The generation and dissipation of solar and galactic magnetic fields

Turbulent diffusion of magnetic field plays an essential role in the generation of magnetic field in most astrophysical bodies. This paper reviews what can be proved, and what can be believed, about the turbulent diffusion of magnetic field. Observations indicate the dissipation of magnetic field at rates that can be understood only in terms of turbulent diffusion. Theory shows that a largescale weak magnetic field diffuses in a turbulent flow in the same way that smoke is mixed throughout the fluid by the turbulence. The small-scale fields (produced from the large-scale field by the turbulence) are limited in their growth by reconnection of field lines at neutral points, so that the turbulent mixing of field and fluid is not halted by them.Altogether, it appears that the mixing of field and fluid in the observed turbulent motions in the Sun and in the Galaxy is unavoidable. Turbulent diffusion causes decay of the general solar fields in a decade or so, and of the galactic field in 10⁸–10⁹ yr. We conclude that continual dynamo action is implied by the observed existence of the fields.This work was supported in part by the National Aeronautics and Space Administration under Grant NGL 14-001-001.     

A 2D numerical study of explosive events in the solar atmosphere

Two-dimensional (2D) compressible magnetohydrodynamic simulations are performed to explore the idea that the asymmetric reconnection between newly emerging intranetwork magnetic field flux and pre-existing network flux causes the explosive events in the solar atmosphere. The dependence of the reconnection rate as a function of time on the density and temperature of the emerging flux are investigated. For a Lundquist number of L _u= 5000 we find that the tearing mode instability can lead to the formation and growth of small magnetic islands. Depending on the temperature and density ratio of the emerging plasma, the magnetic island can be lifted upward and convected out of the top boundary, or is suppressed downward and convected out of the top boundary, or is suppressed downward nad submerged below the bottom boundary. The motions of the magnetic islands with different direction are accompanied respectively with upward or downward high velocity flow which might be associated with the red- and blue-shifted components detected in the explosive events.     

Collisionless Magnetic Reconnection in the Presence of Initial Guide Field

By using the method of 2-dimensional, 3-component full particle simulation, collisionless magnetic reconnection in the presence of various initial guide fields and the Harris current sheet with 1-dimensional initial state are studied. The results show that strong guide fields with B_z0 > 0.5B₀ can evidently alter not only the trajectory of the particles, but also the structure of the electric and velocity fields in the vicinity of the reconnection region, thereby affecting the rate of reconnection and the acceleration of electrons. The generalized Ohm's law is employed to interpret the structural characteristics of the electric fields with various guide fields. Also, via the tracing of the electron beam near he diffusion region, it is revealed that in the 2-D model, for both strong and weak guide fields, the induced electric field perpendicular to the simulation plane at the center of the diffusion region plays the major role in the acceleration of electrons. The contribution of the planar electric field outside the diffusion region is very small.     

Spontaneous Formation of Magnetic Flux Concentrations in Stratified Turbulence

The negative effective magnetic pressure instability discovered recently in direct numerical simulations (DNSs) may play a crucial role in the formation of sunspots and active regions in the Sun and stars. This instability is caused by a negative contribution of turbulence to the effective mean Lorentz force (the sum of turbulent and non-turbulent contributions) and results in the formation of large-scale inhomogeneous magnetic structures from an initially uniform magnetic field. Earlier investigations of this instability in DNSs of stably stratified, externally forced, isothermal hydromagnetic turbulence in the regime of large plasma ?? are now extended into the regime of larger scale separation ratios where the number of turbulent eddies in the computational domain is about 30. Strong spontaneous formation of large-scale magnetic structures is seen even without performing any spatial averaging. These structures encompass many turbulent eddies. The characteristic time of the instability is comparable to the turbulent diffusion time, L ²/?? _t, where ?? _t is the turbulent diffusivity and L is the scale of the domain. DNSs are used to confirm that the effective magnetic pressure does indeed become negative for magnetic field strengths below the equipartition field. The dependence of the effective magnetic pressure on the field strength is characterized by fit parameters that seem to show convergence for larger values of the magnetic Reynolds number.     

