Nonlinear interactions in coronal heating

Characterizing the mechanisms that contribute to the onset of drag increase over micro-grooves (riblets) as the spacing increases is critical to design strategies for riblet-based drag reduction. This study decomposes the roughness function to investigate different mechanisms associated with the breakdown of drag reduction as riblet spacing is increased. We obtain the roughness function through direct numerical simulations (DNS) in a minimal channel and restricted nonlinear (RNL) models. Both the traditional RNL decomposition and an augmented RNL (ARNL) model that includes additional nonlinear interactions are employed as computationally tractable, reduced order representations of the flow field. RNL and ARNL results are compared to those of DNS in minimal channels to investigate the role of the different scale-dependent nonlinear interactions contributing to the roughness function. A comparison of the co-spectra arising from the minimal channel DNS with that from RNL and ARNL simulations indicates that general trends are captured by both reduced order models. However, the additional nonlinearity introduced in the ARNL model produces closer correspondence in the observed structural features of the DNS results. In particular, the ARNL better captures the signatures of the dispersive flow and the texture-coherent fluctuations. There is also a noticeable improvement observed in the profiles of the added stress contributions obtained with the ARNL model versus the RNL model.

The dynamics of magnetic fields in closed regions of solar and stellar coronae are investigated with a reduced magnetohydrodynamic (MHD) model in the framework of Parker scenario for coronal heating. A novel analysis of reduced MHD equilibria shows that their magnetic fields have an asymmetric structure in the axial direction with variation length-scale $z_\ell \sim \ell B_0/b$, where $B_0$ is the intensity of the strong axial guide field, $b$ that of the orthogonal magnetic field component, and $\ell$ the scale of $\mathbf{b}$. Equilibria are then quasi-invariant along the axial direction for variation scales larger than approximatively the loop length $z_\ell \gtrsim L_z$, and increasingly more asymmetric for smaller variation scales $z_\ell \lesssim L_z$. The $critical$ $length$ $z_\ell \sim L_z$ corresponds to the magnetic field intensity threshold $b \sim \ell B_0/L_z$. Magnetic fields stressed by photospheric motions cannot develop strong axial asymmetries. Therefore fields with intensities below such threshold evolve quasi-statically, readjusting to a nearby equilibrium, without developing nonlinear dynamics nor dissipating energy. But stronger fields cannot access their corresponding asymmetric equilibria, hence they are out-of-equilibrium and develop nonlinear dynamics. The subsequent formation of current sheets and energy dissipation is $necessary$ for the magnetic field to relax to equilibrium, since dynamically accessible equilibria have variation scales larger than the loop length $z_\ell \gtrsim L_z$, with intensities smaller than the threshold $b \lesssim \ell B_0/L_z$. The dynamical implications for magnetic fields of interest to solar and stellar coronae are investigated numerically and the impact on coronal physics discussed.

The different electron distributions in the hard and soft spectral states of black hole binaries could be caused by kinetic processes and changing because of varying physical conditions in the corona. In the presence of a magnetic field in the corona, the electron distribution can appear thermal, even when acceleration mechanisms would produce non-thermal distributions. This is due to fast and efficient thermalization through synchrotron self-absorption. In this paper, we have analysed data from 6 yr of observations of Cygnus X-1 with the INTEGRAL observatory and produced 12 high-quality, stacked broad-band hard X-ray spectra representative of the whole range of spectral shapes observed in this source. We then fit these spectra with hybrid thermal/non-thermal Comptonization models and study the evolution of the physical parameters of the accretion flow across the spectral transition. In particular, we use the BELM model to constrain the magnetic field in the corona through its effects on the coronal emission. Indeed, the hot electrons of the X-ray corona produce soft (optical-UV) synchrotron radiation which is then Comptonized and may affect the temperature of the electrons (and thus the slope of the X-ray spectrum) through Compton cooling. We find that in the softer states, the emission is dominated by Comptonization of the disc photons and the magnetic field is at most of the order of 10 6 G. In the harder states, the data are consistent with a pure synchrotron self-Compton model, although a significant contribution of Comptonization of disc photons may not be excluded. If the non-thermal excess observed above a few hundred keV in the hard state is produced in the same region as the bulk of the thermal Comptonization, we obtain an upper limit on the coronal magnetic field of about 10 5 G. If, on the other hand, the non-thermal excess is produced in a different location (such as the jet or a different part of the corona), the constraints on the magnetic field in the hard state are somewhat relaxed and the upper limit rises to 10 7 G. We discuss these constraints in the context of current accretion flow models.

