Helical Dynamo Research Articles

Novel intrinsic helical cores and MHD dynamo flux pumping evidence in DIII-D

Novel intrinsic helical cores and MHD dynamo flux pumping evidence in DIII-D

Helical and non-helical large-scale dynamos in thin accretion discs

ABSTRACT The dynamics of accreting and outgoing flows around compact objects depends crucially on the strengths and configurations of the magnetic fields therein, especially of the large-scale fields that remain coherent beyond turbulence scales. Possible origins of these large-scale magnetic fields include flux advection and disc dynamo actions. However, most numerical simulations have to adopt an initially strong large-scale field rather than allow them to be self-consistently advected or amplified, due to limited computational resources. The situation can be partially cured by using sub-grid models where dynamo actions only reachable at high resolutions are mimicked by artificial terms in low-resolution simulations. In this work, I couple thin-disc models with local shearing-box simulation results to facilitate more realistic sub-grid dynamo implementations. For helical dynamos, detailed spatial profiles of dynamo drivers inferred from local simulations are used, and the non-linear quenching and saturation is constrained by magnetic helicity evolution. In the inner disc region, saturated fields have dipole configurations and the plasma β reaches ≃0.1 to 100, with correlation lengths ≃h in the vertical direction and ≃10 h in the radial direction, where h is the disc scale height. The dynamo cycle period is ≃40 orbital time scale, compatible with previous global simulations. Additionally, I explore two dynamo mechanisms which do not require a net kinetic helicity and have only been studied in shearing-box set-ups. I show that such dynamos are possible in thin accretion discs, but produce field configurations that are incompatible with previous results. I discuss implications for future general-relativistic magnetohydrodynamics simulations.

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.

On large-scale dynamos with stable stratification and the application to stellar radiative zones

Abstract Our understanding of large-scale magnetic fields in stellar radiative zones remains fragmented and incomplete. Such magnetic fields, which must be produced by some form of dynamo mechanism, are thought to dominate angular-momentum transport, making them crucial to stellar evolution. A major difficulty is the effect of stable stratification, which generally suppresses dynamo action. We explore the effects of stable stratification on mean-field dynamo theory with a particular focus on a non-helical large-scale dynamo (LSD) mechanism known as the magnetic shear-current effect. We find that the mechanism is robust to increasing stable stratification as long as the original requirements for its operation are met: a source of shear and non-helical magnetic fluctuations (e.g. from a small-scale dynamo). Both are plausibly sourced in the presence of differential rotation. Our idealized direct numerical simulations, supported by mean-field theory, demonstrate the generation of near equipartition large-scale toroidal fields. Additionally, a scan over magnetic Reynolds number shows no change in the growth or saturation of the LSD, providing good numerical evidence of a dynamo mechanism resilient to catastrophic quenching, which has been an issue for helical dynamos. These properties – the absence of catastrophic quenching and robustness to stable stratification – make the mechanism a plausible candidate for generating in situ large-scale magnetic fields in stellar radiative zones.

Helical turbulent nonlinear dynamo at large magnetic Reynolds numbers

To investigate large-scale magnetic field generation in highly conducting magnetohydrodynamic (MHD) regimes typical of rotating astrophysical systems, we conduct high-resolution MHD simulations in the dynamical regime of large magnetic Reynolds numbers, of turbulence with an inhomogeneous kinetic helicity distribution. We obtain a nonlinear helical dynamo solution, characterized by a weak large-scale magnetic wave slowly propagating over saturated small-scale MHD turbulence, bypassing the catastrophic resistive quenching problem of homogeneous helical dynamos by developing turbulent fluxes of magnetic helicity through the system equator. Hints of fast-reconnection plasmoids are found.

Magnetic Helicity Dissipation and Production in an Ideal MHD Code

Abstract We study a turbulent helical dynamo in a periodic domain by solving the ideal magnetohydrodynamic (MHD) equations with the FLASH code using the divergence-cleaning eight-wave method and compare our results with direct numerical simulations (DNS) using the Pencil Code. At low resolution, FLASH reproduces the DNS results qualitatively by developing the large-scale magnetic field expected from DNS, but at higher resolution, no large-scale magnetic field is obtained. In all those cases in which a large-scale magnetic field is generated, the ideal MHD results yield too little power at small scales. As a consequence, the small-scale current helicity is too small compared with that of the DNS. The resulting net current helicity has then always the wrong sign, and its statistical average also does not approach zero at late times, as expected from the DNS. Our results have implications for astrophysical dynamo simulations of stellar and galactic magnetism using ideal MHD codes.

