On the meaning and inapplicability of the Zeldovich relations of magnetohydrodynamics

Considering a plasma with an initially weak large scale field subject to nonhelical turbulent stirring, Zeldovich (1957), for two‐dimensions, followed by others for three dimensions, have presented formulae of the form 〈b²〉 = f(R_M) . Such “Zeldovich relations” have sometimes been interpreted to provide steady‐state relations between the energy associated with the fluctuating magnetic field and that associated with a large scale or mean field multiplied by a function f that depends on spatial dimension and a magnetic Reynolds number R_M. Here we dissect the origin of these relations and pinpoint pitfalls that show why they are inapplicable to realistic, dynamical MHD turbulence and that they disagree with many numerical simulations. For 2D, we show that when the total magnetic field is determined by a vector potential, the standard Zeldovich relation applies only transiently, characterizing a maximum possible value that the field energy can reach before necessarily decaying. In 3D, we show that the standard Zeldovich relations are derived by balancing subdominant terms. In contrast, balancing the dominant terms shows that the fluctuating field can grow to a value independent of R_M and the initially imposed , as seen in numerical simulations. We also emphasize that these Zeldovich relations of nonhelical turbulence imply nothing about the amount mean field growth in a helical dynamo. In short, by re‐analyzing the origin of the Zeldovich relations, we highlight that they are inapplicable to realistic steady‐states of large R_M MHD turbulence. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydrostatic equilibrium in a magnetized, warped Galactic disc

Hydrostatic equilibrium of the multiphase interstellar medium in the solar vicinity is reconsidered, with the regular and turbulent magnetic fields treated separately. The regular magnetic field strength required to support the gas is consistent with independent estimates, provided that energy equipartition is maintained between turbulence and random magnetic fields. Our results indicate that a mid-plane value of B ₀=4 μG for the regular magnetic field near the Sun leads to more attractive models than B ₀=2 μG . The vertical profiles of both the regular and random magnetic fields contain disc and halo components, the parameters of which we have determined. The layer at 1≲| z |≲4 kpc can be overpressured and an outflow at a speed of about 50 km s⁻¹ may occur there, presumably associated with a Galactic fountain flow, if B ₀≃2 μG .
We show that hydrostatic equilibrium in a warped disc must produce asymmetric density distributions in z , in rough agreement with H  i observations in the outer Galaxy. This asymmetry may be a useful diagnostic of the details of the warping mechanism in the Milky Way and other galaxies. We find indications that gas and magnetic field pressures are different above and below the warped midplane in the outer Galaxy, and quantify the difference in terms of turbulent velocity and/or magnetic field strength.     

Milestones in the Observations of Cosmic Magnetic Fields

Magnetic fields are observed everywhere in the universe. In this review, we concentrate on the observational aspects of the magnetic fields of Galactic and extragalactic objects. Readers can follow the milestones in the observations of cosmic magnetic fields obtained from the most important tracers of magnetic fields, namely, the star-light polarization, the Zeeman effect, the rotation measures (RMs, hereafter) of extragalactic radio sources, the pulsar RMs, radio polarization observations, as well as the newly implemented sub-mm and mm polarization capabilities. The magnetic field of the Galaxy was first discovered in 1949 by optical polarization observations. The local magnetic fields within one or two kpc have been well delineated by starlight polarization data. The polarization observations of diffuse Galactic radio background emission in 1962 confirmed unequivocally the existence of a Galactic magnetic field. The bulk of the present information about the magnetic fields in the Galaxy comes from anal     

Magnetic flux expulsion in powerful superbubble explosions and the α–Ω dynamo

The possibility of magnetic flux expulsion from the Galaxy in superbubble (SB) explosions, important for the α –Ω dynamo, is considered. Special emphasis is put on investigation of the downsliding of the matter from the top of the shell formed by the SB explosion, which is able to influence the kinematics of the shell. It is shown that either Galactic gravity or the development of the Rayleigh–Taylor instabilities in the shell limit the SB expansion, thus making magnetic flux expulsion impossible. The effect of cosmic rays in the shell on the sliding is considered, and it is shown that it is negligible compared with Galactic gravity. Thus the question of the possible mechanism of flux expulsion in the α –Ω dynamo remains open.     

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.     

