Generalized analytical models of Syrovatskii’s current sheet

Two-dimensional stationary magnetic reconnection models that include a thin Syrovatskii-type current sheet and four discontinuous magnetohydrodynamic flows of finite length attached to its endpoints are considered. The flow pattern is not specified but is determined from a self-consistent solution of the problem in the approximation of a strong magnetic field. Generalized analytical solutions that take into account the possibility of a current sheet discontinuity in the region of anomalous plasma resistivity have been found. The global structure of the magnetic field in the reconnection region and its local properties near the current sheet and attached discontinuities are studied. In the reconnection regime in which reverse currents are present in the current sheet, the attached discontinuities are trans-Alfvénic shock waves near the current sheet endpoints. Two types of transitions from nonevolutionary shocks to evolutionary ones along discontinuous flows are shown to be possible, depending on the geometrical model parameters. The relationship between the results obtained and numerical magnetic reconnection experiments is discussed.     

Which electromagnetic equations apply in rotating coordinates?

It was discovered some years ago by Schiff that the equations divE = 4πQ and curlB - (1/c) ∂E/∂t = (4π/c)J for fields in vacuum do not carry over without change from an inertial frame to a frame with rotating axes of space coordinates, even for a region with all velocities of orderv≪c. However, the belief that all four of the field equations are invariant under such conditions is still prevalent and causes misconceptions in physical applications, including astrophysical and geophysical ones. The purpose of the present paper is therefore to call attention to Schiff's discovery, discussing its basis and its extension to fields in material media, and to interpret the additional terms that must be added to the equations in order to obtain valid transformations to rotating axes of coordinates.     

Global magnetic activity in 22-year solar cycles

Properties of even and odd 11-year solar cycles as part of the 22-year magnetic cycle have been studied on the basis of the data on the zonal structure of the large-scale magnetic field, of polar faculae activity cycles, duration of 11-year cycles, high-latitude prominence areas, inclinations of the coronal streamers, velocity of magnetic neutral line migration, and peculiarities of the polar magnetic field reversal. It is shown that the properties of the odd cycle depend on those of the preceding even cycle. The 22-year magnetic cycle, consisting of an even and odd cycle, is a unified dynamic process. The new data obtained show that the poloidal magnetic fieldB(p) of ‘+’ and ‘−’ polarity for the new 22-year magnetic cycle is formed simultaneously, possibly in deep layers of the Sun in the form of a certain magnetic configuration, containing alternating ‘+’ and ‘−’ polarities of the field.     

Nonlinear Force-Free Modeling of Coronal Magnetic Fields Part I: A Quantitative Comparison of Methods

We compare six algorithms for the computation of nonlinear force-free (NLFF) magnetic fields (including optimization, magnetofrictional, Grad–Rubin based, and Green's function-based methods) by evaluating their performance in blind tests on analytical force-free-field models for which boundary conditions are specified either for the entire surface area of a cubic volume or for an extended lower boundary only. Figures of merit are used to compare the input vector field to the resulting model fields. Based on these merit functions, we argue that all algorithms yield NLFF fields that agree best with the input field in the lower central region of the volume, where the field and electrical currents are strongest and the effects of boundary conditions weakest. The NLFF vector fields in the outer domains of the volume depend sensitively on the details of the specified boundary conditions; best agreement is found if the field outside of the model volume is incorporated as part of the model boundary, either as potential field boundaries on the side and top surfaces, or as a potential field in a skirt around the main volume of interest. For input field (B) and modeled field (b), the best method included in our study yields an average relative vector error E_n = 〈 |B−b|〉/〈 |B|〉 of only 0.02 when all sides are specified and 0.14 for the case where only the lower boundary is specified, while the total energy in the magnetic field is approximated to within 2%. The models converge towards the central, strong input field at speeds that differ by a factor of one million per iteration step. The fastest-converging, best-performing model for these analytical test cases is the Wheatland, Sturrock, and Roumeliotis (2000) optimization algorithm as implemented by Wiegelmann (2004).     

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.     

