General Magnetic Field Research Articles

QED effective action in magnetic field backgrounds and electromagnetic duality

In the in-out formalism we advance a method of the inverse scattering matrix for calculating effective actions in pure magnetic field backgrounds. The one-loop effective actions are found in a localized magnetic field of Sauter type and approximately in a general magnetic field by applying the uniform semiclassical approximation. The effective actions exhibit the electromagnetic duality between a constant electric field and a constant magnetic field and between $E(x) = E sech^2 (x/L)$ and $B(x) = B sech^2 (x/L)$.

Evolution of polarization of an electromagnetic wave propagating in magnetized plasma: Comparison between two alternative formalisms

Two alternative formalisms for the analysis of the evolution of electromagnetic wave polarization in a magnetized plasma are considered: the coupled wave formalism and the Stokes vector formalism. The first formalism is developed and extended to more general magnetic field configurations than considered hitherto, thus obtaining a new polarization evolution equation. The two formalisms are compared and their relative merits and limitations are described. In particular, it is shown that the equations for polarization evolution are valid for arbitrarily strong anisotropy induced by the magnetic field and that these equations treat implicitly the effect of coupling between the characteristic waves.

Local and Nonlocal Parallel Heat Transport in General Magnetic Fields

A novel approach for the study of parallel transport in magnetized plasmas is presented. The method avoids numerical pollution issues of grid-based formulations and applies to integrable and chaotic magnetic fields with local or nonlocal parallel closures. In weakly chaotic fields, the method gives the fractal structure of the devil's staircase radial temperature profile. In fully chaotic fields, the temperature exhibits self-similar spatiotemporal evolution with a stretched-exponential scaling function for local closures and an algebraically decaying one for nonlocal closures. It is shown that, for both closures, the effective radial heat transport is incompatible with the quasilinear diffusion model.

Variational symplectic algorithm for guiding center dynamics in the inner magnetosphere

Charged particle dynamics in magnetosphere has temporal and spatial multiscale; therefore, numerical accuracy over a long integration time is required. A variational symplectic integrator (VSI) [H. Qin and X. Guan, Phys. Rev. Lett. 100, 035006 (2008) and H. Qin, X. Guan, and W. M. Tang, Phys. Plasmas 16, 042510 (2009)] for the guiding-center motion of charged particles in general magnetic field is applied to study the dynamics of charged particles in magnetosphere. Instead of discretizing the differential equations of the guiding-center motion, the action of the guiding-center motion is discretized and minimized to obtain the iteration rules for advancing the dynamics. The VSI conserves exactly a discrete Lagrangian symplectic structure and has better numerical properties over a long integration time, compared with standard integrators, such as the standard and adaptive fourth order Runge-Kutta (RK4) methods. Applying the VSI method to guiding-center dynamics in the inner magnetosphere, we can accurately calculate the particles’orbits for an arbitrary long simulating time with good conservation property. When a time-independent convection and corotation electric field is considered, the VSI method can give the accurate single particle orbit, while the RK4 method gives an incorrect orbit due to its intrinsic error accumulation over a long integrating time.

Galactic cosmic ray fluctuation parameter as an indicator of the degree of magnetic field inhomogeneity

The parameter of cosmic ray fluctuations, which indicates the degree of IMF inhomogeneity, was introduced in order to quantitatively describe the dynamics of the galactic cosmic ray (GCR) intensity fluctuations during the geoeffective phases of the 11-year cycle. The 5-min data of the high-latitude neutron monitor at Oulu station (Finland) during cycles 20–23 was used in the calculations. The nonrandom non-Gaussian character of the GCR fluctuation parameter is caused by the nonstationary semiannual variation reflecting the transient nonstationary oscillatory process of sign reversal of the general solar magnetic field. This transient oscillatory process is responsible for the maximal geoeffectiveness and duration of the phase of polarity reversal, which manifests itself in a sharp and deep GCR intensity minimum during the final stage of the field sign reversal. The invariant of the 11-year “amplitude-duration” cycle was confirmed on a new basis: the LF drift of the “low” cycle period was detected, which was observed in an increase in the duration of cycle 23 we anticipated.

