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Parker (1972) first proposed that coronal heating was the necessary outcome of an energy flux caused by the tangling of coronal magnetic field lines by photospheric flows. In this paper we discuss how this model has been modified by subsequent numerical simulations outlining in particular the substantial differences between the "nanoflares" introduced by Parker and "elementary events", defined here as small-scale spatially and temporally isolated heating events resulting from the continuous formation and dissipation of field-aligned current sheets within a coronal loop. We present numerical simulations of the compressible 3-D MHD equations using the HYPERION code. We use two clustering algorithms to investigate the properties of the simulated elementary events: an IDL implementation of a Density-Based Spatial Clustering of Applications with Noise (DBSCAN) technique; and our own Physical Distance Clustering (PDC) algorithm. We identify and track elementary heating events in time, both in temperature and in Joule heating space. For every event we characterize properties such as: density, temperature, volume, aspect ratio, length, thickness, duration and energy. The energies of the events are in the range $10^{18}-10^{21}$ ergs, with durations shorter than 100 seconds. A few events last up to 200 seconds and release energies up to $10^{23}$ ergs. While high temperature are typically located at the flux tube apex, the currents extend all the way to the footpoints. Hence a single elementary event cannot at present be detected. The observed emission is due to the superposition of many elementary events distributed randomly in space and time within the loop., Comment: Astrophysical Journal, in press

We investigate the dynamics of a closed corona cartesian reduced magnetohydrodynamic (MHD) model where photospheric vortices twist the coronal magnetic field lines. We consider two corotating or counter-rotating vortices localized at the center of the photospheric plate, and additionally more corotating vortices that fill the plate entirely. Our investigation is specifically devoted to study the fully nonlinear stage, after the linear stage during which the vortices create laminar and smoothly twisting flux tubes. Our main goal is to understand the dynamics of photospheric vortices twisting the field lines of a coronal magnetic field permeated by finite amplitude broadband fluctuations. We find that depending on the arrangement and handedness of the photospheric vortices an inverse cascade storing a significant amount of magnetic energy may occur or not. In the first case a reservoir of magnetic energy available to large events such as destabilization of a pre-CME configuration develops, while in the second case the outcome is a turbulent heated corona. Although our geometry is simplified our simulations are shown to have relevant implications for coronal dynamics and CME initiation., Comment: 22 pages, 13 figure, submitted to ApJ

The heating of coronal loops is investigated to understand the observational consequences in terms of the thermodynamics and radiative losses from the Sun as well as the magnetized coronae of stars with an outer convective envelope. The dynamics of the Parker coronal heating model are studied for different ratios of the photospheric forcing velocity timescale $t_p$ to the Alfv\'en crossing time along a loop $t_A$. It is shown that for $t_p/t_A \gtrsim$ 10--24 the heating rate and maximum temperature are largest and approximately independent of $t_p/t_A$, leading to a strong emission in X-rays and EUV. On the opposite decreasing $t_p/t_A$ to smaller values leads to lower heating rates and plasma temperatures, and consequently fading high-energy radiative emission once $t_p/t_A \lesssim$ 1--3. The average volumetric loop heating rate is shown to scale as $\ell_p u_p B_0^2/4\pi L^2$, where $\ell_p$ and $u_p$ are respectively the convective granule length-scale and velocity, $B_0$ is the intensity of the strong magnetic field threading the loop, and $L$ the loop length. These findings support a recent hypothesis explaining ultracool dwarf observations of stars with similar magnetic field strength but radically different topologies displaying different radiative emission., Comment: 11 pages, 5 figures, MNRAS (in press)

The evolution of a coronal loop is studied by means of numerical simulations of the fully compressible three-dimensional magnetohydrodynamic equations using the HYPERION code. The footpoints of the loop magnetic field are advected by random motions. As a consequence the magnetic field in the loop is energized and develops turbulent nonlinear dynamics characterized by the continuous formation and dissipation of field-aligned current sheets: energy is deposited at small scales where heating occurs. Dissipation is non-uniformly distributed so that only a fraction of the coronal mass and volume gets heated at any time. Temperature and density are highly structured at scales which, in the solar corona, remain observationally unresolved: the plasma of our simulated loop is multi-thermal, where highly dynamical hotter and cooler plasma strands are scattered throughout the loop at sub-observational scales. Numerical simulations of coronal loops of 50000 km length and axial magnetic field intensities ranging from 0.01 to 0.04 Tesla are presented. To connect these simulations to observations we use the computed number densities and temperatures to synthesize the intensities expected in emission lines typically observed with the Extreme ultraviolet Imaging Spectrometer (EIS) on Hinode. These intensities are used to compute differential emission measure distributions using the Monte Carlo Markov Chain code, which are very similar to those derived from observations of solar active regions. We conclude that coronal heating is found to be strongly intermittent in space and time, with only small portions of the coronal loop being heated: in fact, at any given time, most of the corona is cooling down., Comment: 15 pages, 14 figures, ApJ (in press)