Statistics of magnetic fields for OB stars

Based on an analysis of the catalog of magnetic fields, we have investigated the statistical properties of the mean magnetic fields for OB stars. We show that the mean effective magnetic field B of a star can be used as a statistically significant characteristic of its magnetic field. No correlation has been found between the mean magnetic field strength B and projected rotational velocity of OB stars, which is consistent with the hypothesis about a fossil origin of the magnetic field. We have constructed the magnetic field distribution function for B stars, F(B), that has a power-law dependence on B with an exponent of ≈−1.82. We have found a sharp decrease in the function F(B) for B ⩽ 400 G that may be related to rapid dissipation of weak stellar surface magnetic fields.     

The magnetic structures of electron phase-space holes formed in the electron two-stream instability

It is well known that the parallel cuts of the parallel and perpendicular electric field in electron phase-space holes (electron holes) have bipolar and unipolar structures, respectively. Recently, electron holes in the Earth’s plasma sheet have been observed by THEMIS satellites to have detectable fluctuating magnetic field with regular structures. Du et al. (2011) investigated the evolution of a one-dimensional (1D) electron hole with two-dimensional (2D) electromagnetic particle-in-cell (PIC) simulations in weakly magnetized plasma (Ω _e <ω _pe , where Ω _e and ω _pe are the electron gyrofrequency and electron plasma frequency, respectively), which initially exists in the simulation domain. The electron hole is unstable to the transverse instability and broken into several 2D electron holes. They successfully explained the observations by THEMIS satellites based on the generated magnetic structures associated with these 2D electron holes. In this paper, 2D electromagnetic particle-in-cell (PIC) simulations are performed in the x–y plane to investigate the nonlinear evolution of the electron two-stream instability in weakly magnetized plasma, where the background magnetic field (B₀ = B₀[(e)\vec] _x)(\mathbf{B}_{0} =B_{0}\vec{\mathbf{e}} _{x}) is along the x direction. Several 2D electron holes are formed during the nonlinear evolution, where the parallel cuts of E _x and E _y have bipolar and unipolar structures, respectively. Consistent with the results of Du et al. (2011), we found that the current along the z direction is generated by the electric field drift motion of the trapped electrons in the electron holes due to the existence of E _y , which produces the fluctuating magnetic field δB _x and δB _y in the electron holes. The parallel cuts of δB _x and δB _y in the electron holes have unipolar and bipolar structures, respectively.     

Cross Helicity and Turbulent Magnetic Diffusivity in the Solar Convection Zone

In a density-stratified turbulent medium, the cross helicity 〈u′⋅B′〉 is considered as a result of the interaction of the velocity fluctuations and a large-scale magnetic field. By means of a quasilinear theory and by numerical simulations, we find the cross helicity and the mean vertical magnetic field to be anti-correlated. In the high-conductivity limit the ratio of the helicity and the mean magnetic field equals the ratio of the magnetic eddy diffusivity and the (known) density scale height. The result can be used to predict that the cross helicity at the solar surface will exceed the value of 1 gauss km s⁻¹. Its sign is anti-correlated to that of the radial mean magnetic field. Alternatively, we can use our result to determine the value of the turbulent magnetic diffusivity from observations of the cross helicity.     

On neutral sheets in the solar wind

The structure and dynamics of neutral sheets in the solar wind is examined. The internal magnetic topology of the sheet is argued to be that of thin magnetic tongues greatly distended outward by the expansion inside the sheet. Due to finite conductivity effects, outward flow takes place across field lines but is retarded relative to the ambient solar wind by the reverse J×B force. The sheet thickness as well as the internal transverse magnetic field are found to be proportional to the electrical conductivity to the inverse one third power. Estimating a conductivity appropriate for a current carried largely by the ions perpendicular to the magnetic field, we find sheet dimensions of the order of 500km representative for the inner solar corona. For a radial field of strength 1/2G at 2R , the transverse field there is about 2 × 10^–3G and decreases outward rapidly.The energy release in the form of Joulean dissipation inside the sheet is estimated. It is concluded that ohmic heating in current sheets is not a significant source of energy for the overall solar wind expansion, mainly because these structures occupy only a small percentage of the total coronal volume. However, the local energy release through this mechanism is found to be large - in fact, over 7 times that expected to be supplied by thermal conduction. Therefore, ohmic heating is probably a dominant energy source for the dynamical conditions within the sheet itself.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Evaluation of a Selected Case of the Minimum Dissipative Rate Method for Non-Force-Free Solar Magnetic Field Extrapolations

