitous in the real quiet Sun.

STITCH: A Subgrid-scale Model for Energy Buildup in the Solar Corona

The solar corona routinely exhibits explosive activity, in particular coronal mass ejections and their accompanying eruptive flares, which have global-scale consequences. These events and their smaller counterparts, coronal jets, originate in narrow, sinuous filament channels. The key processes that form and evolve the channels operate on still smaller spatial scales and much longer timescales, culminating in a vast separation of characteristic lengths and times that govern these explosive phenomena. In this article, we describe implementation and tests of an efficient subgrid-scale model for generating eruptive structures in magnetohydrodynamics (MHD) coronal simulations. STITCH—STatistical InjecTion of Condensed Helicity—is a physics-based, reduced representation of helicity condensation: a process wherein small-scale vortical surface convection forms ubiquitous current sheets and pervasive reconnection across the sheets mediates an inverse cascade of magnetic helicity and free energy, thereby forming the filament channels. We have developed a formalism, STITCH, that abstracts these complex processes into a single term in Ohm’s law and the induction equation that directly injects tangential magnetic flux into the low corona. We show that our approach is in very good agreement with a full helicity condensation calculation that treats all of the dynamics explicitly, while enabling substantial reductions in temporal duration and spatial resolution. In addition, we illustrate the flexibility of STITCH at forming localized filament channels and at energizing complex surface flux distributions that have sinuous boundaries. STITCH is simple to implement and computationally efficient, making it a powerful technique for physics-based modeling of solar eruptive events.

Saturation of Turbulent Helical Dynamos.

The presence of large scale magnetic fields in nature is often attributed to the inverse cascade of magnetic helicity driven by turbulent helical dynamos. In this Letter, we show that in turbulent helical dynamos, the inverse flux of magnetic helicity toward the large scales Π_{H} is bounded by |Π_{H}|≤cεk_{η}^{-1}, where ε is the energy injection rate, k_{η} is the Kolmogorov magnetic dissipation wave number, and c an order one constant. Assuming the classical isotropic turbulence scaling, the inverse flux of magnetic helicity Π_{H} decreases at least as a -3/4 power law with the magnetic Reynolds number Rm: |Π_{H}|≤cεℓ_{f}Rm^{-3/4}max[Pm,1]^{1/4}, where Pm is the magnetic Prandtl number and ℓ_{f} the forcing length scale. We demonstrate this scaling with Rm using direct numerical simulations of turbulent dynamos forced at intermediate scales. The results further indicate that nonlinear saturation is achieved by a balance between the inverse cascade and dissipation at domain size scales L for which the saturation value of the magnetic energy is bounded by E_{m}≤cL(εℓ_{f})^{2/3}Rm^{1/4}max[1,Pm]^{1/4}. Numerical simulations also demonstrate this bound. These results are independent of the dynamo mechanism considered. In our setup, they imply that inviscid mechanisms cannot explain large scale magnetic fields and have critical implications for the modeling of astrophysical dynamos.

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 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.

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

Recent progress regarding the noncanonical Hamiltonian formulation of extended magnetohydrodynamics (XMHD), a model with Hall drift and electron inertia, is summarized. The advantages of the Hamiltonian approach are invoked to study some general properties of XMHD turbulence, and to compare them against their ideal MHD counterparts. For instance, the helicity flux transfer rates for XMHD are computed, and Liouville’s theorem for this model is also verified. The latter is used, in conjunction with the absolute equilibrium states, to arrive at the spectra for the invariants, and to determine the direction of the cascades, e.g., generalizations of the well-known ideal MHD inverse cascade of magnetic helicity. After a similar analysis is conducted for XMHD by inspecting second order structure functions and absolute equilibrium states, a couple of interesting results emerge. When cross helicity is taken to be ignorable, the inverse cascade of injected magnetic helicity also occurs in the Hall MHD range—this is shown to be consistent with previous results in the literature. In contrast, in the inertial MHD range, viz at scales smaller than the electron skin depth, all spectral quantities are expected to undergo direct cascading. The consequences and relevance of our results in space and astrophysical plasmas are also briefly discussed.

Large-scale dynamics of magnetic helicity.

In this paper we investigate the dynamics of magnetic helicity in magnetohydrodynamic (MHD) turbulent flows focusing at scales larger than the forcing scale. Our results show a nonlocal inverse cascade of magnetic helicity, which occurs directly from the forcing scale into the largest scales of the magnetic field. We also observe that no magnetic helicity and no energy is transferred to an intermediate range of scales sufficiently smaller than the container size and larger than the forcing scale. Thus, the statistical properties of this range of scales, which increases with scale separation, is shown to be described to a large extent by the zero flux solutions of the absolute statistical equilibrium theory exhibited by the truncated ideal MHD equations.