Plasma rotation in a quasi-symmetric stellarator

The equilibrium plasma rotation in a general toroidal magnetic field is nearly always subsonic and is determined by the requirement that the collisional particle transport should be ambipolar in lowest order in the small-ion-gyroradius expansion. Only in quasi-symmetric fields, where collisional particle transport is intrinsically ambipolar, can the plasma rotate freely and then only in the quasi-symmetry direction. Sonic rotation velocities are allowed in this case. In this paper the effect of rotation in a quasi-axisymmetric field is investigated, and it is found that the symmetry is broken when the rotation speed exceeds the diamagnetic speed appreciably, leading to reappearance of the non-intrinsically ambipolar 1/ν-transport regime. Fortunately, this transport scales with the fourth power of the rotation Mach number and is expected to be modest in most plasmas of interest.

Role of the mean curvature in the geometry of magnetic confinement configurations

Examples are presented of how the geometric notion of the mean curvature is used for general magnetic field configurations and magnetic surfaces. It is shown that the mean magnetic curvature is related to the variation of the absolute value of the magnetic field along its lines. Magnetic surfaces of constant mean curvature are optimum for plasma confinement in multimirror open confinement systems and rippled tori.

A method for finding three-dimensional magnetic skeletons

Magnetic fields are an essential component of a plasma. In many astrophysical, solar, magnetospheric, and laboratory situations the magnetic field in the plasma can be very dynamic and form highly complex structures. One approach to unraveling these structures is to determine the magnetic skeleton of the field, a set of topological features that divide the magnetic field into topologically distinct domains. In general, the features of the magnetic skeleton are difficult to locate, in particular those given by numerical experiments. In this paper, we propose a new set of tools to find the skeleton of general magnetic fields including null points, spines, separatrix surfaces, and separators. This set of tools is found to be considerably better at finding the skeleton than the currently favored methods used in magnetohydrodynamics.

Spin in a general time-varying magnetic field: generalization of the adiabatic factorization of time evolution

An extension of the adiabatic factorization of the time evolution operator is studied for spin in a general time-varying magnetic field B(t). When B(t) changes adiabatically, such a factorization reduces to the product of the geometric operator which embodies the Berry phase phenomenon and a usual dynamical operator. For a general time variation of B(t), there should be a non-adiabatic operator in the factorization that is responsible for non-adiabatic transitions. A simple and explicit expression for the instantaneous angular velocity of the non-adiabatic operator is derived. This is done in a way that is independent of any specific representation of spin. Two classes of simple conditions are given under which the non-adiabatic operator itself can be made explicit. These conditions are used to find new expressions for B(t) that allow the explicit construction of the time evolution operator.

Minimum of 2008 or the “quantum” Sun-2

The prolonged 2007–2009 minimum is a big surprise for solar physics. In order to reveal the causes, we analyze the variability of the general magnetic field (GMF) of the Sun as a star measured by CrAO and five other observatories since 1968 (more than 19000 daily field strengths B were obtained in 41 years). Sharp yearly mean extrema of the negative (S) field took place in 1969, 1990, and 2008, with the third extremum, in contrast to the two previous ones, having coincided with the sunspot minimum. This explains both the long duration of the minimum and the record (over the last 100 years) increase in the length of the Wolf cycle (no. 23) to 12 or more years. The S-field extrema followed with a period of 19.5 ± 1.1 yr—some mean between the 22.1 ± 0.3-yr sunspot cycle, the 18.6-yr saros, and the 19.9-yr Jupiter-Saturn conjunction period. It is pointed out that, for some unclear reason, the negative polarity dominated on the Sun in 1968–2008: the overall mean B = −0.021 ± 0.015 G. The existence of a second Sun that obeys the laws of quantum mechanics is hypothesized. The “quantum” model of the Sun-2 explains many properties of the “classical” Sun-1, including the coronal heating, cyclic activity, periodic variations in GMF, and its sector structure.

