Mean-field dynamo in partially ionized plasmas – I

There are several astrophysical situations where one needs to study the dynamics of magnetic flux in partially ionized turbulent plasmas. In a partially ionized plasma, the magnetic induction is subjected to the ambipolar diffusion and the Hall effect in addition to the usual resistive dissipation. In this paper, we initiate the study of the kinematic dynamo in a partially ionized turbulent plasma. The Hall effect arises from the treatment of the electrons and the ions as two separate fluids and the ambipolar diffusion due to the inclusion of neutrals as the third fluid. It is shown that these non-ideal effects modify the so-called α effect and the turbulent diffusion coefficient β in a rather substantial way. The Hall effect may enhance or quench the dynamo action altogether. The ambipolar diffusion brings in an α which depends on the mean magnetic field. The new correlations embodying the coupling of the charged fluids and the neutral fluid appear in a decisive manner. The turbulence is necessarily magnetohydrodynamic with new spatial and time-scales. The nature of the new correlations is demonstrated by taking the Alfvénic turbulence as an example.     

An explicit scheme for multifluid magnetohydrodynamics

When modelling astrophysical fluid flows, it is often appropriate to discard the canonical magnetohydrodynamic approximation, thereby freeing the magnetic field to diffuse with respect to the bulk velocity field. As a consequence, however, the induction equation can become problematic to solve via standard explicit techniques. In particular, the Hall diffusion term admits fast-moving whistler waves which can impose a vanishing time-step limit.
Within an explicit differencing framework, a multifluid scheme for weakly ionized plasmas is presented which relies upon a new approach to integrating the induction equation efficiently. The first component of this approach is a relatively unknown method of accelerating the integration of parabolic systems by enforcing stability over large compound time-steps rather than over each of the constituent substeps. This method, Super Time-Stepping, proves to be very effective in applying a part of the Hall term up to a known critical value. The excess of the Hall term above this critical value is then included via a new scheme for pure Hall diffusion.     

Kolmogorov dissipation scales in weakly ionized plasmas

In a weakly ionized plasma, the evolution of the magnetic field is described by a 'generalized Ohm's law' that includes the Hall effect and the ambipolar diffusion terms. These terms introduce additional spatial and time-scales which play a decisive role in the cascading and the dissipation mechanisms in magnetohydrodynamic turbulence. We determine the Kolmogorov dissipation scales for the viscous, the resistive and the ambipolar dissipation mechanisms. The plasma, depending on its properties and the energy injection rate, may preferentially select one of these dissipation scales, thus determining the shortest spatial scale of the supposedly self-similar spectral distribution of the magnetic field. The results are illustrated taking the partially ionized part of the solar atmosphere as an example. Thus, the shortest spatial scale of the supposedly self-similar spectral distribution of the solar magnetic field is determined by any of the four dissipation scales given by the viscosity, the Spitzer resistivity (electron–ion collisions), the resistivity due to electron–neutral collisions and the ambipolar diffusivity. It is found that the ambipolar diffusion dominates for reasonably large energy injection rate. The robustness of the magnetic helicity in the partially ionized solar atmosphere would facilitate the formation of self-organized vortical structures.     

The Balbus–Hawley instability in weakly ionized discs

MHD in protostellar discs is modified by the Hall current when the ambipolar diffusion approximation breaks down. Here I examine the Balbus–Hawley (magnetorotational) instability of a weak, vertical magnetic field within a weakly ionized disc. Vertical stratification is neglected, and a linear analysis is undertaken for the case in which the wavevector of the perturbation is parallel to the magnetic field.
The growth rate depends on whether the initial magnetic field is parallel or antiparallel to the angular momentum of the disc. The parallel case is less (more) unstable than the antiparallel case if the Hall current is dominated by negative (positive) species. The less-unstable orientation is stable for χ ≲0.5, where χ is the ratio of a generalized neutral–ion collision frequency to the Keplerian frequency. The other orientation has a formal growth rate of the order of the Keplerian angular frequency even in the limit χ →0! In this limit the wavelength of the fastest-growing mode tends to infinity, so the minimum level of ionization for instability is determined by the requirement that a wavelength fit within a disc scaleheight. In the ambipolar diffusion case, this requires χ > v _A c _s; in the Hall case this imposes a potentially much weaker limit,         

Zones of anomalous activity of the magnetorotational instability in protostellar disks

A criterion for the magnetorotational instability of a protostellar disk in which the dust particles are assumed to be well mixed with the gas over the entire disk volume has been obtained within the framework of Hall magnetohydrodynamics. It is shown that the dusty plasma component affects significantly the Hall current and, under certain conditions, can cause its direction to be reversed compared to the case of a weakly ionized electron-ion plasma. A significant expansion of the range of wave numbers for unstable magnetic fluctuations is a consequence of the Hall current reversal. The spatial localization of the regions of protostellar disks in which not only the long-wavelength Alfvén disturbances but also the short-wavelength ones are subject to the magnetorotational instability is investigated. Possible physical consequences of the presence of anomalously active zones in cold disks for their structure and evolution are pointed out.     

