Introduction
The nature of the mechanisms for heating the solar corona has been one of the most intensely studied problems in solar physics during the past six decades, and yet it remains one of the least understood.
Up until about 1940, it was thought that the temperature of the Sun decreased from the solar surface. However, when it was realized that emission lines seen during total solar eclipses were due to known elements at very high stages of ionisation, solar physicists were confronted with the puzzle of how the temperature in the outer solar atmosphere could pass from the value of about 6000 K in the photosphere to the surprisingly higher 1–2 million degrees (or even more) in the corona.
Since there, the so-called "coronal heating problem" has been restricted by substantial progresses in theoretical modelling, new high-resolution imaging by satellite telescope observations, and more sophisticated data analysis. Nevertheless, a number of important issues must be addressed yet. In particular, the question on whether the plasma heating inside coronal structures is the effect of steady or time-dependent processes, uniform or localised somewhere in space, is still open.
Several theories for coronal heating have been proposed, each of them invoking different physical mechanisms producing different characteristics in the energy deposition, but it has proved difficult to determine which ones, if any, are actually correct. It is clear that a definitive test of any coronal heating model requires a quantitative prediction of observable quantities that is based on a detailed treatment. Plausible theories must be able to tie together a number as large as possible of the observational evidences that have been inferred from the high-quality data delivered by recent solar missions, such as Yohkoh, the Solar and Heliospheric Observatory (SOHO), the Transition Region and Coronal Explorer (TRACE), and Hinode.
EUV and X-ray imaging data from these satellites have revealed that the outer solar atmosphere is highly structured. Non-uniformities are present in both plasma density and temperature and should be a direct result of spatial and temporal variations in the rate of coronal heating. In particular, it has been well established that much of the plasma in the Sun's corona is confined by the magnetic field in distinct, arch-shaped structures known as coronal loops.
Such coronal loops are characterized by different lengths, temperatures, activity levels, and appear to evolve with lifetimes of the order of several hours. The dynamic behaviour as well as the radiative properties of the plasma inside these structures are clear signatures of unknown mechanisms that are responsible for their heating. While early observations of coronal loops were made primarily in soft X-rays and suggested that these structures are in states of quasi-static equilibrium under the action of steady heating, more recent measurements in the EUV have pointed out a number of discrepancies between the predicted features of steady heating models and the characteristics of the observed loops, particularly at low temperatures. It emerges from this picture that coronal loops most probably evolve in response to heating that is strongly time-dependent. Moreover, it is unlikely that any of the suggested heating mechanism provides a continuous, constant supply of heat to the corona.
Theoretical models of coronal loop heating involve as a primary energy source chromospheric footpoint motions or upward leaking Alfvén waves which are dissipated in the corona by a series of possible mechanisms, such as magnetic reconnection, resonant absorption, or phase mixing. Most of these mechanisms produce heating on individual magnetic flux surfaces (i.e., strands) that is impulsive in nature. Therefore, the hypothesis that coronal loops are heated by many tiny, small-scale energetic events, as envisioned by the so-called "nanoflare theory", has recently been advanced. This model postulates that loops are bundles of thin, unresolved magnetic strands that are independently heated by storms of energy pulses.
The concept of nanoflares is appealing for several reasons, but particularly because of its flexibility. Nanoflare-heated loops can in fact achieve quasi-steady conditions if the pulses repeat sufficiently rapidly in each strand, although the heating is indeed time-dependent. Moreover, uniform or non-uniform heating conditions can be obtained if nanoflares are concentrated at a particular location or distributed along the whole structure of the loop.
In this work we analyse the dynamic behaviour of the plasma confined in a magnetic flux-tube and subject to different kinds of heating regimes. We consider both steady and impulsive heating. In the latter case, the particular nanoflare model we use is generic in the sense that it does not specify a particular energy dissipation mechanism. We also examine both the cases of uniform and footpoint-concentrated heating.
The loop plasma evolution in response to these different heating regimes is derived through one-dimensional numerical simulations performed with the Adaptively Refined Godunov Solver (ARGOS) hydro-code, which solves the standard set of hydrodynamic equations for a non-turbulent, compressible fluid flow. ARGOS employs a fully adaptive computational grid by using the Parallel Adaptive Mesh refinement package (PARAMESH). Hence, it is capable of resolving steep gradient regions and/or discontinuities that are likely to develop during the loop plasma evolution.
Our objective is to investigate on whether the variation of the heating properties (i.e., the parameters defining the temporal and spatial dependence of the energy deposition) can affect the hydrodynamic behaviour of the plasma filling the loop and, if so, what kind of signatures produces on the plasma-related observable quantities. The relevant consequences of the different heating regimes on some plasma diagnostics, such as the Doppler shifts of some coronal and transition region spectral lines and the differential emission measure distribution, are therefore discussed in detail, together with the indications that the variety of conditions found in this exploration give on the physical origins of coronal heating and related phenomena.
Our results show that thermal non-equilibrium and, consequently, condensation formation cycles develop along the loop when impulsive heating, with a pulse repetition times lower than the plasma characteristic cooling time, is localised at the loop footpoints and the pulse energy is below a threshold above which the heating balances the radiative losses, thus preventing the catastrophic cooling that triggers the condensation.
Condensations appear to significantly affect spectral lines forming at transition region temperatures, that exhibit rather pronounced red-shifts indicating downflows that trace out the motion of the condensation towards the solar surface. Coronal lines show substantial blue-shifts of their centroids only when impulsive heating has a pulse cadence time longer than the characteristic cooling time of the loop; hence, analysis of these spectral lines could be useful in discriminating between localised heating regimes with high or low nanoflare repetition times.
Conversely, a condensation does not produce observable signatures in the differential emission measure distribution (DEM) because it does not redistribute the plasma over a sufficiently large temperature range. On the other hand, the DEM coronal peak is found sensitive to the pulse cadence time when this is longer or comparable to the plasma cooling time. However, the DEMs derived from our models appear to be unable to reproduce both the transition region and the coronal structure of some observed DEMs with a unique set of heating parameters.
Finally, our simulations could give an explanation of the warm overdense and hot underdense loops observed by TRACE, SOHO, Yohkoh, and Hinode. Overdense conditions at relatively low temperatures are attained by condensation-forming models, while underdense conditions at higher temperatures are generally achieved by impulsive heating models with low cadence times and high nanoflare energies.