This article is the third in a series working towards the construction of a realistic, evolving, non-linear force-free coronal-field model for the solar magnetic carpet. Here, we present preliminary results of 3D time-dependent simulations of the small-scale coronal field of the magnetic carpet. Four simulations are considered, each with the same evolving photospheric boundary condition: a 48-hour time series of synthetic magnetograms produced from the model of Meyer et al. (Solar Phys. 272, 29, 2011). Three simulations include a uniform, overlying coronal magnetic field of differing strength, the fourth simulation includes no overlying field. The build-up, storage, and dissipation of magnetic energy within the simulations is studied. In particular, we study their dependence upon the evolution of the photospheric magnetic field and the strength of the overlying coronal field. We also consider where energy is stored and dissipated within the coronal field. The free magnetic energy built up is found to be more than sufficient to power small-scale, transient phenomena such as nanoflares and X-ray bright points, with the bulk of the free energy found to be stored low down, between 0.5 – 0.8 Mm. The energy dissipated is currently found to be too small to account for the heating of the entire quiet-Sun corona. However, the form and location of energy-dissipation regions qualitatively agree with what is observed on small scales on the Sun. Future MHD modelling using the same synthetic magnetograms may lead to a higher energy release.

Solar magnetic field is believed to play a central role in solar activities and flares, filament eruptions as well as CMEs are due to the magnetic field re-organization and the interaction between the plasma and the field. At present the reliable magnetic field measurements are still confined to a few lower levels like in photosphere and chromosphere. Although IR technique may be applied to observe the coronal field but the technique is not well-established yet. Radio techniques may be applied to diagnose the coronal field but assumptions on radiation mechanisms and propagations are needed. Therefore extrapolation from photospheric data upwards is still the primary method to reconstruction coronal field. Potential field has minimum energy content and a force-free field can provide the required excess energy for energy release like flares, etc. Linear models have undesirable properties and it is expected to consider non-constant-alpha force-free field model. As the recent result indicates that the plasma beta is sandwich-ed distributed above the solar surface (Gary, 2001), care must be taken in modeling the coronal field correctly. As the reconstruction of solar coronal magnetic fields is an open boundary problem, it is desired to apply some technique that can incorporate this property. The boundary element method is a well-established numerical techniques that has been applied to many fields including open-space problems. It has also been applied to solar magnetic field problems for potential, linear force-free field and non-constant-alpha force-free field problems. It may also be extended to consider the non-force-free field problem. Here we introduce the procedure of the boundary element method and show its applications in reconstruction of solar magnetic field problems.

The solar coronal magnetic field is anchored to a complex distribution of photospheric flux consisting of sunspots and magnetic elements. Coronal activity such as flares, eruptions and general heating is often attributed to the manner in which the coronal field responds to photospheric motions. A number of powerful techniques have been developed to characterize the response of the coronal field by describing its topology. According to such analyses, activity will be concentrated around topological features in the coronal field such as separatrices, null points or bald patches. Such topological properties are insensitive to the detailed geometry of the magnetic field and thereby create an analytic tool powerful and robust enough to be useful on complex observations with limited resolution. This article reviews those topological techniques, their developments and applications to observations.

Modeling Solar magnetic field in the corona is very important to understand the solar eruption and heating mechanism. However, the direct measurements of solar magnetic field in the solar corona is still taking a great challenge not only in the high accurate measurements of solar polarization signal, but also in the inversion approach for the Optic-thin condition of corona atmosphere. Existence and uniqueness in the force-free/Non-force-free extrapolation in the solar corona is still unknown. Magnetic helicity, as a topological invariant, has become a key factor in exploring the generation of magnetic fields within the Sun, solar eruptions, and energy transfer processes in interplanetary. We firstly give a short review of modelling magnetic helicity in the solar corona from the observation to simulation. Then we introduce a new method by calculating the potential current in a magnetic-helicity-conservation-decomposed approach to derive the magnetic helicity/energy equivalence of three-dimension magnetic field only based on the photospheric vector magnetogram. We testify our method by using the given magnetic field of Low and Lou (1990) and the difference is very small. Even though, the Lorentz force caused from the calculated magnetic field well explained the strong shearing movements of polarity inversion line (PIL) in the newly emerging active region of NOAA11158. Finally, we apply our method to the observation data and it is also successfully found that the weak/strong loss ratio of magnetic helicity in the solar confined/eruptive solar events.

