Large Magnetic Prandtl Numbers Research Articles

MRI turbulence in vertically stratified accretion discs at large magnetic Prandtl numbers

ABSTRACT The discovery of the first binary neutron star merger, GW170817, has spawned a plethora of global numerical relativity simulations. These simulations are often ideal (with dissipation determined by the grid) and/or axisymmetric (invoking ad hoc mean-field dynamos). However, binary neutron star mergers (similar to X-ray binaries and active galactic nuclei inner discs) are characterized by large magnetic Prandtl numbers, $\rm Pm$, (the ratio of viscosity to resistivity). $\rm Pm$ is a key parameter determining dynamo action and dissipation but it is ill-defined (and likely of order unity) in ideal simulations. To bridge this gap, we investigate the magnetorotational instability (MRI) and associated dynamo at large magnetic Prandtl numbers using fully compressible, three-dimensional, vertically stratified, isothermal simulations of a local patch of a disc. We find that, within the bulk of the disc (z ≲ 2H, where H is the scale-height), the turbulent intensity (parametrized by the stress-to-thermal-pressure ratio α), and the saturated magnetic field energy density, Emag, produced by the MRI dynamo, both scale as a power with Pm at moderate Pm (4 ≲ Pm ≲ 32): Emag ∼ Pm0.74 and α ∼ Pm0.71, respectively. At larger Pm (≳ 32), we find deviations from power-law scaling and the onset of a plateau. Compared to our recent unstratified study, this scaling with Pm becomes weaker further away from the disc mid-plane, where the Parker instability dominates. We perform a thorough spectral analysis to understand the underlying dynamics of small-scale MRI-driven turbulence in the mid-plane and of large-scale Parker-unstable structures in the atmosphere.

Dissipative magnetic structures and scales in small-scale dynamos

ABSTRACT Small-scale dynamos play important roles in modern astrophysics, especially on galactic and extragalactic scales. Owing to dynamo action, purely hydrodynamic Kolmogorov turbulence hardly exists and is often replaced by hydromagnetic turbulence. Understanding the size of dissipative magnetic structures is important in estimating the time-scale of galactic scintillation and other observational and theoretical aspects of interstellar and intergalactic small-scale dynamos. Here we show that, during the kinematic phase of the small-scale dynamo, the cutoff wavenumber of the magnetic energy spectra scales as expected for large magnetic Prandtl numbers, but continues in the same way also for moderately small values – contrary to what is expected. For a critical magnetic Prandtl number of about 0.3, the dissipative and resistive cutoffs are found to occur at the same wavenumber. In the non-linearly saturated regime, the critical magnetic Prandtl number becomes unity. The cutoff scale now has a shallower scaling with magnetic Prandtl number below a value of about three, and a steeper one otherwise compared to the kinematic regime.

Batchelor, Saffman, and Kazantsev spectra in galactic small-scale dynamos

ABSTRACT The magnetic fields in galaxy clusters and probably also in the interstellar medium are believed to be generated by a small-scale dynamo. Theoretically, during its kinematic stage, it is characterized by a Kazantsev spectrum, which peaks at the resistive scale. It is only slightly shallower than the Saffman spectrum that is expected for random and causally connected magnetic fields. Causally disconnected fields have the even steeper Batchelor spectrum. Here, we show that all three spectra are present in the small-scale dynamo. During the kinematic stage, the Batchelor spectrum occurs on scales larger than the energy-carrying scale of the turbulence, and the Kazantsev spectrum on smaller scales within the inertial range of the turbulence – even for a magnetic Prandtl number of unity. In the saturated state, the dynamo develops a Saffman spectrum on large scales, suggestive of the build-up of long-range correlations. At large magnetic Prandtl numbers, elongated structures are seen in synthetic synchrotron emission maps showing the parity-even E polarization. We also observe a significant excess in the E polarization over the parity-odd B polarization at subresistive scales, and a deficiency at larger scales. This finding is at odds with the observed excess in the Galactic microwave foreground emission, which is believed to be associated with larger scales. The E and B polarizations may be highly non-Gaussian and skewed in the kinematic regime of the dynamo. For dust emission, however, the polarized emission is always nearly Gaussian, and the excess in the E polarization is much weaker.