Linear stability analysis of the Hall magnetorotational instability in a spherical domain

We investigate the stability of the Hall‐MHD system and determine its importance for neutron stars at their birth, when they still consist of differentially rotating plasma permeated by extremely strong magnetic fields. We solve the linearised HallMHD equations in a spherical shell threaded by a homogeneous magnetic field. With the fluid/flow coupling and the Hall effect included, the magnetorotational instability and the Hall effect are both acting together. Results differ for magnetic fields aligned with the rotation axis and anti‐parallel magnetic fields. For a positive alignment of the magnetic field the instability grows on a rotational time‐scale for any sufficiently large magnetic Reynolds number. Even the magnetic fields which are stable against the MRI due to the magnetic diffusion are now susceptible to the shear‐Hall instability. In contrast, the negative alignment places strong restrictions on the growth and the magnitude of the fields, hindering the effectiveness of the Hall‐MRI. While non‐axisymmetric modes of the MRI can be suppressed by strong enough rotation, there is no such restriction when the Hall effect is present. The implications for the magnitude and the topology of the magnetic field of a young neutron star may be significant (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Magnetorotational instability in stratified, weakly ionized accretion discs

We present a linear analysis of the vertical structure and growth of the magnetorotational instability in stratified, weakly ionized accretion discs, such as protostellar and quiescent dwarf novae systems. The method includes the effects of the magnetic coupling, the conductivity regime of the fluid and the strength of the magnetic field, which is initially vertical. The conductivity is treated as a tensor and is assumed to be constant with height.
We obtained solutions for the structure and growth rate of global unstable modes for different conductivity regimes, strengths of the initial magnetic field and coupling between ionized and neutral components of the fluid. The envelopes of short-wavelength perturbations are determined by the action of competing local growth rates at different heights, driven by the vertical stratification of the disc. Ambipolar diffusion perturbations peak consistently higher above the midplane than modes including Hall conductivity. For weak coupling, perturbations including the Hall effect grow faster and act over a more extended cross-section of the disc than those obtained using the ambipolar diffusion approximation.
Finally, we derived an approximate criterion for when Hall diffusion determines the growth of the magnetorotational instability. This is satisfied over a wide range of radii in protostellar discs, reducing the extent of the magnetic 'dead zone'. Even if the magnetic coupling is weak, significant accretion may occur close to the midplane, rather than in the surface regions of weakly ionized discs.     

Magnetic field evolution in neutron stars

Neutron stars contain persistent, ordered magnetic fields that are the strongest known in the Universe. However, their magnetic fluxes are similar to those in magnetic A and B stars and white dwarfs, suggesting that flux conservation during gravitational collapse may play an important role in establishing the field, although it might also be modified substantially by early convection, differential rotation, and magnetic instabilities. The equilibrium field configuration, established within hours (at most) of the formation of the star, is likely to be roughly axisymmetric, involving both poloidal and toroidal components. The stable stratification of the neutron star matter (due to its radial composition gradient) probably plays a crucial role in holding this magnetic structure inside the star. The field can evolve on long time scales by processes that overcome the stable stratification, such as weak interactions changing the relative abundances and ambipolar diffusion of charged particles with respect to neutrons. These processes become more effective for stronger magnetic fields, thus naturally explaining the magnetic energy dissipation expected in magnetars, at the same time as the longer-lived, weaker fields in classical and millisecond pulsars. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability and thermal properties of dark cloud L134

Using the results of optical and molecular line observations of the dark cloud L134, some basic cloud parameters are obtained and the stability and energy of the cloud are discussed.It is found that thermal pressure and rotation are unimportant, while internal magnetic field may be effective for supporting the cloud against gravitational collapse. And the cloud could not collapse on the free-fall time scale but on the longer time scale of ambipolar diffusion.The cooling and heating rates in L134 are also calculated. The results show that the work done by gravitation against thermal pressure is not an effective heating source; cosmic rays, however, may provide as much as 20% heating energy required. Calculation shows that internal magnetic energy released through the processes of ambipolar diffusion can supply the most part of the energy required, therefore, it may be the most important source.     