Helical mode interactions and spectral transfer processes in magnetohydrodynamic turbulence

Spectral transfer processes in homogeneous magnetohydrodynamic (MHD) turbulence are investigated analytically by decomposition of the velocity and magnetic fields in Fourier space into helical modes. Steady solutions of the dynamical system which governs the evolution of the helical modes are determined, and a stability analysis of these solutions is carried out. The interpretation of the analysis is that unstable solutions lead to energy transfer between the interacting modes while stable solutions do not. From this, a dependence of possible interscale energy and helicity transfers on the helicities of the interacting modes is derived. As expected from the inverse cascade of magnetic helicity in 3-D MHD turbulence, mode interactions with like helicities lead to transfer of energy and magnetic helicity to smaller wavenumbers. However, some interactions of modes with unlike helicities also contribute to an inverse energy transfer. As such, an inverse energy cascade for non-helical magnetic fields is shown to be possible. Furthermore, it is found that high values of the cross-helicity may have an asymmetric effect on forward and reverse transfer of energy, where forward transfer is more quenched in regions of high cross-helicity than reverse transfer. This conforms with recent observations of solar wind turbulence. For specific helical interactions the relation to dynamo action is established. The present analysis provides new theoretical insights into physical processes where inverse cascade and dynamo action are involved, such as the evolution of cosmological and astrophysical magnetic fields and laboratory plasmas.

Self-similar inverse cascade of magnetic helicity driven by the chiral anomaly

For systems with charged chiral fermions, the imbalance of chirality in the presence of magnetic field generates an electric current - this is the Chiral Magnetic Effect (CME). We study the dynamical real-time evolution of electromagnetic fields coupled by the anomaly to the chiral charge density and the CME current by solving the Maxwell-Chern-Simons equations. We find that the CME induces the inverse cascade of magnetic helicity towards the large distances, and that at late times this cascade becomes self-similar, with universal exponents. We also find that in terms of gauge field topology the inverse cascade represents the transition from linked electric and magnetic fields (Hopfions) to the knotted configuration of magnetic field (Chandrasekhar-Kendall states). The magnetic reconnections are accompanied by the pulses of the CME current directed along the magnetic field lines. We devise an experimental signature of these phenomena in heavy ion collisions, and speculate about implications for condensed matter systems.

INVERSE CASCADE IN IMBALANCED ELECTRON MAGNETOHYDRODYNAMIC TURBULENCE

Electron magnetohydrodynamics (EMHD) provides a fluid-like description of small-scale magnetized plasmas. Balanced EMHD turbulence has been studied for a long time. However, driven imbalanced EMHD turbulence, in which waves moving in one direction (dominant waves) have higher amplitudes than waves moving in the other direction (sub-dominant waves), has not been well studied. In this paper, we numerically study driven three-dimensional imbalanced weak EMHD turbulence. We find the following results. First, in driven imbalanced EMHD turbulence, we clearly observe inverse cascade of magnetic helicity, as well as magnetic energy. This is because magnetic helicity is a conserved quantity and non-zero magnetic helicity is injected into the system in driven imbalanced EMHD turbulence. Second, the magnetic energy spectrum of the dominant waves on scales larger than the energy injection scale does not show a single power-law spectrum, which indicates that the inverse cascade is not a self-similar process. The peak of the spectrum roughly follows a k –3/2 spectrum, which can be explained by a Kolmogorov-type argument for weak turbulence. Third, a small amount of sub-dominant waves is induced by the dominant waves on large scales and the ratio of helicity densities of the dominant and the sub-dominant waves on large scales seems to converge to a certain value.

Primordial magnetic helicity from stochastic electric currents

We study the possibility that primordial magnetic fields generated in the transition between inflation and reheating posses magnetic helicity, $H_M$. The fields are induced by stochastic currents of scalar charged particles created during the mentioned transition. We estimate the rms value of the induced magnetic helicity by computing different four-point SQED Feynman diagrams. For any considered volume, the magnetic flux across its boundaries is in principle non null, which means that the magnetic helicity in those regions is gauge dependent. We use the prescription given by Berger and Field and interpret our result as the difference between two magnetic configurations that coincide in the exterior volume. In this case the magnetic helicity gives only the number of magnetic links inside the considered volume. We calculate a concrete value of $H_M$ for large scales and analyze the distribution of magnetic defects as a function of the scale. Those defects correspond to regular as well as random fields in the considered volume. We find that the fractal dimension of the distribution of topological defects is $D = 1/2$. We also study if the regular fields induced on large scales are helical, finding that they are and that the associated number of magnetic defects is independent of the scale. In this case the fractal dimension is $D=0$. We finally estimate the intensity of fields induced at the horizon scale of reheating, and evolve them until the decoupling of matter and radiation under the hypothesis of inverse cascade of magnetic helicity. The resulting intensity is high enough and the coherence length long enough to have an impact on the subsequent process of structure formation.

