Turbulent Magnetic Diffusivity Research Articles

Budget equations and astrophysical nonlinear mean-field dynamos

Abstract Solar, stellar and galactic large-scale magnetic fields are originated due to a combined action of non-uniform (differential) rotation and helical motions of plasma via mean-field dynamos. Usually, nonlinear mean-field dynamo theories take into account algebraic and dynamic quenching of alpha effect and algebraic quenching of turbulent magnetic diffusivity. However, the theories of the algebraic quenching do not take into account the effect of modification of the source of turbulence by the growing large-scale magnetic field. This phenomenon is due to the dissipation of the strong large-scale magnetic field resulting in an increase of the total turbulent energy. This effect has been studied using the budget equation for the total turbulent energy (which takes into account the feedback of the generated large-scale magnetic field on the background turbulence) for (i) a forced turbulence, (ii) a shear-produced turbulence and (iii) a convective turbulence. As the result of this effect, a nonlinear dynamo number decreases with increase of the large-scale magnetic field, so that that the mean-field αΩ, α2 and α2Ω dynamo instabilities are always saturated by the strong large-scale magnetic field.

Turbulent Processes and Mean-Field Dynamo

Mean-field dynamo theory has important applications in solar physics and galactic magnetism. We discuss some of the many turbulence effects relevant to the generation of large-scale magnetic fields in the solar convection zone. The mean-field description is then used to illustrate the physics of the alpha effect, turbulent pumping, turbulent magnetic diffusivity, and other effects on a modern solar dynamo model. We also discuss how turbulence transport coefficients are derived from local simulations of convection and then used in mean-field models.

Helical correction to turbulent magnetic diffusivity.

The effect of helicity in magnetohydrodynamic turbulence on the effective turbulent magnetic diffusion is considered here. The helical correction to turbulent diffusivity is analytically calculated with the use of the renormalization group approach. In agreement with previous numerical findings, this correction is shown to be negative and proportional to the second power of the magnetic Reynolds number, when the latter is small. In addition, the helical correction to turbulent diffusivity is found to obey a power-law-type dependence on the wave number of the most energetic turbulent eddies, k_{ℓ}, of the form k_{ℓ}^{-10/3}.

Turbulent Magnetic Diffusivity β Effect in a Magnetically Forced System

We have studied the large-scale dynamo forced with helical magnetic energy. Compared to the kinetic forcing process, the magnetic process is not clearly observed nor intuitive. However, it may represent the actual B field amplification in the stellar corona, accretion disk, plasma lab, or other magnetically dominated systems where the strong kinetic effect does not exist. The interaction between the magnetic field and the plasma is essentially nonlinear. However, when the plasma system is driven by helical energy, whether kinetic or magnetic, the nonlinear process can be linearized with pseudotensors a, β and the large-scale magnetic field . Conventionally, the α effect is thought to be the main dynamo effect converting kinetic energy into magnetic energy and transferring it to the large-scale regime. In contrast, β effect has been thought to diffuse magnetic energy. However, these conclusions are not based on the exact definition of α and β. In this paper, instead of the analytic definition of α and β, we derive a semi-analytic equation and apply it to the simulation data. The half analytic and numerical result shows that the averaged α effect is not so important in amplifying the field. Rather, it is the negative β effect combined with the Laplacian (∇2 → −k 2) that plays a key role in the dynamo process. Further, the negative magnetic diffusivity accounts for the attenuation of the plasma kinetic energy in large scales. We discuss this process using the theoretical method and the intuitive field structure model.