Modulational instability of fast magnetosonic waves in a solar plasma

Transverse amplitude modulations of fast magnetosonic waves propagating perpendicular to the background magnetic field are shown to be unstable on a time scale τ ～- λ_∥/V _aφ, if the wave amplitude φ exceeds a critical value, φ _c = C _s/V _a. The slow modes generated by the modulational instability under gravity can propagate along the magnetic field with the characteristic velocity, V _ph = g/2k _∥ V _aφ. The applications of this modulational instability and slow-mode generation mechanism to a solar plasma are discussed.     

Nonlinear excitation of small-scale Alfvén waves by fast waves and plasma heating in the solar atmosphere

We study a nonlinear mechanism for the excitation of kinetic Alfvén waves (KAWs) by fast magneto-acoustic waves (FWs) in the solar atmosphere. Our focus is on the excitation of KAWs that have very small wavelengths in the direction perpendicular to the background magnetic field. Because of their small perpendicular length scales, these waves are very efficient in the energy exchange with plasmas and other waves. We show that the nonlinear coupling of the energy of the finite-amplitude FWs to the small-scale KAWs can be much faster than other dissipation mechanisms for fast wave, such as electron viscous damping, Landau damping, and modulational instability. The nonlinear damping of the FWs due to decay FW = KAW + KAW places a limit on the amplitude of the magnetic field in the fast waves in the solar corona and solar-wind at the level B/B ₀∼10⁻². In turn, the nonlinearly excited small-scale KAWs undergo strong dissipation due to resistive or Landau damping and can provide coronal and solar-wind heating. The transient coronal heating observed by Yohkoh and SOHO may be produced by the kinetic Alfvén waves that are excited by parametric decay of fast waves propagating from the reconnection sites.     

Magnetic reconnection in plasmas

A review of the present status of the theory of magnetic reconnection is given. In strongly collisional plasmas reconnection proceeds via resistive current sheets, i.e. quasi-stationary macroscopic Sweet-Parker sheets at intermediate values of the magnetic Reynolds numberR _m , or mirco-current sheets in MHD turbulence, which develops at highR _m . In hot, dilute plasmas the reconnection dynamics is dominated by nondissipative effects, mainly the Hall term and electron inertia. Reconnection rates are found to depend only on the ion mass, being independent of the electron inertia and the residual dissipation coefficients. Small-scale whistler turbulence is readily excited giving rise to an anomalous electron viscosity. Hence reconnection may be much more rapid than predicted by conventional resistive theory.     

Analytical theory of tachocline transport at mid latitudes

We provide a theory of magnetic diffusion, momentum transport, and mixing in the solar tachocline by considering magnetohydrodynamics (MHD) turbulence on a β plane subject to a large scale shear (provided by the latitudinal differential rotation). In the strong magnetic field regime, we find that the turbulent viscosity and diffusivity are reduced by magnetic fields only, similarly to the two-dimensional MHD case (without Rossby waves). In the weak magnetic field regime, we find a crossover scale (L_R) from a Alfvén dominated regime (on small scales) to a Rossby dominated regime (on large scales). For parameter values typical of the tachocline, L_R is larger than the solar radius so that Rossby waves are unlikely to play an important role in the transport of magnetic field and angular momentum. This is mainly due to the enhancement of magnetic back-reaction by shearing which efficiently generates small scales, thus strong currents. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cosmic ray anisotropies observed late in the decay phase of solar flare events