The minimum dissipative rate (MDR) method for deriving a coronal non-force-free magnetic field solution is partially evaluated. These magnetic field solutions employ a combination of three linear (constant-α) force-free-field solutions with one being a potential field (i.e., α=0). The particular case of the solutions where the other two α’s are of equal magnitude but of opposite sign is examined. This is motivated by studying the SOLIS (Synoptic Optical Long-term Investigation of the Sun (SOLIS), a National Solar Observatory facility) vector magnetograms of AR 10987, which show a global α value consistent with an α=0 value as evaluated by (∇×B) _z /B _z over the region. Typical of the current state of the observing technology, there is no definitive twist for input into the general MDR method. This suggests that the special α case, of two α’s with equal magnitudes and opposite signs, is appropriate given the data. Only for an extensively twisted active region does a dominant, nonzero α normally emerge from a distribution of local values. For a special set of conditions, is it found that (i) the resulting magnetic field is a vertically inflated magnetic field resulting from the electric currents being parallel to the photosphere, similar to the results of Gary and Alexander (Solar Phys. 186:123, 1999), and (ii) for α≈(α _max/2), the Lorentz force per unit volume normalized by the square of the magnetic field is on the order of 1.4×10⁻¹⁰ cm⁻¹. The Lorentz force (F _L) is a factor of ten higher than that of the magnetic force d(B ²/8π)/dz, a component of F _L. The calculated photospheric electric current densities are an order of magnitude smaller than the maximum observed in all active regions. Hence both the Lorentz force density and the generated electric current density seem to be physically consistent with possible solar dynamics. The results imply that the field could be inflated with an overpressure along the neutral line. However, the implementation of this or any other extrapolation method using the electric current density as a lower boundary condition must be done cautiously, with the current magnetography.     

Nonplanar magnetic reconnection associated with conical slow shocks

We study a nonplanar model of magnetic reconnection associated with conical slow shocks, assuming that the shock surfaces are two identical cones with circular cross sections symmetrical about the ±x-axis. In the inflow region upstream of the shocks, two oppositely directed magnetic fields are separated by a current sheet. The model treats the current sheet as a tangential discontinuity and treats shocks and tangential discontinuity as surfaces of zero thickness. The dynamical structure of the global magnetic field in the continuous regions is studied using compressible, non-resistive MHD equations. In the inflow region, nonplanar magnetic field lines first move toward the current sheet. Near the sheet, the middle sections of the field lines become highly flattened, almost parallel to the sheet. Eventually, then oppositely directed field lines merge across the tangential discontinuity between the two shocks, and the magnetic lines are reconnected at the intersection of the shock and the tangential discontinuity. Reconnected magnetic lines are carried away at high speeds by the MHD flow in the outflow region, downstream of the shocks.     

Intensity and Magnetic Field Distribution of Sunspots

We study the relationship between the brightness (I) and magnetic field (B) distributions of sunspots using 272 samples observed at the San Fernando Observatory and the National Solar Observatory, Kitt Peak, whose characteristics varied widely. We find that the I – B relationship has a quadratic form for the spots with magnetic field less than about 2000 G. The slope of the linear part of the I – B curve varies by about a factor of three for different types of spots. In general the slope increases as the spot approaches disk center. The I – B slope does not have a clear dependency on the spot size but the lower limit appears to increase as a function of the ratio of umbra and penumbra area. The I – B slope changes as a function of age of the sunspots. We discuss various sunspot models using these results.     

Magnetic Turbulence and Ion Dynamics in the Magnetotail

The ion dynamics in the Earth's magnetotail is studied in the case when a cross tail electric field E ₀ and reconnection-driven magnetic turbulence are present in the current sheet. The magnetic turbulence observed by the Interball spacecraft is modeled numerically by a power law magnetic fluctuation spectrum. A test particle simulation is performed for the ions, and the distribution function moments are obtained as a function of the magnetic fluctuation level, δB/B ₀, and of the value of the normal component B _n. It appears that even in the presence of magnetic turbulence, the normal component has a marked influence on particle dynamics: the ion bulk velocity along E _y and ion temperature are almost inversely proportional to B _n. The magnetic turbulence causes the current to split in two layers, and the level of magnetic fluctuations needed to have splitting is roughly proportional to B _n. It appears that in the relevant range of parameters, B _n and δB/B ₀ have opposite effects on the current structure and on ion heating. This revised version was published online in July 2006 with corrections to the Cover Date.     

