We review the theory of magnetic reconnection in weakly ionized gases. The theory is relevant to reconnection in the interstellar medium, protostellar and protoplanetary disks, the outer envelopes of cool stars, and a new laboratory experiment. In general, partial ionization introduces three effects beyond the obvious one: increased resistivity due to electron-neutral collisions. First, magnetic neutral sheets are steepened by plasma-neutral drift, setting up the conditions for reconnection. Second, when ion-neutral friction is strong, the effective ion mass is increased by ρ/ρi, the ratio of total to plasma mass density. This reduces the Alfven speed vA by a factor of ρ/ρi and increases the ion skin depth δi by ρ/ρi. As a result, entrainment of neutrals slows MHD reconnection but permits the onset of fast collisionless reconnection at a larger Lundquist number S, or for a longer current sheet, than in the fully ionized plasma case. These effects, taken together, promote fast collisionless reconnection when the ionization fraction is of order 10% to 1%, but reconnection is slowed down for much smaller ionization fractions. Finally, ion-neutral friction can be a strong heating mechanism throughout the inflow and outflow regions. These effects are under study at the Magnetic Reconnection Experiment (MRX).

Hydromagnetic dynamo theory provides the prevailing theoretical description for the origin of magnetic fields in the universe. Here we consider the problem of kinematic, small-scale dynamo action driven by a random, incompressible, non-helical, homogeneous and isotropic flow. In the Kazantsev dynamo model the statistics of the driving flow are assumed to be instantaneously correlated in time. Here we compare the results of the model with the dynamo properties of a simulated flow that has equivalent spatial characteristics as the Kazantsev flow but different temporal statistics. In particular, the simulated flow is a solution of the forced Navier-Stokes equations and hence has a finite correlation time. We find that the Kazantsev model typically predicts a larger magnetic growth rate and a magnetic spectrum that peaks at smaller scales. However, we show that by filtering the diffusivity spectrum at small scales it is possible to bring the growth rates into agreement and simultaneously align the magnetic spectra.

We consider quasi-stationary two-dimensional magnetic reconnection in a partially ionized incompressible plasma. We find that when the plasma is weakly ionized and the collisions between the ions and the neutral particles are significant, the transition to fast collisionless reconnection due to the Hall effect in the generalized Ohm's law is expected to occur at much lower values of the Lundquist number, as compared to a fully ionized plasma case. We estimate that these conditions for fast reconnection are satisfied in molecular clouds and in protostellar disks.

We present a two-fluid magnetohydrodynamics model of quasi-stationary, two-dimensional magnetic reconnection in an incompressible plasma composed of electrons and ions. We found that there are two distinct regimes of slow and fast reconnection. The presence of these two regimes can provide a possible explanation for the initial slow build-up and the subsequent rapid release of magnetic energy frequently observed in cosmic and laboratory plasmas.

In this paper, two theoretical approaches for the calculation of the rate of quasi-stationary, two-dimensional magnetic reconnection with nonuniform anomalous resistivity are considered in the framework of incompressible magnetohydrodynamics (MHD). In the first, 'global' equations approach, the MHD equations are approximately solved for a whole reconnection layer, including the upstream and downstream regions and the layer center. In the second, 'local' equations approach, the equations are solved across the reconnection layer, including only the upstream region and the layer center. Both approaches give the same approximate answer for the reconnection rate. Our theoretical model is in agreement with the results of recent simulations of reconnection with spatially nonuniform resistivity.

Amplification of magnetic field due to kinematic turbulent dynamo action is studied in the regime of small magnetic Prandtl numbers. Such a regime is relevant for planets and stars interiors, as well as for liquid-metal laboratory experiments. A comprehensive analysis based on the Kazantsev-Kraichnan model is reported, which establishes the dynamo threshold and the dynamo growth rates for varying kinetic helicity of turbulent fluctuations. It is proposed that in contrast with the case of large magnetic Prandtl numbers, the kinematic dynamo action at small magnetic Prandtl numbers is significantly affected by kinetic helicity, and it can be made quite efficient with an appropriate choice of the helicity spectrum.

A theoretical model of quasistationary, two-dimensional magnetic reconnection is presented in the framework of incompressible two-fluid magnetohydrodynamics. The results are compared with recent numerical simulations and experiment.

Received 16 December 2008DOI:https://doi.org/10.1103/PhysRevLett.102.029902©2009 American Physical Society

We investigate the structure of magnetic fields amplified by turbulent velocity fluctuations, in the framework of the kinematic Kazantsev–Kraichnan model. We consider Kolmogorov distribution of velocity fluctuations, and assume that both Reynolds number and magnetic Reynolds number are very large. We present the full numerical solution of the model for the spectra and the growth rates of magnetic fluctuations. We consider astrophysically relevant limits of large and small magnetic Prandtl numbers, and address both helical and nonhelical cases.

Thermal conduction along stochastic magnetic field lines may be reduced because the heat conducting electrons become trapped and detrapped between regions of strong magnetic field (magnetic mirrors). We use a simple but realistic model for diffusion of mono-energetic electrons based on the fact that when there is a reduction of diffusion, it is controlled by a subset of the mirrors, the principle mirrors. To obtain the net reduction of the electron thermal conductivity, we average the mono-energetic diffusion over a Maxwellian distribution of electron velocities. As a result, we find that if the magnetic field correlation length in a cluster of galaxies is smaller than the electron mean free path at the thermal speed VT = √ kT/me, then thermal conduction is generally negligible.