On the influence of cooling and heating processes on the Parker instability – II. Numerical simulations

We investigate the Parker instability (PI) influenced by thermal processes in a non-adiabatic, gravitationally stratified interstellar medium and discuss a model including the photoionization heating together with the supplemental heating mechanisms postulated by Reynolds, Haffner and Tufte. A cooling rate due to radiative losses is described by an approximation to the realistic cooling function of Dalgarno and McCray for ionized interstellar gas. An unperturbed initial state of the system simultaneously represents both a magnetohydrostatic and thermal equilibrium, and is thermally stable. We perform a set of 3D numerical magnetohydrodynamic simulations using the zeusmp code. We find that PI developing in the presence of non-adiabatic effects promotes a transition of gas in magnetic valleys to a thermally unstable regime. We find that the region of initially enhanced density due to PI starts to condense more as the result of thermal instability action. The density in this region rises above the classical isothermal limit of two times the equilibrium value at the mid-plane. The maximum density in an evolved system reaches 10–40 times the equilibrium value at the mid-plane, and the structures so formed attain oval shapes. These results lead to the conclusion that PI, operating in the presence of realistic cooling and heating processes, can trigger the formation of dense clouds, which may give rise to giant molecular complexes.     

High-order 2MASS galaxy correlation functions: probing the Gaussianity of the primordial density field

We study Parker instability (PI) operating in a non-adiabatic, gravitationally stratified, interstellar medium. We discuss models with two kinds of heating mechanisms. The first one results from photoionization models. The other, relying on supplemental sources, has been postulated by Reynolds, Haffner & Tufte. The cooling rate, corresponding to radiative losses, is an approximation to the one given by Dalgarno & McCray. An unperturbed state of the system represents a magnetohydrostatic and thermal equilibrium. We perform linear stability analysis by solving the boundary value problem. We find that the maximum growth rate of PI rises for increasing magnitudes of non-adiabatic effects. In the pure photoionization model, the maximum growth rate of the general non-adiabatic case coincides with the isothermal limit. Adding other sources of heat leads to a maximum growth rate that is larger than the one corresponding to the isothermal limit. We find that the influence of the supplemental heating on PI also leads to a decrease in temperature in magnetic valleys. Finally, we conclude that the initial gas cooling due to the action of PI may promote a subsequent onset of thermal instability in magnetic valleys and formation of giant molecular clouds.     

Magnetic fields and the dynamics of spiral galaxies

We investigate the dynamics of magnetic fields in spiral galaxies by performing 3D magnetohydrodynamics simulations of galactic discs subject to a spiral potential using cold gas, warm gas and a two-phase mixture of both. Recent hydrodynamic simulations have demonstrated the formation of interarm spurs as well as spiral arm molecular clouds, provided the interstellar medium model includes a cold H  i phase. We find that the main effect of adding a magnetic field to these calculations is to inhibit the formation of structure in the disc. However, provided a cold phase is included, spurs and spiral arm clumps are still present if β≳ 0.1 in the cold gas. A caveat to the two-phase calculations though is that by assuming a uniform initial distribution, β≳ 10 in the warm gas, emphasizing that models with more consistent initial conditions and thermodynamics are required. Our simulations with only warm gas do not show such structure, irrespective of the magnetic field strength.
Furthermore, we find that the introduction of a cold H  i phase naturally produces the observed degree of disorder in the magnetic field, which is again absent from simulations using only warm gas. Whilst the global magnetic field follows the large-scale gas flow, the magnetic field also contains a substantial random component that is produced by the velocity dispersion induced in the cold gas during the passage through a spiral shock. Without any cold gas, the magnetic field in the warm phase remains relatively well ordered apart from becoming compressed in the spiral shocks. Our results provide a natural explanation for the observed high proportions of disordered magnetic field in spiral galaxies and we thus predict that the relative strengths of the random and ordered components of the magnetic field observed in spiral galaxies will depend on the dynamics of spiral shocks.     