Longitude variations of solar magnetic fields of different intensity in cycle 23 as inferred from the SOHO/MDI data

SOHO/MDI magnetograms have been used to analyze the longitude distribution of the squared solar magnetic field 〈B ²〉 in the activity cycle no. 23. The energy of the magnetic field (〈B ²〉) is shown to change with longitude. However, these variations hardly fit the concept of active longitudes. In the epochs of high solar activity, one can readily see a relationship between longitude variations of the medium-strong ((|B| > 50 G or |B| > 100 G) and relatively weak (|B| ≤ 50 G or |B| ≤ 100 G) fields at all latitudes. In other periods, this relationship is revealed mainly at the latitudes not higher than 30°. The background fields (|B| ≤ 25 G) also display longitude variations, which are, however, not related to those of the strong fields. This makes us think that the fields of solar activity are rather inclusions to the general field than the source of the latter.     

The Plasma and Magnetic Field Characteristics of a Double Discontinuity in Interplanetary Space

A double discontinuity is a rarely observed compound structure composed of a slow shock layer and an adjoining rotational discontinuity layer in the downstream region. In this paper, we report the observations of a double discontinuity detected by Wind on May 15, 1997. This double discontinuity is found to be the front boundary of a magnetic cloud boundary layer. We strictly identify the shock layer and the rotational discontinuity layer by using the high-resolution plasma and magnetic field data from Wind. The observed jump conditions of the upstream and downstream region of the slow shock layer are in good agreement with the Rankine – Hugoniot relations. The flow speeds in the shock frame U _n <V _Acos θ _Bn on both sides of the slow shock layer. In the upstream region, the slow Mach number M _s1=U _n1/V _s1 is 1.95 (above unity), and in the downstream region, the slow Mach number M _s2=U _n2/V _s2 is 0.31 (below unity). Here V _A and V _s represent the Alfvén speed and the local slow magnetosonic speed, respectively, and θ _Bn is the angle between the direction of the magnetic field and the shock normal. The magnetic cloud boundary layer observed by Wind was also detected by Geotail 48 min later when the spacecraft was located outside the bow shock of the magnetosphere. However, Geotail observations showed that its front boundary was no longer a double discontinuity and the rotational discontinuity layer disappeared, indicating that this double discontinuity was unstable when propagating from Wind to Geotail.     

Influence of a chromospheric magnetic field on solar acoustic modes

This paper studies the effect of a magnetic atmosphere on the global solar acoustic oscillations in a simple Cartesian model. First, the influence of the ratio of the coronal and the photospheric temperature τ and the strength of the magnetic field at the base of the corona B_c on the oscillation modes is studied for a convection zone-corona model with a true discontinuity. The ratio τ seems to be an important parameter. Subsequently, the discontinuity is replaced by an intermediate chromospheric layer of thickness L and the effect of the thickness on the frequencies of the acoustic waves is studied. In addition, nonuniformity in the magnetic field, plasma density and temperature in the transition layer gives rise to continuous Alfvén and slow spectra. Modes with characteristic frequencies lying within the range of the continuum may resonantly couple to Alfvén and/or slow waves.     

