Giorgio Einaudi

Energy Release in a Turbulent Corona

Numerical simulations of a two-dimensional section of a coronal loop subject to random magnetic forcing are presented. The forcing models the link between photospheric motions and energy injection in the corona. The results show the highly intermittent spatial distribution of current concentrations generated by the coupling between internal dynamics and external forcing. The total power dissipation is a rapidly varying function of time, with sizable jumps even at low Reynolds numbers, and is caused by the superposition of magnetic dissipation in a number of localized current sheets. Both spatial and temporal intermittency increase with the Reynolds number, suggesting that the turbulent nature of the corona can physically motivate statistical theories of solar activity.

The stability of coronal loops - Finite-length and pressure-profile limits

Results are described from a quickly converging, necessary-and-sufficient, MHD-stability test for coronal-loop models. The primary stabilizing influence arises from magnetic line tying at the photosphere, and this end conditions requires a series expansion of possible loop excitations. The stability boundary is shown to quickly approach a limit as the number of terms increases, providing a critical length for the loop in proportion to its transverse magnetic scale. Several models of force-free-field profiles are tested and the stability behavior of a localized current channel, embedded in an external current-free region, is shown to be superior to that of other, broader, current profiles. Pressure-gradient effects, leading to increased or decreased stability, are shown to be amplified by line tying. Long loops must either conduct low net current, or exhibit an axial-field reversal coexisting with a low-pressure core. The limits on stability depend on the magnetic aspect ratio, the plasma-to-magnetic pressure ratio, and the field orientation at the loop edge. Applications of these results to the structure of coronal loops are described.

Coronal Heating, Weak MHD Turbulence, and Scaling Laws

Long-time high-resolution simulations of the dynamics of a coronal loop in Cartesian geometry are carried out, within the framework of reduced magnetohydrodynamics (RMHD), to understand coronal heating driven by the motion of field lines anchored in the photosphere. We unambiguously identify MHD anisotropic turbulence as the physical mechanism responsible for the transport of energy from the large scales, where energy is injected by photospheric motions, to the small scales, where it is dissipated. As the loop parameters vary, different regimes of turbulence develop: strong turbulence is found for weak axial magnetic fields and long loops, leading to Kolmogorov-like spectra in the perpendicular direction, while weaker and weaker regimes (steeper spectral slopes of total energy) are found for strong axial magnetic fields and short loops. As a consequence we predict that the scaling of the heating rate with axial magnetic field intensity B0, which depends on the spectral index of total energy for given loop parameters, must vary from B for weak fields to B for strong fields at a given aspect ratio. The predicted heating rate is within the lower range of observed active region and quiet-Sun coronal energy losses.

Statistical Properties of Magnetic Activity in the Solar Corona

The long-time statistical behavior of a two-dimensional section of a coronal loop subject to random magnetic forcing is presented. The highly intermittent nature of dissipation is revealed by means of magnetohydrodynamic (MHD) turbulence numerical simulations. Even with a moderate magnetic Reynolds number, intermittency is clearly present in both space and time. The response of the loop to the random forcing, as described either by the time series of the average and maximum energy dissipation or by its spatial distribution at a given time, displays a Gaussian noise component that may be subtracted to define discrete dissipative events. Distribution functions of both maximum and average current dissipation, for the total energy content, the peak activity, and the duration of such events are all shown to display robust scaling laws, with scaling indices δ that vary from δ -1.3 to δ -2.8 for the temporal distribution functions, while δ -2.6 for the overall spatial distribution of dissipative events.

Formation of the slow solar wind in a coronal streamer

We have investigated a magnetohydrodynamic mechanism that accounts for several fundamental properties of the slow solar wind, in particular its variability, latitudinal extent, and bulk acceleration. In view of the well-established association between the streamer belt and the slow wind, our model begins with a simplified representation of a streamer beyond the underlying coronal helmet: a neutral sheet embedded in a plane fluid wake. This wake-neutral sheet configuration is characterized by two parameters that vary with distance from the Sun: the ratio of the cross-stream velocity scale to the neutral sheet width, and the ratio of the typical Alfven velocity to the typical flow speed far from the neutral sheet. Depending on the values of these parameters, our linear theory predicts that three kinds of instability can develop when this system is perturbed: a tearing instability and two ideal fluid instabilities with different cross-stream symmetries (varicose and sinuous). In the innermost, magnetically dominated region beyond the helmet cusp, we find that the streamer is resistively and ideally unstable, evolving from tearing-type reconnection in the linear regime to a nonlinear varicose fluid instability. Traveling magnetic islands are formed which are similar to features recently revealed by the large-angle spectroscopic coronagraph on the joint European Space Agency/NASA Solar and Heliospheric Observatory (SOHO) [Brueckner et al., 1995]. During this process, the center of the wake is accelerated and broadened slightly. Past the Alfven point, where the kinetic energy exceeds the magnetic energy, the tearing mode is suppressed, but an ideal sinuous fluid mode can develop, producing additional acceleration up to typical slow wind speeds and substantial broadening of the wake. Farther from the Sun, the system becomes highly turbulent as a result of the development of ideal secondary instabilities, thus halting the acceleration and producing strong filamentation throughout the core of the wake. We discuss the implications of this model for the origin and evolution of the slow solar wind, and compare the predicted properties with current observations from SOHO.