In this paper, we seek to understand the timescale in which the photospheric motions on the Sun braid coronal magnetic field lines. This is a crucial ingredient for determining the viability of the braiding mechanism for explaining the high temperatures observed in the corona. We study the topological complexity induced in the coronal magnetic field, primarily using plasma motions extracted from magneto-convection simulations. This topological complexity is quantified using the field line winding, finite time topological entropy (FTTE), and passive scalar mixing. With these measures, we contrast mixing efficiencies of the magneto-convection simulation, a benchmark flow known as a “blinking vortex”, and finally photospheric flows inferred from sequences of observed magnetograms using local correlation tracking. While the highly resolved magneto-convection simulations induce a strong degree of field line winding and FTTE, the values obtained from the observations from the plage region are around an order of magnitude smaller. This behavior is carried over to the FTTE. Nevertheless, the results suggest that the photospheric motions induce complex tangling of the coronal field on a timescale of hours.

As a prelude to discussing the interaction of magnetic fields with convection, we first review some general properties of convection in a stratified medium. Granulation, which is the surface manifestation of the major energy carrying convection scales, is a shallow phenomenon. Below the surface, the topology changes to one of filamentary cool downdrafts, immersed in a gently ascending isentropic background. The granular downflows merge into more widely separated downdrafts, on scales of mesogranulation and super-granulation.The local topology and time evolution of the small scale, kilo Gauss, network and facular magnetic field elements are controlled by convection on the scale of granulation. The topology and time evolution of larger scale magnetic field concentrations are controlled by the hierarchical structure of the horizontal components of the large scale velocity field. In sunspots, the small scale magnetic field structure determines the energy balance, the systematic flows and the waves. Below the surface, the small scale structure of the magnetic field may change drastically, with little observable effect at the surface. We discuss results of some recent numerical simulations of sunspot magnetic fields, and some mechanisms that may be relevant in determining the topology of the sub-surface magnetic field. Finally, we discuss the role of active region magnetic fields in the global solar dynamo.

We define the topology of an (axisymmetric) magnetic field, a set of qualitative labels characterizing the connectivity of the lines of force. Under the special continuous deformations of theB-field defined by time evolution under the dynamo equation in the ‘convective’ regime (where the frozen-in behavior dominates the diffusion), this topology is preserved. This theorem should have applications to the study of time-varying magnetic fields in that regime in the case that exact or even approximate solutions are difficult to obtain. A partial generalization to the general case of convection and diffusion is made. As an application, a critique of Hibberd's recent theory of a time-dependent axisymmetric geomagnetic dynamo whose dipole-like field undergoes successive reversals is given.

Abstract. Knowledge of the structure of the coronal magnetic field is important for our understanding of many solar activity phenomena, e.g. flares and CMEs. However, the direct measurement of coronal magnetic fields is not possible with present methods, and therefore the coronal field has to be extrapolated from photospheric measurements. Due to the low plasma beta the coronal magnetic field can usually be assumed to be approximately force free, with electric currents flowing along the magnetic field lines. There are both observational and theoretical reasons which suggest that at least prior to an eruption the coronal magnetic field is in a nonlinear force free state. Unfortunately the computation of nonlinear force free fields is way more difficult than potential or linear force free fields and analytic solutions are not generally available. We discuss several methods which have been proposed to compute nonlinear force free fields and focus particularly on an optimization method which has been suggested recently. We compare the numerical performance of a newly developed numerical code based on the optimization method with the performance of another code based on an MHD relaxation method if both codes are applied to the reconstruction of a semi-analytic nonlinear force-free solution. The optimization method has also been tested for cases where we add random noise to the perfect boundary conditions of the analytic solution, in this way mimicking the more realistic case where the boundary conditions are given by vector magnetogram data. We find that the convergence properties of the optimization method are affected by adding noise to the boundary data and we discuss possibilities to overcome this difficulty.