Turbulent magnetic helicity fluxes in solar convective zone

ABSTRACT Combined action of helical motions of plasma (the kinetic α effect) and non-uniform (differential) rotation is a key dynamo mechanism of solar and galactic large-scale magnetic fields. Dynamics of magnetic helicity of small-scale fields is a crucial mechanism in a non-linear dynamo saturation where turbulent magnetic helicity fluxes allow to avoid catastrophic quenching of the α effect. The convective zone of the Sun and solar-like stars, as well as galactic discs, are the source for production of turbulent magnetic helicity fluxes. In the framework of the mean-field approach and the spectral τ approximation, we derive turbulent magnetic helicity fluxes using the Coulomb gauge in a density-stratified turbulence. The turbulent magnetic helicity fluxes include non-gradient and gradient contributions. The non-gradient magnetic helicity flux is proportional to a non-linear effective velocity (which vanishes in the absence of the density stratification) multiplied by small-scale magnetic helicity, while the gradient contributions describe turbulent magnetic diffusion of the small-scale magnetic helicity. In addition, the turbulent magnetic helicity fluxes contain source terms proportional to the kinetic α effect or its gradients, and also contributions caused by the large-scale shear (solar differential rotation). We have demonstrated that the turbulent magnetic helicity fluxes due to the kinetic α effect and its radial derivative in combination with the non-linear magnetic diffusion of the small-scale magnetic helicity are dominant in the solar convective zone.

Mean-field Analysis on Large-scale Magnetic Fields at High Reynolds Numbers

Solar magnetic fields comprise an 11 yr activity cycle, represented by the number of sunspots. The maintenance of such a solar magnetic field can be attributed to fluid motion in the convection zone, i.e., a dynamo. This study conducts the mean-field analyses of the global solar dynamo simulation presented by Hotta et al. (2016). Although the study succeeds in producing coherent large-scale magnetic fields at high Reynolds numbers, the detailed physics of the maintenance of these fields have not been fully understood. This study extracts the α tensor and the turbulent magnetic diffusivity tensor β through mean-field analyses. The turbulent magnetic diffusivity exhibits a significant decrease toward high Reynolds numbers. The decrease in the turbulent magnetic diffusivity suppresses the energy conversion of large-scale field to small-scale field. This implies that the decrease in the turbulent magnetic diffusivity contributes to the maintenance of a large-scale magnetic field at high Reynolds numbers. A significant downward turbulent pumping is observed; it is enhanced in the weak phase of the large-scale field. This study proposes a cyclic reversal process of a large-scale field, which is dominantly driven by the α effect and is possibly triggered by downward pumping.

Jets from Accretion Disk Dynamos: Consistent Quenching Modes for Dynamo and Resistivity

Astrophysical jets are launched from strongly magnetized systems that host an accretion disk surrounding a central object. The origin of the magnetic field, which is a key component of the launching process, is still an open question. Here we address the question of how the magnetic field required for jet launching is generated and maintained by a dynamo process. By carrying out nonideal MHD simulations (PLUTO code), we investigate how the feedback of the generated magnetic field on the mean-field dynamo affects the disk and jet properties. We find that a stronger quenching of the dynamo leads to a saturation of the magnetic field at a lower disk magnetization. Nevertheless, we find that, while applying different dynamo feedback models, the overall jet properties remain unaffected. We then investigate a feedback model that encompasses a quenching of the magnetic diffusivity. Our modeling considers a more consistent approach for mean-field dynamo modeling simulations, as the magnetic quenching of turbulence should be considered for both a turbulent dynamo and turbulent magnetic diffusivity. We find that, after the magnetic field is saturated, the Blandford–Payne mechanism can work efficiently, leading to more collimated jets, which move, however, with slower speed. We find strong intermittent periods of flaring and knot ejection for low Coriolis numbers. In particular, flux ropes are built up and advected toward the inner disk thereby cutting off the inner disk wind, leading to magnetic field reversals, reconnection and, the emergence of intermittent flares.