We present simulations modeling closed regions of the solar corona threaded by a strong magnetic field where localized photospheric vortical motions twist the coronal field lines. The linear and nonlinear dynamics are investigated in the reduced magnetohydrodynamic regime in Cartesian geometry. Initially the magnetic field lines get twisted and the system becomes unstable to the internal kink mode, confirming and extending previous results. As typical in this kind of investigations, where initial conditions implement smooth fields and flux-tubes, we have neglected fluctuations and the fields are laminar until the instability sets in. But previous investigations indicate that fluctuations, excited by photospheric motions and coronal dynamics, are naturally present at all scales in the coronal fields. Thus, in order to understand the effect of a photospheric vortex on a more realistic corona, we continue the simulations after kink instability sets in, when turbulent fluctuations have already developed in the corona. In the nonlinear stage the system never returns to the simple initial state with ordered twisted field lines, and kink instability does not occur again. Nevertheless field lines get twisted, but in a disordered way, and energy accumulates at large scales through an inverse cascade. This energy can subsequently be released in micro-flares or larger flares, when interaction with neighboring structures occurs or via other mechanisms. The impact on coronal dynamics and CMEs initiation is discussed., Comment: 14 pages, 11 figure

Context. Photospheric motions shuffle the footpoints of the strong axial magnetic field that threads coronal loops giving rise to turbulent nonlinear dynamics characterized by the continuous formation and dissipation of field-aligned current sheets where energy is deposited at small-scales and the heating occurs. Previous studies show that current sheets thickness is orders of magnitude smaller than current state of the art observational resolution (~700 km). Aim. In order to understand coronal heating and interpret correctly observations it is crucial to study the thermodynamics of such a system where energy is deposited at unresolved small-scales. Methods. Fully compressible three-dimensional magnetohydrodynamic simulations are carried out to understand the thermodynamics of coronal heating in the magnetically confined solar corona. Results. We show that temperature is highly structured at scales below observational resolution and nonhomogeneously distributed so that only a fraction of the coronal mass and volume gets heated at each time. Conclusions. This is a multi-thermal system where hotter and cooler plasma strands are found one next to the other also at sub-resolution scales and exhibit a temporal dynamics., Comment: A&A Letter, in press

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 magnetohydrodynamics 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 1D shear flow pattern is present. Initially the magnetic field which developes in the coronal loop is a simple map of the photospheric velocity field. This initial configuration is unstable to a multiple tearing instability which 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-lines footpoints follow but they rather stem from the inherent nonlinear nature of the system., Comment: 14 pages, 10 figures, online animations at http://www.df.unipi.it/~rappazzo/shear/ (non permanent link)

The plasma density enhancements recently observed by the Large-Angle Spectrometric Coronagraph (LASCO) instrument onboard the Solar and Heliospheric Observatory (SOHO) spacecraft have sparked considerable interest. In our previous theoretical study of the formation and initial motion of these density enhancements 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. In the present paper a few spherical geometry effects are included, taking into account either the radial divergence of the magnetic field lines and the average expansion suffered by a parcel of plasma propagating outward, using the Expanding Box Model (EBM), and the diamagnetic force due to the overall magnetic field radial gradients, the so-called melon-seed force. It is found that the values of the acceleration and density contrasts can be in good agreement with LASCO observations, provided the spherical divergence of the magnetic lines starts beyond a critical distance from the Sun and the initial stage of the formation and acceleration of the plasmoid is due to the cartesian evolution of MHD instabilities. This result provides a constraint on the topology of the magnetic field in the coronal streamer., Comment: 15 pages, 10 figures, posting for archival purposes

The Parker or field line tangling model of coronal heating is investigated through long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry within the framework of reduced magnetohydrodynamics (RMHD). Slow photospheric motions induce a Poynting flux which saturates by driving an anisotropic turbulent cascade dominated by magnetic energy and characterized by current sheets elongated along the axial magnetic field. Increasing the value of the axial magnetic field different regimes of MHD turbulence develop with a bearing on coronal heating rates. In physical space magnetic field lines at the scale of convection cells appear only slightly bended in agreement with observations of large loops of current (E)UV and X-ray imagers., Comment: 9 page, 4 figures

To understand the nonlinear dynamics of the Parker scenario for coronal heating, long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry are carried out. A loop is modeled as a box extended along the direction of the strong magnetic field $B_0$ in which the system is embedded. At the top and bottom plates, which represent the photosphere, velocity fields mimicking photospheric motions are imposed. We show that the nonlinear dynamics is described by different regimes of MHD anisotropic turbulence, with spectra characterized by intertial range power laws whose indexes range from Kolmogorov-like values ($\sim 5/3$) up to $\sim 3$. We briefly describe the bearing for coronal heating rates., Comment: 8 pages, 4 figures