MRI turbulence in accretion discs at large magnetic Prandtl numbers

ABSTRACT The effect of large magnetic Prandtl number Pm (the ratio of viscosity to resistivity) on the turbulent transport and energetics of the magnetorotational instability (MRI) is poorly understood, despite the realization of this regime in astrophysical environments as disparate as discs from binary neutron star (BNS) mergers, the inner regions of low-mass X-ray binaries and active galactic nuclei, and the interiors of protoneutron stars. We investigate the MRI dynamo and associated turbulence in the regime Pm > 1 by carrying out fully compressible, 3D MHD-shearing box simulations using the finite-volume code pluto, focusing mostly on the case of Keplerian shear relevant to accretion discs. We find that when the magnetic Reynolds number is kept fixed, the turbulent transport (as parameterized by α, the ratio of stress to thermal pressure) scales with the magnetic Prandtl number as α ∼ Pmδ, with δ ∼ 0.5−0.7 up to Pm ∼ 128. However, this scaling weakens as the magnetic Reynolds number is increased. Importantly, compared to previous studies, we find a new effect at very large Pm – the turbulent energy and stress begin to plateau, no longer depending on Pm. To understand these results we have carried out a detailed analysis of the turbulent dynamics in Fourier space, focusing on the effect of increasing Pm on the transverse cascade – a key non-linear process induced by the disc shear flow that is responsible for the sustenance of MRI turbulence. Finally, we find that α–Pm scaling is sensitive to the box vertical-to-radial aspect ratio, as well as to the background shear.

Evolution of Kinetic and Magnetic Energy in a Large Magnetic Prandtl Number System

Many regions of the universe are in a state of hot, magnetized, and ionized X-ray emitting plasmas. We numerically simulated the energy spectrum of this highly viscous and conductive system. Without magnetic field, the fluctuating plasma motion decays in a relatively large viscous scale l ν (∼1/k ν ). However, the magnetic field extends the viscous scale to the magnetic diffusivity one l η (∼1/k η ) yielding a unique energy spectrum. Numerical simulation shows that kinetic and magnetic energy spectrum are E V ∼ k −3.7 and E M ∼ k −0.85 in the extended viscous scale regime. To explain this extraordinary power law, we set up two simultaneous differential equations for E V and E M and solved them using Eddy Damped Quasi Normal Markovianized approximation. Focusing on the most dominant terms, we analytically derived the spectrum relation consistent with the simulation data. We also simulated the same system with helical energy. The inversely cascaded magnetic energy makes the spectrum steeper. This inverse energy transfer, in addition to the external magnetic field and instabilities, provides us a clue to the diversified spectra characterized by E V ∼ k −3.8 − k −3.07 and E M ∼ k −2.17 − k −0.27 with large magnetic Prandtl number.

Dynamo effect caused by non-stationary turbulence in strongly magnetized, hot, low-density plasma

Context.The large-scale hydromagnetic dynamo mechanism is often assumed to rely on the existence of a resistive phase shift between the kinetic and magnetic components of waves. It is not clear how this mechanism could effectively operate in hot, low-density plasma, where the electrical resistivity is extremely low.Aims.We show that non-stationarity, a common factor in plasma turbulence (e.g., in the interstellar medium caused by random supernova explosions), allows inducing a strong large-scale electromotive force (EMF) by the plasma flow and significantly enhances the dynamo effect.Methods.We adopted the simplest approximation in which plasma evolution is modelled with a set of incompressible magnetohydrodynamic equations with a random, Gaussian, and non-stationary forcing to explicitly demonstrate the role of non-stationarity in the large-scale dynamo process. The EMF was calculated analytically in the limit of large magnetic Prandtl numbers (low magnetic diffusion in comparison with plasma viscosity) and strong magnetic fields for the non-stationary turbulence and the standard diffusive mechanism.Results.We show that the EMF induced by the effects of non-stationarity can typically be expected to dominate the dynamo mechanism (over the diffusive phase shift generation) by several orders of magnitude. This is confirmed by the explicit calculation in two example cases for the interstellar medium in the Milky Way galaxy and hot accretion disks such as those of active galactic nuclei.

Non-Gaussian Generalization of the Kazantsev–Kraichnan Model for a Turbulent Dynamo

Abstract We consider a natural generalization of the Kazantsev–Kraichnan model for a small-scale turbulent dynamo. This generalization takes into account the statistical time asymmetry of a turbulent flow and thus allows one to describe velocity fields with energy cascade. For three-dimensional velocity fields, a generalized Kazantsev equation is derived, and the evolution of the second-order magnetic field correlator is investigated for large but finite magnetic Prandtl numbers. It is shown that as Pr m → ∞, the growth increment tends to the limit known from the T-exponential (Lagrangian deformation) method. Magnetic field generation is shown to be weaker than that in the Gaussian velocity field for any direction of the energy cascade and essentially depends on the Prandtl number.