The dynamics of driven magnetic reconnection in coronal arcades

The dynamics of interacting coronal loops and arcades have recently been highlighted by observations from theYohkoh satellite and may represent a viable mechanism for heating the solar corona. Here such an interaction is studied using two-dimensional resistive magnetohydrodynamic (MHD) simulations. Initial potential field structures evolve in response to imposed photospheric flows. In addition to the anticipated current sheet about theX-point separating the colliding flux systems, significant current layers are found to lie all the way along the separatrices that intersect at theX-point and divide the coronal magnetic field into topologically distinct regions. Shear flows across the separatrices are also observed. Both of these features are shown to be compatible with recent analytical studies of two-dimensional linear steady-state magnetic reconnection, even though the driven system that has been simulated is not strictly ‘open’ in the sense implied by steady-state calculations. The implications for future steady-state models are also discussed. The presence of the neutral point also brings into question any constant-density approximations that have previously been used for such quasi-steady coronal evolution models. This results from the intimate coupling between the neutral point and its separatrices communicated via the gas pressure. In terms of the detailed energetics during the arcade evolution, preliminary results reveal that on the order of 3% of the energy injected by the footpoint motions is lost purely through ohmic dissipation. We would therefore anticipate a local hot spot between the interacting flux systems, and a brightening distributed along the length of any separatrix field lines. Furthermore, as the resistivityη is reduced, the flux annihilation rate and the ohmic dissipation rate are found to scale independently ofη.     

Relationship between Powerful Flares and Dynamic Evolution of the Magnetic Field at the Solar Surface

Powerful flares are closely related to the evolution of the complex magnetic field configuration at the solar surface. The strength of the magnetic field and speed of its evolution are two vital parameters in the study of the change of magnetic field in the solar atmosphere. We propose a dynamic and quantitative depiction of the changes in complexity of the active region: E=u×B, where u is the velocity of the footpoint motion of the magnetic field lines and B is the magnetic field. E represents the dynamic evolution of the velocity field and the magnetic field, shows the sweeping motions of magnetic footpoints, exhibits the buildup process of current, and relates to the changes in nonpotentiality of the active region in the photosphere. It is actually the induced electric field in the photosphere. It can be deduced observationally from velocities computed by the local correlation tracking (LCT) technique and vector magnetic fields derived from vector magnetograms. The relationship between E and ten X-class flares of four active regions (NOAA 10720, 10486, 9077, and 8100) has been studied. It is found that (1) the initial brightenings of flare kernels are roughly located near the inversion lines where the intensities of E are very high, (2) the daily averages of the mean densities of E and its normal component (E _n) decrease after flares for most cases we studied, whereas those of the tangential component of E (E _t) show no obvious regularities before and after flares, and (3) the daily averages of the mean densities of E _t are always higher than those of E _n, which cannot be naturally deduced by the daily averages of the mean densities of B _n and B _t.     

On 17 – 22 January 2005 Events in Space Weather

This contribution is a follow-up to the recent paper of Kuznetsov et al. (Contrib. Astron. Obs. Skalnaté Pleso 36, 85, 2006) on the ground level enhancement (GLE) on 20 January 2005. We focused on a study of Forbush decrease (FD) of 17 – 18 and 21 – 22 January 2005, respectively. The data from the neutron monitor at Lomnicky Štít (1 min counts) and from the Geomagnetic Observatory in Hurbanovo, both in Slovakia, were used as the basis for our investigation. The data on magnetic field and solar wind from GOES 10 and 12, SOHO-CELIAS, ACE and WIND satellites were used for better understanding of the global evolution of the event. The magnetic field is transformed to the RTN (Radial – Tangential – Normal) system where only the disturbed part of the field is compared, i.e., daily variations and a constant part are subtracted. The field reduction method is described. Our results are temporal vector diagrams of variation of all parameters at all positions from where we used the data. The amplitudes of |B| exceed 100 nT and variations during the arrival of the wavefront of CME take place simultaneously at the ground-based station and at GOES satellites. The character of the variations is as if there would be regions with the dominant electric charge of opposite signs, or electric currents with different orientations in the CME. On the basis of the values v _p and n _p and using certain assumptions we determined the mass of CME on 17 January and 21 January, respectively, of 10¹² kg. A decrease of the cosmic ray level runs suddenly (during 10 minutes), starting, however, about two hours after a sudden change of the magnetic field.     

Detailed analysis of fan-shaped jets in three dimensional numerical simulation

We performed three dimensional resistive magnetohydrodynamic simulations to study the magnetic reconnection using an initially shearing magnetic field configuration(force free field with a current sheet in the middle of the computational box).It is shown that there are two types of reconnection jets:the ordinary reconnection jets and fan-shaped jets,which are formed along the guide magnetic field.The fan-shaped jets are significantly different from the ordinary reconnection jets which are ejected by magneti...