Generation of magnetosonic waves and formation of structures in galaxies

We study the generation of magnetosonic waves in galactic gaseous discs taking account of the magnetic field, differential rotation and self-gravity. The special case of perturbations is considered with the wavevector perpendicular to the magnetic field. The necessary condition of the amplification of seed perturbations is the presence of differential rotation and non-vanishing radial component of the magnetic field that can easily be satisfied in galactic discs. Differential rotation stretches the azimuthal field from the radial one and, therefore, we consider the generation of waves on the time-dependent background magnetic field. Basically, an amplification is rather efficient, and seed perturbations become non-linear already after several rotation periods for a wide range of wavelength. The generated magnetosonic waves can be either non-oscillatory or oscillatory depending on the parameters of gas. If perturbations are Jeans stable, then typically non-oscillatory waves are amplified. However, interplay between self-gravity, magnetic field and rotational shear can change qualitatively the classical Jeans instability, so that the latter becomes oscillatory and tends to be suppressed in galaxies.     

Revised equipartition and minimum energy formula for magnetic field strength estimates from radio synchrotron observations

The commonly used classical equipartition or minimum‐energy estimate of total magnetic fields strengths from radio synchrotron intensities is of limited practical use because it is based on the hardly known ratio K of the total energies of cosmic ray protons and electrons and also has inherent problems. We present a revised formula, using the number density ratio K for which we give estimates. For particle acceleration in strong shocks K is about 40 and increases with decreasing shock strength. Our revised estimate for the field strength gives larger values than the classical estimate for flat radio spectra with spectral indices of about 0.5–0.6, but smaller values for steep spectra and total fields stronger than about 10 µG. In very young supernova remnants, for example, the classical estimate may be too large by up to 10×. On the other hand, if energy losses of cosmic ray electrons are important, K increases with particle energy and the equipartition field may be underestimated significantly. Our revised larger equipartition estimates in galaxy clusters and radio lobes are consistent with independent estimates from Faraday rotation measures, while estimates from the ratio between radio synchrotron and X‐ray inverse Compton intensities generally give much weaker fields. This may be explained e.g. by a concentration of the field in filaments. Our revised field strengths may also lead to major revisions of electron lifetimes in jets and radio lobes estimated from the synchrotron break frequency in the radio spectrum. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Statistical properties of polarized radio continuum emission and effects of data processing

Polarized intensity and polarization angles are calculated from Stokes parameters Q and U in a nonlinear way. The statistical properties of polarized emission hold information about the structure of magnetic fields in a large range of scales, but the contributions of different stages of data processing to the statistical properties should first be understood. We use 1.4 GHz polarization data from the Effelsberg 100‐m telescope of emission in the Galactic plane, near the plane and far out of the plane. We analyze the probability distribution function and the wavelet spectrum of the original maps in Stokes parameters Q, U and corresponding PI. Then we apply absolute calibration (i.e. adding the large‐scale emission to the maps in Q and U), subtraction of polarized sources and subtraction of the positive bias in PI due to noise (“denoising”). We show how each procedure affects the statistical properties of the data. We find a complex behavior of the statistical properties for the different regions analyzed which depends largely on the intensity level of polarized emission. Absolute calibration changes the morphology of the polarized structures. The statistical properties change in a complex way: Compact sources in the field flatten the wavelet spectrum over a substantial range. Adding large‐scale emission does not change the spectral slopes in Q and U at small scales, but changes the PI spectrum in a complex way. “Denoising” significantly changes the p.d.f. of PI and raises the entire spectrum. The final spectra are flat in the Galactic plane due to magnetic structures in the ISM, but steeper at high Galactic latitude and in the anticenter. For a reliable study of the statistical properties of magnetic fields and turbulence in the ISM based on radio polarization observations, absolute calibration and source subtraction are required. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamo saturation in direct simulations of the multi‐phase turbulent interstellar medium