No feedback is possible in a small-scale turbulent magnetic field

Evolution of a stochastically homogeneous magnetic field advected by an incompressible turbulent flow with large magnetic Prandtl numbers is considered at scales smaller than the Kolmogorov viscous scale. It is shown that, despite the unlimited growth of the magnetic field, its feedback on the fluid's dynamics remains negligibly small.

Transition from laminar to turbulent dynamo: the effect of varying Prandtl number

We carry out numerical dynamo simulations in a spherical shell to assess the influence of varying Prandtl number on the morphology of the induced magnetic field in both cases when the convection is laminar and turbulent. We work in a regime of large magnetic Prandtl number () due to computational constraints of the full three-dimensional numerical model. We consider dynamos driven by thermal convection in the regimes of small and unit Prandtl numbers. Unlike the laminar convection, in the turbulent regime the morphology of the induced magnetic fields depends only weakly on the Prandtl number. For both values of considered the magnetic field is non-dipolar and small-scale. Together with the previous results of other authors, who found that the morphology of the magnetic field does not change substantially with when is small, the results of our study suggest that the induced magnetic field does not vary substantially with regardless of the strength of the magnetic diffusion.

Reversed Dynamo at Small Scales and Large Magnetic Prandtl Number

Abstract We show that at large magnetic Prandtl numbers, the Lorentz force does work on the flow at small scales and drives fluid motions, whose energy is dissipated viscously. This situation is the opposite of that in a normal dynamo, where the flow does work against the Lorentz force. We compute the spectral conversion rates between kinetic and magnetic energies for several magnetic Prandtl numbers and show that normal (forward) dynamo action occurs on large scales over a progressively narrower range of wavenumbers as the magnetic Prandtl number is increased. At higher wavenumbers, reversed dynamo action occurs, i.e., magnetic energy is converted back into kinetic energy at small scales. We demonstrate this in both direct numerical simulations forced by volume stirring and in large eddy simulations (LESs) of solar convectively driven small-scale dynamos. Low-density plasmas such as stellar coronae tend to have large magnetic Prandtl numbers, i.e., the viscosity is large compared with the magnetic diffusivity. The regime in which viscous dissipation dominates over resistive dissipation for large magnetic Prandtl numbers was also previously found in LESs of the solar corona, i.e., our findings are a more fundamental property of MHD that is not just restricted to dynamos. Viscous energy dissipation is a consequence of positive Lorentz force work, which may partly correspond to particle acceleration in close-to-collisionless plasmas. This is, however, not modeled in the MHD approximation employed. By contrast, resistive energy dissipation on current sheets is expected to be unimportant in stellar coronae.

Ambipolar diffusion in large Prandtl number turbulence

ABSTRACT We study the effects of ambipolar diffusion (AD) on hydromagnetic turbulence. We consider the regime of large magnetic Prandtl number, relevant to the interstellar medium. In most of the cases, we use the single-fluid approximation where the drift velocity between charged and neutral particles is proportional to the Lorentz force. In two cases we also compare with the corresponding two-fluid model, where ionization and recombination are included in the continuity and momentum equations for the neutral and charged species. The magnetic field properties are found to be well represented by the single-fluid approximation. We quantify the effects of AD on total and spectral kinetic and magnetic energies, the ohmic and AD dissipation rates, the statistics of the magnetic field, the current density, and the linear polarization as measured by the rotationally invariant E and B mode polarizations. We show that the kurtosis of the magnetic field decreases with increasing AD. The E mode polarization changes its skewness from positive values for small AD to negative ones for large AD. Even when AD is weak, changes in AD have a marked effect on the skewness and kurtosis of E, and only a weak effect on those of B. These results open the possibility of employing E and B mode polarizations as diagnostic tools for characterizing turbulent properties of the interstellar medium.

Nested polyhedra model of isotropic magnetohydrodynamic turbulence.

A nested polyhedra model has been developed for magnetohydrodynamic turbulence. Driving only the velocity field at large scales with random, divergence-free forcing results in a clear, stationary k^{-5/3} spectrum for both kinetic and magnetic energies. Since the model naturally effaces disparate scale interactions, does not have a guide field, and avoids injecting any sign of helicity by random forcing, the resulting three-dimensional k spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the timescale for the equipartition offset with those of the smallest scales shows a k^{-1/2} scaling.