Generic magnetic field model in poloidal divertor tokamaks in the presence of resonant magnetic perturbations

A generic analytical model for the description of magnetic field lines in poloidal divertor tokamaks in the presence of external resonant magnetic perturbations is proposed. It is based on the Hamiltonian description of magnetic field lines in tokamaks. The safety factor and the spectra of magnetic perturbations are chosen by the requirement to satisfy their generic behaviour near the magnetic separatrix and at the magnetic axis. The field line equations are integrated by the construction of two symplectic and computationally efficient mappings of field lines. The model for internal MHD modes is also proposed. The mapping procedure for field lines which includes the MHD modes is described. It is shown that the numerically calculated diffusion and convection coefficients of field lines are in close agreement with the quasilinear ones. It is found that in the presence of internal MHD modes at the plasma edge the convectional outward transport of field lines may reverse its direction to inward convectional transport.

Interstellar magnetic fields in the Galactic center region

We seek to obtain a picture of the interstellar magnetic field in the Galactic center region that is as clear and complete as possible. To that end, we review the observational knowledge that has built up over the past 25 years on interstellar magnetic fields within ~ 200 pc of the Galactic center. We then critically discuss the various theoretical interpretations and scenarios proposed to explain the existing observations. We also study the possible connections with the general Galactic magnetic field and describe the observational situation in external galaxies. We propose a coherent picture of the magnetic field near the Galactic center, which reconciles some of the seemingly divergent views and which best accounts for the vast body of observations. Our main conclusions are the following. In the diffuse intercloud medium, the large-scale magnetic field is approximately poloidal and its value is generally close to equipartition with cosmic rays (~ 10 microG), except in localized filaments where the field strength can reach ~ 1 mG. In dense interstellar clouds, the magnetic field is approximately horizontal and its value is typically ~ 1 mG.

The magnetosphere of oscillating neutron stars in general relativity

Just as a rotating magnetized neutron star has material pulled away from its surface to populate a magnetosphere, a similar process can occur as a result of neutron-star pulsations rather than rotation. This is of interest in connection with the overall study of neutron star oscillation modes but with a particular focus on the situation for magnetars. Following a previous Newtonian analysis of the production of a force-free magnetosphere in this way Timokhin et al., we present here a corresponding general-relativistic analysis. We give a derivation of the general relativistic Maxwell equations for small-amplitude arbitrary oscillations of a non-rotating neutron star with a generic magnetic field and show that these can be solved analytically under the assumption of low current density in the magnetosphere. We apply our formalism to toroidal oscillations of a neutron star with a dipole magnetic field and find that the low current density approximation is valid for at least half of the oscillation modes, similarly to the Newtonian case. Using an improved formula for the determination of the last closed field line, we calculate the energy losses resulting from toroidal stellar oscillations for all of the modes for which the size of the polar cap is small. We find that general relativistic effects lead to shrinking of the size of the polar cap and an increase in the energy density of the outflowing plasma. These effects act in opposite directions but the net result is that the energy loss from the neutron star is significantly smaller than suggested by the Newtonian treatment.