Exploring helical dynamos with machine learning: Regularized linear regression outperforms ensemble methods

We use ensemble machine learning algorithms to study the evolution of magnetic fields in magnetohydrodynamic (MHD) turbulence that is helically forced. We perform direct numerical simulations of helically forced turbulence using mean field formalism, with electromotive force (EMF) modeled both as a linear and non-linear function of the mean magnetic field and current density. The form of the EMF is determined using regularized linear regression and random forests. We also compare various analytical models to the data using Bayesian inference with Markov chain Monte Carlo (MCMC) sampling. Our results demonstrate that linear regression is largely successful at predicting the EMF and the use of more sophisticated algorithms (random forests, MCMC) do not lead to significant improvement in the fits. We conclude that the data we are looking at is effectively low dimensional and essentially linear. Finally, to encourage further exploration by the community, we provide all of our simulation data and analysis scripts as open source IPYTHON notebooks.

Principle of the Helical and Nonhelical Dynamo and the α Effect in a Field Structure Model

We explain the (non)helical dynamo process using a field-structure model based on magnetic induction equation in an intuitive way. We show how nonhelical kinetic energy converts into magnetic energy and cascades toward smaller eddies in a mechanically forced plasma system. Also, we show how helical magnetic energy is inversely cascaded ($\alpha$ effect) toward large scale magnetic eddies in a mechanically or magnetically forced system. We, then, compare the simulation results with the model qualitatively for the verification of the model. In addition to these intuitive and numerical approaches, we show how to get $\alpha$ and $\beta$ coefficient semi-analytically from the temporally evolving large scale magnetic energy and magnetic helicity.

Role of a continuous MHD dynamo in the formation of 3D equilibria in fusion plasmas

Stationary 3D equilibria can form in fusion plasmas via saturation of magnetohydrodynamic (MHD) instabilities or stimulated by external 3D fields. In these cases the current profile is anomalously broad due to magnetic flux pumping produced by the MHD modes. Flux pumping plays an important role in hybrid tokamak plasmas, maintaining the minimum safety factor above unity and thus removing sawteeth. It also enables steady-state hybrid operation, by redistributing non-inductive current driven near the center by electron cyclotron waves. A validated flux pumping model is not yet available, but it would be necessary to extrapolate hybrid operation to future devices. In this work flux pumping physics is investigated for helical core equilibria stimulated by external 3D fields in DIII-D hybrid plasmas. We show that flux pumping can be produced in a continuous way by an MHD dynamo emf. The same effect maintains helical equilibria in reversed-field pinch (RFP) plasmas. The effective MHD dynamo loop voltage is calculated for experimental 3D equilibrium reconstructions, by balancing Ohm’s law over helical flux surfaces, and is consistent with the expected current redistribution. Similar results are also obtained with more sophisticated nonlinear MHD simulations. The same modelling approach is applied to helical RFP states forming spontaneously in RFX-mod as the plasma current is raised above 0.8–1 MA. This comparison allows to identify the underlying physics common to tokamak and RFP: a helical core displacement modulates parallel current density along flux tubes, which requires a helical electrostatic potential to build up, giving rise to a helical MHD dynamo flow.

Traces of large-scale dynamo action in the kinematic stage

Using direct numerical simulations (DNS) we verify that in the kinematic regime, a turbulent helical dynamo grows in such a way that the magnetic energy spectrum remains to high precision shape-invariant, i.e., at each wavenumber $k$ the spectrum grows with the same growth rate. Signatures of large-scale dynamo action can be identified through the excess of magnetic energy at small $k$, of one of the two oppositely polarized constituents. Also a suitably defined planar average of the magnetic field can be chosen such that its rms value isolates the strength of the mean field. However, these different means of analysis suggest that the strength of the large-scale field diminishes with increasing magnetic Reynolds number ${\rm Re}_{\rm M}$ like ${\rm Re}_{\rm M}^{-1/2}$ for intermediate values and like ${\rm Re}_{\rm M}^{-3/4}$ for larger ones. Both an analysis from the Kazantsev model including helicity and the DNS show that this arises due to the magnetic energy spectrum still peaking at resistive scales, even when helicity is present. As expected, the amplitude of the large-scale field increases with increasing fractional helicity, enabling us to determine the onset of large scale dynamo action and distinguishing it from that of the small-scale dynamo. Our DNS show that, contrary to earlier results for smaller scale separation (only 1.5 instead of now 4), the small-scale dynamo can still be excited at magnetic Prandtl numbers of 0.1 and only moderate values of the magnetic Reynolds numbers ($\sim 160$).