LARGE-SCALE MAGNETIC STRUCTURE FORMATION IN THREE-DIMENSIONAL MAGNETOHYDRODYNAMIC TURBULENCE

The inverse cascade of magnetic helicity in 3D-MHD turbulence is believed to be one of the processes responsible for large scale magnetic structure formation in astrophysical systems. In this work we present an exhaustive set of high resolution direct numerical simulations (DNS) of both forced and decaying 3D-MHD turbulence, to understand this structure formation process. It is first shown that an inverse cascade of magnetic helicity in small-scale driven turbulence does not necessarily generate coherent large-scale magnetic structures. The observed large-scale magnetic field, in this case, is severely perturbed by magnetic fluctuations generated by the small-scale forcing. In the decaying case, coherent large-scale structure form similar to those observed astronomically. Based on the numerical results the formation of large-scale magnetic structures in some astrophysical systems, is suggested to be the consequence of an initial forcing which imparts the necessary turbulent energy into the system, which, after the forcing shuts off, decays to form the large-scale structures. This idea is supported by representative examples e.g. cluster of galaxies.

Role of helicities for the dynamics of turbulent magnetic fields

Investigations of the inverse cascade of magnetic helicity are conducted with pseudospectral, three-dimensional direct numerical simulations of forced and decaying incompressible MHD turbulence. The high-resolution simulations which allow for the necessary scale separation show that the observed self-similar scaling behavior of magnetic helicity and related quantities can only be understood by taking the full nonlinear interplay of velocity and magnetic fluctuations into account. With the help of the eddy-damped quasi-normal Markovian approximation a probably universal relation between kinetic and magnetic helicities is derived which closely resembles the extended definition of the prominent dynamo pseudoscalar α. This unexpected similarity suggests an additional nonlinear quenching mechanism of the current-helicity contribution to α.

Inverse cascade of magnetic helicity in magnetohydrodynamic turbulence

The nonlinear dynamics of magnetic helicity HM, which is responsible for large-scale magnetic structure formation in electrically conducting turbulent media, is investigated in forced and decaying three-dimensional magnetohydrodynamic turbulence. This is done with the help of high-resolution direct numerical simulations and statistical closure theory. The numerically observed spectral scaling of HM is at variance with earlier work using a statistical closure model [Pouquet et al., J. Fluid Mech. 77, 321 (1976)]. By revisiting this theory, a universal dynamical balance relation is found that includes the effects of kinetic helicity as well as kinetic and magnetic energies on the inverse cascade of HM and explains the above-mentioned discrepancy. Consideration of the result in the context of mean-field dynamo theory suggests a nonlinear modification of the α-dynamo effect, which is important in the context of magnetic-field excitation in turbulent plasmas.

SIMULATIONS OF TURBULENT DYNAMOS DRIVEN BY THE MAGNETIC HELICITY FLUX

We present results of numerical simulations of driven magnetohydrodynamic turbulence in a periodic box with an imposed large-scale sinusoidal shear. We produce strong large-scale magnetic fields through dynamo action. This dynamo action is highly dependent on the spectral properties of the forcing function. In particular, we can produce either dynamo or antidynamo action by choosing particular forcing functions for the small-scale turbulence. Our results show that the large-scale electromotive force is well correlated with the local accumulation of eddy-scale magnetic helicity. Conversely, the electromotive force is very weakly correlated with the large-scale magnetic field strength or its product with kinetic helicity. We argue that in these simulations large-scale dynamo action is driven by an inverse cascade of magnetic helicity, ultimately caused by the tendency of anisotropic turbulence to create a magnetic helicity flux.