The Hanle effect in a random magnetic field

Context. The Hanle effect is used to determine weak turbulent magnetic fields in the solar atmosphere, usually assuming that the angular distribution is isotropic, the magnetic field strength constant, and that micro-turbulence holds, i.e. that the magnetic field correlation length is much less than a photon mean free path. Aims. To examine the sensitivity of turbulent magnetic field measurements to these assumptions, we study the dependence of Hanle effect on the magnetic field correlation length, its angular, and strength distributions. Methods. We introduce a fairly general random magnetic field model characterized by a correlation length and a magnetic field vector distribution. Micro-turbulence is recovered when the correlation length goes to zero and macro-turbulence when it goes to infinity. Radiative transfer equations are established for the calculation of the mean Stokes parameters and they are solved numerically by a polarized approximate lambda iteration method. Results. We show that optically thin spectral lines and optically very thick ones are insensitive to the correlation length of the magnetic field, while spectral lines with intermediate optical depths (around 10–100) show some sensitivity to this parameter. The result is interpreted in terms of the mean number of scattering events needed to create the surface polarization. It is shown that the single-scattering approximation holds good for thin and thick lines but may fail for lines with intermediate thickness. The dependence of the polarization on the magnetic field vector probability density function (PDF) is examined in the micro-turbulent limit. A few PDFs with different angular and strength distributions, but equal mean value of the magnetic field, are considered. It is found that the polarization is in general quite sensitive to the shape of the magnetic field strength PDF and somewhat to the angular distribution. Conclusions. The mean field derived from Hanle effect analysis of polarimetric data strongly depends on the choice of the field strength distribution used in the analysis. It is shown that micro-turbulence is in general a safe approximation.

Variational symplectic algorithm for guiding center dynamics and its application in tokamak geometry

A variational symplectic integrator for the guiding center motion of charged particles in general magnetic fields is developed to enable accurate long-time simulation studies of magnetized plasmas. Instead of discretizing the differential equations of the guiding center motion, the action of the guiding center motion is discretized and minimized to obtain the iteration rules for advancing the dynamics. The variational symplectic integrator conserves exactly a discrete Lagrangian symplectic structure and globally bounds the numerical error in energy by a small number for all simulation time steps. Compared with standard integrators, such as the fourth order Runge–Kutta method, the variational symplectic integrator has superior numerical properties over long integration time. For example, in a two-dimensional tokamak geometry, the variational symplectic integrator is able to guarantee the accuracy for both the trapped and transit particle orbits for arbitrarily long simulation time. This is important for modern large-scale simulation studies of fusion plasmas where it is critical to use algorithms with long-term accuracy and fidelity. The variational symplectic integrator is expected to have a wide range of applications.

Effect of active longitudes in cosmic ray flux modulation

Galactic cosmic rays, registered by ground-based neutron monitors, are strongly affected by the heliosphere, i.e., being subjected to solar modulation. Cosmic ray variations are closely related to different solar activity indices and IMF parameters. The longitudinal inhomogeneity of the general solar magnetic field as a star and the manifestation of this inhomogeneity in the magnetic field are considered in the work. It has been established that the longitudinal inhomogeneity of this field, with the dipole distribution of polarities along heliolongitude, mainly contributes to 27-day modulation of galactic cosmic rays.

Analysis on magnetic field dependence of magneto-optical Faraday effect

In this paper, the complicated relations between Faraday rotation and external magnetic field in various circumstances are theoretically deduced using magneto-optical four-level model. A linear field dependence of the Faraday rotation in extremely strong magnetic field or at high temperatures is revealed. Our theory also shows that the Faraday rotation presents an obvious nonlinear field-dependence property in a general magnetic field and at low temperatures. Furthermore, this nonlinear relationship can be simply described as θF=θFs+aHe+bHe2. In addition, a fit to the experimental results in Dy3−xBixFe5O12 (Bi-DyIG) under different magnetic fields is obtained. Meanwhile, it is pointed out that the parameters V1 and V2 have the anisotropy and temperature-dependence characteristic, which will also result in the anisotropic and temperature properties of magneto-optical Faraday effect in Bi-DyIG.