Concurrent interplanetary magnetic field and 0.7–7.6 MeV proton cosmic-ray anisotropy data obtained from instrumentation on Explorers 34 and 41 are examined for five cosmic-ray events in which we observe a persistent eastern-anisotropy phase late in the event (t ? 4 days). The direction of the anisotropy at such times shows remarkable invariance with respect to the direction of the magnetic field (which generally varies throughout the event) and it is also independent of particle species (electrons and protons) and particle speed over the range 0.06 ? β ? 0.56. The anisotropy is from the direction 38.3° ± 2.4° E of the solar radius vector, and is inferred to be orthogonal to the long term, mean interplanetary field direction. Both the amplitude of the anisotropy and the decay time constant show a strong dependence on the magnetic field azimuth. Detailed comparison of the anisotropy and the magnetic field data shows that the simple model of convection plus diffusion parallel to the magnetic field is applicable for this phase of the flare effect. It is demonstrated that contemporary theories do not predict the invariance of the direction as observed, even when the magnetic field is steady; these theories need extension to take into account the magnetic field direction ψ varying from its mean direction ψ _o. It is shown that the late phase anisotropy vector is not expected to be everywhere perpendicular to the mean magnetic field. The suggestion that we are observing kinks in the magnetic field moving radially outwards from the Sun leads to the conclusion that the parallel diffusion coefficient varies as 1/cos² (ψ ? ψ _o). Density gradients in the late decay phase are estimated to be ≈ 700%∣AU for 0.7–7.6 MeV protons. A simple theory reproduces the dependence of the decay time constant on anisotropy; it also leads to a radial density gradient of about 1000%∣AU and diffusion coefficient of 1.3 × 10²⁰ cm² s^?1.     

The double inverse polarity paradigm

A model is presented for the origin of inverse polarity magnetic fields in the perpendicular as well as in the axial direction of quiescent prominences. The model is based on the presence of a discrete coronal arcade structure where magnetic separating surfaces can be identified. On the crossing of these separating surfaces magnetic reconnection driven by photospheric shear and converging motions can create the observed field direction in quiescent prominences.Dedicated to Cornelis de Jager     

Galactic magnetic fields and their interactions with the gas and cosmic rays

(i) The controversy of dynamo or primordial origin of galactic magnetic fields is summarized and extended to show that the dynamo theory appears to fail. However, much more important than their origin are the characteristics of the fields and their interactions with the gas and cosmic rays. (ii) A passive magnetic field frozen into a turbulent plasma is not dissipated like a cloud of smoke (turbulent or eddy diffusion) as believed previously. On the contrary it is amplified exponentially until, within a few eddy periods, either the growing magnetic stresses halt the turbulence or the field becomes chaotic. Even if the Petschek reconnection mechanism operates, the field is always disordered to a scale <0.1L, whereL is the eddy dimension. The investigation may at last provide a semi-quantitative deductive theory of hydromagnetic eddies. (iii) It is concluded that the gas motions observed in our Galaxy are not convective but are mainly hydromagnetic waves or oscillations, with the magnetic field in control. The significance of this result is discussed in connection with the overall gas velocity field, the creation of stars and stellar systems, and with the origin and distribution of cosmic rays.     

Interchange Slip-Running Reconnection and Sweeping SEP Beams

We present a new model to explain how particles (solar energetic particles; SEPs), accelerated at a reconnection site that is not magnetically connected to the Earth, could eventually propagate along the well-connected open flux tube. Our model is based on the results of a low-β resistive magnetohydrodynamics simulation of a three-dimensional line-tied and initially current-free bipole, which is embedded in a non-uniform open potential field. The topology of this configuration is that of an asymmetric coronal null point, with a closed fan surface and an open outer spine. When driven by slow photospheric shearing motions, field lines, initially fully anchored below the fan dome, reconnect at the null point, and jump to the open magnetic domain. This is the standard interchange mode as sketched and calculated in 2D. The key result in 3D is that reconnected open field lines located in the vicinity of the outer spine keep reconnecting continuously, across an open quasi-separatrix layer, as previously identified for non-open-null-point reconnection. The apparent slipping motion of these field lines leads to formation of an extended narrow magnetic flux tube at high altitude. Because of the slip-running reconnection, we conjecture that if energetic particles would be traveling through, or be accelerated inside, the diffusion region, they would be successively injected along continuously reconnecting field lines that are connected farther and farther from the spine. At the scale of the full Sun, owing to the super-radial expansion of field lines below 3?R _⊙, such energetic particles could easily be injected in field lines slipping over significant distances, and could eventually reach the distant flux tube that is well-connected to the Earth.     