The inhomogeneous coronal plasma is a perfect environment for fast growing drift waves. The omnipresence of coronal magnetic loops implies gradients of the equilibrium plasma quantities like the density, magnetic field and temperature. These gradients are responsible for the excitation of drift waves that grow both within the two‐component fluid description (both in the presence of collisions and without it) and within the two‐component kinetic descriptions (due to purely kinetic effects). Some aspects of these phenomena are investigated here. In particular the analysis of the particle dynamics within the growing wave is compared with the corresponding fluid analysis. While both of them predict the stochastic heating, the threshold for the heating obtained from the single particle analysis is higher. The explanation for this effect is given. Also, the effects of the density gradient in the direction perpendicular to the magnetic field vector are investigated within the kinetic theory, in both electrostatic and electromagnetic regimes. The electromagnetic regime implies the coupling of the gradient‐driven drift wave with the Alfvén wave. The growth rates for the two cases are calculated and compared. It is found that, in general, the electrostatic regime is characterized by stronger growth rates, as compared with the electromagnetic perturbations. The released amount of energy density due to this heating should be more dependent on the magnitude of the background magnetic field than on the coupling of the drift and Alfvén waves. The stochastic heating is expected to be much higher in regions with a stronger magnetic field. On the whole, the energy release rate caused by the stochastic heating can be several orders of magnitude above the value presently accepted as necessary for a sustainable coronal heating. The vertical stratification and the very long wavelengths along the magnetic loops imply that a drift‐Alfvén wave, propagating as a twisted structure along the loop, in fact occupies regions with different plasma‐β and, therefore, may have different (electromagnetic‐electrostatic) properties, resulting in different heating rates within just one or two wavelengths.

Using a test-particle simulation, we investigate the effect of large-scale coronal magnetic fields on electron acceleration at an outward-propagating coronal shock with a circular front. The coronal field is approximated by an analytical solution with a streamer-like magnetic field featuring a partially open magnetic field and a current sheet at the equator atop the closed region. We show that the large-scale shock-field configuration, especially the relative curvature of the shock and the magnetic field line across which the shock is sweeping, plays an important role in the efficiency of electron acceleration. At low shock altitudes, when the shock curvature is larger than that of the magnetic field lines, the electrons are mainly accelerated at the shock flanks; at higher altitudes, when the shock curvature is smaller, the electrons are mainly accelerated at the shock nose around the top of closed field lines. The above process reveals the shift of the efficient electron acceleration region along the shock front during its propagation. We also find that, in general, the electron acceleration at the shock flank is not as efficient as that at the top of the closed field because a collapsing magnetic trap can be formed at the top. In addition, we find that the energy spectra of electrons are power-law-like, first hardening then softening with the spectral index varying in a range of −3 to −6. Physical interpretations of the results and implications for the study of solar radio bursts are discussed.

Emerging Flux Regions (EFR’s) on the Sun observed in the Meudon–Mitaka Collaborative Solar Observation Program (1979) are discussed. The rule found earlier (Martres et al. 1973) for the growth (or decay) of the N polarity or the decay (or growth) of the S polarity magnetic field in relation to the photospheric clockwise (or counter-clockwise) vortex motion is reconfirmed for the EFR in the present solar cycle. The rule is emphasized as an indication of active generation of an electric current system by a photospheric motion. The chromospheric manifestations of EFR’s seen in Hα as arch filament systems (AFS’s) and the enhanced brightness are examined with respect to the photospheric magnetic field structure derived from the white light and magnetic observations. An isolated EFR is associated with an AFS which is well coaligned with the photospheric transverse field. The typical EFR configuration may be distorted if the EFR appears in the region of the pre-existing strong field. The observational characteristics of such EFR’s can also be interpreted in terms of the emergence of a nearly potential field associated with the dissipation of an excess magnetic energy. An EFR born in an older EFR shows that the flux loop is low-lying and tends to relax into a potential field configuration.

Dynamics of three-dimensional convection in microgravity crystal growth: g-jitter with steady magnetic fields