Stochastic Reconnection for Large Magnetic Prandtl Numbers

Abstract We consider stochastic magnetic reconnection in high-β plasmas with large magnetic Prandtl numbers, Pr m > 1. For large Pr m , field line stochasticity is suppressed at very small scales, impeding diffusion. In addition, viscosity suppresses very small-scale differential motions and therefore also the local reconnection. Here we consider the effect of high magnetic Prandtl numbers on the global reconnection rate in a turbulent medium and provide a diffusion equation for the magnetic field lines considering both resistive and viscous dissipation. We find that the width of the outflow region is unaffected unless Pr m is exponentially larger than the Reynolds number Re. The ejection velocity of matter from the reconnection region is also unaffected by viscosity unless Re ∼ 1. By these criteria the reconnection rate in typical astrophysical systems is almost independent of viscosity. This remains true for reconnection in quiet environments where current sheet instabilities drive reconnection. However, if Pr m > 1, viscosity can suppress small-scale reconnection events near and below the Kolmogorov or viscous damping scale. This will produce a threshold for the suppression of large-scale reconnection by viscosity when . In any case, for Pr m > 1 this leads to a flattening of the magnetic fluctuation power spectrum, so that its spectral index is ∼−4/3 for length scales between the viscous dissipation scale and eddies larger by roughly . Current numerical simulations are insensitive to this effect. We suggest that the dependence of reconnection on viscosity in these simulations may be due to insufficient resolution for the turbulent inertial range rather than a guide to the large Re limit.

How to form a millisecond magnetar? Magnetic field amplification in protoneutron stars

AbstractExtremely strong magnetic fields of the order of 1015G are required to explain the properties of magnetars, the most magnetic neutron stars. Such a strong magnetic field is expected to play an important role for the dynamics of core-collapse supernovae, and in the presence of rapid rotation may power superluminous supernovae and hypernovae associated to long gamma-ray bursts. The origin of these strong magnetic fields remains, however, obscure and most likely requires an amplification over many orders of magnitude in the protoneutron star. One of the most promising agents is the magnetorotational instability (MRI), which can in principle amplify exponentially fast a weak initial magnetic field to a dynamically relevant strength. We describe our current understanding of the MRI in protoneutron stars and show recent results on its dependence on physical conditions specific to protoneutron stars such as neutrino radiation, strong buoyancy effects and large magnetic Prandtl number.

EXTENSION OF THE MURAM RADIATIVE MHD CODE FOR CORONAL SIMULATIONS

ABSTRACT We present a new version of the MURaM radiative magnetohydrodynamics (MHD) code that allows for simulations spanning from the upper convection zone into the solar corona. We implement the relevant coronal physics in terms of optically thin radiative loss, field aligned heat conduction, and an equilibrium ionization equation of state. We artificially limit the coronal Alfvén and heat conduction speeds to computationally manageable values using an approximation to semi-relativistic MHD with an artificially reduced speed of light (Boris correction). We present example solutions ranging from quiet to active Sun in order to verify the validity of our approach. We quantify the role of numerical diffusivity for the effective coronal heating. We find that the (numerical) magnetic Prandtl number determines the ratio of resistive to viscous heating and that owing to the very large magnetic Prandtl number of the solar corona, heating is expected to happen predominantly through viscous dissipation. We find that reasonable solutions can be obtained with values of the reduced speed of light just marginally larger than the maximum sound speed. Overall this leads to a fully explicit code that can compute the time evolution of the solar corona in response to photospheric driving using numerical time steps not much smaller than 0.1 s. Numerical simulations of the coronal response to flux emergence covering a time span of a few days are well within reach using this approach.

Dynamos at extreme magnetic Prandtl numbers: insights from shell models

ABSTRACTWe present a magnetohydrodynamic (MHD) shell model suitable for computation of various energy fluxes of MHD turbulence for very small and very large magnetic Prandtl numbers Pm; such computations are inaccessible to direct numerical simulations. For small Pm, we observe that both kinetic and magnetic energy spectra scale as k−5/3 in the inertial range, but the dissipative magnetic energy scales as k−11/3exp (− k/kη). Here the kinetic energy at large length scale feeds the large-scale magnetic field that cascades to small-scale magnetic field, which gets dissipated by Joule heating. The large-Pm dynamo has a similar behaviour except that the dissipative kinetic energy scales as k−13/3. For this case, the large-scale velocity field transfers energy to the large-scale magnetic field, which gets transferred to small-scale velocity and magnetic fields; the energy of the small-scale magnetic field also gets transferred to the small-scale velocity field, and the energy thus accumulated is dissipated by the viscous force.