The Parker or field line tangling model of coronal heating is studied comprehensively via long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry within the framework of reduced magnetohydrodynamics (RMHD). Slow photospheric motions induce a Poynting flux which saturates by driving an anisotropic turbulent cascade dominated by magnetic energy. In physical space this corresponds to a magnetic topology where magnetic field lines are barely entangled, nevertheless current sheets (corresponding to the original tangential discontinuities hypothesized by Parker) are continuously formed and dissipated. Current sheets are the result of the nonlinear cascade that transfers energy from the scale of convective motions ($\sim 1,000 km$) down to the dissipative scales, where it is finally converted to heat and/or particle acceleration. Current sheets constitute the dissipative structure of the system, and the associated magnetic reconnection gives rise to impulsive ``bursty'' heating events at the small scales. This picture is consistent with the slender loops observed by state-of-the-art (E)UV and X-ray imagers which, although apparently quiescent, shine bright in these wavelengths with little evidence of entangled features. The different regimes of weak and strong MHD turbulence that develop, and their influence on coronal heating scalings, are shown to depend on the loop parameters, and this dependence is quantitatively characterized: weak turbulence regimes and steeper spectra occur in {\it stronger loop fields} and lead to {\it larger heating rates} than in weak field regions., Comment: 22 pages, 18 figures, uses emulateapj, for mpeg file associated to Figure 17e see (temporarily) http://www.df.unipi.it/~rappazzo/arxiv/jfl.mpg, ApJ, in press

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 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 $B_0$, which depends on the spectral index of total energy for given loop parameters, must vary from $B_0^{3/2}$ for weak fields to $B_0^{2}$ 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., Comment: 4 pages, 5 figures, uses emulateapj, complies with published version

A 3D model intermediate between cellular automata (CA) models and the reduced magnetohydrodynamic (RMHD) equations is presented to simulate solar impulsive events generated along a coronal magnetic loop. The model consists of a set of planes distributed along a magnetic loop between which the information propagates through Alfven waves. Statistical properties in terms of power-laws for energies and durations of dissipative events are obtained, and their agreement with X-ray and UV flares observations is discussed. The existence of observational biases is also discussed., Comment: 11 pages, 9 figures Accepted for publication in Astronomy & Astrophysics

The paper discusses the regime of validity of the theory of dissipative Alfven normal modes presented by Mok and Einaudi (1985) and Einaudi and Mok (1985), which was based on the incompressible closure of the system of ideal MHD equations. Some simple extensions of the earlier results to the compressible case are described. In addition, certain misunderstandings of this work, which have appeared in other papers, are clarified.

Resistive normal mode solutions of the MHD equations are found numerically in a smooth, nonuniform, magnetic field. The (alpha, S) boundary within which normal mode solutions exist is explicitly computed, where alpha is the normalized wavenumber and S the Lundquist number. As an extension of previous analytic results of Mok and Einaudi (1985), the damping rate of these modes is computed to a higher accuracy, and is found to have an a + bS exp -1/3 dependence, where a and b are independent of S.

The effect of resistive dissipation on the propagation of an MHD disturbance in a nonuniform plasma is examined. The present analysis, based on a boundary-layer technique, shows the existence of resistive normal modes with complex eigenfrequencies. The real part of the eigenfrequency is associated with an oscillatory behavior and defines the location in space of the layer where resistivity is important. The dissipation mechanism is responsible for the damping of the wave, in contrast with previous works in which the ideal MHD theory was used.

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.

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 delta-W 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.

Finite-length effects are the primary influence for the MHD stability of coronal loops. This paper gives a new physical description of the photospheric boundary conditions, and the first complete and uniformly convergent initial-perturbation set. A general energy-principle (or variational) stability analysis is then developed. The results of this calculation show that large classes of cylindrical-symmetry perturbations, those with separable contributions to the energy integral, are completely stable. Comparisons are also made with other more restrictive formulations of this problem.

A large-scale potential field with an embedded smaller-scale force-free structure gradient x B equals alpha B is studied in cylindrical geometry. Cases in which alpha goes continuously from a constant value alpha 0 on the axis to zero at large r are considered. Such a choice of alpha (r) produces fields which are realistic (few field reversals) but not completely stable. The MHD-unstable wavenumber regime is found. Since the considered equilibrium field exhibits a certain amount of magnetic shear, resistive instabilities can arise. The growth rates of the tearing mode in the limited MHD-stable region of k space are calculated, showing time-scales much shorter than the resistive decay time.