Compressible Test-field Method and Its Application to Shear Dynamos

In this study, we present a compressible test-field method (CTFM) for computing α-effect and turbulent magnetic diffusivity tensors, as well as those relevant for the mean ponderomotive force and mass source, applied to the full MHD equations. We describe the theoretical background of the method and compare it to the quasi-kinematic test-field method and to the previously studied variant working in simplified MHD (SMHD). We present several test cases using velocity and magnetic fields of the Roberts geometry and also compare with the imposed-field method. We show that, for moderate imposed-field strengths, the nonlinear CTFM (nCTFM) gives results in agreement with the imposed-field method. A comparison of different flavors of the nCTFM in the shear dynamo case also yields agreement up to equipartition field strengths. Some deviations between the CTFM and SMHD variants exist. As a relevant physical application, we study nonhelically forced shear flows, which exhibit large-scale dynamo action, and present a reanalysis of low-Reynolds-number, moderate shear systems, where we previously ignored the pressure gradient in the momentum equation and found no coherent shear-current effect. Another key difference is that in the earlier study we used magnetic forcing to mimic small-scale dynamo action, while here it is self-consistently driven by purely kinetic forcing. The kinematic CTFM with general validity forms the core of our analysis. We still find no coherent shear-current effect, but do recover strong large-scale dynamo action that, according to our analysis, is driven by incoherent effects.

Possible Role of Non-Stationarity of Magnetohydrodynamic Turbulence in Understanding of Geomagnetic Excursions

The natural simplifying assumptions often put forward in the theoretical investigations of the magnetohydrodynamic turbulence are that the turbulent flow is statistically isotropic, homogeneous and stationary. Of course, the natural turbulence in the planetary interiors, such as the liquid core of the Earth is neither, which has important consequences for the dynamics of the planetary magnetic fields generated via the hydromagnetic dynamo mechanism operating in the interiors of the planets. Here we concentrate on the relaxation of the assumption of statistical stationarity of the turbulent flow and study the effect of turbulent wave fields in the Earth’s core, which induces non-stationarity, on the turbulent resistivity in the non-reflectionally symmetric flow and the geodynamo effect. It is shown that the electromotive force, including the so-called α-effect and the turbulent magnetic diffusivity η¯, induced by non-stationary turbulence, evolves slowly in time. However, the turbulent α¯ coefficient, responsible for the dynamo action and η¯ evolve differently in time, thus creating periods of enhanced and suppressed turbulent diffusion and dynamo action somewhat independently. In particular, periods of enhanced α¯ may coincide with periods of suppressed diffusion, leading to a stable and strong field period. On the other hand, it is shown that when enhanced diffusion occurs simultaneously with suppression of the α-effect, this leads to a sharp drop in the intensity of the large-scale field, corresponding to a geomagnetic excursion.

Renormalization group analysis of the magnetohydrodynamic turbulence and dynamo

The magnetohydrodynamic (MHD) turbulence appears in engineering laboratory flows and is a common phenomenon in natural systems, e.g. stellar and planetary interiors and atmospheres and the interstellar medium. The applications in engineering are particularly interesting due to the recent advancement of tokamak devices, reaching very high plasma temperatures, thus giving hope for the production of thermonuclear fusion power. In the case of astrophysical applications, perhaps the main feature of the MHD turbulence is its ability to generate and sustain large-scale and small-scale magnetic fields. However, a crucial effect of the MHD turbulence is also the enhancement of large-scale diffusion via interactions of small-scale pulsations, i.e. the generation of the so-called turbulent viscosity and turbulent magnetic diffusivity, which typically exceed by orders of magnitude their molecular counterparts. The enhanced resistivity plays an important role in the turbulent dynamo process. Estimates of the turbulent electromotive force (EMF), including the so-called $\alpha$ -effect responsible for amplification of the magnetic energy and the turbulent magnetic diffusion are desired. Here, we apply the renormalization group technique to extract the final expression for the turbulent EMF from the fully nonlinear dynamical equations (Navier–Stokes, induction equation). The simplified renormalized set of dynamical equations, including the equations for the means and fluctuations, is derived and the effective turbulent coefficients such as the viscosity, resistivity, the $\alpha$ -coefficient and the Lorentz-force coefficients are explicitly calculated. The results are also used to demonstrate the influence of magnetic fields on energy and helicity spectra of strongly turbulent flows, in particular the magnetic energy spectrum.