Symplectic and energy-conserving algorithms for solving magnetic field trajectories

The exponential splitting of the classical evolution operator yields symplectic integrators if the canonical Hamiltonian is separable. Similar splitting of the noncanonical evolution operator for a charged particle in a magnetic field produces exact energy-conserving algorithms. The latter algorithms evaluate the magnetic field directly with no need of a vector potential and are more stable with far less phase errors than symplectic integrators. For a combined electric and magnetic field, these algorithms from splitting the noncanonical evolution operator are neither fully symplectic nor exactly energy conserving, yet they behave exactly like symplectic algorithms in having qualitatively correct trajectories and bounded periodic energy errors. This work shows that exponential-splitting algorithms of any order for solving particle trajectories in a general electric and magnetic field can be systematically derived by use of the angular momentum operator of quantum mechanics. The use of operator analysis in this work fully comprehends the intertwining interaction between electric and magnetic forces and makes possible the derivation of highly nontrivial integrators.

Self-similar structure of magnetized advection-dominated accretion flows and convection-dominated accretion flows

(Abridged) We study the effects of a global magnetic field on viscously-rotating and vertically-integrated accretion disks around compact objects using a self-similar treatment. We extend Akizuki & Fukue's work (2006) by discussing a general magnetic field with three components ($r, \phi, z$) in advection-dominated accretion flows (ADAFs). We also investigate the effects of a global magnetic field on flows with convection. For these purposes, we first adopt a simple form of the kinematic viscosity $\nu=\alpha c_{s}^{2}/\Omega_{K}$ to study magnetized ADAFs. Then we consider a more realistic model of the kinematic viscosity $\nu=\alpha c_{s}H$, which makes the infall velocity increase but the sound speed and toroidal velocity decrease. We next use two methods to study magnetized flows with convection, i.e., we take the convective coefficient $\alpha_{c}$ as a free parameter to discuss the effects of convection for simplicity. We establish the $\alpha_{c}-\alpha$ relation for magnetized flows using the mixing-length theory and compare this relation with the non-magnetized case. If $\alpha_{c}$ is set as a free parameter, then $|v_{r}|$ and $c_{s}$ increase for a large toroidal magnetic field, while $|v_{r}|$ decreases but $|v_{\phi}|$ increases (or decreases) for a strong and dominated radial (or vertical) magnetic field with increasing $\alpha_{c}$. In addition, the magnetic field makes the $\alpha_{c}-\alpha$ relation be distinct from that of non-magnetized flows, and allows the $\rho\propto r^{-1}$ or $\rho\propto r^{-2}$ structure for magnetized non-accreting convection-dominated accretion flows with $\alpha+g\alpha_{c}< 0$ (where $g$ is the parameter to determine the condition of convective angular momentum transport).

Activity cycle of the giant star of Z Andromedae and its spin period

We have re-analysed the long-term optical light curve (LC) of the symbiotic star Z Andromedae, covering 112 yr of mostly visual observations. Two strictly periodic cycles and one quasiperiodic cycle can be identified in this LC. A P1 = 7550 d quasi-periodicity characterizes the repetition time of the outburst episodes of this symbiotic star. Six such events have been recorded so far. During quiescence states of the system, that is, in time-intervals between outbursts, the LC is clearly modulated by a stable coherent period of P2 = 759.1 d. This is the well-known orbital period of the Z Andromedae binary system that has been measured also spectroscopically. A third coherent period of P3 = 658.4 d is modulating the intense fluctuations in the optical brightness of the system during outbursts. We attribute the trigger of the outburst phenomenon and the clock that drives it, to a solar-type magnetic dynamo cycle that operates in the convection and the outer layers of the giant star of the system. We suggest that the intense surface activity of the giant star during maximum phases of its magnetic cycle is especially enhanced in one or two antipode regions, fixed in the atmosphere of the star and rotating with it. Such spots could be active regions around the North Pole and the South Pole of a general magnetic dipole field of the star. The P3 periodicity is half the beat of the binary orbital period of the system and the spin period of the giant. The latter is then either 482 or 1790 d. If only one pole is active on the surface of the giant, P3 is the beat period itself, and the spin period is 352 d. It could also be 5000 d if the giant is rotating in a retrograde direction. We briefly compare these findings in the LC of Z Andromedae to similar modulations that were identified in the LC of two other prototype symbiotics, BF Cyg and YY Her.