The ordered magnetic field observed via polarised synchrotron emission in nearby disc galaxies can be explained by a mean‐field dynamo operating in the diffuse interstellar medium (ISM). Additionally, vertical‐flux initial conditions are potentially able to influence this dynamo via the occurrence of the magnetorotational instability (MRI). We aim to study the influence of various initial field configurations on the saturated state of the mean‐field dynamo. This is motivated by the observation that different saturation behaviour was previously obtained for different supernova rates. We perform direct numerical simulations (DNS) of three‐dimensional local boxes of the vertically stratified, turbulent interstellar medium, employing shearing‐periodic boundary conditions horizontally. Unlike in our previous work, we also impose a vertical seed magnetic field. We run the simulations until the growth of the magnetic energy becomes negligible. We furthermore perform simulations of equivalent 1D dynamo models, with an algebraic quenching mechanism for the dynamo coefficients. We compare the saturation of the magnetic field in the DNS with the algebraic quenching of a mean‐field dynamo. The final magnetic field strength found in the direct simulation is in excellent agreement with a quenched α) dynamo. For supernova rates representative of the Milky Way, field losses via a Galactic wind are likely responsible for saturation. We conclude that the relative strength of the turbulent and regular magnetic fields in spiral galaxies may depend on the galaxy's star formation rate. We propose that a mean field approach with algebraic quenching may serve as a simple sub‐grid scale model for galaxy evolution simulations including a prescribed feedback from magnetic fields. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy, angular momentum and wave action associated with density waves in a rotating magnetized gas disc

Both fast and slow magnetohydrodynamic (MHD) density waves propagating in a thin rotating magnetized gas disc are investigated. In the tight-winding or WKBJ regime, the radial variation of MHD density-wave amplitude during wave propagation is governed by the conservation of wave action surface density which travels at a relevant radial group speed C _g. The wave energy surface density and the wave angular momentum surface density are related to by = and = m respectively, where is the angular frequency in an inertial frame of reference and the integer m , proportional to the azimuthal wavenumber, corresponds to the number of spiral arms. Consequently, both wave energy and angular momentum are conserved for spiral MHD density waves. For both fast and slow MHD density waves, net wave energy and angular momentum are carried outward or inward for trailing or leading spirals, respectively. The wave angular momentum flux contains separate contributions from gravity torque, advective transport and magnetic torque. While the gravity torque plays an important role, the latter two can be of comparable magnitudes to the former. Similar to the role of gravity torque, the part of MHD wave angular momentum flux by magnetic torque (in the case of either fast or slow MHD density waves) propagates outward or inward for trailing or leading spirals, respectively. From the perspective of global energetics in a magnetized gas sheet in rotation, trailing spiral structures of MHD density waves are preferred over leading ones. With proper qualifications, the generation and maintenance as well as transport properties of MHD density waves in magnetized spiral galaxies are discussed.     

Shattering and coagulation of dust grains in interstellar turbulence

We investigate shattering and coagulation of dust grains in turbulent interstellar medium (ISM). The typical velocity of dust grain as a function of grain size has been calculated for various ISM phases based on a theory of grain dynamics in compressible magnetohydrodynamic turbulence. In this paper, we develop a scheme of grain shattering and coagulation and apply it to turbulent ISM by using the grain velocities predicted by the above turbulence theory. Since large grains tend to acquire large velocity dispersions as shown by earlier studies, large grains tend to be shattered. Large shattering effects are indeed seen in warm ionized medium within a few Myr for grains with radius   a ≳ 10⁻⁶  cm. We also show that shattering in warm neutral medium can limit the largest grain size in ISM  ( a ∼ 2 × 10⁻⁵ cm)  . On the other hand, coagulation tends to modify small grains since it only occurs when the grain velocity is small enough. Coagulation significantly modifies the grain size distribution in dense clouds (DC), where a large fraction of the grains with   a < 10⁻⁶ cm  coagulate in 10 Myr. In fact, the correlation among   R_V   , the carbon bump strength and the ultraviolet slope in the observed Milky Way extinction curves can be explained by the coagulation in DC. It is possible that the grain size distribution in the Milky Way is determined by a combination of all the above effects of shattering and coagulation. Considering that shattering and coagulation in turbulence are effective if dust-to-gas ratio is typically more than ∼1/10 of the Galactic value, the regulation mechanism of grain size distribution should be different between metal-poor and metal-rich environments.     

The relation between magnetic and gas arms in spiral galaxies

The intriguing question of the origin of the arm-like magnetic structures seen between the optical spiral arms of certain spiral galaxies is addressed. Using a two-dimensional approximation to the non-linear disc dynamo equation, it is shown that gas streaming along the arms may produce such a field configuration. Another possibility is a spiral modulation of the turbulent diffusivity, associated with an enhancement of turbulence in the interstellar medium within the arms. The effects of a similar modulation of the alpha-effect are also examined. Finally, the consequences of a non-linear feedback of the large-scale field on the turbulent diffusivity ('η-quenching') are briefly discussed.