Intense bipolar structures from stratified helical dynamos

We perform direct numerical simulations of the equations of magnetohydrodynamics with external random forcing and in the presence of gravity. The domain is divided into two parts: a lower layer where the forcing is helical and an upper layer where the helicity of the forcing is zero with a smooth transition in between. At early times, a large-scale helical dynamo develops in the bottom layer. At later times the dynamo saturates, but the vertical magnetic field continues to develop and rises to form dynamic bipolar structures at the top, which later disappear and reappear. Some of the structures look similar to $\delta$ spots observed in the Sun. This is the first example of magnetic flux concentrations, owing to strong density stratification, from self-consistent dynamo simulations that generate bipolar, super-equipartition strength, magnetic structures whose energy density can exceeds the turbulent kinetic energy by even a factor of ten.

Not much helicity is needed to drive large-scale dynamos

Understanding the in situ amplification of large-scale magnetic fields in turbulent astrophysical rotators has been a core subject of dynamo theory. When turbulent velocities are helical, large-scale dynamos that substantially amplify fields on scales that exceed the turbulent forcing scale arise, but the minimum sufficient fractional kinetic helicity f(h,C) has not been previously well quantified. Using direct numerical simulations for a simple helical dynamo, we show that f(h,C) decreases as the ratio of forcing to large-scale wave numbers k(F)/k(min) increases. From the condition that a large-scale helical dynamo must overcome the back reaction from any nonhelical field on the large scales, we develop a theory that can explain the simulations. For k(F)/k(min)≥8 we find f(h,C)≲3%, implying that very small helicity fractions strongly influence magnetic spectra for even moderate-scale separation.

Threshold shift of helical dynamo induced by stochastic flow fluctuations

We study the generation of magnetic field by a helical flow made of a mean steady flow plus a fluctuating flow assumed to behave like a Gaussian white noise. The steady flow and the fluctuations are both solid body screw motions with constant pitches, respectively, and . The threshold shift induced by fluctuations of small amplitude is calculated analytically as a function of for different values of , m and k being the axial and azimuthal wavenumbers of the magnetic field. For a fixed value of , negative threshold shifts have been found provided belongs to a bounded interval . When there is no mean flow, the analysis is extended to temporal fluctuations modeled by a colored noise. In that case, emphasis is put on the influence of the correlation time on the dynamo onset.

Comparison between turbulent helical dynamo simulations and a non-linear three-scale theory

Progress towards understanding principles of non-linear growth and saturation of large-scale magnetic fields has emerged from comparison of theoretical models that incorporate the evolution of magnetic helicity with numerical simulations for problems that are more idealized than expected in astrophysical circumstances, but still fully non-linear. We carry out a new comparison of this sort for the magnetic field growth from forced isotropic helical turbulence in a periodic box. Previous comparisons between analytic theory and simulations of this problem have shown that a two-scale model compares well with the simulations in agreeing that the driver of large-scale field growth is the difference between kinetic and current helicities associated with the small-scale field, and that the backreaction slows the growth of the large-scale field as the small-scale current helicity grows. However, the two-scale model artificially restricts the small-scale current helicity to reside at the same scale as the driving kinetic helicity. In addition, previous comparisons have focused on the late-time saturation regime and less on the early-time growth regime. Here, we address these issues by comparing a three-scale model to new simulations for both early- and late-time growth regimes. We find that the minimalist extension to three scales provides a better model for the field evolution than the two-scale model because the simulations show that the small-scale current helicity does not reside at the same scale as that of the driving kinetic helicity. The simulations also show that the peak of the small-scale current helicity migrates towards lower wavenumbers as the growth evolves from the fast to saturated growth regimes.

NONLINEAR SMALL-SCALE DYNAMOS AT LOW MAGNETIC PRANDTL NUMBERS

Saturated small-scale dynamo solutions driven by isotropic non-helical turbulence are presented at low magnetic Prandtl numbers Pm down to 0.01. For Pm < 0.1, most of the energy is dissipated via Joule heat and, in agreement with earlier results for helical large-scale dynamos, kinetic energy dissipation is shown to diminish proportional to Pm^{1/2} down to values of 0.1. In agreement with earlier work, there is, in addition to a short Golitsyn k^{-11/3} spectrum near the resistive scale also some evidence for a short k^{-1} spectrum on larger scales. The rms magnetic field strength of the small-scale dynamo is found to depend only weakly on the value of Pm and decreases by about a factor of 2 as Pm is decreased from 1 to 0.01. The possibility of dynamo action at Pm=0.1 in the nonlinear regime is argued to be a consequence of a suppression of the bottleneck seen in the kinetic energy spectrum in the absence of a dynamo and, more generally, a suppression of kinetic energy near the dissipation wavenumber.