Driven waves in a two-fluid plasma

We study the physics of wave propagation in a weakly ionized plasma, as it applies to the formation of multifluid, magnetohydrodynamics (MHD) shock waves. We model the plasma as separate charged and neutral fluids which are coupled by ion–neutral friction. At times much less than the ion–neutral drag time, the fluids are decoupled and so evolve independently. At later times, the evolution is determined by the large inertial mismatch between the charged and neutral particles. The neutral flow continues to evolve independently; the charged flow is driven by and slaved to the neutral flow by friction. We calculate this driven flow analytically by considering the special but realistic case where the charged fluid obeys linearized equations of motion. We carry out an extensive analysis of linear, driven, MHD waves. The physics of driven MHD waves is embodied in certain Green functions which describe wave propagation on short time-scales, ambipolar diffusion on long time-scales and transitional behaviour at intermediate times. By way of illustration, we give an approximate solution for the formation of a multifluid shock during the collision of two identical interstellar clouds. The collision produces forward and reverse J shocks in the neutral fluid and a transient in the charged fluid. The latter rapidly evolves into a pair of magnetic precursors on the J shocks, wherein the ions undergo force-free motion and the magnetic field grows monotonically with time. The flow appears to be self-similar at the time when linear analysis ceases to be valid.     

Effect of magnetic resistivity on the stability of a partially ionized compressible Hall plasma

The dynamic stability of a partially ionized, compressible Hall plasma of finite electrical conductivity has been investigated when the plasma is immersed in a uniform, horizontal magnetic field. Based on the variational principle, which is shown to characterize the problem, the solution has been obtained for a semi-infinite plasma confined between two planes and having an exponential density stratification along the vertical. It is found that the effect of neutral gas friction is stabilizing while magnetic resistivity, Hall currents and compressibility all have destabilizing influence.On leave of absence from Department of Mathematics, University of Jodhpur, Jodhpur, India.     

Thermal instability in a weakly ionized plasma

We revisit the problem of clump formation due to thermal instabilities in a weakly ionized plasma with the help of a linear perturbation analysis, as discussed by Nejad-Asghar & Ghanbari. In the absence of a magnetic field and ambipolar diffusion the characteristic equation reduces to the thermal instability described by Field. We derive the critical wavelengths, which separate the spatial ranges of stability and instability. Contrary to the original analysis of Nejad-Asghar & Ghanbari, perturbations with a wavelength larger than the critical wavelength destabilize the cloud. Moreover, the instability regime of isentropic perturbations is drastically reduced. Isobaric modes with real values of the critical wavelength appear only if the density dependence of the cooling rate is more pronounced than the temperature dependence. Isentropic modes arise only if the power of the density in the cooling rate is smaller than 1/2, which is not fulfilled for CO cooling. We find that ambipolar diffusion is not a dominating heating process in molecular gas.     

Tayler instability with Hall effect in young neutron stars

Collapse calculations indicate that the hot young neutron stars rotate differentially so that strong toroidal magnetic field components should exist in the outer shell where also the Hall effect appears to be important when the Hall parameter = ω_Bτ exceeds unity. The amplitudes of the induced toroidal magnetic fields are limited by the current‐induced Tayler instability. An important characteristics of the Hall effect is its distinct dependence on the sign of the magnetic field. We find for fast rotation that positive (negative) Hall parameters essentially reduce (increase) the stability domain. It is thus concluded that the toroidal field belts in young neutron stars induced by their differential rotation should have different amplitudes in both hemispheres which later are frozen in. Due to the effect of magnetic suppression of the heat conductivity also the brightness of the two hemispheres should be different. As a possible example for our scenario the isolated neutron star RBS 1223 is considered which has been found to exhibit different X‐ray brightness at both hemispheres (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of magnetic equilibria in radio bubbles

Current-carrying flows, in the laboratory and in astrophysical jets, can form remarkably stable magnetic structures. Decades of experience show that such flows often build equilibria that reverse field directions, evolving to a magnetohydrodynamic (MHD) Taylor state, which has remarkable stability properties. We model jets and the magnetic bubbles they build as reversed-field pinch equilibria by assuming the driver current to be stiff in the MHD sense. Taking the jet current as rigid and a fixed function of position, we prove a theorem: that the same, simple MHD stability conditions guarantee stability, even after the jet turns off. This means that magnetic structures harbouring a massive inventory of magnetic energy can persist long after the building jet current has died away. These may be the relic radio 'fossils', 'ghost bubbles' or 'magnetic balloons' found in clusters. These equilibria, which are under magnetic tension, will evolve, retaining the stability properties from that state. The remaining fossil is not a disordered ball of magnetic fields, but a stable structure under tension, able to respond to the slings and arrows of outside forces. Typically their Alfvén speeds greatly exceed the cluster sound speed, and so they can keep out hot cluster plasma, leading to X-ray ghosts. Passing shocks cannot easily destroy them, but can energize and light them up anew at radio frequencies. Bubbles can rise in the hot cluster plasma, perhaps detaching from the parent radio galaxy but stable against Rayleigh–Taylor and other modes.     