The rate of quasistationary, two-dimensional magnetic reconnection is calculated in the framework of incompressible Hall magnetohydrodynamics, which includes the Hall and electron pressure terms in Ohm's law. The Hall-magnetohydrodynamics equations are solved in a local region across the reconnection electron layer, including only the upstream region and the layer center. In the case when the ion inertial length di is larger than the Sweet-Parker reconnection layer thickness, the dimensionless reconnection rate is found to be independent of the electrical resistivity and equal to di/L, where L is the scale length of the external magnetic field in the upstream region outside the electron layer and the ion layer thickness is found to be di.

We investigate magnetic field amplification in a turbulent velocity field with nonzero helicity, in the framework of the kinematic Kazantsev-Kraichnan model. We present the numerical solution of the model for the practically important case of Kolmogorov distribution of velocity fluctuations, with a large magnetic Reynolds number. We find that in contrast to the nonhelical case where growing magnetic fields are described by a few bound eigenmodes concentrated inside the inertial interval of the velocity field, in the helical case the number of bound eigenmodes considerably increases; moreover, new unbound eigenmodes appear. Both bound and unbound eigenmodes contribute to the large-scale magnetic field. This indicates a limited applicability of the conventional alpha model of a large-scale dynamo action, which captures only unbound modes.

In this paper quasistationary, “two-and-a-half-dimensional” magnetic reconnection is studied in the framework of incompressible resistive magnetohydrodynamics. A new theoretical approach for the calculation of the reconnection rate is presented. This approach is based on the local analytical derivations in a thin reconnection layer, and it is applicable to the case when resistivity is anomalous and is an arbitrary function of the electric current and the spatial coordinates. It is found that a quasistationary reconnection rate is fully determined by a particular functional form of the anomalous resistivity and by the local configuration of the magnetic field just outside the reconnection layer. It is also found that, in the special case of constant resistivity, reconnection is Sweet-Parker [Electromagnetic Phenomena, edited by B. Lehnert (Cambridge University Press, New York, 1958), p. 123; Astrophys. J., Suppl. 8, 177 (1963)] and not Petschek [AAS-NASA Symposium on Solar Flares NASA SP5 (National Aeronautics and Space Administration, Washington, D.C., 1964), p. 425].

The prevailing theory for the origin of cosmic magnetic fields is that they have been amplified to their present values by turbulent dynamo inductive action in the protogalactic and galactic medium. Up to now, in the calculation of the turbulent dynamo, it has been customary to assume that there is no back-reaction of the magnetic field on the turbulence, as long as the magnetic energy is less than the turbulent kinetic energy. This assumption leads to the kinematic dynamo theory. However, the applicability of this theory to protogalaxies is rather limited. The reason is that in protogalaxies, the temperature is very high, and the viscosity is dominated by magnetized ions. As the magnetic field strength grows in time, the ion cyclotron time becomes shorter than the ion collision time, and the plasma becomes strongly magnetized. As a result, the ion viscosity becomes the Braginskii viscosity. Thus, in protogalaxies the back-reaction sets in much earlier, at field strengths much lower than those that correspond to field-turbulence energy equipartition, and the turbulent dynamo becomes what we call the magnetized turbulent dynamo. In this paper we lay the theoretical groundwork for the magnetized turbulent dynamo. In particular, we predict that the magnetic energy growth rate in magnetized dynamo theory is up to 10 times larger than that in kinematic dynamo theory. We also briefly discuss how the Braginskii viscosity can aid the development of the inverse cascade of magnetic energy after energy equipartition is reached.

A weak fluctuating magnetic field embedded into a a turbulent conducting medium grows exponentially while its characteristic scale decays. In the interstellar medium and protogalactic plasmas, the magnetic Prandtl number is very large, so a broad spectrum of growing magnetic fluctuations is excited at small (subviscous) scales. The condition for the onset of nonlinear back reaction depends on the structure of the field lines. We study the statistical correlations that are set up in the field pattern and show that the magnetic-field lines possess a folding structure, where most of the scale decrease is due to the field variation across itself (rapid transverse direction reversals), while the scale of the field variation along itself stays approximately constant. Specifically, we find that, though both the magnetic energy and the mean-square curvature of the field lines grow exponentially, the field strength and the field-line curvature are anticorrelated, i.e., the curved field is relatively weak, while the growing field is relatively flat. The detailed analysis of the statistics of the curvature shows that it possesses a stationary limiting distribution with the bulk located at the values of curvature comparable to the characteristic wave number of the velocity field and a power tail extending to large values of curvature where it is eventually cut off by the resistive regularization. The regions of large curvature, therefore, occupy only a small fraction of the total volume of the system. Our theoretical results are corroborated by direct numerical simulations. The implication of the folding effect is that the advent of the Lorentz back reaction occurs when the magnetic energy approaches that of the smallest turbulent eddies. Our results also directly apply to the problem of statistical geometry of the material lines in a random flow.

We study the stochastic coagulation equation using simplified models and efficient Monte Carlo simulations. It is known that (i) runaway growth occurs if the two-body coalescence kernel rises faster than linearly in the mass of the heavier particle; and (ii) for such kernels, runaway is instantaneous in the limit that the number of particles tends to infinity at fixed collision time per particle. Superlinear kernels arise in astrophysical systems where gravitational focusing is important, such as the coalescence of planetesimals to form planets or of stars to form supermassive black holes. We find that the time for all particles to coalesce into a single body decreases as a power of the logarithm of the initial number of particles. Astrophysical implications are briefly discussed.