Generation and effects of electromotive force in turbulent stochastic reconnection

Reconnection is an important process that rules dissipation and diffusion of magnetic energy in plasmas. It is already clear that its rate is enhanced by turbulence and that reconnection itself may increase its stochasticity, but the main mechanism that connects these two effects is still not completely understood. The aim of this work is to identify, from the terms of the electromotive force, the dominant physical process responsible for enhancing the reconnection rate in turbulent plasmas. We employ full three-dimensional numerical simulations of turbulence driven by stochastic reconnection and estimate the production and dissipation of turbulent energy and cross-helicity, the amount of produced residual helicity, and determine the relation between these quantities and the reconnection rate. We observe the development of the electromotive force in the studied models with plasma-β=0.1−2 and the Lundquist number S=10−5−10−4. The turbulent energy and residual helicity develop in the large-scale current sheet, with the latter decreasing the effects of turbulent magnetic diffusion. We demonstrate that the stochastic reconnection, apart from the turbulence, can produce a finite value of cross-helicity (the magnitude of the turbulent cross-helicity to energy of the order of 10−5−10−3). Under this situation, the cross-helicity to turbulent energy ratio, however, has no correlation with the reconnection rate. We show that in this range of magnitude, the cross-helicity is not a necessary condition for fast reconnection to occur. The results suggest that cross-helicity is inherent to turbulent fields, but the reconnection rate enhancement is possibly caused by the effects of magnetic turbulent diffusion and controlled by the residual helicity.

Relativistic Outflows from a GRMHD Mean-field Disk Dynamo

Abstract In this work, we present simulations of thin accretion disks around black holes, in order to investigate a mean-field disk dynamo, using our resistive GRMHD code, which is able to produce a large-scale magnetic flux. We consider a weak seed field in an initially thin disk, a background (turbulent) magnetic diffusivity, and the dynamo action itself. A standard quenching mechanism is applied to mitigate an otherwise exponential increase in the magnetic field. Comparison simulations of an initial Fishbone–Moncrief torus suggest that reconnection may provide another quenching mechanism. The dynamo-generated magnetic flux expands from the disk interior into the disk corona, becomes advected by disk accretion, and fills the axial region of the domain. The dynamo leads to an initially rapid increase in magnetic energy and flux, while for later evolutionary stages the growth stabilizes. Accretion toward the black hole depends strongly on the type of magnetic-field structure that develops. The radial field component supports extraction of angular momentum, and thus accretion. It also sets the conditions for launching a disk wind, initially from the inner disk area. When a strong field engulfs the disk, strong winds are launched, predominantly driven by the pressure gradient of the toroidal field. For rotating black holes, we identify a Poynting flux-dominated jet, driven by the Blandford–Znajek mechanism. This axial Poynting flux is advected from the disk, and therefore accumulates at the expense of the flux carried by the disk wind, which is itself regenerated by the disk dynamo.

Anisotropic β‐effect in the diffusive processes in the rotating magnetoconvection problems

AbstractThe influence of the anisotropic β‐effect in the rotating magnetoconvection problems is studied. Several models of rotating magnetoconvection with different types of anisotropy are presented. A comparison among them is performed. We stress the difference, especially about the magnetoconvection models where there is anisotropy in the turbulent magnetic diffusivity. The influence on convection is thoroughly studied, and its physical meaning is discussed.