Time-dependent configuration changes of the magnetotail and plasma sheet thinning

The configuration of the magnetotail magnetic field has been calculated for a situation where a disruption of a portion of the tail current system develops. The decrease of the current in a localized region of the magnetotail leads to a collapse of the magnetic field in that vicinity. The calculated configuration of the field resembles what is predicted by reconnection models with the field lines moving toward the neutral sheet and then connecting and either moving toward or away from the earth. Associated with this changing magnetic field there is an induced electric field which will then influence the motion of the plasma in the magnetotail via E × B drifts.When the current from X_sm = ?20 to ?40 R_E in the tail is decreasing with a tune-constant of 0.5 h the electric field produced, which is primarily westward, has a maximum value of 0.83 mV m^?1 and produces plasma sheet thinning velocities of 0.3 km s^?1. Higher velocities result for more rapid rates of current decrease, and they agree well with experimental observations. The plasma flows in the sunward direction are, however, much smaller than what has been observed. This is due in part to the inability of the magnetic field model to adequately represent the magnetic field in the immediate vicinity of the neutral sheet. Use of an improved model would give better agreement with the observations.The calculations show that the induced electric field of a time-dependent magnetic field is able to explain certain observed features of the plasma sheet motions. Also, this agreement suggests that the assumption that there is no charge separation contribution to the electric field may be reasonable during situations of large scale and rapid current disruptions in the magnetotail.     

Predictions of Energy and Helicity in Four Major Eruptive Solar Flares

In order to better understand the solar genesis of interplanetary magnetic clouds (MCs), we model the magnetic and topological properties of four large eruptive solar flares and relate them to observations. We use the three-dimensional Minimum Current Corona model (Longcope, 1996, Solar Phys. 169, 91) and observations of pre-flare photospheric magnetic field and flare ribbons to derive values of reconnected magnetic flux, flare energy, flux rope helicity, and orientation of the flux-rope poloidal field. We compare model predictions of those quantities to flare and MC observations, and within the estimated uncertainties of the methods used find the following: The predicted model reconnection fluxes are equal to or lower than the reconnection fluxes inferred from the observed ribbon motions. Both observed and model reconnection fluxes match the MC poloidal fluxes. The predicted flux-rope helicities match the MC helicities. The predicted free energies lie between the observed energies and the estimated total flare luminosities. The direction of the leading edge of the MC’s poloidal field is aligned with the poloidal field of the flux rope in the AR rather than the global dipole field. These findings compel us to believe that magnetic clouds associated with these four solar flares are formed by low-corona magnetic reconnection during the eruption, rather than eruption of pre-existing structures in the corona or formation in the upper corona with participation of the global magnetic field. We also note that since all four flares occurred in active regions without significant pre-flare flux emergence and cancelation, the energy and helicity that we find are stored by shearing and rotating motions, which are sufficient to account for the observed radiative flare energy and MC helicity.     