Inverse cascades andαeffect at a low magnetic Prandtl number

Dynamo action in a fully helical Arn'old-Beltrami-Childress flow is studied using both direct numerical simulations and subgrid modeling. Sufficient scale separation is given in order to allow for large-scale magnetic energy buildup. Growth of magnetic energy obtains down to a magnetic Prandtl number P(M) = R(M)/R(V) close to 0.005, where R(V) and R(M) are the kinetic and magnetic Reynolds numbers. The critical magnetic Reynolds number for dynamo action R(M)(c) seems to saturate at values close to 20. Detailed studies of the dependence of the amplitude of the saturated magnetic energy with P(M) are presented. When P(M) is decreased, numerical experiments are conducted with either R(V) or R(M) kept constant. In the former case, the ratio of magnetic to kinetic energy saturates to a value slightly below unity as P(M) decreases. Examination of energy spectra and structures in real space reveals that quenching of the velocity by a large-scale magnetic field takes place, with an inverse cascade of magnetic helicity and a force-free field at large scale in the saturated regime.

On the Inverse Cascade of Magnetic Helicity

We study the inverse cascade of magnetic helicity in conducting fluids by investigating the detailed transfer of helicity between different spherical shells in Fourier space in direct numerical simulations of three-dimensional magnetohydrodynamics (MHD). Two different numerical simulations are used, one where the system is forced with an electromotive force in the induction equation, and one in which the system is forced mechanically with an ABC flow and the magnetic field is solely sustained by a dynamo action. The magnetic helicity cascade at the initial stages of both simulations is observed to be inverse and local (in scale space) in the large scales, and direct and local in the small scales. When saturation is approached most of the helicity is concentrated in the large scales and the cascade is non-local. Helicity is transfered directly from the forced scales to the largest scales. At the same time, a smaller in amplitude direct cascade is observed from the largest scale to small scales.

Numerical solutions of the three-dimensional magnetohydrodynamicαmodel

We present direct numerical simulations and alpha -model simulations of four familiar three-dimensional magnetohydrodynamic (MHD) turbulence effects: selective decay, dynamic alignment, inverse cascade of magnetic helicity, and the helical dynamo effect. The MHD alpha model is shown to capture the long-wavelength spectra in all these problems, allowing for a significant reduction of computer time and memory at the same kinetic and magnetic Reynolds numbers. In the helical dynamo, not only does the alpha model correctly reproduce the growth rate of magnetic energy during the kinematic regime, it also captures the nonlinear saturation level and the late generation of a large scale magnetic field by the helical turbulence.

The Solar Dynamo: Old, Recent, and New Problems

A number of problems of solar and stellar dynamo theory are briefly reviewed and the current status of possible solutions is discussed. Results of direct numerical simulations are described in view of mean-field dynamo theory and the relation between the α-effect and the inverse cascade of magnetic helicity is highlighted. The possibility of ‘catastrophic’ quenching of the α-effect is explained in terms of the constraint placed by the conservation of magnetic helicity.

Coronal activity from dynamos in astrophysical rotators

We show that a steady mean-field dynamo in astrophysical rotators leads to an outflow of relative magnetic helicity and thus magnetic energy available for particle and wind acceleration in a corona. The connection between energy and magnetic helicity arises because mean-field generation is linked to an inverse cascade of magnetic helicity. To maintain a steady state in large magnetic Reynolds number rotators, there must then be an escape of relative magnetic helicity associated with the mean field, accompanied by an equal and opposite contribution from the fluctuating field. From the helicity flow, a lower limit on the magnetic energy deposited in the corona can be estimated. Steady coronal activity including the dissipation of magnetic energy, and formation of multi-scale helical structures therefore necessarily accompanies an internal dynamo. This highlights the importance of boundary conditions which allow this to occur for non-linear astrophysical dynamo simulations. Our theoretical estimate of the power delivered by a mean-field dynamo is consistent with that inferred from observations to be delivered to the solar corona, the Galactic corona, and Seyfert 1 AGN coronae.

The formation of large-scale structures in supersonic magnetohydrodynamic flows

In this paper we investigate the existence of an inverse cascade of magnetic helicity for statistically steady compressible magnetohydrodynamic flows with rms Mach numbers that go up to unity. This has been done by performing a set of three-dimensional numerical simulations that were carried out using the RIEMANN code for astrophysical fluid and magnetofluid dynamics [D. S. Balsara, Astrophys. J. Suppl. 116, 133 (1988)]. The random forcing is made up of ABC flows corresponding to helical wave patterns at an intermediate scale. Both time histories of integral quantities and Fourier spectra are shown, illustrating the inverse cascade process. The observed build-up of large-scale magnetic helical structures for a wide range of Mach numbers indicates that, in the context of the generation of the magnetic field of the Galaxy where supersonic flows are common, a dynamo mechanism based on the nonlinear dynamics of magnetic helicity may be at work.