On the Existence of Shear-current Effects in Magnetized Burgulence

Abstract The possibility of explaining shear flow dynamos by a magnetic shear-current (MSC) effect is examined via numerical simulations. Our primary diagnostics is the determination of the turbulent magnetic diffusivity tensor η . In our setup, a negative sign of its component η yx is necessary for coherent dynamo action by the SC effect. To be able to measure turbulent transport coefficients from systems with magnetic background turbulence, we present an extension of the test-field method (TFM) applicable to our setup where the pressure gradient is dropped from the momentum equation: the nonlinear TFM (NLTFM). Our momentum equation is related to Burgers’ equation and the resulting flows are referred to as magnetized burgulence. We use both stochastic kinetic and magnetic forcings to mimic cases without and with simultaneous small-scale dynamo action. When we force only kinetically, negative η yx are obtained with exponential growth in both the radial and azimuthal mean magnetic field components. Using magnetokinetic forcing, the field growth is no longer exponential, while NLTFM yields positive η yx . By employing an alternative forcing from which wavevectors whose components correspond to the largest scales are removed, the exponential growth is recovered, but the NLTFM results do not change significantly. Analyzing the dynamo excitation conditions for the coherent SC and incoherent α and SC effects shows that the incoherent effects are the main drivers of the dynamo in the majority of cases. We find no evidence for MSC-effect-driven dynamos in our simulations.

Turbulent viscosity and magnetic Prandtl number from simulations of isotropically forced turbulence

Context.Turbulent diffusion of large-scale flows and magnetic fields plays a major role in many astrophysical systems, such as stellar convection zones and accretion discs.Aims.Our goal is to compute turbulent viscosity and magnetic diffusivity which are relevant for diffusing large-scale flows and magnetic fields, respectively. We also aim to compute their ratio, which is the turbulent magnetic Prandtl number, Pmt, for isotropically forced homogeneous turbulence.Methods.We used simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed large-scale sinusoidal shear flow. Turbulent viscosity was computed either from the resulting Reynolds stress or from the decay rate of the large-scale flow. Turbulent magnetic diffusivity was computed using the test-field method for a microphysical magnetic Prandtl number of unity. The scale dependence of the coefficients was studied by varying the wavenumber of the imposed sinusoidal shear and test fields.Results.We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (Re) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. These results are similar to those obtained earlier for the turbulent magnetic diffusivity. The results for the turbulent transport coefficients appear to converge at sufficiently high values of Re and the scale separation ratio. However, a weak trend is found even at the largest values of Re, suggesting that the turbulence is not in the fully developed regime. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large Re. For small Re we find values between 0.5 and 0.6 but the data are insufficient to draw conclusions regarding asymptotics. We demonstrate that our results are independent of the correlation time of the forcing function.Conclusions.The turbulent magnetic diffusivity is, in general, consistently higher than the turbulent viscosity, which is in qualitative agreement with analytic theories. However, the actual value of Pmtfound from the simulations (≈0.9−0.95) at large Re and large scale separation ratio is higher than any of the analytic predictions (0.4−0.8).

The nature of mean-field generation in three classes of optimal dynamos

In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis (Phys. Rev. Lett., vol. 109, 2012, 251101), we find mean-field dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behaviour is different from that of an$\unicode[STIX]{x1D6FC}^{2}$dynamo, where the two non-vanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a non-uniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, time dependent. The growing mean field can be modelled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chenet al.(J. Fluid Mech., vol. 783, 2015, pp. 23–45), we find, to some extent, similar mean-field dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15 % supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), mean-field dynamo action is found for one-dimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests that negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.

Fossil field decay due to nonlinear tides in massive binaries

Context.Surface magnetic fields have been detected in 5–10% of isolated massive stars, hosting outer radiative envelopes. They are often thought to have a fossil origin, resulting from the stellar formation phase. Yet, magnetic massive stars are scarcer in (close) short-period binaries, as reported by the BinaMIcS (Binarity and Magnetic Interaction in various classes of Stars) Collaboration.Aims.Different physical conditions in the molecular clouds giving birth to isolated stars and binaries are commonly invoked. In addition, we propose that the observed lower magnetic incidence in close binaries may be due to nonlinear tides. Indeed, close binaries are probably prone to tidal instability, a fluid instability growing upon the equilibrium tidal flow via nonlinear effects. Yet, stratified effects have hitherto been largely overlooked.Methods.We theoretically and numerically investigate tidal instability in rapidly rotating, stably stratified fluids permeated by magnetic fields. We use the short-wavelength stability method to propose a comprehensive (local) theory of tidal instability at the linear onset, discussing damping effects. Then, we propose a mixing-length theory for the mixing generated by tidal instability in the nonlinear regime. We successfully assess our theoretical predictions against proof-of-concept, direct numerical simulations. Finally, we compare our predictions with the observations of short-period, double-lined spectroscopic binary systems.Results.Using new analytical results, cross-validated by a direct integration of the stability equations, we show that tidal instability can be generated by nonlinear couplings of inertia-gravity waves with the equilibrium tidal flow in short-period massive binaries, even against the Joule diffusion. In the nonlinear regime, a fossil magnetic field can be dissipated by the turbulent magnetic diffusion induced by the saturated tidal flows.Conclusions.We predict that the turbulent Joule diffusion of fossil fields would occur in a few million years for several short-period massive binaries. Therefore, turbulent tidal flows could explain the observed dearth of some short-period magnetic binaries.