LAGRANGIAN COHERENT STRUCTURES IN NONLINEAR DYNAMOS

Turbulence and chaos play a fundamental role in stellar convective zones through the transportof particles, energy and momentum, and in fast dynamos, through the stretching, twisting and folding of magnetic flux tubes. A particularly revealing way to describe turbulent motions is through the analysis of Lagrangian coherent structures (LCS), which are material lines or surfaces that act as transport barriers in the fluid. We report the detection of Lagrangian coherent structures in helical MHD dynamo simulations with scale separation. In an ABC--flow, two dynamo regimes, a propagating coherent mean--field regime and an intermittent regime, are identified as the magnetic diffusivity is varied. The sharp contrast between the chaotic tangle of attracting and repelling LCS in both regimes permits a unique analysis of the impact of the magnetic field on the velocity field. In a second example, LCS reveal the link between the level of chaotic mixing of the velocity field and the saturation of a large--scale dynamo when the magnetic field exceeds the equipartition value.

Influence of a multiplicative noise on the threshold of helical dynamo

We study the effect of stochastic velocity fluctuations on the generation of magnetic field by a helical flow made of a mean steady flow plus a fluctuating flow assumed to behave like a Gaussian white noise. We calculate analytically, the threshold shift and the frequency shift induced by fluctuations of small amplitude that are geometrically optimized. It is found that the threshold is lowered, provided the steady part of the flow represented by a solid body screw motion with constant pitch satisfies the condition where m and k, respectively, are the axial and azimuthal wavenumbers of the magnetic field such that m = −k = 1. When there is no mean flow we extend the analysis to fluctuations with a finite correlation time and find that the dynamo threshold is higher than for a steady flow with the same geometry.

Spherical single-roll dynamos at large magnetic Reynolds numbers

This paper concerns kinematic helical dynamos in a spherical fluid body surrounded by an insulator. In particular, we examine their behavior in the regime of large magnetic Reynolds number Rm, for which dynamo action is usually concentrated on a simple resonant stream surface. The dynamo eigensolutions are computed numerically for two representative single-roll flows using a compact spherical harmonic decomposition and fourth-order finite differences in radius. These solutions are then compared with the growth rates and eigenfunctions of the Gilbert and Ponty large Rm asymptotic theory [Geophys. Astrophys. Fluid Dyn. 93, 55 (2000)]. We find good agreement between the growth rates when Rm&amp;gt;104 and between the eigenfunctions when Rm&amp;gt;105.

The critical role of magnetic helicity in astrophysical large-scale dynamos

The role of magnetic helicity in astrophysical large-scale dynamos is reviewed and compared with cases where there is no energy supply and an initial magnetic field can only decay. In both cases magnetic energy tends to get redistributed to larger scales. Depending on the efficiency of magnetic helicity fluxes the decay of a helical field can speed up. Likewise, the saturation of a helical dynamo can speed up through magnetic helicity fluxes. The astrophysical importance of these processes is reviewed in the context of the solar dynamo and an estimated upper limit for the magnetic helicity flux of 1046 Mx2/cycle is given.

Toward coupling flow driven and magnetically driven dynamos

Most large-scale dynamo research for astrophysical rotators focuses on interior flow driven helical dynamos (FDHDs), but larger scale coronal fields most directly influence observations. It is thus important to understand the relationship between coronal and interior fields. Coronal field relaxation is actually a type of magnetically dominated helical dynamo (MDHD). MDHDs also occur in fusion plasma devices where they drive a system toward its relaxed state in response to magnetic helicity injection that otherwise drives the system away from this state. Global scale fields of astrophysical rotators and jets are thus plausibly produced by a direct coupling between an interior FDHD and a coronal MDHD, interfaced by magnetic helicity transport through their mutual boundary. Tracking the magnetic helicity also elucidates how both FDHD and MDHDs evolve and saturate. The utility of magnetic helicity is unhampered by its non-gauge invariance since physical fields can always be recovered.