Spontaneous magnetic reconnection

The present review concerns the relevance of collisionless reconnection in the astrophysical context. Emphasis is put on recent developments in theory obtained from collisionless numerical simulations in two and three dimensions. It is stressed that magnetic reconnection is a universal process of particular importance under collisionless conditions, when both collisional and anomalous dissipation are irrelevant. While collisional (resistive) reconnection is a slow, diffusive process, collisionless reconnection is spontaneous. On any astrophysical time scale, it is explosive. It sets on when electric current widths become comparable to the leptonic inertial length in the so-called lepton (electron/positron) “diffusion region”, where leptons de-magnetise. Here, the magnetic field contacts its oppositely directed partner and annihilates. Spontaneous reconnection breaks the original magnetic symmetry, violently releases the stored free energy of the electric current, and causes plasma heating and particle acceleration. Ultimately, the released energy is provided by mechanical motion of either the two colliding magnetised plasmas that generate the current sheet or the internal turbulence cascading down to lepton-scale current filaments. Spontaneous reconnection in such extended current sheets that separate two colliding plasmas results in the generation of many reconnection sites (tearing modes) distributed over the current surface, each consisting of lepton exhausts and jets which are separated by plasmoids. Volume-filling factors of reconnection sites are estimated to be as large as \({<}10^{-5}\) per current sheet. Lepton currents inside exhausts may be strong enough to excite Buneman and, for large thermal pressure anisotropy, also Weibel instabilities. They bifurcate and break off into many small-scale current filaments and magnetic flux ropes exhibiting turbulent magnetic power spectra of very flat power-law shape \(W_b\propto k^{-\alpha }\) in wavenumber k with power becoming as low as \(\alpha \approx 2\). Spontaneous reconnection generates small-scale turbulence. Imposed external turbulence tends to temporarily increase the reconnection rate. Reconnecting ultra-relativistic current sheets decay into large numbers of magnetic flux ropes composed of chains of plasmoids and lepton exhausts. They form highly structured current surfaces, “current carpets”. By including synchrotron radiation losses, one favours tearing-mode reconnection over the drift-kink deformation of the current sheet. Lepton acceleration occurs in the reconnection-electric field in multiple encounters with the exhausts and plasmoids. This is a Fermi-like process. It results in power-law tails on the lepton energy distribution. This effect becomes pronounced in ultra-relativistic reconnection where it yields extremely hard lepton power-law energy spectra approaching \(F(\gamma )\propto \gamma ^{-1}\), with \(\gamma \) the lepton energy. The synchrotron radiation limit becomes substantially exceeded. Relativistic reconnection is a probable generator of current and magnetic turbulence, and a mechanism that produces high-energy radiation. It is also identified as the ultimate dissipation mechanism of the mechanical energy in collisionless magnetohydrodynamic turbulent cascades via lepton-inertial-scale turbulent current filaments. In this case, the volume-filling factor is large. Magnetic turbulence causes strong plasma heating of the entire turbulent volume and violent acceleration via spontaneous lepton-scale reconnection. This may lead to high-energy particle populations filling the whole volume. In this case, it causes non-thermal radiation spectra that span the entire interval from radio waves to gamma rays.     

Evaluating Mean Magnetic Field in Flare Loops

We analyze multiple-wavelength observations of a two-ribbon flare exhibiting apparent expansion motion of the flare ribbons in the lower atmosphere and rising motion of X-ray emission at the top of newly-formed flare loops. We evaluate magnetic reconnection rate in terms of V _r B _r by measuring the ribbon-expansion velocity (V _r) and the chromospheric magnetic field (B _r) swept by the ribbons. We also measure the velocity (V _t) of the apparent rising motion of the loop-top X-ray source, and estimate the mean magnetic field (B _t) at the top of newly-formed flare loops using the relation 〈V _t B _t〉≈〈V _r B _r〉, namely, conservation of reconnection flux along flare loops. For this flare, B _t is found to be 120 and 60 G, respectively, during two emission peaks five minutes apart in the impulsive phase. An estimate of the magnetic field in flare loops is also achieved by analyzing the microwave and hard X-ray spectral observations, yielding B=250 and 120 G at the two emission peaks, respectively. The measured B from the microwave spectrum is an appropriately-weighted value of magnetic field from the loop top to the loop leg. Therefore, the two methods to evaluate coronal magnetic field in flaring loops produce fully-consistent results in this event.     