Nonlinear Magnetohydrodynamic Evolution of Line-tied Coronal Loops

Simulations of the nonlinear evolution of the m = 1 kink mode in magnetic flux tubes with line-tying boundary conditions are presented. The initial structure of the flux tube is intended to model a solar coronal loop that either has evolved quasi-statically through sequences of equilibria with increasing twist due to the application of localized photospheric vortex flows or has emerged with a net current through the photosphere. It is well known that when the twist exceeds a critical value that depends on its radial profile and on the loop length, the loop becomes kink unstable. The nonlinear evolution of the instability is followed using a three-dimensional MHD code in cylindrical geometry, in different types of magnetic field configurations, with the common property that the current is confined within the same radius, so that the magnetic field is potential in the external regions. The differences reside in the net axial current carried by the structure, ranging from a vanishing current (corresponding to an outer axial potential field) to a high current (corresponding to an outer almost azimuthal potential field). It is shown that, during the nonlinear phase of the instability, loops develop current sheets and, consequently, their evolution becomes resistive with the occurrence of magnetic reconnection. The dependence of the topology of the currents at saturation on the initial magnetic structure, the details of the reconnection phenomenon, and the resistive dissipation mechanism are examined. Finally, the impact of the results on the understanding of coronal activity is discussed.

The distribution of flares, statistics of magnetohydrodynamic turbulence and coronal heating

In this paper theoretical evidence in favor of the hypothesis that coronal dissipation occurs in bursts at very small spatial scales is presented. Each individual burst, though unobservable and energetically insignificant, is thought to represent the building block of coronal activity. In this framework, a large number of coherently triggered bursts is what appears as one of the many observed solar atmospheric events (i.e., blinkers, heating events, explosive events, flashes, microflares, flares,…). Histograms of such events, when computed, in terms of total energy, duration and peak luminosity appear to display power-law behavior. Simulations of the energy dissipation in the simplest possible forced magnetohydrodynamic (MHD) system, admitting reconnection events, indeed displays such kind of behavior: dissipative events of varying intensity, size and duration may be defined, whose distributions follow power laws. The meaning of cellular automaton models, introduced to describe the power-law statistics of...

Stability of a diffuse linear pinch with axial boundaries

A formulation of the stability behavior of a finite‐length pinch is presented. A general initial perturbation is expressed as a uniformly convergent sum over a complete discrete k set. A variational calculation is then performed, based on the energy principle, in which the end‐boundary conditions appear as constraints. The requisite Lagrange multipliers mutually couple the elemental periodic excitations. The resulting extended form of δW still admits a proper second‐variation treatment so that the minimization and stability considerations of Newcomb remain applicable. Comparison theorems are discussed as is the relevance of this end‐effect model to the stability of solar coronal loops.

Plasmoid Formation and Acceleration in the Solar Streamer Belt

The dynamical behavior of a configuration consisting of a plane fluid wake flowing in a current sheet embedded in a plasma sheet that is denser than its surroundings is discussed. This configuration is a useful model for a number of structures of astrophysical interest, such as solar coronal streamers, cometary tails, the Earths magnetotail and Galactic center nonthermal filaments. In this paper, the results are applied to the study of the formation and initial motion of the plasma density enhancements observed by the Large-Angle Spectrometric Coronagraph (LASCO) instrument onboard the Solar and Heliospheric Observatory (SOHO) spacecraft. It is found that beyond the helmet cusp of a coronal streamer, the magnetized wake configuration is resistively unstable, that a traveling magnetic island develops at the center of the streamer, and that density enhancements occur within the magnetic islands. As the massive magnetic island travels outward, both its speed and width increase. The island passively traces the acceleration of the inner part of the wake. The values of the acceleration and density contrasts are in good agreement with LASCO observations.

SHEAR PHOTOSPHERIC FORCING AND THE ORIGIN OF TURBULENCE IN CORONAL LOOPS

We present a series of numerical simulations aimed at understanding the nature and origin of turbulence in coronal loops in the framework of the Parker model for coronal heating. A coronal loop is studied via reduced magnetohydrodynamic (MHD) simulations in Cartesian geometry. A uniform and strong magnetic field threads the volume between the two photospheric planes, where a velocity field in the form of a one-dimensional shear flow pattern is present. Initially, the magnetic field that develops in the coronal loop is a simple map of the photospheric velocity field. This initial configuration is unstable to a multiple tearing instability that develops islands with X and O points in the plane orthogonal to the axial field. Once the nonlinear stage sets in the system evolution is characterized by a regime of MHD turbulence dominated by magnetic energy. A well-developed power law in energy spectra is observed and the magnetic field never returns to the simple initial state mapping the photospheric flow. The formation of X and O points in the planes orthogonal to the axial field allows the continued and repeated formation and dissipation of small-scale current sheets where the plasma is heated. We conclude that the observed turbulent dynamics are not induced by the complexity of the pattern that the magnetic field-line footpoints follow but they rather stem from the inherent nonlinear nature of the system.