Hydromagnetic dynamos at the low Ekman and magnetic Prandtl numbers

Abstract Hydromagnetic dynamos are numerically investigated at low Prandtl, Ekman and magnetic Prandtl numbers using the PARODY dynamo code. In all the investigated cases, the generated magnetic fields are dominantly-dipolar. Convection is small-scale and columnar, while the magnetic field maintains its large-scale structure. In this study the generated magnetic field never becomes weak in the polar regions, neither at large magnetic Prandtl numbers (when the magnetic diffusion is weak), nor at low magnetic Prandtl numbers (when the magnetic diffusion is strong), which is a completely different situation to that observed in previous studies. As magnetic fields never become weak in the polar regions, then the magnetic field is always regenerated in the tangent cylinder. At both values of the magnetic Prandtl number, strong polar magnetic upwellings and weaker equatorial upwellings are observed. An occurrence of polar magnetic upwellings is coupled with a regenaration of magnetic fields inside the tangent cylinder and then with a not weakened intensity of magnetic fields in the polar regions. These new results indicate that inertia and viscosity are probably negligible at low Ekman numbers.

Generation of sheet currents by high frequency fast MHD waves

The evolution of fast magnetosonic waves of high frequency propagating into an axisymmetric equilibrium plasma is studied. By using the methods of weakly nonlinear geometrical optics, it is shown that the perturbation travels in the equatorial plane while satisfying a transport equation which enables us to predict the time and location of formation of shock waves. For plasmas of large magnetic Prandtl number, this would result into the creation of sheet currents which may give rise to magnetic reconnection and destruction of the original equilibrium.

Saturation of the magnetorotational instability in the unstratified shearing box with zero net flux: convergence in taller boxes

Previous studies of the nonlinear regime of the magnetorotational instability in one particular type of shearing box model -- unstratified with no net magnetic flux -- find that without explicit dissipation (viscosity and resistivity) the saturation amplitude decreases with increasing numerical resolution. We show that this result is strongly dependent on the vertical aspect ratio of the computational domain $L_z/L_x$. When $L_z/L_x\lesssim 1$, we recover previous results. However, when the vertical domain is extended $L_z/L_x \gtrsim 2.5$, we find the saturation level of the stress is greatly increased (giving a ratio of stress to pressure $\alpha \gtrsim 0.1$), and moreover the results are independent of numerical resolution. Consistent with previous results, we find that saturation of the MRI in this regime is controlled by a cyclic dynamo which generates patches of strong toroidal field that switches sign on scales of $L_x$ in the vertical direction. We speculate that when $L_z/L_x\lesssim 1$, the dynamo is inhibited by the small size of the vertical domain, leading to the puzzling dependence of saturation amplitude on resolution. We show that previous toy models developed to explain the MRI dynamo are consistent with our results, and that the cyclic pattern of toroidal fields observed in stratified shearing box simulations (leading to the so-called butterfly diagram) may also be related. In tall boxes the saturation amplitude is insensitive to whether or not explicit dissipation is included in the calculations, at least for large magnetic Reynolds and Prandtl number. Finally, we show MRI turbulence in tall domains has a smaller critical $\rm{Pm}_c$, and an extended lifetime compared to $L_z/L_x\lesssim 1$ boxes.

Saturation of the turbulent dynamo.

The origin of strong magnetic fields in the Universe can be explained by amplifying weak seed fields via turbulent motions on small spatial scales and subsequently transporting the magnetic energy to larger scales. This process is known as the turbulent dynamo and depends on the properties of turbulence, i.e., on the hydrodynamical Reynolds number and the compressibility of the gas, and on the magnetic diffusivity. While we know the growth rate of the magnetic energy in the linear regime, the saturation level, i.e., the ratio of magnetic energy to turbulent kinetic energy that can be reached, is not known from analytical calculations. In this paper we present a scale-dependent saturation model based on an effective turbulent resistivity which is determined by the turnover time scale of turbulent eddies and the magnetic energy density. The magnetic resistivity increases compared to the Spitzer value and the effective scale on which the magnetic energy spectrum is at its maximum moves to larger spatial scales. This process ends when the peak reaches a characteristic wave number k☆ which is determined by the critical magnetic Reynolds number. The saturation level of the dynamo also depends on the type of turbulence and differs for the limits of large and small magnetic Prandtl numbers Pm. With our model we find saturation levels between 43.8% and 1.3% for Pm≫1 and between 2.43% and 0.135% for Pm≪1, where the higher values refer to incompressible turbulence and the lower ones to highly compressible turbulence.