Compressibility in turbulent magnetohydrodynamics and passive scalar transport: mean-field theory

We develop a mean-field theory of compressibility effects in turbulent magnetohydrodynamics and passive scalar transport using the quasi-linear approximation and the spectral $\unicode[STIX]{x1D70F}$-approach. We find that compressibility decreases the $\unicode[STIX]{x1D6FC}$ effect and the turbulent magnetic diffusivity both at small and large magnetic Reynolds numbers, $Rm$. Similarly, compressibility decreases the turbulent diffusivity for passive scalars both at small and large Péclet numbers, $Pe$. On the other hand, compressibility does not affect the effective pumping velocity of the magnetic field for large $Rm$, but it decreases it for small $Rm$. Density stratification causes turbulent pumping of passive scalars, but it is found to become weaker with increasing compressibility. No such pumping effect exists for magnetic fields. However, compressibility results in a new passive scalar pumping effect from regions of low to high turbulent intensity both for small and large Péclet numbers. It can be interpreted as compressible turbophoresis of non-inertial particles and gaseous admixtures, while the classical turbophoresis effect exists only for inertial particles and causes them to be pumped to regions with lower turbulent intensity.

Turbulent diffusion in the photosphere as observational constraint on dynamo theories

AbstractWe utilized line-of-sight magnetograms acquired by HMI/SDO to derive the value of turbulent magnetic diffusivity in undisturbed photosphere. Two areas, a coronal hole area (CH) and an area a super-granulation pattern, SG, were analyzed. The behavior of the turbulent diffusion coefficient on time scales of 1000-40000 s and spatial scales of 500-6000 km was explored. Small magnetic elements in both CH and SG areas disperse in the same way and they are more mobile than the large elements. The regime of super-diffusivity is found for small elements (the turbulent diffusion coefficient K growths from 100 to 300 km2 s−1). Large magnetic elements disperse differently in the CH and SG areas. Comparison of these results with the previously published shows that there is a tendency of saturation of the diffusion coefficient on large scales, i.e., the turbulent regime of super-diffusivity gradually ceases so that normal diffusion with a constant value of K ≈ 500 km2 s−1 might be observed on time scales longer than a day. The results show that the turbulent diffusivity should not be considered in modeling as a scalar, the flux- and scale-dependence is obvious.

Magnetic Field Amplification in Supernova Remnants

Abstract Based on the new findings on the turbulent dynamo in Xu & Lazarian, we examine the magnetic field amplification in the context of supernova remnants. Due to the strong ion-neutral collisional damping in the weakly ionized interstellar medium, the dynamo in the preshock turbulence remains in the damping kinematic regime, which leads to a linear-in-time growth of the magnetic field strength. The resultant magnetic field structure enables effective diffusion upstream and shock acceleration of cosmic rays to energies above the “knee.” Differently, the nonlinear dynamo in the postshock turbulence leads to a linear-in-time growth of the magnetic energy due to the turbulent magnetic diffusion. Given a weak initial field strength in the postshock region, the magnetic field saturates at a significant distance from the shock front as a result of the inefficiency of the nonlinear dynamo. This result is in a good agreement with existing numerical simulations and well explains the X-ray spots detected far behind the shock front.