Formation of Small-Scale Condensations in the Molecular Clouds via Thermal Instability

A systematic study of the linear thermal instability of a self-gravitating magnetic molecular cloud is carried out for the case when the unperturbed background is subject to local expansion or contraction. We consider the ambipolar diffusion, or ion-neutral friction on the perturbed states. In this way, we obtain a non-dimensional characteristic equation that reduces to the prior characteristic equation in the non-gravitating stationary background. By parametric manipulation of this characteristic equation, we conclude that there are, not only oblate condensation forming solutions, but also prolate solutions according to local expansion or contraction of the background. We obtain the conditions for existence of the Field lengths that thermal instability in the molecular clouds can occur. If these conditions establish, small-scale condensations in the form of spherical, oblate, or prolate shape may be produced via thermal instability.     

Can the turbulent galactic dynamo generate large-scale magnetic fields?

Large-scale magnetic fields in galaxies are thought to be generated by a turbulent dynamo. However, the same turbulence also leads to a small-scale dynamo which generates magnetic noise at a more rapid rate. The efficiency of the large-scale dynamo depends on how this noise saturates. We examine this issue, taking into account ambipolar drift, which obtains in a galaxy with significant neutral gas. We argue as follows.
(i) The small-scale dynamo generated field does not fill the volume, but is concentrated into intermittent rope-like structures. The flux ropes are curved on the turbulent eddy scales. Their thickness is set by the diffusive scale determined by the effective ambipolar diffusion.
(ii) For a largely neutral galactic gas, the small-scale dynamo saturates, as a result of inefficient random stretching, when the peak field in a flux rope has grown to a few times the equipartition value.
(iii) The average energy density in the saturated small-scale field is subequipartition, since it does not fill the volume.
(iv) Such fields neither drain significant energy from the turbulence nor convert eddy motion of the turbulence on the outer scale into wave-like motion. The diffusive effects needed for the large-scale dynamo operation are then preserved until the large-scale field itself grows to near equipartition levels.     

Ambipolar diffusion in the solar atmosphere

The process of non-linear ambipolar diffusion in the region overlying the solar surface can be an effective mechanism for producing sharp magnetic structures and current sheets. These may be the sites responsible for the occurrence of connectivity of magnetic field lines, and the subsequent explosive input of energy for heating of some of the features in the atmosphere of the Sun..     

Primordial magnetic fields in the post-recombination era and early reionization

We explore the ways in which primordial magnetic fields influence the thermal and ionization history of the post-recombination Universe. After recombination, the Universe becomes mostly neutral, resulting also in a sharp drop in the radiative viscosity. Primordial magnetic fields can then dissipate their energy into the intergalactic medium via ambipolar diffusion and, for small enough scales, by generating decaying magnetohydrodynamics turbulence. These processes can significantly modify the thermal and ionization history of the post-recombination Universe. We show that the dissipation effects of magnetic fields, which redshifts to a present value   B ₀= 3 × 10⁻⁹ G  smoothed on the magnetic Jeans scale and below, can give rise to Thomson scattering optical depths  τ≳ 0.1  , although not in the range of redshifts needed to explain the recent Wilkinson Microwave Anisotropy Probe ( WMAP ) polarization observations. We also study the possibility that primordial fields could induce the formation of subgalactic structures for   z ≳ 15  . We show that early structure formation induced by nanoGauss magnetic fields is potentially capable of producing the early reionization implied by the WMAP data. Future cosmic microwave background observations will be very useful to probe the modified ionization histories produced by primordial magnetic field evolution and constrain their strength.     

Localized magnetic reconnection as a cause of extraplanar diffuse ionized gas in the Galactic halo

Many observations indicate the occurrence of ionized gas in the distant haloes of galaxies (including our own). Since photoionization by stars (mainly O stars, young stars or evolved low-mass stars depending on the kind of galaxy) does not seem to be exclusively responsible for the ionization of the hydrogen filaments that should otherwise cool fast and recombine quickly, the question arises which extra energy source can produce the quasi-stationary ionization. We show that stationary localized magnetic reconnection in current filaments may contribute to the ionization of the extraplanar halo gas. In these filaments magnetic energy is dissipated. Consequently, the ionized as well as the neutral component is heated and re-ionized on a time-scale significantly shorter than the recombination time-scale. The amount of energy required for efficient re-ionization can in principle easily be provided by the free magnetic energy. We present quasi-static models that are characterized by plasma temperatures and densities that agree well with the observed values for the diffuse ionized gas component of the interstellar medium. Plasma–neutral gas fluid simulations are made to show that the recombination-induced dynamical reconnection process indeed works in a self-regulatory way.