Study of Ribbon Separation and Magnetic Reconnection Rates in a Two-Ribbon Flare

In this paper, we present a study of the correlation between the speed of flare ribbon separation and the magnetic flux density during the 10 April 2001 solar flare. The study includes the section of the neutral line containing the flare core and its peripheral area. This event shows clear two-ribbon structure and inhomogeneous magnetic fields along the ribbons, so the spatial correlation and distribution of the flare and magnetic parameters can be studied. A weak negative correlation is found between the ribbon separation speed (V _r) and the longitudinal magnetic flux density (B _z ). This correlation is the weakest around the peak of the flare. Spatially, the correlation is also weakest at the positions of the hard X-ray (HXR) sources. In addition, we estimate the magnetic reconnection rate (electric field strength in the reconnection region E _rec) by combining the speed of flare ribbons and the longitudinal magnetic flux density. During flare evolution, the time profiles of the magnetic reconnection rate are similar to that of the ribbon separation speed, and the speeds of ribbon separation are relatively slow in the strong magnetic fields (i.e., V _r is negatively correlated with B _z ). However, along the flare ribbons, E _rec fluctuates in a small range except near the HXR source. A localized enhancement of the reconnection rate corresponds to the position of the HXR source.     

Evolution of Relative Magnetic Helicity: Method of Computation and Its Application to a Simulated Solar Corona above an Active Region

For a better understanding of solar magnetic field evolution it is appropriate to evaluate the magnetic helicity based on observations and to compare it with numerical simulation results. We have developed a method for calculating the vector potential of a magnetic field given in a finite volume; the method requires the magnetic flux to be balanced on all the side boundaries of the considered volume. Our method uses a fast Laplace/Poisson solver to obtain the vector potentials for a given magnetic field and for the corresponding potential (current-free) field. This allows an efficient calculation of the relative magnetic helicity in a finite 3D volume. We tested our approach on a theoretical model (Low and Lou, Astrophys. J. 352, 343, 1990) and also applied our method to the magnetic field above active region NOAA 8210 obtained by a photospheric-data-driven MHD model. We found that the amount of accumulated relative magnetic helicity coincides well with the relative helicity inflow through the boundaries in the ideal and non-ideal cases. The temporal evolution of relative magnetic helicity is consistent with that of magnetic energy. The maximum value of normalized helicity, H _m/Φ²=0.0298, is reached just before a drastic energy release by magnetic reconnection. This value is close to the corresponding value inferred from the formula that connects the magnetic flux and the accumulated magnetic helicity based on the observations of solar active regions.     

Magnetic reconnection rate and spectral index for two double-ribbon flares

We explore the correlations between the inferred reconnection rate and hard X-ray spectral hardness for two double-ribbon flares on 2003 November 1 and 2005 May 17 in this paper. The magnetic reconnection rate of φ _rec is derived from the time rate of change of the product between the normal magnetic field and the newly brightened ribbon area of TRACE 1600 Å observations. And the spectral index of γ is derived from RHESSI hard X-ray data. Both events show a soft-hard-soft spectral behaviour in the rise-peak-decay phases. An anti-correlated behaviour is found between the time evolution of reconnection rate and spectral index. From the regression analysis, we obtain a negative power-law dependence and quantify the relationship between these two quantities. This is consistent with the simulation results before, and further confirms the importance of magnetic reconnection for the electron acceleration in solar flares.     

Velocity-Space Proton Diffusion in the Solar Wind Turbulence

We study the velocity-space quasi-linear diffusion of the solar wind protons driven by oblique Alfvén turbulence at proton kinetic scales. Turbulent fluctuations at these scales possess the properties of kinetic Alfvén waves (KAWs) that are efficient in Cherenkov-resonant interactions. The proton diffusion proceeds via Cherenkov kicks and forms a quasi-linear plateau – the nonthermal proton tail in the velocity distribution function (VDF). The tails extend in velocity space along the mean magnetic field from 1 to (1.5?–?3) V _A, depending on the spectral break position, on the turbulence amplitude at the spectral break, and on the spectral slope after the break. The most favorable conditions for the tail generation occur in the regions where the proton thermal and Alfvén velocities are about equal, V _Tp/V _A≈1. The estimated formation times are within 1?–?2 h for typical tails at 1 AU, which is much shorter than the solar wind expansion time. Our results suggest that the nonthermal proton tails, observed in situ at all heliocentric distances >?0.3 AU, are formed locally in the solar wind by the KAW turbulence. We also suggest that the bump-on-tail features – proton beams, often seen in the proton VDFs, can be formed at a later evolutional stage of the nonthermal tails by the time-of-flight effects.